Adaptive model following control ofelectrohydraulic velocity control systems subjectedto unknown disturbances

J.S. Yun, PhDH.S. Cho, PhD

Indexing term: Servomotors, Adaptive control, Control systems

Abstract: This paper considers the velocitycontrol problem of nonlinear hydraulic servosystems subjected to unknown and time-varyingexternal load disturbances. Hydraulically operatedprocesses are usually represented by a hydraulicactuator-load system whose dynamic character-istics are complex and highly nonlinear, owingeither to the flow-pressure relationships of thehydraulic system or to a load system motion itself.Furthermore, these characteristics are sometimesunknown owing to the uncertainty in externalload disturbances to the processes. Therefore, theconventional approach to the controller design ofthese systems may not assure satisfactory controlperformance. To obtain better performance anadaptive model following control scheme wasderived based upon Lyapunov's direct method[8]. In order to deal with the uncertainties thatare associated with the plant dynamics and theunknown disturbances, this method uses a smallultimate bound of the state error as an adap-tation criterion. A series of simulation studies wereperformed to demonstrate the effectiveness of thiscontroller. The results show that the proposedAMFC is fairly robust to unknown and time-varying external load disturbances, yieldingimproved performance characteristics when com-pared with a suboptimal PID controller with con-stant feedback gain.

List of symbols

A = piston ram area, m2

Ap, Am = system matrix of plant and modelB = viscous damping coefficient, Ns/mBp, Bm = input matrix of plant and modelCt = total leakage coefficient, m3/s/(N/m2)D = linearized parameter of nonlinear external load

disturbance, Ns/mF, = external load disturbance, Ni = input current, mA

Paper 5898D (C8/C9), first received 9th October 1986 and in revisedform 10th March 1987Dr. Cho is with the Department of Production Engineering, KoreaAdvanced Institute of Science and Technology, PO Box 150, Chongry-ang, Seoul, Korea

Dr. Yun is with the Spent Fuel Management Division, Korea AdvancedEnergy Research Institute, PO Box 7, Daeduck-Danji, Choong-Nam,Korea

MPiPs

QiruVt

xp,X

F,

= pressure gain of servo valve, m3/s/(N/m2)= flow gain of servo valve, m3/s/mA= total mass of piston and load, kg= load pressure, N/m2

= supply pressure, N/m2

= load flow rate, m3/s= reference input, m/s= control input, mA= total volume of valve and cylinder chamber,

m3

= state variable of plant and model= velocity of the piston, m/s= desired velocity of the piston, m/s= bulk modulus of fluid, N/m2

= maximum eigenvalue of P matrix= minimum eigenvalue of Q matrix= natural frequency of hydraulic system and

model, rad/s= maximum boundary of system matrix and

input matrix= maximum boundary of state variable xm and

reference input r= damping ratio of hydraulic system and model,

dimensionless

1 Introduction

Most processes requiring a large processing force areoperated by hydraulic servo systems. These hydraulicallyoperated processes are usually represented by hydraulicactuator-load systems whose dynamic characteristics arecomplex and highly nonlinear, owing either to the flow-pressure properties of the hydraulic system or to the loadsystem motion itself. The conventional approach to thecontroller design of these systems is to linearise thesystem dynamics in the vicinity of an operating point ofinterest. Based upon the linearised system dynamics aconstant gain feedback control has usually been adoptedfor the process control. From the design point of view,however, this approach needs further improvements inthe following problem areas:

(a) Since the system parameters (such as supply pres-sure, inertia mass, viscous friction of the actuator, and oiltemperature) vary during operation, these parametercharacteristics should be considered in the controllerdesign.

(b) The controller gain parameter has to be redesignedrepeatedly when the processes operate under differentconditions. In practice the process is often operatedunder a wide range of operating conditions and in thiscase the conventional controller design approach

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988 149

becomes ineffective, even causing instability of the systemoperation.

(c) In many processes certain forms of external dis-turbance usually influence the hydraulic servo system.These disturbance loads are often uncertain due to thecomplex interaction between the process dynamics andthe hydraulic actuator motion. Due to the uncertaintythe conventional controller design approach may notensure satisfactory control performances.

In the case of the parameter variation several investiga-tors [1-5] applied a model reference adaptive controlscheme (MRAC) to compensate for the time-varyingeffect: they dealt with a positioning control problem ofthe linearised hydraulic system which has no externalload disturbance, but extensive research effort that treatsfully a wide range of operating conditions and uncertainload disturbances has not yet been made.

Therefore, the objective of this paper is to solve suchremaining problems. An adaptive model followingcontrol scheme (AMFC) is presented for the problem ofcontrolling the velocity of hydraulic systems subjected tounknown or uncertain external load disturbances. It hasthe potential of eliminating or reducing the requirementsfor designing at a number of different operating points,and of yielding good velocity response under all oper-ating conditions. Lyapunov's direct method [6, 7] whichis specifically intended to deal with nonlinear time-varying systems, is the basis for the controller synthesistechnique. This technique utilises an ultimate boundof uncertain parameters for the design of the adaptivecontroller [8-11]. The control system is thus designed tohold the velocity of the hydraulic servo system to trackthat of an ideal model, which is specified by the designer.

2 Problem description

In the problem of the position control of hydraulicsystems with no external load disturbance, a nominaloperating point can be assumed constant, since a controlvalve operates in the vicinity of a zero control input anda zero load pressure [12]. Such an assumption cannot besatisfied in the case of a velocity control problem, sincethe operating point varies in the wide range of operatingconditions according to the velocity of the actuator. Inthe case where uncertain external load disturbances arepresent, the operating point of the control valve as wellas the linearised parameters become unknown. Therefore,the linearised approach cannot be applied to the velocitycontrol problem of the hydraulic system.

The electrohydraulic velocity control system con-sidered herein consists of a power cylinder and a servovalve as shown in Fig. 1. The piston of the power cylin-der is subjected to an arbitrary unknown external dis-turbance (F,) which acts in the opposite direction to themovement of the piston. The objective of the control is tokeep the velocity (x) of the hydraulic system following thedesired trajectory (that of the model) as closely as pos-sible, regardless of the operating points and the unknowndisturbances.

The relation between the input current (i) and the loadflow rate (Q,) of the servo valve is governed by the wellknown orifice law and given by

(1)6, = KiJ(Ps - sgn (/)

where the input current is constrained by

Ps is the supply pressure and P, is the load pressureacross the cylinder. The continuity equation of the servovalve and the cylinder chamber yields

(2)

actuator

pump andmotor

tank

Fig. 1 Schematic diagram of electrohydraulic velocity control system

where A is the piston ram area, Ct is the total leakagecoefficient, Vt is the total volume of the valve and thecylinder chamber, /? is the bulk modulus of the oil and xis the velocity of the piston. Under the assumption thatCoulomb friction between the piston and its sleeve is neg-ligible, the equation of motion of the piston is given by

x = Mx + Bx (3)

where B is the viscous damping coefficient. The externalload disturbance (F,) is assumed to be dependent uponthe velocity of the piston, and slowly time-varying

F{=f(x) (4)

The above form of the external load disturbance can fre-quently be found in industrial processes [13, 14] whichuse the hydraulic servo system. For example, the externaldisturbances in the ring-rolling machine and in thehydraulic injection moulding machine are present in theabove form (eqn. 4) through the reaction force caused bythe interaction between the velocity of the hydraulicservo system and the process dynamics. To achieve satis-factory process control, which guarantees good quality offinal products in such machines, the process controllerdesign has to be based upon an exact process dynamicmodel which can describe all the physical phenomenaassociated with the process. This modelling problem is anextremely difficult task, since the mechanism of themanufacturing process is quite complicated and largelyuncertain. Therfore, there exist the uncertainties in eqn. 4and these necessitate developments of an adaptivecontrol of the process which takes them into consider-ation.

For the application of the adaptive law, the linearisedeqn. 1 obtained for an arbitrary operating point yields

l = Kqi-KcPl (5)

where Kq =

150

„ . . - sgn (UP,,), Kc = K | im | /{Zj(Pa— sgn (iJ-P/,,,)} and * means the nominal operating

point. Under the assumption that eqn. 4 is differentiablewith x, the linearised equation is given by

F, = Dx (6)

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988

where D = df/dx \im. The above equation indicates thatthe external load disturbance acts as a linear damperwhose damping coefficient varies with the operatingvelocity. As shown in eqns. 5 and 6 the linearised para-meters (Kq, Kc) are subject to variation of the operatingpoints and the coefficient (D) is uncertain due to theunknown external disturbance. Using eqns. 2, 3, 5 and 6and defining a state vector (xp) as [x, 3c] and the controlinput (u) as i, the following state equation can be derived[5,15]:

(7)x = Apx Bpu

where the control input is constrained by

PM\ (B + D)

U 4̂

(Or, =

and Ke = Kc + Ct. The effect of the unknown dis-turbance on the hydraulic system is included in (p . In theabove equation £p and Kq are also unknown and aresubject to variation of the operating points, the linearisedparameters (Kq, Kc) and the unknown disturbanceparameter, D.

3 Derivation of the adaptive control law

The structure of the model following control system con-sidered here is shown in Fig. 2. The model is assumed to

F,(X)

u

r

1hydraulic

servo system

adaptivecontroller

model

X-P

e

Fig. 2 Block diagram of adaptive model following control system

generate a desired output vector for a given referenceinput. The hydraulic servo system has unknown para-meters and is controlled by a feedback signal that is gen-erated by the adaptive controller. The design of theadaptive controller makes use of the signal synthesismethod and determines the adaptive control signal, u(t),as a function of the output error vector, e(t). The objec-tive of the design procedure is to synthesise u such thatthe state variables of the hydraulic servo system given ineqn. 7 follow those of a model described by the timeinvariant state equations

where xm e R2 is the model state vector, Am and Bm arethe time invariant model matrices, and r is the referenceinput. The adaptation criterion may be the asymptoticstability of the generalised error vector, but in this paperthe ultimate bound of the state error vector [8] isused for the subsequent developments. Let the unknownhydraulic system dynamics and the reference input bebounded such that

Bp-Bn

\r\ for i = 1, 2

(9)

(10)

(11)

where api and ami are the ith column vector of Ap and y4m

matrices, respectively, and || • || represents the Euclideannorm. The problem of concern is then as follows: Giventhe plant (eqn. 7), the reference model (eqn. 8), and theassumptions stated in eqns. 9-11, to design an adaptivecontroller which forces the plant state xp to follow themodel state xm.

3.1 Selection of the modelThe model matrixes Am and Bm were obtained by linear-ising the hydraulic system about an operating point forwhich the linearised dynamic has a desirable transientresponse. Considering that the unknown disturbanceinfluences the damping coefficient of the system as shownin eqn. 7, the following form of the state equation of themodel can be selected

xm = (12)

where com and Cm c a n be chosen, depending upon the

desired time response. Once a>m and Cm are selected, Am

and Bm matrices in eqn. 8 are automatically fixed. Now inorder to relate the inertia load of the plant to the modelfrequency (caj, and the unknown varying parameter (£p)to the damping coefficient of the model ((m), the non-dimensional parameters Zx and Z2 are defined as

Z1=(Op/0)m, Z2=Cp/Cn (13)

3.2 Synthesis of adaptive control lawLet the state error (e) be xm — xp. If Am xp is added toand subtracted from the right side of eqn. 7 and theresulting equation subtracted from eqn. 8, an auxiliarydynamic system defining the state error vector is given by

e = Ame + Bmr - Bpu + {Am - Ap}xf (14)

Let Q be an arbitrary positive definite matrix. Then for astable model matrix Am, P which is a solution of

AlP=-Q (15)

xm = Amx Bmr (8)

is also a positive definite matrix. We define the Lyapunovfunction as the following positive definite quadratic form,

V{e) = eTPe (16)

Utilising eqns. 14-16, V(e) is given by

V(e) = -eTQe + 2eTP\_Bmr -Bpu+{Am- Ap}Xp]

(17)

Defining a new matrix (Q) which is related to the systemmatrix (Ap) as follows;

Q = Q - 2PAA (18)

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988 151

where A A = Ap — Am and using eqn. 18, V{e) is rewrittenas

V(e) = -e T Qe + lerP\Bmr-Bpu- AAXm] (19)

Now, the main problem is to ensure that V(e) is negativedefinite for all e. To satisfy this condition, Q must bedesigned to be positive definite, and the control input(u) has to be chosen to make the second term on the rightside of eqn. 19 to be negative definite outside a suitablysmall region about e = 0. We will discuss the method ofchoosing u before discussing the design problem of thematrix Q. If matrices Bp and AAp were known, and B ~1

existed for all x , u and t, then u could be chosen as~1

(20)

This control would yield perfect model following, in thesense that it would cause e -* 0 as t -+ oo. However, sincethe matrices mentioned previously are not known for thisproblem, the ultimate bound of error (e) [8-11] isemployed as a stable adaptive criterion, instead of theasymptotic stability of the generalised error. This ulti-mate bound means that the error trajectory, startingfrom some value e(t0), will decrease and go into somebounded region Qs, and that e(t), which at some time t ^t0 is in Qs, will never thereafter leave fis. The theorems[8,16] used in this paper are summarised as follows:

Let Q be a closed bounded set and fis be the set of allpoints whose distance from Q is less than s, and Qc andQs be the complements of 0. and fis, respectively. IfV(e(t), t) ^ - e < 0 for all t ^ t0 and for all e(t) e Qc theneach solution of eqn. 14 is ultimately inside Qs where Qs

is such a set that

V(e(tl), h)<V{e{t2\t2), (21)

The design objective is then to obtain an adaptive con-troller u as a function of e, so as to make Qs as small aspossible. Therefore, the problem of concern is to synthe-sise u so as to make V(e) negative definite, at least inthe region Qc, where the above theorem will guaranteethe bounded stability of the error differential equation(eqn. 14) in the region Qc

s. To satisfy the adaptation cri-terion it will be reasonable to consider the mini-malisation of V(e) with respect to u. However, since onlybounds of the uncertainties (eqns. 9-11) are known, adirect minimalisation is not feasible. In the following, aminimalisation of V{e) for the worst possible uncer-tainties that are associated with Bp will be considered

minimise maximise V(e) (22)

Substituting eqns. 7, 12 and 13 into eqn. 19, V(e) is givenby

V(e) = -eTQe + 2/c[co^{r - Z\(KjA)u]

+ co2m(Zl - l)xpl + 2Cm • <om{ZxZ2 - l}xp2] (23)

where fe is pi2ei + Viiei a n d P12 and p22 are the com-

ponents of P

P12P22

For the solution of eqns. 22 and 23 the followingmin-max problem is considered;

minimise maximise [ —fe(Z\a)2KJA)u~]

+ maximiseKg

= minimise [—U

x {feco2m(l - Z\

= minimise [ -fe w2m u + ? I fe I I«I]

u

The inner maximisation in eqn. 24 is achieved for

(24)

where £b is

maximiseKa

— Z\ KJA~\.

(25)

(26)

Considering that the control input is bounded as shownin eqn. 7, for the solution of eqn. 24, which is also thesolution of eqn. 22 the control input is given by

(27)

- u

The direct application of the min-max solution obtainedabove to the adaptive control function is effective as faras the small ultimate error region is concerned. Thismethod may be referred to as a high gain model follow-ing control scheme (HGMFC) [10]. However, for somesituations it acts as a bang-bang control with high fre-quency [8]. To avoid such a discontinuity in the control

•5b | fe l -9

§b|fe

-umax

Fig. 3 The adaptive control u with respect tofea)%,

function, in this work the adaptive control given in eqn.27 is employed, by introducing a linear portion to thecontrol obtained from the min-max solution. This isshown in Fig. 3 and gives

if

if(28)

if -tb\f.\>f.a>l>-Zb\f.\-0max

if -

152 IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988

Under the adaptive control law given in eqn. 28 and forthe positive Q matrix, the error system defined by eqn. 14is ultimately bounded and the error set of Q and fis in thetheorem is given by

Q = (29)

or

= <e{t)\\\e\\<2kl

= \e(t)\eT(t)Pe(t)^s,

s = maximise eT(t)Pe(t) > (30)e(r)efl

where XQ is the minimum eigenvalue of Q matrix, kp isthe maximum eigenvalue of P matrix, £x is the maximumvalue of || xm ||, £a, £b and £ are given in eqns. 9-11, e isan arbitrary small number and b is a positive number.

"^ + ten + q22Km)K

of Kq are given by

• Cp)) (33)

Once the Cp value is selected, the £b and Zx value can bedetermined from eqns. 31-33.

As discussed in the preceding Section, the P and Qmatrices must be determined to be _ positive definite,and to satisfy the condition that the Q matrix (given ineqn. 18) is also positive definite. Let positive symmetricmatrix Q be

(34)

Then all the successive principal minors of Q matrix arepositive; i.e.,

qn>0,\Q\=qliq22-q2

12>0 (35)

Using eqns. 12 and 34, P matrix can be obtained asfollows

(36)

where Ka is <o2 and Kg is 2Cm com. The conditions that P matrix should be positive definite are given by

te 0

Substituting the above P and Q matrices and eqns. 7 and 12 into eqn. 18, the Q matrix is given by

ql2

+ <?22KJ*J1 " Z\) q22 - ten + <?22*J{1 " Zv

(37)

(38)

For the detailed derivation of eqns. 29 and 30 see Refer-ence 8. In order to apply the adaptive control law givenin eqn. 28 to the hydraulic system, 9, £b and the P matrixmust be determined. The 6 value must be carefully deter-mined in consideration of the instability problem and thesteady state error [8, 10]. The £b value must be deter-mined by solution of eqn. 26, and the P matrix must bedetermined so as to make the Q and Q matrixes given ineqns. 15 and 18 positive definite.

3.3 Determination of f and the P matrixExamination of the solution for u in eqn. 28 shows thatthe synthesised control input becomes zero at any errorintervals for £b larger than wl. Therefore, £b must bechosen to be less than co2. Since the Kq is a positiveparameter, Kq<olJA must be less than 1 for any operatingconditions. For such conditions, the £h value and therange of Zv value are easily determined as follows:

J (31)In order to determine the minimum and maximum valuesof the unknown parameter Kq in eqn. 5, the followingphysical consideration can be utilised;

\P,\/PS^CP (32)

where Cp is an arbitrary positive number less than 1.Using eqns. 5 and 32, the minimum and maximum values

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988

Then, the conditions for the successive principal minorsof the Q matrix to be positive are given by

«n=«uZi>0 (39)

- toil +

-qll{l-ZlZ2}K(/KJ>0

Now, the problem is to determine Q matrix so as tosatisfy the inequalities given in eqns. 35, 37 and 39 simul-taneously. There can be several mathematical solutionswhich satisfy the above inequalities. However, the simplesolution such as qxl > 0, g22 > 0 and q12 = 0 can beused for the solution of eqns. 35 and 37. Then theremaining problem is to determine the positive numbersqlt and #22 so as to satisfy the inequality given in eqn. 39regardless of the unknown parameter Z 2 . Substitutingqi2 = 0 into inequality eqn. 39, the condition for \Q\ tobe positive is given by

161 = + ^22X1 - ZXZ2)} > 0

(40)

or422

Therefore, qtl and g22 must be determined to satisfy theworst possible condition of the above inequality, and this

153

condition is given by

'2min (41)

In order to determine the Z2min in the above inequality(eqn. 41) the minimum values of Ke and D in eqn. 7 mustbe considered. The minimum value of Ke can be easilydetermined from eqns. 5 and 7 and is given by theleakage coefficient (C,). The minimum value of D can beobtained from the following physical consideration;

D > 0 (42)

Substituting Ke = Ct and D = 0 into eqn. 7, and usingeqn. 13, Z2min is obtained by

y _^2min —

c, (43)CmA\J Vt

Substituting Z2min and Zx into eqn. 41, the qtl and <j22values can be determined.

4 Simulation study

4.1 Simulation procedureA series of simulation studies were made, for variousuncertain external load disturbances, to evaluate the per-formance of the adaptive model following controller(AMFC). These results were compared with those of aconventional suboptimal PID controller. The nonlineareqns. 1-4 were used for the mathematical model of theplant. The system parameters used in this simulationstudy are listed in Table 1. Also, the design parameters,such as Cp, com, Cm, €b, 0, P and Q are listed in Table 2

Table 1 : System parameters

Parameter Value Dimension

/w"

1.52 x8.82 x2.24 x1.62 x155.34 x6.86 x1.79 x3.43 x2

io

106

10-3

108

m2

N-sec/mm5/sec/Nm4/sec/(mA-7N)mAkgN/m2

m3

N/m2

dimensionless

Table 2: Design parameters

Parameter Value Dimension

P

P =[76.62[o.OO5

0.666661.1165.440.721946.1100

0.0050.0024

dimensionlessdimensionlessrad/secdimensionlessdimensionlessdimensionless

2740

J Land these parameters have been chosen so as to satisfythe conditions given in eqns. 31, 32 and 41. A specialremark must be addressed with regards to the selection ofthe 6 value. To choose this value several simulationstudies were made by changing the 0 value, from 100 tooo. Although the results have not been shown here dueto space limitation, they were found to influence thestability and the error characteristics of the velocity re-sponse. In this simulation the 6 value was fixed at d = 100.

In order to investigate the effect of the uncertain exter-nal load disturbance on the hydraulic system, eqn. 4 was

replaced by the following nonlinear equation [13,14];

F, = Cxn (45)

where C and n are positive constants. Using the systemparameters and the design parameters, the dynamicequations were solved by a 4th order Runge-Kutta algo-rithm with a step size At = 50 fis. Throughout the simu-lations the reference velocity input was kept at0.0015 m/s.

4.2 Results and discussionsFrom the results of the simulation, the velocity responsesof the AMFC were obtained for various external loaddisturbances. These responses are plotted in Fig. 4. for

0.020

0.015

0.010

0.005

0«] 0.020£-•0.015

iT 0.010

I 0.005

00.020 r

I

0.015J- .-"•

0.010^//

0.005ft"

_L i i

f;

0.0A 0.08 0.12 0.16time, t

d

0.20

Fig. 4 The variation of the velocity response owing to the variation ofdisturbance

model+ - + AMFC

Subopt(a) c = 4.9 x 106

(b)c = 0(c) c = 9.8 x 106

(d) c = 9.8 x 106

various C values in eqn. 45. Also, those of the conven-tional suboptimal controller are plotted for comparisonpurposes. The suboptimal controller adopted herein wasa proportional plus integral plus derivative (PID) con-troller, whose gains were optimised so as to minimise aperformance index given by the ISE (integral squareerror) criterion. This minimization problem was solvedby the Rosenbrock algorithm [17] which utilises thedirect search method. For a detailed optimisation pro-cedure see Reference 13. As a result of the optimisationthe suboptimal PID controller was found to be

(46)

154 IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988

In this optimisation the reference velocity (xd) was kept at0.0015 m/s while the external load disturbance (F, = 4.9x 106x) were used. As shown in Fig. 4 the response of

the model (solid line) shows satisfactory response charac-teristics such as good damping, fast response and nosteady state error. For the case of C = 4.9 x 106 (Fig. 4a),the velocity response of the AMFC slightly lags behindthat of the model and a small amount of steady stateerror, about 0.8% of the desired value (0.0015 m/s), per-sists. On the other hand, the suboptimal controller yieldsslightly faster response characteristics than the AMFCand shows no steady state error. The performance of thederived AMFC is observed to be inferior to that of thesuboptimal controller, since, in this case, the suboptimalcontroller has been designed especially for the known dis-turbance, C = 4.9 x 106. For the case of no disturbance(C = 0, Fig. 4b) the velocity response of the AMFC isalmost identical to that of the model except for the smallamount of steady state error. However, the suboptimalcontroller designed for the case of C = 4.9 x 106 showslarge oscillation in this case. The oscillation is caused bythe small damping ratio ({p, eqn. 7) of the hydraulicsystem itself [3, 18]. Therefore, the suboptimal gainsmust be adjusted in order to reduce such oscillatory phe-nomena. However, online adjustment of these gainsseems to be very difficult, since the external load dis-turbance is unknown a priori. As the magnitude of thedisturbance increases (C = 9.8 x 106, Fig. 4c), the veloc-ity response of the AMFC lags further behind that of themodel during the transient period. This is due to the factthat the control input is saturated, as can be seen in Fig.4d. Compared with the AMFC, the suboptimal controllershows larger steady state error. From these figures it isalso observed that the velocity responses of the AMFCshow slower response characteristics as the magnitude(C) of the disturbance increases. This trend can beexplained as follows: as the magnitude of disturbanceincreases, the damping ratio (£p) also increases, causinglarger differences between the system matrices, the plant(Ap) and the model (AJ (see eqn. 7). Thus, the maximumbound value (£fl) given in eqn. 9 increases, resulting in alarger error bound value (|| e ||), as given in eqn. 29.

In order to investigate the effectiveness of the AMFCfor a time-varying external load disturbance, the simula-tion study was performed for the case of a high frequencydisturbance. The C value was substituted by4.9 x 106 + 0.98 x 106 sin (40 nt). In Fig. 5 the result isplotted and compared with that of the suboptimal con-

0.016

: 0.012

0.008

0.004

0.04 0.08 0.12time(t),s

0.16 0.20

Fig. 5 Velocity response for the time-varying disturbancemodel

+ - + AMFCSubopt

c = 4.9 x t06 + 0.98 x 106 sin (40 nt)

troller. The velocity response of the AMFC slightly lagsbehind that of the model during the transient period, andshows small oscillatory phenomena at steady state due tothe high frequency disturbance. The suboptimal control-ler shows very large oscillation which may be a cause ofthe dangerous instability of hydraulic systems. This com-parative result indicates that the derived AMFC is veryeffective for the time-varying external load disturbance.

Based upon the results presented above, it can be con-cluded that the suboptimal constant controller cannotguarantee satisfactory response for the uncertain dis-turbances, although it yields good performance for aknown disturbance. On the other hand, the response ofthe AMFC shows satisfactory results for the disturbanceswith unknown as well as time-varying characteristics.

5 Concluding remarks

An adaptive model following control method has beenapplied to the velocity control of a nonlinear hydraulicservo system subjected to unknown external load dis-turbances. The method is based upon the Lyapunov'sdirect method and employs an ultimate bound of thestate error as an adaptation criterion in order to dealwith the uncertainties that are associated with the plantdynamics and the unknown disturbances. The adaptivemodel following control system (AMFC) thus designed ismade to track the desired velocity of the model as closeas possible. The results obtained from a series of simula-tion studies show that the proposed AMFC is fairlyrobust to unknown, time-varying external load dis-turbances and yields good performance when following adesired model response.

6 References

1 PORTER, B., and TATNALL, M.L.: 'Performance characteristicsof an adaptive hydraulic servo-mechanism', Int. J. Control, 1970, 11,pp. 741-757

2 HESSE, H.: 'Load adaptive electrohydraulic position controlsystems with near time optimal response', Proceedings of 3rd Inter-national Fluid Power Symposium, Turin, Italy, May 1973, pp. El. l-E1.12

3 KULKARNI, M.M., TRIVEDI, D.B., and CHANDRASEKHAR,J.: 'An adaptive control of an electro-hydraulic position controlsystem', Proceedings of American Control Conference, 1984, pp.443-448

4 CHO, S.H., and LEE, K.I.: 'An adaptive control scheme for theelectro-hydraulic position control system using the second orderlinear model (in Korean)', Proceedings of the Korean Society ofMechanical Engineering, 1985, pp. 309-314

5 CHEN, J.S., OU, Y.L., LI, Y.P., and LU, Z.M.: The further study ofhigh performance adaptive controller for hydraulic servo-systems',Proceedings of International Conference on Fluid Power Transmis-sion and Control, September 1985, pp. 163-175

6 KALMAN, R.E., and BERTRAM, J.E.: 'Control system analysisand design via the second method of Lyapunov, I. continuous-timesystems', Trans. ASME Ser. D, 1960, pp. 373-393

7 KALMAN, R.E., and BERTRAM, J.E.: 'Control system analysisand design via the second method of Lyapunov, II. discrete-timesystems', Trans. ASME Ser. D, 1960, pp. 394-400

8 NIKIFORUK, P.N., GUPTA, M.M., and TAMURA, K.: Thedesign of a signal synthesis model reference adaptive control systemfor linear unknown plants', Trans. ASME Ser. G, J. Dyn. Syst. Meas.& Control, 1977, 99, pp. 123-129

9 LEITMANN, G.: 'Guaranteed asymptotic stability for some linearsystems with bounded uncertainties', Trans. ASME. J. Dyn. Syst.Meas. & Control, 1979,101, pp. 212-216

10 MONOPOLI, R.V.: 'Model following control of gas turbineengines', Trans. ASME. J. Dyn. Syst. Meas. & Control, 1981,103, pp.285-289

11 GUTMAN, S.: 'Uncertain dynamical systems — a Lyapunovmin-max approach', IEEE Trans., 1979, AC-24, pp. 437-443

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12 TAKAHASHI, K.: 'Application of the model reference adaptivecontrol technique to an electro-hydraulic servosystem', Proceedingsof International Conference on Fluid Power Transmission andControl, September 1985, pp. 68-87

13 YUN, J.S., and CHO, H.S.: 'A suboptimal design approach to thering diameter control for ring rolling processes', Trans. ASME. J.Dyn. Syst., Meas. & Control, 1985,107, pp. 207-212

14 CHO, Y.J., CHO, H.S., and LEE, CO.: 'Optimal open-loop controlof the mould filling process for injection moulding machines',Optimal Control Appl. & Methods, 1983, 4, pp. 1-12

15 MERRIT, H.E.: 'Hydraulic control system' (John Wiley & Sons,New York, 1967)

16 LASALLE, J., and LEFSCHTZ, S.: 'Stability by Liapunov's directmethod with application', (Academic Press, 1961)

17 KUESTER, J.L., and MIZE, L.J.: 'Optimisation with Fortran',(McGraw-Hill, New York, 1973) pp. 320-330

18 FAULHABER, S., BULTGES, H., and RAKE, H.: 'Multiloop con-trols and state feedback controls for weakly damped hydraulicdrives', International Federation of Automatic Control, 1984, 2, pp.271-276

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