Adaptive control systems gang feng rogelio lozano

8 Optimal adaptive tracking for nonlinear systems 18410.5 Output feedback inverse control 269 10.7 Designs for multivariable systems 27510.8 Designs for nonlinear dynamics 281 11 Stable

Adaptive Control

Systems GANG FENG and ROGELIO LOZANO

OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI

An imprint ofButterworth-Heinemann

Linacre House, Jordan Hill, Oxford OX2 8DP

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A member ofthe Reed Elsevier plc group

First published 1999

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Typeset by David Gregson Associates, Beccles, Su€olk

Printed in Great Britain by Biddles Ltd, Guildford, Surrey

8 Optimal adaptive tracking for nonlinear systems 184

10.5 Output feedback inverse control 269

10.7 Designs for multivariable systems 27510.8 Designs for nonlinear dynamics 281

11 Stable multi-input multi-output adaptive fuzzy/neural control 287

11.3 Indirect adaptive control 296

12 Adaptive robust control scheme with an application to PM

Appendix E De®nitions of 2, 3, '2, and '3 326

List of contributors

Amit Ailon

Department ofElectrical and Computer Engineering

Ben Gurion University ofthe Negev

Israel

Anuradha M Annaswamy

Adaptive Control Laboratory

Department ofMechanical Engineering

Massachusetts Institute ofTechnology

Cambridge

MA02139

USA

Chiang-Ju Chien

Department ofElectronic Engineering

Hua Fan University

Taipei

Taiwan

ROC

Aniruddha Datta

Department ofElectrical Engineering

Texas A & M University

College Station

TX 77843±3128, USA

Dimitrios Dimogianopoulos

Universite de Technologie de Compiegne

HEUDIASYC UMR 6599 CNRS-UTC BP 20.529

60200 Compiegne

France

Gang Feng

School ofElectical Engineering

University ofNew South Wales

Department ofElectrical and Computer Engineering

The University ofNewcastle

Department ofElectrical Engineerng

National University ofSingapore

10 Kent Ridge Crescent

Singapore 119260

Y A Jiang

School ofElectrical Engineering

University ofNew South Wales

Sydney

NSW 2052

USA

Department ofElectrical Engineering

National University ofSingapore

10 Kent Ridge Crescent

Department ofElectrical Engineering

National University ofSingapore

Singapore 119260

Rogelio Lozano

Universite de Technologie de Compiegne

HEUDIASYC UMR 6599 CNRS-UTC BP 20.529

60200 Compiegne

France

Department ofElectrical Engineering

The Ohio State University

Department ofElectrical Engineering

The Ohio State University

Department ofElectrical Engineeing

Texas A&M University

College Station

TX 77843-3128

USA

J X Xu

Department ofElectrical Engineering

National University ofSingapore

10 Kent Ridge Crescent

Singapore 119260

Department ofElectrical and Computer Engineering

The University ofNewcastle

NSW 2308

Australia

R Zmood

Department ofCommunication and Electrical Engineering

Royal Melbourne Institute ofTechnology

Melbourne

Victoria 3001

Australia

Adaptive control has been extensively investigated and developed in boththeory and application during the past few decades, and it is still a very activeresearch ®eld In the earlier stage, most studies in adaptive control concen-trated on linear systems A remarkable development ofthe adaptive controltheory is the resolution ofthe so-called ideal problem, that is, the proofthatseveral adaptive control systems are globally stable under certain ideal con-ditions Then the robustness issues ofadaptive control with respect to non-ideal conditions such as external disturbances and unmodelled dynamics wereaddressed which resulted in many di€erent robust adaptive control algorithms.These robust algorithms include dead zone, normalization, "-modi®cation, e1-modi®cation among many others At the same time, extensive study has beencarried out for reducing a priori knowledge of the systems and improving thetransient performance of adaptive control systems Most recently, adaptivecontrol ofnonlinear systems has received great attention and a number ofsigni®cant results have been obtained

In this book, we have compiled some ofthe most recent developments ofadaptive control for both linear and nonlinear systems from leading worldresearchers in the ®eld These include various robust techniques, performanceenhancement techniques, techniques with less a priori knowledge, adaptiveswitching techniques, nonlinear adaptive control techniques and intelligentadaptive control techniques Each technique described has been developed toprovide a practical solution to a real-life problem This volume will thereforenot only advance the ®eld ofadaptive control as an area ofstudy, but will alsoshow how the potential ofthis technology can be realized and o€er signi®cantbene®ts

The ®rst contribution in this book is `Adaptive internal model control' by A.Datta and L Xing It develops a systematic theory for the design and analysisofadaptive internal model control schemes The ubiquitous certainty equiva-

lence principle ofadaptive control is used to combine a robust adaptive lawwith robust internal model controllers to obtain adaptive internal modelcontrol schemes which can be proven to be robustly stable Speci®c controllerstructures considered include those ofthe model reference, partial poleplacement, and H2 and H1 optimal control types The results here not onlyprovide a theoretical basis for analytically justifying some of the reportedindustrial successes ofexisting adaptive internal model control schemes butalso open up the possibility ofsynthesizing new ones by simply combining arobust adaptive law with a robust internal model controller structure.The next contribution is `An algorithm for robust direct adaptive controlwith less prior knowledge' by G Feng, Y A Jiang and R Zmood It discussesseveral approaches to minimizing a priori knowledge required on the unknownplants for robust adaptive control It takes a discrete time robust directadaptive control algorithm with a dead zone as an example It shows thatfor a class of unmodelled dynamics and bounded disturbances, no knowledgeofthe parameters ofthe upper bounding function on the unmodelled dynamicsand disturbances is required a priori Furthermore it shows that a correctionprocedure can be employed in the least squares estimation algorithm so that noknowledge ofthe lower bound on the leading coecient ofthe plant numeratorpolynomial is required to achieve the singularity free adaptive control law Theglobal stability and convergence results ofthe algorithm are established.The next contribution is `Adaptive variable structure control' by C J.Chiang and Lichen Fu A uni®ed algorithm is presented to develop the variablestructure MRAC for an SISO system with unmodelled dynamics and outputmeasurement noises The proposed algorithm solves the robustness andperformance problem of the traditional MRAC with arbitrary relativedegree It is shown that without any persistent excitation the output trackingerror can be driven to zero for relative degree-one plants and driven to a smallresidual set asymptotically for plants with any higher relative degree.Furthermore, under suitable choice ofinitial conditions on control parameters,the tracking performance can be improved, which is hardly achievable by thetraditional MRAC schemes, especially for plants with uncertainties.

The next contribution is `Indirect adaptive periodic control' by D.Dimogianopoulos, R Lozano and A Ailon This new, indirect adaptivecontrol method is based on a lifted representation of the plant which can bestabilized using a simple performant periodic control scheme The controllerparameters computation involves the inverse ofthe controllability/observa-bility matrix Potential singularities ofthis matrix are avoided by means ofanappropriate estimates modi®cation This estimates transformation is linked tothe covariance matrix properties and hence it preserves the convergenceproperties ofthe original estimates This modi®cation involves the singularvalue decomposition ofthe controllability/observability matrix's estimate Ascompared to previous studies in the subject the controller proposed here does

not require the frequent introduction of periodic n-length sequences ofzeroinputs Therefore the new controller is such that the system is almost alwaysoperating in closed loop which should lead to better performancecharacteristics.

The next contribution is `Adaptive stabilization ofuncertain discrete-timesystems via switching control: the method oflocalization' by P V.Zhivoglyadov, R Middleton and M Fu It presents a new systematic switchingcontrol approach to adaptive stabilization ofuncertain discrete-time systems.The approach is based on a method oflocalization which is conceptuallydi€erent from supervisory adaptive control schemes and other existing switch-ing control schemes The proposed approach allows for slow parameterdrifting, infrequent large parameter jumps and unknown bound on exogenousdisturbances The unique feature of the localization-based switching adaptivecontrol proposed here is its rapid model falsi®cation capability In the LTI casethis is manifested in the ability of the switching controller to quickly converge

to a suitable stabilizing controller It is believed that the approach is applicable

to a wide class oflinear time invariant and time-varying systems with goodtransient performance

The next contribution is `Adaptive nonlinear control: passivation and smallgain techniques' by Z P Jiang and D Hill It proposes methods to system-atically design stabilizing adaptive controllers for new classes of nonlinearsystems by using passivation and small gain techniques It is shown that for aclass oflinearly parametrized nonlinear systems with only unknown param-eters, the concept ofadaptive passivation can be used to unify and extend mostofthe known adaptive nonlinear control algorithms based on Lyapunovmethods A novel recursive robust adaptive control method by means ofbackstepping and small gain techniques is also developed to generate a newclass ofadaptive nonlinear controllers with robustness to nonlinear un-modelled dynamics

The next contribution is `Active identi®cation for control of discrete-timeuncertain nonlinear systems' by J Zhao and I Kanellakopoulos A novelapproach is proposed to remove the restrictive growth conditions ofthenonlinearities and to yield global stability and tracking for systems that can

be transformed into an output-feedback canonical form The main novelties ofthe design are (i) the temporal and algorithmic separation ofthe parameterestimation task from the control task and (ii) the development of an activeidenti®cation procedure, which uses the control input to actively drive thesystem state to points in the state space that allow the orthogonalizedprojection estimator to acquire all the necessary information about theunknown parameters It is proved that the proposed algorithm guaranteescomplete identi®cation in a ®nite time interval and global stability andtracking

The next contribution is `Optimal adaptive tracking for nonlinear systems'

by M KrsÏtic and Z H Li In this chapter an `inverse optimal' adaptivetracking problem for nonlinear systems with unknown parameters is de®nedand solved The basis ofthe proposed method is an adaptive tracking controlLyapunov function (atclf) whose existence guarantees the solvability of theinverse optimal problem The controllers designed are not ofcertaintyequivalence type Even in the linear case they would not be a result ofsolving

a Riccati equation for a given value of the parameter estimate Inverseoptimality is combined with backstepping to design a new class ofadaptivecontrollers for strict-feedback systems These controllers solve a problem leftopen in the previous adaptive backstepping designs ± getting transient per-formance bounds that include an estimate of control e€ort

The next contribution is `Stable adaptive systems in the presence linear parameterization' by A M Annaswamy and A P Loh This chapteraddresses the problem ofadaptive control when the unknown parametersoccur nonlinearly in a dynamic system The traditional approach used inlinearly parameterized systems employs a gradient-search principle in estimat-ing the unknown parameters Such an approach is not sucient for nonlinearlyparametrized systems Instead, a new algorithm based on a min±max optimiza-tion scheme is developed to address nonlinearly parametrized adaptive systems

ofnon-It is shown that this algorithm results in globally stable closed loop systemswhen the states ofthe plant are accessible for measurement

The next contribution is `Adaptive inverse for actuator compensation' by G.Tao A general adaptive inverse approach is developed for control of plantswith actuator imperfections caused by nonsmooth nonlinearities such as dead-zone, backlash, hysteresis and other piecewise-linear characteristics Anadaptive inverse is employed for cancelling the e€ect of an unknown actuatornonlinearity, and a linear feedback control law is used for controlling thedynamics ofa known linear or smooth nonlinear part following the actuatornonlinearity State feedback and output feedback control designs are presentedwhich all lead to linearly parametrized error models suitable for the develop-ment ofadaptive laws to update the inverse parameters This approachsuggests that control systems with commonly used linear or nonlinear feedbackcontrollers such as those with an LQ, model reference, PID, pole placement orother dynamic compensation design can be combined with an adaptive inversefor improving system tracking performance despite the presence of actuatorimperfections

The next contribution is `Stable multi-input multi-output adaptive fuzzy/neural control' by R OrdoÂnÄez and K Passino In this chapter, stable direct andindirect adaptive controllers are presented which use Takagi±Sugeno fuzzysystems, conventional fuzzy systems, or a class of neural networks to provideasymptotic tracking of a reference signal vector for a class of continuous timemulti-input multi-output (MIMO) square nonlinear plants with poorly under-

stood dynamics The direct adaptive scheme allows for the inclusion of a prioriknowledge about the control input in terms ofexact mathematical equations orlinguistics, while the indirect adaptive controller permits the explicit use ofequations to represent portions ofthe plant dynamics It is shown that with orwithout such knowledge the adaptive schemes can `learn' how to control theplant, provide for bounded internal signals, and achieve asymptotically stabletracking ofthe reference inputs No initialization condition needs to beimposed on the controllers, and convergence ofthe tracking error to zero isguaranteed.

The ®nal contribution is `Adaptive robust control scheme with an tion to PM synchronous motors' by J X Xu, Q W Jia and T H Lee A new,adaptive, robust control scheme for a class of nonlinear uncertain dynamicalsystems is presented To reduce the robust control gain and widen theapplication scope ofadaptive techniques, the system uncertainties are classi®edinto two di€erent categories: the structured and nonstructured uncertaintieswith partially known bounding functions The structured uncertainty isestimated with adaptation and compensated Meanwhile, the adaptive robustmethod is applied to deal with the non-structured uncertainty by estimatingunknown parameters in the upper bounding function It is shown that the newcontrol scheme guarantees the uniform boundedness of the system and assuresthe tracking error entering an arbitrarily designated zone in a ®nite time Thee€ectiveness ofthe proposed method is demonstrated by the application to PMsynchronous motors

applica-Adaptive internal model

1.1 Introduction

Internal model control (IMC) schemes, where the controller implementationincludes an explicit model of the plant, continue to enjoy widespreadpopularity in industrial process control applications [1] Such schemes canguarantee internal stability for only open loop stable plants; since most plantsencountered in process control are anyway open loop stable, this really doesnot impose any signi®cant restriction

As already mentioned, the main feature of IMC is that its implementationrequires an explicit model of the plant to be used as part of the controller.When the plant itself happens to be unknown, or the plant parameters varyslowly with time due to ageing, no such model is directly available a priori andone has to resort to identi®cation techniques to come up with an appropriateplant model on-line Several empirical studies, e.g [2], [3] have demonstratedthe feasibility of such an approach However, what is, by and large, lacking inthe process control literature is the availability of results with solid theoreticalguarantees of stability and performance.

Motivated by this fact, in [4], [5], we presented designs of adaptive IMCschemes with provable guarantees of stability and robustness The scheme in [4]involved on-line adaptation of only the internal model while in [5], in addition

to adapting the internal model on-line, the IMC parameter was chosen in acertainty equivalence fashion to pointwise optimize an H2 performance index

In this chapter, it is shown that the approach of [5] can be adapted to designand analyse a class of adaptive H1 optimal control schemes that are likely toarise in process control applications This class speci®cally consists of those H1

norm minimization problems that involve only one interpolation constraint.Additionally, we reinterpret the scheme of [4] as an adaptive `partial' pole-placement control scheme and consider the design and analysis of a modelreference adaptive control scheme based on the IMC structure In other words,this chapter considers the design and analysis of popular adaptive controlschemes from the literature within the context of the IMC con®guration Asingle, uni®ed, analysis procedure, applicable to each of the schemes con-sidered, is also presented

The chapter is organized as follows In Section 1.2, we present severalnonadaptive control schemes utilizing the IMC con®guration Their adaptivecertainty equivalence versions are presented in Section 1.3 A uni®ed stabilityand robustness analysis encompassing all of the schemes of Section 1.3 ispresented in Section 1.4 In Section 1.5, we present simulation examples todemonstrate the ecacy of our adaptive IMC designs Section 1.6 concludesthe chapter by summarizing the main results and outlining their expectedsigni®cance

1.2 Internal model control (IMC) schemes:

known parameters

In this section, we present several nonadaptive control schemes utilizing theIMC structure To this end, we consider the IMC con®guration for a stable

plant P…s† as shown in Figure 1.1 The IMC controller consists of a stable

`IMC parameter' Q…s† and a model of the plant which is usually referred to as the `internal model' It can be shown [1, 4] that if the plant P…s† is stable and

the internal model is an exact replica of the plant, then the stability of the IMCparameter is equivalent to the internal stability of the con®guration in Figure1.1 Indeed, the IMC parameter is really the Youla parameter [6] that appears

in a special case of the YJBK parametrization of all stabilizing controllers [4]

Because of this, internal stability is assured as long as Q…s† is chosen to be any

stable rational transfer function We now show that di€erent choices of stable

Q…s† lead to some familiar control schemes.

1.2.1 Partial pole placement control

From Figure 1.1, it is clear that if the internal model is an exact replica of theplant, then there is no feedback signal in the loop Consequently the poles ofthe closed loop system are made up of the open loop poles of the plant and the

poles of the IMC parameter Q…s† Thus, in this case, a `complete' pole

placement as in traditional pole placement control schemes is not possible

Instead, one can only choose the poles of the IMC parameter Q…s† to be in

some desired locations in the left half plane while leaving the remaining poles at

the plant open loop pole locations Such a control scheme, where Q…s† is

chosen to inject an additional set of poles at some desired locations in thecomplex plane, is referred to as `partial' pole placement

1.2.2 Model reference control

The objective in model reference control is to design a di€erentiator-free

controller so that the output y of the controlled plant P…s† asymptotically

tracks the output of a stable reference model Wm…s† for all piecewise

continuous reference input signals r…t† In order to meet the control objective,

we make the following assumptions which are by now standard in the modelreference control literature:

(M1) The plant P…s† is minimum phase; and

(M2) The relative degree of the reference model transfer function Wm…s† is

greater than or equal to that of the plant transfer function P…s†.

Assumption (M1) above is necessary for ensuring internal stability sincesatisfaction of the model reference control objective requires cancellation ofthe plant zeros Assumption (M2), on the other hand, permits the design of adi€erentiator-free controller to meet the control objective If assumptions (M1)and (M2) are satis®ed, it is easy to verify from Figure 1.1 that the choice

Q…s† ˆ Wm…s†P 1…s† …1:1†

for the IMC parameter guarantees the satisfaction of the model referencecontrol objective in the ideal case, i.e in the absence of plant modelling errors

1.2.3 H2 optimal control

In H2 optimal control, one chooses Q…s† to minimize the L2 norm of the

tracking error r y provided r y 2 L2 From Figure 1.1, we obtain

y ˆ P…s†Q…s†‰rŠ

) r y ˆ ‰1 P…s†Q…s†Š‰rŠ

)

Z 1

0 …r…† y…††2d ˆ k‰1 P…s†Q…s†ŠR…s†k … 2†2(using Parseval's Theorem)

where R…s† is the Laplace transform of r…t† and k
…s†k2 denotes the standard

H2 norm Thus the mathematical problem of interest here is to choose Q…s† to minimize k‰1 P…s†Q…s†ŠR…s†k2 The following theorem gives the analytical

expression for the minimizing Q…s† The detailed derivation can be found in [1] Theorem 2.1 Let P…s† be the stable plant to be controlled and let R…s† be the Laplace Transform of the external input signal r…t†1 Suppose that R…s† has no

poles in the open right half plane2and that there exists at least one choice, say

Q0…s†, of the stable IMC parameter Q…s† such that ‰1 P…s†Q0…s†ŠR…s† is

stable3 Let zp1; zp2; ; zpl be the open right half plane zeros of P…s† and de®ne

the Blaschke product4

2This assumption is reasonable since otherwise the external input would beunbounded

3The ®nal construction of the H2optimal controller serves as proof for the existence

of a Q0…s† with such properties.

4Here …
† denotes complex conjugation.

where PM…s† is minimum phase Similarly, let zr 1; zr 2; ; zr k be the open right

half plane zeros of R…s† and de®ne the Blashcke product

Q…s† in (1.2), the Q…s† to be implemented is given by

where F…s† is the stable IMC ®lter The design of the IMC ®lter for H2optimal

control depends on the choice of the input R…s† Although this design is carried

out in a somewhat ad hoc fashion, care is taken to ensure that the originalasymptotic tracking properties of the controller are preserved This is because

otherwise ‰1 P…s†Q…s†ŠR…s† may no longer be a function in H2 As a speci®cexample, suppose that the system is of Type 1.5Then, a possible choice for theIMC ®lter to ensure retention of asymptotic tracking properties is

F…s† ˆ 1

…s ‡ 1†n,  > 0 where n is chosen to be a large enough positive

integer to make Q…s† proper As shown in [1], the parameter  represents a

trade-o€ between tracking performance and robustness to modelling errors

1.2.4 H1 optimal control

The sensitivity function S…s† and the complementary sensitivity function T…s† for the IMC con®guration in Figure 1.1 are given by S…s† ˆ 1 P…s†Q…s† and

T…s† ˆ P…s†Q…s† respectively [1] Since the plant P…s† is open loop stable, it

follows that the H1 norm of the complementary sensitivity function T…s† can

be made arbitrarily small by simply choosing Q…s† ˆ1k and letting k tend to

5Other system types can also be handled as in [1]

in®nity Thus minimizing the H1 norm of T…s† does not make much sense

since the in®mum value of zero is unattainable

On the other hand, if we consider the weighted sensitivity minimization

problem where we seek to minimize kW…s†S…s†k 1 for some stable, minimum

phase, rational weighting transfer function W…s†, then we have an interesting

H1 minimization problem, i.e choose a stable Q…s† to minimize

kW…s†‰1 P…s†Q…s†Šk 1.The solution to this problem depends on the number

of open right half plane zeros of the plant P…s† and involves the use of Nevanlinna±Pick interpolation when the plant P…s† has more than one right

half plane zero [7] However, when the plant has only one right half plane zero

b1 and none on the imaginary axis, there is only one interpolation constraintand the closed form solution is given by [7]

Q…s† ˆ ‰1 W…b W…s†1† ŠP 1…s† …1:4†

Fortunately, this case covers a large number of process control applicationswhere plants are typically modelled as minimum phase ®rst or second ordertransfer functions with time delays Since approximating a delay using a ®rstorder Pade approximation introduces one right half plane zero, the resultingrational approximation will satisfy the one right half plane zero assumption.Remark 2.2 As in the case of H2optimal control, the optimal Q…s† de®ned by

(1.4) is usually improper This situation can be handled as in Remark 2.1 so

that the Q…s† to be implemented becomes

where F…s† is a stable IMC ®lter In this case, however, there is more freedom

in the choice of F…s† since the H 1optimal controller (1.4) does not necessarilyguarantee any asymptotic tracking properties to start with

1.2.5 Robustness to uncertainties (small gain theorem)

In the next section, we will be combining the above schemes with a robustadaptive law to obtain adaptive IMC schemes If the above IMC schemes areunable to tolerate uncertainty in the case where all the plant parameters areknown, then there is little or no hope that certainty equivalence designs based

on them will do any better when additionally the plant parameters areunknown and have to be estimated using an adaptive law Accordingly, wenow establish the robustness of the nonadaptive IMC schemes to the presence

of plant modelling errors Without any loss of generality let us suppose that theuncertainty is of the multiplicative type, i.e

P…s† ˆ P0…s†…1 ‡ 
m…s†† …1:6†

where P0…s† is the modelled part of the plant and 
m…s† is a stable

multi-plicative uncertainty such that P0…s†
m…s†is strictly proper Then we can state

the following robustness result which follows immediately from the small gaintheorem [8] A detailed proof can also be found in [1]

Theorem 2.2 Suppose P0…s† and Q…s† are stable transfer functions so that the

IMC con®guration in Figure 1.1 is stable for P…s† ˆ P0…s† Then the IMC

con®guration with the actual plant given by (1.6) is still stable provided

 2 ‰0;   † where   ˆ kP 1

0…s†Q…s†
m…s†k 1.

1.3 Adaptive internal model control schemes

In order to implement the IMC-based controllers of the last section, the plantmust be known a priori so that the `internal model' can be designed and the

IMC parameter Q…s† calculated When the plant itself is unknown, the

IMC-based controllers cannot be implemented In this case, the natural approach tofollow is to retain the same controller structure as in Figure 1.1, with theinternal model being adapted on-line based on some kind of parameter

estimation mechanism, and the IMC parameter Q…s† being updated pointwise

using one of the above control laws This is the standard certainty equivalenceapproach of adaptive control and results in what are called adaptive internalmodel control schemes Although such adaptive IMC schemes have beenempirically studied inthe literature, e.g [2, 3], our objective here is to developadaptive IMC schemes with provable guarantees of stability and robustness

To this end, we assume that the stable plant to be controlled is described by

P…s† ˆZR0…s†

0…s† ‰1 ‡ 
m…s†Š;  > 0 …1:7†

where R0…s† is a monic Hurwitz polynomial of degree n; Z0…s† is a polynomial

of degree l with l < n; ZR0…s†

0…s†represents the modelled part of the plant; and


m…s† is a stable multiplicative uncertainty such that ZR0…s†

0…s†
m…s† is strictly

proper We next present the design of the robust adaptive law which is carriedout using a standard approach from the robust adaptive control literature [9]

1.3.1 Design of the robust adaptive law

We start with the plant equation

y ˆZR0…s†

0…s† ‰1 ‡ 
m…s†Š‰uŠ;  > 0 …1:8†

where u, y are the plant input and output signals This equation can berewritten as

R0…s†‰yŠ ˆ Z0…s†‰uŠ ‡ 
m…s†Z0…s†‰uŠ

Filtering both sides by 1…s†, where …s† is an arbitrary, monic, Hurwitzpolynomial of degree n, we obtain

y ˆ …s† R …s†0…s† ‰yŠ ‡Z…s†0…s† ‰uŠ ‡
m…s† …s†Z0…s† ‰uŠ …1:9†

The above equation can be rewritten as

 ˆD
m…s†Z0…s†

…s† ‰uŠ …1:11†

Equation (1.10) is exactly in the form of the linear parametric model withmodelling error for which a large class of robust adaptive laws can bedeveloped In particular, using the gradient method with normalization andparameter projection, we obtain the following robust adaptive law [9]

is a known compact convex set containing

 ; Pr‰
Š is the standard projection operator which guarantees that the eter estimate …t† does not exit the set Cand 0> 0 is a constant chosen so that

param-
m…s†, …s† are analytic in Re‰sŠ 1 20 This choice of o, of course,necessitates some a priori knowledge about the stability margin of the

unmodelled dynamics, an assumption which has by now become fairlystandard in the robust adaptive control literature [9] The robust adaptiveIMC schemes are obtained by replacing the internal model in Figure 1.1 by

that obtained from equation (1.14), and the IMC parameters Q…s† by

time-varying operators which implement the certainty equivalence versions of thecontroller structures considered in the last section The design of these certaintyequivalence controllers is discussed next

1.3.2 Certainty equivalence control laws

We ®rst outline the steps involved in designing a general certainty equivalenceadaptive IMC scheme Thereafter, additional simpli®cations or complexitiesthat result from the use of a particular control law will be discussed

Step 1: First use the parameter estimate …t† obtained from the robust

adaptive law (1.12)±(1.16) to generate estimates of the numerator anddenominator polynomials for the modelled part of the plant6

Step 2: Using the frozen time plant ^P…s; t† ˆZ^^0…s; t†

R0…s; t†, calculate the

appro-priate ^Q…s; t† using the results developed in Section 1.2.

Step 3: Express ^Q…s; t† as ^ Q…s; t† ˆQ^^n…s; t†

Qd…s; t† where ^Qn…s; t† and ^Qd…s; t† are

time-varying polynomials with ^Qd…s; t† being monic.

Step 4: Choose 1…s† to be an arbitrary monic Hurwitz polynomial of degree

equal to that of ^Qd…s; t†, and let this degree be denoted by nd Step 5: The certainty equivalence control law is given by

u ˆ qT

d…t†and 1…s†

1…s† ‰uŠ ‡ qTn…t†and…s†

1…s† ‰r "m2Š …1:17†

where qd…t† is the vector of coecients of 1…s† Q^d…s; t†; qn…t† is the

vector of coecients of Q^n…s; t†; an d…s† ˆ ‰sn d; sn d 1; ; 1ŠT and

Figure 1.2 We now proceed to discuss the simpli®cations or additionalcomplexities that result from the use of each of the controller structurespresented in Section 1.2.

1.3.2.1 Partial adaptive pole placement

In this case, the design of the IMC parameter does not depend on the estimated

plant Indeed, Q…s† is a ®xed stable transfer function and not a time-varying

operator so that we essentially recover the scheme presented in [4].Consequently, this scheme admits a simpler stability analysis as in [4] althoughthe general analysis procedure to be presented in the next section is alsoapplicable

1.3.2.2 Model reference adaptive control

In this case from (1.1), we see that the ^Q…s; t† in Step 2 of the certainty

equivalence design becomes

^

Q…s; t† ˆ Wm…s† ^P…s; t† 1 …1:18†

Our stability analysis to be presented in the next section is based on results inthe area of slowly time-varying systems In order for these results to beapplicable, it is required that the operator ^Q…s; t† be pointwise stable and

also that the degree of ^Qd…s; t† in Step 3 of the certainty equivalence design not

change with time These two requirements can be satis®ed as follows:

The pointwise stability of ^Q…s; t† can be guaranteed by ensuring that the

frozen time estimated plant is minimum phase, i.e ^Z0…s; t† is Hurwitz stable

for every ®xed t To guarantee such a property for ^Z0…s; t†, the projection set

C in (1.12)±(1.16) is chosen so that 8  2 C, the corresponding

projection set C cannot be speci®ed as a single convex set, results fromhysteresis switching using a ®nite number of convex sets [11] can be used The degree of ^Qd…s; t† can also be rendered time invariant by ensuring that

the leading coecient of ^Z0…s; t† is not allowed to pass through zero This

feature can be built into the adaptive law by assuming some knowledgeabout the sign and a lower bound on the absolute value of the leadingcoecient of Z0…s† Projection techniques, appropriately utilizing this knowl-

edge, are by now standard in the adaptive control literature [12]

We will therefore assume that for IMC-based model reference adaptive

control, the set C has been suitably chosen to guarantee that the estimate

…t† obtained from (1.12)±(1.16) actually satis®es both of the properties

mentioned above

1.3.2.3 Adaptive H2 optimal control

In this case, ^Q…s; t† is obtained by substituting ^P 1

M…s; t†, ^B 1

P …s; t† into the

right-hand side of (1.3) where ^PM…s; t† is the minimum phase portion of ^ P…s; t† and

corre-P …s; t† are removed, and F…s† is an IMC ®lter used

to force ^Q…s; t† to be proper As will be seen in the next section, speci®cally

Lemma 4.1, the degree of ^Qd…s; t† in Step 3 of the certainty equivalence design

can be kept constant using a single ®xed F…s† provided the leading coecient of

^

Z0…s; t† is not allowed to pass through zero Additionally ^Z0…s; t† should not

have any zeros on the imaginary axis A parameter projection modi®cation, as

in the case of model reference adaptive control, can be incorporated into theadaptive law (1.12)±(1.16) to guarantee both of these properties

1.3.2.4 Adaptive H1 optimal control

In this case, ^Q…s; t† is obtained by substituting ^ P…s; t† into the right-hand side of

where ^b1is the open right half plane zero of ^Z0…s; t† and F…s† is the IMC ®lter.

Since (1.20) assumes the presence of only one open right half plane zero, theestimated polynomial ^Z0…s; t† must have only one open right half plane zero

and none on the imaginary axis Additionally the leading coecient of ^Z0…s; t†

should not be allowed to pass through zero so that the degree of ^Qd…s; t† in Step

3 of the certainty equivalence design can be kept ®xed using a single ®xed F…s†.

Once again, both of these properties can be guaranteed by the adaptive law by

appropriately choosing the set C

Remark 3.1 The actual construction of the sets C for adaptive modelreference, adaptive H2 and adaptive H1 optimal control may not bestraightforward especially for higher order plants However, this is a well-known problem that arises in any certainty equivalence control scheme based

on the estimated plant and is really not a drawback associated with the IMCdesign methodology Although from time to time a lot of possible solutions tothis problem have been proposed in the adaptive literature, it would be fair tosay that, by and large, no satisfactory solution is currently available

1.4 Stability and robustness analysis

Before embarking on the stability and robustness analysis for the adaptiveIMC schemes just proposed, we ®rst introduce some de®nitions [9, 4] and stateand prove two lemmas which play a pivotal role in the subsequent analysis

De®nition 4.1 For any signal x : ‰0; 1† ! Rn, xtdenotes the truncation of x

to the interval ‰0; tŠ and is de®ned as

The k…
†tk2 represents the exponentially weighted L2 norm of the signal

truncated to ‰0; tŠ When  ˆ 0 and t ˆ 1, k…:†tk2 becomes the usual L2

norm and will be denoted by k:k2 It can be shown that k:k2 satis®es theusual properties of the vector norm

De®nition 4.3 Consider the signals x : ‰0; 1† ! Rn, y : ‰0; 1† ! R ‡ and theset

for some c  0 and 8 t; T  0 We say that x is y-small in the mean if x 2 S…y†.

Lemma 4.1 In each of the adaptive IMC schemes presented in the last section,the degree of ^Qd…s; t† in Step 3 of the certainty equivalence design can be made

time invariant Furthermore, for the adaptive H2 and H1 designs, this can be

done using a single ®xed F…s†.

Proof The proof of this lemma is relatively straightforward except in the case

of adaptive H2optimal control Accordingly, we ®rst discuss the simpler casesbefore giving a detailed treatment of the more involved one

For adaptive partial pole placement, the time invariance of the degree of

^

Qd…s; t† follows trivially from the fact that the IMC parameter in this case is

time invariant For model reference adaptive control, the fact that the leadingcoecient of ^Z0…s; t† is not allowed to pass through zero guarantees that the

degree of ^Qd…s; t† is time invariant Finally, for adaptive H 1 optimal control,the result follows from the fact that the leading coecient of ^Z0…s; t† is not

allowed to pass through zero

We now present the detailed proof for the case of adaptive H2 optimalcontrol Let nr; mr be the degrees of the denominator and numerator

polynomials respectively of R…s† Then, in the expression for ^ Q…s; t† in (1.19),

where n  nr, strict inequality being attained when some of the poles of

RM…s† coincide with some of the stable zeros of ^B 1

P …s; t† Moreover, in any

case, the nth order denominator polynomial of ‰ ^B 1

P …s; t†RM…s†Š  is a factor ofthe nrth order numerator polynomial of R 1

M…s† Thus for the ^ Q…s; t† given in (1.19), if we disregard F…s†, then the degree of the numerator polynomial is

n ‡ nr 1 while that of the denominator polynomial is l ‡ mr4 n ‡ nr 1.Hence, the degree of ^Qd…s; t† in Step 3 of the certainty equivalence design can

be kept ®xed at …n ‡ nr 1†, and this can be achieved with a single ®xed F…s† of relative degree n l ‡ nr mr 1, provided that the leading coecient of

^

Z0…s; t† is appropriately constrained.

Remark 4.1 Lemma 4.1 tells us that the degree of each of the certaintyequivalence controllers presented in the last section can be made timeinvariant This is important because, as we will see, it makes it possible tocarry out the analysis using standard state-space results on slowly time-varyingsystems

Lemma 4.2 At any ®xed time t, the coecients of ^Qd…s; t†, ^Qn…s; t†, and hence

the vectors qd…t†, qn…t†, are continuous functions of the estimate …t†.

Proof Once again, the proof of this lemma is relatively straightforward except

in the case of adaptive H2 optimal control Accordingly, we ®rst discuss thesimpler cases before giving a detailed treatment of the more involved one

For the case of adaptive partial pole placement control, the continuity

follows trivially from the fact that the IMC parameter is independent of …t†.

For model reference adaptive control, the continuity is immediate from (1.18)and the fact that the leading coecient of ^Z0…s; t† is not allowed to pass

through zero Finally for adaptive H1 optimal control, we note that the righthalf plane zero ^b1 of ^Z0…s; t† is a continuous function of …t† This is a

consequence of the fact that the degree of ^Z0…s; t† cannot drop since its leading

coecient is not allowed to pass through zero The desired continuity nowfollows from (1.20)

We now present the detailed proof for the H2optimal control case Since theleading coecient of ^Z0…s; t† has been constrained so as not to pass through

zero then, for any ®xed t, the roots of ^Z0…s; t† are continuous functions of …t†.

Hence, it follows that the coecients of the numerator and denominator

polynomials of‰ ^PM…s; t†Š 1ˆ ‰ ^BP…s; t†Š‰ ^ P…s; t†Š 1 are continuous functions of

…t† Moreover, ‰‰ ^BP…s; t†Š 1RM…s†Š  is the sum of the residues of

‰ ^BP…s; t†Š 1RM…s† at the poles of RM…s†, which clearly depends continuously

on …t† (through the factor ‰ ^BP…s; t†Š 1) Since F…s† is ®xed and independent of

, it follows from (1.19) that the coecients of ^Qd…s; t†, ^Qn…s; t† depend

continuously on …t†.

Remark 4.2 Lemma 4.2 is important because it allows one to translate slow

variation of the estimated parameter vector …t† to slow variation of the

controller parameters Since the stability and robustness proofs of mostadaptive schemes rely on results from the stability of slowly time-varyingsystems, establishing continuity of the controller parameters as a function ofthe estimated plant parameters (which are known to vary slowly) is a crucialingredient of the analysis

The following theorem describes the stability and robustness properties of theadaptive IMC schemes presented in this chapter

Theorem 4.1 Consider the plant (1.8) subject to the robust adaptive IMCcontrol law (1.12)±(1.16), (1.17), where (1.17) corresponds to any one of the

adaptive IMC schemes considered in the last section and r…t† is a bounded external signal Then, 9   > 0 such that 8  2 ‰0;   †, all the signals in the

closed loop system are uniformly bounded and the error y ^y 2 S cm222

forsome c > 07

7In the rest of this chapter, `c' is the generic symbol for a positive constant The exactvalue of such a constant can be determined (for a quantitative robustness result) as in[13, 9] However, for the qualitative presentation here, the exact values of theseconstants are not important

Proof The proof is obtained by combining the properties of the robustadaptive law (1.12)±(1.16) with the properties of the IMC-based controllerstructure We ®rst analyse the properties of the adaptive law.

From (1.10), (1.13) and (1.14), we obtain

 12"2m2‡12m222 (completing the squares) …1:24†

From (1.11), (1.15), (1.16), using Lemma 2.1 (Equation (7)) in [4], it followsthat m 2 L 1 Now, the parameter projection guarantees that V; ~;  2 L 1

Hence integrating both sides of (1.24) from t to t ‡ T, we obtain

This completes the analysis

of the properties of the robust adaptive law To complete the stability proof, wenow turn to the properties of the IMC-based controller structure

The certainty equivalence control law (1.17) can be rewritten as

the above equation can be rewritten as

37775

B ˆD

00

01

26664

37775

Since the time-varying polynomial ^Qd…s; t† is pointwise Hurwitz, it follows that

for any ®xed t, the eigenvalues of A…t† are in the open left half plane Moreover,

since the coecients of ^Qd…s; t† are continuous functions of …t† (Lemma 4.2)

and …t† 2 C, a compact set, it follows that 9 s > 0 such that

m2 L 1 , it follows from Lemma 3.1 in [9] that 9  1 > 0 such that

8  2 ‰0;   †, the equilibrium state xeˆ 0 of _x ˆ A…t†x is exponentially

stable, i.e there exist c0; p0 > 0 such that the state transition matrix …t; †

corresponding to the homogeneous part of (1.26) satis®es

k…t; †k  c0e p 0…t † 8 t   …1:27†

From the identity u ˆ1…s†

1…s† ‰uŠ, it is easy to see that the control input u can be

Also, using (1.28) in the plant equation (1.8), we obtain

Since "m 2 S m222

andm is bounded, it follows using Lemma 2.2 in [4] that

9   2 …0;  

1† such that 8  2 ‰0;   †, mf 2 L 1, which in turn implies that

m 2 L 1 Since m,m are bounded, it follows that ,  2 L 1 Thus

"m2 ˆ ~T ‡  is also bounded so that from (1.26), we obtain X 2 L 1

From (1.28), (1.29), we can now conclude that u, y 2 L 1 This establishes theboundedness of all the closed loop signals in the adaptive IMC scheme Since

Remark 4.3 The robust adaptive IMC schemes of this chapter recover theperformance properties of the ideal case if the modelling error disappears, i.e

we can show that if  ˆ 0 then y ^y ! 0 as t ! 1 This can be established

using standard arguments from the robust adaptive control literature, and is a

consequence of the use of parameter projection as the robustifying tion in the adaptive law [9] An alternative robustifying modi®cation which canguarantee a similar property is the switching- modi®cation [14].

im-(1.17), with …0† ˆ ‰ 1:0; 2:0; 3:0; 1:0ŠT and all other initial conditions set to

zero, we obtained the plots in Figure 1.3 for r…t† ˆ 1:0 and r…t† ˆ sin…0:2t† From these plots, it is clear that y…t† tracks …s2‡ s ‡ 1†…s ‡ 4† s ‡ 2 ‰rŠ quite well.

Let us now consider the design of an adaptive model reference controlscheme for the same plant where the reference model is given by

Wm…s† ˆs2‡ 2s ‡ 11 The adaptive law (1.12)±(1.16) must now guaranteethat the estimated plant is pointwise minimum phase, to ensure which, we

now choose the set Cas Cˆ ‰ 5:0; 5:0Š  ‰ 4:0; 4:0Š  ‰0:1; 6:0Š  ‰0:1; 6:0Š.

All the other design parameters are exactly the same as before except that now(1.17) implements the IMC control law (1.18) and 1…s† ˆ s3‡ 2s2‡ 2s ‡ 2.

The resulting plots are shown in Figure 1.4 for r…t† ˆ 1:0 and r…t† ˆ sin…0:2t†.

From these plots, it is clear that the adaptive IMC scheme does achieve modelfollowing

The modelled part of the plant we have considered so far is minimum phasewhich would not lead to an interesting H2 or H1 optimal control problem.Thus, for H2 and H1 optimal control, we consider the plant (1.7) with

Z0…s† ˆ s ‡ 1, R0…s† ˆ s2‡ 3s ‡ 2,
mˆ s ‡ 1 s ‡ 3 and  ˆ 0:01 Choosing

0 2‡ 2s ‡ 2, 1…s† ˆ s2‡ 2s ‡ 2, Cˆ ‰ 5:0; 5:0Š

‰ 4:0; 4:0Š  ‰ 6:0; 0:1Š  ‰ 6:0; 6:0Š, F…s† ˆ 1

…s ‡ 1†2and implementing the

adaptive H2 optimal control scheme (1.12)±(1.16), (1.17), with

…0† ˆ ‰2:0; 2:0; 2:0; 2:0ŠT and all other initial conditions set to zero, we

obtained the plot shown in Figure 1.5 From Figure 1.5, it is clear that y…t†

asymptotically tracks r…t† quite well Note that the projection set C here hasbeen chosen to ensure that the degree of ^Z0…s; t† does not drop.

Finally, we simulated an H1 optimal controller for the same plant used forthe H2 design.The weighting W…s† was chosen as W…s† ˆ s ‡ 0:010:01 and the set

Cwas taken as Cˆ ‰ 5:0; 5:0Š  ‰ 4:0; 4:0Š  ‰ 6:0; 0:1Š  ‰0:1; 6:0Š This

choice of C ensures that the estimated plant has one and only one right halfplane zero Keeping all the other design parameters the same as in the H2

optimal control case and choosing r…t† ˆ 1:0 and r…t† ˆ 0:8 sin…0:2t†, we

obtained the plots shown in Figure 1.6 From these plots, we see that theadaptive H1-optimal controller does produce reasonably good tracking

time t (second)

y−system outputdesired output r(t)=sin(0.2t)

Figure 1.3 PPAC IMC simulation