Adaptive Control System Part 13 doc

10.8 Designs for nonlinear dynamicsThe adaptive inverse approach developed in Sections 10.4±10.6 can also beapplied to systems with nonsmooth nonlinearities at the inputs of smoothnonlin

10.8 Designs for nonlinear dynamics

The adaptive inverse approach developed in Sections 10.4±10.6 can also beapplied to systems with nonsmooth nonlinearities at the inputs of smoothnonlinear dynamics [16, 17] State feedback and output feedback adaptiveinverse control schemes may be designed for some special cases of a nonlinearplant

_x…t† ˆ f …x…t†† ‡ g…x…t††u…t†; u…t† ˆ N…v…t††

where f …x† 2 Rn, g…x† 2 Rn, and h…x† 2 R are smooth functions of x 2 Rn,

N…
† represents an unknown nonsmooth actuator uncertainty as a dead-zone, backlash, hysteresis or piecewise-linear characteristic, and v…t† 2 R is the control input, while u…t† is not accessible for either control or measurement.

The main idea for the control of such plants is to use an adaptive inverse

to cancel the e€ect of the unknown nonlinearity N…
† so that feedback control schemes designed for (10.128) without N…
†, with the help of c NI…
†, can be applied to (10.128) with N…
†, to achieve desired system performance To

illustrate this idea, we present an adaptive inverse design [16] for a order parametric-strict-feedback nonlinear plant [6] with an actuator non-

and b1< j0…x†j < b2, for some positive constants b1, b2 and 8x 2 R3

To develop an adaptive inverse controller for (10.130), we assume that the

nonlinearity N…
† is parametrized by   2 Rn  as in (10.2) and its inverse cNI…
†

is parametrized by  2 Rn , an estimate of , as in (10.10), with the statedproperties Then, the adaptive backstepping method [6] can be combined with

an adaptive inverse cNI…
† to control the plant (10.130), in a three-step design: Step 1: Let the desired output be r…t† to be tracked by the plant output y…t†,

with bounded derivatives r…k† …t†, k ˆ 1; 2; 3 De®ning z1ˆ x1 r and

z2ˆ x2 1, where 1 is a design signal to be determined, we have

_z1ˆ z2‡ 1‡  T

Adaptive Control Systems 281

to be stabilized by 1 with respect to the partial Lyapunov function

@x1'1

 Suggested by (10.137), we choose thesecond stabilizing function 2 as

sk2 Then we can rewrite (10.137) as

282 Adaptive inverse for actuator compensation

error in (10.11) The expression (10.145), with the need of parameter projection

for …t† for implementing an inverse c NI…
†, suggests the adaptive update law for

…t†:

initialized by (10.35) and projected with f …t† in (10.36) for g…t† ˆ z30!.This adaptive inverse control scheme consisting of (10.129), (10.141),(10.142) and (10.146) has some desired properties First, the modi®cation(10.139) ensures that fs…s†…s 

s†Ts 0, and the parameter projection (10.36) ensure that …   †T 1f  0 and i…t† 2 ‰a

i; b

iŠ, i ˆ 1; ; n, for

Adaptive Control Systems 283

zk2 L 1 , k ˆ 1; 2; 3 Since z1ˆ x1 r is bounded, we have x12 L 1and hence'1…x1† 2 L 1 It follows from (10.134) that 12 L 1 By z2 ˆ x2 1, we have

x22 L 1 and hence '2…x1; x2† 2 L 1 It follows from (10.138) that 22 L 1.Similarly, x32 L 1, and ud 2 L 1 in (10.141) Finally, v…t† in (10.129) and u…t†

in (10.128) are bounded Therefore, all closed loop system signals are bounded.Similarly, an output feedback adaptive inverse control scheme can bedeveloped for the nonlinear plant (10.128) in an output-feedback canonicalform with actuator nonlinearities [17] For such a control scheme, a stateobserver [6] is needed to obtain a state estimate for implementing a feedbackcontrol law to generate ud…t† as the input to an adaptive inverse:

v…t† ˆ c NI…ud…t††, to cancel an actuator nonlinearity: u…t† ˆ N…v…t††.

10.9 Concluding remarks

Thus far, we have presented a general adaptive inverse approach for control ofplants with unknown nonsmooth actuator nonlinearities such as dead zone,backlash, hysteresis and other piecewise-linear characteristics This approachcombines an adaptive inverse with a feedback control law The adaptive inverse

is to cancel the e€ect of the actuator nonlinearity, while the feedback controllaw can be designed as if the actuator nonlinearity were absent Forparametrizable actuator nonlinearities which have parametrizable inverses,state or output feedback adaptive inverse controllers were developed whichled to linearly parametrized error models suitable for the developments ofgradient projection or Lyapunov adaptive laws for updating the inverseparameters

The adaptive inverse approach can be viewed as an algorithm-basedcompensation approach for cancelling unknown actuator nonlinearitiescaused by component imperfections As shown in this chapter, this approachcan be incorporated with existing control designs such as model reference, PID,pole placement, linear quadratic, backstepping and other dynamic compensa-tion techniques An adaptive inverse can be added into a control system loopwithout the need to change a feedback control design for a known linear ornonlinear dynamics following the actuator nonlinearity

Improvements of system tracking performance by an adaptive inverse have

284 Adaptive inverse for actuator compensation

been shown by simulation results Some signal boundedness properties ofadaptive laws and closed loop systems have been established However, despitethe existence of a true inverse which completely cancels the actuator non-linearity, an analytical proof of a tracking error convergent to zero with anadaptive inverse is still not available for a general adaptive inverse controldesign Moreover, adaptive inverse control designs for systems with unknownmultivariable or more general nonlinear dynamics are still open issues underinvestigation.

of Paraplegics', IEEE Trans Autom Cont., Vol 36, No 6, 683±691

[4] Ioannou, P A and Sun, J (1995) Robust Adaptive Control Prentice-Hall.[5] Krasnoselskii, M A and Pokrovskii, A V (1983) Systems with Hysteresis.Springer-Verlag

[6] KrsticÂ, M., Kanellakopoulos, I and KokotovicÂ, P V (1995) Nonlinear andAdaptive Control Design John Wiley & Sons

[7] Merritt, H E (1967) Hydraulic Control Systems John Wiley & Sons

[8] Physik Instrumente (1990) Products for Micropositioning Catalogue 108±12,Edition E

[9] Rao, S S (1984) Optimization: Theory and Applications Wiley Eastern

[10] Recker, D (1993) `Adaptive Control of Systems Containing Piecewise LinearNonlinearities', Ph.D Thesis, University of Illinois, Urbana

[11] Tao, G and Ioannou, P A (1992) `Stability and Robustness of MultivariableModel Reference Adaptive Control Schemes', in Advances in Robust ControlSystems Techniques and Applications, Vol 53, 99±123 Academic Press

[12] Tao, G and KokotovicÂ, P V (1996) Adaptive Control of Systems with Actuator andSensor Nonlinearities John Wiley & Sons

[13] Tao, G and Ling, Y (1997) `Parameter Estimation for Coupled MultivariableError Models', Proceedings of the 1997 American Control Conference, 1934±1938,Albuquerque, NM

[14] Tao, G and Tian, M (1995) `Design of Adaptive Dead-zone Inverse forNonminimum Phase Plants', Proceedings of the 1995 American ControlConference, 2059±2063, Seattle, WA

[15] Tao, G and Tian, M (1995) `Discrete-time Adaptive Control of Systems withMulti-segment Piecewise-linear Nonlinearities', Proceedings of the 1995 AmericanControl Conference, 3019±3024, Seattle, WA

Adaptive Control Systems 285

[16] Tian, M and Tao, G (1996) `Adaptive Control of a Class of Nonlinear Systemswith Unknown Dead-zones', Proceedings of the 13th World Congress of IFAC, Vol.

E, 209±213, San Francisco, CA

[17] Tian, M and Tao, G (1997) `Adaptive Dead-zone Compensation for Feedback Canonical Systems', International Journal of Control, Vol 67, No 5,791±812

Output-[18] Truxal, J G (1958) Control Engineers' Handbook McGraw-Hill

286 Adaptive inverse for actuator compensation

of the plant dynamics We prove that with or without such knowledge theadaptive schemes can `learn' how to control the plant, provide for boundedinternal signals, and achieve asymptotically stable tracking of the referenceinputs We do not impose any initialization conditions on the controllers, andguarantee convergence of the tracking error to zero.

11.1 Introduction

Fuzzy systems and neural networks-based control methodologies haveemerged in recent years as a promising way to approach nonlinear controlproblems Fuzzy control, in particular, has had an impact in the controlcommunity because of the simple approach it provides to use heuristic controlknowledge for nonlinear control problems However, in the more complicatedsituations where the plant parameters are subject to perturbations, or when thedynamics of the system are too complex to be characterized reliably by anexplicit mathematical model, adaptive schemes have been introduced that

gather data from on-line operation and use adaptation heuristics to matically determine the parameters of the controller See, for example, thetechniques in [1]±[7]; to date, no stability conditions have been provided forthese approaches Recently, several stable adaptive fuzzy control schemes havebeen introduced [8]±[12] Moreover, closely related neural control approacheshave been studied [13]±[18].

auto-In the above techniques, emphasis is placed on control of input output (SISO) plants (except for [4], which can be readily applied to MIMOplants as it is done in [5, 6], but lacks a stability analysis) In [19], adaptivecontrol of MIMOsystems using multilayer neural networks is studied Theauthors consider feedback linearizable, continuous-time systems with generalrelative degree, and utilize neural networks to develop an indirect adaptivescheme These results are further studied and summarized in [20] The scheme

single-in [19] requires the assumptions that the tracksingle-ing and neural network eter errors are initially bounded and suciently small, and they provideconvergence results for the tracking errors to ®xed neighbourhoods of theorigin

param-In this chapter we present direct [21] and indirect [22] adaptive controllers forMIMOplants with poorly understood dynamics or plants subjected to param-eter disturbances, which are based on the results in [8] We use Takagi±Sugenofuzzy systems or a class of neural networks with two hidden layers as the basis

of our control schemes We consider a general class of square MIMOsystemsdecouplable via static nonlinear state feedback and obtain asymptotic con-vergence of the tracking errors to zero, and boundedness of the parametererrors, as well as state boundedness provided the zero dynamics of the plant areexponentially attractive The stability results do not depend on any initializa-tion conditions, and we allow for the inclusion in the control algorithm of

a priori heuristic or mathematical knowledge about what the control inputshould be, in the direct case, or about the plant dynamics, in the indirect case.Note that while the indirect approach is a fairly simple extension of thecorresponding single-input single-output case in [8], the direct adaptive case

is not The direct adaptive method turns out to require more restrictiveassumptions than the indirect case, but is perhaps of more interest because,

as far as we are aware, no other direct adaptive methodology with stabilityproof for the class of MIMOsystems we consider here has been presented inthe literature The results in this chapter are nonlocal in the sense that they areglobal whenever the change of coordinates involved in the feedback lineariza-tion of the MIMOsystem is global

The chapter is organized as follows In Section 11.2 we introduce the MIMOdirect adaptive controller and give a proof of the stability results In Section11.3 we outline the MIMOindirect adaptive controller, giving just a shortsketch of the proof, since it is a relatively simple extension of the results in [8]

In Section 11.4 we present simulation results of the direct adaptive method

288 Stable multi-input multi-output adaptive fuzzy/neural control

applied, ®rst, to a nonlinear di€erential equation that satis®es all controllerassumptions, as an illustration of the method, and then to a two-link robot.The robot is an interesting practical application, and it is of special interest herebecause it does not satisfy all assumptions of the controller; however, we showhow the method can be made to work in spite of this fact In Section 11.5 weprovide the concluding remarks.

11.2 Direct adaptive control

Consider the MIMOsquare nonlinear plant (i.e a plant with as many inputs asoutputs [23, 24]) given by

where X ˆ ‰x1; ; xnŠT2 Rn is the state vector, U :ˆ ‰u1; ; upŠT2 Rp is the

control input vector, Y :ˆ ‰y1; ; ypŠT2 Rp is the output vector, and

f ; gi; hi; i ˆ 1; ; p are smooth functions If the system is feedback linearizable

[24] by static state feedback and has a well-de®ned vector relative degree

r :ˆ ‰r1; ; rpŠT, where the ri's are the smallest integers such that at least one ofthe inputs appears in y…ri†

i , input±output di€erential equations of the system aregiven by

with at least one of the Lgj…Lr i 1

f hi† 6ˆ 0 (note that Lfh…X† : Rn! R is the Lie

derivative of h with respect to f , given by Lfh…X† ˆ@h

@Xf …X†† De®ne, for

convenience, i…X† :ˆ Lr i

fhi and i j…X† :ˆ Lgj…Lfr i 1hi† In this way, we may

rewrite the plant's input±output equation as

375

|‚‚‚‚{z‚‚‚‚}

Y…r† …t†

ˆ

1

p

264

375

375

|‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚}

B…X;t†

u1

up

264

375

U ˆ B 1… A ‡ m† …11:4†

(note that, for convenience, we are dropping the references to the independentvariables except where clari®cation is required), where the term

m ˆ ‰1; ; pŠT is an input to the linearized plant dynamics In order for U

to be well de®ned, we need the following assumption:

(P1) Plant Assumption

The matrix B as de®ned above is nonsingular, i.e B 1 exists and has

bounded norm for all X 2 Sx; t  0, where Sx2 Rn is some compact set

of allowable state trajectories This is equivalent to assuming

The plant is feedback linearizable by static state feedback; it has a general

vector relative degree r ˆ ‰r1; ; rpŠT, and its zero dynamics are tially attractive (please refer to [24] for a review on the concept of zerodynamics and static state feedback of square MIMOsystems) We alsoassume the state vector X to be available for measurement

exponen-Our goal is to identify the unknown control function (11.4) using fuzzysystems Here we will use generalized Takagi±Sugeno (T±S) fuzzy systems withcentre average defuzzi®cation To brie¯y present the notation, take a fuzzy

system denoted by ~f…X; W† (in our context, X could be thought of as the state

vector, and W as a vector of possibly exogenous signals) Then,

~f…X; W† ˆPPRiˆ1cii

R

iˆ1i Here, singleton fuzzi®cation of the input vectors

X ˆ ‰x1; ; xnŠT, W ˆ ‰w1; ; wqŠT is assumed; the fuzzy system has Rrules, and i is the value of the membership function for the premise of theith rule given the inputs X, W It is assumed that the fuzzy system is

constructed in such a way that 0  i  1 and PR

iˆ1i6ˆ 0 for all X 2 Rn,

W 2 Rq The parameter ci is the consequent of the ith rule, which in thischapter will be taken as a linear combination of Lipschitz continuousfunctions, zk…X† 2 R; k ˆ 1; ; m 1, so that ciˆ ai;0‡ ai;1z1…X† ‡


‡

ai;m 1zm 1…X†; i ˆ 1; ; R De®ne

290 Stable multi-input multi-output adaptive fuzzy/neural control

3775

Then, the nonlinear equation that describes the fuzzy system can be written as

~f…X; W† ˆ zT…X†A…X; W† (notice that standard fuzzy systems may be treated

as special cases of this more general representation) It was shown in [8] thatthe T±S model can represent a class of two-layer neural networks and manystandard fuzzy systems Note that while  may depend on both X and W and isbounded for any value they may take, z depends on X only This allows us toimpose no restrictions on W to guarantee boundedness of the fuzzy system

We will represent the ith component of the ideal control (11.4), i ˆ 1; ; p,

do not restrict the sizes of Sxand Sw; however, the sup in (11.8) is assumed toexist) As a result of the proof we will be able to determine that X actuallyremains within a compact subset of Sx Note that the ideal control law (11.4) is

a function not only of the states, but also of m, which may depend on variablesother than the states (as will be described below) The vector W is provided toaccount for this dependence The term uk i represents a known part of the idealcontrol input, which may be available to the designer through knowledge of theplant or expertise If it is not available, this term may be set equal to zero withall the properties of the adaptive controller still holding The only restriction on

uk is that it must be bounded Note that an appropriate use of uk may help to

Adaptive Control Systems 291

signi®cantly improve the performance of the controller, even though inprinciple it has no e€ect on stability Thus, the fuzzy system approximation

The desired reference trajectories ym iare ritimes continuously di€erentiable,with ymi; ; y…ri†

m i measurable and bounded, for i ˆ 1; ; p.

We de®ne the output errors eoi:ˆ ymi yi De®ne also the error signals

i ˆ 1; ; p, are picked so that the transfer functions are stable Let the ith

component of the parameter m in (11.4) be given by i:ˆ y …ri†

m i ‡ iesi‡ esi,where i> 0 is a constant Consider the control law

so that

292 Stable multi-input multi-output adaptive fuzzy/neural control

[12] Tao, G and KokotovicÂ, P V (1996) Adaptive Control of Systems... error convergent to zero with anadaptive inverse is still not available for a general adaptive inverse controldesign Moreover, adaptive inverse control designs for systems with unknownmultivariable... `Discrete-time Adaptive Control of Systems withMulti-segment Piecewise-linear Nonlinearities'', Proceedings of the 1995 AmericanControl Conference, 3019±3024, Seattle, WA

Adaptive Control Systems 285