Adaptive Control System Part 9 potx

`Adaptive Output-feedback Control of a Class of Discrete-time Nonlinear Systems', Proceedings of the 1993 American ControlConference, San Francisco, CA, 1359±1364.. non-an adaptive track

t 1 Then (A.3) can be reorganized as

On the other hand, from the assumption we have T

t 1 implies that ~Tt v ˆ 0, which can be

rewritten as ^Ttv ˆ Tv On the other hand, v 2 S0

t 1 implies that v 2 S0

t‡l 1,since S0

[4] Song, Y and Grizzle, J W (1993) `Adaptive Output-feedback Control of a Class

of Discrete-time Nonlinear Systems', Proceedings of the 1993 American ControlConference, San Francisco, CA, 1359±1364

[5] Yeh, P.-C and KokotovicÂ, P V (1995) `Adaptive Control of a Class of NonlinearDiscrete-time Systems', International Journal of Control, Vol 62, 302±324

182 Active identi®cation for control of discrete-time uncertain nonlinear systems

[6] Yeh, P.-C and KokotovicÂ, P V (1995) `Adaptive Output-feedback Design for aClass of Nonlinear discrete-time Systems', IEEE Transactions on AutomaticControl, vol 40, 1663±1668.

[7] Goodwin, G C and Sin, K S (1984) Adaptive Filtering, Prediction and Control,Prentice-Hall, Englewood Cli€s, NJ

[8] Chen, H F and Guo, L (1991) Identi®cation and Stochastic Adaptive Control,BirkhaÈuser, Boston, MA

[9] Guo, L and Wei, C (1996) `Global Stability/Instability of LS-based Discrete-timeAdaptive Nonlinear Control', Preprints of the 13th IFAC World Congress, SanFrancisco, CA, July, Vol K, 277±282

[10] Guo, L (1997) `On Critical Stability of Discrete-time Adaptive NonlinearControl', IEEE Transactions on Automatic Control, Vol 42, 1488±1499

[11] Kanellakopoulos, I (1994) `A Discrete-time Adaptive Nonlinear System', IEEETransactions on Automatic Control, Vol 39, 2362±2364

[12] AÊstroÈm, K J and Wittenmark, B (1984) Computer Controlled Systems, Hall, Englewood Cli€s, NJ

Prentice-[13] NesÏicÂ, D and Mareels, I M Y (1998) `Dead Beat Controllability of PolynomialSystems: Symbolic Computation Approaches', IEEE Transactions on AutomaticControl, Vol 43, 162±175

[14] Zhao, J and Kanellakopoulos, I (1997) `Adaptive Control of Discrete-time feedback Nonlinear Systems', Proceedings of the 1997 American ControlConference, Albuquerque, NM, June, 828±832

Strict-[15] Zhao, J and Kanellakopoulos, I (1997) `Adaptive Control of Discrete-timeOutput-feedback Nonlinear Systems', Proceedings of the 36th Conference onDecision and Control, San Diego, CA, December, 4326±4331

[16] Feldbaum, A A (1965) Optimal Control Systems, Academic Press, NY

non-an adaptive tracking control Lyapunov function (atclf) whose existenceguarantees the solvability of the inverse optimal problem The controllersdesigned in this chapter are not of certainty equivalence type Even in the linearcase they would not be a result of solving a Riccati equation for a given value

of the parameter estimate Our abandoning of the CE approach is motivated

by the fact that, in general, this approach does not lead to optimality of thecontroller with respect to the overall plant-estimator system, even though boththe estimator and the controller may be optimal as separate entities Ourcontrollers, instead, compensate for the e€ect of parameter adaptationtransients in order to achieve optimality of the overall system

We combine inverse optimality with backstepping to design a new class ofadaptive controllers for strict-feedback systems These controllers solve aproblem left open in the previous adaptive backstepping designs ± gettingtransient performance bounds that include an estimate of control e€ort, which

is the ®rst such result in the adaptive control literature

8.1 Introduction

Because of the burden that the Hamilton±Jacobi±Bellman (HJB) pde's impose

on the problem of optimal control of nonlinear systems, the e€orts made overthe last few years in control of nonlinear systems with uncertainties (adaptiveand robust, see, e.g., Krstic et al., 1995; Marino and Tomei, 1995; and thereferences therein) have been focused on achieving stability rather thanoptimality Recently, Freeman and Kokotovic (1996a, b) revived the interest

in the optimal control problem by showing that the solvability of the (robust)stabilization problem implies the solvability of the (robust) inverse optimalcontrol problem Further extensive results on inverse optimal nonlinear stabil-ization were presented by Sepulchre et al (1997)

The di€erence between the direct and the inverse optimal control problems isthat the former seeks a controller that minimizes a given cost, while the latter isconcerned with ®nding a controller that minimizes some `meaningful' cost Inthe inverse optimal approach, a controller is designed by using a controlLyapunov function (clf) obtained from solving the stabilization problem Theclf employed in the inverse optimal design is, in fact, a solution to the HJB pdewith a meaningful cost

In this chapter we formulate and solve the inverse optimal adaptive trackingproblem for nonlinear systems We focus on the tracking rather than the (set-point) regulation problem because, even when a bound on the parametricuncertainty is known, tracking cannot (in general) be achieved using robusttechniques ± adaptation is necessary to achieve tracking The cost functional inour inverse optimal problem includes integral penalty on both the trackingerror state and control, as well as a penalty on the terminal value of theparameter estimation error To solve the inverse optimal adaptive trackingproblem we expand upon the concept of adaptive control Lyapunov functions(aclf) introduced in our earlier paper (Krstic and KokotovicÂ, 1995) and used

to solve the adaptive stabilization problem

Previous e€orts to design adaptive `linear-quadratic' controllers (see, e.g.,Ioannou and Sun, 1995) were based on the certainty equivalence principle: aparameter estimate computed on the basis of a gradient or least-squares updatelaw is substituted into a control law based on a Riccati equation solved for thatvalue of the parameter estimate Even though both the estimator and thecontroller independently possess optimality properties, when combined, theyfail to exhibit optimality (and even stability becomes dicult to prove) becausethe controller `ignores' the time-varying e€ect of adaptation In contrast, theLyapunov-based approach presented in this chapter results in controllers thatcompensate for the e€ect of adaptation

A special class of systems for which we constructively solve the inverseoptimal adaptive tracking problem in this chapter are the parametric strict-feedback systems, a representative member of a broader class of systems dealtwith in Krstic et al (1995), which includes feedback linearizable systems and, inparticular, linear systems A number of adaptive designs for parametric strict-feedback systems are available, however, none of them is optimal In this

chapter we present a new design which is optimal with respect to a meaningfulcost We also improve upon the existing transient performance results Thetransient performance results achieved with the tuning functions design inKrstic et al (1995), even though the strongest such results in the adaptivecontrol literature, still provide only performance estimates on the trackingerror but not on control e€ort (the control is allowed to be large to achievegood tracking performance) The inverse optimal design in this chapter solvesthe open problem of incorporating control e€ort in the performance bounds.The optimal adaptive control problem posed here is not entirely dissimilarfrom the problem posed in the award-winning paper of Didinsky and Bas°ar(1994) and solved using their cost-to-come method The di€erence is twofold:(a) our approach does not require the inclusion of a noise term in the plantmodel in order to be able to design a parameter estimator, (b) while Didinskyand Bas°ar (1994) only go as far as to derive a Hamilton±Jacobi±Isaacsequation whose solution would yield an optimal controller, we actually solveour HJB equation and obtain inverse optimal controllers for strict-feedbacksystems A nice marriage of the work of Didinsky and Bas°ar (1994) and thebackstepping design in Krstic et al (1995) was brought out in the paper by Panand Bas°ar (1996) who solved an adaptive disturbance attenuation problem forstrict-feedback systems Their cost, however, does not impose a penalty oncontrol e€ort.

This chapter is organized as follows In Section 8.2, we pose the adaptivetracking problem (without optimality) The solution to this problem is given inSections 8.3 and 8.4 which generalize the results of Krstic and KokotovicÂ(1995) Then in Section 8.5 we pose and solve the inverse optimal problem forgeneral nonlinear systems assuming the existence of an adaptive trackingLyapunov function (atclf) A constructive method for designing atclf's based

on backstepping is presented in Section 8.6, and then used to solve the inverseoptimal adaptive tracking problem for strict-feedback systems in Section 8.7 Asummary of the transient performance analysis is given in Section 8.8

8.2 Problem statement: adaptive tracking

We consider the problem of global tracking for systems of the form

_x ˆ f …x† ‡ F…x† ‡ g…x†u

where x 2 Rn, u 2 R, the mappings f …x†, F…x†, g…x† and h…x† are smooth, and 

is a constant unknown parameter vector which can take any value in Rp Tomake tracking possible in the presence of an unknown parameter, we make thefollowing key assumption

186 Optimal adaptive tracking for nonlinear systems

(A1) For a given smooth function yr…t†, there exist functions …t; † and r…t; †

yr…t† ˆ h  …t; † by the objective of tracking yr…t† ˆ h  …t; ^…t††, where ^…t†

is a time function Ð an estimate of  customary in adaptive control

Consider the signal xr…t† ˆ …t; ^…t†† which is governed by

(With a slight abuse of notation, we will write g…x† also as g…t; e; ^†.) The global

tracking problem is then transformed into the problem of global stabilization

of the error system (8.5) This problem is, in general, not solvable with static

feedback This is obvious in the scalar case n ˆ p ˆ 1 where, even in the case

yr…t† ˆ xr…t†  0, a control law u ˆ …x† independent of  would have the impossible task to satisfy x‰ f …x† ‡ F…x† ‡ g…x†…x†Š < 0 for all x 6ˆ 0 and all

 2 R Therefore, we seek dynamic feedback controllers to stabilize system (8.5)

for all 

De®nition 2.1 The adaptive tracking problem for system (8.1) is solvable if

(A1) is satis®ed and there exist a function ~…t; e; ^† smooth on

R‡  …Rnn f0g†  Rp with ~…t; 0; ^†  0, a smooth function …t; e; ^†, and a positive de®nite symmetric p  p matrix , such that the dynamic controller

guarantees that the equilibrium e ˆ 0; ~ ˆ 0 of the system (8.5) is globally stable and e…t† ! 0 as t ! 1 for any value of the unknown parameter  2 Rp

8.3 Adaptive tracking and atclf's

Our approach is to replace the problem of adaptive stabilization of (8.5) by aproblem of nonadaptive stabilization of a modi®ed system This allows us tostudy adaptive stabilization in the Sontag±Artstein framework of controlLyapunov functions (clf) (Sontag, 1983; Artstein, 1983; Sontag, 1989).De®nition 3.1 A smooth function Va: R‡  Rn Rp! R ‡, positive de®nite,decrescent, and proper (radially unbounded) in e (uniformly in t) for each , iscalled an adaptive tracking control Lyapunov function (atclf) for (8.1) (oralternatively, an adaptive control Lyapunov function (aclf) for (8.5)), if (A1) issatis®ed and there exists a positive de®nite symmetric matrix 2 R pp such

that for each  2 Rp, Va…t; e; † is a clf for the modi®ed nonadaptive system

Since these terms are present only when is nonzero, the role of these terms is

to account for the e€ect of adaptation Since Va…t; e; † has a minimum at e ˆ 0 for all t and , the modi®cation terms vanish at the e ˆ 0, so e ˆ 0 is an

equilibrium of (8.9)

We now show how to design an adaptive controller (8.7)±(8.8) when an atclf

is known

Theorem 3.1 The following two statements are equivalent:

(1) There exists a triple …~; Va; † such that ~…t; e; † globally uniformly asymptotically stabilizes (8.9) at e ˆ 0 for each  2 Rp with respect to theLyapunov function Va…t; e; †.

(2) There exists an atclf Va…t; e; † for (8.1).

Moreover, if an atclf Va…t; e; † exists, then the adaptive tracking problem for

(8.1) is solvable

Proof …1 ) 2† Obvious because 1 implies that there exists a continuous

function W : R‡  Rn Rp! R ‡, positive de®nite in e (uniformly in t) foreach , such that

…2 ) 1† The proof of this part is based on Sontag's formula (Sontag, 1989).

We assume that Va is an atclf for (8.1), that is, a clf for (8.9) Sontag's formulaapplied to (8.9) gives a control law smooth on R‡  …Rnn f0g†  Rp:

called the small control property (Sontag, 1989): for each  2 Rp and for any

" > 0 there is a  > 0 such that, if e 6ˆ 0 satis®es jej  , then there is some ~u with j~uj  " such that

Assuming the existence of an atclf we now show that the adaptive tracking

problem for (8.1) is solvable Since …2 ) 1†, there exists a triple …~; Va; † and a

function W such that (8.13) is satis®ed Consider the Lyapunov functioncandidate

problem for (8.1) is solvable.

The adaptive controller constructed in the proof of Theorem 3.1 consists of a

control law ~u ˆ ~…t; e; ^† given by (8.14), and an update law _^ ˆ …t; e; ^† with (8.21) The control law ~…t; e; † is stabilizing for the modi®ed system (8.9) but

may not be stabilizing for the original system (8.5) However, as the proof of

Theorem 3.1 shows, its certainty equivalence form ~…t; e; ^† is an adaptive

globally stabilizing control law for the original system (8.5) The modi®edsystem `anticipates' parameter estimation transients, which results in incorpor-ating the tuning function  in the control law ~ Indeed, the formula (8.14) for ~depends on  via

y ˆ x1

…8:25†

In light of (8.1), f …x† ˆ ‰x2; 0ŠT, F…x† ˆ ‰'…x1†; 0ŠT, g…x† ˆ ‰0; 1ŠT For any

given C2 function yr…t†, the function …t; † ˆ ‰1…t†; 2…t; †ŠT is given by

1…t† ˆ yr…t† and 2…t; † ˆ _yr…t† '…yr†T, and the reference input is

r…t; † ˆ yr…t† @'…yr†

@yr  _yr…t† Hence Assumption (A1) is satis®ed.

With the signal xr…t† ˆ …t; ^† and the tracking error e ˆ x xr, we get theerror system

where z1ˆ e1, z2ˆ c1e1‡ e2‡ ~'T, and c1; c2> 0, globally uniformly

asymp-totically stabilizes (8.27) at e ˆ 0 with respect to Vaˆ1

2…z2‡ z2† with

W…t; e; † ˆ c1z2

1‡ c2z2

2 By Theorem 3.1, the adaptive tracking problem for

(8.25) is solved with the control law ~u ˆ ~…t; e; ^† and the update law

_^ˆ …t;e; ^† ˆ '…t;e; ^†1; c1‡@ ~'T

@e1^z …8:29†

As it is always the case in adaptive control, in the proof of Theorem 3.1 we

used a Lyapunov function V…t; e; ^† given by (8.19), which is quadratic in the

parameter error  ^ The quadratic form is suggested by the linear dence of (8.5) on , and the fact that  cannot be used for feedback We willnow show that the quadratic form of (8.19) is both necessary and sucient forthe existence of an atclf

depen-De®nition 3.2 The adaptive quadratic tracking problem for (8.1) is solvable ifthe adaptive tracking problem for (8.1) is solvable and, in addition, there exist asmooth function Va…t; e; † positive de®nite, decrescent, and proper in e (uniformly in t) for each , and a continuous function W…t; e; † positive

de®nite in e (uniformly in t) for each , such that the derivative of (8.19) alongthe solutions of (8.5), (8.7), (8.8) is given by (8.22)

Corollary 3.1 The adaptive quadratic tracking problem for the system (8.1) issolvable if and only if there exists an atclf Va…t; e; †.

192 Optimal adaptive tracking for nonlinear systems