Adaptive Control System Part 14 pptx

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12 Adaptive robust control scheme with an application

in this chapter is only partially known with unknown parameters Moreover,the bounding function is convex to the set of unknown parameters, i.e thebounding function is no longer linear in parameters The structured uncer-tainty is estimated with adaptation and compensated Meanwhile, the adaptiverobust method is applied to deal with the non structured uncertainty byestimating unknown parameters in the upper bounding function The -modi®cation scheme [1] is used to cease parameter adaptation in accordancewith the adaptive robust control law The backstepping method [2] is alsoadopted in this chapter to deal with a system not in the parametric±purefeedback form, which is usually necessary for the application of backsteppingcontrol scheme The new control scheme guarantees the uniform boundedness

of the system and at the same time, the tracking error enters an arbitrarilydesignated zone in a ®nite time The e€ectiveness of the proposed method isdemonstrated by the application to PMsynchronous motors

12.1 Introduction

Numerous adaptive robust control algorithms for systems containing tainties have been developed [1]±[11] In [3] variable structure control with anadaptive law is developed for an uncertain input±output linearizable nonlinearsystem, where linearity-in-parameter condition for uncertainties is assumed.The unknown gain of the upper bounding function is estimated and updated byadaptation law so that the sliding condition can be met and the error statereaches the sliding surface and stays on it To deal with a class of nonlinearsystems with partially known uncertainties, in [4] an adaptive law using a deadzone and a hysteresis function is proposed to guarantee both the uniformboundedness of all the closed loop signals and uniform ultimate boundedness

uncer-of the system states In both control schemes, it is assumed that the systemuncertainties are bounded by a bounding function which is a product of a set ofknown functions and unknown positive constants The objective of adaptation

is to estimate these unknown constants

In [1], a new adaptive robust control scheme is developed for a class ofnonlinear uncertain systems with both parameter uncertainties and exogenousdisturbances Including the categories of system uncertainties in [3] and [4] asits subsets, it is assumed that the disturbances are bounded by a known upperbounding function Furthermore, the input distribution matrix is assumed to

be constant but unknown

In this chapter we proposed a continuous adaptive robust control schemewhich is the extension of [1] in the sense that more general classes of nonlinearuncertain dynamical systems are under consideration The unknown inputdistribution matrix of the system input can be state dependent here instead ofbeing a constant matrix in [1] To reduce the robust control gain and widen theapplication scope of adaptive techniques, the system uncertainties are supposed

to be composed of two di€erent categories: the ®rst can be separated andexpressed as the product of known function of states and a set of unknownconstants, and the other category is not separable but with partially knownbounding functions It is further assumed that the bounding function is convex

to the set of unknown parameters, i.e the bounding function is no longer linear

in parameters The ®rst category of uncertainties is dealt with by means of thewell-used adaptive control method Meanwhile an adaptive robust method isapplied to deal with the second category of uncertainties, where the unknownparameters in the upper bounding function are estimated with adaptation Itshould also be noted that the backstepping method [2] is adopted in thischapter to deal with a system not in the parametric±pure feedback form, which

is usually necessary for the application of a backstepping control scheme.The proposed method is further applied to a permanent magnet synchronous(PMS) motor, which is a typical nonlinear control system The dynamics of thePMsynchronous motor can be presented by a dynamic electrical subsystem

and a mechanical subsystem, which are nonlinear di€erential equations.Strictly speaking, most control methods for permanent magnet synchronousmotors are only locally stable because the d-axis current is assumed to be zeroand the design procedure is based on the reduced model In this chapter,instead of only zeroing d-axis current, the extra d-axis control input voltage isused to deal with the nonlinear coupling part of the dynamics as well.This chapter is organized as follows Section 12.2 describes the class ofnonlinear uncertain systems to be controlled Section 12.3 gives the designprocedure of the adaptive robust control and the stability analysis Section 12.4describes the application of the proposed control method to the PMsynchro-nous motors.

where xiˆ ‰xi1; xi2; ; xiniŠ > 2 Rn i; i ˆ 0; 1; 2; are the measurable state

vec-tors of the system, where n0ˆ n1 and n0‡ n1‡ n2 ˆ n; x 2 Rn is de®ned as

x ˆ ‰x >

0; x>

1; x>

2Š >; uiˆ ‰ui1; ui2; ; uiniŠ > 2 Rn i, i ˆ 1; 2, are the control inputs

of the system; p 2 P is an unknown system parameter vector and P is the set of

admissible system parameters; fi2 Rn i, i ˆ 0; 1; 2, are nonlinear function

vectors; gi2 Rn i , i ˆ 0; 1; 2, and 4g02 Rn 0n2, 4gi2 Rn i, i ˆ 1; 2, are

non-linear uncertain function vectors of the state x, unknown parameter p, time t aswell as a set of random variables ! Here we make the following assumptions:(A1) f0…t†, f1…x; t† and f2…x; t† are known nonlinear function vectors The

matrices Bi…p†, i ˆ 0; 1; 2, are unknown but positive de®nite.

where …
† indicates the eigenvalues of `
'.

(A3) The structured uncertainty gi2 Rn i, i ˆ 0; 1; 2, are nonlinear function

vectors which can be expressed as

310 Adaptive robust control scheme: an application to PM synchronous motors

known function vectors The nonstructured uncertainty 4gi…x; p; !; t†,

i ˆ 0; 1; 2, are bounded such that

where jj
jj represents the Euclidean norm for vectors and the spectral norm for matrices; D is a compact subset of Rn in which the solution of (12.1)±(12.3)uniquely exists with respect to the given desired state trajectory xd…t†.

vectors qi2 P Here d i…x; qi; t† is di€erentiable and convex to qi, that is

di…x; qi2; t† d…x; qi1; t†  …qi2 qi1† >@d

@qi

The control objective is to ®nd suitable control inputs u1and u2for the state

x0 to track the desired trajectory xd…t† 2 Rn 0, where xd is continuouslydi€erentiable

Remark 2.1 The sub system (12.1) has x1as its input However, it is not in theparametric±pure feedback form due to the existence of the nonlinear uncertain

term 4g0…x; p; !; t†x2 Thus the well-used backstepping design needs to berevised to deal with the dynamical system (12.1)±(12.3)

Remark 2.2 It should be noted that gi…x; p; t† can be absorbed into

shown through the following example Assume that the structured uncertainty

is g ˆ 11‡ 22 with actual values 1 ˆ a, 2ˆ a and a is an unknown

constant Assume that the nonlinear function 2ˆ 1‡ 4, where

1jj; jj2jjŠ > ˆ ‰jjajj; jjajjŠ > and jj1 ˆ ‰jj1jj; jj2jjŠ > The upperbound

> This implies that the actual

uncertainty g ˆ a4 has been ampli®ed to its normed product

jjajj
jj4jj even if the estimates converge to the true values On the contrary,

if the uncertainty is expressed by (12.5), the unknown parameters to be

estimated is ‰a; aŠ > This means that, when the estimated parameters are nearthe true values, the estimated uncertainty of g will be able to approach the

actual uncertainty a4.

12.3 Adaptive robust control with l-modi®cation

The adaptive robust technique is combined with backstepping method in thissection to develop a controller which guarantees the global boundedness of thesystem The design procedures are presented in detail as follows

De®ne the measured state tracking error vector as

1 is de®ned as

xref

where K0 is a gain matrix ^0 and ^0 are the estimates of 0 and 0

respectively The ®rst order derivative of xref

where ^qi; ^i; ^i are the estimates of qi; i; i, i ˆ 0; 1; 2, respectively.

The control law ui, i ˆ 1; 2, are chosen to be

1 ; t†; The corresponding adaptive

laws are de®ned as

where ij; j ˆ 1; 2; 3 are positive de®nite matrices chosen to be

ij; j ˆ 1; 2; 3, which constitute the -modi®cation scheme, are de®ned as

Theorem 3.1 By properly choosing the control gain matrix, the proposedadaptive robust control law (12.20)±(12.24) ensures that the system trajectoryenters the set E0 in a ®nite time Moreover, the tracking errors as well as theparameter estimation errors are bounded by the set

matrix A respectively, and " and  are positive values to be de®ned later.Proof The following positive de®nite function V is selected

Take the derivatives of V1, V2 and V3 along the trajectory of the dynamicsystem (12.13)±(12.15), we have (See Appendices A±C)

where K ˆ diag ‰K0; K1; K2Š, and " ˆP2iˆ0"d i‡P2iˆ1"v i

By choosing K such that

continuous function We can show that there exists a constant 0 < "0

0< "0suchthat (see Appendix E)

12.4 Application to PM synchronous motors

Model of permanent magnet synchronous motor A permanent magnet chronous motor (PMSM) is described by the following subsystems: (1) adynamic mechanical subsystem, which for the purposes of this discussionincludes a single-link robot manipulator and the motor rotor; (2) a dynamicelectrical subsystem which includes all of the motor's relevant electrical e€ects

and ud are the input control voltages; Idand Iqare the motor armature current;

R is the stator resistance; Ld and Lq are the self-inductances; J is the inertiaangular momentum; and f is the ¯ux due to permanent magnet For the aboveelectromechanical model, we assume that the true states (i.e., , !, Id and Iq) areall measurable This model is obtained by using circuits theory principles and aparticular dq reference frame The control objective is to develop a linkposition tracking controller for the electromechanical dynamics of (12.40)±(12.43) despite parametric uncertainty In this chapter we assume that all themotor parameters are unknown

Remark 4.1 In most existing control schemes for PMsynchronous motors,the controllers are designed based on the following reduced model

uq Obviously, based on the reduced model, the control design will result inonly a locally stable controller.

Control Design For a given desired tracking state d…t†, de®ne a quantity z0

to be

where d…t† is at least twice continuously di€erentiable Di€erentiating (12.47),

multiplying by J and substituting the mechanical subsystem dynamics of(12.41), yields

where the unknown constant parameter vectors 2, 3 and 4 and the known

318 Adaptive robust control scheme: an application to PM synchronous motors

regression vectors '2, '3 and '4 are de®ned as (see Appendix E for thederivation of 2, 3, '2 and '3)

Note that in practical applications, the parameters J and T are constants butmay vary in a wide range due to the variation of payload On the other hand,the unknown motor parameters have fewer deviations from its rated values(nominal values) in comparison with that of load Therefore, it would be moreappropriate for us to deal with the unknown parameters J and T by usingadaptive techniques and treat the bounded motor parameters by using robustmethods In this way, referring to (12.20)±(12.24) the control inputs with thecorresponding adaptive laws are given as

12.3 Adaptive robust control with l-modi®cation

The adaptive robust technique is combined with backstepping method in thissection to develop a controller which guarantees... as

Theorem 3.1 By properly choosing the control gain matrix, the proposedadaptive robust control law (12.20)±(12.24) ensures that the system trajectoryenters the set E0 in... following subsystems: (1) adynamic mechanical subsystem, which for the purposes of this discussionincludes a single-link robot manipulator and the motor rotor; (2) a dynamicelectrical subsystem which