robust adaptive control for a class of mimo nonlinear systems - usc

728 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 5, MAY 2003

Robust Adaptive Control for a Class of MIMONonlinear Systems With Guaranteed

Error BoundsHaojian Xu and Petros A. Ioannou, Fellow, IEEE

Abstract—The design of stabilizing controllers for mul-tiple-input–multiple-output (MIMO) nonlinear plants withunknown nonlinearities is a challenging problem. The high dimen-sionality coupled with the inability to identify the nonlinearitiesonline or offline accurately motivates the design of stabilizingcontrollers based on approximations or on approximate estimatesof the plant nonlinearities that are simple enough to be generatedin real time. The price paid in such case, could be lack of theo-retical guarantees for global stability, and nonzero tracking orregulation error at steady state. In this paper, a nonlinear robustadaptive control algorithm is designed and analyzed for a classof MIMO nonlinear systems with unknown nonlinearities. Theproposed control scheme provides a general approach to bypassthe stabilizability problem where the estimated plant becomesuncontrollable without any restrictive assumptions. The controlleris continuous and guarantees closed-loop semiglobal stability andconvergence of the tracking error to a small residual set. The sizeof the tracking error at steady state can be specifieda priori andguaranteed by choosing certain design parameters. A procedurefor choosing these parameters is presented. The properties of theproposed control algorithm are demonstrated using simulations.

Index Terms—Adaptive control, neural networks, nonlinear sys-tems, robustness, switching functions.

I. INTRODUCTION

T HE traditional way of designing feedback control systemsis based on the use of linear time invariant (LTI) models for

the plant. Offline frequency domain techniques could be used tofit such an LTI model to experimental data and identify its pa-rameters. In the case, where the parameters of the LTI modelchange with time, gain scheduling, online parameter identifica-tion, adaptive control, robust control techniques, etc. were de-veloped over the years to address such situations [1]. The re-liance on LTI models for control design purposes often puts lim-itations on the performance improvement that could be achievedfor the plant under consideration. For example, if the plant con-sists of strong nonlinearities, its approximation by an LTI model,may considerably reduce the region of attraction in the presenceof disturbances and other modeling uncertainties. During the re-cent years, considerable research efforts have been made to deal

Manuscript received November 15, 2001; revised September 23, 2002. Rec-ommended by Associate Editor A. Datta. This work was supported in part by theNational Science Foundation under Grant ECS-9877193 and in part by a col-laborative linkage grant from the NATO Cooperative Science and TechnologySub-Programme.

The authors are with the Department of Electrical Engineering Systems,University of Southern California Los Angeles, CA 90089 USA (email:[email protected]).

Digital Object Identifier 10.1109/TAC.2003.811250

with the design of stabilizing controllers for classes of nonlinearplants. These efforts are described in detail in a recent surveypaper [2], where a very elegant and informative historical per-spective of the evolution of nonlinear control design is presentedand discussed. Most of the recent efforts (e.g., surveyed in [2]and [3]–[5]) on nonlinear control design assumed that the plantnonlinearities are known. The case where the plant nonlineari-ties are products of unknown constant parameters with knownnonlinearities gave rise to a number of adaptive control tech-niques [1], [6]–[18]. For a class of single-input–single-output(SISO) nonlinear systems, an adaptive control scheme based ona min–max optimization approach is proposed in [19]–[21] byassuming that the nonlinear functions are convex/concave (asopposed to linear assumed in almost every adaptive system) withrespect to the unknown parameters. The results are extended toinclude general parameterizations for a class of SISO systemswith a triangular structure [22].

Neural approximation together with adaptive control tech-niques have been proposed by a number of investigators[23]–[45] for controlling nonlinear plants with unknownnonlinear functions. In this case, neural networks are usedas approximation models of the unknown nonlinearities. Thecontrol design is then based on the neural network modelrather than the actual system. Most of these studies arefocused on the SISO systems [23]–[35]. The control of mul-tiple-input–multiple-output (MIMO) nonlinear systems withunknown nonlinear functions introduces additional complexi-ties and is considered by several investigators [36]–[44]. Theproblems that arise in the control of MIMO nonlinear systemsand up to date contributions are described here.

Consider the class of MIMO systems described by the fol-lowing differential equation:

(1)

where is the output vector;; ;

withis the state vector available for measurement;

is the control vector; and ,are continuous unknown functions of the state. A sufficientcondition for controllability is that , , wheredenotes the minimum singular value of the matrix . Forcontrol design purposesis calculated based on the estimatedplant which has the same form as the unknown plant, i.e.,

(2)

0018-9286/03$17.00 © 2003 IEEE

XU AND IOANNOU: ROBUST ADAPTIVE CONTROL FOR A CLASS OF MIMO NONLINEAR SYSTEMS 729

where , are the online estimate of , ,respectively. Since is calculated based on the estimated plant,for the control law to exist we require , , .The online estimators that generate, have to guarantee that

, , . In fact for computational purposes and uni-form boundedness it is required that , , ,where is a small constant. This requirement is not guaranteedby the usual estimators, for each time, unless special modifica-tions are introduced and in some cases additional assumptionsare made. Several attempts have been made to deal with thisso called “stabilizability problem” in the linear as well as non-linear case [1]. For example in [36], a neural-net controller isanalyzed for a class of th-order MIMO system whereis a constant identity matrix. Several other investigators assumethat is a known matrix [37]. Most of the previous studies[18], [38]–[40] for the MIMO case are dealing with the con-trol of robots, whose dynamics is a special case of (1). By usingsome special properties, e.g., skew-symmetric property, the sta-bilizability problem is bypassed. These control schemes, how-ever, cannot extend to the general class of MIMO nonlinear sys-tems described by (1). In [15], the controllability of the esti-mated plant (2) is assumed to be true at each timeeven thoughno guarantees are provided that this is the case. This assump-tion was relaxed using a robust discontinuous control law byassuming that is positive definite for all , a lower boundof the norm of and an upper bound of the norm ofare known. However, the discontinuous control leads to a non-linear system with discontinuities and the existence and unique-ness of solutions of the closed-loop system cannot be guaran-teed [45]. Furthermore, the discontinuous control law may causechattering at certain boundaries with adverse effects on perfor-mance. In [16] and [17], is assumed to contain some un-known constant vector as where is linearwith respective to . In this case, no modeling errors are in-cluded in the model for . It is assumed that a convex set

is knowna priori such that implies . Pro-jection is used to guarantee , . However, ingeneral it is difficult if at all possible to establish a convex set

with even in the linear case [1]. In [41], the stabiliz-ability problem was discussed without any stability analysis. In[42] and [43], each element of is approximated by neuralnetworks of the form whereis an approximated function constructed by neural networks,

, , is a constant parameter vector corre-sponding to some unknown weights of the neural network, and

denotes the approximation error. The weights are es-timated on line generating , the estimate of at eachtime , which, in turn, is used to generate and there-

fore , the estimate of at each time. It is assumed that is close to the actual value

and is updated slowly by choosing small adaptive gains.Based on this condition it is established that stays in-side an invariant set where . However, sinceis unknown and does not have any physical meaning, the as-sumption that is close to the desired values is difficultif at all possible to guarantee. In [44], an indirect adaptive con-trol algorithm based on fuzzy-neural systems is proposed for

the class of systems (1) provided convex setscan be con-structed such that and for all

, , where denotes themaximum and the minimum singular value of re-spectively and , are two positive constants. The esti-mated weights of the fuzzy/neural systems are then guaranteedto remain inside by using projection. However, such sets

are difficult to construct in general even if the weights wereknown let alone to know thema priori where such weights arecompletely unknown. If sets are constructed to be convexin the parameter space, there is no guarantee that the unknown

that corresponds to the “optimal” approximation belongs tothese sets. If the possibility of instability or bad per-formance cannot be excluded. In [44], it was shown that ifis strictly diagonally dominant with known lower and upperbounds for the main diagonal entries, and if the first deriva-tives of the main diagonal entries in are upper boundedby some known functions and the upper bounds of all off-diag-onal elements , , , are known, theMIMO system can be decomposed into a set of SISO subsys-tems. Then a controller of each subsystem can be modeled by afuzzy system plus a discontinuous robust term. A direct fuzzyadaptive control scheme is designed for each SISO subsystemprovided the fuzzy model is upper bounded by a known contin-uous function. All off-diagonal entries are treated as modelingerrors.

Another important issue in adaptive nonlinear control withunknown nonlinearities is that of tracking error performance.By performance in this context we mean the size of the re-gion of attraction for signal boundedness and the size of thetracking or regulation error at steady state. Performance issuessuch as transient behavior are difficult to establish analyticallyeven in the case of known nonlinearities and is not addressedin most of nonlinear control literature at least analytically. Inmost papers on adaptive nonlinear control with unknown non-linearities, signal boundedness is established first for some re-gion of attraction within which the assumed neural approxima-tions are valid. Signal boundedness then implies that the approx-imation or modeling error is also bounded. The upper bound forthe tracking or regulation error is shown to be of the order ofthe bound on the modeling or approximation error [14], [17],[27]–[29], [41]. In some cases, the approximation error is as-sumed to be upper bounded by some known nonlinearities [14],[28] leading to an upper bound for the tracking error at steadystate that is a function of some design parameters. In [30], [31],[42], and [43], the tracking error is shown to converge inside asmall residual set that depends on bounds of unknown signalsand design parameters. In [15] and [35]–[40], the tracking errormay be made smaller by increasing the control gain. In all thesecases, it is no clear, from the analysis how changes in the de-sign parameters to improve tracking performance will affect theregion of attraction for signal boundedness. Furthermore, theupper bound for the tracking error cannot be easily computedand, therefore, the controller gain cannot be designeda priorito achieve a desired tracking error bound at steady state.

In this paper, we develop a robust adaptive control schemefor the class of MIMO nonlinear linearizable systems with un-


known nonlinearities described by (1). The only assumptionsmade are that the unknown nonlinear functions are continuous,and a sufficient condition for controllability is satisfied. Wepropose a control law that bypasses the stabilizability problemfor the MIMO system. The estimate of the unknown matrix

is replaced by the estimate of a scalar function. The adap-tive laws use a new continuous switching function to guaranteeclosed-loop system stability and convergence of the trackingerror even in the case where the estimated plant loses controlla-bility. A dead zone technique for the MIMO case is incorporatedin the adaptive law in order to guarantee closed loop stabilityand robustness with respect to modeling errors. The proposedscheme guarantees closed loop semiglobal stability and conver-gence of the tracking error to a residual set whose size dependson design parameters that can be chosena priori. For any givendesired upper bound for the tracking error at steady state, ourapproach provides a procedure for choosing the design param-eters to meet the tracking error bound.

This paper is organized as follows: In Section II, the problemstatement and preliminaries are presented. In Section III, we de-sign a nonlinear controller for the case where the approximationmodel is assumed known. In Section IV, we present the proposedadaptive control scheme is discussed. In Section V, the theoret-ical results are applied to the control of a two-link robot. Finally,Section VI includes the conclusions. Throughout this paper,indicates the absolute value, and indicates the Euclideanvector norm.

II. PROBLEM STATEMENT AND PRELIMINARIES

Consider the MIMO nonlinear system described by (1), i.e.,

...

(3)

where ,with ,

is the overall state vector, , , are theinputs and , , are the outputs of thesystem. The nonlinear functions and , ,are assumed to be continuous functions. The problem is todesign the control input such thatthe outputs of the system track the desired tra-jectories , respectively, as close as possible.

System (3) can also be written in the compact form

(4)

where,

......

...

We make the following assumptions.

Assumption 1:The matrix is knownto be either uniformly positive definite or uniformly negativedefinite for all where is a compact set, i.e.,

(5)

where represents the smallest singular value of the matrixinside the bracket and is its lower bound.

Assumption 1 guarantees that the nonlinear system (3) is uni-formly strongly controllable.

Assumption 2:The desired trajectories ,, are known bounded functions of time with

bounded known derivatives.Assumption 3:The state of the system is available for mea-

surement.Assumption 4:The functions and ,

are continuous functions but otherwisecompletely unknown.

Let us first consider the case where and ,, are completely known and examine whether we can

meet the control objective. This is a reasonable step to take sinceif we cannot meet the control objective in the case of knownnonlinearities, it is unlikely that we will do so in the case ofunknown nonlinearities.

We define the following error metric , that describes thedesired dynamics of the error system:

...

(6)

where are positive constants to be selected. It fol-lows from (6) that for we have a set of linear dif-ferential equations whose solutions , , con-verges to zero with time constant . In addition all thederivatives of up to also converge to zero [3], [4].It follows from (6) that

......

......

. . ....

... (7)

where

...

(8)

and , , are coefficients of thebinomial expansion of the corresponding error metric in (6).


Equation (7) can be written in the compact form:

(9)

where,

Let us consider the Lyapunov-like function

(10)

Then, the time derivative of is given by

(11)

If we now choose the control law so that

(12)

then it can be shown that . If is invertible,then the control law [3]

(13)

where , and , ,guarantees

(14)

which implies that and, therefore, converge to zeroexponentially fast.

The calculation of the inverse of in the control law (13)can be avoided if we use Assumption 1 and the following lemmato modify the control law (13).

Lemma 1: Let us define , whereand is the unit vector. If satis-

fies Assumption 1, then we have and we have.

Proof: The proof is presented in Appendix A.Instead of (13), let us now consider the control law

(15)

where is a design constant. Using Lemma 1, we canrewrite (15) as

(16)

The control law (16) guarantees that

(17)

which in turn implies that exponentially fast

and, therefore, and all its derivatives up to convergeto zero exponentially fast.

In the case where and are unknown nonlinearfunctions, the control law (16) can no longer be used. We as-

sume that the nonlinear functions and can be ap-proximated by a general one layer neural network [25]–[29],[33], [46]–[48] on a compact set as

(18a)

(18b)

(18c)

(18d)

where , are selected basis functions, , areunknown constant parameters, and, are the number of thenodes, respectively. The neural network approximation errors

, , are given by

(19a)

(19b)

(19c)

where .Let ,

where . By “optimal” approximation, wemean the weights , are chosen tominimize , for all , respectively,i.e.,

(20a)

(20b)

Assumption 5:The approximation errors , areupper bounded by some known constants , ,

, over the compact set , i.e.,

(21a)

(21b)

From (21b), the approximation error is upper boundedby a known constant over the compact set , i.e.,

(21c)

We should note that we do not require the approximation er-rors to arbitrarily small. It is also assumed that the basis func-tions, number of nodes, , , , are specifiedby the designer, and the only unknowns are the output weights.


As shown in [27], [33], [46]–[48], and the references therein,different basis functions can be used to satisfy Assumption 5.

III. N ONLINEAR CONTROL FOR THECASE OF KNOWN

APPROXIMATION NONLINEARITIES

Let us now assume that the approximation functions ,instead of the actual ones and are known.

In this case, replacing and with , inthe control law (16) will be a straightforward approach. The sta-bility and performance analysis of the closed-loop system, how-ever, is difficult due to the approximation errors ,being different than zero. In order to deal with the approxima-tion error, we modify (16) and propose the control law

(22)

(23)

if

if(24)

where ; , ,are small design constants, and is a constant chosenby the designer. The small linear boundary layer characterizedby is used to smooth out the control discontinuity and avoidpossible singularities in calculating .

The design parameter can be incorporated into as adeadzone width by defining a new function as

sat (25)

satifif

(26)

The properties of are described by the following lemma.Lemma 2: The function defined in (25) has the following

properties:

if (27a)

if (27b)

Proof: From (25), we have when whichalso implies that for . If ,

, then .The following theorem establishes the stability and perfor-

mance properties of the closed-loop system with (22) as the con-trol law.

Theorem 1: Consider the system (3) and the control law (22).Assume that Assumptions 1–5 hold. If the lower boundof

satifies the condition , then there exists a pos-itive constant , a set , and design constants

that satisfy, such that for all , all sig-

nals in the closed-loop system are bounded and the tracking er-rors converge to the residual set

, as exponentially fast.

Proof: Let us consider the following Lyapunov-like func-tion:

(28)

Note that implies that and . Thetime derivative is given by

if (29a)

if (29b)

In the following proof, we only consider the region ,where . Rewrite in (22) as

(30)

(31)

Using Lemma 2, we have

(32)

where . The termin (32) is a modeling error term representing the

effect of the approximation errors , . Using the

inequality , it follows that

(33)

Using , we have

(34)

where is defined as

(35)

Using (32) and (34), becomes

(36)


Let us now choose the design constants, , , so thatthe following inequalities are satisfied:

(37a)

(37b)

(37c)

(37d)

Then, it follows that

(38)

We now need to establish that such design constants exist tosatisfy (37a)–(d). In (37a)

(39)

Assume that

(40)

then (37a) is satisfied. Given (40), we design, , , tosatisfy

(41a)

(41b)

(41c)

Given that the aforementioned inequalities are satisfied, we have

if (42a)

if (42b)

The results (42a)–(b) hold under Assumptions 1 and 5, i.e.,for . Therefore, need to establish that the proposed con-trol law and initial conditions do not forceto get out of the set

at any point in time .Let . Sup-

pose , where is a known compact set, and, where is defined as

(43)

Consider the set

(44)

where is chosen as the largest constant such that. Then, for all , it follows

from (28) and (42a)–(b) that is bounded from above byfor all , which in turn implies that , .From (43) implies and, therefore, cannotleave at any time . From (28) and (42b), it follows that

converges to zero exponentially fast which in turn impliesthat converges to the residual set ex-ponentially fast [1]. The boundedness ofand all signals in theclosed loop follows from the boundedness of , , .

Inside the residual set we also have which, in turn,implies that , , as [4],[10].

In the case where , and areunknown, the control law (22) cannot be used. In this case,we follow the certainty equivalence (CE) principle [1] and re-place the unknown functions , in (22) with theirestimates. Let the estimates of the unknown functions ,

, at time be formed as

(45a)

(45b)

(45c)

(45d)

where and are the estimates of and at time, respectively, to be generated by some adaptive law.

The difference between the estimated and actual parametervalues results in the estimation errors

(46a)

(46b)

(46c)

(46d)

where

(47a)

(47b)

are the parameter errors. For simplicity, let us define

, where

, which will be used later in the discussion.Given the estimates and we can use the

CE approach to come up with an initial guess for the adaptivecontrol law

(48)

where , and design an adap-tive law for generating the parameter estimates ,and, therefore, and so that the overall systemis stable and the tracking error converges to a small residual setwith time. However, it is well known in adaptive control that theestimate cannot be guaranteed to be away from zerofor any given time . This implies that cannot be guaranteedto be bounded uniformly with time. Therefore, the CE control


law (48) cannot be used to stabilize the closed-loop system forthe case where the estimated plant loses its controllability, i.e.,

at some time.In Section IV, we modify the control law (48) to bypass this

stabilizability problem [1] and guarantee stability and perfor-mance for the case and are unknown.

IV. ROBUST ADAPTIVE CONTROL SCHEME

Let us consider (3) and control problem solved in Section IIIfor the case of known nonlinearities. In this section, we assumethat the nonlinear functions in (3) are unknown and design acontrol law to meet the control objective. In order to take carethe case where the estimated plant becomes uncontrollable atsome points in time, we modify the CE control law (48) as

(49)

where is a small design constant. By design, the controllaw in (49) is free of singularities since

, , . Therefore, the proposed controller overcomesthe difficulty encountered in many adaptive control laws wherethe identified model becomes uncontrollable at some points intime. It is also interesting to note that with the samespeed as . Thus, when the estimateapproaches zero, the control input remains bounded and alsoreduces to zero. In other words, in such case it is pointless tocontrol what appears to the controller as uncontrollable plant.This design is critical since the potential loss of controllabilityhas been the main drawback of many nonlinear adaptive controlalgorithms that are based on inverse dynamics. The nonlinearterms , are used to compensate for theeffect of the approximation errors , and effect ofthe design constant .

The control law (49) can be rewritten in the compact form

(50)

where

(51)

provides some insight into the action of the controller. The con-trol vector is expressed as a normalized directional vectorscaled by the control effort . represents the direction ofthe error metric vector . The control vector can be viewedas apportioning the total control effort in different directions.The components, corresponding to large values in, have rela-tively larger control energy than those components with smallervalues. Intuitively, this suggests that the bigger control energyis directed to those with higher values.

Fig. 1. Continuous switching function�(t).

The adaptive laws for generating the estimates , ,, are as follows:

(52a)

(52b)

sign

where

(53)

, , , are daptive gains chosenby the designer, is a design parameter, signis the signfunction (sign , if and sign , otherwise).Then sign if is positive definiteand sign if is negative definite.

is a switching function defined as

if

ifif

(54)where is a design parameter used to avoid discontinuityin . A continuous switching function as shown in Fig. 1,instead of a discontinuous one, is used to guarantee that the re-sulting differential equation representing the closed-loop systemsatisfies the conditions for existence and uniqueness of solu-tions [45]. The constant is defined in (21c) and representsthe upper bound in the approximation of with .

The properties of the switching function are describedby the following Lemma 3 and 4.

Lemma 3: Using the switching function , the followinginequality holds:

sign (55)

Proof: For or , (55) holds. Ifand , we have and

. This results in

(56)

Note that

(57)


Applying (56) together with the fact implies thatthe sign of is always the opposite of sign for

and (55) follows.Lemma 4: Let us define

(58a)

(58b)

(58c)

Then, the following inequality holds:

(59)

where is given by

(60)

Proof: The proof is presented in the Appendix B.The properties of the overall control law (49) and (52a)–(52b)

are described by the following theorem.Theorem 2: Consider the system (3), the control law (49),

and the adaptive laws (52a)–(b). Assume that Assumptions 1–5hold. If the lower bound of satisfies the condition

for some small positive constants ,there exist positive constants , sets

, and , where ,

and design constants , that satisfy

, such that for alland all , all signals in the closed-loop system arebounded and the tracking errors converge to the residualset , as

.Proof: Let us consider the following Lyapunov-like func-

tion:

(61)

Using the adaptive laws (52a)–(b), we can establish that

if (62a)

if (62b)

In the following proof, we only consider the region .Rewrite in (51) as

(63)

Using Lemma 2 and (63), we obtain

(64)

Using the identities

(65a)

(65b)

where , it follows that

(66)

The term in (66) is a modeling error termrepresenting the effect of the design parameterand the ap-proximation error . The price paid to avoid the singularityin control law (49) is that the design parameterappears as adisturbance in the closed-loop system. The modeling error term


will become dominant when the estimate.

From (66), the first term in can be described as

(67)

Applying Lemma 4, the absolute value of the modeling errorterm in (67) can be expressed as

(68)

Using

(69)

gives that

(70)

Applying (70) into (67) gives

(71)

Let us now consider the second term of in (62b). Usingthe adaptive law (52a), we have

(72)

Finally, we consider the last term of in (62b). Using theadaptive law (52b), we have

sign (73)

By applying Lemma 3 to (73), the last term of can be rep-resented by

sign

(74)

Combining (71)–(74), is finally given by

(75)

Let us now choose the design constants, , , , sothat the following inequalities are satisfied:

(76a)

(76b)

(76c)

(76d)

(76e)

(76f)

It follows that

(77)

We now need to establish that such design constants exist tosatisfy (76a)–(f). In (76a)

(78)

Assume that

(79)

then (76a) is satisfied. This inequality depends on the value ofwhich is a characteristic of the nonlinear system. It suggests

that the design constant has to be chosen relatively small anda sufficient number of nodes in have to be used in orderto make small. Given (79), we design , , , , tosatisfy

(80a)


(80b)

(80c)

(80d)

Given that (80a)–(d) are satisfied, we have

if (81a)

if (81b)

The results (81a)–(b) hold under the assumption forall . We need to establish that for and

, where , , need to be specified, we have .Suppose , , and . The set isdefined as

(82)

Consider the set

(83)

where is chosen as the largest constant for which. Then, for all , ,

where , it follows from (61) and (81a)–(b) thatis bounded from above by for all , which implies

that , . Then, it follows from (82) forall . Therefore, cannot leave at any time .

The properties of together with imply thatand, therefore, , , are bounded for all ,i.e., , , . This, in turn, implies that ,

are bounded and has a limit, i.e., .

Using the fact that for and (81b), we have

which implies that . From , ,, it follows that all signals are bounded which im-

plies that . From and , we haveas which implies that converges to the

residual set [1]. Inside the residual set,we also have which in turn implies that

, as [4], [10].

A. Design Parameter Procedure

The design parameters can be chosen to guarantee that thetracking error is within a desired prespecified bound at steadystate by using the following procedure provided the lower bound

related to the controllability of the plant, and , , theupper bounds for the approximation errors are knowna priori.

1) Using , the upper bound of the approximation error, check if the lower bound of satisfies

. If so, choose the design parameters, such that. If not then the number of nodes ,

, of the neural network for has tobe increased in order to obtain a smaller approximationerror bound .

2) Set the desired upper bound for the tracking error of eachoutput at steady state equal to and choose ,to satisfy the bound.

3) Calculate , , using (58a)–(c) and the knowledgeof , , , , i.e.,

, ,.

4) Choose in the adaptive law (52b) such that.

5) Choose , , such that ,, .

Remark 2: It is worth noting that the ratio, ,gives an upper bound for the approximation error thatcan be tolerated by the closed-loop system. Largerimpliesthat the closed-loop system can tolerate larger approximationerror . On the other hand, small requires more accu-rate approximation of by , which normally impliesmore nodes in the neural network for .

Remark 3: The tracking errors at steady state are guaran-teed to converge inside the residual set ,

, whose size depends on the design parameter.The smaller the design parameteris, the smaller the trackingerror is guaranteed to be at steady state. The design parameteris inversely proportional to the gain . For a given , the gain

depends on the approximation error, . From (80a)–(d),it is easy to check that is a critical parameter used to deter-mine the design parameters , , . Let us rewrite as

, which is a function of the3 ratios , , . For the design parameter ,we require . Using the expressions

, ,it follows that , depend on the ratios , . From

, we have , . Thesmaller ratios , , imply that the smaller valuesof the design parameters , , , can be chosen. How-ever, a smaller value for requires a better approximationof the unknown function , which implies a higher orderneural network. A smaller value for may imply a largercontrol input when becomes smaller. A smaller valuefor may imply a faster switching action. Therefore, thereis a tradeoff between the design parameters,, , , , andthe ratios, , , . These tradeoffs can be takeninto account in the design parameter procedure presented abovein order to improve the properties of the closed-loop system fur-ther.

Remark 4: The -modification termsign has been incorporated in

the adaptive law (52b) to ensure stability and robustness. In theexpression for , the modeling error termappears because of the design parameter,, and theapproximation error . This term may become dominantin the case where approaches zero. This implies the signalsin the closed-loop system may become unbounded when

. The -modification is activated to guarantee stabilitywhen the estimate becomes smaller than the lower boundof . In fact, it guarantees the convergence of the trackingerror even if approaches zero since it appears as a negativeterm in the derivative of the Lyapunov-like function to cancelthe effect of the modeling error . Therefore,this special -modification ensures robustness whereas the


classical -modification is used to avoid the estimate of theparameters to drift to infinity [1]. From the adaptive learningpoint of view, when the -modification is activated, ,

, will increase along the direction of theactual . Thus, the special-modification could be viewedas a soft projection algorithm used to prevent fromgoing to zero.

Remark 5: The estimate of the scalar function isobtained by using the estimate of each element in , i.e.,

. However, we should note that we can approximate theunknown scalar function directly instead of the unknownmatrix . The price paid in this case is that the dimension of

is doubled that of the state . However, for theregulation problem, can be replaced by .

Remark 6: If does not satisfy Assumption 1, there ex-ists a known matrix such that satisfies As-sumption 1. The control law in this case will be modified as

(84)

where .

V. SIMULATION RESULTS

We demonstrate the performance of the proposed adaptivecontrol system using a dynamical model of a planar, two-link,articulated robotic manipulator [4], [42]. The dynamics of thisrobotic system are nonlinear with strong coupling between thetwo degrees of freedom. The equations of motion in terms ofthe generalized coordinates and , representing the angularpositions of joints 1 and 2 and applied torquesand at thesejoints is given by

(85)

where

with

The numerical values used for simulation purposes are

The robot initially is at rest at the position , . Itis desired to determine control inputs and such that

and follow a desired trajectory defined by

(86)

Fig. 2. Tracking error response of link 1 during the first 2 s. The dotted linesindicate the required error bound of�1 degree.

with the tracking performance defined by

(87)

Since the inertia matrix is positive definite, (85) can bewritten as

(88)

The previous equation is in the general form of (3) with. Because the workspace is a closed set, it is

easy to show that both and are uniformly positivedefinite for all positions in the robot’s workspace. In thissimulation, the lower bound of the minimum singular valueof is . Assuming a completely unknownnonlinear plant, one layer radial basis neural network with abasis function , where

, are design parameters, was used to approximate theunknown functions over the compact set

with 5760nodes and over the compact setwith 300 nodes. The upper bounds for the approximationerrors are estimated using offline training as ,

and , i.e., and. Following the design procedure, the condition

is satisfied. Then, the valuesof , are chosen to be 0.002, 0.01, respectively, such that

. Taking ,, , the desired bounds for the steady state

tracking errors are satisfied, i.e.,and . Using (58a)–(c), we obtain

, , . Based on conditions(80a)–(d), we choose , , and

. In this simulation, the system is assumed initiallyat rest and the initial conditions for the parameter estimatesare taken to be zero, reflecting the fact that the system iscompletely unknown. Fig. 2 and 3 show the simulation results


Fig. 3. Tracking error response of link 2 during the first 2 s. The dotted linesindicate the required error bound of�1 degree.

Fig. 4. Action of the switch function�(t) during the first 2 s.

for the tracking errors. Fig. 4 shows the action of the switchingfunction .

The simulation results in Figs. 2 and 3 demonstrate the theoryand calculated steady-state error bounds. Let us now change thetracking error performance requirements to the following de-sired tracking error bounds:

(89)

In this case, we use the design inequalities (80a)–(d) to choose, , and , leading to

and . As demon-strated in Figs. 5 and 6 the algorithm meets the new error boundrequirements at steady state. We should also note that as shownin Figs. 4 and 7 the switching function reaches a steadystate in a short period of time after which no more switchingtakes place.

VI. CONCLUSION

In this paper, we consider the control problem of a class ofnonlinear MIMO with unknown nonlinearities. The nonlin-

Fig. 5. Tracking error response of link 1 during the first 2 s. The dotted linesindicate the required error bounds�0.5 degree.

Fig. 6. Tracking error response of link 2 during the first 2 s. The dotted linesindicate the required error bounds�0.5 degree.

Fig. 7. Action of the switching function�(t) during the first 0.5 s.

earities are assumed to be continuous functions and as suchcan be approximated and estimated online using a single-layerneural network. A nonlinear robust adaptive control scheme


is designed and analyzed. The adaptive laws use a continuousswitching function instead of complicated projection tech-niques that requirea priori knowledge of unknown parameters.The controller guarantees closed-loop semiglobal stability andconvergence of the tracking error to a small residual set evenin the case where the estimated plant loses controllability. Thesemiglobal stability is characterized by a region of attractionfor stability whose size depends on the compact set used toapproximate the nonlinear functions of the plant. The size ofthe residual set for the tracking error depends solely on designparameters, which can be chosen to meet desired upper boundsfor the tracking error. A design procedure is presented whichshows how to choose the various parameters in the control lawin order to guarantee that the steady-state tracking error arewithin prespecified bounds.

APPENDIX

A. Proof of Lemma 1

Let us write as the sum of a symmetric matrix and askew-symmetric matrix as

(A1)

This gives

(A2)

Because the quadratic form associated with a skew-symmetricmatrix is always zero. Let be the unit vector. Then

(A3)Since is uniformly symmetric definite,all eigenvectors are orthogonal and all eigenvalues are real. Let

indicate a set of unit eigenvectors of the spanof . are orthonormaland form a set of basis in space . Thus, the unit vectorcan be expressed as a linear combination of

(A4)where are corresponding coefficients and

.Let be the eigenvalues of

. Applying (A4) into (A3) givesthat

(A5)

where

(A6)

is a linear combination of all eigenvalues of the matrixand is real for all .

From (A6), it follows that

(A7)

B. Proof of Lemma 4

For , (59) holds. In the following proof, we onlyconsider the case where .

Let us start with the following equation:

(B1)

From (63), we obtain

(B2)

Substituting into (B2), itfollows that

(B3)

Therefore, the scalar function can be expressed as

(B4)Applying (B4) together with the fact

gives

(B5)

From the expression of (53) and (60), the scalar functioncan be written as

(B6)

From (B6), we have

(B7)

and

(B8)

From (B8), can be expressed as

(B9)


Therefore, we have

(B10)

The absolute value of the first term on the right-hand side of(B1) can be written as

(B11)Applying (B5) and (B10) into (B11) gives

(B12)

The second term on the right-hand side of (B1) can be de-scribed as

(B13)

Using the fact only if, we have

(B14)

Combining the results of (B12) and (B14), it follows that

(B15)

where , , are defined in (58a)–(c).

ACKNOWLEDGMENT

The authors would like to thank Prof. E. Kosmatopoulos ofthe Technical University of Crete, Prof. V. Kuntsevich, Prof.V. Gubarev, Dr. L. Zhiteckij, and Dr. N. Aksinov of the SpaceResearch Institute of the Ukranian National Academy for manyuseful discussions on adaptive control and the results of thispaper that took place as a result of a joint collaborative projectsupported by NATO.

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Haojian Xu received the B.S. degree from Nan-jing University of Aeronautics and Astronautics,Nanjing, China, and the M.S. and Ph.D. degreesin electrical engineering from the University ofSouthern California, Los Angeles, in 1990, 1999,and 2002, respectively.

From 1990 to 1997, he worked as an Engineer inthe Beijing Aerospace Automatic Control Institute,China Academy of Launch Vehicle Technology, Bei-jing, China. He has served as a reviewer for variousjournals and conferences. His current research inter-

ests include robust adaptive control, nonlinear system control, control of un-known systems using neural networks, and system identification and control ofhard disk drive systems.

Petros A. Ioannou (S’80–M’83–SM’89–F’94)received the B.Sc. degree (first class honors) fromUniversity College, London, U.K., and the M.S.and Ph.D. degrees from the University of Illinois,Urbana, in 1978, 1980, and 1982, respectively.

In 1982, he joined the Department of ElectricalEngineering Systems, University of SouthernCalifornia, Los Angeles, where he is currently aProfessor and the Director of the Center of AdvancedTransportation Technologies. His research interests

are in the areas of adaptive control, neural networks, nonlinear systems,vehicle dynamics and control, intelligent transportation systems, and marinetransportation. He was Visiting Professor at the University of Newcastle,Australia, in the fall 1988, the Technical University of Crete in summer 1992,and served as the Dean of the School of Pure and Applied Science at theUniversity of Cyprus in 1995. He is the author/coauthor of five books andover 150 research papers in the area of controls, neural networks, nonlineardynamical systems and intelligent transportation systems.

Dr. Ioannou was a recipient of the Outstanding IEEE Transactions PaperAward in 1984, and the recipient of a 1985 Presidential Young InvestigatorAward.

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