Adaptive Control Theory and Applications · 2019. 8. 7. · The applications of adaptive control to...

Adaptive Control Theory and Applications

Guest Editors: Chengyu Cao, Lili Ma, and Yunjun Xu

Journal of Control Science and Engineering

Journal of Control Science and Engineering


Guest Editors: Chengyu Cao, Lili Ma, and Yunjun Xu

Copyright © 2012 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Journal of Control Science and Engineering.” All articles are open access articles distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, providedthe original work is properly cited.

Editorial Board

Edwin K. P. Chong, USARicardo Dunia, USANicola Elia, USAPeilin Fu, USABijoy K. Ghosh, USAF. Gordillo, SpainShinji Hara, Japan

Seul Jung, Republic of KoreaVladimir Kharitonov, RussiaJames Lam, Hong KongDerong Liu, ChinaTomas McKelvey, SwedenSilviu-Iulian Niculescu, FranceYoshito Ohta, Japan

Yang Shi, CanadaZoltan Szabo, HungaryOnur Toker, TurkeyXiaofan Wang, ChinaJianliang Wang, SingaporeWen Yu, MexicoMohamed A. Zribi, Kuwait

Contents

Adaptive Control Theory and Applications, Chengyu Cao, Lili Ma, and Yunjun XuVolume 2012, Article ID 827353, 2 pages

Discrete Model Reference Adaptive Control System for Automatic Profiling Machine, Peng Song,Guo-kai Xu, and Xiu-chun ZhaoVolume 2012, Article ID 964242, 5 pages

Nonlinear Dynamic Model-Based Adaptive Control of a Solenoid-Valve System, DongBin Lee,Peiman Naseradinmousavi, and C. NatarajVolume 2012, Article ID 846458, 13 pages

General Form of Model-Free Control Law and Convergence Analyzing, Xiuying Li, Guanghui Wang,and Zhigang HanVolume 2012, Article ID 750373, 7 pages

Adaptive Control for Nonlinear Systems with Time-Varying Control Gain,Alejandro Rincon and Fabiola AnguloVolume 2012, Article ID 269346, 9 pages

Adaptive Control for a Class of Nonlinear System with Redistributed Models, Haisen Ke and Jiang LiVolume 2012, Article ID 409139, 6 pages

Adaptive Impedance Control to Enhance Human Skill on a Haptic Interface System,Satoshi Suzuki and Katsuhisa FurutaVolume 2012, Article ID 365067, 10 pages

Adaptive Control Allocation in the Presence of Actuator Failures, Yu Liu and Luis G. CrespoVolume 2012, Article ID 502149, 16 pages

Pilot-Induced Oscillation Suppression by Using L1 Adaptive Control, Chuan Wang, Michael Santone,and Chengyu CaoVolume 2012, Article ID 394791, 7 pages

Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2012, Article ID 827353, 2 pagesdoi:10.1155/2012/827353

Editorial


Chengyu Cao,1 Lili Ma,2 and Yunjun Xu3

1 Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-4077, USA2 Electronics and Mechanical Department, Wentworth Institute of Technology, Boston, MA 02115, USA3 Department of Mechanical, Materials, and Aerospace Engineering, University of Central Florida, Orlando, FL 32816, USA

Correspondence should be addressed to Chengyu Cao, [email protected]

Received 15 August 2012; Accepted 15 August 2012

Copyright © 2012 Chengyu Cao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Adaptive control is an active field in the design of control sys-tems to deal with uncertainties. The key difference betweenadaptive controllers and linear controllers is the adaptivecontroller’s ability to adjust itself to handle unknown modeluncertainties. Adaptive control is roughly divided into twocategories: direct and indirect. Indirect methods estimate theparameters in the plant and further use the estimated modelinformation to adjust the controller. Direct methods are oneswherein the estimated parameters are those directly used inthe adaptive controller.

Recently, much effort has been placed in adaptive controlin both theory and applications. Theory-wise, new controllerdesign techniques are introduced to handle nonlinear andtime-varying uncertainties. Broader systems with larger non-linear uncertainties can be covered by these developments.As a result, adaptive control finds use in various real worldapplications. This special session generalizes some of thelatest results of adaptive control in both theory and appli-cations. After a thorough review process, 8 papers wereselected. The papers in this special section include the fol-lowing.

The paper entitled “Adaptive control for nonlinear systemswith time-varying control gain” by A. Rincon and F. Angulo, ascheme for nonlinear plants with time-varying control gainsand time-varying plant coefficients is proposed and appliedon a plant model consisting of a Brunovsky type modelwith polynomials as approximators. The methodology hasbeen applied to the speed control of a permanent magnetsynchronous motor (PMSM) and proper tracking resultshave been achieved.

The paper entitled “Adaptive impedance control to enhancehuman skill on a haptic interface system” by S. Suzuki and

K. Furuta, adaptive assistive control for a haptic interfacesystem is proposed. An adaptive mechanism derived from aLyapunov candidate function is used to tune an impedanceof the virtual model for the haptic device according tothe identified operator’s characteristics for enhanced perfor-mance. It was verified that the operator’s characteristics canbe estimated and further enhanced.

The paper entitled “Pilot-induced oscillation suppressionby using L1 adaptive control” by C. Wang and C. Cao, wherepilot-induced oscillation (PIO) is a phenomenon that occursin both flight tests and operational aircrafts. In this paper, theL1 adaptive controller has been introduced to suppress PIO,which is caused by rate limiting and pure time delay. Dueto its architecture, the L1 adaptive controller will achieve adesired response with fast adaptation. The simulation resultsindicate that the L1 adaptive control is efficient in solving thiskind of problem.

In the paper entitled “Adaptive control for a class ofnonlinear system with redistributed models” by H. Ke, and J. Li,a novel multiple model adaptive controller for a classof nonlinear system in parameter-strict-feedback form isproposed. It not only improves the transient performancesignificantly, but also guarantees the stability of all the statesof the closed-loop system. A simulation example is proposedto illustrate the effectiveness of the developed multiple modeladaptive controller.

The paper entitled “Adaptive Control Allocation in thePresence of Actuator Failures” by Y. Liu and L. G. Crespoproposes a control allocation framework, where a feedbackadaptive signal is designed for a group of redundant actua-tors and is then adaptively allocated among all group mem-bers. In the adaptive control allocation structure, cooperative

2 Journal of Control Science and Engineering

actuators are grouped and treated as an equivalent controleffector. Two adaptive control allocation algorithms aredeveloped.

The implementation and effectiveness of the strategiesproposed is demonstrated in detail using several examples.

In the paper entitled “General form of model-free controllaw and convergence analyzing” by X. Li and Z. Han, thegeneral form of model free control law is introduced andits convergence is analyzed. First, the necessity to improvethe basic form of the model-free control law is explainedand the functional combination method as the approach ofimprovement is presented. Then, a series of sufficient con-ditions required for the convergence are given. The analysisdemonstrates that these conditions can be satisfied easily inengineering practice.

In the paper entitled “Nonlinear dynamic model-basedadaptive control of a solenoid-valve system” by D. Lee, P.Naseradinmousavi, and C. Nataraj, a nonlinear model-basedadaptive control approach is proposed for a solenoid-valvesystem. The challenge is that solenoids and butterfly valveshave uncertainties in multiple parameters in the model,which makes it difficult for the system to adjust to theenvironment. These kinds of valves have different varyingphysical attributes such as stroke, size, and weight; uncertainparameters including inertia, damping, and torque coeffi-cients; and various kinds of operational parameters such aspipe diameters and flow velocities. These uncertainties arefurther complicated because of the solenoid and butterflyvalve nonlinear dynamic models. The main contribution ofthis research is the application of adaptive control theoryand Lyapunov type stability approach to design a controllerfor a dynamic model of the solenoid-valve system in thepresence of those uncertainties. Numerical simulation resultsare shown to demonstrate good performance of the proposednonlinear model-based adaptive approach. Also shown arecomparisons of the performance of the same solenoid-valvesystem with a nonadaptive method.

The paper entitled “Discrete model reference adaptivecontrol system for automatic profiling machine” by P. Song, G.K. Xu and X. C. Zhao discusses that an automatic profilingmachine is a system which has a high degree of parametervariation and high frequency of transient process. In thispaper, a discrete model reference adaptive control system isapplied on an automatic profiling machine. The results ofsimulation show that the adaptive control system has thedesired dynamic performance.

Chengyu CaoLili Ma

Yunjun Xu


Research Article

Discrete Model Reference Adaptive Control System forAutomatic Profiling Machine

Peng Song, Guo-kai Xu, and Xiu-chun Zhao

College of Electromechanical and Information Engineering, Dalian Nationalities University, Dalian 116605, China

Correspondence should be addressed to Peng Song, [email protected]

Received 3 December 2011; Revised 15 May 2012; Accepted 6 June 2012

Academic Editor: Yunjun Xu

Copyright © 2012 Peng Song et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Automatic profiling machine is a movement system that has a high degree of parameter variation and high frequency of transientprocess, and it requires an accurate control in time. In this paper, the discrete model reference adaptive control system of automaticprofiling machine is discussed. Firstly, the model of automatic profiling machine is presented according to the parameters of DCmotor. Then the design of the discrete model reference adaptive control is proposed, and the control rules are proven. The resultsof simulation show that adaptive control system has favorable dynamic performances.

1. Introduction

Automatic profiling machine is a movement system whichhas a high degree of parameter variation and high frequencyof transient process, and it requires an accurate control intime. The traditional linear control methods, such as PID,cannot meet present needs or requirements of advancedperformance. The adaptive control not only solves theproblem that the control plant cannot be observed directly,but also improved the abilities of resisting interference [1].The applications of adaptive control to movement system arewidespread [2–4], but it is infrequent in automatic profilingmachine. A practical adaptive controlling scheme is proposedfor automatic profiling machine in [5]. In this paper, adiscrete model reference adaptive control (MRAC) methodis applied to automatic profiling machine. The discretecontrol method is convenient for program and contributesto application of adaptive control theory in practice.

2. The Mathematics Model

The control system is double closed-loop control systemwhose current loop is PI control and speed loop is MRAC.The DC motor parameters [6] of automatic profilingmachine are shown in Table 1. The equivalent plant of speed

loop is consisted of current loop and motor, and the transferfunction is

G(s) = 2377011.5s2 + 2252.87s + 5747.23

. (1)

3. The Design of MRAC System

The difference equation of the plant is

A(z−1)y(k) = z−1B

(z−1)u(k), (2)

where

A(z−1) = 1−

n∑

i=1

aiz−i, B

(z−1) =

m∑

i=0

biz−i, (3)

y(k) and u(k) are the output and input of the plant,respectively; z−1 is delay operator; k is the discrete-timevariable.

The difference equation of the reference model is

Am(z−1)ym(k) = z−1Bm

(z−1)r(k), (4)

where

Am(z−1) = 1−

n∑

i=1

aiz−i, Bm

(z−1) =

m∑

i=0

biz−i, (5)


z−1

Am(z−1)

Bm(z−1)

ym(k)

H(z−1)

r(k) e(k)u(k)

++

+

+−

−

−

F( −1)

1

G(z−1) A(z−1)

z−1 B(z−1)

y(k)

Adaptive algorithms

z

Figure 1: Block diagram of MRAC system.

Table 1: The parameters of DC motor.

Parameter Unit Value

Rated voltage V 100

Rated current A 6

Rated speed r/min 1000

Motor constant V/r/min 0.1

Electromagnetic time constant s 0.00246

Mechanical time constant s 0.078

ym(k) and r(k) are the output and input of the referencemodel.

The output error and its prediction are given by

e(k) = y(k)− ym(k),

e◦(k) = y◦(k)− ym(k),(6)

where e◦(k) and the other variables with “◦” represent thepredictions.

The structure of the adaptive control system is shown inFigure 1 where

H(z−1) =

n∑

i=1

hi(k)z−i+1,

G(z−1) =

m∑

i=0

gi(k)z−i,

F(z−1) =

n∑

i=1

fi(k)z−i+1.

(7)

From Figure 1, the following relationship can be obtained

Bm(z−1)r(k) = H

(z−1)ym(k) + F

(z−1)e(k) +G

(z−1)u(k).

(8)

Introduc (8) into (4):

Am(z−1)ym(k)

= z−1[H(z−1)ym(k) + F

(z−1)e(k) +G

(z−1)u(k)

].

(9)

Subtract (9) from (2):

A(z−1)e(k)

= [Am(z−1)− A(z−1)− z−1H

(z−1)]ym(k)

+[z−1B

(z−1)− z−1G

(z−1)]u(k)

− z−1F(z−1)e(k).

(10)

According to the Hyperstability theory, the discretesystem control laws [1] are

hIi (k) = hIi (k − 1) + λie(k)ym(k − i),

hPi (k) = μie(k)ym(k − i),(11)

where

i = 1, 2, . . . ,n; λi > 0; μi ≥ −λi2 ,

gIi (k) = gIi (k − 1) + ρie(k)u(k − i− 1),

gPi (k) = σie(k)u(k − i− 1),

(12)

where

i = 1, 2, . . . ,m; ρi > 0; σi ≥ −ρi

2,

f Pi (k) = qie(k)e(k − i),

f Ii (k) = f Ii (k − 1) + lie(k)e(k − i),

(13)

where

i = 1, 2, . . . ,n; li > 0; qi ≥ − li2 . (14)

Journal of Control Science and Engineering 3

Zero-orderhold5 Zero-order

hold4

Zero-orderhold3

Transfer Fcn2

2377011.5

Step1

Scope6

Scope5

Scope1

Discrete filter3

Discrete filter2

1

Discrete filter

1

Out10.1475 + 0.1451z−1

0.1475 + 0.1451z−1 z−1

+−

+

−

−

1 − 1.95z−1 + 0.9512z−2

ym

ym

H(ze

e

e

e e

)

H(z)

u

u

u

1

1/G(z)

e0

F(z)

F(z)

s2 + 2252.87s + 5747.23

Figure 2: Simulation block diagram of MRAC system.

E(k) cannot be found directly in the operations above, soit can be replaced by e◦(k). According to (10), e(k) becomes

e(k) =n∑

i=1

[ai − fi(k)

]e(k − i)

+n∑

i=1

[ai − ai − hi

(k)]ym(k − i)

+m∑

i=0

[bi − gi(k)

]u(k − i− 1).

(15)

The prediction error can be gained:

e◦(k) =n∑

i=1

[ai − f Ii (k − 1)

]e(k − i)

+n∑

i=1

[ai − ai − hIi

(k − 1

)]ym(k − i)

+m∑

i=0

[bi − gIi(k − 1)

]u(k − i− 1).

(16)

Subtract (16) from (15) and link (11)∼(13); the functionbecomes

e(k)− e◦(k) = −⎡

⎣n∑

i=1

(li + qi

)e2(k − i)

+n∑

i=0

(λi + μi

)y2m(k − i)

+m∑

i=0

(ρi + σi

)u2(k − i− 1)

⎤

⎦e(k).

(17)

Equation (17) becomes (18) by calculating,

e(k) = e◦(k)(1 +

∑ni=1

(li + qi

)e2(k − i) +

∑ni=0

(λi + μi

)y2m(k − i) +

∑mi=0

(ρi + σi

)u2(k − i− 1)

) . (18)

4. Simulation Studies

The reference model takes the form as follows:

z−1Bm(z−1

)

Am(z−1)= 0.1475z−1 + 0.1451z−2

1− 1.95z−1 + 0.9512z−2. (19)

Make simulation according to the analysis above bySIMULINK. The structure of the adaptive control system isshown in Figure 2, and the parameters value are shown inTable 2.

The simulation results are shown in Figures 3–5. To makea calculation, the percentage overshoot is 6%, the rise timeis 1 s, the settling time is 2 s (Δ = 0.02). Figure 3 is shownthat the maximal error value is less than 40 r/min, so thecontrol plant could better track reference model. Input theinterference signal into the front of plant transfer functionbetween 10 s and 20 s. The simulation results (in Figure 4)indicate that the control plant could be stable in 2 seconds.Input white noise into the reference input of plant not thereference input of model, and the result is shown in Figure 5.


0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

t (s)

n(r

/min

)

(a) Reference model

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

t (s)

n(r

/min

)

(b) Control plant

0 1 2 3 4 5 6 7 8 9 10−40

−30

−20

−10

0

10

20

30

40

t (s)

n(r

/min

)

(c) Error figure

Figure 3: Output figures of simulation.

0 5 10 15 20 25 300

200

400

600

800

1000

1200

t (s)

n(r

/min

)

(a) Control plant

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t (s)

n(r

/min

)

(b) Interference signal

0 5 10 15 20 25 30−40

−30

−20

−10

0

10

20

30

40

50

60

t (s)

n(r

/min

)

(c) Error figure

Figure 4: Output figure with interference signal.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

200

400

600

800

1000

1200

With white noiseWithout white noise

t (s)

n(r

/min

)

Figure 5: Output figures with white noise.

Table 2: Parameter values of MRAC system.

Parameter Value Parameter Value

hI1(0) 0 hI2(0) 0

λ1 0.1 λ2 0.01

gI1(0) 1 gI2(0) 0

ρ1 3000 ρ2 0.01

f I1 (0) 0 f I2 (0) 0

l1 1 l2 0.1

Obviously the figure has no visible variation. So the controlsystem has better abilities of resisting interference.

5. Conclusion

The discrete model reference adaptive control system ofautomatic profiling machine is discussed in this paper. Theresults of simulation show that adaptive control system hasfavorable dynamic performance. The discrete design methodis easy to realize by computer. The work in this paper willlay a foundation for the application of adaptive control inpractice.

Acknowledgments

This paper was supported by the National Science and Tech-nology Support Program Project (2009BAH41B05) and theFundamental Research Funds for the Central Universities.

References

[1] S. C. Wu and Z. Q. Wu, Adaptive Control, China Machine Press,Beijing, China, 2005.

[2] G. K. Xu and X. C. Zhao, The Adaptive Control of Drive SystemFor HS2000 Electric Vehicle, The Research and Development ofChina Electric Vehicle, Beijing Institute of Technology Press,Beijing, China, 2005.

[3] P. Song, W. Q. Long, G. K. Xu, and X. C. Zhao, “Researchon discrete-time model reference adaptive control of electricvehicle driven motor,” Journal of System Simulation, vol. 19, no.18, pp. 4261–4264, 2007.

[4] G. K. Xu, L. R. You, X. C. Zhao, P. Song, and C. H. Liu, “Aresearch on adaptive speed controller of a 4M gantry planningmachine,” Control Theory and Applications, vol. 24, no. 5, pp.846–850, 2007.

[5] L. R. You, F. Qiao, and G. K. Xu, “Adaptive controller for auto-matic profiling machines,” Control Theory and Applications, vol.21, no. 6, pp. 1025–1028, 2004.

[6] G. K. Xu, Z. Y. Chu, and S. C. Wu, The Adaptive ControllerDesign For Movement System, Chinese Technology Press, Bei-jing, 2004.


Research Article

Nonlinear Dynamic Model-Based Adaptive Control ofa Solenoid-Valve System

DongBin Lee, Peiman Naseradinmousavi, and C. Nataraj

Department of Mechanical Engineering and Center for Nonlinear Dynamics & Control (CENDAC), Villanova University,Villanova, PA 19085, USA

Correspondence should be addressed to DongBin Lee, [email protected]

Received 6 December 2011; Revised 8 March 2012; Accepted 30 March 2012

Academic Editor: Lili Ma

Copyright © 2012 DongBin Lee et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a nonlinear model-based adaptive control approach is proposed for a solenoid-valve system. The challenge is thatsolenoids and butterfly valves have uncertainties in multiple parameters in the nonlinear model; various kinds of physical appear-ance such as size and stroke, dynamic parameters including inertia, damping, and torque coefficients, and operational parametersespecially, pipe diameters and flow velocities. These uncertainties are making the system not only difficult to adjust to the environ-ment, but also further complicated to develop the appropriate control approach for meeting the system objectives. The main con-tribution of this research is the application of adaptive control theory and Lyapunov-type stability approach to design a controllerfor a dynamic model of the solenoid-valve system in the presence of those uncertainties. The control objectives such as set-pointregulation, parameter compensation, and stability are supposed to be simultaneously accomplished. The error signals are first for-mulated based on the nonlinear dynamic models and then the control input is developed using the Lyapunov stability-type analysisto obtain the error bounded while overcoming the uncertainties. The parameter groups are updated by adaptation laws using aprojection algorithm. Numerical simulation results are shown to demonstrate good performance of the proposed nonlinear model-based adaptive approach and to compare the performance of the same solenoid-valve system with a non-adaptive method as well.

1. Introduction

In order to achieve advanced automation [1] in systemssuch as marine vessels or ship-based machinery system [2],solenoid actuators and valves are often used [3] to increasesurvivability and capability. One typical type of actuatordriven by solenoids is shown in Figure 1, which is operated bythe electromagnetic force. The electric-driven solenoid valvesystem [4] and its sophisticated control can provide high lev-els of automation in large systems. The useful function of thesolenoid-valve, once an electrical signal (current or voltage)is applied, is to activate a mechanical motion such as dis-placement or rotation via the solenoid magnetic forces andtorques. The proportional solenoids normally require inte-grated electronics for controlling the plunger to give such asignal. Hydrodynamic torque of a butterfly valve comprisesthe core knowledge of fluid valve system design [5], andit is known that most of the valves in real systems havestrongly nonlinear characteristics between the force and

displacement [6, 7]. The use of an intelligent approach [8, 9]such as adaptive, robust, optimal, or nonlinear control of theactuator-valve machinery systems will benefit a wide spec-trum of nonlinear systems, compensating for nonlinearities[10] and dynamic characteristics. This approach will not onlydecrease the amount of cost and casualties but also improvethe performance of the mechatronic system. To investigatethe particular application, it is important to emphasizethe nonlinear dynamic modeling analysis of such actuator-valve systems because the accuracy and reliability of thesesystems depend highly on the mathematical system modeling[11] and its validation. In [12], the authors developed andanalyzed the nonlinear dynamic model of a solenoid-valvesystem; the reader is also referred to [13, 14] for recent mod-eling and analysis of solenoid actuators.

This paper will focus on model-based nonlinear adaptivecontrol of an actuator-butterfly valve. The solenoid-valvesystem is described based on the exact model knowledgeof the system. Figure 1 shows the integrated system, which


Pinion

Stem

Disk

Plu

nge

r

Coi

l

Coi

l

Yoke

k B1

V0 Dp

VJ

x

α ∼ 5

Figure 1: System Configuration.

consists of an electric-driven solenoid and a butterfly valve.The valve operates by solenoids that use a magnetic coilto move a movable plunger connected with the valve stemby means of a gear train and linkage. The control inputis designed by substituting the current signal from themodel of the electromagnetic force, pulling the plunger, andthen controlling the angular position of the butterfly valve.The system has uncertainties in multiple parameters in thedynamic model, which requires the system to continuouslyadjust to the environment and consequently requires adap-tation for sustainability and capability. The integrated systemis highly nonlinear in addition to its parameter uncertainties.Hence, an adaptation law is proposed [8, 9, 17] and anadaptive control method is developed for the solenoid-valvesystem in multiple parametric uncertainties. A closed-loopstable controller is designed for the set-point trajectorytracking by introducing a Lyapunov-based stability analysis[9] based on the error signals of the nonlinear solenoid-valvesystem. The numerical results in the simulation are used forinitial verification and performance evaluation.

2. Model-Based Nonlinear System

2.1. System Model. The dynamic equations of motion of theplunger and butterfly valve are given by [15]

mx + B1x + kx = Fmag − Fc,

J a + B2α = rgFc − Ttot,(1)

where x(t) is the displacement of the solenoid plunger, α(t) isthe angle of butterfly disk, and Ttot(t) is the sum of the hydro-dynamic and bearing torques expressed as Ttot = Tb + Th.The bearing torque is given as Tb = (π/8)μDsD2

pΔPvCR(α),

where the bearing toque coefficient, CR(α) =√C2L + C2

D =√

(1.1 4√

sin((α/90)3180))2 + ( 3√

cosα)2, is obtained from the

valve modeling (see [5, 15]) and the two subterms, lifting

force CL = 1.1 4√

sin((α/90)3180) and drag force CD = 3√

cosα,are nonlinear functions of the valve angle rotation α(t). Thehydrodynamic torque Th is obtained by reviewing three-dimensional hydrodynamic torque coefficient based on [7,16] as Th = (8/3π)ρD3

pV2OTc(α)[(VJ /VO)(α)]2, where both

Tc(α) and (VJ /VO)(α) depend on the closing angle (α) of thebutterfly valve and D3

p term is a nonlinear term accordingto the pipe size. Solving the two equations in (1) with thecontact force, Fc(t), and substituting the magnetic force,Fmag(t), into the equation yields

(

m +J

r2g

)

rg x +

(

B1 +B2

r2g

)

rg x + krgx

= rgC2N2

2(C1 + C2x)2 i2 − Ttot,

(2)

where the magnetic force Fmag = (C2N2/2(C1 + C2x)2)i2

used to lift the plunger of the solenoid actuator. The actuatoris a current-controlled solenoid [15], proportional to thesquare of the current i(t), and C1, C2 are reluctances ofthe magnetic paths, obtained from the geometry of electricactuator [4, 15]. It is assumed that the pinion and the valveare moving at the same speed, that is, the gear ratio is 1 : 1 andx(t) = rgα is the simple geometric relationship between thedisplacement of the pinion and the valve angle. Hence, thecurrent source i2(t) is substituted for designing the closed-loop control input, u(t), and then the following equation isobtained:

(

m +J

r2g

)

rg x +

(

B1 +B2

r2g

)

rg x + krgx

= rgC2N2

2(C1 + C2x)2 u− Ttot.

(3)

For the subsequent controller design, multiplying (3)with the inverse term of the control input, 2(C1 + C2x)2/(rgC2N2), yields a compact form of the dynamic equation as

M(x, θ)x + C(x, θ)x +D(x, θ)x = u− B(Tb + Th), (4)

where θ describes a lumped expression of parametersobtained from (3) and (4), where each parameter is shownin Table 2 and the substituted terms are defined as follows:

M(x, θ) =(

m +J

r2g

)

rgB, B = 2(C1 + C2x)2

rgC2N2,

C(x, θ) =(

B1 +B2

r2g

)

rgB, D(x, θ) = krgB.

(5)

2.2. Error Signals Formulation. The following set-point con-trol approach is used. Let xd(t) define the set-point trajectoryand then the error can be defined as

e ≡ xd − x, e = xd − x, e = xd − x, (6)


0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

70

80

90

α◦ :

val

ve r

otat

ion

an

gle

(deg

)

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 2: Valve rotation angle α(t) (adaptive approach).

where xd(t) and xd(t) are the first and second time derivativesof xd(t), which are assumed to be bounded. Premultiplyinge(t) in the last error signal of (6) with M(x, θ) yieldsM(x, θ)e = M(x, θ)xd −M(x, θ)x. Substituting M(x, θ)x in(4) into the above equation produces

M(x, θ)e = M(x, θ)xd + C(x, θ)x +D(x, θ)x − u

+ B(Tb + Th).(7)

A filtered error signal and its derivative are defined as

r ≡ e + λ1e, r = e + λ1e, (8)

where λ1 ∈ �+ is a positive adjustable control gain. Multi-plying (8) with M(x, θ) and then substituting for M(x, θ)e(t)in (7) yields

M(x, θ)r = M(x, θ)xd + C(x, θ)x +D(x, θ)x − u

+ B(Tb + Th) + M(x, θ)λ1e +1r2ge − 1

r2ge,

(9)

where the last term e(t)/r2g is added and subtracted for fur-

ther development of the control design based on Lyapunov’smethod.

3. Lyapunov-Based Adaptive Feedback Control

Let V(t) be a Lyapunov candidate function

V = 12

(rTMr + eTe + eTα eα + ΘTΓ−1Θ

), (10)

where the last term of the Lyapunov candidate function, Γ =γIp×p, is a constant diagonal matrix with the gain value γ,Ip×p is a p× p identity matrix, and the parameter estimation

error, Θ, is defined as Θ = Θ− Θ, where Θ ∈ �p is a known

Table 1: Regression and parameter estimation terms.

No. Terms No. Terms

W101 xd W113 CR(α)x

W102 xdx W114 Tc(α)[VJ(α)/VO]2x2

W103 xdx2 W115 CR(α)x2

W104 x W116 xe

W105 xx W117 xe

W106 xx2 W118 xxe

W107 x W119 xxe

W108 x2 W120 e

W109 x3 W121 λ1e

W110 Tc(α)[VJ (α)/VO]2 W122 λ1xe

W111 CR(α) W123 λ1x2e

W112 Tc(α)[VJ(α)/VO]2x — —

Θ101 Ms2C21/(C2N2) Θ113 Tb14C1/(rg N2)

Θ102 Ms4C1/N2 Θ114 Th12C2/(rg N2)

Θ103 Ms2C2/N2 Θ115 Tb12C2/(rg N2)

Θ104 Cs2C21/(C2N2) Θ116 Ms2C1/N2

Θ105 Cs4C1/N2 Θ117 Ms2C1/N2

Θ106 Cs2C2/N2 Θ118 Ms2C2/N2

Θ107 k2C21/(C2N2) Θ119 Ms2C2/N2

Θ108 k4C1/N2 Θ120 1/r2g

Θ109 k2C2/N2 Θ121 Ms2C21/(C2N2)

Θ110 Th12C21/(rg C2N2) Θ122 Ms4C1/N2

Θ111 Tb12C21/(rg C2N2) Θ123 Ms2C2/N2

Θ112 Th14C1/(rg N2) — —

Table 2: List of simulation parameters.

m J rg B1 B2 k N ΔPv

0.1 1.04e−6 1e−2 10 20 4e2 8.8e3 0.5

C1 C2 ρ Dp VO μ Ds p

1.57e6 6.32e8 1e3 5′′ ∼ 8′′ 3.7 0.1 0.1 23

constant parameter vector and Θ ∈ �p is the estimatedconstant parameter vector (see Table 1). Differentiating (10)yields

V = rTMr +12rTMr + eT e +

1r2geT e − ΘTΓ−1 ˙Θ, (11)

where the time derivative of the inertia matrix is obtainedas M = (m + J/r2

g )(4(C1 + C2x)/N2)x, where B = (4(C1 +C2x)/rgN2)x, eTα eα = (1/r2

g )eT e as eα ≡ αd − α = xd/rg −x/rg = e/rg , eα ≡ αd − α = xd/rg − x/rg = e/rg , in which theerror signals of the valve angle, eα(t), can be defined using

the geometric relationship and ˙Θ(t) = − ˙Θ comes from the

definition of Θ.

4 Journal of Control Science and Engineeringx:

plu

nge

r di

spla

cem

ent

(m)

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

00 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 3: Plunger displacement x(t) (adaptive approach).

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

70

80

90

α◦ :

val

ve r

otat

ion

an

gle

(deg

)

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 4: Valve rotation angle α(t) (no adaptation scheme).

3.1. Design of Control Input. Substituting M(x, θ)r(t) into(11) yields

V = rT(

Mxd + Cx +Dx − u + B(Tb + Th) + Mλ1e +1r2ge

)

− (e + e)Ter2g

+12rTMr + eT e +

1r2geT e − ΘTΓ−1 ˙Θ,

(12)

where the last term in (9) premultiplied by r(t) came out ofthe parenthesis in (12) and is used for the definition of r(t)

0.015

0.01

0.005

00 0.5 1 1.5 2 2.5 3 3.5 4

x: p

lun

ger

disp

lace

men

t (m

)

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 5: Plunger displacement x(t) (no adaptation scheme).

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

−0.01

r(t)

: filt

ered

err

or

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 6: Filtered Error r(t) (adaptive approach).

in (8). Then, combining the parameterized terms in (12) andsubstituting them into WΘ yields

WΘ =Mxd + Cx +Dx + B(Tb + Th) + Mλ1e +1r2ge +

12Mr,

(13)

where W(xd, x, x, r,α, e, e) ∈ �1×p is a known regressionvector, which is shown in the left side of Table 1 via theprocess given later (see (17)) and Θ as the nominal value ofthe lumped parameter vector. Rearranging (12) produces

V = rT(WΘ− u)− eTe

r2g

+ eT e − ΘTΓ−1 ˙Θ, (14)


0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

−0.01

e(t)

: dis

plac

emen

t er

ror

(m)

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 7: Displacement tracking error e(t) (adaptive approach).

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

−0.01

x′: t

he

rate

of

disp

lace

men

t (m

/s)

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 8: The rate of displacement x(t).

where eTe/r2g is canceled in the last second term in (12),

having the opposite sign because they are scalar, eTe = eT e.The control input can be designed based on Lyapunov

stability analysis, making V negative definite to be shown inthe end, as

u =WΘ + k1r + e, (15)

where r(t) is a feedback error term, k1 is a positive constantas the control gain, e(t) is another feedback error term addedto cancel the term having the opposite sign, eT e, outside theparenthesis by utilizing the definition of r(t) given in (8), andWΘ captures the uncertainties associated with the elements

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

0

50

100

150

u(t

): c

ontr

ol in

put

(A2)

Figure 9: Control input u(t) equivalent to square current i2(t)[A2].

0

1

2

3

4

5

6

7

8×104

Fm

ag: e

lect

rom

agn

etic

forc

e (N

)

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 10: Electromagnetic force Fmag(t).

of M, C, D, B(Tb1 +Th1), Mλ1, 1/r2g , and M, which is defined

as

WΘ = Mxd+Cx+Dx+B

(

Tb1CR(α)+Th1Tc(α)[VJ

VO(α)]2)

+ Mλ1e+1r2ge+

12

˙Mr,

(16)

where the estimated parameter sets are given as M(x, θ) =(m + J/r2

g )rg B, where B = 2(C1 + C2x)2/rg C2N2, C(x, θ) =

(B1 + B2/r2g )rg B, D(x, θ) = krg B, Tb1 = (π/8)μDsD2

pΔPv,


0

50

100

150

200

250

Tt=Th

+Tb: t

otal

torq

ue

(Nm

)

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 11: Total torque Tt(t).

and Th1 = (8/3π)ρD3pV

2O. In order to develop the estimate

parameter vector Θ in (16), we need to first define theregression and the estimate terms. Thus, the first term, Mxd,in (16) can be defined as

Mxd =(

m +J

r2g

)

rg2(C1 + C2x

)2

rg C2N2xd

= Ms2C2

1

C2N2xd + Ms

4C1

N2xxd + Ms

2C2

N2x2xd

=[xd xxd x2xd

]

︸︷︷︸W101∼W103

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ms2C2

1

C2N2

Ms4C1

N2

Ms2C2

N2

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸Θ101∼Θ103

,

(17)

where Ms = (m + J/r2g ), Cs = (B1 + B2/r2

g ) (this is shown in

Table 1), and rg , C1, or C2 are canceled. W101 ∼W103 are the

measurable regression terms and Θ101 ∼ Θ103 is the estimat-ed parameters, defined in Table 1, respectively. Similarly to(17), the rest of the terms in (16) are also given in Table 1.

3.2. Online Adaptation Laws for Parameter Updates. Thefollowing is constructed to define the known upper andlower bounds but with an unknown parameter of Θ(t) in thesense that

Θj≤ Θ j(t) ≤ Θ j , (18)

0

0.005

0.01

0.015

0.02

0.025

0.03

Tc(α

): to

rqu

e co

effici

ents

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 12: Hydrodynamic torque coefficient Tc(α).

0

20

40

60

80

100

120

140

160

VJ/VO

(α):

th

e ra

tio

of m

ean

flow

an

d je

t ve

loci

ties

0 0.5 1 1.5 2 2.5 3 3.5 4

= 5= 6

78

Time (s)

Dp

Dp

==

Dp

Dp

Figure 13: The ratio of VJ and VO: (VJ /VO)(α).

where Θ j(t) are the estimated parameters as shown in Table 1

and Θj

and Θ j are the lower and upper bounds of the

estimated parameters, respectively, which will be set to the

amount of percentage of their true values. The vector ˙Θ j(t)

is designed to update using a projection-based algorithm as

˙Θ j = Proj

{ΓWTr, Θ j

}, (19)


0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

1.1

1.2

1.3

1.4

1.5×10−5 θ101

8.944

8.946

8.948

8.95

8.952

θ102×10−3

1.8027

1.8027

1.8027

1.8027

1.8027

θ103

20.1068

20.1069

20.107

20.1071θ104

Time (s) Time (s)

Time (s)Time (s)

= 5= 6

78

Dp

Dp

==

Dp

Dp

= 5= 6

78

Dp

Dp

==

Dp

Dp

Figure 14: Parameter estimates: Θ101 ∼ Θ104.

where Proj{·} is the projection operator [8] and each lumpedparameter is adaptively updated using the adaptation laws[17] for online estimation of unknown parameter as follows:

Proj{

˙Θ j

}= Proj

{ΓWTr, Θ j

}

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ΓWTr if Θ j > Θj, Θ j < Θ j ,

ΓWTr if Θ j = Θj, if ΓWTr > 0,

ΓWTr if Θ j = Θ j , if ΓWTr ≤ 0,

0 elsewhere.(20)

Thus, substituting u(t) in (15) into V in (13) yields

V = rTWΘ− rTk1r − (e + e)Te − eTe

r2g

+ eT e − ΘTΓ−1 ˙Θ

= − rTk1r −(

1 +1r2g

)

eTe + ΘT{WTr − Γ−1 ˙Θ

}.

(21)

Here, WΘ is defined as

WΘ = M(xd + λ1e) + Cx + Dx + BTh1Tc(α)VJ

VO(α)

+ BTb1CR(α) + rg e +12˜Mr,

(22)

where M = M − M, C = C − C, D = D − D, BTh1 =BTh1 − BTh1, BTb1 = BTb1 − BTb1, rg = 1/r2

g − 1/r2g ,


Time (s)

Time (s)

Time (s)

Time (s)

0 1 2 3 4

0 1 2 3 4

0 1 2 3 4

0 2 4

1.6207

1.6207

1.6207

1.6207

1.6207

×104θ105

3.2659

3.2659

3.2659

×106θ106

0.04

0.0401

0.0402

0.0403

0.0404θ107

32.4127

32.4128

32.4128

32.4128

32.4128

θ108

= 5= 6

78

Dp

Dp

==

Dp

Dp

= 5= 6

78

Dp

Dp

==

Dp

Dp


and ˜M = M − M, in which Th1 = (8/3π)ρD3pV

2O and Tb1 =

(π/8)μDsD2pΔPv. Actually, Θ(t) can be expressed, for exam-

ple, using the definitions of Θ in (16), as follows: owing to thesubsequent adaptation law, the time derivative of V(t) yieldsa negative definite function except the origin and upperbound by

V ≤ −k1‖r‖2 −(

1 +1r2g

)

‖e‖2, (23)

which can be written as

V ≤ −k2‖z‖2, (24)

where k2 = min{k1, (1 + 1/r2g )} is a positive constant and

z =[rT , eT

]T. (25)

Using Barbalat’s lemma [18], the set-point tracking error‖z(t)‖ → 0, thus ‖r(t)‖ → 0 and ‖e(t)‖ → 0 as t → ∞.

Remark 1. According to the analysis from (11) to (25) ofV , the property of V , and the control law of (15) with theparameter updates of (19) and the projection-based methodof update laws of (20), it is straightforward to derive aconclusion that the tracking error vector z(t) in (25) is drivento zero. Thus, the set-point errors r(t), e(t), and eα(t) alsovanish and the parameter estimation error vector Θ in (10)is bounded where Θ, defined after (10), is bounded due tothe projection-based update method and the constant knownparameter, Θ. Owing to the bounds of r(t), e(t) in (8), e(t)is bounded and then x(t), resulting in α(t), and x(t) arebounded, respectively, where all desired trajectories such asxd(t) and xd(t) are assumed to be bounded. M(·), B(·), C(·),and D(·) matrices in (4) are bounded because θ (due toΘ) and x(t) are bounded, and B and M after (11) are alsobounded owing to the bounds of x(t). Ttot is thus boundedbecause α(t) is bounded. Hence, W(·) and the control inputu(t) are bounded and, thus, the current is bounded. Thisleads to the boundedness of x(t) in the dynamic model given


0 1 2 3 4

0 1 2 3 4 0 2 4

0 1 2 3 4

6531.5313

6531.5313

6531.5313

6531.5313

×10−8

θ109 θ110

θ111 θ112

Time (s) Time (s)

Time (s) Time (s)

= 5= 6

78

Dp

Dp

==

Dp

Dp

= 5= 6

78

Dp

Dp

==

Dp

Dp

0.2

0.4

0.6

0.8

1

0.5

1

1.5

2

2.5

0

200

400

600

800


in (4), which enables the boundedness of e(t) in (6), and thenthe set-point tracking error dynamics r(t) in (8) is bounded.Therefore, we can conclude that all signals are bounded.

Remark 2. WΘ in (16) where the known regression W(xd, x,x, r,α, e, e) ∈ �1×23 terms and the parameter estimates Θ ∈�23 are given in this system as in Table 1.

4. Simulation Results

Based on the dynamic model in (4), the numerical sim-ulation is performed to verify the proposed controllerswith consistently changing parameter values. The parametervalues can be divided into two categories: operational valuessuch as Dp and VO and uncertain values such as B1, B2,and μ. After the flow velocity VO is kept on 3.7 [m/s], thepipe diameter Dp shown in Table 2 is used to vary from 5inches to 8 inches as Dp = {5.0, 6.0, 7.0, 8.0} [in], whereDp = {0.1270, 0.1524, 0.1778, 0.2032}[m], by assuming thatthe pipe or transmission lines are different according totheir applications. With each Dp size, the variations of all

the parameters are set to 30% for the simulation resultsprovided here. The determined parameter vectors Tc(α) and[VJ /VO](α) of the butterfly valve model are borrowed from[16] for the simulation given as look-up tables from theexperimental data. The amount of upper and lower variationof the unknown parameter sets is 30% of their real values.The control gains were chosen selectively as γ = 10, k1 =250, and λ1 = 1.0 for all cases. MATLAB and Simulink areused for the simulation. The desired set-point distance of thesolenoid, xd(t), is given as A · (1 − e−Bt), where A = 0.0148and B = 5.

A typical parameter set for this simulation is given byTable 2. Figures 2 and 3 show the rotation of the valve angleand the actual displacement of the plunger, respectively. Thisadaptive control approach shows better results compared tothe results obtained from the previous research [15] usingnonadaptive method, which are shown in Figures 4 and 5 forthe displacement of the plunger and the angle of the butterflyvalve. The developed mathematical model is the same asthat of adaptive method based on the nonlinear models andthe new specific approach in this paper is that the error


0

0.5

1

1.5

2

0

0.5

1

1.5

2

1

2

3

4

4.4725

4.473

4.4735

4.474

0 1 2 3 4

0 1 2 3 4 0 2 4

0 1 2 3 4

×10−5 ×105

×10−3 ×10−3

θ113 θ114

θ115 θ116

Time (s) Time (s)

Time (s) Time (s)

= 5= 6

78

Dp

Dp

==

Dp

Dp

= 5= 6

78

Dp

Dp

==

Dp

Dp


signals are formulated by introducing the desired trajectoryto reducing the displacement error in the adaptive methodwhile overcoming the complicated parametric uncertainties.And also the response of the adaptive method shows that themotion process is more consistent in both variables, showingoverdamped phenomena and short rising times, and thenit quickly reaches the steady state. The actual movement issmoother and faster while all the parameters are varying 30%in each flow velocity.

Figures 6 and 7 are the filtered error r(t), defined in (8),and set-point error e(t), given in (6), respectively. Figure 8shows the rate of the displacement to Figure 3. From Figure 8,all figures are presented without the notation of the adaptiveapproach. Figure 9 shows the control input u(t) of the sole-noid actuator for each pipe diameter, Dp, designed in (15)with the nonlinear adaptive controller by substituting thesquare of the current, i2(t). The electromagnetic force Fmag(t)given in (2) and the total torque Tt(t) given in (1) by sum-ming up the hydrodynamic and bearing torques are plotted in Figures 10 and 11, respectively. The torque coefficient Tc(t)

shown in Figure 12 and the ratio of input and output jetvelocities shown in Figure 13 are changed for every 5◦ of thebutterfly angle. As the strokes are increased by the controlinput, the angles get larger and then accordingly the valueof the inlet jet velocity increases, which affects the slowermotion of the strokes and angles but the variables (strokesand angles) reach the steady state and the ratio as well.

The challenge is that most parameter terms in Table 1 arecombined and lumped together due to the dynamic modelin the presence of parameter uncertainty and the model iscomplicated owing to the control objectives, set-point regu-lation, and parametric adaptation. As given in the right sideof Table 1, the unknown bounded parameter estimate vectorΘ ∈ �23 is shown in Figures 14, 15, 16, 17, 18, and 19. It canbe seen that the parameters such as 101, 110, 111, 113, 116,118, and 121 are quickly updated in the form of premultipli-cation by its regression term, W and go to steady state. Thus,they are more parametric-centric terms and related to thefiltered error r(t). The parameters such as 102, 103, 104, 105,117, 119, and 122 are updated according to the motion of


4.473

4.473

4.473

4.4731

1.8027

1.8027

1.8027

1.8027

1.8027

1.8027

1.8027

1.8027

1.8027

1

1

1

1

1

0 2 4 0 1 2 3 4

0 1 2 3 40 1 2 3 4

×104

×10−3 θ117 θ118

θ119 θ120

Time (s) Time (s)

Time (s) Time (s)

= 5= 6

78

Dp

Dp

==

Dp

Dp

= 5= 6

78

Dp

Dp

==

Dp

Dp


the angle and stroke of the solenoid-butterfly valve system.Some parameters such as 106, 112, 114, and 115 are notupdated but other parameters such as 107, 108, and 120 keepgetting updated. Any parameters changing their values mustbe related to the control objective because these terms areincorporated into the control input in the form of the esti-mates.

5. Conclusion and Future Work

For developing advanced automation systems such as ship-based hydraulic systems, a typical solenoid-butterfly valve,which is driven by electromagnetic, fluid mechanics, andhydrodynamic forces and torques, is chosen as a continuationof previous research and an adaptive stable control approachwith adaptation laws is developed accounting for uncertain-ties in multiple parameters on the nonlinear dynamic model.A stable adaptive controller of the solenoid-valve system isdesigned positioning the angle of the butterfly valve via aLyapunov-based approach. The approach yields bounded

error while adapting to the environment in the presence ofcomplex uncertainties such as different physical appearances,uncertain parameters, operational characteristics, and para-metric nonlinear dynamic models.

The parameter estimation for the unknown boundedparameters is performed using a projection algorithm whoseoutput yields the upper and lower bounds. Numericalsimulation is used to verify the performance of the proposedapproach to show its effectiveness by comparing to thesame dynamic model without adaptation from the previousresearch; when compared to the nonadaptive method, theresponses of the plunger displacement and the rotatingangle are steadier, smoother, and faster. Future work willbe focused on demonstrating the results of hardware-in-the-loop or experiments for the nonlinear solenoid valve systemas well as applying the suggested adaptive method based onLyapunov-based control approach to the real-world system.Further research on developing control techniques usingrobust or optimal method would be continued to overcomenonlinearities such as hysteresis or nonlinear dynamics.


1.1

1.2

1.3

1.4

1.5

8.9459

8.9459

8.9459

8.9459

8.9459

1.8027

1.8027

1.8027

1.8027

1.8027

0 2 4 0 1 2 3 4

0 0.5 1 1.5 2 2.5 3 3.5 4

×10−3×10−5 θ121 θ122

θ123

Time (s) Time (s)

Time (s)

= 5= 6

78

Dp

Dp

==

Dp

Dp


Nomenclature

B1, B2: Damping coefficients of thesolenoid and butterfly valve,respectively (Ns/m)

C1, C2: Reluctances of the magnetic pathsobtained from plunger geometry ofsolenoid actuator

CR(α): Bearing torque coefficient as afunction of the valve angle (α)

Ds: Stem diameter (m)Dp: Pipe diameter (inch or m)Fmag: Magnetic force (N)Fc: Contact force or resultant force

Fr(N)i: Current of solenoid actuator (A)J : Inertia moment (kgm2)k: Spring stiffness (N/m)m: Mass of solenoid plunger (kg)μ: Friction coefficient of bearing areaN : Number of turns of the coilΔPv: Valve differential pressure (psi)p: Number of estimatesrg : Radius of pinion gear (m)ρ: Fluid mass density (kg/m3)u: Control input (A2)

Tc(α): Hydrodynamic torque coefficient as afunction of the valve angle (α)

Tb, Th: Bearing and hydrodynamic torques,respectively (Nm)

VJ , VO, (VJ /VO)(α): Jet velocity, mean flow velocity, andtheir ratio as a function of the valveangle (α), respectively.

Acknowledgments

This research is supported by the Office of Naval Research(N00014-08-1-0435), which the authors gratefully acknowl-edge. Thanks are in particular due to Mr. Anthony SemanIII. The authors would also like to thank Dr. Stephen Mastroand Mr. Frank Ferrese of Naval Surface Warfare Center(NSWC, Philadelphia) for help with many aspects of thepaper. They are grateful to the anonymous reviewer forcritical comments, which led to substantial improvement ofthe paper.

References

[1] A. Seman, “Adaptive automation for machinery control,” inProceedings of the Office of Naval Research (ONR) Presentation,2007.


[2] R. Hughes, S. Balestrini, K. Kelly, N. Weston, and D. Mavris,“Modeling of an integrated reconfigurable intelligent system(IRIS) for ship design,” in Proceedings of the ASNE Ships & ShipSystems Symposium, 2006.

[3] P. Tich, V. Mark, P. Vrba, and M. Pechoucek, “Chilled watersystem automation,” Rockwell Automation Case Study Report,2005.

[4] J. R. Brauer, Magnetic Actuators and Sensors, Wiley IEEE Press,Hoboken, NY, USA, 2006.

[5] Z. Leutwyier and H. Dalton, “A CFD study of the flow field,resultant force, and aerodynamic torque on a symmetric diskbutterfly valve in a compressible fluid,” Journal of PressureVessel Technology, vol. 130, Article ID 021302, pp. 1–10, 2008.

[6] T. Sarpkaya, “Oblique impact of a bounded stream on a planelamina,” Journal of the Franklin Institute, vol. 267, no. 3, pp.229–242, 1959.

[7] T. Sarpkaya, “Torque and cavitation characteristics of butterflyvalves,” Journal of Applied Mechanics, vol. 29, pp. 511–518,1961.

[8] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinearand Adaptive Control Design, Series on Adaptive and LearningSystems for Signal Processing, John Wiley & Sons, New York,NY, USA, 1st edition, 1995.

[9] D. B. Lee, C. Nataraj, and P. Naseradinmousavi, “Nonlinearmodel-based adaptive control of a solenoid-valve system,” inProceedings of the ASME Dynamic Systems and Control Con-ference (DSCC ’10), pp. 1–8, Boston, Mass, USA, September2010.

[10] J. Pohl, M. Sethson, P. Krus, and J.-O. Palmberg, “Modellingand validation of a fast switching valve intended for combus-tion engine valve trains,” Proceedings of the Institution of Mech-anical Engineers. Part I: Journal of Systems and Control Engi-neering, vol. 216, no. 2, pp. 105–116, 2002.

[11] C. Nataraj, Vibration of Mechanical Systems, Cengage, 1st edi-tion, 2011.

[12] P. Naseradinmousavi and C. Nataraj, “Nonlinear mathemati-cal modeling of butterfly valves driven by solenoid actuators,”Applied Mathematical Modelling, vol. 35, no. 5, pp. 2324–2335,2011.

[13] M. K. Zavarehi, P. D. Lawrence, and F. Sassani, “Nonlinearmodeling and validation of solenoid-controlled pilot-operat-ed servovalves,” IEEE/ASME Transactions on Mechatronics, vol.4, no. 3, pp. 324–334, 1999.

[14] R. R. Chladny, C. R. Koch, and A. F. Lynch, “Modeling auto-motive gas-exchange solenoid valve actuators,” IEEE Transac-tions on Magnetics, vol. 41, no. 3, pp. 1155–1162, 2005.

[15] C. Nataraj and P. Mousavi, “Nonlinear analysis of solenoidactuators and butterfly valve systems,” in Proceedings of the14th International Ship Control Systems Symposium, pp. 1–8,Ottawa, Canada, September 2009.

[16] J. Y. Park and M. K. Chung, “Study on hydrodynamic torqueof a butterfly valve,” Journal of Fluids Engineering, Transactionsof the ASME, vol. 128, no. 1, pp. 190–195, 2006.

[17] J. B. Pomet and L. Praly, “Adaptive nonlinear regulation: esti-mation from the Lyapunov equation,” IEEE Transactions onAutomatic Control, vol. 37, no. 6, pp. 729–740, 1992.

[18] H. K. Khalil, Nonlinear Systems, Prentice Hall, Upper SaddleRiver, NJ, USA, 3rd edition, 2002.


Research Article

General Form of Model-Free Control Law andConvergence Analyzing

Xiuying Li,1, 2 Guanghui Wang,1 and Zhigang Han1, 2

1 Key Laboratory of Electronic Engineering, College of Heilongjiang Province Heilongjiang University, Harbin 150080, China2 Department of Automation, School of Electronic Engineering, Heilongjiang University, Harbin 150080, China

Correspondence should be addressed to Zhigang Han, [email protected]

Received 31 July 2011; Revised 25 March 2012; Accepted 12 April 2012

Academic Editor: Chengyu Cao

Copyright © 2012 Xiuying Li et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The general form of model-free control law is introduced, and its convergence is analyzed. Firstly, the necessity to improve the basicform of model free control law is explained, and the functional combination method as the approach of improvement is presented.Then, a series of sufficient conditions of convergence are given. The analysis denotes that these conditions can be satisfied easily inthe engineering practice.

1. Introduction

The model-free control method we proposed is of greatsignificance in theory and practical applications. The overallframework to design the model-free control law is given in[1, 2], where the following characteristics are also described.Firstly, it is an adaptive control law not only in parameterbut also in structure. Secondly, it is a nonlinear controllerby breaking through the linear framework constraints ofPID [3]. Thirdly, it is designed by the new thought ofmodeling and control, that is, “the integration of modelingand control” approach [4]. Since the model-free controlmethod has been put forward, it has been applied to the oilrefining [5, 6], chemical [7, 8], power [9], glass [10], andother industries. References [11, 12] give a brief overviewof its application and progress. In the actual context of theindustrial process, the unit control issue of complex systemsis analyzed in [13], the cascade form and its applicationeffects are analyzed in [14], and the control functions ofmodel-free control law are analyzed in [15–17]. It has beenillustrated that the model-free control law has a strongability of antidisturbance [18] and the ability to overcomethe large time-delay [19], as well as a strong decouplingperformance [20]. In general, the practical application resultsare richer than theory research of model-free control law. It is

imperative to do more in-depth theoretical study for furtherapplication of this method.

We published the article [4] with the topic of “theintegration approach of modeling and adaptive control” in2004, in which we put forward the “universal model” and“the basic form of model-free control law” under the ideaof the integration of modeling and control and analyzedits convergence. The control results are satisfactory forsimple objects, but not pleasing for complex. So in theengineering application, the model-free control law is usednot in its basic form, but some improved forms. Theseimproved forms of model-free control law have receivedmore satisfactory results for the objects difficult to control.To lay the theoretical basis for practical applications, it isnecessary to give the theoretical analysis for these improvedforms. This paper is written under such circumstances.

For the universality of the theoretical research, all theimproved model-free control law can be written in a unifiedform, which called as the general form of model-free controllaw.

In this paper, the necessity and method of the generalform of model-free control law are discussed firstly, thenthe convergence of the control law is analyzed, and someconvergent conditions are also given. It can be seen ascontinued and expanded of the reference [4].


2. Necessity and Approach of ImprovingModel-Free Control Law

In the literature [4], we had discussed the integrationapproach of modeling and adaptive control and obtained theintegration framework of modeling and feedback control forthe systems without model, which are given as follows.

(1) Based on observation data and the universal model ofsystem S

y(k)− y(k − 1) = ϕ(k − 1)T{u(k − 1)− u(k − 2)}, (1)

we can get the estimation value ϕ(k − 1) of ϕ(k − 1)by using some appropriate estimation method.

(2) To look for the one step prediction value ϕ∗(k) ofϕ(k − 1), a simple way is to take

ϕ∗(k) = ϕ(k − 1). (2)

We still write ϕ∗(k) as ϕ(k) for seeking control law,and based on universal model, we can derive thecontrol law:

u(k) = u(k − 1) +λk

a +∥∥ϕ(k)

∥∥2 ϕ(k)

{y0 − y(k)

}. (3)

(3) Under the effect of control law (3), we can obtain anew output y(k + 1) of system S. And then we get agroup of new data {y(k + 1),u(k)}.

On the basis of this new group data, we repeat theprocedure (1), (2), and (3), can also get the new data {y(k +2),u(k + 1)}, and continue to do it. It can be proven that, aslong as the system S satisfies certain conditions, the outputy(k) of system S will be gradually approaching to y0 undersuch procedures.

We call the control law (3) as the basic form of model-freecontrol law. In (3), λk is named as the control parameter, a isa small positive number, which can avoid the denominatorbeing zero, ϕ(k) is the estimation value of the characteristicparameter ϕ(k).

It is not difficult to find in practical application, thecontrol results for the simple objects of control law (3) aresatisfied, but the results for more complex objects cannot besatisfactory. This also can be seen by the following examples.

Consider the system described by the following model:

y(k) = 1.3y(k − 1)− 0.42y(k − 2) + 0.5u(k − 1). (4)

It is controlled by using the basic form of model-freecontrol law (3). Let λk = 1, ϕ(k) = 100,A = 1, and thesetting value is y0 = 70. The control result is shown inFigure 1.

When the system is changed into

y(k) = 1.3y(k − 1)− 0.31y(k − 2) + 0.5u(k − 1), (5)

we still use the aforementioned control law to control thissystem (parameters remain the same). The result is shown inFigure 2.

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

80

90

100

y(k)

u(k)

k (step)

y0

Figure 1: Control result of system (4).

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

y(k)

u(k)

k (step)

y0


This case shows that, when the performance of controlledsystem changes, the control result by using the control law (3)can not be satisfactory.

If the system is changed into

y(k) = 1.3y(k − 1)− 0.28y(k − 2) + 0.5u(k − 1), (6)

then the control result is divergence under the effect ofcontrol law (3) (graphics is omitted). This confirms the abovejudgment. Thus it is necessary to improve the control law (3).

The way of improvement is to add functional combina-tion partD[Yk−n

k−1 ,Uk−mk−1 , θ, k] on the basic form of model-free

control law.The functional combination part D[Yk−n

k−1 ,Uk−mk−1 , θ, k] is

obtained by the functional combination approach for con-troller’s design [1, 2]. The starting point is that the controlleris derived from the controlled object’s functional requests tothe control law but does not depends on the mathematicalmodel of the controlled object. The controlled system(object) has some basic requirements to the control law,typically such as: “overcoming deviation” and “convergenceaccelerating”, which can be explained from Figure 2. Thedifference between the maximum value of system respondsand the set value be smaller, the performance of the system isthe better. We call this function of control law as “overcoming


deviation”. Moreover, the time of the system achieved theset value be shorter, the convergence of the system is thebetter. We call this function of control law as “convergenceaccelerating”.

So the procedure of controller’s design is to find the rep-resentations of these functions and combine them in someproper way. Practice shows that there will be great differencesamong the controllers’ performances if these representationsof the functions and the combination modes are different. Sothe basic contents of the functional combination method are(1) to find all kinds of functions that the controlled objectrequests, which are called as element function set; (2) togive each basic function (called element function) a kindof representation, best of all in the algorithmic form; (3) tocombine these functions in some optimal way.

In the application scene, we call the procedures to findthe proper functional combination as the configuration ofmodel-free control law. For a different controlled object, theconfiguration process is also different; thus we can obtain themodel-free control law applied to the controlled object.

The improved one is called as the general form of model-free control law. There are two expressions of it.

(1) The form easy to analyze:

u(k) = u(k − 1) +λk

a +∥∥ϕ(k)

∥∥2 ϕ(k)

×{A +D

(Yk−nk−1 ,Uk−m

k−1 , θ, k)}(

y0 − y(k)),

(7)

where D(· · · ) is a proper function, which denotesthe functional combination part of control law, and:

Yk−nk−1 =

{y0, y(k − 1), . . . . . . , y(k − n)

}

Uk−mk−1 = {u(k − 1), . . . . . . ,u(k −m)}

(8)

a is a small positive number, A and θ are theconfiguration parameters, A > 0, θ is a nonnegativevector, that is, all of its components are nonnegative.m and n are positive integers.

(2) The form easy to apply:

u(k) = u(k − 1) +λk

a +∥∥ϕ(k)

∥∥2 ϕ(k)

{A(y0 − y(k)

)+G


k−1 , θ, k)}

,

(9)

where G(· · · ) is a proper function, and it is thefunctional combination part of the control law.

Obviously the two forms can mutually convert. Thispaper will analyze the control law (7). Therefore, assumethat the function D[Yk−n

k−1 ,Uk−mk−1 , θ, k] satisfies the following

conditions.

(1) When θ = 0, we have

D[Yk−nk−1 ,Uk−m

k−1 , 0, k]= 0. (10)

0 100 200 300 400 500 600 700 800 900 10000

10

20

30

40

50

60

70

80

90

100

y(k)

u(k)

k (step)

y0


(2) D[Yk−nk−1 ,Uk−m

k−1 , θ, k] is a continuous function of θ,and the continuity to other variable is consistent.

To control the system,

y(k) = 1.3y(k − 1)− 0.31y(k − 2) + 0.5u(k − 1) (11)

by using the general form of model-free control law, we stilllet λk = 1, ϕ(k) = 100 and A = 1 and select the appropriateparameter θ. The control result is shown in Figure 3.

It can be seen that the functional combination part playsa huge role in improving the control performance. Thisfurther confirms the necessity to improve the basic form ofmodel-free control law.

Further, Let us give an application example. We haveintroduced a model-free control scheme with functionalcombinations for the propylene flow system in the carbonylcompose reactor of fourth-octanol device of the secondchemical plant of the Daqing Petrochemical Company. Thecontrolled system is a complex system with large time delayand strong disturbances. Under the control of DCS of theHoneywell Corporation, many sets of this system cannotbe controlled steadily. We change the original DCS withmodel-free control method and have obtained good effectsin practice.

The 24-hour scene record plan before and after themodel–free control law applied is given in Figure 4, fromwhich we can see that the model-free control method canobtain a good control performance.

We had analyzed the composition of functional com-bination part in [21]. In this paper, we will focus on theconvergence after adding functional combination part to thebasic form of model-free control law.

3. Convergence Analysis of General Form ofModel-Free Control Law

We will need the following lemma in the discussion.


Figure 4: The scene record plan of model-free control method.

Lemma 1. For B > 0, there exists the constant vector θ0 > 0,such that when the configuration parameter θ satisfies 0 ≤θ < θ0, the functional combination part D[Yk−n

k−1 ,Uk−mk−1 , θ, k]

constantly holds:

B >∣∣∣D(Yk−nk−1 ,Uk−m

k−1 , θ, k)∣∣∣. (12)

In fact, from conditions (1) and (2) that the functionD[Yk−n

k−1 ,Uk−mk−1 , θ, k] satisfied, it can be concluded that is for

Yk−nk−1 ,Uk−m

k−1 , and k, we consistently have

�imθ→ 0


k−1 , θ, k]= 0. (13)

So for B > 0, there must exist η > 0, such that when‖θ‖ < η, we constantly have


k−1 , θ, k)∣∣∣. (14)

Then (12) holds as long as we choose θ0 to satisfy ‖θ0‖ <η. We call the configuration parameter θ that satisfies theabove lemma as being useable.

In order to design the model-free control law, we havediscussed the so-called linearization problem of nonlineardynamic systems in [22], where we illuminate that thefollowing nonlinear model

y(k) = f[Yk−nk−1 ,u(k − 1),Uk−m

k−2 , k]

(15)

can be described in real time by the following universalmodel:

y(k)− y(k − 1) = ϕ(k − 1)T{u(k − 1)− u(k − 2)}, (16)

combining with the control law in the sense of the input-output equivalence. Obviously we can rewrite the aboveformula as the following:

y(k + 1)− y(k) = ϕ(k)T{u(k)− u(k − 1)}. (17)

According to the universal model, we can prove thefollowing convergence theorem.

Theorem 2. Consider the control law:

u(k) = u(k − 1) +λk

a +∥∥ϕ(k)

∥∥2 ϕ(k)

×{A +D


k−1 , θ, k)}(

y0 − y(k)).

(18)

Suppose that there exists N > 0, such that, when k > N , ϕ(k)and its estimation value ϕ(k) constantly satisfy the relation:

ϕ(k)− ϕ(k) = ε(k)

− α

2≤ ε(k)τ ϕ(k) ≤ α

2,

α ≤ ∥∥ϕ(k)∥∥2 ≤ β,

(19)

where α and β are both positive constants. Then there existsthe constant λk in the control law (18), such that for A > 0,when the configuration parameter θ is useable, the output of thecontrolled system under the effect of this control law constantlyholds:

�imh→∞

y(k + h) = y0. (20)

Proof. In the control law (18), let

F(k) = A +D(Yk−nk−1 ,Uk−m

k−1 , θ, k). (21)

Because the configuration parameter θ is useable, for thegiven B > 0 (A > B), there exists the constant vector θ0 > 0such that, when the configuration parameter θ satisfies 0 ≤θ < θ0, we have


k−1 , θ, k)∣∣∣; (22)

that is,

B > D(Yk−nk−1 ,Uk−m

k−1 , θ, k)> −B. (23)

Properly choose A to satisfy A− B = ρ > 0; then we have

A + B > A +D(Yk−nk−1 ,Uk−m

k−1 , θ, k)> A− B = ρ. (24)

Let F(k) = A +D(Yk−nk−1 ,Uk−m

k−1 , θ, k); thus

A + B > F(k) > ρ > 0. (25)

Now the control law (18) is changed into

u(k) = u(k − 1) +λkF(k)

a +∥∥ϕ(k)

∥∥2 ϕ(k)

{y0 − y(k)

}. (26)

Let

Δk = ϕ(k)Tϕ(k)

a +∥∥ϕ(k)

∥∥2 . (27)

Notice that

Δk = ϕ(k)Tϕ(k)

a +∥∥ϕ(k)

∥∥2 =∥∥ϕ(k)

∥∥2 + ε(k)Tϕ(k)

a +∥∥ϕ(k)

∥∥2 . (28)

Therefore

0 ≤∥∥ϕ(k)

∥∥2 − α/2∥∥ϕ(k)

∥∥2 + a

≤ Δk ≤∥∥ϕ(k)

∥∥2 + α/2∥∥ϕ(k)

∥∥2 + a

. (29)


We can derive from (17) and (25) that

y(k + 1)− y(k) = ϕ(k)T[u(k)− u(k − 1)]

= ϕ(k)TλkF(k)

a +∥∥ϕ(k)

∥∥2 ϕ(k)

{y0 − y(k)

}

= λkF(k)Δk{y0 − y(k)

}.

(30)

Consequently we have

y0 − y(k + 1) = y0 − y(k)− λkF(k)Δk{y0 − y(k)

}

= (1− λkF(k)Δk){y0 − y(k)

}.

(31)

Generally, for the positive integer h, we have

∣∣y0 − y(k + h)∣∣ =

h−1∏

j=0

(1− λk+ jF(k)Δk+ j

)∣∣y0 − y(k)∣∣.

(32)

So as long as λk+ jF(k) ≥ 0, we can obtain∣∣y0 − y(k + h)

∣∣

≤ ∣∣y0 − y(k)∣∣h−1∏

j=0

∣∣∣∣∣1− λk+ jF(k)

∥∥ϕ(k)

∥∥2 − α/2

∥∥ϕ(k)

∥∥2 + a

∣∣∣∣∣

≤ ∣∣y0 − y(k)∣∣h−1∏

j=0

∣∣∣∣∣1− λk+ jF(k)

α− α/2β + a

∣∣∣∣∣

= ∣∣y0 − y(k)∣∣h−1∏

j=0

∣∣∣∣∣1− λk+ jF(k)

α

2(β + a

)

∣∣∣∣∣

≤ ∣∣y0 − y(k)∣∣h−1∏

j=0

∣∣∣∣∣1− λk+ j

αρ

2(β + a

)

∣∣∣∣∣.

(33)

It can be seen that, as long as λk+ j = λ satisfying λ = (β +a)/αρ, we have

∣∣y0 − y(k + h)

∣∣ ≤ ∣∣y0 − y(k)

∣∣(

1− 12

)h

= ∣∣y0 − y(k)∣∣(

12

)h−→ 0 (h −→ ∞);

(34)

that is,

�imh→∞

y(k + h) = y0. (35)

The proof is completed.

Theorem 3. Consider the control law:

u(k) = u(k − 1) +λk

a +∥∥ϕ(k)

∥∥2 ϕ(k)

×{A +D


k−1 , θ, k)}(

y0 − y(k)).

(36)

Suppose that there exist the constants δ > 0, β > 0 and N > 0,such that, when k > N , ϕ(k) and its estimation value ϕ(k)constantly satisfy

δ ≤ ϕ(k) ≤ β,

δ ≤ ϕ(k) ≤ β.(37)

Then there exists the constant λk in the control law (36), suchthat for A > 0, when the configuration parameter θ is useable,the output of the system under the effect of the control lawconstantly holds:

�imh→∞

y(k + h) = y0. (38)

Proof. Let

F(k) = A +D(Yk−nk−1 ,Uk−m

k−1 , θ, k). (39)

Notice that

Δk = ϕ(k)Tϕ(k)

a +∥∥ϕ(k)

∥∥2 . (40)

So we have

δ2

a + β2≤ Δk ≤ β2

a + δ2. (41)

From the control law (36) and the universal model (17),we can easily obtain

y0 − y(k + h) =h−1∏

j=0


)(y0 − y(k)

). (42)

Therefore

∣∣y0 − y(k + h)

∣∣ =

h−1∏

j=0

∣∣∣1− λk+ jF(k)Δk+ j

∣∣∣∣∣y0 − y(k)

∣∣.

(43)

As long as λk+ j > 0, we can derive from the above formulathat

∣∣y0 − y(k + h)∣∣ ≤ ∣∣y0 − y(k)

∣∣h−1∏

j=0

∣∣∣∣∣1− λk+ jF(k)

δ2

a + β2

∣∣∣∣∣.

(44)

Recall the inequality (25); that is,

A + B > F(k) > ρ > 0; (45)

then we have

λk+ jρ < λk+ jF(k) < λk+ j(A + B). (46)

Notice the formula (41), we can further have

λk+ jρδ2

a + β2≤ λk+ jF(k)Δk ≤ λk+ j(A + B)

β2

a + δ2. (47)


Thus it can be seen that, as long as we choose λk = λ as

λ = a + β2

3ρδ2, (48)

we can obtain

∣∣y0 − y(k + h)∣∣ =

h−1∏

j=0


)∣∣y0 − y(k)∣∣

<h−1∏

j=0

(1− 1

3

)∣∣y0 − y(k)

∣∣ =

(23

)h

× ∣∣y0 − y(k)∣∣.

(49)

So we have

�imh→∞

∣∣y0 − y(k + h)∣∣ = 0; (50)

that is,

�imh→∞

y(k + h) = y0. (51)

The proof is completed.

4. Analysis of Convergent Conditions andProblem Discussion

Here we will analyze the significance of the conditions inthe convergence theorem and discuss the general applicationform of model-free control law. Two results are given onthe convergence, and it can be seen that these results areonly the sufficient conditions. If we suppose that the inputis one dimension as well as output, these conditions can beconcluded into the following.

(1) There exists N > 0, such that when k > N , ϕ(k) andits estimation value ϕ(k) permanently hold:

ϕ(k)− ϕ(k) = ε(k),

a

2≤ ε(k)ϕ(k) ≤ a

2,

a ≤ ∥∥ϕ(k)∥∥2 ≤ b.

(52)

(2) There exist constants δ > 0, β > 0 and N > 0, suchthat, when k > N , ϕ(k) and its estimation value ϕ(k)permanently hold:

δ ≤ ϕ(k) ≤ β,

δ ≤ ϕ(k) ≤ β.(53)

Each of conditions (1) and (2) is sufficient to make thecontrol law (36) converge.

According to [4], ϕ(k) is the approximate of (∂y(k +1))/(∂u(k)) in a sense, and ϕ(k) is the estimation value

of ϕ(k). In the engineering, ϕ(k) is actually the grads(derivation) of the output y(k + 1) to the input u(k)of the system, so the engineering practical significance ofconditions (1) and (2) is obvious.

About condition (1) that the functionD[Yk−nk−1 ,Uk−m

k−1 ,θ, k]satisfies, because θ = 0 means that the function D[Yk−n

k−1 ,Uk−mk−1 ,θ, k] does not contain any element function, we have


k−1 , θ, k]= 0. (54)

About condition (2), because D[Yk−nk−1 ,Uk−m

k−1 , θ, k] is com-posed of the mathematical expressions of element functionsby their algebraic operation, it is a continuous function ofθ, and the continuity is consistent to the remaining variable.This requirement is natural.

We will preliminarily discuss the other form easy to applyof model-free control law. The expression is

u(k) = u(k − 1) +λk

a +∥∥ϕ(k)

∥∥2 ϕ(k)

×{A(y0 − y(k)

)+G


k−1 , θ, k)}.

(55)

Considering the form easy to analyze, the expression isgiven by

u(k) = u(k − 1) +λk

a +∥∥ϕ(k)

∥∥2 ϕ(k)

×{A +D


k−1 , θ, k)}(

y0 − y(k)).

(56)

It can be seen that as long as we let


k−1 , θ, k](y0 − y(k)

) = G[Yk−nk−1 ,Uk−m

k−1 , θ, k]

(57)

the control law (56) can be changed into (55), butG[Yk−n

k−1 ,Uk−mk−1 , θ, k] need not to be expressed by the form:


k−1 , θ, k](y0 − y(k)

). (58)

So form (55) has more universality than form (56). Of coursethe convergence of form (55) should be analyzed, but it is notcarried out in this paper.

5. Conclusion

Model-free control law is designed by using the traditionalPID controller for reference, which does not depend onthe mathematical model of controlled object. Through thenew understanding of the P, I, D control functions, weproposed the functional combination idea for controller’sdesign. We can obtain the general form of model-free controllaw by adding the functional combination part on the basicform. The improved model-free control law is a nonlinearcontroller, which breaks through the linear combinationframework of PID. It is widely used in the actual productionprocess control. Some complex systems that are not well


controlled by PID can basically realize stability control byusing general form of model-free control law.

In this paper, we have introduced the general form ofmodel-free control law and analyzed the convergence of oneform. It is shown that the conditions of the convergencetheorem can be satisfied in the engineering. There are someother researches on convergence of model-free control law[23, 24], but in general, the results of convergence we haveobtained are still preliminary. This shows that the theoryresearch of model-free control law is behind its appliedresearch currently.

Acknowledgments

This work is supported by the Key Laboratory of ElectronicsEngineering, College of Heilongjiang Province Foundationunder Grant DZZD20100023, and by the Student InnovationLaboratory Foundation of Heilongjiang University underGrant CX11143.

References

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[9] P. Ma, W. Li, G. W. Zheng, S. Y. Zhang, and W. W. Ning, “Mainsteam temperature control system based on MFAC,” ElectricPower Science and Engineering, no. 1, pp. 19–21, 2006.

[10] X. X. Jiang, H. Wu, G. Zeng, and Z. G. Han, “Application studyof model free control method,” Control Engineering of China,vol. 14, no. 1, pp. 24–26, 2007.

[11] Z. G. Han, “The application of model free controller,” ControlEngineering of China, vol. 9, no. 4, pp. 22–25, 2004.

[12] Z. G. Han, “The progress of the theory and applicationof model free controller,” Techniques and Application ofAutomation, vol. 23, no. 2, pp. 13–16, 2004.

[13] Z. G. Han, “The approach of functional combination ofthe controller design for large complex system,” ControlEngineering of China, vol. 11, no. 2, pp. 103–107, 2004.

[14] Z. G. Han and G. Q. Wang, “Cascade scheme of model free

control law and its application,” Acta Automatica Sinica, vol.32, no. 3, pp. 345–352, 2006.

[15] A. P. Jiang, X. Y. Li, and H. Wu, “Control function analysis ofmodel free control law,” Control Engineering of China, vol. 14,no. 1, pp. 14–17, 2007.

[16] X. Y. Li, Y. Shen, and Z. G. Han, “Optimal design of controllerbased on functional combination,” in Proceedings of the 7thAsian Control Conference (ASCC ’09), pp. 1375–1380, HongKong, August 2009.

[17] J. Y. Xue, L. Tu, and Z. G. Han, “Performance analysis of modelfree control method,” Control Engineering of China, vol. 16, no.5, pp. 531–542, 2009.

[18] Q. B. Luo, X. Y. Li, and Z. G. Han, “Analysis of disturbance-resistant ability of model free control method,” Journal ofSystem Simulation, vol. 20, no. 13, pp. 3472–3476, 2008.

[19] J. Y. Xue, H. Wu, and Z. G. Han, “On study of model freecontrol system applied to complex large scale time delaysystems,” Techniques and Application of Automation, vol. 23,no. 4, pp. 1–6, 2004.

[20] Z. G. Han, A. P. Jiang, and G. Q. Wang, “Study on controlof multivariable coupling systems with model free controlmethod,” Control and Decision, vol. 19, no. 10, pp. 1155–1162,2004.

[21] A. P. Jiang, X. Y. Li, and Z. G. Han, “Constituting and analyzingthe functional combination of model free control law,” ControlEngineering of China, vol. 13, no. 5, pp. 494–497, 2006.

[22] Z. G. Han, A. P. Jiang, and H. Q. Wang, Adaptive Identification,Prediction and Control-the Multi-Level Recursive Approach,Heilongjiang Education Press, Heilongjiang, China, 1995.

[23] Z. G. Yu and J. Zhao, “Convergence analysis of model freecontrol law with primary pattern,” Control Engineering ofChina, vol. 16, no. 2, pp. 130–132, 2009.

[24] T. Z. Zhang and Z. G. Han, “The convergence analysis of thegeneral form of model free controller,” Electric Machines andControl, vol. 10, no. 3, pp. 333–340, 2006.


Research Article

Adaptive Control for Nonlinear Systems withTime-Varying Control Gain

Alejandro Rincon1 and Fabiola Angulo2

1 Programa de Ingenierıa Ambiental, Facultad de Ingenierıa y Arquitectura, Universidad Catolica de Manizales, Carrena 23 No. 60-30,Manizales 170002, Colombia

2 Departamento de Ingenierıa Electrica, Facultad de Ingenierıa y Arquitectura, Universidad Nacional de Colombia, Sede Manizales,Electronica y Computacion, Percepcion y Control Inteligente, Bloque Q, Campus La Nubia, Manizales 170003, Colombia

Correspondence should be addressed to Fabiola Angulo, [email protected]

Received 21 November 2011; Accepted 11 April 2012


Copyright © 2012 A. Rincon and F. Angulo. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We propose a scheme for nonlinear plants with time-varying control gain and time-varying plant coefficients, on the basis of aplant model consisting of a Brunovsky-type model with polynomials as approximators. We develop an adaptive robust controlscheme for this plant, under the following assumptions: (i) the plant terms involve time-varying but bounded coefficients, beingits upper bound unknown; (ii) the control gain is unknown, not necessarily bounded, and only its signum is known. To achieverobustness, we use a combination of robustifying control inputs and dead zone-type update laws. We apply this methodology tothe speed control of a permanent magnet synchronous motor (PMSM), and we achieve proper tracking results.

1. Introduction

Nonlinear behavior is difficult to model accurately, renderingcontroller design cumbersome. An approach to handle this isto use linear models, which have local validity in the statespace, that is, different values of the parameter set wouldbe required for each region in the state space. A secondapproach is to use a plant model in either the Brunovskyform defined in [1], or the parametric-pure feedback formdefined in [2], using the so-called “function approximationtechniques”, such as locally weighted statistical learning [3],fuzzy sets [4], Fourier series [5], orthonormal functions,[6] or neural networks [7]. In the last case, state and timedependent terms, known as “basis functions,” are used torepresent the nonlinear and time-varying behaviour. Someidentification or learning methods are used to perform theapproximation, resulting in a negligible modelling error ifthere are a sufficient number of basis functions [3]. Insome cases, a Brunovsky form model of the plant is used,assuming that there is full-state measurement and that thebasis functions are multiplied by unknown constants. Then,

a sliding surface model reference adaptive control (SSMRAC)is devised, handling the residual approximation error bymeans of auxiliary robustifying inputs [7].

Common drawbacks of the above-mentioned schemesare

(i) the convergence of the tracking error to some smallresidual set is not ensured in a strict sense anddepends on the value of the approximation error [7];

(ii) upper or lower bounds of the plant coefficients arerequired to be known [8];

(iii) discontinuous auxiliary inputs are used, which maylead to chattering [5];

(iv) high enough gains are used, which require excessivevalues [6].

Due to environment changes, the coefficients of the plantmodel may experience time-varying but bounded behavior[9, 10]. Both, the control gain and other coefficients mayexhibit this behavior. The time-varying behavior of coeffi-cients different from the control gain may be handled by


means of robustness techniques. See for instance the robustMRAC scheme in [10], whose drawback is that projection-or σ-type update laws are used, and hence upper boundsof the plant coefficients are required to be known, in orderto achieve the convergence of the tracking error to someresidual set of user-defined size. The time-varying controlgain may be handled by means of robustness techniques[11] or Nussbaum gain technique [12], a projection-typeupdate law is used, such that a lower bound of the controlgain is required to be known. In [11], a σ type updatelaw is used, such that bounds of the plant coefficients arerequired to be known. In summary, the main drawback ofusing robustness techniques to handle varying control gainis that upper or lower bounds of some plant coefficients arerequired to be known. On the other hand, the Nussbaumgain technique relaxes this requirement. The main drawbackis that the upper bound of the transient behavior of the state Sis significantly altered with respect to that of the disturbance-free case: the value of this bound depends on time integralsof the terms that comprise the Nussbaum function. Anotherdrawback is that the control gain is usually required to beupper-bounded by some constant.

The scheme of [13] achieves adequate robustness prop-erties. Mainly, upper bounds of the plant coefficients are notrequired to be known. The essential element of the techniqueis to define the quadratic forms related to the sliding surfaceS in terms of a dead zone function of S rather than in termsof S. This scheme has the following benefits.

(Ri) The tracking error converges to a residual set whosesize is of the user’s choice;

(Rii) known upper bounds of the plant coefficients are notrequired, such that high enough gains are not used;

(Riii) all the closed-loop signals are bounded (parameterdrifting is avoided);

(Riv) auxiliary control signals are not discontinuous interms of both the tracking error and the slidingsurface, hence input chattering is avoided;

(Rv) upper bounds of time-varying but bounded coeffi-cients are not required to be known.

The disadvantages of [13] are (i) the design is only developedfor systems with hysteresis nonlinearities in the actuator; (ii)the time-varying control gain is tackled by means of theNussbaum gain method, which entails higher complexity ofthe Lyapunov analysis; (iii) the control gain is assumed to betime-varying but bounded.

In contrast to the approaches of [3, 4, 14–16], we proposethat an adequate regression model for highly nonlinear sys-tems may be obtained from a Brunovsky type model, insert-ing polynomials to approximate the nonlinear behavior, andconsidering also: (i) time-varying but bounded behavior ofsome plant coefficients; (ii) unknown control gain, time-varying, and not necessarily bounded. We develop a robustadaptive scheme for this plant, achieving benefits (Ri) to(Rv). If we compare the robust technique developed here

with the technique developed in [13], the main differences,which are also contributions, are

(Rvi) we consider unknown time-varying control gain,not necessarily bounded, not restricted to actuatornonlinearities;

(Rvii) we consider time-varying but bounded behavior ofsome plant terms;

(Rviii) we tackle the control gain by means of robustnesstechniques, which gives a simpler design in compari-son with the Nussbaum technique.

This paper is organized as follows. The outline of thescheme is given in Section 2. The function approximation isdiscussed in Section 3. The plant regression model and thestatement of the problem are established in Section 4. Thecontrol and update laws are derived in Section 5. The controlalgorithm and its stability are presented in Section 6. Anapplication of the scheme to a PMSM is given in Section 7.Finally, the conclusions are presented in Section 8.

2. Outline of the Scheme

We propose the use of polynomials to approximate thenonlinear behavior, taking into account the fact that polyno-mials are universal approximators for continuous functions,according to [17]. We consider a Brunovsky-form plantmodel, into which we introduce the polynomial terms.

We devise a robust adaptive controller for this plant,achieving benefits (Ri) to (Rix) mentioned in the intro-duction. We use the SSMRAC method stated in [18], asthe basic framework for designing the control and updatelaws. To handle the effect of modelling error and time-varying behavior of plant terms, we use a robust continuouscontrol input, whose magnitude is adjusted, and a dead zone-type update law. The resulting controller has the followingfeatures: (i) the magnitude of the control input is adjustedto cope with the unknown upper bounds of the time-varying coefficients; (ii) the control input is proportional tosome continuous function of the sliding surface S, so thatchattering is avoided; (iii) an inactivation is introduced inall the update laws, which occurs when the sliding surface Sreaches a target region. If we compare this with projection-type update laws, it has the advantage of not requiring theupper bounds of the plant parameters.

For the stability analysis, we use a truncated version ofthe quadratic form related to the sliding surface, denoted byVs. The validity of this technique, including the conditionsof the Lyapunov function, is stated in [19]. It is worthnoticing that the standard conditions of the Lyapunov theoryfor time-variable systems, shown in [18] or [20], are notsatisfied because of the truncation. We prove the asymptoticconvergence of the tracking error in a rigorous manner bymeans of Barbalat’s Lemma. To that end, we redefine theexpression of V as an inequality in terms of Vs.


3. Function ApproximationBased on Polynomials

According to [17, page 116], a continuous real-valued func-tion f (x), where x ∈ D, D ⊂ Rn, being D a compact set, maybe approximated by a polynomial function g in the intervalx ∈ D, with some finite error, being g defined as

g(x, θo) =∑

k1,k2,...,kn

θo[ j]xk11 x

k22 · · · xknn : θo ∈ Ω, Ω ⊂ Rp,

(1)

where Ω is a convex set and θo[ j] means the jth element ofthe vector θo. Thus, f (x) can be expressed in terms of g asfollows (cf. [17] page 122):

f (x) = g(x, θa) + εo(x), |εo(x)| < ε,

θa ={

θo ∈ Ω : θo = arg minθo

(

supx∈D

∣∣∣ f (x)− g(x, θo)

∣∣∣

)}

,

(2)

where ε is a positive constant, and it is known as approxi-mation error, θa is an ideal parameter set and g(x, θa) is anideal representation of f (x). The polynomial can be linearlyparameterized with respect to its coefficients:

f (x) = φ�θa + εo(x), (3)

where φ contains polynomial terms, whereas θa containsconstants. In this work, f represents the nonlinear part of thedynamical equation. We will handle the unknown constantvector θa by means of adjustment rules and εo by means ofrobust techniques.

4. Plant Model and Problem Statement

We assume that the dynamical nonlinear system can be re-presented by a Brunovsky type model, as defined in [1], withpolynomial functions:

y(n) = −cn−1y(n−1) − · · · − c1 y − co y + f (x) + bu, (4)

x =[y, y, . . . , y(n−1)

]�, (5)

where y(t) ∈ R1 is the system output u(t) ∈ R1 is theinput and the coefficients cn−1, . . ., and co are unknownand time-varying but bounded. The relative degree n maybe established by means of previous step response analysis.Inserting the function (3) into (4) gives

y(n) = bu + φ�θa + d, (6)

d = εo(x)− cn−1y(n−1) − · · · − c1 y − co y,

|cn−1| ≤ μn−1, . . . , |c1| ≤ μ1,

|co| ≤ μn, |εo(x)| ≤ μn+1 = ε.(7)

We make the following assumptions:

(Ai) the entries of θa and the terms μ1, . . ., and μn+1 areconstant and unknown;

(Aii) the control gain b varies with respect to the state x, ortime, so that it satisfies the following:

(i) 0 < bm ≤ |b| ≤ fb, ∀t ≥ to, (8)

(ii) sign(b) constant, ∀t ≥ to, (9)

where the value of sign(b) is known, bm isan unknown positive constant, and fb is anunknown positive function of time or x. Noticethat condition (8) implies that b is always dif-ferent from zero;

(Aiii) the entries of the vector x are available for measure-ment;

(Aiv) the entries of the vector φ are known functionsof x;

(Av) the value of the desired trajectory yd(t) and its

derivatives y(n−1)d , . . . , yd is bounded.

The desired output yd is specified in terms of a boundedexternal command r(t) as follows:

y(n)d + am,n−1y

(n−1)d + · · · + am,o yd = am,or, (10)

where am,n−1, . . . , am,o are constant coefficients prespecifiedby the user, such that the polynomial K(p) is Hurwitz withat least one real root, being K(p) defined as K(p) = p(n) +am,n−1p(n−1) + · · · + am,o. The external reference signal r(t)must be bounded. The objective of the MRAC design is toformulate a control law u(t) such that the tracking errore(t) = y(t) − yd(t) asymptotically converges to the residualset De, defined as follows:

De = {e : |e| ≤ Cbe}, (11)

where Cbe is a user-defined bound.

5. The Control and Update Laws

Let S the dynamics imposes over the tracking error given by

S = (p + λ)n−1

e = pn−1e + λn−2pn−2e + · · · + λ1e + λoe,

(12)

where λ is a positive constant chosen by the user. Havingdefined S, we establish the dynamic equation for S bydifferentiating with respect to time:

S = pne + λn−2pn−1e + · · · + λ1e + λoe, (13)

S = y(n) − y(n)d + λn−2p

n−1e + · · · + λ1e + λoe, (14)

S = y(n) + ϕa, (15)

ϕa = −y(n)d + λn−2p

n−1e + · · · + λ1e + λoe. (16)


We insert the expression for y(n) of (6) into (15):

S = bu + θ�a φ + d + ϕa. (17)

Define

u∗ = −θ�a φ − ϕa − amS = ϕ�θ∗,

θ∗ = [θ�a , 1]�,

(18)

ϕ = [−φ�,−ϕa − amS]�, (19)

where ϕa is defined in (16), am is a positive constant. Weexpress (17) in terms of u∗ as follows:

S = −amS + bu + d − u∗ = −amS + bu + d − ϕ�θ∗. (20)

Let

l = n + 1, (21)

c∗ = 2 + l

2Cbvs

(1

2√bmn

)2

(22)

Cbvs �(

12

)(λn−1Cbe

)2, (23)

f1 =∣∣ y∣∣, . . . , fl−2 =

∣∣∣y(n−1)

∣∣∣,

fl−1 =∣∣y∣∣, fl = 1,

μ1 = max{|c1|}, . . . , μl−2 = max{|cn−1|},μl−1 = max{|co|}, μl = ε.

(24)

Now, we multiply (20) by S and add and subtract the termc∗S2:

SS = −amS2 + bSu + Sd − Sϕ�θ∗ + c∗S2 − c∗S2, (25)

where the terms c∗S2 − c∗S2, −Sϕ�θ∗, and Sd can be ex-pressed in terms of adjustment errors:

(i) c∗S2 − c∗S2 = −c∗S2 − cS2 + cS2, c = c − c∗;

(26)

(ii) − Sϕ�θ∗ = Sϕ�θ − Sϕ�θ, θ = θ − θ∗;(27)

(iii) Sd≤(−1)c1|S| f1 +c1|S| f1 +· · · +(−1)cl|S| fl+cl|S| fl,(28)

c1 = c1 − μ1, . . . , cl = cl − μl, (29)

where c, θ, c1, . . . , cl are adjusted parameters whose updatelaws will be defined later. Inserting the above properties into(25) gives

SS ≤ −amS2 − cS2 + Sϕ�θ + (−1)c1|S| f1+ · · · + (−1)cl|S| fl + cS2 − Sϕ�θ + c1|S| f1+ · · · + cl|S| fl + bSu− c∗S2.

(30)

If b were constant, we would choose the control u so as tocancel the terms involving updated parameters c, θ, c1, . . . , cl.Since b is varying, we chose u to attenuate the effect ofadjusted parameters, being the remaining error rejected by−c∗S2:

u = (−1) sgn(b)c2S3 + (−1) sgn(b)(ϕ�θ

)2S

+ (−1) sgn(b)c21 f

21 S + · · · + (−1) sgn(b)c2

l f2l S,

(31)

where c, θ, c1, . . . , cl are adjusted parameters whose updatelaws will be defined later. Replacing the above control lawinto (20) and (30) gives

S = −amS − |b|c2S3 − |b|(ϕ�θ

)2S

− |b|c21 f

21 S + · · · − |b|c2

l f2l S + d − ϕ�θ∗,

(32)

SS ≤ −amS2 − cS2 + Sϕ�θ + (−1)c1|S| f1+ · · · + (−1)cl|S| fl + cS2 − Sϕ�θ + c1|S| f1+ · · · +cl|S| fl − |b|c2S4 −|b|

(ϕ�θ

)2S2 −|b|c2

1 f2

1 S2

− · · · − |b|c2l f

2l S2 − c∗S2,

(33)

where the last terms of the above equation satisfy the fol-lowing inequality (see the proof in Appendix A):

cS2 − Sϕ�θ + c1|S| f1 + · · · + cl|S| fl − |b|c2S4

− |b|(ϕ�θ

)2S2 − |b|c2

1 f2

1 S2 + · · · − |b|c2l f

2l S2

− c∗S2 ≤ 0 if S2 ≥ 2Cbvs.

(34)

Equation (34) expresses the attenuation of the effect of theadjusted parameters. Substituting (34) into (33) gives

SS ≤ −amS2 − cS2 + Sϕ�θ − c1|S| f1− · · · − cl|S| fl if S2 ≥ 2Cbvs,

(35)

or equivalently,

SS ≤ −amS2 − cS2 + Sϕ�θ − |S|C�d f if S2 ≥ 2Cbvs,(36)

Cd = [c1, . . . , cl]�, f = [ f1, . . . , fl

]�. (37)

We choose the following update laws:

˙c ={γcS2 if S2 ≥ 2Cbvs,

0 otherwise,

˙Cd ={Γd f |S| if S2 ≥ 2Cbvs,

0 otherwise,

˙θ =

{−ΓϕS if S2 ≥ 2Cbvs,

0 otherwise,

(38)

where S is defined in (12), Cbvs in (23), whereas γc, and thediagonal entries of Γ, Γd are positive constants chosen by theuser, being Γ and Γd diagonal matrices.


6. The Control Algorithm

Now we recall the equations corresponding to the controllerand establish the tracking convergence theorem. The controllaw is given by (31); the update laws are given by (38); theterms ϕ and S are defined in (19) and (12); ϕa is defined in(16); Cbvs is defined in (23) and f is defined in (37). Theadjusted parameters c1, . . . , cl, required to compute u, are theentries of the vector Cd, that is, Cd = [c1, . . . , cl]

�.

6.1. Theorem: Boundedness and Tracking Convergence. If thecontroller designed in Section 5 is applied to the plant (6),then the tracking error e(t) converges to De asymptotically,and the closed-loop signals remain bounded.

Proof. Now we proceed to analyze the stability of the con-trolled system using the direct Lyapunov method and theBarbalat’s Lemma. First, we establish the boundedness of theclosed-loop signals on the basis of the time derivative of theLyapunov function. Then, we establish the convergence ofthe tracking error to the target region De defined in (11),on the basis of the Barbalat’s Lemma. The validity of theprocedure can be stated using the theory developed andapplied in [18, 19, 21, 22].

The closed-loop dynamics is given by (32), and (38). Wedefine the following truncated quadratic form, on the basisof the truncation presented in [19]:

Vs ={Vs if Vs ≥ Cbvs,

Cbvs if Vs < Cbvs,(39)

Vs �(

12

)S2, (40)

where Cbvs is defined in (23). The form Vs has the followingproperties:

−Vs < −Vs + Cbvs < 0 if Vs > Cbvs,

−Vs < −Vs + Cbvs = 0 if Vs = Cbvs,

−Vs ≤ 0 = −Vs + Cbvs = 0 if Vs < Cbvs.

(41)

The states of the closed-loop system may be grouped in thefollowing vector:

x =[S, c, C�d , θ�

]�, (42)

where c, Cd, and θ are defined in (26), (37), (27). We definethe following function:

V(x(t)) = Vs +Vθ +Vc +Vd,

Vθ =(

12

)θ�Γ−1θ, Vc =

(12

)γ−1c c2,

Vd =(

12

)C�d Γ

−1d Cd.

(43)

Notice that V(x) does not satisfy standard conditions ofLyapunov theory for time-variable systems, shown in [18].It satisfies the conditions of [19] instead, which we use toprove the boundedness of the closed-loop signals and the

convergence of the tracking error to the residual set De (11).The time derivative of Vs along the trajectory (32) is

Vs = SS = −amS2 − |b|c2S4 − |b|(ϕ�θ

)2S2

+ (−1)|b|c21 f

21 S2 + · · · + (−1)|b|c2

l f2l S2

+ S(d − ϕ�θ∗) ≤ −amS2 − cS2 + Sϕ�θ

+ (−1)|S|C�d f if Vs ≥ Cbvs,

(44)

where the last inequality is obtained from (36). The timederivative of Vs can be derived from (39) and expressed onthe basis of the above equation:

Vs

=⎧⎨

⎩Vs ≤ −amS2 − cS2 +Sϕ�θ+(−1)|S|C�d f if Vs ≥ Cbvs,

0 if Vs < Cbvs.

(45)

The time derivative Vθ + Vc + Vd along trajectories (38) is

Vθ + Vc + Vd =⎧⎨

⎩

(−1)Sθ�ϕ + cS2 + Cd f |S| if Vs ≥ Cbvs,

0 otherwise.

(46)

Therefore, V is given by

V = Vs + Vθ + Vc + Vd

={Vs + Vθ + Vc + Vd ≤ −amS2 if Vs ≥ Cbvs,

0 if Vs < Cbvs.

(47)

The above equation can be rewritten in terms of Vs definedin (40):

V ≤ −2amVs if Vs ≥ Cbvs,

V = 0 if Vs < Cbvs.(48)

This implies that V ≤ 0 for all t ≥ to. Thus, according to

[19], all the closed-loop signals are bounded, that is, (S, θ, c,Cd) ∈ L∞, or equivalently, x ∈ L∞, where x is defined in (42).As a consequence, (V , Vs, Vs, Vθ , Vc, Vd) ∈ L∞, accordingto the definitions in (39) and (43). The boundedness of ximplies (e, . . ., e(n−1)) ∈ L∞ according to (12); hence, S ∈L∞ [22], and V s ∈ L∞. To establish the convergence of thetracking error, we begin by expressing the equation ahead interms of Vs, according to the definition of Vs in (39) and theproperties of (41):

V ≤ −2amVs < −2amVs + 2amCbvs ≤ 0 if Vs ≥ Cbvs,

V = 0 = 2am(−Vs + Cbvs

)= 0 if Vs < Cbvs,

=⇒ V ≤ −2amVs + 2amCbvs ≤ 0.(49)

We reorganize the above equation and integrate as follows:

2am

∫ t

to

(Vs − Cbvs

)dτ +V(t) ≤ V(to). (50)


Thus, (Vs−Cbvs) ∈ L1. Recall that (Vs, Vs) ∈ L∞. Thus, (Vs−Cbvs) ∈ L∞ ∩ L1, (d/dt)(Vs − Cbvs) ∈ L∞. By invoking theBarbalat Lemma [23], we obtain (Vs − Cbvs) → 0 as t → ∞.In turn, this implies that Vs → Dvs, being Dvs defined as

Dvs = {Vs : Vs ≤ Cbvs}, (51)

where Cbvs is defined in (23). S can be expressed in terms ofVs, on the basis of the definition of Vs in (40):

S =√

2Vs. (52)

In the following, we analyze the convergence of S. Takinginto account the definition of Cbvs in (23), the fact that Vs

converges to Dvs defined in (51), and (52) the convergence ofS is:

limt→∞S = Ds, Ds = {S : |S| ≤ Cbs},

Cbs =√

2Cbvs = λn−1Cbe.(53)

To analyze the convergence of e, we begin by expressing e interms of S, according to (12):

e = 1(p + λ

)n−1 S. (54)

The convergence of the tracking error may be established onthe basis of (53) and (54) [18, 21]:

limt→∞e(t) = De, De = {e : |e| ≤ Cbe}, (55)

which means that the tracking error e converges to a residualset whose size is of the user’s choice.

7. Application to a Permanent MagnetSynchronous Motor

A PMSM is a kind of highly efficient and high-poweredmotor. The benefits of the PMSM are discussed in [24,25]. The PMSM has the following difficulties [26]: (i) it ishighly nonlinear; (ii) the parameters of the physical modelexperience unknown time-varying behavior, for example,the stator resistance R and the friction coefficient B; (iii)unknown external disturbances appear, that is, the loadtorque disturbance TL. Moreover, by varying the permanentmagnet flux λa f the state ω exhibits a pitchfork bifurcation[27]. All the mentioned facts lead to response deteriorationof many controllers, specially for high-speed and high-precision tasks in real applications [28].

In view of the complex behavior, with parameters varyingwith respect to time and state plane, the scheme developedin this work is suitable. We apply the developed scheme bysimulation, to a PMSM whose model and parameters arepresented in [29]. Therein, the possible manipulated inputsare ud and uq; and id, iq, and ω are possible outputs. Wechoose the motor angular frequency ω as the output to becontrolled and the d-axis voltage ud as the control input, thatis, y = ω, u = ud, υd = 0. A similar choice is made in [24],where an adaptive controller is derived for a PMSM. After

performing a step response analysis, the variable ω exhibits asecond-order behavior when a step change is introduced inud. This second order behavior was already noticed in [29].Thus, we use the regression model (6) with n = 2. We cansummarize the basic features of the plant as

n = 2, sgn(b) = +1,

l = 3, f1 =∣∣ y∣∣, f2 =

∣∣y∣∣,

f3 = 1, =⇒ f = [∣∣ y∣∣,∣∣y∣∣, 1

]�,

φ = [y2, y3]�, ϕa = − yd + λoe,

=⇒ ϕ = [−φ�,−ϕa − amS]�.

(56)

We used a factorization with terms y2 and y3 to take intoaccount the presence of the pitchfork bifurcation. Accordingto [30] the normal form of the pitchfork bifurcation gives adescription of the system behavior in a tight neighborhoodof the bifurcation point. For the case of the pitchforkbifurcation, the normal form includes the terms y2 andy3. Although the system usually works in different regionsfar from the bifurcation point, we wanted to include thebehavior in the neighborhood of the bifurcation point. Thus,we included the terms of the normal form for the pitchforkbifurcation [30], that is, y2 and y3. The terms 1 and y arenot included in φ because they are already present in d. Theremaining parameters of the controller are defined on thebasis of the parameters in (56):

e = y − yd, S = (p + λ)e = e + λe =⇒ λo = λ,

Cbvs =(

12

)(λCbe)

2,

u = (−1) sgn(b)c2S3 + (−1) sgn(b)(ϕθ)2

S

+ (−1) sgn(b)S(c2

1 f2

1 + c22 f

22 + c2

3 f2

3

),

= (−1) sgn(b)S(c2S2 +

(ϕθ)2

+ c21 f

21 + c2

2 f2

2 + c23 f

23

).

(57)

In addition, we choose:

λ = 3, am = 10, am,1 = 70,

am,o = 1225 (for the reference model), Cbe = 0.1,

γc = 20, Γd = diag[20, 20, 20],

Γ = diag[0.4, 0.4, 0.003].(58)

The results are shown in Figure 1. The simulation showsexpected results: all the closed-loop signals remain bounded,and transient values of the tracking error remain in a smallinterval. It is worthy of note that the transient values ofe depend on its initial value and the initial values of theadjustment errors. To show the effect of the time-varyingbehavior of the model coefficients, we change the statorresistance R from 1.4 to 1.7Ω at the time instant 0.6 sec.Results are shown in Figure 2.

In addition, we consider the variation of the dampingconstant B from 0.00038818 to 0.00046582 Nm/(rad/s) at the


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−10

0

10

20

30

ω

Time

Rotor speedReference

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−10

0

10

20

30

uq

Time

(b)

Figure 1: Transient behavior of the output and control input.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.1

−0.05

0

0.05

0.1

e

Time

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−10

0

10

20

30

uq

Time

(b)

Figure 2: Performance of the tracking error and control inputvarying the resistance value.

instant 0.6 sec. Results are shown in Figure 3. Notice thatin the three cases the final value of |e| is less than Cbe =0.1. Moreover, the control input u belongs to the interval[−200 200] V. Notice in Figures 2 and 3 that the effect ofthe disturbance on the tracking error is almost negligible.Nevertheless, the control input experiences a large variation,as can be seen in Figure 3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.1

−0.05

0

0.05

0.1

e

Time

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−50

0

50

100

150

uq

Time

(b)

Figure 3: Performance of the tracking error and control inputvarying the damping constant value.

8. Conclusions

In this work, we have proposed a control scheme for highlynonlinear plants, based on a simple plant model with poly-nomial approximators, which provides an adequate descrip-tion of transient behavior. It is worth noticing the fact thatbenefits Ri to Rviii (Section 2) are achieved at the same time,with minimal requirements on the knowledge of the plant.Indeed, the relative degree and the signum of the control gaincan be established by a previous step response analysis. Manytechniques are combined at the same time: sliding surfaceMRAC, dead zone-type update law, robustifying auxiliarycontrol, approximation techniques, and truncation of thequadratic forms.

The disadvantage of polynomials as approximators is thatthey may be less accurate than other techniques, for example,neural networks or fuzzy sets, leading to higher approxi-mation error. Since the approximation error is bounded, itcan be handled by means of robustness techniques withoutrequiring the upper bound to be known. Moreover, weconsidered the coefficients of the terms y, . . . , y(n−1) as time-varying but bounded, with constant and unknown bounds.Then, we handled this by means of robust control, withoutrequiring the upper bounds to be known.

We handled the time-varying behavior of the controlgain by means of robustness techniques, without using theNussbaum gain method. The redefinition of the plant termsin terms of adjustment errors and adjustment parametersis a fundamental step. The resulting expression allows astraightforward definition of the control law. The variationof the control gain implies that the terms involving adjustedparameters cannot be cancelled. Rather, we attenuate itseffect by means of squared terms and handle the residualerror by means of an additional control term that is onlya function of the sliding surface. The resulting expression


for V , V , and the design is simpler in comparison with theNussbaum technique.

Appendix

Proof of (34)

As the first step, we factorize several summands of (34) andapply the property (8):

− |b|c2S4 + cS2 ≤ −bmc2S4 + cS2

=−[√

bmcS2 − 1

2√bm

]2

+

(1

2√bm

)2

≤(

1

2√bm

)2

.

(A.1)

Likewise, we obtain:

(−1)|b|(ϕ�θ

)2S2 − Sϕ�θ ≤

(1

2√bm

)2

,

(−1)|b|c21 f

21 S2 + c1|S| f1 ≤

(1

2√bm

)2

,

...

(−1)|b|c2l f

2l S2 + cl|S| fl ≤

(1

2√bm

)2

.

(A.2)

By adding (A.1) and (A.2) we obtain:

− |b|c2S4 + (−1)|b|(ϕ�θ

)2S2 + (−1)|b|c2

1 f2

1 S2

+ · · · + (−1)|b|c2l f

2l S2 + cS2 − Sϕ�θ + c1|S| f1

+ · · · + cl|S| fl ≤ (2 + l)

(1

2√bm

)2

.

(A.3)

As a second step, we use the definition of c∗ in (22) to rewritethe term −c∗S2:

−c∗S2 = −(2 + l)S2

2Cbvs

(1

2√bm

)2

≤ −(2 + l)

(1

2√bm

)2

if S2 ≥ 2Cbvs.

(A.4)

As the third step, we add (A.3) and (A.4):

− |b|c2S4 + (−1)|b|(ϕ�θ

)2S2 + (−1)|b|c2

1 f2

1 S2

+ · · · + (−1)|b|c2l f

2l S2 + cS2 − Sϕ�θ + c1|S| f1

+ · · · + cl|S| fl − c∗S2 ≤ 0 if S2 ≥ 2Cbvs,

(A.5)

which is (34).

Acknowledgments

This work was partially supported by Universidad Nacionalde Colombia-Manizales, project 12475, Vicerrectorıa deInvestigacion, DIMA, resolution number VR-2185.

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Research Article

Adaptive Control for a Class of Nonlinear System withRedistributed Models

Haisen Ke and Jiang Li

College of Mechanical and Electrical Engineering, China Jiliang University, Zhejiang, Hangzhou 310018, China

Correspondence should be addressed to Haisen Ke, [email protected]

Received 3 November 2011; Accepted 11 April 2012


Copyright © 2012 H. Ke and J. Li. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Multiple model adaptive control has been investigated extensively during the last ten years in which the “switching” or “switchingand tuning” have emerged as the mainly approaches. It is the “switching” that can improve the transient performance to someextent and also make it difficult to analyze the stability of the system with multiple models adaptive controller. Towards this goal,this paper develops a novel multiple models adaptive controller for a class of nonlinear system in parameter-strict-feedback formwhich not only improves the transient performance significantly, but also guarantees the stability of all the states of the closed-loopsystem. A simulation example is proposed to illustrate the effectiveness of the developed multiple models adaptive controller.

1. Introductions

The multiple model adaptive control was introduced tocope with the large parametric uncertainty [1] which alwaysresults in large and oscillatory responses or even instablewhen using the classical adaptive control methods. Themultiple models adaptive control [1–7] employing both fixedmodel and adaptive model have been used to identify thecharacteristics of the plants, and numerous methods arecurrently available for controlling such plant satisfactorily.However, the methods mainly focus on the linear timeinvariant plant [1, 2, 4–6]. The multiple models adaptivecontroller for nonlinear system is firstly considered in [8],which uses a direct parameter update law to guaranteethe stability of the closed-loop system. Then, Ciliz andCezayirli [9] proposes a different nonlinear multiple modelsadaptive control which require the condition of persistenceof excitation, so that the unknown parameter can be eval-uated at the very beginning. Recently, an indirect multiplemodels adaptive control was developed in [7] which alsodemonstrated the global asymptotic stability of the closed-loop switching system.

As illustrated in the literature that the “switching” (to theclosest model) based on the index of performance results infast response, and tuning (from the closet model) improvesthe identification and control errors on a slower time scale,

which have the assumption that there are abundant modelsavailable. Otherwise, the results may be improved less ifthe number of the identification models is not adequate toachieve the satisfactory response.

In this paper, a novel multiple models adaptive controlwas considered for the nonlinear system in parameter-strict-feedback form, which retains the advantages of themultiple models adaptive controller, meanwhile facilitatethe procedure to analyze and synthesize the controller ofthe closed-loop system. The approach developed here inwhich the multiple models adaptive controller are used toplay a significantly larger role in the decision making role,results in substantial improvement in performance. Besides,we also reduce the number of the identification models byredistributing the candidate models even as the system is inoperation.

2. Problem Formulation

Consider the multiple models adaptive control of the follow-ing nonlinear parameter-strict-feedback (PSF) system:

xi = xi+1 + ϕTi (xi)θ, 1 ≤ i ≤ n− 1,

xn = β(x)u + ϕTn (x)θ,

y = x1,

(1)


where xi = [x1, . . . , xi]T ∈ Ri and x ∈ Rn are the state,

u ∈ R is the control input, θ ∈ Rp is an unknown parametervector belonging to a known compact set S. The functionsϕi(xi) and β(x) are known smooth functions with β(x) /= 0,for all x ∈ Rn. The focus of this paper is to improve thetransient performance in the presence of large parametricuncertainties.

One easly way to improve the transient performance maybe choosing sufficiently large high-frequency parametersin the conventional backstepping adaptive control design.Unfortunately, the control efforts can also be very largesimultaneously [7]. Alternately, in cope with these difficul-ties, “switching” or “switching and tuning” have emerged asthe leading methods during the last decade.

3. Multiple Models Adaptive Controller Design

In order to ensure the stability and transient performanceof the system with larger parametric uncertainty, and con-sequently the boundedness of the state x(t), the well-established results from the classical adaptive control cannotbe used directly. Our multiple models adaptive controllercontains N parallel operating identification models on whichthe control law and the adaptive law are based. For improvingthe transient performance, it is necessary to distribute the

initial estimate values of the unknown parameter {θ j(0)}Nj=1uniformly in the compact set S to which the unknown

parameter θ belongs. Therefore, at least one θ j(0) is close toθ, consequently there must exists one or more identificationmodels in its neighborhood. Since adaptive control canperform well when parametric errors are small, it is naturallythat the controller developed on the jth identificationmodel can stabilize the system with satisfactorily transientperformance.

3.1. Multiple Identification Models. We will run in parallelN identification models with the same structure which takethe different initial parameter estimate values {θ j(0)}Nj=1uniformly distributed in the compact set S to which theunknown parameter belongs. We first introduce the filters asfollows:

ξ0 =(

A0 − λΞ(x)ΞT(x)P)(ξ0 − x

)+ f (x,u), ξ0 ∈ Rn,

(2)

ξ =(

A0 − λΞ(x)ΞT(x)P)ξ + Ξ(x), ξ ∈ Rn×p, (3)

where

f (x,u) =[x2 . . . xn β(x)u

]T,

Ξ(x) =[ϕ1(x1) . . . ϕn(x)

]T.

(4)

λ > 0, and A0 is a Hurwitz matrix such that the Lyapunovequation: PA0 + AT

0 P = −I has a positive definite solution P.

Define

e = x− ξ0 − ξθ, (5)

e j = x − ξ0 − ξθ j , j = 1, . . . ,N , (6)

θ j = θ − θ j , j = 1, . . . ,N. (7)

It can be derived from (1)–(7) that

˙e =(

A0 − λΞ(x)ΞT(x)P)

e, (8)

e j = ξθ j + e, j = 1, . . . ,N. (9)

Since e converges to zero exponentially, (9) are calledidentification error equations.

3.2. Controller Design. The controller design involves Nmodels at total and is developed as [10], which can guaranteethe asymptotic tracking when there is not identificationerror and avoid the finite time escape phenomenon whenthere exists bounded identification error. Now, the firstidentification model’s adaptive controller is given by

u1 =[α1,n

(x, θ1, yr , . . . , ynr

)]

β(x), (10)

where yr is the reference signal to be tracked and α1,n can berecursively designed by

zi = xi − α1,i−1

(x1, . . . , xi−1, θ1, yr , . . . yi−1

r

), (11)

α1,i = −zi−1 − c1,izi −wT1,iθ1 + yir − s1,izi

+i−1∑

k=1

(∂α1,i−1

∂xkxk+1 +

∂α1,i−1

∂yk−1r

ykr

)

,(12)

w1,i

(x1, . . . xi, θ1, yr , . . . , yi−1

r

)= ϕi −

i−1∑

k=1

∂α1,i−1

∂xkϕk, (13)

s1,i = k1,i∣∣w1,i

∣∣2 + g1,i

∣∣∣∣∣∂α1,i−1

∂θ1

∣∣∣∣∣

2

, (14)

We choose

V1 = 12

n∑

i=1

z2i . (15)


The time derivative of V1, computed with (10)–(14), isgiven by

V1 = −n∑

i=1

c1,iz2i +

n∑

i=1

(

wT1,iθ1 − ∂α1,i−1

∂θ1

˙θ1

)

zi

−n∑

i=1

⎛

⎝k1,i∣∣w1,i

∣∣2 + g1,i

∣∣∣∣∣∂α1,i−1

∂θ1

∣∣∣∣∣

2⎞

⎠z2i

≤ −n∑

i=1

c1,iz2i −

n∑

i=1

k1,i

∣∣∣∣∣w1,izi − 1

2k1,iθ1

∣∣∣∣∣

2

−n∑

i=1

g1,i

∣∣∣∣∣∂α1,i−1

∂θ1

zi − 12g1,i

˙θ1

∣∣∣∣∣

2

+n∑

i=1

14k1,i

∣∣∣θ1

∣∣∣

2+

n∑

i=1

14g1,i

∣∣∣∣

˙θ1

∣∣∣∣

2

≤ −n∑

i=1

c1,iz2i +

n∑

i=1

14k1,i

∣∣∣θ1

∣∣∣

2+

n∑

i=1

14g1,i

∣∣∣∣

˙θ1

∣∣∣∣

2

,

(16)

with c1,i, k1,i, g1,i being designed parameters. Equation (16)implies the boundedness of the states of zi, 1 ≤ i ≤ n,and which in turn indicates the boundedness of the statesof xi, 1 ≤ i ≤ n and control u1 on the conditions of θ and˙θ are bounded which will be proved later. The rest of N-1controllers can be designed and analyzed similarly which canalso guarantee the boundedness of the states of zi, 1 ≤ i ≤ n,and which in turn indicates the boundedness of the states ofxi, 1 ≤ i ≤ n and control uj , j = 1, . . . ,N .

3.3. Construction of Equivalent Control. In this section, thecrucial point is that the transient performance can beimproved significantly, and at the same time the switchingbetween the identification models can be avoided. Besides,the information provided by all the identification models isto be utilized efficiently. For the complement of the goalsmentioned, instead of using the estimate values of the modelwith the minimum of performance criterion to reinitial anadaptive controller, a convex combination of all the N modelsis used to generate the control of the plant as

u =N∑

j=1

γjuj , (17)

and the adaptive update law as

˙θ j = Γ

ξTe j

1 + v|ξ|2 , Γ = ΓT > 0, v > 0, j = 1, . . . ,N , (18)

where γj are nonnegative values satisfying∑N

j=1 γj = 1, and γj

can be calculated from

γj =(

1/J j)

∑Nj=1

(1/J j

) , (19)

where J j is the performance indices of the form:

J j(t) = αe2j (t) + β

∫ t

t0e2j (τ)dτ, a ≥ 0, β > 0, (20)

with t0 can be reset when the identification models isredistributed.

3.4. Redistribution of the Identification Models. In this sec-tion, the goal is that the transient performance can beimproved significantly as far smaller numbers of the identifi-cation models as possible. As is illustrated in the literature,the classical adaptive control can cope with the control oflinear time invariant system with unknown parameters andachieve satisfactory closed-loop objective only if the plantparametric uncertainty is small. So if the number of theidentification models that can be used is abundantly large,the “switching” or “switching and tuning” scheme may acton satisfactorily. Otherwise, the multiple models adaptivecontrol cannot work as expected when the numbers ofidentification models available is relatively smaller comparedwith the size of the uncertainty region. Inspired by the“switching” techniques [11–13], we consider the methodin which the location of the identification models can beredistributed. From (8) and (9), it can be concluded that thee = 0 can be achieved by choosing the initial values of ξ0 andξ as long as the initial state x0 is known or there exists T > 0such that

e j = ξθ j , j = 1, . . . ,N , t > T. (21)

It is obviously that the errors e j and θ j , j = 1, . . . ,Nare linearly related. This implies that the index of theperformance J j(t) is a quadratic function of the unknown

parameter vector θ j . Since ξTξ is not negative definite, itfollows that the performance indices of all the models aremerely points on a time-varying quadratic surface, whoseminimum corresponds to the plant indicating the mostlycloset identification modelMj (corresponds to the parameter

θ j). So we can redistribute the other (N − 1) modelsMk (k /= j) as

θk =√Jk

√Jk +

√J jθ j +

√J j

√Jk +

√J jθk. (22)

By introducing the minimum of interval time Tmin into ourswitching scheme to ensure a finite number of switching.

4. Stability Analysis

Theorem 1. Suppose the multiple models adaptive controller(17) and adaptive law (18) presented in this paper is appliedto system (1). Then, for all initial conditions, all closed-loopstates are bounded on [0,∞), and asymptotic tracking can beachieved, that is, Limt→∞z(t) = 0 or y(t) = yr(t) as t → ∞.

Proof. Since all N models are identical structure and onlywith different initial estimate parameters, it follows that eachcontroller acts on the system is only different from each otherat the weight (each of the controllers can be designed with thesame structure and designed parameters).

When we choose the whole candidate Lyapunov functionas

V = 12

n∑

i=1

z2i . (23)


It is obvious that (23) can be divided into

V = 12

n∑

i=1

z2i =

N∑

j=1

γj12

n∑

i=1

z2i =

N∑

j=1

γjVj. (24)

As illustrated by (16), each of the controllers can guaranteethe boundedness of the states of zi, 1 ≤ i ≤ n at its portion,which accompanied with the control (17), and weightingcoefficient (19) can establish the boundedness of the statesof zi, 1 ≤ i ≤ n.

Next, we prove the controller (17) and adaptive law (18)can also guarantee the asymptotic tracking of the closed-loopsystem states. It can be computed from (3) that

d

dt

(ξPξT

)= −ξξT − 2λξPΞTΞPξT + ξPΞT + ΞPξT

= −ξξT−2λ(ΞPξT − 1

2λI)T(

ΞPξT− 12λ

I)

+1

2λ,

(25)

which shows ξ is bounded regardless of the state x. Let V j =(θ

T

j Γ−1θ j + eT e)/2, it can be derived that

V j = −eTj(

e j − e)

1 + v|ξ|2 + eT(

A0 − λΞ(x)ΞT(x)P)

e

≤ −34

eTj e j

1 + v|ξ|2 ,

(26)

without loss of generality, we can design the parametersatisfies A0 − λΞ(x)ΞT(x)P > I , I is a unit matrix. Therefore,e, θ j , j = 1, . . . ,N are all bounded, which companied with

the boundedness of ξ, further yields from e j = ξθ j + ethat e j is bounded. It can be also concluded from (26)that e j is squarely integrable on [0,∞). Furthermore, we

can also conclude from (18) that˙θ j is bounded, which can

accomplish the assumption that it is bounded. We can nowgive the asymptotically tracking control analysis.

The time derivate of identification error is given by

e j =(

A0 − λΞΞTP)

e j + Ξθ j − ξ˙θ j . (27)

Due to the boundedness of all the closed-loop systemstates e j , ˙e j , j = 1, . . . ,N are also bounded, so byBarbalat’s lemma, we must have Limt→∞e j(t) = 0 and since

Limt→∞∫ tt1 e j(τ)dτ = limt→∞e j(t) − e j(t1) < ∞, we further

have Limt→∞e j(t) = 0. Then, it can be concluded from (18)

that Limt→∞˙θ j = 0 which accompanied with (27) implies

Limt→∞Ξθ j = 0 and in turn leads to

Limt→∞Nj

(z, θ, yr

)Ξ θ j = 0, (28)

where

Nj

(z, θ j , yr

)=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0

−∂αj,1∂x1

1 0

.... . . 0

−∂αj,n−1

∂x1−∂αj,n−1

∂x21

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (29)

By direct calculating, the differentiation of z with parametriccontroller uj can be described, in z coordination, by

z = −n∑

i=1

cj,izi +Nj

(z, θ, yr

)Ξθ j −

∂αj,i−1

∂θ j

˙θ j

−n∑

i=1

⎛

⎜⎝kj,i

∣∣∣wj,i

∣∣∣

2+ gj,i

∣∣∣∣∣∣

∂αj,i−1

∂θ j

∣∣∣∣∣∣

2⎞

⎟⎠zi.

(30)

From (28), accompanied with Limt→∞˙θ j = 0 and the

designed parameters are all positive, it can be easily con-cluded that Limt→∞z(t) = 0, and thus Limt→∞z1(t) =Limt→∞(y(t)− yr(t)) = 0. The proof is completed.

5. Simulation

Consider the following second-order nonlinear system:

x1 = x2 + θ1x1 + θ2x21,

x2 = u,

y(t) = x1(t),

(31)

where θ1 ∈ [1, 5] and θ2 ∈ [1, 40] are unknownparameters. The output y(t) = x1(t) is to asymptoticallytrack the reference signal yr(t) = sin 2t.

In simulation, the parametric controller is developed as(10)–(14) and (17)–(19) with v = 0, Γ = 5, cj,1 = cj,2 = 4,kj,1 = kj,2 = gj,2 = 0.1, j = 1, . . . ,N , α = β = 1, Tmin is 5units of time. Since in (31), the unknown parameter appearsonly in the first equation, the filter can be constructed as [1]to reduce filter dynamic order:

ξ0 = −c(ξ0 − x

)+ x2, ξ0 ∈ R1,

ξ = −cξ +[x1, x2

1

], ξ ∈ R1×2,

(32)

where c = 10.The unknown parameter is [θ1, θ2] = [4.4, 38.5]; the

number of the multiple identification models is N = 4; forconvenience to comparison with [7], the initial plant state is[x1(0), x2(0)] = [0.5,−10]; the same initial filter states are

ξ0 = 0.5, ξ =[

0 0]

, and the initial estimate parameters

for model 1, model 2, model 3, and model 4 are θ1(0) =[1, 1]T , θ2(0) = [1, 5]T , θ3(0) = [5, 1]T , θ4(0) = [5, 40]T ,respectively. Figures 1–4 depict the simulation results.

These simulation results clearly showed that the multiplemodels adaptive controller presented in this paper guaran-tees the boundedness of all the states in the closed-loopsystem and achieves the asymptotic tracking of the output.

Figure 1 is the output y(t), which demonstrates that themultiple models adaptive controller developed in this paperhas the similar property as shown in [7] and is significantlybetter than using the classical adaptive control. Figures 2and 3 are the control inputs which show that the multiplemodel adaptive control can reduce the maximum controlinput dramatically. Besides, it seems to conclude that themultiple model adaptive control proposed in this paper hasthe similar property and so the trajectory is nearly to overlap


0 5 10 15 20

0

1

2

3

−1

Time (s)

Figure 1: Output y(t). dash-dotted line for the classical adaptivecontrol, dashed line for the multiple model case (N = 200) as in[7], and solid line for the multiple identification model developedin this paper.

0 0.01 0.02 0.03 0.04 0.05 0.06

0

0.5

1

−1

−0.5

×105

Time (s)

Figure 2: Control u(t) on the time interval [0, 0.06]. dash-dottedline for the classical adaptive control, dashed line for the multiplemodel case (N = 200) as in [7], and solid line for the multipleidentification model developed in this paper.

2 4 6 8 10 12 14 16 18 20

0

100

200

300

−300

−200

−100

0.6

Time (s)

Figure 3: Control u(t) on the time interval [0.6, 20]. dash-dottedline for the classical adaptive control, dashed line for the multiplemodel case (N = 200) as in [7], and solid line for the multipleidentification model in this paper.

from Figures 1–3 because the method in [7] uses moreidentification models than ours. Figure 4 is the trajectoryof the redistribution of the identification models which canfind the most suitable identification model and enhance thetransient performance.

1 2 3 4 50

10

20

30

40

Time (s)

Figure 4: The redistribution trajectory of the identification models.

0 2 4 6 8 10

0

0.5

1

1.5

−1.5

−0.5

−1

Time (s)

Figure 5: Output y(t). dashed line for the multiple model case(N = 4) as in [7], solid line for the multiple identification modeldeveloped in this paper.

0 0.01 0.02 0.03 0.04 0.05 0.06

0

200

400

−600

−200

−400

Figure 6: Control u(t) on the time interval [0, 0.06]. dashed linefor the multiple model case (N = 4) in [7], and solid line for themultiple model developed in this paper.

Next, we can compare the approach presented in thispaper with the method developed in [7] with the multipleidentification models (N = 200) is set to (N = 4), which isthe same identification models used in our approach. Figures5–7 depict the simulation results.

Figure 5 is the output y(t) with the multiple modelsadaptive controller, which shows the approach developedin this paper is superior to the method presented in [7].Figures 6 and 7 are the control inputs which show that themultiple model adaptive control developed in this paper has


0.06 1 2 3 4 5 6 7 8 9 10

0

50

100

−200

−150

−100

−50

Time (s)

Figure 7: Control u(t) on the time interval [0.06, 10]. dashed linefor the multiple model case (N = 4) in [7], and solid line for themultiple model developed in this paper.

better properties than the method presented in [7] which hasswitching and larger control input.

6. Conclusions

In this paper, a novel multiple models adaptive controllerwas developed for a class of nonlinear systems. The multiplemodels technique was used to describe the most appropriatemodel at different environments. If the number of theidentification models that can be used is abundantly large,the “switching” or “switching and tuning” scheme may acton satisfactorily. Otherwise, the multiple models adaptivecontrol cannot work as expected when the number of iden-tification models available is relatively small compared withthe size of the uncertainty region. So we consider the methodin which the location of the identification models can beredistributed. Unlike previous results, we do not require aswitching scheme to guarantee the most appropriate modelto be switched into the controller design which can simplifythe analysis of the stability of the closed-loop system.

Acknowledgments

The authors would like to thank the reviewers and theeditors for their helpful and insightful comments for furtherimprovement of the quality of this work. The work is sup-ported by the National Natural Science Foundation of China(nos. 60674023 and 60905034) and the Zhejiang ProvincialNatural Science Foundation of China (no. Y12F030119).

References

[1] K. S. Narendra and J. Balakrishnan, “Improving transientresponse of adaptive control systems using multiple modelsand switching,” IEEE Transactions on Automatic Control, vol.39, no. 9, pp. 1861–1866, 1994.

[2] K. S. Narendra and J. Balakrishnan, “Adaptive control usingmultiple models,” IEEE Transactions on Automatic Control, vol.42, no. 2, pp. 171–187, 1997.

[3] J. P. Hespanha, D. Liberzon, and A. S. Morse, “Overcoming thelimitations of adaptive control by means of logic-based

switching,” Systems and Control Letters, vol. 49, no. 1, pp. 49–65, 2003.

[4] Z. Han and K. S. Narendra, “Multiple adaptive models for con-trol,” in 2010 49th IEEE Conference on Decision and Control,CDC 2010, pp. 60–65, usa, December 2010.

[5] K. S. Narendra and Z. Han, “The changing face of adaptivecontrol: the use of multiple models,” Annual Reviews in Con-trol, vol. 35, no. 1, pp. 1–12, 2011.

[6] K. S. Narendra and K. George, “Adaptive control of simplenonlinear systems using multiple models,” in 2002 AmericanControl Conference, pp. 1779–1784, usa, May 2002.

[7] X. D. Ye, “Nonlinear adaptive control using multiple identifi-cation models,” Systems and Control Letters, vol. 57, no. 7, pp.578–584, 2008.

[8] K. S. Narendra and K. George, “Adaptive control of simplenonlinear systems using multiple models,” in 2002 AmericanControl Conference, pp. 1779–1784, usa, May 2002.

[9] M. K. Ciliz and A. Cezayirli, “Increased transient performancefor the adaptive control of feedback linearizable systems usingmultiple models,” International Journal of Control, vol. 79, no.10, pp. 1205–1215, 2006.

[10] M. Krstic and P. V. Kokotovic, “Adaptive nonlinear design withcontroller-identifier separation and swapping,” IEEE Transac-tions on Automatic Control, vol. 40, no. 3, pp. 426–440, 1995.

[11] H. S. Ke and X. D. Ye, “Robust adaptive controller design fora class of nonlinear systems with unknown high frequencygains,” Journal of Zhejiang University, vol. 7, no. 3, pp. 315–320, 2006.

[12] P. Bashivan and A. Fatehi, “Improved switching for multiplemodel adaptive controller in noisy environment,” Journal ofProcess Control, vol. 22, no. 2, pp. 390–396, 2012.

[13] H. H. Xiong and S. Y. Li, “Satisfying optimal control ofswitching multiple models based on mixed logic dynamics,”International Journal of Modelling, Identification and Control,vol. 10, no. 1-2, pp. 175–180, 2010.


Research Article

Adaptive Impedance Control to Enhance Human Skill on a HapticInterface System

Satoshi Suzuki and Katsuhisa Furuta

Department of Robotics and Mechatronics, School of Science and Technology for Future Life, Tokyo Denki University, 5 Asahi-Chou,Senju, Adachi-Ku, Tokyo 120-8551, Japan

Correspondence should be addressed to Satoshi Suzuki, [email protected]

Received 5 December 2011; Revised 9 March 2012; Accepted 30 March 2012


Copyright © 2012 S. Suzuki and K. Furuta. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Adaptive assistive control for a haptic interface system is proposed in the present paper. The assistive control system consists ofthree subsystems: a servo controller to match the response of the controlled machine to the virtual model, an online identifier ofthe operator’s control characteristics, and a variable dynamics control using adaptive mechanism. The adaptive mechanism tunesan impedance of the virtual model for the haptic device according to the identified operator’s characteristics so as to enhance theoperator’s control performance. The adaptive law is derived by utilizing a Lyapunov candidate function. Using a haptic interfacedevice composed by a xy-stage, an effectiveness of the proposed control method was evaluated experimentally. As a result, it wasconfirmed that the operator’s characteristics can be estimated sufficiently and that performance of the operation was enhanced bythe variable dynamics assistive control.

1. Introduction

An impedance control is a key technology of the force/motion control for any mechanical systems such as anactive vehicle suspension, a power steering system, a ma-chining and handling by manipulators, and a tele-operationsystem. Since dynamics of such controlled mechanism canbe adjusted by changing the virtual impedance model,this method is effective to adapt to an ever-changingenvironment and conditions. Also a biological system hasacquired similar strategy of the variable impedance controlin the course of an evolution. It is known that an impedanceof a musculoskeletal system is changed dynamically duringwalking, running, and moving the hand [1]. Therefore,variable impedance methods have been studied for artifi-cial legs/orthosis [2] and material-handling machines [3,4]. Parameters of those impedance control methods are,however, often tuned empirically and intuitively; hence,the system designers have to adjust them according toindividuals. Due to this issue, users sometimes have toadapt themselves to the controlled machine when the tuningcondition given by the designer is not adequate for the user.

To resolve this paradox, the following approach is ideal;the control characteristics of each user are identified duringthe operation, and then control of the machine is adjustedadaptively according to the identified user’s characteristics.While several similar approaches concerning online variableimpedance control are reported, troublesome processes suchas a training phase [5] or an empirical tuning for differenttypes of motion [6] are required. Since these approachesare not real adaptive control, a realization of an automatictuning mechanism without intervention of system designersis expected. Therefore, the present paper presents a designprocedure of true adaptive variable impedance control foran assisting system which is considered with the followingproperties based on the previous method presented in [7]:

(a) adaptivity to control characteristics of individualuser,

(b) derivation of adaptive law for variable impedancetuning based on an adaptive control theory.

Main purpose of the control design proposed here is a devel-opment of an adaptive tuning law of the machine dynamics,


of which parameters are fixed in an ordinary mechatronicssystem, in order to enhance manipulation performance ofwhole of a human-machine system. And, the purposes of thepresent study are as follows:

(c) experimental evaluation,

(d) confirmation of the benefit and issue.

Item (a) is realized with an on-line identification based onan assumed model of the human controller. Concerning item(b), an adaptive control law to adjust impedance parametersof the virtual model is derived to ensure the stability andperformance of the whole human-machine system. Item (c),evaluation, was performed using a haptic interface devicethrough a point-to-point operation task. Issues and analysisdenoted at item (d) are discussed based on the results of theexperiment.

This paper is organized as follows. In Section 2, a conceptof the adaptive impedance control and its background arementioned. The haptic interface system and task whichwere used in the experiment are explained there. Section 3explains a procedure of the presented assistive control, and itstheoretical proof is given there. Section 4 shows results of theexperiment and analysis to confirm the effectiveness of thepresented method. Last Section 5 presents the conclusion.

2. Human Assistive System

2.1. Concept of Adaptive Impedance Control. In order todesign an assistive mechanism for user’s manipulation in ahuman-in-the-loop system, an adequate human modelingand a feasible assistive control are required. Human mod-eling has been studied in the field of control engineeringfrom its early beginnings such as a linear servo control model[8], a PID-base time-variant model [9], and an optimalcontrol model [10]. Those models can express a humanbehavior well for each assumed situation; however, it isinadequate to explain the learning process of a user fromthe beginner phase to the expert phase. To find adequatemodel which can treat a human adaptability, it is adequateto refer the voluntary motion control of a musculoskeletalsystem. The reason is that such model is formulated toexplain a process of a human development, and most popularmodel is a feedback-error-learning model [11, 12]. Thismodel can be utilized to explain the learning of an externalunknown dynamics since such an external system can bethought as an extension of our body. On the learning processof the external dynamics, a delay has to be consideredbecause it concerns the stability and performance of wholehuman-machine system. The delay arises certainly at thevisual processing and at the neural transmission between abrain and sensory receptor/muscles. Such undesirable effectgiven by the delay is compensated by an internal feedbackcompensation using an efferent neuron and by a delaycompensation mechanism which is explained by the Smithpredictor theory [13]. Additionally, as shown in Kleinman’sresearch of the dynamics of a pilot [10], a human (i.e.,pilot) has a high ability to compensate the delay in theresponse of a vehicle. Hence, if a time-delay effect inside the

Reference

Human

++ +

Inverse modelP−1(s)

Feedbackcontroller−

Delaycompensator

MachineP(s)

Learning of

the dynamics

Figure 1: Human model inside a human-machine system.

machine side system does not change, it is expected that thedelay in the human-machine system can be compensatedrelatively easily. Therefore, the block diagram shown inFigure 1 appears adequate for designing of a human-machineassist system. The model mentions that human learns theunknown dynamics of the machine and uses the identifiedmodel as an inverse model for the manipulation of themachine.

The concept shown in Figure 1 indicates that differenceof the machine’s dynamics affects indirectly to the learningof the operator. If the operated machine can change its owndynamics characteristics so as to be learned by individualoperator without difficulty, performance of the operationwould be enhanced. Therefore, in order to enhance theoperator’s performance, as shown in Figure 2, an originaldynamics of the machine is replaced to a virtual dynamicsmodel from the operator’s side by making a local loopfeedback with a virtual internal model control. In short, theimpedance of the virtual dynamics model is modified so as todecrease an error which relates with each task performance.To summarize this discussion, the following three functionsare required to realize aforementioned adaptive impedancecontrol.

Step 1. Virtual internal model (VIM) control.

Step 2. Online identification of the operator.

Step 3. Adaptive mechanism to tune the VIM.

The VIM control for Step 1 is realized by making a localservo system that tracks the output of a virtual impedancemodel. The servo control input law is designed usingLinear-Quadratic Regulator (LQR). Identification for Step 2is performed by assuming a parametric model of operator’scontrol characteristics. Concerning Step 3, the adaptivemechanism is designed by changing the impedance param-eters of the VIM obtained at the Step 3 after derivation ofthe adaptive law of the VIM model based on a Lyapunov-likefunction. Details of this process are explained in Section 3.

2.2. Experimental System. A haptic interface system, which isshown in Figure 3, was used to evaluate the adaptive assistivecontrol presented in Section 3. The haptic device consists ofa two degree-of-freedom planer xy-stage, produced by NSKcorporation, and a real-time CG monitor programmed byvisual C++. The xy-stage is driven by two linear direct drive


Reference

Human

++ + +

Inverse modelP−1(s)

Feedbackcontroller

− −Delay

compensator

MachineP(s)

Virtualmodel

Identifier

Servocontroller

Adaptive assist system

Figure 2: Structure of human assistive system with the adaptive impedance control.

Gripxy-stage

Force sensor

Operation monitor

Figure 3: Haptic interface devices.

−0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Target

Pointer

(synchronized to

grip’s position)

Yax

is(m

)

X-Y table location

Figure 4: Operation monitor of the haptic interface system.

motors. The operator moves a grip attached to the xy-stageand a pointer displayed on the CG monitor (see Figure 4) isalso moved according to the position of the grip. Operator’shand force is measured by a 6-axis force sensor embeddedbetween the grip and the stage. The x- and y-axes of thestage do not effect each other because of a mechanicallyindependent design. Computations of the control of the

stage and the CG displaying are executed by a PC/AT 3 GHzcomputer under real-time scheduling control. The controlinterval is 2 ms, and the movable range is about 62 mm inboth x and y directions.

2.3. Task of Manipulation. Since a point-to-point (PTP)manipulation is a popular task both in a daily life andin an industrial situation, the PTP-task was adopted forverification of the presented method. The PTP tasks wererepeated by changing the target’s position at random so asto keep the distance constant from each last target to the nextone. As soon as the target is displayed on the monitor, theoperator moves the pointer to the target by manipulatingthe grip of the xy-stage. When position of the pointer iskept inside the target circle for 3 seconds, one PTP motion(one trial) is finished, and then a new next target circle isdisplayed at random. To reduce the fatigue of the participant,ten-second rest was given to the participant after every fivetrials.

3. Design of Adaptive Impedance Control

3.1. Virtual Internal Model Control (Step 1). A procedureto apply the virtual internal model (VIM) control [14] tothe xy-stage is mentioned in this section. Any mechan-ical mechanism includes nonlinearity caused by friction,variances of viscosity, and unknown dynamics; hence, it isdifficult to apply a linear system control theory to actualmachine without any nonlinear compensation. Since VIM iseffective to suppress inherent characteristics of mechanicalcomponents such as frictions, an adaptive control for linearsystems can produce an effect. Although the haptic systemused in the experiment has two degree-of-freedom motions,controllers of the x- and y-axes can be designed separatelythanks to the mechanical independence; hence, subscriptsof x and y are omitted in later explanation. Variables andparameters of the haptic device model are shown in Figure5 and Table 1.

The block diagram of the virtual internal model controlis shown in Figure 6 and the related variables and parametersare summarized in Table 2. Dynamic equations of the stageand the virtual model are expressed as follows:

mpxp + dpxp = fh + fa, (1)

mrxr + dr xr = fh. (2)


xpx

dpx

mpx

fhy fh

fhxxpy

dpy

mpy

Grip

Stage

Figure 5: Model of the xy-stage.

Table 1: Parameters and variables of haptic interface device.

Variables/parameters Unit Meanings

xp∗ [m] Position of a grip

fh∗ [N] Force from an operator

fa∗ [N] Force from an actuator

mp∗ [kg] Mass of a stage

dp∗ [Ns/m] Viscosity of a stage

∗: x or y for x- and y-axis.

Table 2: Parameters and variables for virtual model control.

Variables/parameters Unit Meanings

xr [m] Position of a virtual model

er [m] Error (=: xr − xp)

mr [kg] Mass of a virtual model

dr [Ns/m] Viscosity of a virtual model

Defining an error as er := xr−xp, (1) and (2) are transformedinto

d

dt

⎡⎣erer

⎤⎦ =⎡⎣0 1

0 0

⎤⎦⎡⎣erer

⎤⎦ +

⎡⎣0

1

⎤⎦uu := −dr xr + fh

mr+dpxp − fh − fa

mp.

(3)

Minimizing the error defined by (3) makes the stage conformto a response of the virtual model described by (2). Tocompensate steady-state error, the integral variable

∫er is

taken into consideration in the state vector as follows:

d

dtz = Az + Bu, (4)

where

A :=

⎡⎢⎢⎢⎣0 1 0

0 0 1

0 0 0

⎤⎥⎥⎥⎦, B :=

⎡⎢⎢⎢⎣0

0

1

⎤⎥⎥⎥⎦, z :=

⎡⎢⎢⎢⎣∫er

er

er

⎤⎥⎥⎥⎦. (5)

The control law is calculated using an LQR method with thequadratic criterion:

J =∫∞

0

(zTQz + uTRu

)dt, (6)

where a positive semi-definite Q ∈ R3 × 3 and a positivedefinite R ∈ R1 × 1 are weighting matrices. The input is givenas

u = −Fz, F := R−1BTP, (7)

where P is a symmetry positive-definite matrix of a RiccatiAlgebraic Equation given by

PA + ATP − PBR−1BTP + Q = 0. (8)

Since (7) is expanded as

−Fz = −dr xr + fhmr

+dpxp − fh − fa

mp, (9)

a final form of the control law is obtained as follows:

fa =mp

mr

(fh − dr xr

)+ dpxp − fh −mpFz. (10)

For the actual apparatus used in the experiment, parametersof VIM were specified as mrx = mry = 50 [kg] and drx =dry = 50 [Ns/m] to intentionally obtain a slightly difficultmanipulation feeling as a training test with considerationof input range of the actuators. The weighting matricesin (6) for the LQR servo design were decided as Q =diag (150, 1.12 × 107, 5100) and R = 1. As a result, the feed-back gain matrix was obtained as F = [10.9, 3007.9, 100.7].

3.2. Online Identification of Human Control Characteristics(Step 2). Human control characteristics are complex becausevarious kinds of compensators, such as an oculomotorcontrol, a proprioceptive control, and a neuromuscularcontrol, are related to each other [13]. There is, however, afairly large body of data that can be explained by a linearmodel plus time delay [15] when an operation conditionis limited. One of most famous models supporting suchlinear model assumption is a crossover model. This modelinsists that a frequency transfer function of a skilled operatorin a man-machine system adapts to make the total systemkept unchanged under a variation of the controlled systemdynamics. In other words, human changes own control char-acteristics so that a closed-loop transfer function of wholehuman-machine system becomes a first-order system at awide frequency band and the human plays a role of simplelinear model. Also in previous study of the present authors,an identification analysis of the skilled operator’s frequencycharacteristics showed an existence of the cross-over modelthrough a juggling task using a haptic test device [16]. Hence,whole system relating a voluntary motion is simplified intothe three components in this study: a linear controller insidea brain, a neuromuscular dynamics, and reaction time delay.After a learning of the machine dynamics is sufficientlyfinished, the human can be considered as a simple feedbackcontroller which moves the grip to the target position bywatching the monitor in case of the PTP task. Finally, a blockdiagram of a visual voluntary motion control is assumed asa feedback model as shown in Figure 8. In the figure, r is areference position for a pointer, eh is an error between the


fh

Humanforce

Forcesensor

1(ms+b)s

Virtualmodel

xr+ +

+erLQRservo−

DD-motor

fa

Plant

xy-stage

xp

Figure 6: Block diagram of virtual internal model control.

target and the present pointer, and uh is an input computedby a brain controller. Here, the plant block is a virtualxy-stage of which impedance property is adjusted by theVIM control. The neuromuscular dynamics can often beapproximated by a first-order lag [17] and a simplest humancontroller is a PD controller [9]; hence, the human transferfunction, G′h(s), is assumed in this study as

G′h(s) = Kds + Kp

Ts + 1e−Ls, (11)

where Kp,Kd,T , and L are a proportional gain of thehuman brain controller, the differential gain of it, a timeconstant of the neuromuscular system, and reaction timedelay, respectively. As discussed in Section 2.1, compensationof the delay factor is necessary not only for voluntary motioncontrol in a human but also for an adequate human-machinesystem, and a human has an excellent ability to compensatethe delay effect. And an influence of the delay to the controlcharacteristics of the whole human-machine system dependson a response speed of the machine and the task condition.Therefore, in the present study, the response delay of partici-pants was investigated as a preliminary experiment using theVIM control which was designed at Section 3.1. Participantaged 22 years was requested to execute the PTP manipulationhundred times. The time that the pointer begins moving justafter the new target circle was displayed on the monitor wascounted as the response delay. Figure 7 shows the changeof the measured time delay. The dots represent measuredvalues, and the solid lines express an approximated third-order polynomial fitting curve from the measured data. Thisgraph shows no conclusive relationship between time delayand the number of trials, and the value is almost constant atabout 0.4 second. Additional nine participants showed sametendency, and significant difference between individuals wasnot confirmed. For this reason, it was expected that thesimple data shift would be sufficient to compensate the delayeffect in the identification for the present study. Therefore,the time delay factor described in (11) was omitted forthe identification by shifting the measured data for the0.4 second as a rest time, and the following model wasconsidered for later process:

Gh(s) = Kds + Kp

Ts + 1. (12)

Applying a bilinear transformation

s � 2Δ· 1− z−1

1 + z−1(13)

to (12) yields the following discrete impulse transfer functionGh[z]:

Gh[z] = b1z−1 + b0

a1z−1 + 1, (14)

a1 := −2T + Δ

2T + Δ, (15)

b0 := 2Kd + KpΔ

2T + Δ, (16)

b1 := KpΔ− 2Kd

2T + Δ, (17)

where Δ is a sampling interval. From (15)–(17), followingequations are derived:

T = Δ

21− a1

1 + a1,

Kp = 2T + Δ

2Δ(b0 + b1),

Kd = 2T + Δ

4(b0 − b1).

(18)

If a0, b0, and b1 are identified from the input/output responsedata, characteristic parameters of the human controller canbe derived using (18). These parameters are used in a designof the next variable dynamics assistive controller.

3.3. Variable Dynamics Assist Control (Step 3). An assistivecontrol proposed in this paper changes dynamics of the inter-nal model on-line depending on operator’s characteristics. Ablock diagram of the assistive control is shown in Figure 9(a).In the figure, r, y, e, v, and f are a positional reference,a position of the stage, the error, an output of a braincontroller, and a force generated by the hand, respectively. Itis assumed that (a) Kp,Kd, and T are time-slowing changingparameters and that (b) parameters of a virtual machine m

and b can be tuned, because the assumption (b) is realized by

the VIM control, that is, m and b are adjustable parameters inthis scheme. Figure 9(a) expresses a general human-machinesystem that includes a human controller (Kp + Kd)/(Ts +

1) =: C and a plant 1/(ms + b)s =: P for virtual hapticinterface device. It can be considered conversely that thesystem consists of a plant C changing slowly the parameters(Kp,Kd, and T) and the controller P having directly variable

coefficients (m and b), as shown in Figure 9(b).Note that the output y of new controller P cannot be

changed arbitrarily and that only tuning of the controller’s


0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1Variance of response delay

(Number of trial)

(s)

Measured

Interpolated curve

Figure 7: Variance of response delay.

coefficients is possible. Moreover, transformation of theblock diagram shown in Figure 9(b) yields a general feedbackform, as shown in Figure 9(c). In the following, in orderto avoid misunderstanding owing to habits, characters forvariables x and u are used instead of f and e, respectively.Then, the following equations are obtained:

x(s) = Kp + Kds

Ts + 1u(s), (19)

u(s) = 1(ms + b

)se′(s), (20)

e′(s) := r′(s)− x(s), (21)

r′(s) :=(ms + b

)s · r(s). (22)

The purpose of the PTP task is a tracking such thaty → r in the original block diagram shown in Figure 9(a).This means that e → 0 (in Figure 9(b)), that is, u → 0 (inFigure 9(c)); then (20) indicates that e′ → 0 as t → ∞. First,choosing a Lyapnov candidate V as V := (1/2)e′(t)2, thecondition of convergence is investigated. It can be consideredthat a closed-loop system shown in Figure 9(c) is almoststable under the assumptions of (a) and (b); hence, it is notalways necessary that dV/dt < 0 holds for keeping the

stability. Second, an update law for m and b is derived usinga Lyapunov-like analysis. If a step input is chosen for r(t)for the PTP motion, the response of r′(t) defined by (22)becomes almost impulse shape. The moment of t = 0is, however, not important practically because the purposeof the control is an enhancement of the performance ofthe motion by making the tracking error small whichoccurs mainly by the positioning near the target position. Inaddition, the impulsive response converges into zero rapidly,hence, an approximation as dr′(t)/dt � 0 holds if t � 0.

Then, the time-derivative of V can be approximated and canbe transformed as follows:

d

dtV(t) = e′(t)

d

dte′(t) � −e′(t) d

dtx(t) (t > 0)

= −e′(t) ddt

L−1

⎡⎣Kp + Kds

Ts + 11(

ms + b)se′(s)

⎤⎦

= −e′(t)L−1

⎡⎣Kp + Kds

Ts + 11(

ms + b)e′(s)

⎤⎦= −e′(t)L−1

[Kp − Kd/T

bT − m· 1s + 1/T

e′(s)

+Kp − Kdb/m

m− T b· 1

s + b/me′(s)

⎤⎦

= −e′(t)⎧⎨⎩Kp − Kd/T

bT − m· φ(t,

1T

)

+Kp − Kdb/m

m− T b· φ(t,b

m

)⎫⎬⎭,

(23)

where the function φ(t,α) is defined as

φ(t,α) :=∫ t

0e−α(t−τ) · e′(τ)dτ. (24)

Since it is necessary for each term in (23) to be negative inorder to satisfy dV/dt < 0 as long as possible, the followingconditions are considered:

Kp − Kd/T

bT − m> (<)0 if e′(t)φ

(t,

1T

)> (<)0, (25)

Kp − Kdb/m

m− T b> (<)0 if e′(t)φ

(t,b

m

)> (<)0. (26)

Conversely, if parameters do not fulfill the previous in-

equality conditions, variable parameters m and b are tunedso as the unsatisfied condition will be recovered. Now, thefollowing intermediate variables are introduced:

δ1 := η1 ·(bT − m

),

δ2 := η2 ·(m− T b

),

η1 := sgn(Kp − Kd

T

)· sgn

{e′(t)φ

(t,

1T

)},

η2 := sgn

(Kp − Kdb

m

)· sgn

{e′(t)φ

(t,b

m

)}.

(27)

By checking signs of a numerator and a denominator of

(25) and signs of m and b, the following update law can beconsidered:

b[t + Δ]←− b[t] + k1σ(δ1)η1 · |e|,m[t + Δ]←− m[t]− k2σ(δ1)η1 · |e|,

(28)


r

Human model

+Delay

visual, recog.−

Delayadjust

eh ehController

uhMuscle

fhPlant

xp

Identifier T , Kp , Kd

Figure 8: A human control model and its identification.

r

Controller C

e

Plant P

+ 1(ms+b)s

Purposeof control

y → r

y1

Ts+1

A fKp + Kds

−

(a)

r e+

1(ms+b)s

Purpose

of control−Kp+Kd sTs+1

e → 0

f

y

(b)

r + 1(ms+b)s

Purpose

of control−

(ms + b)sr e

Refrencemodifier

New controller P

e(= u)

New plant C

f (= x)

u→ 0

(c)

Figure 9: Transformation of block diagrams for variable dynamicscontrol.

where k1 and k2 are positive constant parameters, Δ is acontrol interval, brackets in previous equations mean adiscrete-time point, and a function σ is defined as

σ(δ) =⎧⎨⎩0, δ > 0,

|δ|, δ < 0.(29)

The other update law is derived from (26) in same manner asfollows:

b[t + Δ]←− b[t]− k3σ(δ2)η2 · |e|,m[t + Δ]←− m[t] + k4σ(δ2)η2 · |e|,

(30)

where k3 and k4 are positive constants. Equations (28)–(30)are summarized into the following parameter update law.

⎡⎣ bm

⎤⎦[t+Δ]

=⎡⎣ bm

⎤⎦[t]

+

⎡⎣ k1 −k3

−k2 k4

⎤⎦⎡⎣σ(δ1)η1

σ(δ2)η2

⎤⎦. (31)

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.005

0.01

0.015

0.02

0.025PTP response

Time (s)

Actual response By identified modelReference position

−0.005

Posi

tion

(x)

(m

)

Figure 10: Comparison of PTP responses.

On the implementation, these parameters are updated underthe following practical limit to avoid an input saturation ofactual actuators:

b < b < b, m < m < m, (32)

where b, b,m, and m are constant. Here parameters ki arechosen as they satisfy k1k4 − k2k3 /= 0. Integral computationdescribed in (24) is executed by using the following alterna-tive online recursive computation:

φ[t,α] = e−αΔφ[t − Δ,α] + e′[t]Δ. (33)

Since (22) cannot be computed directly, an approximation as

(ms + b)s � (ms + b)s/(0.01s + 1)2 is used, and the responseis computed by the Eular integration with the state-spacemodel which is derived with a controllable canonical form.Kp,Kd, and T are identified on every PTP motion and areupdated according to an appropriateness of the identificationresult.

4. Experimental Result and Analysis

4.1. Online Identification of Human Controller. For a designof the VIM control of the xy-stage, the initial parameters

were chosen as m[0] = 50 [kg], and b[0] = 50 [Ns/m]. Inputinformation for the identification was chosen as an error


10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3Settling time

Number of trials

(s)

No adaptive impedence ctrl.

Adaptive impedence ctrl.

Figure 11: Evolution of settling time.

10 20 30 40 50 60 70 800.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05Accumulated error

Number of trials

Adaptive impedence ctrl.

No adaptive impedence ctrl.

(—)

Figure 12: Evolution of accumulated error.

between the current position and the target one. Mea-surement value of force filtered through a 36 Hz LPF wasused as output information for the identification. Thetime-delay effect was compensated by sifting the measuredinput signal at every PTP motion. Measured data wasdecimated by a factor of 10 for an identification; in short, anidentification sampling time is 20 ms to suppress oscillationin the identified parameters. One result of the identificationis shown in Figure 10. The solid curve is a simulated stepresponse that was computed using an identified humancontroller model and virtual dynamics model of the stage.Those identified parameters were Kp = 779.0, Kd = 288.0,

and T = 0.18, and the time delay was treated as L = 0.406in the simulation. Since the response of the identified modelresembles to the actual response, it can confirmed that theidentification process was reasonable.

4.2. Verification of Assistive Effect. Since a key point of theproposed assistive method is to increase performance of the

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

80

90

100Change of tuned parameters

Time (s)

Virtual mass mVirtual viscosity b

(kg)

(N

s/m

)

Figure 13: Evolution of tuned parameters.

10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

Time constant: T

Number of trials

(s)

Figure 14: Change of identified T .

operator’s manipulation by adjusting the machine dynamics,we investigated whether the performance of an operator whowas used to the PTP operation without the adaptive controlcould be increased with the proposed adaptive control. Fromthis aim, before the presented assistive control was applied toa participant, sufficient training was given to become a skilledoperator using the haptic device tuned with fixed parameterswhich were same initial values on the assistive control. Asa result of this preliminary training, it was confirmed thatthe performance of the participant became good and didnot indicate no further improvement by checking the settlingtime on the PTP operation.

For the assistive control, parameters of the update law in(31) were chosen as k1 = k4 = 1 × 10−4, and k2 = k3 =2 × 10−4. Since even the expert showed perturbation in theperformance at the beginning of several trials, the normalVIM control was executed from the first trial and the adaptive


10 20 30 40 50 60 70 800

200

400

600

800

1000

1200

Number of trials

Gain: Kp

(—)

Figure 15: Change of identified Kp.

10 20 30 40 50 60 70 800

100

200

300

400

500

Number of trials

Gain: Kd

(—)

Figure 16: Change of identified Kd .

impedance control was activated after 50 seconds (about10 trials). Figure 11 shows an evolution of the settling timefrom when the new target was displayed on the monitortill when the pointer was reached into the target circle.When three seconds passed after the pointer was kept stayinginside the target circle on the monitor, it was judged thatthe pointer had been moved to the target by the operation.Values of the y-axis in the figure show the settling time thatdoes not include three seconds. The solid line shows theresult of the adaptive impedance control, and the dotted lineshows the other result obtained by nonadaptive impedancecontrol before the participant did not yet try the adaptiveimpedance control. Each line shows the evolution of trendcomputed by the moving average computation against fivePTP tasks. While the nonadaptive impedance control caseshows roughly steady state of 1.8 seconds after 50 trials, theother adaptive impedance control case shows a decrease to

about 1.1 seconds. In short, speed of the PTP motion wasimproved by the adaptive impedance control.

Figure 12 shows an evolution of the accumulation errors∫ |e(t)|dt till each settling time at each trial. Similarly, thesolid and dotted lines show the results of the adaptiveimpedance control and nonadaptive control, respectively.It can be confirmed that the accumulation error was alsodecreasing in case of the adaptive control. Both Figures 12and 11 demonstrated effects of the presented method.

Figure 13 shows change of tuned parameters m and b. Atthe beginning of trials, these values were constant becausethe assistive control was activated after 50 seconds. After 150seconds, m was saturated at the lower limit that was specifiedfor the safety. The reason of this nonconvergence is that theupdate law (31) cannot guarantee to stop the update of theparameters since the law was designed so as to make thetracking error be zero. This practical issue can be avoided byintroducing a dead-zone against small error against theupdate law.

Finally, transitions of the identified parameters of theoperator’s control model, T ,Kp, andKd, are shown in Figures14, 15, and 16, respectively. Identified parameter variesduring till 30 trials. Transitions of their moving averagesare comparatively flat at period of 30–80 trials exceptrapid change due to large outlier in 53rd trial. Although itis difficult to find tendency of change of the identifiedcharacteristics, it was confirmed that their moving averagesof parameters T ,Kp, and Kd are almost constant after 60trials of when the tracking error keeps small in Figures 11and 12. Their constant values do not differ much from theirinitial values which are ones before the activation of theadaptive control law. In short, it can be considered that totalperformance was increased by changing the machine sidemainly without imposition of large change in human side.This supposition is not authentic since it is not demonstratedby statistical analysis with sufficient number of participants.These are future work.

Results of the experiment, however, showed that theproposed assistive control approach works well, and it can besaid that the total performance of whole human-machinesystem can be enhanced by changing the dynamics of themachine itself.

5. Conclusion

For a force-feedback haptic interface system, an adaptiveimpedance assistive control to enhance the manipulationperformance was proposed. The strategy consists of anidentification of an operator’s control characteristics andan adaptive online tuning of the dynamic property of themachine. The tuning is executed by changing impedanceparameters of a virtual internal model for the machine.The adaptive law of the tuning was derived by utilizing aLyapunov stability concept. Using a haptic 2-DOF inter-face device, it was demonstrated that proposed adaptiveimpedance assistive control worked effectively. The tuninglaw, however, cannot guarantee a convergence to a steadystate without reaching to the safety limit yet since the


presented algorithm was designed by only focusing on anenhancement of the performance of human manipulationwithout consideration of the machine limit such as an inputsaturation and frequency bandwidth. This practical issue willbe resolved by introducing a tradeoff computation between aperformance and requirement of the machine side such as anenergy consumption. In the present study, however, a basicstrategy for the design of human assistive system could beshown; hence, we would like to treat such practical issues infuture.

Acknowledgments

This work is supported by the Grant-in-Aid for 21st Century(Center of Excellence) COE Program in Ministry of Educa-tion, Culture, Sports, Science and Technology. Preparation ofthe experimental system and the experiment were supportedby Keiichi Kurihara and other participants who embracedthe authors requests kindly. The authors are grateful to allof them for supporting this work.

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Research Article

Adaptive Control Allocation in the Presence of Actuator Failures

Yu Liu and Luis G. Crespo

National Institute of Aerospace, Hampton, VA 23666, USA

Correspondence should be addressed to Yu Liu, [email protected]

Received 3 November 2011; Revised 17 February 2012; Accepted 2 March 2012

Academic Editor: Yunjun Xu

Copyright © 2012 Y. Liu and L. G. Crespo. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper proposes a control allocation framework where a feedback adaptive signal is designed for a group of redundant actuatorsand then it is adaptively allocated among all group members. In the adaptive control allocation structure, cooperative actuatorsare grouped and treated as an equivalent control effector. A state feedback adaptive control signal is designed for the equivalenteffector and adaptively allocated to the member actuators. Two adaptive control allocation algorithms, guaranteeing closed-loopstability and asymptotic state tracking when partial and total loss of control effectiveness occur, are developed. Proper groupingof the actuators reduces the controller complexity without reducing their efficacy. The implementation and effectiveness of thestrategies proposed is demonstrated in detail using several examples.

1. Introduction

Actuator redundancy is highly desirable for fault-tolerantcontrol. This redundancy yields multiple ways to implementthe forces and moments prescribed by the controller. How-ever, this freedom creates the need for properly allocatingthe control inputs among all individual actuators. Whilemultiple actuator configurations do generate the desiredforces and moments, some of them may yield unintendedoutcomes, for example, the effect of some control surfacedeflections counteracts the effect of other ones. Redundancymanagement is the need for properly allocating the controlinputs among functionally redundant actuators when someof them may not be fully functional.

The purpose of control allocation is to distribute thecontrol signals to the available actuators such that the desiredmoments and forces are efficiently generated. Traditionalcontrol allocation methods include explicit ganging [1],daisy chaining [2], pseudinverse [3, 4], and error andcontrol minimization [5–10]. Explicit ganging performscontrol allocation by finding a fixed relation between thedesired control moments and forces and the designedcontrol signals. Multiple actuators (e.g., two aileron surfaces)can be combined to generate the desired effects. Daisychaining allocates inputs in a prioritized fashion. It utilizes

the actuators in sequence to generate certain effect. If anactuator is unable to generate such an effect, say due toactuator saturation, the next actuator in the sequence willbe used. The pseudoinverse approach, which accounts forinput saturation and failure, performs control allocationby solving a linear optimization problem in real time.Error and control minimization is another common controlallocation approach. This approach minimizes the errorbetween the desired and generated control moments subjectto control constraints. Several approaches can accommodatefor actuator failure and saturation compensation, providedthat the actuator failure or saturation has been identified.Thus one obvious drawback of these approaches is thatthey require a fault detection system. These systems, whichare usually complex, require the characterization of severalfailure modes a priori. Furthermore, they may requiresolving optimization problems in real time; thus they cansubstantially increase the software and hardware demands ofthe flight control system.

Adaptive control, on the other hand, does not requireknowing which controllers have failed nor the class orseverity of the failure. This is due to its ability to changecontrol parameters according to the existing flight con-dition. Due to parameter adaptation, it is also able toaccommodate for parametric uncertainties in the vehicle


dynamics. Substantial developments in adaptive control foractuator failures have been made in the last decade [11,12]. Adaptive control’s ability to seamlessly compensate foractuator failures requires for the system’s built-in actuationredundancy to be sufficient. This is usually described as arank condition on the input matrix B [11, 13]. To take advan-tage of all redundancy available, one common approachin multivariable adaptive control is to generate a controlsignal for each control surface. Such an approach endows thecontroller the maximum degree of freedom to compensatefor failure. When certain control surfaces are stuck or havereduced control effectiveness, the remaining control surfaceswill adaptively cooperate until a new combination of inputsfor the remaining control surfaces is found. This will occurautomatically without knowing which surfaces have failed,or when such failures occur. Although adaptive controlensures closed-loop stability and tracking performance,it does not differentiate between control generation andcontrol allocation, and the resulting actuation scheme maybe unacceptable. For instance, separately designed controlsignals for multiple control surface segments may cancel eachother’s effects, for example, it has been observed that thesteady-state deflection of both rudder segments for a directadaptive control had opposite angles, resulting in a wingsleveled flight with increased drag.

The lack of control allocation in the current directadaptive control framework motivates this research effort.In this study, we aim at separating control generation fromcontrol allocation. In the adaptive control allocation frame-work, a key step is to combine redundant control surfacessimilar to explicit ganging. For each group of combinedactuators, we design an adaptive control signal, which is thenallocated among group members by an adaptive gain. If nofailure occurs, a nonadaptive control allocation scheme setin advance is enforced. In the presence of uncertainty ofactuator failure, the adaptive flight controller modifies theallocation of input accordingly.

The structure of the adaptive control allocation frame-work is illustrated in Figure 1. This is a simple aircraft controlexample with the elevator controlling the pitch motion.The aircraft longitudinal state, denoted as x(t), should trackthe state of a reference system for a given reference inputr(t). The elevator consists of four segments, namely, leftoutboard, left inboard, right outboard, and right inboardsegments. For pitch control, they can be grouped togetherand considered as an equivalent elevator by adding the fourcolumns of the input matrix. A “virtual” elevator signalv0 is generated by the adaptive controller for the desiredpitching maneuver. This elevator signal is then allocated bythe adaptive allocation gains αi(t), i = 1, . . . , 4. The resultingelevator signals v0i(t) = αi(t)v0(t), i = 1, . . . , 4 will be fedto the four elevator segments. The allocation gains can beupdated on-line based on the knowledge of the nominalplant and v0 to mitigate the uncertainties of actuator failures.Conversely, in a fixed allocation scheme αi are constant, forexample, αi = 1/4, i = 1, . . . , 4.

One advantage of this adaptive control allocation struc-ture is the ease in solving problems such as the counteractingactuation. To remedy this problem, the signs of the adaptive

allocation gains αi can be made the same, say, via theprojection algorithm [14] so that the allocated control signalscannot go in opposite directions. Another advantage of thisstructure is the reduction of the controller complexity. Forinstance, let us consider a state feedback adaptive controldesign for an n-state system with m controls. The numberof controller parameters to be updated is m × n. If theproblem is solved by having all actuators in one group,there are m + n adaptive parameters. For example, if wetry to stabilize a 5-state system with 3 controls, the totaladaptive parameters using a direct model reference adaptivecontroller would be 15. With a grouping of control inputsfor which the system remains controllable (i.e., the systemis controllable for the control input matrix associated withthe grouping), the total number of adaptive parametersrequired is 8 (1 × 5 gain vector and 3 α’s). The largerthe family of redundant actuators, the bigger the benefit.Another advantage of this approach is that the virtual controlsignal can be designed using a nonadaptive (fixed) statefeedback gain, and failure compensation is achieved by onlyadapting the control allocation gains α’s. This implies thatthe proposed structure could be added to a conventionallydesigned state feedback controllers without having to usefault detection and isolation.

In this paper, we develop the mathematical foundation ofthis adaptive control allocation structure for a single groupof actuators. Two adaptive control allocation algorithmsare presented for both loss of effectiveness and constant-magnitude actuator failures. Technical issues such as designconditions, adaptive law designs, and stability analysis areaddressed. The proposed schemes are shown to guaranteestability and asymptotic state tracking in the presence ofunknown failures. Simulation-based examples are used todemonstrate the strategy proposed.

The paper is organized as follows. Section 2 presentsthe adaptive control allocation algorithm for loss of controleffectiveness. This is followed by Section 3, where a schemefor constant-magnitude actuator failure compensation isdeveloped in Section 3. Finally, a few concluding remarksclose the paper.

2. Adaptive Control Allocation Design forLoss of Effectiveness Failures

2.1. Problem Formulation

2.1.1. Plant. Consider the linear time-invariant (LTI) system

x(t) = Ax(t) + Bu(t), (1)

where x ∈ Rn and u ∈ Rm are the system state and the controlinput. The matricesA ∈ Rn×n and B ∈ Rn×m are constant andknown. The matrix B is the control gain matrix for the groupof actuators.

The control signal u(t) can be expressed as

u(t) = Λv(t), (2)

where v(t) is the allocated control signal (i.e., input to theactuator) and Λ is a piecewise constant uncertain diagonal


Longitudinal dynamics

LOB

LIB

ROB

RIB elevator

elevator

elevator

elevator

x(t)r(t) v0Controller

Control generation

Control allocation

α1(t)

α2(t)

α3(t)

α4(t)

α1υ0

α2υ0

α3υ0

α4υ0

Figure 1: Aircraft pitch control with adaptive control allocation.

control effectiveness matrix with Λ = diag{λ1, λ2, . . . , λm},and 0 < λi ≤ 1, i = 1, . . . ,m. The ith actuator is fullyfunctional when λi = 1 and has a loss of effectivenessfailure when 0 < λi < 1. For the design in this section, weassume that λi /= 0, that is, no actuator outage occurs. For thereminder of this section we assume that Λ is always positivedefinite.

The control input v(t) = [v1(t), v2(t), . . . , vm(t)]� is givenby

vj(t) = αj(t)v0(t), j = 1, 2, . . . ,m, (3)

where v0(t) is a control signal designed for the group,and αj(t) is the adaptive allocation gain for jth actuator.We assume that a nominal allocation scheme has beenprescribed. This scheme will be enforced under nominaloperating conditions. The nominal allocation gain vector isdenoted as α∗ ∈ Rm. We also define the equivalent controlgain vector as b0 � Bα∗. The vector b0 can be seen as theequivalent control gain matrix for the equivalent controleffector representing the actuator group. We further assumethat the pair (A, b0) is stabilizable.

2.1.2. Reference Model. For the adaptive control, the desiredclosed-loop dynamics is given by

xm(t) = Amxm(t) + Bmr(t), (4)

where Am ∈ Rn×n is a Hurwitz matrix, and Bm ∈ Rn. Thesignal r(t) ∈ R is a bounded piecewise continuous referenceinput, and xm(t) is the desired state. For a given symmetricpositive definite matrix Q ∈ Rn×n, there exists a unique P ∈Rn×n that satisfies

PAm + A�mP = −Q, P = P� > 0. (5)

For the adaptive control design, we need the followingstandard plant model matching condition.

Assumption 1. There exist constant K∗1 ∈ Rn, K∗2 ∈ R suchthat

A + b0K∗�1 = Am, b0K

∗2 = Bm. (6)

2.1.3. Control Objective. The control objective is to designthe virtual control signal v0(t) and adaptive allocation gainsαj , j = 1, . . . ,m, such that all the closed-loop signals arebounded and the system state x(t) tracks the desired statexm(t) asymptotically in the presence of uncertain controleffectiveness Λ.

2.2. Adaptive Control Allocation Design

2.2.1. Nominal Controller. The plant model matching con-dition in Assumption 1 indicates the existence of a nominalcontroller v∗(t) for the system without failures and a nom-inal constant allocation gain vector α∗ such that the closed-loop response is identical to that of the reference model whenthe responses of any unmatched initial conditions vanishexponentially. This signal takes the form

v∗0 (t) = K∗�1 x(t) + K∗2 r(t) � θ∗�ω(t),

v∗(t) = α∗v∗0 (t),(7)

where θ∗ � [K∗�1 ,K∗2 ]� ∈ Rn+1, and ω(t) = [x�(t), r(t)]�.The above state feedback control v∗0 (t) together with aprespecified distribution α∗ ensures the state tracking errore(t) = x(t)− xm(t) approaches zero exponentially.

2.2.2. Adaptive Controller. When a failure occurs, the allo-cation gain will be adaptively adjusted to accommodate forfailure, but α∗ may no longer be effective. For this, we usethe adaptive versions of control signal and allocation gain

v0(t) = K�1 (t)x(t) + K2(t)r(t) = θ�(t)ω(t),

v(t) = α(t)v0(t),(8)

where K1(t) and K2(t) are estimates of K∗1 and K∗2 , andθ(t) = [K�1 (t),K2(t)]� ∈ Rn+1. The updated α(t) =[α1(t), . . . ,αm(t)]� is an estimate of α∗.


2.2.3. Error Dynamics. With the plant and control in (1) and(2), nominal controller in (7), and adaptive controller in (8),the closed-loop dynamics can be expressed as

x(t) = Ax(t) + BΛα(t)v0(t)

= Ax(t) + Bα∗v0(t) + Bα(t)v0(t)

= Ax(t) + b0v0(t) + Bα(t)v0(t)

= Ax(t) + b0v∗0 (t) + b0θ

�(t)ω(t) + Bα(t)v0(t)

= (A + b0K∗�1

)x(t) + b0K

∗2 r(t) + b0θ

�(t)ω(t)

+ Bα(t)v0(t)

= Amx(t) + Bmr(t) + b0θ�(t)ω(t) + Bα(t)v0(t),

(9)

where α(t) � Λα(t)− α∗(t) and θ(t) = θ(t)− θ∗.From the closed-loop dynamics in (9) and reference

model in (4), the error dynamics can be obtained as

e(t) = Ame(t) + b0θ�(t)ω(t) + Bα(t)v0(t). (10)

It can be seen that the error dynamics in (10) is suitable foradaptive law design in that the latter half of its right-hand

side is linear in θ(t) and α(t).

2.2.4. Adaptive Laws. Based on the error dynamics in (10),we can design the adaptive laws for the control parameterθ(t) and allocation gain α(t) as

θ(t) = −Γθω(t)e�(t)Pb0, (11)

α(t) = −ΓαB�Pe(t)v0(t), (12)

where Γθ and Γα are symmetric positive definite matrices andP is determined by (5). From the adaptive laws, we can seethat the controller parameters of the virtual control signalfor the actuator group are updated using the information ofthe equivalent control gain vector b0. The allocation gains areupdated with the B matrix since successful allocation of thevirtual control signal to each actuator requires the knowledgeof each column of B.

The properties of the adaptive control allocation schemecan be summarized in the following theorem whose proof ispresented in the appendix.

Theorem 1. For the system in (1), the adaptive controller andallocation scheme in (8), and the adaptive laws in (11) and(12) guarantee that the all the closed-loop signals are boundedand limt→∞[x(t)−xm(t)] = 0 in the presence of uncertain lossof effectiveness actuator failures in (2).

Remark 2. From the definition of α(t) in (9), we can seethat the adaptation of α(t) is essential for compensating theuncertainties in Λ, while K1(t) and K2(t) can be fixed to theirnominal values K∗1 and K∗2 . In this case, with θ(t) = 0 in(10) and (A.1), the closed-loop stability and asymptotic statetracking results still hold.

2.3. Examples

2.3.1. Linear Plants. Two case studies based on linearizedaircraft models are presented next.

Plant. Consider the linearized lateral dynamic model of alarge transport aircraft flying in a steady wings-level cruisecondition with u = 778 ft/s [15]. The aircraft model is

x = Ax + Bu, (13)

x = [vb, pb, rb,φ,ψ]�, u = [δa, δr]

�. (14)

The state includes the lateral velocity vb (ft/s), roll rate pb(rad/s), yaw rate rb (rad/s) (all in body-axis frame), roll angleφ (rad), and yaw angle ψ (rad). The control inputs are aileronδa and rudder δr deflections (deg). The system matrices Aand B are

A =

⎡

⎢⎢⎢⎢⎢⎣

−0.129 28.328 −774.92 32.145 0−0.012 −1.4419 0.9409 0 00.004 −0.0409 −0.1757 −0.0001 0

0 1 0.0372 0 00 0 1.0007 0 0

⎤

⎥⎥⎥⎥⎥⎦

,

B =

⎡

⎢⎢⎢⎢⎢⎣

0.0542 0.46690.0443 0.02000.0025 −0.0382

0 00 0

⎤

⎥⎥⎥⎥⎥⎦.

(15)

Hereafter we will group the aileron and rudder into a singlegroup. Note that these surfaces have related but differentfunctionalities. If adaptive control signals are designedseparately for the two control inputs with the state feedbackstructure, the total updated parameters would be (5+1)×2 =12. They include 5 state feedback parameters and 1 feed-forward parameter (for tracking the reference input) for eachcontrol surface. The adaptive control allocation scheme, onthe other hand, requires 5 + 1 + 2 = 8 parameters since onlyone adaptive control signal is designed and allocated withtwo updated allocation gains.

Nominal Parameters. The nominal allocation gain vector isselected as α∗ = [8.6103,−1]�, thus the equivalent controlgain vector is b0 = Bα∗ = [0, 0.3614, 0.0594, 0, 0]�. Thenominal allocation gain is chosen so that the first element inb0, that is, the control gain for the lateral acceleration is zero.The purpose of this choice is to attain a coordinated turn.

The nominal controller is designed based on the LQRapproach for (A, b0). The resulting gains are

K∗1 = [−0.8963, 22.1655,−73.6645, 28.8488, 4.0825]�,

K∗2 = 1.(16)

Reference Model. For this example, the reference model ischosen as the closed-loop dynamics with the above LQRcontroller, that is,

Am = A + b0K∗�1 , Bm = b0K

∗2 . (17)


The reference input to the reference model is chosen as r(t) =2.1376 for t ≥ 0, which leads to a desired state trajectory withsteady-state values

xm(∞) = [0, 0, 0, 0, 0.5236]�. (18)

This reference command yields a right turn with a change inthe yaw angle of 30 degrees.

Actuator Failure. We will consider the 80% loss in controleffectiveness:

u1(t) = 0.2v1(t), for t ≥ 10 seconds, (19)

where u1(t) is the output of the aileron actuator and v1(t) isthe designed control input to aileron.

Simulation Results. The following cases will be studied.

(i) Case 1: adaptive allocation of the adaptive controldesigned in Section 2.2 ((11) and (12)).

(ii) Case 2: adaptive allocation (as in (12)) of a nonadap-tive control signal with fixed gains (as in (16)).

Case 1. The time history of the system state under failure andthe desired state is shown in Figure 2. It can be seen that theasymptotic state tracking is achieved after failure.

Figure 3 shows the designed control signal v0(t), allo-cated control signals v(t) and actuator outputs u(t). Thedesigned control signal is allocated by the adaptive allocationgains. The actuator outputs are different from the allocatedcontrol signals due to the failure.

The adaptive parameters are shown in Figure 4. Wecan see that K2(t) also plays an important role in thecompensation of actuator failures. The adaptive allocationgains are shown in Figure 5. Both allocations gains adaptimmediately after the failure occurs and settle to a newcombination of steady-state values that, together with K2(t)and K1(t), guarantee state tracking after failure.

Case 2. The system state and desired state are shown inFigure 6. Despite the uncertain actuator failure, asymptoticstate tracking is achieved. The designed control signal,allocated control signals and the actuator outputs are shownin Figure 7. Similar to the previous case, the allocated controlsignals and the actuator outputs are not identical due to theactuator failure. The allocation gains are shown in Figure 8.The allocation gains are updated autonomously for failurecompensation.

In this simulation case, we have achieved similar resultsto Case 1 with a slightly degraded transient response (seeFigures 2 and 6). A possible explanation for it is that the con-troller parameters are fixed in this case and cannot contributeto the failure compensation and trajectory tracking as theydo in Case 1. However, closed-loop stability and asymptoticstate tracking are achieved even though only the allocationgains are adapted. The successful demonstration of failurecompensation in Case 2 implies that the proposed adaptiveallocation unit can be added to a control loop having any

02

velo

city

(ft

/s)

Late

ral

010

012

rate

(de

g/s)

Yaw

rate

(de

g/s)

Rol

l

05

10

angl

e (d

eg)

Rol

l

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

02040

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

angl

e (d

eg)

Yaw

−2

−10

Figure 2: Time history of plant state (solid) and reference modelstate (dashed) (Case 1).

40

012

V

irtu

al c

ontr

ol

in

put

v 0 (

deg)

05

1015

inpu

ts v

(de

g)

Act

uat

or

AileronRudder

0 5 10 15 20 25 30 35

400 5 10 15 20 25 30 35

400 5 10 15 20 25 30 35

05

1015

Time (s)

Time (s)

Time (s)

outp

uts

u (

deg)

Act

uat

or

−1

−10−5

−5

Figure 3: Time history of control signals and actuator outputs(Case 1).

state feedback controller. The added adaptive allocation tothe nonadaptive controller ensures closed-loop stability andasymptotic state tracking despite uncertain actuator failures.

2.3.2. NASA Generic Transport Model. In this section, weapply the control allocation strategy to the NASA GenericTransport Model (GTM). The NASA GTM is a high-fidelitymodel of the NASA AirSTAR UAV testbed [16, 17]. Thepurpose of this example is to show that this adaptive scheme,with the grouping of actuators having different physical


0 5 10 15 20 25 30 35 40

0204060

Time (s)

0 5 10 15 20 25 30 35 40

Time (s)

0.75

0.8

0.85

K1(t

)K

2(t

)

−20−40

Figure 4: Time history of controller parameters (Case 1).

0 5 10 15 20 25 30 35 40

0

2

4

6

8

10

12

Time (s)

Allo

cati

on g

ain

α(t

)

−2

α1(t): gain for aileron

α2(t): gain for rudder

Figure 5: Time history of allocation gains (Case 1).

02

010

012

05

10

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

02040

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

−2

−10

velo

city

(ft

/s)

Late

ral

rate

(de

g/s)

Yaw

rate

(de

g/s)

Rol

l

angl

e (d

eg)

Rol

lan

gle

(deg

)Ya

w

Figure 6: Time history of plant state (solid) and reference modelstate (dashed) (Case 2).

0123

inpu

t v

(de

g)

Vir

tual

con

trol

01020

inpu

ts v

(de

g)A

ctu

ator

AileronRudder

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0

0

5 10 15 20 25 30 35 40

05

101520

Time (s)

Time (s)

Time (s)

outp

uts

u (

deg)

Act

uat

or

−1

−10

−5

Figure 7: Time history of control signals and actuator outputs(Case 2).

0 5 10 15 20 25 30 35 40

0

2

4

6

8

10

Time (s)

Allo

cati

on g

ain

(t

)

α1(t): gain for aileron

α2(t): gain for rudder

α

−2

Figure 8: Time history of allocation gains (Case 2).

functions, can be applied to the nonlinear plant to achieveclosed-loop stability and asymptotic tracking.

LTI Model. For this simulation study, we trim and linearizethe GTM at a wings-level flight for an aerodynamic speed of92.09 knots. The same states and controls of (14) are used.

Flight Conditions. As in the previous example, the aircraftis commanded to turn right from the initial wings-levelhorizontal flight. The turn starts at 10 seconds and at steadystate the heading angle will increase 60 degrees.

Actuator Failure. We let the left aileron lose 90% of itseffectiveness at 12 seconds. The failure magnitude and itsonset time instant are unknown to the controller. The controlobjective is for the aircraft to achieve an accurate turn in thepresence of the failure.


−101

−100

10

−505

10

02040

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

050

100

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

velo

city

v (

ft/s

)La

tera

lra

te r

(de

g/s)

Yaw

rate

p (

deg/

s)R

oll

angl

e (d

eg)

Rol

lan

gle

(deg

)Ya

w

Figure 9: Time history of lateral states (solid) and reference modelstates (dashed).

100

150

200

Forw

ard

velo

city

u (

ft/s

)

68

10

Ver

tica

lve

loci

ty w

(ft

/s)

0

5

Pit

chra

te q

(de

g/s)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 50

010

Pit

ch a

ngl

e

−5

−10θ(d

eg)

Figure 10: Time history of longitudinal states.

Simulation Results. The simulation results are shown inFigures 9–14. Figure 9 shows the relevant states of the ref-erence model and of the plant. The states track the referencetrajectories asymptotically in spite of the disturbance causedby the failure. The yaw angle is shown to reach 60 degreesaccurately. This accurate turning is also shown in Figure 11by the ground track of the aircraft.

The longitudinal states are shown in Figure 10. Thereare fluctuations in the longitudinal states since they are notcontrolled by aileron and rudder.

0 1 2 3 4 5 6

0

1

2

3

4

5

6

West-east displacement (ft)

Sou

th-n

orth

dis

plac

emen

t (f

t)

×104

×104

−1−1

Figure 11: Ground track of the aircraft.

inpu

t v

(de

g)

0 5 10 15 20 25 30 35 40 45 50

0 5 10 15 20 25 30 35 40 45 500

Vir

tual

con

trol

Time (s)

Time (s)

inpu

ts v

(de

g)A

ctu

ator

AileronRudder

0.5

0

−0.5

−1

−1.5

0.040.02

00.02−−

0.04−0.06−0.08−0.1−0.12

Figure 12: Time history of virtual control signal and actuatorinputs.

Figures 12 and 13 show the time history of the virtualcontrol signal, allocated control signals, and actual aileronand rudder deflections. The discontinuity in Figure 13 at thetop is a consequence of failure.

Figure 14 shows the control allocation gains and con-troller parameters. The parameters adjust autonomouslyafter the failure occurs. Note that the allocation gains start toupdate at the beginning of the turn at t = 10 seconds, beforethe failure occurs. The reason for this phenomenon is thatthe allocation gains are improving the tracking performancebeyond of what the adaptive gains can do alone.

3. Adaptive Control Allocation Design forConstant Failures

3.1. Problem Formulation. When constant failures occur, thecontrol signal can be rewritten as [11]

u(t) = v(t) + σ f (u− v(t)), (20)


defl

ecti

ons

(deg

)

Act

ual

aile

ron

Left aileron

Right aileron

0 5 10 15 20 25 30 35 40 45 50

Time (s)

0 5 10 15 20 25 30 35 40 45 50

Time (s) de

flec

tion

s (d

eg)

Act

ual

ru

dder

Upper rudderLower rudder

1

0.5

0

−0.5

−1

−1.5

0.040.02

0−0.02−0.04−0.06−0.08−0.1

Figure 13: Time history of actuator deflections.

0 10 20 30 40 5012.52

12.54

12.56

12.58

12.6

12.62

12.64

for

aile

ron

Time (s)

Time (s) Time (s)

0 10 20 30 40 500.68

0.7

0.72

0.74

0.76

0.78

0.8

for

rudd

er

0 10 20 30 40 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Ada

ptiv

e co

ntr

olle

r pa

ram

eter

sαα

−0.1

Figure 14: Time history of allocation gains and adaptive parameters.

where u = [u1, . . . ,um]� is the failure vector whose elementsare unknown constants, and σ f represents the failure pattern,which is defined as

σ f = diag{σ1, σ2, . . . , σm}, (21)

with σi = 1 if the ith actuator has failed, that is, ui = ui,and σi = 0 otherwise. The failures are assumed to occurinstantaneously, that is, σi are piecewise constant function oftime. An example of such actuator failures is when a controlsurface (such as the rudder or an aileron) is stuck at someunknown fixed angle at an unknown time instant. This type


of failures could be caused by failed hydraulic systems ormechanical linkages.

The plant dynamics can then be rewritten as

x(t) = Ax(t) + B(I − σ f

)v(t) + Bσ f u. (22)

The constant failure u introduces an uncertain disturbancethat needs to be accommodated for. The control objective forthis adaptive control allocation scheme is to design v(t) toguarantee closed-loop stability and asymptotic state trackingwhen uncertain constant failures occur.

For adaptive control of constant actuator failures, suffi-cient built-in actuation redundancy is required. The redun-dancy condition is described in the following assumption.

Assumption 2. The rank of B matrix satisfies that rank[B] =1, and there is at least one operable actuator in the system.

The rank condition characterizes the redundancy ofactuation which is necessary for a successful constant failurecompensation. This rank condition suggests that the systemremains controllable after a failure, and the effect of theconstant failure can be properly matched and canceled by theallocated control signals through other columns of B. Basedon this condition, there can be up to m− 1 constant actuatorfailures at any given time.

3.1.1. Nominal Controller. Consider the nominal controllerstructure

v∗0 (t) = K∗�1 x(t) + K∗2 r(t) + K∗3 � θ∗�ω(t),

v∗(t) = α∗v∗0 (t),(23)

where θ∗ = [K∗�1 ,K∗2 ,K∗3 ]� and ω = [x�(t), r(t), 1]�.

When there is no failure in the system, K∗1 , K∗2 , and α∗

are chosen as in (6), and K∗3 is set to be zero. In thisway, the controller ensures the match between the referencemodel and the nominal plant without failures. Similar to theprevious section, we also define Bα∗ = b0 which is known inadvance for controller design.

Next we will show that the controller in (23) attainsmodel matching when failures occur. When there are failuresin the system, K∗3 cannot generally be zero, and a new set ofallocation gains, denoted as α∗, may be needed. From (22)and (23) we obtain

x = Ax + B(I − σ f

)α∗(K∗�1 x + K∗2 r + K∗3

)+ Bσ f u. (24)

We assume that at most p actuators can fail with p ≤ m − 1so that there is at least one operable actuator left. Define anindex set for failed actuators as F = {i1, . . . , ip} such thatσk = 1 for any k ∈ F . Then B(I − σ f )α∗ can be expressed as

B(I − σ f

)α∗ =

∑

j /∈F

bjα∗j , (25)

where bj is the jth column of B and α∗j is the jth element ofα∗.

Based on the rank condition in Assumption 2, we knowthat the columns of B are linearly dependent, that is, for any

two columns of B, bi and bj , bi = ci jb j where ci j is a constantscalar. Thus we know that the vector b0 = Bα∗, which is thelinear combination of all columns of B, is also parallel to anycolumn in B. So for each column bk in B, k = 1, 2, . . . ,m, wecan find a scalar ck satisfying bk = ckb0. Therefore (25) canbe expressed as

B(I − σ f

)α∗ =

∑

j /∈F

bjα∗j =

∑

j /∈F

cjα∗j b0. (26)

If α∗j , j /∈ F are chosen such that

∑

j /∈F

cjα∗j = 1, (27)

then we may obtain

B(I − σ f

)α∗ = b0. (28)

One possible choice for α∗j , j /∈ F is

α∗j =1

cj(m− p

) . (29)

Equation (28) indicates that, under constant failures, theplant model condition in (6) can still be satisfied. Note thatfor the system without failures

Bα∗ =m∑

j=1

α∗j b j =m∑

j=1

cjα∗j b0 = b0 (30)

which implies that

m∑

j=1

c jα∗j = 1. (31)

Comparing (27) and (31), we can see that α∗ is generallydifferent from α∗.

For Bσ f u, we can also get

Bσ f u =∑

k∈F

bkuk =∑

k∈F

ckukb0 � d∗b0. (32)

With (28) and (32), (24) can be expressed as

x(t) = Ax(t) + b0K∗�1 x(t) + b0K

∗2 r(t) + b0K

∗3 + d∗b0. (33)

By choosing K∗3 = −d∗, we have b0K∗3 = −Bσ f u, and (33)

can be reduced to

x(t) = Ax(t) + b0K∗�1 x(t) + b0K

∗2 r(t)

= (A + b0K∗�1

)x(t) + b0K

∗2 r(t).

(34)

Therefore when failures are present, a nominal controllercan always be found, with K∗1 and K∗2 specified in (6), α∗

characterized in (27), andK∗3 = −d∗ ensures that the closed-loop system is stable under constant failures, and the stateconverges to the desired state xm in (4) exponentially.


3.2. Adaptive Control Allocation Design

3.2.1. Adaptive Controller. Due to the uncertain nature ofthe failures, the controller parameters must be adjusted.Consider the adaptive control allocation scheme

v0(t) = K�1 (t)x(t) + K2(t)r(t) + K3(t) � θ�

(t)ω(t),v(t) = α(t)v0(t),

(35)

where K1(t), K2(t), and K3(t) are the estimates of K∗1 , K∗2 ,and K∗3 in (23). The signal ω(t) = [x�(t), r(t), 1]�. With theadaptive controller in (35) and the plant dynamics in (22),the closed-loop system is

x(t) = Ax(t) + B(I − σ f

)α(t)

× [K�1 (t)x(t) + K2(t)r(t) + K3(t)]

+ Bσ f u.(36)

3.2.2. Error Dynamics. For this adaptive control schemedesign, we define α(t) � α(t)− α∗, and (36) becomes

x(t) = Ax(t) + B(I − σ f

)α(t)v0(t) + Bσ f u

= Ax(t) + B(I − σ f

)α(t)v0(t) + B

(I − σ f

)α∗v0(t)

+ Bσ f u

= Ax(t) + B(I − σ f

)α(t)v0(t) + B

(I − σ f

)α∗v∗0 (t)

+ B(I − σ f

)α∗v0(t) + Bσ f u,

(37)

where v0(t) = (θ(t)− θ∗)�ω(t) � θ�(t)ω(t).With (28), B(I − σ f )α∗v∗0 (t) in (37) becomes

B(I − σ f

)α∗v∗0 (t) = b0v

∗0 (t) = b0K

∗�1 x(t) + b0K

∗2 r(t)

+ b0K∗3 .

(38)

With (38), (37), and the plant model matching condition in(6), we have

x(t) = Amx(t) + Bmr(t) + B(I − σ f

)α(t)v0(t) + b0θ

�(t)ω(t).

(39)

From the closed-loop dynamics in (39) and the referencemodel in (4), we obtain the error dynamics

e(t) = Ame(t) + B(I − σ f

)α(t)v0(t) + b0θ

�(t)ω(t). (40)

3.2.3. Adaptive Laws. From the error dynamics in (40), thefollowing adaptive laws are derived:

θ(t) = −Γθω(t)e�(t)Pb0, (41)

α j(t) = −γje�(t)Pbjv0(t), j = 1, 2, . . . ,m, (42)

where bj is the jth column of B, γj > 0 and Γθ = Γ�θ > 0 areadaptive gains.

The following theorem summarizes the properties of theadaptive control allocation scheme and the proof is providedin the Appendix:

−100

1020

TAS

(ft/

s)

−2−1

01

AO

A (

deg)

−10−5

05

(deg

/s)

Pit

ch r

ate

0 5 10 15 20 25 30−5

0

5

Time (s)

Time (s)

Time (s)

Time (s)

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

(deg

)P

itch

an

gle

Figure 15: Time history of plant state (solid) and reference modelstate (dashed).

12345

elev

ator

(de

g)D

esig

ned

05

10

eleLOBeleLIB

eleROB

eleRIB

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

05

10

Time (s)

outp

uts

(de

g)

Ele

vato

rin

puts

(de

g)

Ele

vato

r

−5

−5

Figure 16: Time history of control signals and actuator outputs.

Theorem 3. The adaptive control allocation scheme in (35)with the adaptive laws in (41) and (42) applied to the plantin (22) in the presence of constant failures guarantees that allclosed-loop signals are bounded and limt→∞(x(t)−xm(t)) = 0.

3.3. Example

3.3.1. Linear Plant. Consider the longitudinal LTI model ofthe NASA GTM given by

x = Ax + Bv,

x = [VT ,αa, q, θ]�,

v = [δelob, δelib, δerob, δerib]�,

(43)

where the state includes the true airspeed VT (ft/s), angle ofattack αa (rad), pitch rate q (rad/s), and pitch angle θ (rad).The control inputs are the deflections of the four elevator


150155160165170175

Forw

ard

0102030

Pit

ch r

ate

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

010

Pit

ch a

ngl

eve

loci

ty (

ft/s

)(d

eg/s

)(d

eg)

Time (s)

−10−20

−10−20−30

Figure 17: Time history of longitudinal states (solid) and reference model states (dashed).

05

Rol

l rat

e

012

Yaw

rat

e

05

10

Rol

l an

gle

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

024

Yaw

an

gle

(deg

/s)

(deg

/s)

(deg

)(d

eg)

−5−10

−5−10

−2

−2−1

−4

Figure 18: Time history of lateral states (solid) and reference model states (dashed).

segments: left outboard elevator δelob, left inboard elevatorδelib, right outboard elevator δerob, and right inboard elevatorδerib (deg). The system matrices A and B are

A =

⎡

⎢⎢⎢⎣

−0.0450 −8.9632 0.0349 −32.1740−0.0035 −2.7429 0.9514 0−0.0056 −42.6233 −3.5616 0

0 0 1 0

⎤

⎥⎥⎥⎦

,

B =

⎡

⎢⎢⎢⎣

−0.0110 −0.0110 −0.0110 −0.0110−0.0012 −0.0012 −0.0012 −0.0012−0.1962 −0.1962 −0.1962 −0.1962

0 0 0 0

⎤

⎥⎥⎥⎦.

(44)

Obviously, rank[B] = 1. For the simulation study, we willinclude the four elevator surfaces into one group, for whichan elevator control signal will be designed and allocated.

Nominal Parameters. The nominal allocation gain α∗ ischosen as α∗ = [0.25, 0.25, 0.25, 0.25]� and b0 = Bα∗ =[−0.011,−0.0012,−0.1962, 0]� so the deflection of the four

segments will be the same if no failure occurs. The nominalcontroller is designed using the LQR approach for (A, b0).The resulting gains are

K∗1 = [0.1494, 25.0761,−3.2841,−27.1293]�, K∗2 = 1.(45)

Reference Model. Similar to the simulation study inSection 2, the reference model is chosen as the closed-loopdynamics of the LQR controller, that is,

Am = A + b0K∗�1 , Bm = b0K

∗2 . (46)

The reference input to the reference model is chosen as r(t) =4.1841 for t ≥ 0, which leads to a reference trajectory withsteady-state values

xm(∞) = [10,−0.0139, 0,−0.0110]�, (47)

whose physical meaning is that the aircraft speed is increasedby 10 ft/s; its angle of attack is reduced by 0.0139 rad; and itspitch angle is reduced by 0.0110 rad.


00.5

1

01

23

500

1000

1500

2000

Alt

itu

de (

ft)

1 0 0.5 1

0

0.5

1

1.5

2

2.5

3

0 50 100 150400

600

800

1000

1200

1400

1600

1800

2000

Time (s)

Alt

itu

de (

ft)

West-east displacement (ft)

West–east displacement (ft)×104×104

×105

×105 −−1− 1−

0.5−0.5

−0.5

Sou

th-n

orth

dis

plac

emen

t (f

t)

South–north displacement (ft)

Figure 19: Flight trajectory, ground track, and altitude of the aircraft.

−1012

elev

ator

(de

g)D

esig

ned

−5

−5

05

10

inpu

ts (

deg)

Ele

vato

rs

LOBLIB

ROBRIB

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

05

1015

defl

ecti

ons

(deg

)E

leva

tors

Figure 20: Time history of lumped and allocated elevators andelevator deflections.

Actuator Failure. The left outboard elevator is stuck at −5degrees after 1 second, that is,

u1(t) = −5 deg, for t ≥ t f second, (48)

where t f = 1 second. The signal u1(t) is the output of the leftoutboard elevator. For t > t f the elevator does not respondto the elevator input v1(t).

Simulation Results. The results are shown in Figures 15 and16. From Figure 15, we can see that the plant state tracksthe reference state after failure. Figure 16 shows the designedelevator signal, elevator inputs, and elevator outputs. Thefailure occurs at t f = 1 second and the other elevators canbe seen to accommodate the failure with the adaptation.

3.3.2. NASA Generic Transport Model. Here we apply theadaptive control scheme to the nonlinear NASA GTM.

LTI Model for Controller Design. For adaptive control design,we consider a LTI model for a wings levelled flight having a3 deg angle of attack at trim. Similar to the linear simulationstudy, we study the compensation for constant elevatorfailures and will consider multiple elevator failures in whichboth the lateral and longitudinal are active. The state andcontrol vectors are given by

x = [u, q, θ, p, r,φ,ψ],

u = [δelob, δelib, δerob, δerib, δtl, δtr, δal, δar, δru, δrl],(49)

where δelob, δelib, δerob, and δerib are the four elevatorsegments, δtl and δtr are the two engine throttles, δal andδar are left and right aileron segments, and δru and δrl areupper and lower rudder segments. The actuators will begrouped into elevator, engine, aileron, and rudder groups. Avirtual control signal is designed for each group and allocatedto its members adaptively. The design is based on that inSection 3.2 and is similar to that in the linear simulationstudy. We include lateral states and lateral actuators to


1

0.5

0

0.5

Des

ign

ed a

lero

n (

deg)

0 50 100 150

Time (s)

1.5−−

−

(a)

2

0

1

2

Aile

ron

defl

ecti

ons

(deg

)

Left aileronRight aileron

0 50 100 150

Time (s)

1−−

(b)

0

0.2

Des

ign

ed r

udd

er (

deg)

0 50 100 150

Time (s)

0.6

0.4

0.2

−

−−−

0.8

(c)

0.6

0.4

0.2

0

0.2

Ru

dder

def

ecti

ons

(deg

)Upper rudder

Lower rudder

0 50 100 150

Time (s)

−

−−−

0.8

(d)

0 50 100 150

0

5

10

Des

ign

ed t

hro

ttle

(%

)

Time (s)

5−

(e)

0 50 100 150

Time (s)

0

5

10

15

thro

ttle

s (%

)

Allo

cate

d en

gin

e

Left engine

Right engine

5−

(f)

Figure 21: Time history of the inputs and outputs of ailerons, rudders, and engines.

regulate the disturbance of the elevator failures propagatedto the lateral dynamics.

Flight Conditions. A set of commands that aim to make theaircraft climb at 4-degree pitch angle is applied at 5 seconds.The nominal parameters and reference inputs are obtainedas in the linear simulation example.

Actuator Failures. The right outboard elevator is locked at10 degrees at 10 seconds and the right inboard elevator at 5degrees at 20 seconds. The control objective is to maintainthe climbing flight in the presence of these failures.

The simulation results are shown in Figures 17–23.Figure 17 shows the time history of the longitudinal statesand reference model states. It can be seen that the tracking ofthe desired longitudinal attitude is accurately achieved.

The lateral states are shown in Figure 18. Some distur-bances can be observed after the occurrence of asymmetric

elevator failures. These disturbances are regulated by theailerons and asymptotic tracking of the lateral states can bealso achieved.

Figure 19 shows the flight trajectory, ground track, andaltitude of the aircraft. From the ground track plot, we cansee the effect of asymmetric failure on the lateral dynamics.The deviation is corrected by the lateral actuators. Figure 20shows the lumped elevator signal, allocated elevator signals,and the actual elevator deflections. The elevator failuresappear as constant (step signals) on the lower plot, andthe designed and allocated elevator signals are shown toaccommodate for the failures after their occurrence. Theinput signals and outputs of other actuators are shown inFigure 21. Note the actuation of the aileron and rudderrequired to compensate for the asymmetric failure. The timehistory of the control allocation gains and some selectedadaptive parameters are shown in Figures 22 and 23. It canbe seen that the allocation gains and controller parametersare updated autonomously to ensure the desired flight.


0 50 100 1507

7.5

8

8.5

for

elev

ator

s

Time (s)

LOBLIB

ROBRIB

α

(a)

0 50 100 1500.99

1

1.01

1.02

1.03

1.04

for

aile

ron

s

Time (s)

Left aileronRight aileron

α

(b)

0 50 100 1500.995

1

1.005

1.01

1.015

1.02

1.025

for

rudd

ers

Time (s)

Upper rudderLower rudder

α

(c)

0 50 100 1500.9

1

1.1

1.2

1.3

1.4

for

engi

nes

Time (s)

Left engineRight engine

α

(d)

Figure 22: Time history of control allocation gains.

para

met

ers

of K

Sele

cted

para

met

ers

of K

Sele

cted

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

para

met

ers

of K

Sele

cted

Time (s)

Time (s)

Time (s)

0.10

12

3

−0.1−0.2−0.3−0.4

0.10.080.060.040.02

0.030.020.01

0−0.01

Figure 23: Time history of selected adaptive controller parameters.

4. Conclusions

A novel adaptive control allocation framework is proposedherein. The adaptive allocation scheme includes an adaptivecontrol signal and a control allocation unit with adaptivelyupdated allocation gains. Two adaptive control allocation

algorithms have been proposed for the compensation ofuncertain failures. The proposed algorithms have beenshown to guarantee closed-loop stability and asymptoticstate tracking. It has also been shown that the proposedadaptive control allocation framework reduces the controllercomplexity with proper grouping of the actuators. In this


framework, the control signal to be adaptively allocatedcan be actuated by nonadaptive controllers. The simulationresults demonstrate the performance of the proposed algo-rithms and their applicability to aircraft flight control. Somefuture research topics in this direction include the extensionof the adaptive control allocation framework to systems withmultiple groups of actuators and the strict enforcement ofa pure allocation, that is, exactly enforcing the designedcontrol signal with

∑αi = 1.

Appendix

Proof of Theorem 1. When a failure occurs, the control effec-tiveness matrix Λ has a discontinuity. It would lead to a finitejump in α(t) as well, which could result in a a finite jumpin the error dynamics. Let us assume that there are l failuresoccurring in the system, and the actuator failures occur attime instants tk, with tk < tk+1, k = 1, 2, . . . ,N . For theclosed-loop stability and state tracking analysis, we choosethe following Lyapunov-like function:

V(t) = 12e�(t)Pe(t) +

12α�(t)Γ−1

α Λ−1α(t) +12θ�(t)Γ−1

θ θ(t),

(A.1)

for each time interval (tk, tk+1), k = 0, 1, . . . ,N with t0 = 0and tN+1 = ∞. V(t) is thus discontinuous with finite jumpsat tk, k = 1, . . . ,N . Taking the time derivative of V(t) andsubstituting the adaptive laws in (11) and (12) into the resultfor each (tk, tk+1), we obtain

V(t) = e�(t)Pe(t) + α�(t)Γ−1α α(t) + θ�(t)Γ−1

θ θ(t)

= e�(t)P[Ame(t) + b0θ

�(t)ω(t) + Bα(t)v0(t)]

− α�(t)B�Pe(t)v0(t)− θ�(t)ω(t)e�(t)Pb0

= −12e�Qe ≤ 0.

(A.2)

Thus we can conclude that for each (tk, tk+1), k = 0, 1, . . . ,N ,V(t) is bounded. Since V(t) only has finite jumps at tk,k = 1, . . . ,N , we can conclude that V(t) is bounded fort ∈ [0,∞), and e(t) ∈ L∞, α(t) ∈ L∞, θ(t) ∈ L∞, α(t) ∈ L∞,θ(t) ∈ L∞, x(t) ∈ L∞, and ω(t) ∈ L∞. Integrating both sidesof (A.2), we can obtain

V(t+k)−V

(t−k+1

)= 1

2

∫ tk+1

tk e�(τ)Q(τ)e(τ)dτ. (A.3)

For N + 1 intervals: [0, t1), (t1, t2), . . ., (tN−1, tN ), and (tN ,∞),(A.3) holds. Summing both sides of (A.3) for k = 0, 1, . . . ,N ,

we obtain

12

∫∞

0e�(τ)Q(τ)e(τ)dτ = V(0)−V(t−1

)+V

(t+1)

−V(t−2)

+V(t+2)−· · ·−V

(t−k)

+V(t+k)−· · ·−V(t−N

)+V

(t+N)

−V(∞)

= V(0) +N∑

i=1

[V(t+i)−V(t−i

)]

−V(∞) <∞,(A.4)

because the jumps V(t+i ) − V(t−i ) are finite and the numberN of jumps is also finite. Thus we have e(t) ∈ L2. We can alsoconclude from (10) that e(t) ∈ L∞. So from e(t) ∈ L2 ∩ L∞,and e(t) ∈ L∞, we have lim0 → ∞e(t) = 0.

Proof of Theorem 3. In Section 3.1, we have assumed thatthere are at most p ≤ m − 1 constant actuator failures anddefined an index set for failed actuators as F = {i1, . . . , ip}such that σk = 1 for all k ∈ F . Here we further assume thatthe failures occur at instants tk, with tk < tk+1, k = 1, 2, . . . ,Nwith 1 ≤ N ≤ p. The number of failure instants maybe smaller than the total number of failures since multiplefailures may happen at the same time. For the stability proof,we choose the following Lyapunov-like function

V(t) = 12e�(t)Pe(t) +

12

∑

i /∈F

γ−1i α2

i (t) +12θ�(t)Γ−1

θ θ(t),

(A.5)

for each time interval (tk, tk+1), k = 0, 1, . . . ,N , with t0 =0 and tN+1 = ∞. The time derivative in each time interval(tk, tk+1) is

V(t) = e�(t)Pe(t) +∑

i /∈F

γ−1i αi(t)αi(t) + θ�(t)Γ−1

θ θ(t)

= e(t)�PAme(t) + e�(t)PB(I − σ f

)α(t)v0(t)

+ e�(t)Pb0θ�(t)ω(t)

− e�(t)Pv0(t)∑

i /∈F

αi(t)bi − θ�(t)ω(t)e�(t)Pb0

= e�(t)PAme(t) = −12e�(t)Qe(t) ≤ 0

(A.6)

with the fact that B(I − σ f )α(t) = ∑i /∈F αi(t)bi. Following

a similar approach to the stability analysis in Section 2, wecan conclude that for any t ∈ [0,∞), V(t) ∈ L∞, e(t) ∈L∞, x(t) ∈ L∞, ω(t) ∈ L∞, θ(t) ∈ L∞, v0(t) ∈ L∞, e(t) ∈L2. Since V(t) only includes αi(t) with i /∈ F , we can onlyconclude the boundedness of αi(t) and αi(t) for i /∈ F . Toshow the boundedness of αj(t), j ∈ F , we note that for anyj ∈ F

αj(t) = αj(0)−∫ t

0γje

�(τ)Pv0(τ)bj dτ, (A.7)


based on the adaptive law in (42). Note that all the columnsof B are linearly dependent based on the rank condition inAssumption 2. So we can have

bj = c∗j bk, ∀ j ∈ F , (A.8)

where bk is the column of B that corresponds to an arbitraryhealthy actuator, that is, k /∈ F , and c∗j is a nonzero constant.Equation (A.7) can thus be expressed as

αj(t)= αj(0)−∫ t

0γje

�(t)Pv0(t)c∗j bk dt

= αj(0)− γjγkc∗j

∫ t

0γke

�(t)Pv0(t)bk dt

= αj(0) +γjγkc∗j

∫ t

0

(−γke�(t)Pv0(t)bk)dt

= αj(0) +γjγkc∗j

∫ t

0αk(t)dt

= αj(0) +γjγkc∗j [αk(t)− αk(0)], ∀ j ∈ F .

(A.9)

Since αk(t), k /∈ F has been proved to be bounded, we haveαj(t) ∈ L∞, for j ∈ F .

We can further obtain that e(t) ∈ L∞ from (40). Withe(t) ∈ L∞ ∩ L2 and e(t) ∈ L∞, we can have limt→∞e(t) =0.

Acknowledgments

This work was supported by the NRA NNX08AC62A of theIRAC project of NASA. The authors would like to thank Drs.Suresh M. Joshi and Sean P. Kenny at the NASA LangleyResearch Center for their valuable comments.

References

[1] M. W. Oppenheimer, D. B. Doman, and M. A. Bolender, “Con-trol allocation for over-actuated systems,” in Proceedings ofthe 14th Mediterranean Conference on Control and Automation(MED ’06), pp. 1–6, June 2006.

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Research Article

Pilot-Induced Oscillation Suppression byUsing L1 Adaptive Control

Chuan Wang, Michael Santone, and Chengyu Cao

Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-3139, USA

Correspondence should be addressed to Chuan Wang, [email protected]

Received 12 December 2011; Accepted 24 February 2012


Copyright © 2012 Chuan Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Despite significant technical advances, pilot-induced oscillation (PIO) continues to occur in both flight tests and operationalaircrafts. Such a phenomenon has led to significant research activities that aim to alleviate this problem. In this paper, the L1

adaptive controller has been introduced to suppress the PIO, which is caused by rate limiting and pure time delay. Due to itsarchitecture, the L1 adaptive controller will achieve a desired response with fast adaptation. The analysis of PIO and its suppressionby L1 adaptive controller are presented in detail in the paper. The simulation results indicate that the L1 adaptive control is efficientin solving this kind of problem.

1. Introduction

The high performance demands of modern aircrafts, espe-cially highly maneuverable military jets, require the imple-mentation of advanced control systems. The use of a modernelectronic flight control system could provide great potentialfor improvement in the aircraft’s performance. In spite ofall the improvements, a significant handling quality problemarose with the introduction of electronic flight controlsystem: pilot-induced oscillation, also known as pilot in-the-loop oscillation.

Pilot-induced oscillations are described as pilot-aircraftdynamic couplings, which could lead to instability in the sys-tems [1]. Both previous and current research has attemptedto explain, predict, and avoid these oscillations. Almost everymodern aircraft has experienced PIO, which is well knownby the public for the catastrophic event it caused, such as theYF-22 [2] and Olympic Airways Falcon 900. The occurrenceof such events has led to significant research activities thatare intended to alleviate the negative effects due to PIO.Despite the research efforts made, PIOs continue to occur,and reports of PIOs on operational aircrafts are increasing.

The focus of this paper is to suppress the pilot-inducedoscillation caused by both rate limiting and pure time delay

by using the L1 adaptive controller. The effects of rate limit-ing and other system nonlinearities are considered to be themain factors that result in the occurrence of PIOs [3–5].Some of the existing methods, to some extent, could handlethe PIO caused by rate limiting, but not necessarily workwith the presence of pure time delay in the actuator model.The L1 adaptive control is known for its fast adaptation andsmooth control implementation due to its powerful controlarchitecture [6, 7]. With proper design, the L1 adaptivecontroller will make the system respond in the desiredmanner.

The objective of this paper is to design the L1 adaptivecontroller to make the inner loop of the system respondaccording to a given first-order system, while suppressingthe PIO phenomenon caused by both rate limiting and puretime delay in the pilot dynamic model. Section 2 gives a briefintroduction of pilot-induced oscillation and pilot model. InSection 3, the design of an inner loop adaptive controller tosuppress the PIO and to track desired response is given. Theinner loop adaptive controller cannot handle the adverselyhigh pilot command inputs because they are the commandsto the inner closed loop system. The inner loop controlledby L1 adaptive controller solely tracks the command inputsfrom the pilots.


2. Pilot-Induced Oscillation

Pilot-induced oscillations can be regarded as a closed-loopinstability of the pilot-aircraft loop, and it often happenswhen the pilot proves to be unable to adapt himself to a sud-den change of the vehicle dynamics during a high demandingflight task. PIOs are complex interactions between the pilotand the aircraft dynamics.

2.1. Types of PIOs. According to the report by NationalResearch Council (NRC) Committee in 1997 [8], PIOs canbe separated into three categories.

(i) Category I PIO: Linear pilot-vehicle system oscil-lation. These PIOs result from linear phenomena such asexcessive time delay and excessive phase loss between thepilot’s control input and the aircraft response. They arethe simplest to model, understand, and prevent. Beforethe introduction of fly-by-wire technology, almost all PIOswere this type, and a large amount of research has beenconducted on them. But they are currently the least commonin operational flight. The main causes of Category I PIOare excessive lags due to time delays and various digitalfilter dynamics in the flight control system. These effectslead to a high frequency phase roll-off in the frequencyresponse. There are roughly two groups of design criteriafor preventing Category I PIO. One is according to theflight control stability, and the other is the handling qualityrequirements. In the first group of design criteria, the openloop aircraft frequency response needs to be checked as wellas the phase and magnitude margin. If the phase rate of thesystem is too high, a small increase in frequency will result ina strong additional phase delay. This is usually related withtime delay in the whole system.

(ii) Category II PIO: Quasi-linear events with somenonlinear contribution, such as rate or position limiting.For the most part, these PIOs can be modeled as linearevents, with an identifiable nonlinear contribution that maybe treated separately. Figure 1 shows two typical positionswhere rate-limiting blocks are installed in the flight controlsystem. The block after the pilot model is installed to preventthe system from receiving a high input rate by the pilot. Theother block after the controller is used to protect the actuatoragainst overload. In the case of rate limiting, nonlinearsystem theory should be utilized for the analysis of the flightsystem, such as the Lyapunov theory and the phase planemethod.

The majority of aircraft crashes due to PIOs are causedby Category II PIO through activating actuator rate limits.When the rate limiter is saturated, phase lag occurs and theaircraft dynamics change suddenly, which cause PIO andthreaten the flight safety. Rate limiting adds additional phaselag, increasing the delay between the pilot input and theaircraft response. This tends to make the pilot compensatewith faster responses, often worsening the situation. Ratelimiting also reduces the gain, which the pilot interpretsas a lack of control response and therefore makes largercommand inputs, again making the situation worse.

(iii) Category III PIO: Nonlinear PIOs with transients.Such events are difficult to recognize and rarely occur but are

always severe. Mode switching or rapid changes in effectivevehicle characteristics could be reasons for this type ofPIO. Fortunately, these events that result in nonoscillatorydivergence and loss of control of the aircraft are rare.

2.2. Rate Limiting. Rate limiting of the actuator is one ofmain factors that lead to pilot-induced oscillation. Actuatorrate limiting occurs when pilot input command errorrequires a higher rate than the actuator can actually provide.A simplified model of a rate-limited actuator is shown inFigure 2.

Rate limiting adds additional phase lag, increasing thedelay between the pilot input and aircraft response. Thistends to make the pilot compensate with faster responses,which will often worsen the situation.

2.3. Pilot Modeling. With rate limiting considered, anotherimportant factor that causes the PIO phenomenon is thepilot model, which is also regarded as the weakest pointin the analysis due to its high nonlinearity and complexity.However, there are several specific pilot behavioral patterns[9] that could be used to analyze the cause of these PIOs.The pilot model is the source factor that distinguishes severePIO problems from most aircraft feedback control designproblems. The difference resides in unique human propertiesrelated to the adaptive characteristics of the human pilot.

Pilots exhibit peculiar transitions in the organizationalstructure of the pilot vehicle system. These transitionscan involve both the pilot’s compensation and effectivearchitecture of the pilot’s control strategy. The human pilotdynamics can be roughly separated into the following types.

(i) Compensatory behavior: essentially, the pilot hasgenerated a lead to cancel out the lag in the aircraft model.However, the higher frequency lags of the pilot model can beapproximated at the lower frequencies by the pure time delay.

(ii) Pursuit behavior: the introduction of the pursuitbehavior permits an open-loop control in conjunction withthe compensatory closed-loop error correcting action. Thepilot model of both compensatory behavior and pursuitbehavior will be superior to that where only compensatoryoperations are possible.

(iii) Precognitive behavior: this kind of pilot modelgives a higher level of control performance. Based on theknowledge of the system dynamics, the pilot model willgenerate proper control signals at the right time so as to resultin machine outputs that are almost as desired. This operationalso appears in company with compensatory behavior aswell as pursuit behavior. Most highly skilled movements willautomatically fall into this category.

3. PIO Suppression by L1 Adaptive Control

3.1. System Modeling. In most PIO cases that have happenedin recent years, rate- and position-limiting phenomenaappear. Hence, numerous research works have been con-ducted to analyze rate and position limiting [10–12]. In thissituation, PIOs can be explained by limit cycles occurring in anonlinear system where the nonlinearities cause a sustained,


Set pointPilot

model Controller Actuator Aircraft

Sensor

Figure 1: Rate limiting in flight control systems.

In 1Gain 1

1

Saturation IntegratorOut 1

11/τ+− 1/s

Figure 2: Rate limit actuator dynamic model.

constant amplitude oscillation. In those studies, rate limitingof the actuator is considered as one possible cause for PIOs.There are some other factors, such as relatively high pilotcommand gain and lag in the pilot model, contributing tothe onset of PIOs. Rate limiting is commonly met in aviationpractice. Hence, if an adaptive inner loop can compensatefor the nonlinearities caused by rate limiting and eliminatethe limit-cycle oscillations, it can be considered that the PIOis suppressed for such commonly seen situations. We applythe L1 adaptive controller on the F-14 model taken from [10]in order to test this idea. A model of the Grumman AircraftCompany (GAC) F-14 was developed for this investigation.The approximated model in [10] is compared with data ofa real F-14 that has severe PIOs during flight tests. Therate-limiting phenomenon we considered is due to suddenoff-nominal conditions such as hydraulic pressure failure asin the F-14 example. Therefore, it is hard to estimate thenonlinearity caused by rate limiting in advance, like mostother research work has done. Adaptive control is needed forsuch situations.

The control structure is shown in Figure 3. It is alongitudinal dynamic model. The system states are [αθ q],and control input is the elevator. The L1 adaptive innerloop controller is a pitch-rate augmentation system, whichprovides the desired dynamics for the [α q] subsystem. Theouter loop takes the θ feedback signal and injects it into thepilot model. The pilot model outputs a pitch rate commandfor the inner loop to achieve the pitch attitude control (the θangle). The pilot model is given by

p(s) = K(s + β

)e−τs, (1)

which describes the compensatory behavior of the pilot.

3.2. Delay Margin Detection. Consider the Pilot modeldescribed by (1); we assume the desired response of the innerloop is given by

gc(s) = 1Tps + 1

. (2)

With the Pilot model, we could derive the transfer functionof the closed-loop as

g(s) = p(s)gc(s)p(s)gc(s) + 1

= K(s + β

)e−τs

Tps + 1 + K(s + β

)e−τs

. (3)

In the PIO problem concerned, the pure time delaydeviates between 0.2 and 0.3 second. The time delay term canbe approximated by e−τs = (1−(τs/2))/(1+(τs/2)) accordingto Pade approximation. Equation (3) can be further writtenas

g(s)

= K(s + β

)1 + τs/2

(Tps + 1

)(1 + τs/2) + K

(s + β

)(1− τs/2)

e−τs

= (Kτ/2)s2 +(1+τβK/2

)s+βK

((Tp−K

)τ/2

)s2 +

(Tp+K−τ/2−τβK/2

)s+1 +Kβ

e−τs,

(4)

according to Routh stability criterion, we can get thefollowing inequalities:

Tp > K ,

1 + Kβ > 0,

Tp > Kτβ/2− τ/2− K.(5)


Pilot Actuator

Sensor

Aircraftlongitudinal

dynamics1/s− −

−

−

kα

α

e kIq

kq

θ

αF

θc δeu1 u

Figure 3: Control structure.

Control law

Adaptive law State predictor

Plant

Filter

r(t)

x(t) u(t) x(t)

x(t)

σ(t)x(t)

Figure 4: L1 adaptive controller architecture.

Set point Pilot model Desired systemtransfer function

Figure 5: PIO delay margin detection architecture.

If the parameter Tp chosen is satisfied with unequalconditions in (5), the system will achieve the desired responsewith stability condition. From many PIO studies, the onset ofPIO often was accompanied by high pilot command inputs.If some off-nominal conditions happen, such as hydraulicfailure of the actuators in F-14 PIO accidents, the ratelimiting comes into play. When the L1 inner loop adaptivecontroller is applied to this model, it can compensate for theunknown effects caused by rate limiting and time delay, nomatter whether they are linear or nonlinear. The design ofthe L1 adaptive controller does not need a priori informationfor the rate limits of the actuator.

3.3. L1 Adaptive Control Design to Achieve Desired Response.Consider the L1 architecture given in Figure 4, the L1

adaptive control [13] could be divided into three parts: theadaptive law, the state predictor, and the control law.

State Predictor Design. The state predictor of the aircraft lon-gitudinal dynamic system in Figure 3 is designed as follows:

˙x(t) = Amx(t) + Bu(t) + σx(t), x(0) = x0, (6)

where Am ∈ R3×3, B ∈ R3×1,

˙x(t) =⎡

⎢⎣θ(t)q(t)α(t)

⎤

⎥⎦, σx(t) =

⎡

⎢⎣σ1(t)σ2(t)σ3(t)

⎤

⎥⎦. (7)

Matrix A should be Hurwitz to make sure of the stability ofthe model. σ can be divided into two parts, the modelmatched part and the model unmatched part

σ = Bσm + Bσum, (8)

where B is the null space of B. σm is the matched part, and σumindicates the unmatched part. σm and σum can be calculatedas follows:

[σmσum

]

=[B B

]−1σ . (9)

Adaptive Law Design. Assume the system dynamics is as fol-lows:

x(t) = Ax(t) + Bu + σ(t). (10)

Given any T > 0, we have

Φx(T) =∫ T

0eAm(T−τ)dτ. (11)

Letting x(t) = x(t) − x(t), the adaptive law for σx(t) is givenby

σx(t) = σx(iT), t ∈ [iT , (i + 1)T],

σx(iT) = −Φ−1x (T)μx(iT), i = 0, 1, 2, . . . ,

(12)


0 10 20 30

−0.5

0

0.5

0 10 20 300

0.05

0.1

0.15

0 10 20 300

0.05

0.1

0.15

0 10 20 300

0.05

0.1

0 10 20 300

0.05

0.1

0 10 20 300

0.05

0.1

Figure 6: system responses with increasing Tp.

where Φ(T) is defined above, and

μx(iT) = eAmT x(iT), i = 1, 2, . . .

σx(t) = −Φ−1x (T)eAmT x(t).

(13)

Control Law Design. The output estimation of the system isgiven by

y = Cx. (14)

The control signal can be calculated by

u = ur + um + uum,

ur = − r

CA−1m B

,

um = −σm,

uum =[C(SI − Am)−1B

]−1C(SI − Am)−1Bσum.

(15)

ur is the control signal used to track the reference signal withdesired response designed aforementioned. um and uum arecontrol signals used to cancel the uncertainty part of thesystem, where um is the matched control signal and uum isthe unmatched one.

4. Simulation Study

In general, rate limiting results in an amplitude reductionand a significant added phase lag. With the introduction of

pure time delay into the pilot model, the closed-loop systemcould even become unstable.

Based on the delay margin detection architecture shownin Figure 5, the time variable in the desired system responsemodel can be obtained from the result derived in theprevious section. There is another alternative method thatcould be used for the detection of the delay margin bysearching for the optimal value of Tp in (2). With anincrease in Tp, the system response will change graduallyin accordance. At some critical point, the oscillation willdisappear and the system response will converge to the set-point value instead. If we further increase the time variable,the damping ratio of the closed-loop system will increaseas well, which will affect the system’s transient responseincluding overshoot, rise time and so forth (Figure 6).

In Figure 7, the F-14 system responses of different timeconstants Tp are given. If the time variable Tp is small asshown in Figure 7(a), the system responds rapidly, but theovershot is also very large. With increasing Tp, the overshootdecreases, but transient performance reduces as well. Thus,we need to build a balance between the overshoot and therise time, which can be simply implemented by introducingan objective function as

min J = ω1δ + ω2Tr , (16)

where ω1,ω2 are weighting coefficients and δ,Tr representthe overshoot and rise time, respectively.


0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16θ

(a) PIO elimination (Tp = 10)

0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

θ

(b) PIO elimination (Tp = 50)

0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

θ

(c) PIO elimination (Tp = 100)

0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

θ

(d) PIO elimination (Tp = 200)

Figure 7: PIO elimination with different desired responses.

5. Conclusion and Future Research

This paper proposes an alternative method for suppressingthe pilot-induced oscillation phenomenon caused by ratelimiting and time delay in the pilot dynamic model. Thedelay margin is detected at the beginning, and the criticalpart of the method is design the L1 adaptive controller toachieve the desired response of the system.

To further improve the performance of this method, weneed to consider the following issues. First, certain criterionneed to be built to determine the choice of Tp. The systemresponse can be evaluated by stability, overshoot, and risetime, and so forth. For the robustness of this method, a time-varying pilot model should be considered.

Acknowledgment

The authors would like to thank Dr. Jiang Wang from ZonaInc. for his useful suggestions and help.

References

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[3] D. G. Mitchell, B. A. Kish, and J. S. Seo, “A flight investigationof pilot-induced oscillation due to rate limiting,” in Proceed-ings of the IEEE Aerospace Conference, vol. 3, pp. 59–74, March1998.

[4] S. L. Gatley, M. C. Turner, I. Postlethwaite, and A. Kumar,“A comparison of rate-limit compensation schemes for pilot-inducedoscillation avoidance,” Aerospace Science and Tech-nology, vol. 10, no. 1, pp. 37–47, 2006.

[5] Y. Yildiz and I. V. Kolmanovsky, “A control allocation tech-nique to recover from pilot-induced oscillations (capio) due toactuator rate limiting,” in Proceedinjgs of the American ControlConference (ACC ’10), pp. 516–523, July 2010.

[6] C. Cao and N. Hovakimyan, “L1 adaptive controller for a classof systems with unknown nonlinearities: part I,” in Proceedings


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[8] National Research Council (U.S.). Committee on the Effectsof Aircraft-Pilot Coupling on Flight Safety, Aviation Safety andPilot Control: Understanding and Preventing Unfavorable Pilot-Vehicle Interactions, National Academies Press, Washigton,DC, USA, 1997.

[9] D. McRuer, U. S. N. Aeronautics, S. A. Scientific, and T.I. Program, Pilot-Induced Oscillations and Human DynamicBehavior, vol. 4683, NASA, 1995.

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[12] D. H. Klyde and D. G. Mitchell, “A pio case study—lessons learned through analysis,” in Proceedings of the AIAAAtmospheric Flight Mechanics Conference and Exhibit, pp. 1–17, 2005.

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