Semiglobal ISpS Disturbance Attenuation With Output Tracking via Direct Adaptive Design

IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 18, NO. 4, JULY 2007 1129

Semiglobal ISpS Disturbance Attenuation WithOutput Tracking via Direct Adaptive Design

Shuzhi Sam Ge, Fellow, IEEE, and Thanh-Trung Han

Abstract—Direct adaptive partial state feedback control ispresented to achieve semiglobally input-to-state practically stable(ISpS) disturbance attenuation with output tracking for a classof uncertain time-varying nonlinear systems in which the unmea-sured dynamics do not possess a constant disturbance attenuationlevel (CDAL). Identifying a necessary condition for the existenceof a CDAL, direct adaptive neural networks (NNs) control isdeveloped, where the universal approximation property of NNsand the domination design are employed together to overcomethe difficulties due to the lack of state information, unknownsystem nonlinearities, and unknown state-dependent disturbanceattenuation gain. The proposed method is coherent in the sensethat it is applicable to the case in which a CDAL exists.

Index Terms—Adaptive control, disturbance attenuation,input-to-state stability (ISS), neural network (NN) parametriza-tion, semiglobal solutions.

I. INTRODUCTION

ORIGINATING from the work of Willems [1], extensiveresearch has been carried out for almost disturbance

decoupling and disturbance attenuation for certain classes ofnonlinear systems [2]–[20]. In this paper, disturbance attenua-tion with output tracking is investigated for a class of nonlinearsystems with appended dynamics. The usual assumptions inthe literature are relaxed to allow a broader class of systems. Inparticular, the appended dynamics may not possess a constantdisturbance attenuation level (CDAL) and the virtual controlcoefficients may be functions of both measured and unmeasuredstate variables.

Systemswith appendeddynamicscanbeused to modelvariouspractical systems such as electric motors with external load [21],space robotic systems with external payload orcatchinga movingobject [22], and systems with flexible structure [23]. Control de-sign for systems with appended dynamics has attracted much at-tentionrecently[7], [14], [16], [24]–[27]. Ingeneral,systemswithappended dynamics can be described by

(1)

where is the state of the appended dynamics -subsystem,is the state of -subsystem, is the input of -sub-system, is the system input, is the time-varying parameter,

Manuscript received December 5, 2005; revised August 25, 2006; acceptedFebruary 5, 2007. This work was supported in part by the Natural Science Foun-dation of China (NSFC) under Grant 60428304.

The authors are with the Department of Electrical and Computer Engi-neering, National University of Singapore, Singapore 117576, Singapore(e-mail: [email protected]).

Digital Object Identifier 10.1109/TNN.2007.899159

is the disturbance, and , , and are functions of ap-propriate dimensions. Under different conditions, various con-trol designs have been investigated for classes of systems (1) inthe literature [7], [14], [16], [17], [24], [26]. Existing control de-signs typicallyrequire the -subsystemtoenjoycertainpropertiesincluding input-to-state stability (ISS) with respect to[26], [28] and possessing a constant disturbance attenuation gain[13], [20]. The desired properties of the closed-loop system areachieved by either matching the controls to the system nonlin-earities using the terms , where s are compo-nent functions of ,or including aknown function arising frombounding functions of s [25], [26]. Accordingly, the controlcoefficients sareusually required tobeeither independentofunmeasured variables or bounded by functions of measured vari-ables. Using the dissipativity theory, robust stabilization was in-vestigated for a class of cascade-connected systems in the form of(1) with the absence of time-varying parameter [27], where thecontrol coefficients may depend on unmeasured variable . Fortime-varying systems (1), where the control coefficients may de-pend on the unmeasured variable and the input to -subsystemis the whole state of -subsystem, i.e., , control design re-mainsopenandischallengingtothewidelyusedLyapunov-baseddesignmethodology. In this paper, the difficulties associatedwiththeappearanceofunmeasuredvariable in controlcoefficientsandthe appearance of the whole state of -subsystem as input to theappendeddynamicsareovercomebythecombinationof thedom-ination design [26] and changing supply function technique [29].Though the control could be constructed using dominating func-tions of measured variables, the dominating functions are un-known or very hard to determine even for known system modelin practice, thus neural network (NN) control is called upon tosolve this problem.

For nonlinear systems, achieving disturbance attenuation isof theoretical, challenging, and practical importance. The clas-sical setting in solving the problem of disturbance attenuation isto satisfy the so-called Hamilton–Jacobi–Isaccs (HJI) inequality[13]. The satisfaction of the HJI inequality is governed by notonly the system dynamics, but also the disturbance attenuationlevel (DAL), a factor characterizing the disturbance attenuationquality. Recently, it was shown that the constant DAL cannot bemade arbitrarily small for certain classes of systems, partiallybecause the disturbance enters the -dynamics [13], [14], [30].As a consequence, significant effort has been devoted to deter-mine an optimal value of CDAL [13], [14], [30]. However, it wasshown by examples that CDAL may not exist for some systems[19]. Solutions to this difficulty are to address disturbance at-tenuation using the concepts of practical disturbance attenuation[19] and ISS disturbance attenuation [31]. Though the concept

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1130 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 18, NO. 4, JULY 2007

of ISS disturbance attenuation seems to be more general thanthe classical concept of disturbance attenuation, it remainsopen whether or not ISS disturbance attenuation is achievablewhen CDAL does not exist. In this paper, for a class of sys-tems described by (1), we present a condition under which onlystate-dependent DAL for -subsystem may exist. Then, fromthe condition that is necessary but not sufficient for achievingeither disturbance attenuation or ISS-like disturbance atten-uation, we further show the existence of a state-dependent DALfor -subsystem. Once again, as the control could be constructedusing this state-dependent DAL that is generally unknown orvery hard to determine even for known system model, NN con-trol is essential to solve this difficulty.

The usual approach in dealing with systems which have un-known and time-varying parameters is to use the certainty equiv-alence principle to design robust controllers under the assump-tion of boundedness of parameter variations [12], [27], [32],[33]. In this paper, we consider the time-varying parameter asscheduling variable to exploit the internal time-varying structureof the system for better control performance. Though the classof systems is a generalization of the classes of systems consid-ered in [34] and [35], the higher derivatives of the need not tobe available for feedback, and hence, the number of NN inputsis reduced for the ease of computation.

For system (1), control design may depend on functions,e.g., unknown state-dependent DAL and unknown dominatingfunctions, that are unknown or very hard to determine evenfor known system model. This makes adaptive approxima-tion-based control become the most promising approach toachieve the solution. For systems with unknown nonlinearities,universal approximation over a compact set using neural/fuzzyparametrization, together with the adaptive control technique,has been widely accepted as a mature methodology in con-trol system design [36]–[48]. Motivated by the adaptive NNdesign developed in [49]–[52], novel control is presented inthis paper, which can handle challenging technical problemsof the following: 1) the appearance of unmeasured variables

in the control coefficients, 2) the appearance of the wholesystem state in appended dynamics, and 3) the unknownstate-dependent DAL for -subsystem. We will show that theproblem is only solvable using functional approximation-basedtechnique—NN control is only used to demonstrate the mainidea. In particular, semiglobal and practical solution rather thanglobal asymptotical solution is achieved.

The paper is organized as follows. In Section II, we identifya necessary condition for the existence of a CDAL for systemswith affine disturbance. Then, we present assumptions and de-rive a dissipation inequality for the appended dynamics. Basedon this preparatory work, Section III presents a systematic de-sign procedure achieving input-to-state practically stable (ISpS)disturbance attenuation with output tracking. Section IV con-siders a fourth-order time-varying nonlinear system. We showthat a CDAL for this system cannot be found. By applying thedeveloped design method to that system, the achieved simula-tion results show the effectiveness of the proposed controller.

Notation: Let , , , and , respectively, denotethe absolute value for scalars, the two-norm for matrices, Eu-clidean norm for vectors, the Frobenius norm for matrices, and

for continuous functions .We use the abbreviation p.d.p. for positive–definite and properfunctions. The argument of the functions will be omitted when-ever no confusion can arise from the context.

Terminology [53]: A function is said to beof class if it is continuous, strictly increasing, and is zero atzero; it is of if it is of class and unbounded. A function

is of class if, for each fixed , thefunction is a class function, and, for each , it isdecreasing to zero as .

Definition 1.2: A system with inputand time-varying parameter is semiglobally ISpS [with re-spect to ] uniformly with respect to if for each compact set

and each measurable essentially bounded input definedfor all , there exist a compact set , a class function

, a class function , and a positive number such that,for any initial condition , the associated solutionexists for all , does not exceed , and satisfies

When and are whole state space, the property is then calledinput-to-state practical stability with respect to .

Lemma 1.1: For system with inputand time-varying parameter , assume that there existsa smooth function satisfying

where and are class functions, such that the timederivative of along the trajectory of the system satisfies

for some class functions and , and some positiveconstant . Then, the system is ISpS uniformly with respect to

with the corresponding being a class of .Proof: It is omitted as it is an obvious extension of the

well-known results [28], [53], [54].

II. PROBLEM FORMULATION AND PRELIMINARIES

Consider the class of single-input–single-output (SISO) time-varying nonlinear systems described by

(2)

where and are the control input andsystem output, respectively, and

are the measured states, , ,and are the unmeasured states, are -dimen-sional disturbances, the variable is considered as anadditional input or a possible exogenous scheduling variable,the functions and are considered as interpolatingfunctions that introduce the time-varying nature of the system,

GE AND HAN: SEMIGLOBAL ISpS DISTURBANCE ATTENUATION WITH OUTPUT TRACKING 1131

and the functions , , , , and areunknown functions of appropriate dimensions.

The class of systems (2) is a generalization to systems withappended dynamics and disturbance input of the class of sys-tems considered in [34], [35], and [55], where the functions smay be considered as the interpolating functions in which theplant dynamic structure changes according to the schedulingvariable , and the system (2) may be considered as multiplemodels where acts as the switching variable and the switchingbetween models occurs smoothly (see [55] for more discussion.)

Given a bounded reference trajectory and a positiveconstant , the control objective is to design adaptive control forsystem (2) achieving ISpS disturbance attenuation with outputtracking as follows:

1) all the state variables of the resulting closed-loop systemare uniformly ultimately bounded for bounded disturbance

;2) the inequality

(3)

holds for some class function , , somepositive constant , and some positive constant that de-pends only on the initial state of the closed-loop system.

The performance index (3) gives us some physical insight.The integrals on the left-hand side and the right-hand side of(3) are the -norms of the tracking error and dis-turbance , respectively. is a constant thatdepends on the initial state of the (closed-loop) system. From thepassivation theory, it says that the energy accumulated at timeof the tracking error signal does not exceed the initial energy ofthe system plus part of the energy of the disturbance. The dis-turbance attenuation performance is achieved in the sense thatthe less disturbance energy is transmitted to the tracking errorsignal, the more robust the tracking performance to the distur-bance is. In view of (3), the larger the design parameter is, thebetter the disturbance attenuation becomes. However, as will beshown later, the cost is a larger control effort.

Remark 2.1: In fact, NNs-based control achieves the con-trol objective practically due to the inherent NN approximationerrors. Here, we consider the NN approximation errors, repre-sented by the constant , as the design disturbances. Thus, theperformance index (3) reflects the attenuation of both externaldisturbances s and design disturbance .

For system (2), the disturbance attenuation property of theoverall system is governed by the disturbance attenuation prop-erty of the -dynamics. Because the desired trajectory of theoutput is given a priori, and other measured state variables aredriven in such a way that the output is forced to track the giventrajectory, the measured variables should be viewed as distur-bance inputs to the -dynamics. Accordingly, certain stabilityproperties of the -dynamics are needed to make the problemsolvable. Let us consider a general description of -dynamics

(4)

where , , and represent s, s, and s,respectively.

Definition 2.1: System (4) is said to satisfy the ISS distur-bance attenuation condition if there exists a smooth p.d.p. func-tion satisfying

where and are class functions, such that

(5)

for some class functions and and some function.

Definition 2.2: System (4) is said to satisfy the disturbanceattenuation condition, if it satisfies the ISS disturbance attenua-tion condition for for some constant .

The satisfaction of disturbance attenuation condition is abasic requirement that has been widely used in achievingdisturbance attenuation for certain systems with zero dynamics[13]–[15], [20]. However, this condition does not hold in generalas will be detailed in Proposition 2.1 for system (4) satisfying

disturbance attenuation condition.For clarity, define the following function:

(6)

which apparently is a function of both and , rather than aconstant.

Proposition 2.1: Consider system (4) satisfying the dis-turbance attenuation condition. Then, holds forall .

Proof: According to Definition 2.2, there exist a smoothp.d.p. function and a constant such that

(7)

As (7) holds for arbitrary , it is equivalent to

(8)

Opening the square, (8) becomes

(9)

which is obviously equivalent to

(10)

Thus, (7) is equivalent to (10). That is to say (10) always holdfor system satisfying disturbance attenuation condition.

For each , since is of class and is positive,exists, and hence, is well defined. Clearly,

, , then (10) implies

(11)

which is the conclusion of the proposition.FromProposition2.1,weknowthat disturbanceattenuation

for system (4) is not solvable if, for every pair of smooth p.d.p.


function and positive constant , we have forsome . Since the -subsys-tems of system (2) are in the form (4), disturbance attenuationfor system (2) is not solvable if there is any -dynamics to whichthe corresponding -function, i.e., , is un-bounded in for any smooth p.d.p. function . However,(11) suggests that a possible solution is to relax to be a functionof . In the following, we will show that, for system (4), distur-bance attenuation with being a function of is also a necessarycondition for ISS disturbance attenuation.

Lemma 2.1 [26], [27]: For any continuous functionwhere and , there are smooth scalar-valuefunctions , , , and suchthat

and

Proposition 2.2: Suppose that the system (4) satisfies the ISSdisturbance attenuation condition for some smooth p.d.p. func-tion , some class functions and , and somepositive function . Then, there exist a class function

, a nonvanishing function , and a positive func-tion such that

(12)

Proof: According to Definition 2.1, (5) holds for system(4) satisfying ISS disturbance attenuation condition. From (5),letting , we have

(13)

By Lemma 2.1, there exist two smooth positive functionsand such that . Letting

and adding to bothsides of (13) yield

(14)where and . Moreover, (14) stillholds if the function , isused instead of . Hence, the conclusion is obtainedas (12) is equivalent to (14).

In view of Definition 2.2 and Proposition 2.2, condition (12)is quiet weak since it is necessary for either disturbance at-tenuation or ISS disturbance attenuation. This paper achievesthe control objective by requiring -subsystems to satisfy (12)as follows.

Assumption 2.1: For each , there exists a smoothp.d.p. function satisfying

for some class functions and such that,for each

for some class functions , some nonvanishingfunctions , and some smooth positive functions

, where . In addition, for eachand , there exists a class function such that

(15)

and exists, .Remark 2.2: As discussed previously, -subsystems must

possess appropriate properties to enable the solvability of theproblem. Assumption 2.1 simply requires each -subsystem tobe dissipative with the supply rate

. Obviously, this assumption is weaker than as-sumptions in the existing results [13], [20], where s are as-sumed to be constants. Note that (15) always holds forbeing constant as is of class . Condition (15) expressesthat the dissipation rate of the -subsystem is fast enough withrespect to its disturbance attenuation gain.

For convenience, let

where “ ” stands for , , or , and “ ” stands for or . Foreach , the time derivative of is

(16)

where , , and

For each compact subset of , let

,, and

.Assumption 2.2: For each two compact sets

and , there exist real numbers, , and such that, for each

and

for all , , and .


Assumption 2.3: The time derivative of the exogenous vari-able is bounded on in the sense that there is a possiblyunknown constant such that , , and isavailable for feedback. In addition, there exist positive numbers

and such that , ,, and .

Note that Assumption 2.2 implies that the affine terms inthe system dynamics have bounded gains and bounded ratesof change. With the continuity assumption on s, s, and

s, Assumptions 2.2 and 2.3 always hold within compactsets.

Remark 2.3: Theoretically, we may not use for feedbackas the boundedness over compact sets of is enough for con-trol design developed in Section III. However, we would notconsider this as an advantage since may have the practicalmeaning of acting as the scheduling variable or switching vari-able that are available for control design. In addition, using forfeedback means that more information on the dynamic structureof the system is available and, as a result, better control per-formance may be obtained. Thus, we will use for feedbackwithout weakening the results.

On the compact sets and , consider the following func-tions:

(17)

where , are smooth monotone nonde-creasing functions to be designed. Their constructions will beprovided in the proof of Proposition 3.1.

Proposition 2.3: For functions s satisfying Assumption2.1 and functions s defined by (17) with s being smoothmonotone nondecreasing functions, there exist class func-tions and such that

(18)

Proof: Clearly, we have

(19)

Let

(20)

Using (19), it is straightforward to show that anddefined by (20) satisfy (18).

For each and , by Lemma 2.1,there exist smooth functions and such that

. By the smoothness and

compactness properties of and , respectively, thereexists a smooth function such that

Thus, with the convention that and , foreach and , we have

for all , . The time derivative ofsatisfies

(21)

Noting that , (21) can be rewritten as

In the following, let us consider different cases for the pair.

Case 1) . Clearly

Case 2) . Since is of class, we have

Using Assumption 2.1, this leads to


Since , will be designed in the proof of Propo-sition 3.1 to be positive monotone nondecreasing functions, wehave

Let . Com-bining both cases, we have

Following the same procedure for the pairsand , we arrive at

(22)

with and

.Note that the previous argument is also valid for , where

, . Multiplying both sides of (22)with , we have

(23)

Define the function . Taking thesum of both sides of (23) over all values of , we obtain

(24)

where

(25)

Note that is bounded whenever is bounded andthe superscript is to indicate that and depend onthe choice of .

Assumption 2.4: The reference signal is available formeasurement and there exists an unknown constant suchthat

Lemma 2.2 [26], [27]: For any function ,there are smooth functions and ,and real numbers and such that

and

III. NEURAL-BASED ADAPTIVE CONTROLLER DESIGN

As pointed out in [56], one of the main difficulties arisesfrom the uncertain affine terms s. When these terms are un-known, approximation-based control using feedback lineariza-tion may lead to the controller singularity problem. There are anumber of alternative ways to deal with this difficulty [33]–[35],[56], [57]. Following the understanding of approximation-basedcontrol in [57], we are interested in designing adaptive NN con-trol that, for bounded initial conditions, guarantees the bound-edness of all the signals in the closed-loop system provided thatthe NN is chosen to cover a compact set of sufficiently largesize. For the rest of the derivation, let us introduce the followingtwo technical lemmas.

Lemma 3.1 [50]: Given a three-layer NN

(26)

with being the input vector, andand

being the first-to-second and the second-to-third layer weights,respectively, where they represent the NN approximation of asmooth function over a compact set

(27)

where is the approximation error. Letting and be theestimates of and , respectively, the estimation error canbe expressed as

where ,, and are defined to be the neural weights

estimation errors. with

and the residual term is bounded by


Lemma 3.2: Let and be smooth functionssatisfying

for some class functions , , , and . Then, thereexists class functions and such that

(28)

Proof: Let , whichis obviously a class function. Note that

and is a class function for any classfunction . By the property of the class functions that

, we have

Similarly, we have

(29)

where . Thus, we have (28).Assumption 3.1: Over the compact set , the NN approx-

imation error is bounded. In addition,the ideal NN weights and are bounded, i.e., and

are finite.This is a mild assumption since it always holds for continuous

functions [49]. The control design is aimed at satisfying condi-tions of Lemma 1.1 on the appropriate compact sets. We em-phasize that ISpS disturbance attenuation obtained in this wayallows any bounded and possibly persistent disturbances s.This is slightly more general than the standard disturbance at-tenuation problems [5], [13].

As usual, we use the convention that if is some unknownconstant, then denotes its estimate and denotesthe estimation error. Let be the tracking error.Consider functions derived in (25). Using Lemmas 2.1and 2.2, we have

(30)

for some smooth functions , , and , andsome constant . Since is bounded and depends on

only, is also bounded. This together with (30) leadsto the existence of smooth functions and constantssuch that

(31)

Remark 3.1: The control design presented here does not fixa priori the compact set over which the NN approximationsare employed and allows avoiding the problem of driving thesystem state back to the set , a necessary task to validate theNN approximation [48]. As will be shown in the proof of The-

orem 3.1, for each set of initial values of state variables and NNparameters that belong to a compact set , there is a compactset , whose size is determined by these initial values and de-sign parameters and is independent of NN parameters, such thatthe adaptive controller designed based on NN approximationsover keeps all the closed-loop system state variables withinthis set. Therefore, our design needs not to specify either the set

of initial values or the set over which NN approximationsare employed. The simple fact is that for each set , there ex-ists a compact set over which NN approximations areemployed and the controller is designed to keep the system stateso it does not escape , so that NN approximations are valid forall .

Note that by Assumption 2.2, for each ,is continuous and is bounded away from zero, and hence, itssign is unchanged with respect to its arguments. Without loss ofgenerality, we assume further that s are positive functions.The design procedure requires steps as follows.

Step 1: The time derivative of is

(32)

Consider the following Lyapunov function candidate

where is a design constant.By Assumption 2.2, it is clear that

Using Proposition 2.3 and applying Lemma 3.2 forand , it is verified

that is a smooth p.d.p. function uniformly with respect to .Note that . Using (24) and (32), we have

(33)

From Assumptions 2.2 and 2.4, we have

and (34)


Substituting (34) into (33), we obtain

(35)

By Lemma 2.1, there exist two positive functionsand such that

Using Young’s inequality, we have

(36)

Let be a smooth function such that

Using Lemma 2.1 and Assumption 2.4 and replacing by, we have

(37)

where and are smooth functions.Using (37), the combined use of Cauchy–Schwartz and

Young’s inequalities leads to

(38)

Substituting (36) and (38) into (35) while noting that

we arrive at

(39)

Consider the first desired virtual control law

(40)

where is a smooth positive–definite function that will bespecified at the final step. Note that, in (40), is replaced by

to consider as a function of , , and .Let

(41)

Redefining , adding to, and then sub-tracting it from the right-hand side of (39), we obtain

(42)

Since is formed in part by unknown functions and un-known parameters, it cannot be implemented in practice. Toovercome this difficulty, adaptive control combined with adap-tive NN control are employed. Let

(43)

The desired control becomes

(44)

By smoothness assumption and universal approximation the-orem, and can be approximated by -node three-layer NNs

(45)

where and are the input vectorsand and are the NN approximation errors.


Remark 3.2: Better results can be obtained when moreproperties of the underlying system are exploited [56]. Thecontrol structure (44) is relevant to the idea of using the usualadaptive control to handle the unknown constants and usingadaptive NN control to handle the unknown functions. TheNN approximations (45) of unknown functions are employedseparately for and to keep as an adjustable parameter.In addition, to exploit the system structure and to reflect thetime-varying nature of the underlying system in (2), the NNstructure for in (45) is used with playing therole of interpolator between local controllers of the form

[34], [35].

Remark 3.3: If is chosen such that , we canremove from (44) with the implication that is con-tained in . However, it will be shown that there is an unknownquantity, namely, in (50), arising from the use of NN ap-proximation. Thus, to obtain better performance, we deal with

and the arising unknown quantity as independent parame-ters. An additional benefit in this way is that it allows to bedesigned freely.

Since the optimal NN weights , , , and and the

actual values of and are not known a priori, considerthe following approximation of the ideal virtual control law :

(46)where

(47)

are the estimates of the ideal NN approximations (45), andand , namely the leakage terms, play the role of eliminatingthe square terms resulting from the use of estimated neuralweights instead of unknown ideal ones. Now, adding toand subtracting it from the right-hand side of (42) and using(40) and (41), we have

(48)

Based on Lemma 3.1 and Assumption 2.3, using Young’s in-equality, it is straightforward that

(49)

In view of (43) and (45), the NN parameters and

are independent of control design. Thus, the real numbersatisfying

(50)can be used for control design. Using Young’s inequality, wehave

(51)

Let

(52)


Substituting (49) and (51) into (48) and using (50) and (52),we have

(53)

From (53), we see that the following leakage terms:

(54)

cause to satisfy

(55)

However, because is unknown, is uncertain also. Fortu-nately, it is possible to use the usual adaptive control to handle thisunknown parameter. Furthermore, it is reasonable to incorporate

to by considering as a whole.Togetherwith the previous derivation, this consideration and (46) suggestthe following design for the first virtual control law:

(56)

with being redesigned as

(57)

Since the role of is the same as that of , we state thatthe redesigned virtual control law (56) with given by (54)and given by (57) leads to the same inequality (55) withbeing in place of .

Now, we are ready to design parameter update laws. Considerthe Lyapunov function candidate

(58)

where and are the so-called adaptation gain and adap-tation gain matrices, respectively. Using the redesigned virtualcontrol law (56) and (57) and the corresponding inequality (55)with being in place of , we have

(59)

From (59), it is obvious that the following adaptation laws:

(60)


with being small design parameters give

(61)

Completing the squares, we obtain

(62)

Define the constant

(63)

Substituting (62) and (63) into (61), we have

(64)

This completes the first step of the design procedure. Let. The inductive step is performed as follows.

Step : Suppose that at step , the followingoccurs.

1) A set of virtual control laws comprising of defined by(56) and , , defined by

(65)

where , defined by (57) and ,, defined by

(66)

are the leakage terms, defined by (47) and ,, defined by

(67)

are the outputs of -node NNs whose inputs are

where

(68)

2) A set of adaptation laws comprising of (60) and

(69)

such that the Lyapunov candidates comprising of de-fined in (58) and , , defined by

(70)

satisfy

(71)

for some positive functions , , and someconstant .


We will show that (71) also holds for . In view of(65)–(69), is a function of , , , ,

, and ; hence, the time derivative of is

(72)

where and are given by (68) for .Remark 3.4: Note that and defined by (68) are func-

tions of known variables, and hence, they are computable andcan be used for NN inputs. The introduction of these variablesis to reduce the number of NN inputs, and hence, the size ofNNs are reduced considerably.

Remark 3.5: In view of (65) and (68), , , andare continuous functions of , , and . Thus, whenever

the state of the closed-loop system , , and belongsto the compact set , the variables , , and are,respectively, within appropriate compact sets , ,and . Therefore, the NN parameters in the function ap-proximation (81) do not change for , , and varyingwithin , and hence, the adaptive control can be employed.

Consider the following Lyapunov function candidate:

(73)

From Assumption 2.1, definitions of and , Proposi-tion 2.3, and Lemma 3.2, it is verified that is smooth positivedefinite and proper uniformly with respect to . Using (16) and(72), we have

(74)

where we have defined

(75)

Note that is replaced by to consider asa function of and instead of . By Lemma 2.1, thereexist two positive functions and such that

Using Young’s inequality, we have

(76)

Let be a smooth function such that

By Lemma 2.1, there exist smooth functions andsuch that

(77)

Based on (77), a similar application of theCauchy–Schwartz’s inequality together with Young’sinequality as in (38) results in

(78)

By Assumptions 2.2–2.4, we have the following inequalities:

(79)


where . With the notation (75), collecting(76), (78), and (79) to substitute into (74), we obtain

Thus, choosing the th desired control law as

we arrive at

(80)

Let and

The desired control becomes

Consider the following approximation of by a -nodethree-layer NN:

where is the computableinput of the NN and is the NN approximation error. Asthe optimal values of NN weights are not known a priori,

is uncertain. For this reason, we present the followingapproximation of :

and consider the th virtual control law in the form

where is the leakage term to be designed [see (85)] andis an adaptive element used to handle the unknown constantarising from the use of NN approximation. Addingto the right-hand side of (80), we have

(81)

Using Lemma 3.1 and Assumption 2.3, it is straightforwardthat

Let

(82)Consider the Lyapunov function candidate

(83)

Using (81) and (82), we have

(84)

Let and use as its estimate. From (84),following the same procedure that achieves (59)–(64) in Step 1,


it is known that by applying the leakage term

(85)and the adaptation laws (69) with , the time derivative of

satisfies

(86)

where we have defined

(87)

Substituting (71) with and (24) with into(86), we have

(88)

Referring to (31), we define

(89)

By Young’s inequality, we have

(90)

Let

(91)

Substituting (89)–(91) into (88), it is known that the inductiveconclusion holds at step . This completes the inductive step ofthe design procedure.

Proposition 3.1: For each , there are sets of func-tions and , and a set of constants ,

, such that

(92)with .

Proof: In view of (92), the constructions of s shouldbe based on s. However, s are formed in part by func-tions that depend on s. To avoid the circular argument, thestructure of s is exploited and the proof is inductive back-wardly from to as follows.

1) Initial Step: Referring to (91), let

By Lemma 2.1 and the smoothness of , we have

for some smooth positive functions and .By definition, is independent of . Hence,

is independent of as well. Letbe any monotone nondecreasing function that satisfies

Since is of class , outside the ball, we have

This results in

(93)

Let be any constant satisfying. Such constant exists due to the

continuity of and the compactness of .Obviously, we have

(94)Combining (93) and (94), we have

(95)


Having (95), it is straightforward that

where . This con-cludes that the Proposition 3.1 holds with .

2) Inductive Step: Assume that (92) holds at ,i.e.,

(96)

for some functions , , ,and some constants , . We will showthat (92) also holds at . Let

By Lemma 2.1, we have

for some smooth positive functions and .By definition, , and hence, , is indepen-dent of . Let be any monotone nonde-creasing function that satisfies

Following the same procedure as the Initial Step, we have

(97)

with and beingsome constant. Adding (96) and (97) side-by-side gives the in-ductive conclusion.

At , as . Thus, we have theconclusion of the proposition.

Theorem 3.1: For the uncertain time-varying nonlinearsystem (2), under Assumptions 2.1–2.4 and 3.1, the problem ofISpS disturbance attenuation with output tracking is semiglob-ally practically solvable by partial state feedback.

Proof: Following the presented design procedure, at thefinal step, the actual control input appears and Proposition 3.1holds. Let

(98)

Substituting (92) and (98) into (71) with , we obtain

(99)

Note that, by Assumption 2.1 and by construction of s, wehave

(100)Let

(101)

For the sake of simplicity, let . Substituting(100) into (99) and using notation (101), we have

(102)

Using Proposition 2.3 and applying Lemma 3.2 to the lowerbound functions

and

and upper bound functions

and

it is verified that there exist class functions andsuch that the function satisfies

Thus, the closed-loop system presented by -dynamics satis-fies the assumption of Lemma 1.1 provided that, for all ,


Fig. 1. Compact sets in Theorem 3.1.

remains in some compact set over which the NN approx-imations are employed. According to the good understanding ofapproximation-based control technique in [57], the size of sucha compact set depends on the initial state . Assumethat the system states are initiated in a compact set .Whenever (102) holds, by Lemma 1.1, we have

(103)

with being class function, being class func-tion, and being a class functions of . In view of (103),let us consider the compact set

(104)whose size is determined by , , and . Let

(105)where is a constant. Obviously, depends on the initial state

and design parameters only and we can make being asubset of by adjusting the design parameter large enoughsuch that . This gives rise to remaining infor all . Accordingly, the assumption of Lemma 1.1 issatisfied over the compact set . Consequently, it is concludedthat the closed-loop system with state is semiglobally ISpSwith respect to . Because the boundedness of impliesthe boundedness of s, s and parameter estimations, thefirst requirement of ISpS disturbance attenuation holds. By inte-grating both sides of (99) from zero to , the second requirement(3) follows immediately. Thus, ISpS disturbance attenuation forsystem (2) is solved semiglobally.

Remark 3.6: In Theorem 3.1, there are three compact sets:, , and . The relationship among these compact sets is

illustrated in Fig. 1. Each initial values of system state givesrise to a compact set defined by (105). The NN approxima-tions employed on this compact set and determine the com-pact set defined by (104). In view of (104) and (105), if thedesign parameter is not large enough, then may be strictlycontained in and the system state may escape the set

. This give rise to the invalidation of the derivation in Theorem3.1. However, if is large enough, then is a strict subset of

and, in view of (103), the system state remains in forall . Thus, the derivation in Theorem 3.1 is totally valid.However, since the parameter depends on the NN parameters

that is generally unknown due to the uncertainty of system non-linearities, there would no explicit formula for calculating ,and hence, the choice of may be experimental through simu-lations.

IV. SIMULATION STUDY

To illustrate the proposed neural-based adaptive control de-sign procedure, we consider the output tracking problem for thefollowing system:

(106)

where , , , , , , and are system coefficientsthat depend on the output of the following exogenous dynamicsystem:

(107)

with command input switching between 1 and 1,, and for and

, otherwise. The reference signal is givenby . To illustrate the use of interpolatingoperation, we assume the coefficients , , and to be splineapproximations of the sample points given in the upper part ofTable I. Consider the interpolation functions

(108)

where the centers and the spreads , , are givenin the lower part of Table I. Let “ ” stand for , , or , and let

be approximated by a function of the form

where the optimal values are given in Table II. By virtue ofthe approximating ability of adaptive NNs, we can leave thesevalues to be unknown.

Proposition 4.1: The inequality

(109)

does not hold for any positive–definite and proper function, any class function , any positive–definite

function , and any positive constant , whenever, , and are all positive.

Proof: See the Appendix.Proposition 4.1 indicates that we will fail if we try to

find a CDAL for system (106). However, it is not hard tocheck from Table II that it satisfies Assumption 2.1 with


TABLE ISAMPLE DATA AND CHARACTERISTICS OF INTERPOLATION FUNCTIONS

TABLE IIOPTIMAL COEFFICIENTS FOR INTERPOLATION FUNCTIONS

Fig. 2. Command signal and the output of the exogenous system.

, ,, , , and

.Note that the previous consideration is analytical. The im-

plementation of the developed method does not require theknowledge of Table II, , , and , ,

. In addition, since the system nonlinearities arecontinuous, it is verified that and are lowerbounded by 1, and and are upper boundedon compact sets. Thus, the system (106) satisfies Assump-tion 2.2. Moreover, from (107) and (108), it is verified that

Fig. 3. Disturbances w and w .

Assumption 2.3 is fulfilled. Note that the reference signalsatisfies Assumption 2.4 as well.

Following the proposed design procedure, a neural-basedadaptive controller for system (106) is constructed aftertwo steps. At each step, the NNs , ,

, contain seven hidden nodes, i.e., .The activation function is taken aswith . The design parameters of the previous controllerare , ,

, , , and

. The NN weights and are all initialized

to 0. The initial values of state variables andexogenous variables are and ,respectively.

The simulation results in Figs. 1–3 demonstrate the effec-tiveness of the developed method. Although the exogenous dy-namics are very fast in comparison with those of the referencesignal and the switching command signal, the tracking perfor-mance is well obtained. The ISpS property of the closed-loopsystem has been illustrated. At the time instant 20 s, thedisturbances reach its maximum value. Accord-ingly, more control effort is required to maintain output tracking(see Figs. 4 and 5). After 20 s, the disturbance decreases andsmaller control effort is presented to maintain output tracking. Ithas been illustrated that the tracking performance is robust withrespect to the disturbance in the sense that the tracking error ismaintained in a small region while the disturbance varies overa wide range. The boundedness of other state variables , ,and and control signal are shown in Figs. 5 and 6.

V. CONCLUSION

This paper has offered a solution for one of the long standingproblems of nonlinear control. The disturbance attenuation isachieved in the sense of the ISpS notion. This notion was in-troduced recently in [31] wherein the stabilization purpose wasobtained by full state feedback design. The developed methodallows output tracking for systems with unmeasured dynamicsthat does not possess a CDAL. Although the ISpS disturbance


Fig. 4. Tracking performance and error time diagram.

Fig. 5. Control input.

attenuation is achieved for the overall system, the design methoddoes not require that the -subsystem must be ISpS with re-spect to disturbance. Only the mild Assumption 2.1, which iseasy to check, is needed. It has been proven that if the distur-bance is bounded, the proposed design method guarantees thesemiglobal uniform ultimate boundedness of the closed-loopsignals and the system output tracks the reference signal withthe tracking error staying within a bounded region that can bealtered by design parameters.

APPENDIX

PROOF OF PROPOSITION 4.1

Suppose that (109) holds for some p.d.p. function ,some class function , some positive–definite func-tion , and some positive constant at some value of

Fig. 6. States z , z , and x .

such that , , and are all positive. Setting, (109) becomes

(110)

Let and .It is easy to verify that (110) is equivalent to

(111)


Let

From (111), we have

(112)

Let . It is not hard to derive from(112) that

(113)

Replacing the functions and in (113) by theiractual definitions, we have

(114)

Integrating both sides of (114) with respect to yields

(115)

As is a bounded function, (115) implies thatis bounded. This is a contradiction.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their valuable comments that improved the quality and thepresentation of this paper.

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Shuzhi Sam Ge (S’90–M’92–SM’00–F’06) re-ceived the B.Sc. degree in control engineering fromBeijing University of Aeronautics and Astronautics(BUAA), Beijing, China, in 1986 and the Ph.D.degree and the Diploma of Imperial College (DIC),both in robotics, from the Imperial College ofScience, Technology and Medicine, London, U.K.,in 1993.

He is a Full Professor at the Department of Elec-trical and Computer Engineering, the National Uni-versity of Singapore, Singapore. He is the Head of

Social Robotics Lab, Interactive Digital Media Institute, the National Univer-sity of Singapore. He is the founding Director of Personal e-Motion Pte Ltd. Heprovides technical consultation to industrial and government agencies. He hascoauthored three books: Adaptive Neural Network Control of Robotic Manip-ulators (Singapore: World Scientific, 1998), Stable Adaptive Neural NetworkControl (New York: Kluwer, 2001) and Switched Linear Systems: Control andDesign (New York: Springer-Verlag, 2005). He edited a book Autonomous Mo-bile Robots: Sensing, Control, Decision Making and Applications (New York:Taylor & Francis, 2006) and wrote over 300 international journal and confer-ence papers. His current research interests include social robotics, multimediafusion, adaptive control, and intelligent systems.

Dr. Ge has served as an Associate Editor for a number of flagship jour-nals including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, theIEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, the IEEETRANSACTIONS ON NEURAL NETWORKS, and Automatica. He also serves as anEditor of the Taylor & Francis Automation and Control Engineering Series. Heis an elected member of Board of Governors, IEEE Control Systems Society.

Thanh-Trung Han received the B.Sc. degree inautomatic control from Hanoi University of Tech-nology, Hanoi, Vietnam, in 2002. Currently, heis working towards the Ph.D. degree in automaticcontrol at the Department of Electrical and ComputerEngineering, the National University of Singapore,Singapore.

From 2002 to 2004, he was a Lecturer at the De-partment of Automatic Control, Hanoi University ofTechnology. He is currently involved in research onqualitative theory of switched systems and adaptive

and intelligent control of switched and time-delay systems. His research inter-ests are in the field of control and systems theory, with emphasis on stabilityanalysis and control of dynamical systems.

Semiglobal ISpS Disturbance Attenuation With Output Tracking via Direct Adaptive Design

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