1310 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 8, AUGUST 2002

We assume that the parameters ofF1 represent the actuator model andthat we are not free to modify these.

IV. CONCLUSION

We have considered a discrete-time linear system under nested sat-uration and derived conditions to determine whether a given ellipsoidis invariant. With the aid of these conditions, we developed analysisand design methods for closed-loop stability. The analysis and designmethods were then applied to a linear system under a nested saturationcontrol law, and to a linear system subject to simultaneous actuatormagnitude and rate saturation.

REFERENCES

[1] A. J. Bateman, “Stability analysis and control design for linear systemssubject to nested saturation,” M.S. thesis, Univ. Virginia, Charlottesville,VA, May 2002.

[2] A. Bateman and Z. Lin, “An analysis and design method for linear sys-tems under nested saturation,” inProc. 2002 Amer. Control Conf., An-chorage, AK, 2002, pp. 1198–1203.

[3] J. Berg, K. Hammett, C. Schwartz, and S. Banda, “An analysis of thedestabilizing effect of daisy chained rate-limited actuators,”IEEE Trans.Contr. Syst. Technol., vol. 4, pp. 171–176, Mar. 1996.

[4] F. Blanchini, “Set invariance in control—A survey,”Automatica, vol. 35,pp. 1747–1767, 1999.

[5] E. Davison and E. Kurak, “A computational method for determiningquadratic Lyapunov functions for nonlinear systems,”Automatica, vol.7, pp. 627–636, 1971.

[6] E. Gilbert and K. Tan, “Linear systems with state and control constraints:The theory and application of maximal output admissible sets,”IEEETrans. Automat. Contr., vol. 36, pp. 1008–1020, Sept. 1991.

[7] H. Hindi and S. Boyd, “Analysis of linear systems with saturation usingconvex optimization,” inProc. 37th IEEE Conf. Decision and Control,Tampa, FL, 1998, pp. 903–908.

[8] T. Hu and Z. Lin,Control Systems With Actuator Saturation: Analysisand Design. Boston, MA: Birkhäuser, 2001.

[9] T. Hu, Z. Lin, and B. Chen, “An analysis and design method for linearsystems subject to actuator saturation and disturbance,”Automatica, vol.38, no. 2, pp. 351–359, 2002.

[10] , “Analysis and design for discrete-time linear systems subject toactuator saturation,”Syst. Control Lett., vol. 45, no. 2, pp. 97–112, 2002.

[11] H. Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice-Hall,1996.

[12] K. Loparo and G. Blankenship, “Estimating the domain of attraction ofnonlinear feedback systems,”IEEE Trans. Automat. Contr., vol. AC-23,pp. 602–607, Apr. 1978.

[13] T. Nguyen and F. Jabbari, “Output feedback controllers for disturbanceattenuation with actuator amplitude and rate saturation,”Automatica,vol. 36, no. 9, pp. 1339–1346, 2000.

[14] C. Pittet, S. Tarbouriech, and C. Burgat, “Stability regions for linear sys-tems with saturating controls via circle and Popov criteria,” inProc. 36thIEEE Conf. Decision and Control, 1997, pp. 4518–4523.

[15] H. Sussmann, E. Sontag, and Y. Yang, “A general result on the stabiliza-tion of linear systems using bounded controls,”IEEE Trans. Automat.Contr., vol. 39, pp. 2411–2425, Dec. 1994.

[16] Y. Yang, E. D. Sontag, and H. J. Sussmann, “Global stabilization oflinear discrete-time systems with bounded feedback,”Syst. ControlLett., vol. 30, pp. 273–281, 1997.

[17] A. Teel, “Global stabilization and restricted tracking for multiple inte-grators with bounded controls,”Syst. Control Lett., vol. 18, pp. 165–171,1992.

[18] F. Tyan and D. Bernstein, “Dynamic output feedback compensation forlinear systems with independent amplitude and rate saturation,”Int. J.Control, vol. 67, no. 1, pp. 89–116, 2001.

[19] A. Vanelli and M. Vidyasagar, “Maximal Lyapunov functions and do-main of attraction for autonomous nonlinear systems,”Automatica, vol.21, pp. 69–80, 1985.

[20] S. Weissenberger, “Application of results from the absolute stabilityto the computation of finite stability domains,”IEEE Trans. Automat.Contr., vol. AC-13, pp. 124–125, Feb. 1968.

A Novel Error Observer-Based Adaptive Output FeedbackApproach for Control of Uncertain Systems

Naira Hovakimyan, Flavio Nardi, and Anthony J. Calise

Abstract—We develop an adaptive output feedback control methodologyfor nonaffine in control uncertain systems having full relative degree. Givena smooth reference trajectory, the objective is to design a controller thatforces the system measurement to track it with bounded errors. A linearin parameters neural network is introduced as an adaptive signal. A simplelinear observer is proposed to generate an error signal for the adaptive laws.Ultimate boundedness is shown through Lyapunov’s direct method. Simu-lations of a nonlinear second-order system illustrate the theoretical results.

Index Terms—Adaptive output feedback, error observer, neuralnetworks, nonlinear control, uncertain systems.

I. INTRODUCTION

Research on adaptive output feedback control of uncertain nonlinearsystems is of paramount importance today, particularly considering thegrowing interest in the use of unconventional control devices such aspiezoelectric actuators and synthetic jets. Systems that employ suchdevices, such as flexible robot arms, aeroelastic structures, flow, andcombustion processes, to name a few, are typically very poorly mod-eled.

Output feedback control of full relative degree, affine nonlinear sys-tems was introduced in [1]. The methodology employs a high gain ob-server to estimate the unmeasured states. A solution to the output feed-back stabilization problem for systems in which nonlinearities dependonly upon the available measurement was given in [2]. References [3]and [4] present backstepping-based approaches to adaptive output feed-back control of uncertain systems linear with respect to the unknownparameters. An extension of these methods can be found in [5].

For adaptive observer design, the condition of linear dependenceupon unknown parameters has been relaxed by introducing a linearlyparameterized neural network (NN) in the observer structure [6]. Adap-tive output feedback control using a high gain observer and radial basisfunction NNs has also been proposed in [7] for affine nonlinear systemsrepresented by input–output models. Another method that involves de-sign of an adaptive observer using function approximators and back-stepping control can be found in [8]. However, this result is limited tosystems that can be transformed to output feedback form, i.e., systemsin which the output nonlinearities depend only upon the measurement.

In this note, we consider single-input–single-output (SISO) non-affine in control uncertain systems for which the output has fullrelative degree, and we propose an adaptive output feedback controlmethodology that uses a linear error observer. Using an approximatefeedback linearizing control law, the nonlinear dynamics are invertedand the resulting tracking-error dynamics arealmost linear. Takingadvantage of this fact, we propose alinear observer for thealmostlinear error dynamics. The estimates provided by this observer areused as an error signal in the adaptation laws and as inputs to the NN.Ultimate boundedness is shown through Lyapunov’s direct method.

Manuscript received October 2, 2001; revised March 27, 2002. Recom-mended by Associate Editor A. Datta. This work was supported by the AirForce Office of Scientific Research under Grant F4960-98-1-0437.

N. Hovakimyan and A. J. Calise are with the School of Aerospace Engi-neering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail:naira.hovakimyan@ae.gatech.edu; anthony.calise@ae.gatech.edu).

F. Nardi is with Fiat Auto R&D, Farmington Hills, MI 48331–3473 USA(e-mail: nardi@fiatrd.com).

Publisher Item Identifier 10.1109/TAC.2002.800766.

0018-9286/02$17.00 © 2002 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 8, AUGUST 2002 1311

The assumption regarding full relative degree can be relaxed, but itconsiderably complicates the presentation [9]. We choose insteadto invoke this assumption so as to focus on the central idea of theobserver design.

This note is organized as follows. In Section II, we present theproblem formulation. In Section III, we develop the controller struc-ture and derive the error dynamics. The observer design is presented inSection IV. In Section V, we define the adaptive element and adaptationlaws. Section VI contains the stability analysis. Section VII providesa collection of remarks concerning the conditions in Section VI. Anillustrative example is treated in Section VIII. Section IX summarizesthe note.

II. PROBLEM FORMULATION

Consider the following observable nonlinear SISO system:

_xxx = fff(xxx; u) y = g(xxx) (1)

wherexxx 2 Rn is the state of the plant,u 2 R, andy 2 R are the input(control) and output (measurement), respectively.

Assumption 1:The functionsfff : Rn+1 7! Rn andg: Rn 7! Rare sufficiently smooth, and the output has full relative degreen for all(xxx; u) 2 � R, where � Rn.

Then following [10], there exists a mapping that transforms thesystem in (1) into the so-called normal form

_�i = �i+1; i = 1; . . . ; n� 1

_�n =h(���; u)

�1 = y (2)

where h(���; u) = L(n)fff g are the Lie derivatives, and��� =

[ �1 . . . �n ]T .The control objective is to synthesize anoutputfeedback control law,

such thaty(t) tracks a smooth bounded reference trajectoryyc(t) withbounded errors. The functionsfff andg may be approximately knownor unknown.

III. CONTROLLER DESIGN AND ERRORDYNAMICS

A. Feedback Linearization and Control Design

Feedback linearization is approximated by defining the followingcontrol input signal:

u = h�1(y; v) (3)

wherev is commonly referred to as a pseudocontrol. The functionh(y; u) represents any available approximation ofh(���; u) that is in-vertible with respect to its second argument. It may be constructed froman approximate linear model. Additional requirements onh(y; u) willbe specified in Assumption 2.

Based on Assumption 1 and using (3), the output dynamics in (2)can be expressed as

y(n) = v +� (4)

where

�(���; v) = h ���; h�1(�1; v) � h �1; h�1(�1; v) (5)

is the inversion error. The pseudocontrol is chosen to have the form

v�= y(n)c + vdc � vad + v (6)

wherey(n)c is thenth derivative of the input signal, generated by a stablecommand filter,vdc is the output of a linear dynamic compensator,vadis the adaptive control signal, andv is a robustifying term [11].

With (6), the dynamics in (4) reduce to

y(n) = y(n)c + vdc � vad + v +�: (7)

From (5), notice that� depends onvad throughv. We will designvad to approximately cancel�. Therefore, the following assumption isintroduced to guarantee the existence and uniqueness of a solution forvad.

Assumption 2:The mappingvad 7! � is a contraction over theentire input domain of interest.

A contraction is defined by the condition:j@�=@vadj < 1. Using(5), this reduces to

@�

@vad=

@ h� h

@u

@u

@v

@v

@vad=

@h=@u

@h=@u� 1 < 1: (8)

Condition (8) is equivalent to the following requirements onh:

• sgn(@h=@u) = sgn(@h=@u);• 1 > j@h=@uj > j@h=@uj=2 > 0.

The first condition means that when choosing the approximate modelh, the sign of the control effectiveness is known. The second conditionplaces a lower bound on the estimate of the control effectiveness in (3).It implies that it is better to overestimate the true value.

B. Dynamic Compensator Design and Error Dynamics

Define the output tracking error ase�= yc � y. Then the dynamics

in (7) can be rewritten as

e(n) = �vdc + vad � v ��: (9)

For the case� = 0, the adaptive termsvad � v are not required andthe error dynamics in (9) reduce to

e(n) = �vdc: (10)

The following dynamic compensator is introduced to stabilize the dy-namics in (10):

_��� =Ac��� + bce

vdc = cc��� + dce dim��� = n� 1: (11)

Notice, that��� needs to be at least of dimension(n� 1) [12]. This fol-lows from the fact that (10) corresponds to error dynamics that haven poles at the origin. One could elect to design a compensator of di-mension� n as well. In the sequel, we will assume that the minimumdimension is chosen.

Returning to (9), the vectore = [ e _e � � � e(n�1) ]T togetherwith the compensator state��� will obey the following dynamics, here-after (with a slight abuse of language) referred to as the tracking-errordynamics:

_EEE =AEEE + b [vad � v ��]

zzz =CEEE (12)

whereEEE�= [ eT ���T ]T , b = [bT 0 ]T

A =A� dcbc �bcc

bcc Ac

C =c 0

0 I(13)

1312 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 8, AUGUST 2002

A =

0 1 0 � � � 0

0 0 1 0

......

. . .

0 0 0 � � � 0

b =

0

0

...

1

c =

1

0

...

0

T

andzzz is the vector of available signals. Notice thatAc; bc; cc; dc in(11) should be designed such thatA is Hurwitz.

IV. DESIGN AND ANALYSIS OF AN OBSERVER FOR THE

ERRORDYNAMICS

For the full-state feedback case [13], [14], Lyapunov like stabilityanalysis of the error dynamics results in update laws for the adaptivecontrol parameters in terms of the state error vector. In the output feed-back setup in [15] and [16], an adaptive state observer has been devel-oped for the nonlinear plant to provide estimates of the states neededin the linear controller and the adaptation laws. However, the stabilityanalysis was limited to second-order systems with position measure-ments. To relax these assumptions, we propose a linear observer forthe tracking-error dynamics in (12), assuming that the adaptive part ofthe control signal can compensate for the inversion error. This observerprovides estimates of the unavailable error signals for the update lawsof the adaptive parameters that will be presented in Section VI.

A minimal-order observer of dimensionn� 2 may be designed forthe dynamics in (12), however, to streamline the subsequent stabilityanalysis, in what follows, we consider the case of a full-order observerof dimension2n�1. To this end, consider the following linear observerfor the tracking-error dynamics in (12):

_EEE =AEEE +K (zzz � zzz)

zzz =CEEE (14)

whereK is a gain matrix chosen such that~A�= A � KC is asymp-

totically stable, andzzz is defined in (12). Introduce the observer errorsignal ~EEE

�= EEE �EEE and write the observer error dynamics

_~EEE = ~A ~EEE � b [vad � v ��]

~zzz =C ~EEE (15)

where~zzz = zzz � zzz.

V. ADAPTIVE ELEMENT

Following [17], given a compact setD � Rn+1 and��, the modelinversion error�(���; v) can be approximated overD by a neural net-work

�(���; v) =WWWT���(���) + �(���) (16)

where

����= [ �1 � � � �n v ]T 2 D; j�j � �

�: (17)

Designvad

vad�= WWW

T���(���) dim WWW = N (18)

whereWWW , being the estimates ofWWW , are updated according to the fol-lowing adaptation law:

_WWW = � WWW 2��� (���) EEE

TP2b+ �WWW WWW �WWW 0 (19)

in which WWW 0 is the vector of initial values (or guess) of the NNweights, WWW > 0; �WWW > 0, P2 is the solution of the LyapunovequationA TP2 + P2A = �Q2, for someQ2 > 0, and ��� is an

implementable input vector to the NN on the compact set���, definedas���

�= [ �1 � � � �n v ]T 2 ���, �i = Ei + y

(i�1)c ; i = 1; . . . ; n.

Notice that in (18) there is an algebraic loop, since��� by definitiondepends uponvad throughv. This algebraic loop has at least one fixedpoint solution as long as���(�) is made up of bounded basis functions.

The robustifying term is designed as follows:

v = sgn 2EEETP2b (20)

where the adaptive gain is updated according to the following adap-tation law:

_ = 2EEE

TP2b sgn 2EEE

TP2b � � � 0 (21)

in which 0 is an initial value (or guess) of , > 0; � > 0.Using (16), (18), and (20), the tracking-error dynamics in (12) and theobserver error dynamics in (15) can be written as

_EEE =AEEE + b ~WWWT���+WWW

T ~���� � � sgn 2EEETP2b (22)

_~EEE = ~A~EEE � b ~WWWT���+WWW

T ~���� � � sgn 2EEETP2b (23)

where ~WWW = WWW � WWW , ��� = ���(���), ~����= ���(���) � ���(���). Notice that

for radial basis and many other activation functions that satisfyj�ij �1; i = 1; . . . ; N , there exists an upper bound over the setD

k���(���)k � � � = max���2D

k���(���)k (24)

where� remains of order one, even ifN is large. With this, we havethe following upper bound:

WWWT ~��� � 2kWWWk�: (25)

VI. STABILITY ANALYSIS

In this section, we show through Lyapunov’s direct method that thecomposite error vector��� = [EEET ~EEE

T ~WWWT ~ ]T is ultimately

bounded, where~ = � , �= �� + 2�kWWWk. Notice that��� can be

viewed as a function of the state variables���; ���; EEE; ZZZ , the commandvectoryyyc, and a constant vectorZZZ

��� = FFF ���; ���; EEE; ZZZ; yyyc; ZZZ (26)

whereZZZ = [ WWWT

]T , ZZZ = [WWWT ]T .The relation in (26) represents a mapping from the original domains

of the arguments to the space of the error variablesFFF : ��� � ��� �EEE � ZZZ � yyy � ZZZ ! ��� . Recall that (17) introduces the setDover which the NN approximation is valid. From (17), it follows that

��� 2 D , ��� 2 ��� v 2 v: (27)

Also notice that, since the observer in (14) is driven by the outputtracking error~y = yc � y and compensator states���, having��� 2 ��� ,

yyyc 2 yyy , ��� 2 ��� , implies thatEEE 2 EEE , the latter being a compactset.

According to (6)

v = Fv ���; EEE; ZZZ; yyyc (28)

whereFv : 7! v; = ��� � EEE � ZZZ � yyy .Thus, (26)–(28) ensure that��� is a bounded set. Introduce the largest

ball which is included in��� in the error space

BR�= f���jk���k � Rg; R > 0: (29)

For every��� 2 BR, we have��� 2 D; ZZZ 2 ZZZ , where bothD andZZZare bounded sets.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 8, AUGUST 2002 1313

Assumption 3:Assume

R > TMTm

� (30)

whereTM andTm are the maximum and minimum eigenvalues of thefollowing matrix:

T�= 1

2

2P1 0 0 0

0 2P2 0 0

0 0 �1WWW IN�N 0

0 0 0 �1

(31)

and

= maxZ + 4(� )2

�min(Q1)� 2;

Z + 4(� )2

�min(Q2);

Z + 4(� )2

���2 �+ 12

Z�=�WWW2

kWWW �WWW 0k2 + � 2j � 0j2

� = 12min(�WWW ; � ) > �2 �+ 1

2

��= P1b+ P2b :

P1 is the solution of

~ATP1 + P1 ~A = �Q1

for someQ1 > 0 with minimum eigenvalue�min(Q1) > 2.Theorem 1: Consider the system (1) together with the observer dy-

namics (14), the NN weight and adaptive gain adaptation laws (19),(21). If the initial errors belong to the set (34), defined in the following,then the feedback control law given by (3) and (6) guarantees that theerror signalsEEE; ~EEE; ~WWW; ~ are ultimately bounded.

Proof: Consider the following Lyapunov function candidate:

V = ~EEETP1 ~EEE +EEETP2EEE + 1

2~WWWT �1WWW ~WWW + 1

2~ �1

~ : (32)

The derivative ofV along (22), (23), (19), and (21), with the definitionsof EEE = EEE � ~EEE, = + ~ , can be expressed as

_V =�~EEETQ1

~EEE �EEETQ2EEE � 2~EEET(P1 + P2)b

� ~WWWT���+WWWT ~���� �� sgn 2EEE

TP2b

� 2EEETP2b � �WWW T ~���+ sgn 2EEE

TP2b

� �WWW ~WWWT

WWW �WWW 0 � ~ � � 0 :

Using (25), the fact thatj�j � � sgn� = 0, the following propertyfor vectors ~WWW

T(WWW �WWW 0) = (1=2)k ~WWWk2 + (1=2)kWWW �WWW 0k2 �

(1=2)kWWW � WWW 0k2, and completing squares twice, the upper boundreduces to [18]

_V � �[�min(Q1)� 2] ~EEE2

� �min(Q2)kEEEk2

� ���2 �+ 12 ~ZZZ

2

+ Z + 4(� )2:

Either of the following conditions:

~EEE >Z + 4(� )2

�min(Q1)� 2kEEEk > Z + 4(� )2

�min(Q2)

~ZZZ >Z + 4(� )2

���2 �+ 12 (33)

will render _V < 0 outside the compact set:B = f��� 2 BRjk���k � g.Notice from (30) thatB 2 BR. Consider the Lyapunov functioncandidate in (32) and write it as:V = ���TT��� . Let � be themaximum value of the Lyapunov functionV on the edge ofB :�

�= maxk���k= V = 2TM . Introduce the set: = f���jV � �g.

Let � be the minimum value of the Lyapunov functionV on the edgeof BR: �

�= mink���k=R V = R2Tm. Define the set

� = f��� 2 BRjV � �g: (34)

The condition in (30) ensures that � � and, thus, ultimate bound-edness of��� .

VII. COMMENTS

1) If fff andg in (1) are unknown, for (3) we may useh = ku subjectto the conditions onk specified in Assumption 2.

2) The ultimate bound for the tracking error and its estimate, as in-dicated in (33), can be made small by choosing largeQis withoutproportionally increasing thePis. Unfortunately, this requiresincreasing the compensator gains in (11), which may lead to apeaking phenomenon [7]. However, Theorem 1 remains valid aslong as Assumptions 1, 2, and 3 hold.

3) Assumption 3 may be interpreted as implying bothan upper and lower bound for the adaptation gains.Define

�= max( WWW ; ),

�= min( WWW ; )

and ��= max(�max (P1); �max(P2)) and

��= min(�min(P1); �min(P2)), where �(�) denotes the

eigenvalue value. Then an upper bound for the adaptation gainsresults when2� > 1 and2� > 1, for which the relationin (30) reduces to < R2=( 2�). A lower bound for theadaptation gains results when2� < 1 and2� < 1, for which(30) reduces to > 2=(R2�).

VIII. SIMULATION RESULTS

To illustrate the performance of the proposed adaptive controller, weconsider the following second-order nonaffine in control system:

_x1

_x2=

0 1

�1 2

x1

x2+

0

1

u

juj+ 0:1� 2x21x2

0

1

y = x1:

The open-loop system is unstable. No approximate knowledge ofh(�; �) in (2) is assumed, which amounts to lettingh(�1; u) = u.In using this approximation, we have ignored the dependence of theinverse function on the states of the dynamics. The command signalsyc andy(2)c are generated by processing a series of steps through asecond-order command filter with natural frequency of 1 rad/s anddamping of 0.5. The dynamic compensator was designed to place theclosed loop poles of the error dynamics in (10) at�2:9; �1:03 � j.A Gaussian radial basis function (RBF) NN with three neurons anda bias term was used in the adaptive control. The functional form foreach RBF neuron was defined by

�i(���) = e�(������� ) (������� )=� ; � =p2; i = 1; 2; 3:

(35)The centers���c ; i = 1; 2; 3, were randomly selected over a grid ofpossible values for the vector���. All of the NN inputs were normalizedusing an estimate for their maximum values. The NN adaptation rateis set to WWW = 35. The other design parameters are:Q2 = I; �WWW =0:01; � = 0:01; = 0:001. The poles of the observer have been setto be five times faster than those of the closed-loop error system. Fig. 1compares the tracking performances without and with adaptation. The

1314 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 8, AUGUST 2002

Fig. 1. Tracking (dashed) without and (dash-dotted) with NN.

steady-state error can be eliminated by including an integral term in thelinear part of the controller [19].

IX. SUMMARY

A new approach has been proposed for adaptive output feedbackcontrol of uncertain nonlinear systems using neural networks. Asimple linear observer was introduced to estimate the derivatives of thetracking error. These estimates are used as inputs to the neural networkand in the adaptation laws as an error signal. Ultimate boundedness ofall error signals was proven by Lyapunov’s direct method. Simulationsof a second-order system illustrated the theoretical results.

ACKNOWLEDGMENT

The authors would like to thank M. Idan from Technion, Israel, foruseful discussions. The authors would also like to thank the anonymousreviewers for their comments.

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Cheap Control Performance of a Class ofNonright-Invertible Nonlinear Systems

Julio H. Braslavsky, Rick H. Middleton, and Jim S. Freudenberg

Abstract—For strict-feedback nonlinear systems, this note shows that itis impossible to reduce to zero the optimal cost in the regulation of morestates than the number of control inputs in the system, even using unre-stricted control effort. By constructing a near-optimal cheap control law,we characterize the infimum value of the optimal regulation cost as the op-timal value of a reduced-order regulator problem. We illustrate our resultswith an example on the optimal control of a magnetic suspension system.

Index Terms—Optimal control, performance limitations, strict feedbacksystems.

I. INTRODUCTION

This note studies the cheap control performance of a class of op-timal regulators for nonlinear systems in strict-feedback form. In theoptimal cheap regulator problem, a small positive parameter" scalesthe penalty on the control effort in the performance cost. As" ! 0,the control effort becomes free, and the optimal regulator approaches

Manuscript received November 14, 2000; revised July 11, 2001. Recom-mended by Associate Editor H. Wang. The work of J. H. Braslavsky and R. H.Middleton was supported by the Centre for Integrated Dynamics and Control(CIDAC), The University of Newcastle, Callaghan, Australia. The work of J. S.Freudenberg was supported in part by the National Science Foundation underGrants ECS-9414822 and ECS-9810242 and in part by Ford Motor Company.

J. H. Braslavsky is with the Department of Science and Technology,Universidad Nacional de Quilmes, Bernal 1876, Argentina (e-mail:jbrasla@unq.edu.ar).

R. H. Middleton is with the Department of Electrical and Computer Engi-neering, The University of Newcastle, Callaghan NSW 2308, Australia (e-mail:rick@ecemail.newcastle.edu.au).

J. S. Freudenberg is with the Department of Electrical Engineering and Com-puter Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail:jfr@eecs.umich.edu).

Publisher Item Identifier 10.1109/TAC.2002.800762.

0018-9286/02$17.00 © 2002 IEEE