Adaptive control of a class of nonlinear time-varying system s with multiple models

Koshy GEORGE,Karpagavalli SUBRAMANIAN

1.PES Centre for Intelligent System s;Department of Telecommunication Engineering,PES Institute of Technology,Bangalore,India;

2.PES Centre for Intelligent System s;Department of Electronics and communication Engineering,PES University,Bangalore,India

1 Introduction

Time-variations in dynamical system s occur in many contexts.Several physical system s,for example,can be modelled as a Hill equation with a sinusoidal parametric excitation resulting in the canonical form of the Mathieu equation.These include vibrations of stretched elliptical membranes,elliptical waveguides,gravitationally stabilised earth-pointing satellites,and rolling motion of ships[1,2].Several researchers have addressed the identification and control of such system s[3-10].

Time-variations may be classified as either slow or fast,and can be known or unknown.It is possible to use traditional adaptive control w hen the unknown parameters vary slow ly with respect to time[11-13].Robust control methods are used when these variations are bounded.For known variations,frequency domain stability criteria for dynamical system s with a single memoryless time-varying nonlinearity in the feedback path are provided in[14].An inversion based control law is proposed in[15]for exact output tracking.For nonlinear system s(that can be transform ed into the strict feedback form)with unmodelled time-varying parameters,a robust state-feedback control is designed in[16].Adaptive control is used in[17]when these system s are linear in the unknown time-varying parameters.Backstepping method has been used as well for nonlinear time-varying systems in the strict feedback form;for instance[18]and[19].

Some attempts have been made to use com putational intelligence techniques to model and control the time-variations.Adaptive control with radial basis function network is proposed in[20].Adaptive fuzzy based and neural-network based H∞controllers are designed in[21]foruncertain nonlinear time-varyingsystem s.The application exam p le isa robotic system with sinusoidally varying parameters.The nonlinearities in a continuous stirred tank chemical reactor are modelled using both a radial basis function network and a multilayer feed forward neural network in[22]with the form er network yielding better results.Two sinusoidally varying parameters are considered here.It was demonstrated in[23]that a time-varying dynamical system described by a set of unknown maps(linear or nonlinear)sequenced periodically can be identified and controlled using the multiple models with switching and tuning(MMST)adaptive control methodology.

Essentially,for slow and sufficiently smooth parametric time-variations,it is possible to use traditional adaptive controlto adaptivecontrol to achieve output tracking and robust control is preferred when these variations are within reasonably small bounds.However,as system s become more comp lex it is possible to experience large and rapid changes in the parameters.examples include mechanical processes with large variations in load,actuator or subsystem failures,and transition control tasks in chemical processes.Our concern in this paper is to adaptively control a class of nonlinear time-varying system s linear in the unknown parameters.The time-variations of interest are discontinuous in nature in that the parameters can sw itch between different values.

Adaptive control of system s has a rich literature.The proof of stability for continuous-time linear time invariant system s was independently arrived at in[24]and[25].Adaptive control has been utilised in diverse applications;recent uses in robotics include[26]and[27].The adaptive control of a class of nonlinear systems s with fixed but unknown set of parameters was dealt with in[28].Whilst extending the MMST methodology to this class of system s,the authors in[28]also demonstrated its efficacy by show ing the resulting controller perform s satisfactorily even when there is a sudden change in the parameters caused by a fault.In contrast,adaptive control with a single identification model leads to signals that are unbounded in the presence of such changes.

The goal of this paper is to extend adaptive control with multiple models to the class of systems wherein the unknown parameters change rapid ly and frequently.Specifically,we first show that the MMST approach of[28]can be applied to this class of system s.However,as will be evident later,this technique requires a rather large number of identification models.Therefore,the second,and more important,objective is to show that multiple models with second level adaptation proposed in[29]can be applied to similar scenario but with relatively very few models.To our best know ledge this application is novel.

This paper is organised as follow s:For the class of nonlinear system s considered in this paper(Section 2),we first present adaptive control using a single identification model in Section 3.We then describe the multiple model approach in Section 4.Multiple models with switching and tuning,and multiple models with second level adaptation techniques are discussed.Simulation results are provided in Section 5.

2 The problem

Consider the class of linearly-param etrised nonlinear system s represented by the following differential equation:

Define the state vectorwithand so on.Thus,

whereAis the matrix in companion form,andb=(0 ···0 1)T∈ Rn.In what follow s,it is assumed thaty(t)and its derivatives are measurable,and hence the statex(t).The origin of the unforced system is clearly unstable.Since the linearised model in the neighbourhood of the origin is controllable one can design local controllers.Indeed,for known θ designing globally asymptotically stable controllers such thatasis rather trivial.Herexmis the state of the reference model

whereAmis in com panion form and is Hurw itz,andr(t)is a piecewise-continuous bounded function.(Due to its inherent property,the existence of a vectork∈Rnsuch thatAm=A+bkTis Hurw itz is guaranteed.)

Our interest in this paper is when θ is both unknown and time-varying.For this paper we assume that these time-variations are step changes in the values of the parameters with unknown step sizes,and that there is a minimum time-interval between these changes.The latter allows for the possibility of the transients to settle down before the next set of changes in the parameters.We,how ever,insist that the parameter vector remain in the interior of Sθf(wàn)or all timet.Our goal here is to determine a control lawuthat achieves the following despite these changes in the parameter vector:i)All the signals in the closed loop system are bounded,and ii)the state of the plant(2)asymptotically track the desired statexmof(3).We propose an adaptive solution based on multiple models.Towards this we first describe the adaptive solution for an unknown but fixed θ.

3 Adaptive control using a single model

Consider the plant(2)with θ(t)≡ θ;i.e.,θ is constant but unknown.We first choose the control inputu(t)as

where the signalv(t)is yet to bedesigned.The dynamics of the system can therefore be described as follow s:

for some appropriatek∈Rn.Consider the identification model

whereis an estimate of θ at instantt.Here,the stateof(6)can be considered as an estimate of the stateof the given dynamical system(5).We have the following result:

Theorem 1Consider the nonlinear system described by(5)and let(6)be the identification model.If

whereis the identification error,which satisfies the Lyapunov equation

for someQ=QT＞ 0 and γ ＞ 0,and if the input is chosen as

then,all signals are bounded andas

The proof of this result is based on standard arguments[30]and is described in[28].From(5)and(6),we have

Evidently,

along any trajectory,given the adaptive law(7).Clearly,and.with the chosen control law(8),the identification model is described by the differential equations

which is similar to that of the reference model.Accordingly,This ensures thatandx∈L∞,and in turnBy Barbalat’s lemm a,we concludeast→∞.It then follow s that

4 Adaptive control using multiple models

In the context of continuous-time linear time-invariant system swith unknow n parameters,globally stableadaptive controllers proposed in the 1980s by[24]and[25]often resulted in unacceptably poor transient performances when the initial parametric estimates?θ(0)were quite different from their true values.Larger the values of these errors,the larger are the transients in the system,even when the given dynamical system is linear and time-invariant.It has been the experience[28]that the signals can grow unbounded for the class of system s described by(2)for fixed θ despite theoretically proved to be globally asymptotically stable.This may be attributed to the errors that accumulate due to finite precision arithmetic serving as a secondary input to the nonlinear system.

The methodology of multiple models with switching and tuning was introduced in[31]to address this issue.Subsequently,multiple models with second level adaptation was proposed in[32-35]for further improvement of the transient response.Essentially,in both approaches,Midentification models are initiated in the parameter space.Nonetheless,control is based on one identification model.Whilst one of theMidentification models is chosen as per a specified criterion in the former technique,control is based on the so-called virtual model in the latter technique.In what follows,these methods are described in the context of adaptively controlling the class of nonlinear system s considered in this paper.

4.1 multiple models with switching and tuning

In the 1990s,Narendra and Balakrishnan introduced in[31]the multiple models with switching and tuning(MMST)methodology to address the issue of poor transient response that often resulted in the adaptive control of linear time-invariant system s.The principal idea is to randomly initiate a number of models in the parametric space with the expectation that at least one of the models is sufficiently close(in some sense)to the given plant.Consequently,the initial errors in the parametric estimates are ensured to be sufficiently small thereby making the transient response acceptable.

Although the concepts of multiple models,switching,and tuning have been independently explored by several researchers,the idea of combining these for the purpose of improving transient response was first proposed in[31].Since then,this notion was systematically and rigorously developed into a methodology.The proofs of stability w ere provided for continuous-and discretetime linear time-invariant system s.Both deterministic and stochastic system s were dealt with.It was extended to a class of nonlinear system s in[28].The initial developm ents are described by Narendra et al.in[36].This paperalso discussed examplesystem s thatre quired MMST for adaptive control to be effective.O ther applications include interference cancellation,blind source separation when the sources of all signals are m oving,active noise control,simultaneous identification of multiple plants,and robust performance[37].multiple models with switching was used in[38]to make model predictive control robust to modelling uncertainties.

The architecture for the adaptive control of a controlled process using the MMST methodology was introduced in[31].It consists ofMidentification models operating in parallel to the given dynamical plant.These models have a structure similar to(6)and described as follow s:

1≤j≤M.TheMidentification models(10)and the plant(5)have the same input signalv(t).Since it is assumed that the state of the plant is measurable,the initial states are taken asThe initial parametric estimatesare uniform ly distributed in Sθas in this paper θ changes with respect to time.

We define the identification error corresponding to each model aswhereis the state of thejth model in(10).Consider the set of candidate Lyapunov functions

whereand γj＞0 for 1≤j≤M.Along any trajectoryif

wherePis as defined earlier.It follow s thatandfor all 1≤j≤M.A proper choice of the control law ensures that the plant state tracks the desired state asymptotically.

Associated with thejth model in(10)is the signal

The strategy of the MMST methodology is to determine at every instant the model whose identification error is the least according to a criterion,and to use the corresponding signal(12)as the control input to the plant and all the identification models.In this paper,w e choose

as the performance index associated with thejth model.At every instantt,thelth model is chosen such that

Here,we assume that there is a minimum intervalTmin＞0 between sw itches.The control input is then

We have the following result.

Theorem 2Consider the nonlinear system described by(5),and let theMidentificationmodelsbe described by(10).If these parameters are adjusted according to(11)and thecontrollersaresw itched based on(14)such that there is a minim um intervalTmin＞0 between sw itches,all the signals are bounded and

We summ arise here the proof given in[28].Suppose that according to(14),the chosen model at instant τ is thelth one.There is no change in the model in the inter-.During this interval it can readily be observed that thelth identification model has the same description as the reference model:

Thus,Accordingly,over the intervaland hence.It then follows thatast→∞.Moreover,since

and

it is evident that

where.That is,(16)is satisfied at every instant for somel.Therefore,asis bounded for allt≥0 and 1≤j≤M.In turn,this im plies that the statex∈L∞,and hencefor 1≤j≤M.We conclude thatast→∞,and hence from(16),asymptotically.

Comm entsi)The MMST methodology is based on distributing identification models in the parameter space.Larger the diam eter of the com pact set Sθ,the larger the number of these models to ensure reasonable transient performance.This is a fundamental issue with this approach.It was suggested in[39]that it is sufficient to use 2 adaptive models and the remaining fixed models in order to reduce the com putational complexity.The fixed models are used to determine the approximate location(in the parameter space)of the plant.One adaptive model is initialised with the parameters of that fixed model closest to the plant.The second adaptive model evolves freely and is necessary to prove the stability of the overall system.It may be noted these situations were considered in[28]for the class of nonlinear systems described by(1)with fixed θ.ii)Even if there areMidentification models gathering information about the plant,at any instanttthe model is chosen in accordance to(14),and the control action is based on it.Thus,the information gathered byM-1 models is discarded.

4.2 Mu ltip le models with second level adap tation

multiple models with second level adaptation(MMSLA)was introduced in[32-35]in order to meaningfully use the information gathered by allMidentification models.(Our description of this approach follow s very closely these references.)Consider theMidentification models described by(10).The initial state is chosen as.However,the initial parametric estimatesare chosen such that the parametric spacewhere K0is the convex hull of the set

Comm entsi)Since θ∈Rp,it im plies that onlyM=p+1 models are required to satisfy this property.This number can be rather sm all com pared to the number of models required in the MMST approach.ii)Even though the theoretical lower bound isp+1,in practice one choosesM＞p+1 to improve the transient performance.iii)TheseMmodels are referred to as prim ary models.

It then follow s that

For a similar set of candidate Lyapunov functions

w hereand γj＞0 for 1≤j≤M,it straightforwardly follows that along any trajectoryif

w here γj＞0 andPis as defined earlier.Thus,andfor all 1≤j≤M.We have the following result.

Theorem 3For the nonlinear plant described by(5)consider theMidentification models described by(10).Suppose that the parameters of the models are ad justed in accordance to(18)then all the signals are bounded andif the control action is taken as

The proof of this result is similar to the previous one.The dynamics of the virtual model is clearly

For the control input(19),we have

Thus.Moreover,it is evident that

where.Accordingly,

Therefore,and henceThus,we conclude thatand henceasymptotically.

Accordingly,

Thus,θ is the invariant point within K0.

Define the following

Thus,Eα =0.Since the coefficients αjs are unknown,we can set up an estimation model as follow s:

whereis the estimate of α and is ad justed in accordance with the adaptive law

where

Ifit follow s that

Therefores are bounded and piecewise differentiable signals satisfying the condition

The identification models for multiple models with second level adaptation are described by the following differential equations:

1≤j≤M,where λ ＞ 0 is a parameter suitably chosen,and the last term introduced to address the stability issue.

Theorem 4For the nonlinear plant described by(5)consider theMidentification models described by(24).Suppose that the parameters of the models are ad justed in accordance to

where γj＞ 0,then all the signals are bounded andif the control action is taken as

where

andare bounded and piecewise differentiable signals satisfying the condition

The proof given here follow s the work of Narendra and his co-workers.We first note that

for some λ0＞ 0.Define the normalised error signal

Choose a set of candidate Lyapunov functionswherePis as defined earlier.Along any trajectory it can be shown that

Choosing λ＞λ0,it follows thatfor allj.Clearly,

Now,suppose that the state of the plantxgrow s unbounded.It then follow s thatandgrow at the same rate.This im p lies thatgrow at a smaller rate.Further,grow at the same rate as the state of the plant,and so is its derivative.In turn,it im p lies thatgrows at a smaller rate.Therefore,grows at the same rate asx,and henceecat a smaller rate thanx,leading to a contradiction.Therefore,x∈L∞.Finally,using similar arguments,we haveasand henceasymptotically.

5 Sim ulation exam ples

In this section,we discuss the application of multiple models to adaptively control the class of nonlinear system s with unknown parameters which abruptly change in time.We com pare MMST and MM-SLA methodologies in this context.Two plants are considered here.Both are third order system s with matched uncertainties but with different number of unknown parameters.The first plant(denoted as Plant A in the sequel)is taken from[28]where the performance of MMST methodology was shown to be superior to the single model case:

The parameters are known to lie in the com pact set Sθ=[-2,2]×[-3,3]×[-4,4].The goal here is to ensure that the statex(t)of this plant track the statexm(t)of the following reference model:

where

and the gains are assumed to bek=(k1k2k3)T=(-3-5-4)T.We consider two possible changes of the plant parameter vector θ;the parameter vectors for Case I(denoted θ(1))and Case II(denoted θ(2))are as follows:

Evidently,θ(2)changes more rapid ly than θ(1).

The second plant(denoted as Plant B)is described by the following set of differential equations:

The parameters are know n to lie in the com pact set Sθ=[-2,2]× [-3,3]×[-4,4]×[-5,5]×[-6,6].For simplicity,the reference model for this is(28)with the bounded input(29)and the gain vectorksame as chosen earlier.We again consider tw o possible changes of the plant parameter vector θ;the parameter vectors for Case III(denoted θ(3))and Case IV(denoted θ(4))are as follows:

The purpose of the first simulation experiment is to show that a nonlinear plant with the description(27)can adaptively be controlled with a single identification model when the plant parameters are fixed,and the initial estimates of the plant parameter vector are sufficiently near the true values.The results are shown in Fig.1 for

and

The three states are respectively shown in Fig.1(a)-(c).From the tracking error of the first state show n in Fig.1(d),it is quite clear that the performance of the adaptive controller is acceptable.When the initial estimate ?θ(0)is poor,the performance can be quite bad,and quite often the signals become unbounded even though the controller was theoretically shown to be globally asymptotically stable.This experience is described in greater detail in[28],where the source of instability was attributed to the approximation error due to finite precision arithmetic acting as a secondary noise input to the system.It was further demonstrated in[28]that a single sudden change in the parameter vector may also lead to instability in that the signals grow unbounded.In this paper we are dealing with multiple changes in the parameters.Clearly,the adaptive controller with a single identification model is not the ideal solution in such scenario.(Experience indicates that a careful choice of the plant parameter vector with such changes in the parameters and proper initialisation of the identification model can result in all signals are bounded.However,the tracking response is typically quite poor.)

Fig.1 Comparing the states of the reference model(D)with the states of(27)with fixed parameters(A)using a single identification model.(a)

We now present the results of experiments with multiple models for Plant A and θ(1)(i.e.,Case I).The states of the plant when it is controlled using MMST methodology is shown in Fig.2.We recall thatMadaptive identification models(described by(10))are initialised in the parameter space and at every instant the model that is closest to the plant is chosen in accordance with(14).The control input(15)is then determined using this particular model.In order to obtain meaningful results for the class of nonlinear systems considered here, the number of adaptive identification models is typically large.For the particular case shown in Fig.2,315 models were uniform ly distributed in the com pact set Sθ.As mentioned earlier,at every instant the information gathered by 314 models is discarded with this approach.

In contrast to the MMST technique,the MM-SLA approach requires only 12 models.(Theoretically,only 4 models are required with this approach.How ever,the transient response is rather poor.It has been the experience that larger the number of models com pared to this lower bound the better the transient response.)The simulation results are show n in Fig.3.From a comparison of these response plots with the corresponding ones shown in Fig.2,it is clear that comparable performance is obtained with very few models.(Indeed,a careful comparison of Fig.2(d)and 3(d)reveal that the performance with the MM-SLA approach is better.)We recall that the control is determined from the virtual model which is the convex combination of the identification models.

The simulation results for Plant B with faster variation in the parameters(i.e.,Case II)are shown in Figs.4 and 5 respectively for controllers based on MMST and MMSLA.The number of models used is the same as that for Case I;i.e.,315 and 12 respectively for the MMST and MM-SLA approaches.The performances of both approaches are com parable with the latter technique providing smaller transients.As the time-variations are faster,the performance of both methods are marginally poorer when com pared to that shown in Figs.2 and 3.Evidently,only fewer models are required w hen all the information is used for purposes of control.

Fig.2 Com paring the states of the reference model(D)with the states of Plant A and Case I(A)using MMST.(a)x1.(b)x2.(c)x3.(d)e=x1-x m,1.

Fig.3 Comparing the states of the reference model(D)with the states of Plant A and Case I(A)usingMM-SLA.(a)x1.(b)x2.(c)x3.(d)e=x1-x m,1.

Fig.4 Comparing the states of the reference model(D)with the states of Plant A and Case II(A)using MMST.(a)x1.(b)x2.(c)x3.(d)e=x1-x m,1.

with an increase in the number of unknown parameters,the required number of adaptive identification models becom es larger,and the necessary com putational power is prohibitively high needing high-end processors with large on-board memory.This is the curse of dimensionality.Thus,for Plant B(which has five unknown parameters)the number of models required is larger than 500 which could not be processed in typical computing machines.In contrast,the MM-SLA approach requiresvery few models.Indeed,the simulation results for Plant B with parameters θ(3)(Case III)and θ(4)(Case IV)respectively are shown in Figs.6 and 7.These results are obtained with m erely 12 models.Evidently,the performance in both cases is quite acceptable.

Fig.5 Comparing the states of the reference model(D)with the states of Plant A and Case II(A)using MM-SLA.(a)x1.(b)x2.(c)x3.(d)e=x1-x m,1.

Fig.6 Comparison between the states of the reference model(D)and the states of Plant B and Case III(A)using MM-SLA.(a)x1.(b)x2.(c)x3.(d)e=x1-x m,1.

Fig.7 Comparison between the states of the reference model(D)and the states of Plant B and Case IV(A)using MM-SLA.(a)x1.(b)x2.(c)x3.(d)e=x1-x m,1.

6 Conclusions

Adaptive control with multiple models is extended to a class of nonlinear time-varying system s.Two approaches are considered here:the multiple models with switching and tuning methodology, and multiple models with second level adaptation.For the class of system s considered here,both approaches provide satisfactory transient performance.However,the former technique requires a rather large number of models and hence perhaps non-imp lementable using conventional computational resources.In contrast,when the information gathered by all the identification models are considered for control action,the number of models required is relatively very few.Thus,the approach of multiple models with second level adaptation is perhaps the way forward for time-varying nonlinear system s.

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