Adaptive Controller Design For Tracking And Disturbance

1066 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 8, AUGUST 1998

Adaptive Controller Design for Trackingand Disturbance Attenuation in Parametric

Strict-Feedback Nonlinear SystemsZigang Pan,Member, IEEE, and Tamer Basar, Fellow, IEEE

Abstract—The authors develop a systematic procedure forobtaining robust adaptive controllers that achieve asymptotictracking and disturbance attenuation for a class of nonlinearsystems that are described in the parametric strict-feedback formand are subject to additional exogenous disturbance inputs. Theirapproach to adaptive control is performance-based, where theobjective for the controller design is not only to find an adaptivecontroller, but also to construct an appropriate cost functional,compatible with desired asymptotic tracking and disturbanceattenuation specifications, with respect to which the adaptivecontroller is “worst case optimal.” In this respect, they also departfrom the standard worst case (robust) controller design paradigmwhere the performance index is fixedpriori .

Three main ingredients of the paper are the backsteppingmethodology, worst case identification schemes, and singularperturbations analysis. Under full state measurements, closed-form expressions have been obtained for an adaptive controllerand the corresponding value function, where the latter satisfies aHamilton–Jacobi–Isaacs equation (or inequality) associated withthe underlying cost function, thereby leading to satisfaction of adissipation inequality for the former. An important by-product ofthe analysis is the finding that the adaptive controllers that meetthe dual specifications of asymptotic tracking and disturbanceattenuation are generally not certainty-equivalent, but are asymp-totically so as the measure quantifying the designer’s confidencein the parameter estimate goes to infinity. To illustrate the mainresults, the authors include a numerical example involving athird-order system.

Index Terms—Adaptive control, backstepping, disturbance at-tenuation, nonlinear systems, tracking.

I. INTRODUCTION

T HE DESIGN of adaptive controllers for parametric un-certain linear or nonlinear systems has been one of

the most researched topics in control theory for the pasttwo decades. For linear systems, adaptive controller designshave been centered on the certainty-equivalence principle [1],[2], where the controller structure is borrowed intact from a

Manuscript received April 19, 1996; revised April 25, 1997. Recommendedby Associate Editor, A. A. Stoorvogel. This work was supported in part bythe U.S. Department of Energy under Grant DOE-DEFG-02-94ER13939 andin part by the National Science Foundation under Grant NSF-ECS-93-12807.

Z. Pan is with the Department of Electrical Engineering, PolytechnicUniversity, Brooklyn, NY 11201 USA.

T. Basar is with the Department of Electrical and Computer Engineeringand Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801USA.

Publisher Item Identifier S 0018-9286(98)05811-5.

design with known parameter values and implemented usingidentified values for these parameters. Many success storieshave been reported in achieving global boundedness of internalstates and asymptotic performance of the system output usingthis approach in both stochastic and deterministic (noise-free)settings [3]–[8]. However, establishment of counterparts ofthese results for general nonlinear systems, using the certainty-equivalence approach, has been quite elusive, with successesreported initially mainly for the case when the nonlinearitiessatisfy some global linear growth conditions [9]. For nonlinearsystems with severe nonlinearities, a breakthrough took placeafter the much celebrated characterization of the class offeedback linearizable systems[10], [11]. A pioneering workin this area has been [12], which presented a systematicdesign paradigm based on the novelintegrator backsteppingmethod to globally adaptively stabilize a subclass of feedbacklinearizable systems described inparametric strict-feedbackform. This approach was further refined in [13], where over-parameterization was removed and was generalized in [14] toa larger class of nonlinear systems. It has also been appliedto decentralized systems [15], as well as to nonholonomicnonlinear systems [16]. For an up-to-date list of referenceson the development of the backstepping approach, we referthe reader to [17].

Intuitively, an adaptive controller design uses (and generatesonline) more information on the system uncertainties than non-adaptive designs and therefore should lead to controllers withbetter robustness properties. In spite of this, many adaptivecontrollers have been shown to exhibit undesirable robustnessproperties [18]–[20].Nonrobustnessof an adaptive controllercould lead to inferior transient behavior and burstiness inthe closed-loop system under external disturbance inputs. Toovercome these difficulties, various modifications to earlierdesigns have been proposed to robustify the adaptive controllerdesign, for both linear and nonlinear systems [21]–[23], butthese still fall short of addressing directly the disturbanceattenuation property for the adaptive controller design.

General objectives of a robust adaptive controller designare (and should be) to improve transient performance, attaina finite (acceptable) level of disturbance attenuation, and sus-tain unmodeled dynamics. These are precisely the objectivesthat have motivated the study of the -optimal controlproblem for linear systems (with known parameters), whichhas more recently been extended to the nonlinear framework

0018–9286/98$10.00 1998 IEEE

PAN AND BASAR: ADAPTIVE CONTROLLER DESIGN 1067

[24]–[31], motivated by the differential game approach [32].It would therefore be natural to cast a robust adaptive controlproblem in the framework of nonlinear optimal control,where specific measures of asymptotic tracking, transientbehavior, and disturbance attenuation can all be incorporatedinto a single cost functional. This has in fact been donerecently in the context of parameter identification (for lin-ear and nonlinear systems) [33], [34]—a study that has ledto a new class of robust identifiers that guarantee desiredachievable levels of disturbance attenuation. The structure ofthese worst case identifiers resembles that of a least squaresidentifier, except for the presence of additional state esti-mate dynamics and an extra negative-definite term in thedifferential equation for the error covariance matrix. The factthat these have been obtained as the result of aworst caseoptimization process, leading to satisfaction of a dissipationinequality, makes them an ideal candidate to use in any cost-optimization-based adaptive controller design—which is whatwe do here.

Accordingly, this paper studies robust adaptive controllerdesign using the worst case design methodology. To obtainexplicit formulas for the controller, we consider the special(but important) class of nonlinear systems that is described inthe parametric strict-feedback form, which we further take tobe subject to additional affine exogenous disturbance inputs.The design specifications for the robust adaptive controller areasymptotic tracking of a given reference signal and achieve-ment of a desired level of disturbance attenuation over theentire time interval which would then translate intomuch improved transient response. We present a systematicdesign paradigm, which leads to robust adaptive controllerswith the following three appealing features: 1) asymptoticconvergence to certainty-equivalent controllers as the iden-tification error covariance approaches zero; 2) utilization ofrobust parameter identification schemes as basic buildingblocks; 3) attenuation of exogenous disturbance inputs to anydesired level of performance over the entire time interval

Departing from the standard robust control setup, our ob-jective for the controller design includes the characterizationof an appropriate cost functional, compatible with the givenasymptotic tracking and disturbance attenuation specifications,under which the controller designed satisfies a dissipationinequality, or equivalently, ensures a zero upper value fora particular zero-sum differential game. The cost functionincludes a positive-definite quadratic weighting on the trackingerror (and possibly also weighting on internal states) and anegative-definite quadratic weighting on the exogenous inputs,whose ratio reflects the desired disturbance attenuation levelfor the closed-loop system. The freedom in the choice of thecost function allows us to extend the backstepping methodol-ogy of [12] and apply it to this robust adaptive control problem.We make use of the identifier designs obtained in [33] as a firststep to the robust adaptive control design. Because of the two-time-scale structure of the worst case identifier, the controllerdesign is based on the quasi-steady-state dynamics of theidentifier dynamics, which are precisely the dynamics underthe less restrictive measurement scheme that allows additional

access to the derivative of the state variables for feedback. Inthis case, we show that one of the causes of the nonrobustnessof a certainty-equivalent controller design is the fact that,in the closed-loop, the disturbance inputs enter the systemthrough the differentiation of the parameter estimates, alsoknown as theswapping termin the adaptive control literature.A singular perturbations analysis is then naturally employedto establish the robustness of the adaptive controller, when thetwo-time-scale identifier dynamics are utilized. The closed-loop system admits a value function that can be expressedin closed form and satisfies a Hamilton–Jacobi–Isaacs (HJI)equation (or inequality) associated with the underlying costfunction, thereby guaranteeing a desired level of performancefor the adaptive controller. Three main ingredients of thepaper that pervade the derivation and the analyses are thebackstepping methodology, worst case identification schemes,and singular perturbations analysis. A numerical example isincluded in Section IV to illustrate the theoretical findingsof the paper. With minor modifications, the design paradigmpresented here can be applied to a class of minimum phaseparametric uncertain linear systems, as briefly discussed in theconclusions section.

The problem of asymptotic tracking and disturbance atten-uation for parametric strict-feedback nonlinear systems hasbeen addressed also in the recent book [35]. However, theapproach adopted there is considerably different from theone developed here and is related to the tuning functionapproach introduced earlier in [13]. Here, we make use ofthe robust identification schemes derived in [33], and as aresult of this combined identification and control design, theadaptive controller obtained is only asymptotically certaintyequivalent, and it requires a smaller control magnitude toachieve the same level of disturbance attenuation. On the topicof transient performance, we should mention the two recentreferences [36] and [37] which have focused on the analysisof existing adaptive control algorithms, rather than on thedesign of controllers with respect to an optimality criterion,which puts direct weight on the transient performance of theclosed-loop system.

The balance of the paper is organized as follows. In thenext section, we provide a precise formulation for the non-linear adaptive tracking and disturbance attenuation problemto be studied in the paper. In Section III, we first presenta backstepping design tool for asymptotically tracking andsimultaneously disturbance attenuating controllers for a two-level nonlinear system; then, we make repeated use of thisbackstepping tool to design robust adaptive controllers usingthe robust identifiers derived earlier in [33] and establish theirrobustness with respect to exogenous disturbances. A numeri-cal example that involves a third-order nonlinear system witha single unknown parameter is presented in Section IV, whichclearly illustrates many appealing features of the designedcontroller. The paper ends with the concluding remarks ofSection V and two Appendixes, the first of which presentssome derivations that lead to the identifier dynamics used inthe adaptive controller design, based on the results of [33]; thesecond one provides relevant expressions for a robust adaptivecontroller designed for a modified version of the main model.


II. PROBLEM FORMULATION

Motivated by the results and formulation of [12] (and also[23]), we consider in this paper a class of single-input/single-output (SISO) nonlinear systems, in the followingnoise-proneparametric strict-feedback form:

(1a)...

...

(1b)

(1c)

(1d)

Here, is the -dimensional state vector,with initial state being ; is the scalar control input;

is the -dimensional exogenous input(disturbance) where is of dimension ; isthe scalar output; is an -dimensional vectorof unknown parameters of the system whereis of dimension

; and the nonlinear functions andare known and satisfy the triangular structure

depicted in (1). Note that, as compared with the parametricstrict-feedback form introduced in [12], the above systemfurther incorporates additional additive disturbance inputs,where the nonlinear functions multiplying the disturbanceterms are also in triangular form. We should further note thatthe above form of the nonlinear system is block diagonalin terms of the disturbance and the parameter vector

—this specific structure being essential for the applicabilityof the backstepping design procedure for the derivation of anadaptive controller when the parameter vectoris unknown.In order to bring an original system into the noise-proneparametric strict-feedback form as above, one may have totreat any single parameter that enters (1) at different integrationstages as different parameters, which would then clearly leadto overparameterization of the plant; one may also have to treatany single disturbance that enters the dynamics (1) at differentintegration stages as independent disturbances, which againwould lead to an additional level of conservatism.

For the nonlinear system (1), we make the following twobasic assumptions as a starting point of our study.

Assumption A1:The nonlinear functions and aretimes continuously differentiable in all their arguments

(or simply are ), The nonlinear functionsand are in all their arguments.Assumption A2:There exists a positive constantsuch that

Assumption A1guarantees the smoothness of the nonlinearfunctions and which is required forour design procedure.Assumption A2requires the disturbanceto enter every channel of the nonlinear system (1), whichis needed to avoid singularity in the identification of theparameters when full state measurement is available.

Associated with (1), we are given a reference trajectorythat the output of the system,, is to track. We make the

following smoothness assumptions on this reference trajectory.Assumption A3:The reference trajectory is times con-

tinuously differentiable, where the signal and the derivativesare uniformly bounded, i.e., for some

and

with The signal and its first derivatives areavailable for control design.

For future reference, we denote the vectorby

The uncertainty, both intrinsic as well as exogenous tothe system, is the triple that is the initialstate, the true values of the unknown parameters, and thedriving disturbance. Since we are interested in the worst caseperformance, the exogenous input can be taken to beany open-loop time function, as in the case of -optimalcontrol problems. In view of the results of [33], we taketo belong to some subset of all uniformly boundedtime functions, and the uncertainty triple tobelong to which is taken as an appropriate subset of

to be specified later.The objective of the controller design is to force the

output to track the reference signal asymptotically, whileattenuating the effect of the exogenous input (disturbance)the initial condition and the unknown parameter vector

A precise statement of this objective is now given below.Definition II.1: A controller is said to beasymptotically

tracking with disturbance attenuation level if there existnonnegative functions and such thatfor all the following dissipation inequality holds:

(2)

Here, denotes the Euclidean norm, isthe initial guess for the unknown parameters, and thedimensional matrix is the quadratic weighting of errorbetween the true value ofand the initial guess quantifyingour level of confidence in the initial guess.

We will take to be of block diagonal structure:

block diagonal

where is of dimensions and corresponds to ourlevel of confidence in the initial guess

An important point to note here is that in the performancefunction (2), there is no weighting on the control input. Hence,any attenuation level can be achieved by allowing the


magnitude of the control input to increase asdecreases. Thesmaller the value of is, the better will be the disturbancerejection property, but at the expense of a larger controleffort. Any controller that achieves the above objective hasthe following property, for any :

Hence, the norm of the tracking error is always smaller thantimes the norm of the disturbance input plus a constant

that depends only on the initial states of the system.We will consider the class ofstate trajectory-feedback

controllers where is piecewise continuousin and Lipschitz continuous in the state trajectory Wewill refer to this measurement scheme underlying the givencontroller as theunknown parameter full state information(UPFSI), to differentiate it from the one where also thetrajectory of the derivative of is available for controlpurposes—which we will refer to as theunknown parameterfull state with derivative information(UPFSDI); we will haveoccasion to use this measurement scheme at a crucial interme-diate step in the derivation of the robust adaptive controller.For a more in-depth discussion of the controller design underthese two measurement schemes, as well as a third (moreexpanded) measurement scheme that allows the controller tohave access to the true value of the parameter vector—calledknown parameter full state information(KPFSI)—we refer thereader to [38].

We now conclude this section by introducing some notationand convention that we will adopt throughout the paper.The vector will denote some transformedstate variables; the vector will denote theestimate of the parameter vectorwith being the estimateof the vector is then theestimation error any function symbol with an “over bar”will denote a function defined in terms of the transformed statevariables, such as denoting the equivalent form of function(of ) in terms of transformed state variableFor any matrix

the vector is formed by stacking up its column vectors.For any functional depending on a parameter vectordenotes the partial derivative which is a row vector.

III. A DAPTIVE CONTROLLER DESIGN

We first present a backstepping lemma that will be usedrepeatedly in the controller design stage. The intuition behindthis result is that if a desired disturbance attenuation levelcan be achieved through a virtual control input, then the sameattenuation level can be achieved using the actual controllaw which generates the virtual control input through a first-order dynamics. As compared with the existing backsteppingalgorithms, the lemma below provides a recursive solution

for achieving disturbance attenuation without the necessityof increasing the guaranteed attenuation level at each stepof recursion and takes advantage of the achieved disturbanceattenuation property in the original design. Therefore, thisalgorithm leads to a controller with smaller controller gain,while guaranteeing the same level of robustness with respectto the disturbance.

Lemma III.1: Consider a noise perturbed nonlinear systemgiven by

(3a)

(3b)

where is dimensional, is scalar, is dimensional,and the functions and are smooth with

for any Suppose that with pickedas the virtual control input to the subsystem dynamics (3a),and with some arbitrary but fixed nonnegative-definitefunction, there exists a control law such that thefollowing HJI inequality is satisfied by a nonnegative-definitevalue function :

(4)

Then, there exists a control law and a nonnegative-definitevalue function for the overall system suchthat the following HJI inequality is satisfied for any desirednonnegative-definite where

(5)

Proof: First we observe that satisfies the HJI inequal-ity (4) if and only if it satisfies, along with the dynamics (3a),the following inequality:

Now, to prove the lemma, we introduce a new disturbanceand a new state variable

In terms of these quantities, the system


dynamics can be expressed as follows:

Introduce a candidate value function for the overall system

Differentiation of this function yields the inequality

where the control is given by

By the same observation as at the beginning of the proofabove, and some manipulations, we obtain that the inequalityfor above, evaluated along the full system dynamics, isequivalent to the HJI inequality (5). This completes the proofof the lemma.

We now proceed to the derivation of the adaptive controllerdesign, by first obtaining a (worst case) parameter identifierfor (using the framework and results of [33]) and thendeveloping the control law using a backstepping procedure.

Worst case identifiers for in (1), with guaranteed dis-turbance attenuation bounds, can be derived by making useof the general approach and results of [33], as outlined inAppendix A. These identifiers are parameterized in terms of

nonnegative-definite matrices whereis of dimensions and may depend on the variables

, i.e.,

Note that the dependence of the’s on the internal statesof the identifier is in a lower triangular form. Now, in termsof these matrices and a small design parameterthe -identifier for to be denoted by is generated by the

following1:

(6a)

(6b)

(6c)

where the functions are defined by

(7)

For small values of the identifier (6) exhibits a two-time-scale behavior. The coordinate transformation that yieldsthe standard singularly perturbed form is given by

The identifier dynamicsin this coordinate system is given by

(8a)

(8b)

(8c)

For ease of reference, we introduce the notationand The quasi-steady-state

behavior of the identifier is given by the following dynamics,where 2:

(9a)

(9b)

We note here that the quasi-steady-state dynamics corre-spond precisely to the dynamics of the identifier obtainedunder the UPFSDI measurement scheme where the derivativeof the state variables are also available for feedback; seeAppendix A. The specific structure of the limiting identifier(9) enables us to employ backstepping tools in the derivationof a controller for the overall system that uses derivative in-formation, i.e., a robust adaptive controller under the UPFSDImeasurement scheme. As it will become clear shortly, in theactual implementation, we will retain the structure of the robustcontrol law but will replace the identifier part with the UPFSIidentifier (6). This leads to the desired UPFSI adaptive controllaw, whose robustness with respect to exogenous disturbancesis then proven using a singular perturbations analysis underadditionalAssumptions A4 and A5,to be introduced shortly.

Using the UPFSDI identifier (9), the identification errorobeys the following dynamics:

where

1This identifier was derived in [33] as a limiting case of another full-order identifier for a noise-perturbed measurement scheme, using singularperturbations analysis. It was called there approximate noise-perturbed full-state information (NPFSI) identifier.

2This is obtained by setting the LHS of (8c) tozero, solving for ei; andsubstituting it into (8a) by also making use of the original system dynamics.


Introduce a candidate value function associated with theidentifier

(10)

whose derivative is given by

where The left-hand side (LHS) of (2)can then be equivalently written as follows, by adding theidentically zero function :

Thus, the attenuation problem with respect tohas beeneffectively converted to an attenuation problem with respectto the equivalent disturbance In terms of the systemdynamics (1) and the identifier (9) satisfy, for

(11a)

(11b)

(11c)

Thus settling the issue of design of the robust identifier, wenow move on to controller design underAssumptions A1–A3,which involves steps of integrator backstepping by repeatedapplication of Lemma III.1. Let be any fixed, desiredlevel of disturbance attenuation.

Step 1: At this step, we consider the combined statesas and state as in the statement of LemmaIII.1. We choose as the value function for underthe virtual control input which allows the desired virtualcontrol law to be chosen as zero. For notational consistency,we again let Following the steps of Lemma III.1, thederivative of then satisfies

Choose a nonnegative function which isagain a design parameter. We introduce

Then, the dynamics for can be rewritten as

Introduce a value function

the derivative of which, along the dynamics of is

This completes the first step of the design.Note that at this stage, the nonlinear functions anddo not depend on the covariance statesHence, if

the adaptive controller is a certainty-equivalent controller. But,for the general case, this certainty equivalence property willonly hold asymptotically as as we willshortly see.

Step : Assume that from the previous steps wehave the following structures:

(12a)

(12b)

(12c)

(12d)

(12e)

(12f)

(12g)

(12h)


At this step, we first evaluate the dynamics of

Following the procedure of Lemma III.1, we will constructa value function to include the state Fix a design parameter

which is a nonnegative function, according to

The dynamics of can be rewritten as

where we have introduced the following functions:

(13a)

(13b)

(13c)

(13d)

(13e)

The functions and depend on the variables—a prop-

erty that is consistent with the corresponding hypotheses in(12).

Introduce the value function for this step as

After some algebraic manipulations, we obtain

and this completes theth design step. Again, the defini-tions above are consistent with the corresponding inductionhypotheses in (12), and the backstepping process can becontinued from to using the same methodology.At step the actual control appears explicitly in the stateequation for which allows us to complete the controllerdesign process.

Remark III.1: The above derivation quite naturally takesinto account the disturbance inputs that enter the dynamicsof through differentiation with respect to the parameterestimates —known in the adaptive control liter-ature asswapping terms. The standard certainty-equivalentcontroller design, however, does not take the presence ofthese terms into account. A significance of this fact is thatthese disturbance inputs are amplified by nonlinear gains

which gener-ally disrupt the disturbance attenuation property of the systemwhen ’s are not zero—an observation that sheds light on thepoor transient behavior of certainty-equivalent controllers.

Step : First choose a design parameter which is anonnegative function, according to

Define the functions and as in (13),with the index set to Then, the state dynamics for aregiven by

This can further be rewritten as

with the control law defined as

(14)

For this final step, the value function for the closed-loopsystem is given by


whose derivative can be evaluated to be

This completes the backstepping design process for theUPFSDI case. The closed-loop system under the control law(14) and the identifier (9) is now described collectively by thefollowing set of differential equations:

(15a)

(15b)

(15c)...

...

(15d)

(15e)

(15f)

(15g)

(15h)

(15i)

Introduce the vector

(16)

In terms of this notation, the set of (15) can be written incompact form as

(17)

where the nonlinear functions and are defined accord-ingly. The derivative of is given by

and hence satisfies the HJI equation

(18)

where This implies thatthe control law (14) is asymptotically tracking with disturbanceattenuation level

To remove the dependence of the identifier on the derivativeinformation of the state variables we will supply the con-troller (14) with the and that are generatedby the UPFSI identifier (6). To guarantee the robustness ofthe resulting closed-loop system, we invoke the following twoassumptions.

Assumption A4:There exists a positive constant suchthat for some

where the matrices ’s and ’s are functions ofand the inequalities

hold for all values of these variables.Assumption A5:There exists a constant such that

for all values of their arguments.Using the special structure of the matrices’s, as given in

A4, the identifier dynamics can be expressed as

(19a)

(19b)

(19c)

This then makes the identifier prescribed here correspondto the worst case identifier obtained in [33, Section 7],for the time-varying parameter case.

Remark III.2: Assumption A4on the specific form of thedesign matrices ’s is introduced above for two reasons.On the one hand, with this specific structure of’s, thecovariance matrices ’s are bounded away from zero (frombelow) uniformly. Therefore, it is no longer necessary to usecovariance resetting to enable the identifier to track the slowlytime-varying parameters. This benefit is also evident from thefact that the proposed identifier structure corresponds to theworst case identifier for the time-varying parameter case, asin [33]. Furthermore, the convergence rate of the parameterestimates is guaranteed to be exponential as long as the’sare uniformly upper bounded (which is the persistency ofexcitation condition in this context), which is a critical stepin the proof of the main result of this section, as we willshortly see. On the other hand, when the design matrices’sare positive definite, it then mandates persistent excitationsin the measurements to keep the’s uniformly boundedfrom above. In fact, a simple choice of letting ’s be time-invariant positive-definite matrices exposes thedynamics topossible finite escapes in the absence of sufficient excitation.Accordingly, the positive-definite components, ’s, of ’sare scaled by ’s. Consequently, the covariance matriceswill be bounded above for an arbitrary level of excitation,even though the precise upper bounds will in general dependon the level of excitation in the closed-loop system.


In view of the corresponding result of [33, Th. 9], weconsider the following set of admissible uncertainty triples,for an arbitrary positive constant :

(20)

We can employ a singular perturbation analysis to establishthe following robustness property of the proposed adaptivecontroller under the above class of uncertainties.

Theorem III.1: Consider the nonlinear system (1), and letbe fixed. LetAssumptions A1–A5hold, and the set

be defined as in (20) for some Then we have thefollowing.

1) There exists a positive scalar such that for allthe control law defined by (14), with identi-

fier (19), achieves asymptotic tracking with disturbanceattenuation level for any uncertainty triple in the set

Furthermore, the closed-loop signals andare uniformly bounded on

2) For any uncertainty triple in the set such thatthe expanded state vector con-

verges to zero as for any

Proof: Introduce the value functions and definedexactly as before. Let again denote the expanded state vectordefined as in (16), and let nonlinear functionsand bedefined exactly as in (17). In terms of state variablesandthe closed-loop system under the control law (14) and identifier(19) can be expressed as

where

......

...

.... ..

......

It is not necessary here to explicitly write down the analyticforms of the functions and which turn out tobe quite complicated. It is important to note, however, thatthese nonlinear functions all belong to and further satisfythe following two relationships:

These two algebraic relationships are obtained using the factthat the slow manifold of the above singularly perturbeddynamics is exactly the dynamics (17). These nonlinear sin-gularly perturbed dynamics are linear in the fast state variable

and the disturbance input Time-scale decompositionand robust control of this class of systems has been studiedextensively in the earlier paper [39]. Motivated by the results

of that paper, we introduce here another function,associated with the fast dynamics:

Let the overall value function be Then,the derivative of along the closed-loop trajectory can bewritten as (after some lengthy algebraic manipulations)

where the partial differential equation (18) has been used inthe derivation, and the function is defined as

Consider the following time-varying set:

(21)

where the positive scalar is sufficiently large such that

and

The significance of the first inequality above will be seenshortly.

The first statement of the theorem can be established byproving the following two statements.

1) If the state of the closed-loop system, be-longs to for any then the controllerachieves asymptotic tracking with disturbance attenu-ation level for any uncertainty triple in and

is uniformly bounded, for sufficiently smallvalues of

2) If the closed-loop system starts in then itbelongs to for any and for sufficientlysmall values of


Toward proving statement 1), suppose thatfor any which implies that

Then, there exists a constant such that forall since can be expressed as a continuous func-tion of and which themselves are uniformly bounded.Under the workingAssumption A3,corresponding to anyuncertainty triple in there exists a constant suchthat Furthermore, the constant can be chosento be independent of the parameterbecause of the diagonalstructure of the dynamics for and Assumption A2.Becauseof the uniform boundedness of the covariance matrices

’s are strictly larger than or equal to, in matrix sense, thesolutions of the following Riccati differential equations:

where is a positive constant such that

Hence, there exists a positive constant which generallydepends on the value of the constantsuch that

The covariance matrices ’s are then upper and lowerbounded in the following fashion:

over the entire time intervalUtilizing the above bounds on and ’s, there further

exist positive scalar constants such that thefollowing growth conditions are satisfied on for any

and any uncertainty triple in :

and, for each

where denotes any continuous nonlinear function, of nodirect interest to us in this derivation. These inequalities implythat, for sufficiently small values of

for any uncertainty triple in the setLet functions and be such that

Then the performance inequality (2) is satisfied forsufficiently small, and this establishes statement 1).

For statement 2), let us fix any and anysuch that Then the result will follow if we can

prove that for any uncertainty triple in By thedefinition of the function and the fact that wehave either or In the former case,we have the following set of inequalities, under the working


Assumptions A4 and A5:

which yields the nonpositiveness of the derivative of Inthe latter case, the uniform bounds onand imply that

and further for sufficiently smallvalues of This establishes statement 2) and therefore thefirst statement of the theorem.

For any uncertainty triple in such that the

signals and allbelong to for sufficiently small values of Furthermore,their derivatives are uniformly bounded for any fixed valueof Hence, these signals converge to zero asymptotically.This verifies the second statement of the theorem, and thuscompletes its proof.

Remark III.3: We note that the controller (14) depends onthe covariance information of the identifier and therefore isnot a certainty-equivalent controller. Yet, it is asymptoticallyequivalent to a controller designed with fixed parameters, as

On the other hand, with the designparameter matrices ’s chosen as inA4, the error covariancematrices ’s become uniformly bounded from below by aconstant positive-definite matrix when the system stateisuniformly bounded from above. The controller (14) will notconverge to a certainty equivalent controller in this case. Whenthe covariance matrices ’s, as well as their derivatives,are small (in the spectral radius sense), the behavior of thecontroller is close to that of the certainty-equivalent one.

Remark III.4: Consider the class of nonlinear systems

(22a)...

...

(22b)

(22c)

(22d)

where the same disturbanceenters all subsystems, but thenonlinear functions and have only as their arguments.For the results of this section to be directly applicable tothis class of systems, overparameterization and overconser-vativeness have to be introduced in order to transform thesystem into the noise-prone parametric strict-feedback from(1). This can be avoided, however, by utilizing the specialstructure of the system, where the functions and

depend only on and are To avoid

singularity in identification, we must assume that the set ofvectors are linearly independent for allThe parameter identifier for this class of systems dependsonly on the variable which then allows for a backsteppingprocedure similar to the one described in Lemma III.1 to beemployed in the design of an asymptotically tracking anddisturbance attenuating controller. See Appendix B for thedesign equations for this class of systems, and Section IV fora numerical example involving this type of a nonlinear system.

IV. A N EXAMPLE

To illustrate the design procedure developed in the previoussections, we consider a third-order nonlinear system withan unknown scalar parameter To further corroborate thestatement of Remark III.4, the system is taken to be of thespecial form (22), for which the relevant design equations canbe found in Appendix B

For this system, the reference signal is generated by athird-order internal model with transfer function

In state space, the reference model can be represented by

where is a command signal, taken to be a step function,

The initial states for the plant and reference model are

The true value of the parameteris taken to be 1. The desiredattenuation level with respect to the disturbanceis takento be 3.

KPFSI Case: As a benchmark for comparison, wefirst designed a robust disturbance attenuating controller

which uses full state measurements andfull knowledge of the parameter This was done byemploying recursively the backstepping procedure describedin Lemma III.1, and by picking the design parameterwhich is the guaranteed level of disturbance attenuation.Due to its complexity, we are not including here the explicitform of the controller.


UPFSDI Case: In this case, the parameteris unknown,but the derivative of the state is available. We took the initialguess to be with a confidence level of Thedesign parameter was taken to be zero. Note that this choiceof the parameter corresponds to a standard least squaresidentifier. The identifier for the system in this case is given by

where the initial states are set to andUsing the expressions given in Appendix B, we arrived

at an asymptotically tracking controller with disturbanceattenuation level 3.

UPFSI Case: The parameter is again unknown. Theonly measurement available is the state history Inthis case, the design parameterwas chosen, according toAssumption A4, to be where the parameters

and were both fixed at 0.1. Since the system outputis to asymptotically track 1, there is enough excitation

in the system to keep from escaping to infinity for alarge class of disturbance signals. When the outputtracks 1exactly, the covariance will converge tobut not zero. We took the controller as designed in theUPFSDI case. The additional design parameterwas fixedat 0.05. In this case, the identifier is given by, after somealgebraic manipulations

where the initial conditions are and

Fig. 1. State trajectory of the reference model.

The design process for the above three controllers consistedof programming in Mathematica, to arrive at the controllerformula, followed by implementation in Matlab codes, andfinally simulation of the closed-loop system using Simulink.The state trajectory of the reference model can be seen inFig. 1.

We simulated the closed-loop system, under each of thethree controllers, against two sets of disturbance inputs. First,the disturbances were set to zero, to demonstrate the commandfollowing property of the controller. Next, the disturbanceinputs were picked as follows:

Band limited white noise signal

with power 0.01 and sample rate 5 Hz

The system response under the KPFSI controller is depictedin Fig. 2. When the disturbances are set to zero, the outputtracks the reference signal exactly (the difference is in theorder of 10 ). Under sinusoidal and white noise disturbanceinputs, the system output still follows the reference signal,with the tracking error bounded by 0.4.

The system response under the UPFSDI controller is de-picted in Fig. 3. Despite the large initial error of the parameterestimate, the system has a fair transient response which settlesin less than 5 s. The parameter estimateconverges to thetrue value faster in the noisy case than in the noise-free case.This is due to the increased level of excitation for the noisysystem. After the transient, the performance of the controller issimilar to that in the KPFSI case for both sets of disturbances.The transient performance depicted here appears to be worsethan that of the UPFSI case. This is because of the relativelyslow convergence rate of the least squares identifier. As willbe pointed out later, in the UPFSI case the controller makesuse of the additionala priori excitation information about thesystem, which leads to performance improvement.


(a) (b)

Fig. 2. System response under the KPFSI controller. (a) System outputy and (b) control inputu: Solid line for noise-free case; dash line for noisy case.

(a) (b)

(c) (d)

Fig. 3. System response under the UPFSDI controller. (a) System outputy; (b) control inputu; (c) parameter estimate�; and (d) error covariance�:

Solid line for noise-free case; dash line for noisy case.

The system response under the UPFSI controller is depictedin Fig. 4. The closed-loop system has a very good transientperformance because of the fast convergence rate and robust-ness of the identifier. It is important to note that this desirable

transient performance is made possible by utilizing thea prioriinformation of the excitation level of the system. The choice ofthe parameter also limits the set of admissible uncertainties.The performance of the controller is very similar to that in


(a) (b)

(c) (d)

Fig. 4. System response under the UPFSI controller. (a) System outputy; (b) control inputu; (c) parameter estimate�; and (d) error covariance�:

Solid line for noise-free case; dash line for noisy case.

the UPFSDI case in steady state, which validates the singularperturbations analysis of Theorem III.1.

This example clearly illustrates the effectiveness of thecontroller design tool developed in this paper. In the face of anopen-loop unstable plant (with possibility of a finite escape),the controller designed achieves both asymptotic tracking anddisturbance attenuation with a moderate control effort.

V. CONCLUSIONS

For a class of SISO nonlinear systems described in noise-prone parametric strict-feedback form, we have developeddesign tools that lead to an explicit construction for a classof (robust adaptive) controllers that asymptotically track agiven reference signal and achieve prespecified disturbanceattenuation levels with respect to exogenous system inputs.We have presented an explicit design paradigm that leads torobust adaptive controllers with the following three appealingfeatures:

1) asymptotic convergence to certainty-equivalent con-trollers as the identification error covariance approacheszero (in the UPFSDI case);

2) utilization of robust parameter identification schemes asbasic building blocks;

3) attenuation of exogenous disturbance inputs to desiredperformance levels over the time interval

The design procedure developed is based on worst caseidentification, the integrator backstepping methodology, andsingular perturbations analysis. The closed-loop system isshown to admit a closed-form value function that satisfiesan associated HJI inequality, thereby guaranteeing a desiredlevel of performance for the adaptive controller. We haveshown that the certainty-equivalence principle holds in thestrict sense only for first-order systems, whereas for higherorder nonlinear systems it holds only asymptotically, as theconfidence in the parameter estimates reaches infinity. A nu-merical example, included in Section IV, clearly demonstratesthe superior performance of the controller designed.

Viewed as an -control problem, we have here one ofthe rare situations where there exists an explicit solution toa genuine nonlinear problem with partial information (notethat, the parameters, which also constitute state, are notdirectly measurable). The controllers are nonlinear and are


parametrically defined with one of these parameters being theprespecified level of disturbance attenuation.

An immediate extension of the results developed here wouldbe to the class of input–output linearizable systems whose zerodynamics are bounded-input/bounded-state stable with respectto control, disturbance, and the state variableswhere is the relative degree. This includes, in particular,the minimum phase parametric uncertain linear systems.

The general results of this paper can also be immediatelyextended to the time-varying parameter case. In this case,the parameter vector would, for example, be generated bydynamics such as

where is an dimensional unknown exogenous input to thesystem. The performance criterion (2) would then include anegative cost penalty associated with the disturbance:

A parameter identifier appropriate for this case can be foundin [33], which again facilitates a backstepping design for anasymptotically tracking and disturbance attenuating controllerfor the system.

Future research on this topic lies in several directions. Oneof these is the generalization of these results to the outputfeedback measurements scheme. Two possible subcases are:

1) only the output trajectory is available for feedback;2) only noisy measurements are available.

Another direction would be the investigation of the case whenthere is an additional prescribed weighting with respect to thecontrol input in the performance criterion (2), as in [40] butwith the special nonlinear structure of the model adopted here.Yet a further direction of study would be to study robustnessof these controllers to unmodeled dynamics.

APPENDIX ADERIVATION OF WORST CASE IDENTIFIERS

In this Appendix, we present the derivations that lead to theUPFSI and UPFSDI identifiers used in Section III, as well asthose used in the next section (Appendix B).

First, we derive the identifier in the UPFSDI case, thatis when both state and derivative information are available.For the nonlinear system (1), we consider the following costfunction (to be minimized):

(23)

where ’s are the parameter estimates to be designed,isa positive scalar, and ’s are the positive-definite designparameter matrices.

When both the state and its derivative are available forfeedback, we view as the dynamic equation and(1) as the set of measurements. Then following thecost-to-come function analysis that led to [33, Th. 1], it is ratherstraightforward to derive (9) as the worst case identifierminimizing (23). It should be noted that the worst casecovariance matrices ’s here correspond to in thenotation of [33]. We also note that, in the present case, we havethe following counterpart of [33, eq. (11)] as the cost-to-comefunction:

where ’s are defined by (7), and ’s satisfy the differentialequations (9a). The choice of then yields the identifier(9).

In the UPFSI case, on the other hand, the identifier (6)is obtained by the following process. First, we study theidentification design under noise perturbed full state measure-ments—a measurement scheme that is informationally inferiorto the UPFSI measurement. In other words, we first assumethat we have the measurement equation

where is a scalar measurement noise affecting theth statevariable. Under the UPFSI measurement scheme, we simplyhave This observation then allows usto rewrite the system dynamics as

......

where we have replaced ’s by ’s on the right-hand side(RHS) of the state equation (1). We associate with this systemthe cost function


where and the weighting matrixdepends on rather than

The solution to this worst case identification problem canbe obtained from [33], as a special case of Remark 4 of thatreference, under the correspondence

...

...

...

This is referred to in [33] as the full-order identifier. To arriveat a reduced-order identifier, we follow the development of[33] and replace the worst case covariance matrix with itslimiting solution as We work here with the inverseof the worst case covariance matrix. First partition it into2 2 subblocks, compatible with the partitioning ofto reveal the two-time-scale property as in [33, eq. (37)].Setting in the resulting equations yields the quasi-steady-state dynamics for the worst case covariance matrix. Usingthis quasi-steady-state dynamics with the state and parameterestimate dynamics yields the identifier (6).

A line of reasoning similar to the above yields the identifiers(24) and (25), under UPFSI and UPFSDI measurements,respectively, for the special class of nonlinear systems con-sidered in Remark III.4.

APPENDIX BDERIVATION UNDER CORRELATED DISTURBANCES

WITHOUT OVERPARAMETERIZATION

In this appendix, we present explicit design equations for thespecial class of nonlinear systems considered in Remark III.4.We start by noting that in view of [33, Th. 7], and thediscussion of Appendix A, a relevant identifier in this caseis given by

(24a)

(24b)

(24c)

where

is an dimensional positive-definite matrix to be chosenby the designer, and is a small design parameter.

The quasi-steady-state dynamics of the identifier, asare given by

(25a)

(25b)

which is exactly the identifier under the UPFSDI measurementscheme.

Because of the simplified structure of the above quasi-steady-state dynamics of the identifier, the controller designwill be based on these dynamics. After such a controller isobtained, the estimate and the covariance are replacedby those generated by the UPFSI identifier (24), to form animplementable UPFSI adaptive controller. The performanceand robustness of this UPFSI adaptive controller can then beestablished by using a singular perturbations analysis.

Under the UPFSDI identifier, the identification errorsatisfies the following dynamics:

whereIntroduce a candidate value function associated with the

identifier

the derivative of which is given by

Using this, we can (as in Section III) convert the attenuationproblem with respect to to one with respect to In termsof dynamics (22) can be rewritten as

Hence, a backstepping design process, which is similar to theone introduced in Section III, can be carried out here withoutthe overparameterization and overconservativeness associatedwith the standard form (1). This again involves repeated useof Lemma III.1.

Fix nonnegative functions which are thedesign parameters:

We can then define the following functions recursively, forwhere and are introduced for

notational consistency


Then, the adaptive controller is given by

(26)

and the value function for the closed-loop system becomes

which satisfies a corresponding HJI equality.The implementable adaptive controller is formed by com-

bining the control law (26) with the UPFSI identifier (24).As in Section III, we will make some structural assump-tions on the weighting matrix to guarantee the robustnessof the closed-loop adaptive system. The parameter matrix

is selected to be of the form

whereUsing this structure, the dynamics for the covariance matrixcan be rewritten as

which is again recognized to be the identifier for the time-varying parameter case (see [33, Th. 9]).

The admissible uncertainty set is then defined as follows,for an arbitrary constant :

The counterpart of Theorem III.1 here can be establishedfor this controller (as in the earlier case) under additionalsmoothness assumptions on the nonlinear functionsandas delineated in Remark III.4.

This completes the derivation of the disturbance-attenuatingadaptive controllers for the class of nonlinear systems (22).

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Zigang Pan (S’92–M’96) received the B.S. degreein automatic control from Shanghai Jiao Tong Uni-versity in 1990 and the M.S. and Ph.D. degrees inelectrical engineering from University of Illinois,Urbana-Champaign, in 1992 and 1996, respectively.

In 1996, he was a Research Engineer at theCenter for Control Engineering and Computationat the University of California, Santa Barbara. Inthe same year, he joined the Polytechnic University,Brooklyn, NY, as an Assistant Professor in theDepartment of Electrical Engineering. His current

research interests include perturbation theory for robust control systems,multivariable optimality-guided robust parameter identification, multivariableoptimality-guided robust adaptive control, and applications of adaptive controlsystems.

Dr. Pan was a coauthor of a paper (with T. Ba¸sar) that received the 1995George Axelby Best Paper Award. He is a member of SIAM, the AMS, andPhi Kappa Phi Honor Society.

Tamer Basar (S’71–M’73–SM’79–F’83) receivedthe B.S.E.E. degree from Robert College in 1969,and the M.S., M.Phil., and Ph.D. degrees in engi-neering and applied science from Yale University,New Haven, CT, in 1972.

After being at Harvard University, Marmara Re-search Institute, and Bogazici University, he joinedthe University of Illinois, Urbana-Champaign, in1981, where he is currently the Fredric and Eliz-abeth Nearing Professor of Electrical and Com-puter Engineering. He has spent sabbatical years at

Twente University of Technology, the Netherlands, 1978–1979, and INRIA,France, 1987–1988 and 1994–1995. He has authored or coauthored over 150journal articles and book chapters and also numerous conference publications,in the general areas of optimal, robust and nonlinear control, dynamic games,stochastic control, estimation theory, stochastic processes, information theory,and mathematical economics. He is coauthor of the textsDynamic Noncoop-erative Game Theory(New York: Academic, 1982, 1995) andH1-OptimalControl and Related Minimax Design Problems(Boston, MA: Birkhauser,1991, 1995).

Dr. Basar carries memberships in several scientific organizations, amongwhich are SIAM, SEDC, and ISDG. Currently, he is the Deputy Editor-in-Chief of the IFAC journalAutomatica, the Managing Editor of theAnnalsof the International Society of Dynamic Games, and Associate Editor ofvarious journals, including theJournal of Economic Dynamics and ControlandSystems & Control Letters. He is also a Vice-President of the IEEE ControlSystems Society and a member of the IEEE Publications Board. Among someof the recent awards and recognitions he has received are: the Medal of Scienceof Turkey (1993), Distinguished Member Award of the IEEE Control SystemsSociety (1993), and the George S. Axelby Outstanding Paper Award of thesame society (1995) for a 1994 paper he coauthored (with Z. Pan).

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Adaptive Controller Design For Tracking And Disturbance