Identification and Control of Dynamical Systems Using Neural Networks

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where x ( t ) 2 [ x l ( t ) , 2 ( t ) ,[ u , ( t > , 2 ( t ) , ** - - , y m ) l T epresent a p input m output system of or-der n with u i ) representing the inputs, xi ) the statevariables, and y i ( t ) he outputs of the system. nd \kare static nonlinear maps defined as 9 R x Rp -+R and\k :Rfl Rm. The vector x ) denotes the state of thesystem at time t and is determined by the state at time toC t and the input u defined over the interval [ t o , ) . Theoutput y ( ) is determined completely by the state of thesystem at time t . Equation (2) is referred to as the input-state-output representation of the system. In this paper wewill be concerned with discrete-time systems which canbe represented by difference equations corresponding tothe differential equations given in (2). These take the form

- - , x , , ( t ) l T ,u ( t ), u p ( t ) l T nd Y ( t ) [ Y l ( t ) ,Y 2 ( t ) ,

x ( k + 1 ) = 9 [ x ( k ) ,u ( k ) ]Y ( k ) = W k ) ] ( 3 )

where u .), x .), and y ( .) are discrete time sequences.Most of the results presented can, however, be extendedto continuous time systems as well. If the system de-scribed by (3) is assumed to be linear and time invariant,the equations governing its behavior can be expressed asx k + 1 ) = A+), + B u ( k )

Y ( k ) = W k ) 4 )where A , B, a n d C a r e ( n X n ) , ( n X p ) , and ( m X n )matrices, respectively. The system is then parameterizedby the triple { C , A , B } . The theory of linear time-invari-ant systems, when C , A , and B are known, is very welldeveloped and concepts such as controllability, stability,and observability of such system s have been studied ex-tensively in the past three decades. Methods for deter-mining the control input U ( . to optimize a performancecriterion are also well known. The tractability of thesedifferent problems may be ultimately traced to the factthat they can be reduced to the solution of n linear equa-tions in n unknowns. In contrast to this, the problems in-volving nonlinear equations of the form (3), where thefunctions 9 nd P re known, result in nonlinear alge-braic equations for the solution of which similar powerfulmethods do not exist. Consequently, as shown in the fol-lowing sections, several assumptions have to be made tomake the problems analytically tractable.C. Identijication and Control

I . Identijication: When the functions 9 nd \k in (3),or the m atrices A , B, and C in 4), are unknown, the prob-lem of identification of the unknown system (referred toas the plant in the following sections) arises [12]. Thiscan be formally stated as follows [11:The input and output of a time-invariant, causal dis-crete-time dynamical plant are u ( .) and y,,( .), respec-tively, where u ( . is a uniformly bounded function oftime. T he plant is assumed to be stable with a known pa-rameterization but with unknown values of the parame-

q x w )odel M-r P3)dentification ypF--t-kmodel C k )(a) (b)

Fig. 1. (a) Identification. (b) Model reference adaptive control.

ters. T he objective is to construct a su itable identificationmodel (Fig. l(a)) which when subjected to the same inputu ( k ) as the plant, produces an output j j p ( k ) which ap-proximates y p ( k ) n the sense described by (1).2. Control: Control theory de als with the analysis andsynthesis of dynamical systems in which one or morevariables are kept within prescribed limits. If the func-tions 9 nd \k in (3) are known, the problem of control isto design a controller which generates the desired controlinput u k ) based on all the information available at thatinstant k. While a vast body of frequency and time-do-main techniques exist for the synthesis of controllers forlinear systems of the form described in 4) ith A, B, andC known, similar methods do not exist for nonlinear sys-tems, even when the functions 9 . , ) and P . are spec-ified. In the last three dec ades there has been a grea t dealof interest in the control of plants when uncertainty existsregarding the dynam ics of the plant [13. To assure m ath-ematical tractability, most of the effort has been directedtowards the adaptive control of linear time-invariant plantswith unknown parameters. Our interest in this pap er liesprimarily in the identification and control of unknownnonlinear dynamical systems.Adaptive systems which make explicit use of modelsfor control have been studied extensively. Such systemsare commonly referred to as model reference adaptivecontrol (MRAC) systems. The implicit assumption in theformulation of the MRAC problem is that the designer issufficiently f amiliar with the plant u nder consideration sothat he can specify the desired behavior of the plant interms of the output of a reference model. The MRACproblem can be qualitatively stated as follows (Fig. l(b)) .a. Model reference adaptive control: A plant P withan input-output pair { u k), y p ( k ) } s given. A stablereference model M is specified by its input-output pair{ r ( k ) , y , ( k ) } where r : N -+ R is a bounded function.The output y m ( k ) s the desired output of the plant. Theaim is to determine the control input u ( k ) or all k oso that

for so me specified constant E 0.As described earlier, the choice of the identificationmodel (i.e., its parameterization) and the method of ad-justing its parameters based on the identification errorei ( k ) constitute the two principal parts of the identifica-tion problem. Determining the controller structure, andadjusting its parameters to minimize the erro r between the


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output of the plant and the desired output, represent thecorresponding parts of the control problem. In Section11-C-3, some well-known m ethods for setting up an id en-tification model and a controller structure for a linear plantas well as th e adjustment of identification and control pa-rameters are described . Following this, in Section 11-C-4,the problems encountered in th e identification and controlof nonlinear dynamical system s are briefly presented.3. Linear Systems: For linear time-invariant plantswith unknown p arame ters, the generation of identificationmodels are currently well known. For a single-input sin-gle-output (SISO) controllable and ob servable plant, th ematrix A and the vectors B and C in (4) can be chosen insuch a fashion that the plant equation can be w ritten asn - l m -

where ai and Pi are constant unknown parameters. A sim-ilar representation is also possible for the multi-inputmulti-output (MIMO) case. This implies that the outputat t ime k + 1 is a linear combination of the past values ofboth the input and the output. Equation 5 ) motivates thechoice of the following identification models:n - 1j p k + 1 = c & i ( k ) j p ( k- )i = O

m -+ c B j k ) U k - J )j = O(Parallel model )

n - lj p k + 1 = c & i ( k ) y p ( k - )i = O

m -+ e- j ( k ) U ( k - )j = O(Series-parallel model ) (7 )

where & i ( i = 0, 1 , * - , n - 1 ) and B j j = 0, 1 ,. . . , m - 1 ) are adjustable parameters. The output ofthe parallel identification model (6) at t ime k + 1 is j p k+ 1) and is a linear combination of its past values as wellas those of the input. In the series-parallel m odel, j p k +1 ) is a linear combination of the past values of the inputand output of the plant. T o generate stab le adaptive laws,the series-parallel m odel is found to be preferable. In sucha case, a typical adaptive algorithm has the form

, m T will be denoted by p a?d7 an-1 PO,that of, the identification model [&o, * - 9 b n - 1 9 PO,* * , P m - i I T b y B .Linear time-invariant plants which are controllable canbe shown to be stabilizable by linear state feedback. Thisfact has been used to desig n adaptive controllers for suchplants. For example, if an upper bound on the order ofthe plant is known, the control input can be generated asa linear combination of the past values of the input andoutput respectively. If 8 ( k ) represents the control param-eter vector, it can be shown that a constant vector t9* ex-ists such that when 8 ( k ) = t9* the plant togeth er with thecontroller has the same input-output characteristics as thereference model. Adaptive algorithms for adjusting t9 k )in a stable fashion are now w ell known and have th e gen-eral form shown in (8).4 . Nonlinear Sysrems: The importance of controllabil-ity and observability in the formulation of the identifica-tion and control problems for linear systems is evidentfrom the discussion in Section 11-C-3. Other well-knownresults in linear systems theory are also called upon tochoose a reference model as well as a suitable parameter-ization of the plant and to assure the exis tence of a desiredcontroller. In recent years a number of authors have ad-dressed issues such as controllability, observability, feed-back stabilization, and observer design for nonlinear sys-tems [ 1 3 ] - [ 1 6 ] . In spite of such attempts constructiveprocedures, similar to those available for l inear systems,do not exist for nonlinear systems. Hence, the choice ofidentification and controller models for nonlinear plantsis a formidable problem and successful identification andcontrol has to depend upon several strong assumptions re-garding the input-output behavio r of the plant. For ex-ample, if a SISO system is represented by the equation( 3 ) , we shall assume that the state of the system can bereconstructed from n measurements of the input and out-put. More precisely, y p ( k ) = \ k [ x ( k ) ] ,y p ( k + 1 ) =* [ * [ x ( k ) , u ( k ) ] ] , - , y p ( k + n - 1 ) = * [ a [ ** [ * [ x ( k ) , u ( k ) l , u ( k + 111, * * * 7 u ( k + n - 2111yield n nonlinear equations in n unknowns x ( k ) f u ( k ) ,y p ( k + n - 1 arespecified and we shall assume that for any set of valuesof u ( k ) in a compact region in U, a unique solution tothe above problem ex ists. T his permits identification pro-cedures to be proposed for nonlinear systems along linessimilar to those in the linear case.is known in (3) and the statevector is accessible, th e determination of u . for the plant

*

, u ( k + n - 2 ) , y p ( k ) ,

Even when th e function&i(k + 1 to have a desired trajectory is an equally difficult problem.Hence, for the generation of the control input, the exis-tence of suitable inverse operators hav e to be assumed. Ifa controller structure is assumed to generate the input

u . , further assumptions have to be made to assure theexistence of a constant control parameter vector to achievethe desired objective. All these indicate that considerableprogress in nonlinear control theory will be needed to ob-tain rigorous solutions to the identification and controlproblems.

= & ( k ) - 9e ( k + l ) y p ( k - )

1 + clLd y : ( k - ) + u 2 k - )(')

where 9 > 0 determines the step size. In the followingdiscussions, th e constant vector of plant parameters [ao,


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In spite of the above comments, the linear models de-scribed in Section 11-C-3 motivate the choice of structuresfor identifiers and controllers in the nonlinear case. It isin these structures that we shall incorporate neural net-works as described in Sections V and VI. A variety ofconsiderations discussed in Section I11 reveal that bothmultilayer neural networks as well as recurrent networks,which are currently being extensively studied, will featureas subsystems in the desig n of identifiers and controllersfor nonlinear dynamical systems.

111. MULTILAYERND RECU RREN TETWORKSThe assumptions that have to be made to assure wellposed problems using models suggested in Sections V andVI are closely related to the properties of multilayer andrecurrent networks. In this section , we describe briefly thetwo classes of neural networks an d indicate why a unifiedtreatment of the two may b e warranted to deal with morecomplex sys tems in the future.

A. Multi layer NetworksA typical multilayer network with an input layer, anoutput layer, and two hidden layers is shown in Fig. 2.For convenience we denote this in block diagram form asshown in Fig. 3with three weight matrices W , W 2 , ndW 3 nd a diagonal nonlinear operato r r with identical sig-moidal elements y [ i . e . , y x) = 1 - e P x / l + e- ] fol-lowing each of the weight matrices. E ach layer of the net-work can then be represented by the operator

N ~ [ u ] r[wL] (9)and the input-output mapping of the multilayer networkcan be represented by

y = N [ u ] = r [ W 3 r [ W 2 I [ W 1 u ] ] ]N 3 N 2 N I [ u ] .(10)

In practice, multilayer networks h ave been used success-fully in pattern recognition problems [2]-[5]. The weightsof the network W , W 2 , nd W 3 re adjusted as describedin Section IV to m inimize a suitable function of the erro re between the output y of the network and a desired outputyd . This results in the m apping function N [U ] realized bythe network, mapping vectors into corresponding outputclasses. Generally a discontinuous mapping such as anearest neighbor rule is used at the last stage to map theinput sets into points in the range space corresponding tooutput classes. From a systems theoretic point of view,multilayer networks can b e considered as versatile nonlin-ear maps with the elements of the weight matrices as pa-rameters. In the following sections we shall use the terms"weights" and "parameters" interchangeably.B. Recurrent Networks

Recurrent networks, introduced in the works of Hop-field [6] and discussed qu ite extensively in the literature,provide an alternative approach to pattern recognition.One version of the network suggested by Hopfield con-

h p t Hidden Hidden Outputpattern layer layer layerFig. 2 . A three layer neural network.

Fig. 3 . A block diagram representation of a three layer network.

sists of a single layer network N I , ncluded in feedbackconfiguration, with a time delay (Figs. 4 nd 5 ) . Such anetwork represents a discrete-time dynamical system andcan be described by~ ( k 1 ) = N I [ x ( k ) ] , ~ ( 0 ) XO.

Given an initial value xo, he dynamical system evolvesto an equilibrium state if NI s suitably ch osen. T he set ofinitial conditions in the neighborhood of xo which con-verge to the same equilibrium state is then identified withthat state. The term "associative memory" is used to de-scribe such systems. Recently, both continuous-time anddiscrete-time recurrent networks have been studied withconstant inputs [ 1 7 ] . The inputs rather than the initialconditions represent the patterns to be classified in thiscase. In the continuous-time case, the dynamic system inthe feedback path has a diagonal transfer matrix withidentical elements l / ( s + a) long the diagonal. Thesystem is then represented by the equationX = -U + N ~ [ x ] I (11)

so that x ( t ) E R s the state of the system at time t , andthe constant vector I E W is the input.C. A Unijied Approach

In spite of the seeming differences between the two ap-proaches to pattern recognition using neural networks, itis clear that a close relation exists between them. Recur-rent networks with or without constant inputs are merelynonlinear dynamical system s and the asymptotic behaviorof such systems depends both on the initial conditions aswell as the specific input used. In both cases, this dependscritically on the nonlinear map represented by the neuralnetwork used in the feedback loop. For example, whenno input is used, the equilibrium state of the recurrentnetwork in the discrete case is merely the fixed point ofthe mapping N I . Thus the existence of a fixed point, theconditions under which it is unique, the maximum num-ber of fixed points that can be achieved in a given networkare all relevant to both multilayer and recurrent networks.Much of the current literature deals with such problems[18] and for mathematical tractability most of them as-

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Fig. 4. The Hopfield network.rT;;TFig. 5 . Block diagram representation of the Hopfield network.

sume that recurrent networks contain only single layernetworks (i .e ., NI . . As mentioned earlier, inputs whenthey exist are assumed to be constant. Recently, two layerrecurrent networks have also been considered [19] andmore general forms of recurrent networks can be con-structed by including multilayer networks in the feedbackloop [20]. In spite of the interesting ideas that have beenpresented in these papers, ou r understanding of such sys-tems is still far from complete. In the identification andcontrol problems considered in Sections V and VI, mul-tilayer networks are used in c ascade and feedback config-urations and the inputs to such models are functions oftime.D . Generalized Neural Networks

From the above discus sion, it follows that the basic ele-ments in a multilayer network is the mapping N , [ . ] =r [ W1 , while the addition of the time delay element z - 'in the feedback path (Fig. 5 ) results in a recurrent net-work. In fact, general recurrent networks can be con-structed composed of only the basic operations of 1 ) de-lay, 2) summation, and 3) the nonlinear operator Ni [ . .In continuous-time networks, the delay operator is re-placed by an integrator. In some cases (as in (11)) mul-tiplication by a constant is also allowed . Hence such net-works are nonlinear feedback system s which consist onlyof elements N I I , in addition to the usual operationsfound in linear systems.Since arbitrary linear time-invariant dynamical systemscan be constructed using the operations of summation,

Fig. 6. (a) Representation 1. (b) Representation 2 . (c) Representation 3 .(d) Representation 4.

multiplication by a constant and time delay, the class ofnonlinear dynamical system s that can be generated usinggeneralized neural networks can be represented in termsof transfer matrices of linear systems [i.e., W z ) ] ndnonlinear operators N [ . . Fig. 6shows these operatorsconnected in cascad e and feedback in fo ur configurationswhich represent the building blocks for more complexsystems. T he superscript notation N' is used in the figuresto distinguish between different multilayer networks in anyspecific representation.From the discussion of generalized neural networks, itfollows that the mapping properties of Ni [ . and conse-quently N [ . (as defined in (10)) play a central role in allanalytical studies of such networks. It has recently beenshown in [21], using the Stone-Weierstrass theorem , thata two layer network with an arbitrarily large number ofnodes in the hidden laye r can approximate any continuousfunctionfE C(R, W) over a compa ct subset of W. Thisprovides the motivation to assume that the class of gen-eralized networks described is adequate to deal with alarge class of problems in nonlinear systems theory. Infact, all the structures used in Section V and VI for theconstruction of identification and controller models aregeneralized neural networks and are closely related to theconfigurations shown in Fig. 6.For ease of discussion inthe rest of the paper, we shall denote the class of functionsgenerated by a network containing N layers by the symbol3ZT,i2,. r i N + , . Such a network has i, inputs, i N + loutputsand ( N - 1 ) sets of nodes in the hidden layers, each con-taining i 2 , i 3 , - , N nodes, respectively.

IV. BACKPROPAGATIONN STATICND DYNAMICSYSTEMSIn both static identification (e.g., pattern recognition)and dynamic system identification of the type treated inthis paper, if neural networks are used, the objective is todetermin e an adaptive algorithm o r rule which adjusts theparameters of the network based on a given set of input-output pairs. If the weights of the networks are considered

as elements of a p arameter vector 8 , the learning processinvolves the determination of the vector e* which opti-mizes a performance function J based on the output error.Back propagation is the most commonly used method for


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mult,plications- V w ,J - V w J - V w J

Fig. 7 . Architecture for back propagation

= \k, ( k ) E R m . + and \k, are Jacobian matrices andthe vector represents the partial derivative of + withrespect to 8. Equation (12) represents the linearized equa-tions of the nonlinear system around the nominal trajec-tory and input. If A ( k ) , v k), and C ( k ) can be com-puted, w ( k ) , the partial deriva tive of y with respect to ecan be obtained as the output of a dynamic sensitivitymodel.In the previous section, generalized neural networkswere defined and four representations of such networkswith dynamical syste ms and multilayer neural networksconnected in series and feedback were shown in Fig. 6.Since complex dynamical sy stems can be expressed interms of these f our representations, the back-propagationmethod can be extended to such systems if the partial de-rivative of the ou tputs with respect to the parameters canbe determined for each of the representations. In the fol-lowing we indicate briefly how (12) can be specialized tothese four cases. In all cases it is assumed that the partialderivative of the output of a multilayer neural networkwith respect to one of the parameters can be computedusing static back propagation and can be realized as theoutput of the netwark in Fig. 7 .In representation 1 , the desired output Yd ( k ) as well asthe error e ( k ) 2 (k) - yd(k) are functions of time.Representation 1 is the simplest situation that can arise indynamical systems. This is because

where 0 is a typical parameter of the network N. Sincedu/aOj can be com puted at every instant using static backpropagation, de ( k ) dej can be realized as the output of adynamical system W ( z )whose inputs are the partial de-rivatives generated.In representation 2 , the determination of the gradient isrendered more com plex by the presence of neural networkN' . If 0, is a typical param eter of N I , the partial derivative

ae (k)/ae, is computed by static back propagation. How-ever, if 0, is a typical parameter of N2ay - c ay1 au/-ae, / av, ae,

Since av/ae, can be computed using the method de-scribed in representation l and ayl/av can be obtained bystatic back propagation, the product of the two yield thepartial derivative of the signal yI with respect to the pa-rameter 0,.Representation 3 shows a neural network connected infeedback with a transfer matrix W ( z ) . The input to thenonlinear feedback sys tem is a vector U ( k ) . If 0, is a typ-ical parameter of then eura l letw ork , the aim is to deter-mine th e derivatives ay,(k)/aO, f o r i = 1, 2, * * * , m andall k 2 0. We observ e here for_the first time a situationnot encountered earlier, in th g ayI k)_/ae, is the solutionof a difference equation, i.e., ayI (k) /ae, is affected by itsown past values

In (13), ay/aej is a vector and aN[ v]/au and aN[ u]/aOjare the Jacobian matrix and a vecto r, respectively, whichare evaluated around th e nominal trajectory. Hence it rep-re se cs a linearized difference equation in the variablesputed at every instant of time, the desired partial deriva-tives can be generated a s the output o_fa d_ynamical systemshown in Fig. 8(a) (th e bar notation ay/aOj is used in (13)to distinguish between ay/ , and aN[ v]/aej).In the final representation, the feedback system is pre-ceded by a neural network N2. The presence of N2 doesnot affect the computation of the partial derivatives of theoutput with respect to the parameters of NI. However, if0 is a typical p arameter of N2, t can be shown that ay/aejcan be obtained as

&/aej. Since aN[v]/au and aN[v]/dej can be com-

or alternately it can be represented as the output of thedynamical system shown in Fig. 8(b) whose inputs can becomputed at every instant of time.In all the problems of identification and control that wewill be concerned with in the following s ections, the ma-trix W z ) s diagonal and consists only of elements of theform (i. e., a delay of di nits). F urther since dynamicback propagation is considerably more involved than staticback propagation, the structure of the identificationmodels is chosen, wherever possible, so that the latter canbe used. The models of back propagation developed herecan be applied to general control problems where neuralnetworks and lin ear dynamical systems are interconnectedin arbitrary configurations and where static back propa-gation cannot be justified. F or further details the reader is


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Fig. 9. Representation of SISO plants. (a) Model I. (b) Model 11. c) Model111. (d) Model 1V.

The m odels described thus far are for the representationof discrete-time plants. Continuous-time analogs of thesemodels can be described by differential equations, asstated in Section 11. Wh ile we shall deal exclusively withdiscrete-time systems, the same methods also carry overto the continuous time case.B. Identijication

The problem of identification, as described in Section11-C, consists of setting up a suitably parameterized iden-tification model and adjusting the parameters of the modelto optimize a performance function based on the err or be-tween the plant and the identification model outputs. Sincethe nonlinear functions in the representation of the plantare assumed to belong to a known class 37. . . . I N + , inthe domain of interest, the structure of the identificationmodel is chosen to be identical to that of the plant. By

assumption, weight matrices of the neural networks in theidentification m odel exist so that, f or the sam e initial con-ditions, both plant and model have the same output forany specified input. Hence the identification procedureconsists in adjusting the parameters of the neural net-works in the model using the method described in SectionIV based on the error between the plant and model out-puts. However, as shown in what follows, suitable pre-cautions have to be taken to ensure that the procedure re-sults in convergence of the identification mod el parametersto their desired values.1 Parallel Identijication Model: Fig. 10(a) shows aplant which can be represented by Model I with n = 2and m = 1. To identify the plant one can assume the

structure of the identification model shown in Fig. 10(a)and described b y the equation


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(a) (b)Fig. 10. (a) Parallel identification model. (b) Series-parallel identificationmodel.

As mentioned in Section 11-C-3, this is referred to as aparallel model. Identification then involves the estimationof the parameters &* as well as the weights of the neuralnetwork using dynam ic back propagation based on the er-ror e ( k ) between the model output j p ( k ) nd the actual

From the assumptions made earlier, the plant isbounded-input bounded-output (BIBO) stable in the pres-ence of an input (in the assumed class). Hence, all thesignals in the plant are uniformly bounded. In contrast tothis, th e stability of the identification model as describedhere with a neural network cannot be assured and has tobe proved. Hence if a parallel model is used, there is noguarantee that the parameters will converge or that theoutput error will tend to zero. In spite of two decades ofwork, conditions under which the parallel model param-eters will converge even in the linear case ar e at presentunknown. Hence, for plant representations using ModelsI-IV, the following identification model, known a s theseries-parallel mode l, is used.2. Series-Parallel Model: In contrast to the parallelmodel described above, in the series-parallel model theoutput of the plant (rather than the identification model)is fed back into the identification model as shown in Fi g.10(b). This implies that in this case the identificationmodel has the form

output yp ( k ) .

j p k + 1) = & o y p ( k ) + & l y p ( k - 1) + N [ u ( k ) ] .W e shall use the same procedure with all the four modelsdescribed earlier. Th e series-parallel identification m odelcorresponding to a plant represented by M odel IV has theform shown in Fig . 11.TD L in Fig. 11denotes a tappeddelay line whose output vector has for its elements thedelayed values of the input signal. Hence the past valuesof the input and the output of the plant form the inputvector to a neural network whose output j p ( k ) corre-sponds to the estimate of the plant output at any instantof t ime k . Th e series-parallel model enjoys several advan-

Nonlinear

Neuralnetwork

Fig. 11. Identification of nonlinear plants using neural networks

tages over the parallel model. Since the plant is assumedto be BIBO stable, all the signals used in the identificationprocedure (i.e., inputs to the neural networks) arebounded. Further, since no feedback loop exists in themodel, static back propagation can be used to adjust theparameters reducing the computational overhead substan-tially. Finally, assuming that the output error tends to asmall value asymptotically so that y p ( k ) = j p k ) , he se-ries-parallel model may be replaced by a parallel modelwithout serious consequences. This has practical impli-cations if the identification model is to be used off line.In view of the above considerations the series-parallelmodel is used in all the simulations in this paper.C . Simulation Results

In this section simulation results of nonlinear plantidentification using the models suggested earlier are pre-sented. Six exam ples are presented w here the prior infor-mation available dictates the choice of one of the Mode lsI-IV. Each exam ple is chosen to emphasize a specificpoint. In the first five examp les, the series-parallel modelis used to identify the given plant and static back-propa-gation is used to adjust parameters of the neural networks.A final example is used to indicate how dynamic backpropagation m ay be used in identification problems. Dueto space limitations, only the principal results are pre-sented here. T he reader interested in further details is re-ferred to [27]-129).


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154 . I . I . I . 5 . , . , . , . ,

I300 400 BOO 800 700(a) (b)

Fig. 12. Example 1: (a) Outputs of the plant and identification model whenadaptation stops at k = 500. (b) Response of plant and identificationmodel after identification using a random input.

I . Example I : The plant to be identified is governedy p ( k + 1 ) = 0 . 3 y p ( k )+ 0 . 6 y p ( k- 1 + f ~ ( k ) ]

where the unknown function has the form f U ) = 0 . 6 sin( n u ) + 0 . 3 sin (37ru) + 0.1 sin (5nu). From (15), i t isclear that the unforced lin ear system is asymptotically sta-ble and hence any bounded input results in a bounded out-put. In ord er to identify the plant, a series-parallel modelgoverned by the difference equation

j p ( k + 1) = 0 . 3 y p ( k )+ 0 . 6 y p ( k- 1 + N [ u ( k ) ]was used. The weights in the neural network were ad-justed at every instant of time (Ti 1) using static backpropagation. The neural network belonged to the classX ; , 2 ~ , 1 0 , ~nd the gradient method employed a step sizeof 17 = 0.2 5. T he input to the plant and the model was asinusoid U ( k ) = sin (2 nk/ 25 0). As seen from Fig. 12(a),the output of the model follows the output of the plantalmost immediately but fails to do so when the adaptationprocess is stopped at k = 500, indicating that the identi-fication of the plant is not complete. Hence the identifi-cation procedure was continued for 50 000 time stepsusing a random input whose am plitude was uniformly dis-tributed in the interval [ - , 1 ] at the end of which theadaptation was terminated. Fig. 12(b) shows the outputsof the plant and the trained model. Th e nonlinear functionin the plant in this case is f [ U ] = u3 + 0 . 3 ~ ~0.4~.As can be seen from the figure, the identification error issmall even when the input is changed to a sum of twosinusoids U ( k ) = s in ( 2 n k / 2 5 0 ) + s in (2nk /25 ) a t k =250.2. Example 2: The plant to be identified is describedby the second-order difference equationwhere

by the differen ce equation

( 1 5 )

Yp(k + 1 ) = f Y p ( k ) , Y p ( k- I ) ] +f [ Y p ( k ) ,Yp(k - I ) ]

This corresponds to Model 11. A series-parallel identifierof the type discussed earlier is used to identify the plantfrom input-output data and is described by the equationj p k + 1) = N [ Y p ( k L Y p ( k - 01 + u k) (17)

where N is a neural network with N E The iden-tification process involves the adjustment of the weightsof N using back propagation.Some prior information concerning the input-outputbehavior of the plant is needed before identification canbe undertaken. This includes the number of equilibriumstates of the unforced system and their stability proper-ties, the compact set U to which the input belongs andwhether the plant output is bounded for this class of in-puts. Also, it is assumed that the mapping N can approx-imate f ver the desired d omain.a. Equilibrium states of the unforced system: Theequilibrium states of the unforced system y p ( k + 1 ) =f [ y , ( k ) , y p ( k - l ) ] withfas defined in ( 1 6 ) are (0, 0)a nd (2 , 2 ) , respectively, in the state space. This impliesthat while in equilibrium without an input, the output ofthe plant is either the sequence { 0 } or the sequence { 2 } .Furth er, for any input U ( k ) , the output of the plantis uniformly bounded for initial conditions (0, 0) a nd (2 ,2) and satisfies the inequality I y p k ) 3.Assuming different initial conditions in the state spaceand with zero input, the weights of the neural networkwere adjusted so that the error e ( k + 1 ) = y p ( k + 1 -N [ p ( k ) , p ( k - 1 )] is minimized. When the weightsconverged to constant values, the equation j p k + 1 ) =N [ p ( k ) , p k - 1 ) ] was simulated for initial conditionswithin a radius of 4. he identified system was found tohave the sa me trajectories a s the plant fo r the same initialconditions. The behavior of the plant and the identifiedmodel for different initial conditions are shown in Fig . 13.It must be em phasize d here that in practice the initial con-ditions of the plant cannot be chosen at the discretion ofthe designer and must be realized only by using differentinputs to the plant.b. Identijication: While the neural network realizedabove can be used in the identification model, a separate


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(a) (b)Fig. 13. Example 2: Behavior of the unforced system. (a) Actual plant.(b) Identified model of the plant.

u- mtn[h k/25]simulation was carried out using both inputs and outputsand a series-parallel model. T he input U ( k ) was assumedto be an i.i.d. random signal uniformly distributed in theinterval [ -2, 21 and a step size of 9 = 0.25 was used inthe gradient method. The weights in the neural networkwere adjusted at intervals of five steps using the gradientof E t = , - , e 2 ( i . Fig. 14shows the outputs of the plantand the model after the identification procedure was ter-minated at k = 100 0003. Example 3: In Example 2, the input is seen to occurlinearly in the difference equation describing the plant. In

4

this example the plant is described by Model II Ia nd hasthe form -2 0 2 o D u M wFig. 14. Example 2: Outputs of the plant and the identification m odel.

which corresponds t o f [ y p ( k ) ] = y p ( k ) / ( 1 + y P ( k l 2 )a n d g [ u ( k ) ] = u 3 ( k ) n 14). A series-parallel identifi-cation model consists of two neural networks Nf nd N gbelonging to 32:,20, and can b e described by the differ-ence equation

The estimates f and g are obtained by using neural net-works Nf nd N g . The weights in the neural networks wereadjusted at every instant of time using a step size of 7 =0. 1 and was continued fo r 100 000 time steps. Since theinput was a random input in interval [ -2 , 21 , g approx-imates g only o ver this interval. Sinc e this in turn re ultsin the variation of y p over the interval [ - 0, l o ] , f ap-proximates f over the latter interval. Th e functions f andg as well as f and g over their respective domains areshown in Fig. 15(a) and (b). In Fig. 15(c), the outputs ofthe plant as well as the identification model for an inputU ( k ) = s in ( 2 a k / 2 5 ) + sin 2nk/10) are shown and areseen to be indistinguishable.4. Example 4: The same methods used for identifica-tion of plants in examples 1-3 can be used when the un-known plants are known to belong to Model IV. In this

example, the plant is assumed to be of the formY p(k + 1 ) = f [ Y p ( k ) , Y , ( k -

Y p k - 2 ) ,4 k - 111where the unknown function f has the form

In the identification mode l, a neural network N belongingto the class '3Z:.20, I is used to approximate the functionf Fig. 16 shows the output of the plant and the modelwhen the identification procedure was carried out for100 000 steps using a random input signal uniformly dis-tributed in the interval [ - , 1 and a step size of 9 =0.25. As mentioned ea rlier, during th e identification pro-cess a series-parallel model is used, but after the identi-fication process is terminated the performance of themodel is studied using a parallel model. In Fig. 16 ,theinput to the plant and the identified model is given byu k ) = sin (27rk/250) for k 00 and u k ) = 0.8 sin( 2 a k / 2 5 0 ) + 0 .2 s in (2nk /25 ) fo r k > 500.5. Example 5: In this example, it is shown that thesame methods used to identify SIS O plants can be used to


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0 . e r . . . . , . . . . l . . . . , . , , ,0.4

;2' 0 . 24 0 . 051\ 0.4a.2 -0.2

V

-10 -6 5 10

DYNAMICAL SYSTEMS

1 0 , . I . . , . , , ,c 1

0 2 0 4 0 D D 1 0 0(C)

Fig. 15. Example 3: (a) Plots of the functions f and f . (b) Plots of thefunctions g and g. (c) Outputs of the plant and the identification model.

0 . 5

.9 0 . 0'c(

-0.5

200 400 MM 800Fig. 16. Example 4: Identification of Model IV.

identify MIMO plants as well. The plant is described bythe equations

JThis corresponds to the multivariable version of Model11. The series-parallel identification model consists of two

I

17

neural networks NI and N 2 nd is described by th e equa-tion

The identification procedure was carried o ut for 100 000time steps using a ste p size of 7 = 0.1 with random inputsu I k ) nd u2 k) niformly distributed in the interval [ - ,11. Th e responses of the plant and the identification modelfor a vec tor input [s in (2ak/25) , cos ( 2 a k / 2 5 ) I T a r eshown in Fig. 17 .Comment: In examples 1 , 3, 4, nd 5 the adjustmentof the parameters was carried out by computing the gra-dient of e ( k ) at instant k while in exam ple 2 adjustmentswere based o n the gr adient of an error function evaluatedover an interval of length 5.While from a theoretical pointof view it is preferable to use a larger interval to definethe error function, very little improvement was observedin the simulations. This accounts for the fact that in ex-amples 3, 4 , and 5 adjustments were based on the instan-taneous rather than an average error signal.

6. Example 6: In examples 1-5, a series-parallel iden-tification model was used and hence the parameters of theneural networks were adjusted using the static back-prop-agation method. In this example, we consider a simple


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0 20 40 80 100 0 20 U Io 80 100(a) (b)

Fig. 17. Example 5: Responses of the plant and the identification model.

1

0

-3

Fig. 18. Example 6: (a) Outputs of plant and identification model. (b)f[u]and N [ ] for u E [ - , 11.

first order nonlinear system which is identified using thedynam ic back-propagation method discussed in SectionIV. The nonlinear plant is described by the differenceequationYp(k + 1 ) = 0.8YJk) + f [ u ( k ) Iw here the func t io n f [u ] = ( U - 0 . 8 ) u ( u + 0.5) is un-known. However, it is assumed that f an be approxi-mated to the desired degree of accuracy by a multilayerneural network.The identification model used is described by the dif-ference equationj p ( k + 1) = 0 .8 jp (k ) + N [ u ( k ) ]

and the neural network belonged to the class '32~.20.10,1.The model ch osen corresponds to representation 1 in Sec-tion IV (refer to Fig. 6(a)). The objective is to adjust atotal of 261 weights in the neural network so that e ( k )= j p k ) - yp(k ) 0 asymptotically. Defining the per-formance criterion to be minimized as J = ( 1 / 2 T )cf k-T + e 2 ( ), the partial derivative of with respectto a weight 8, in the neural network can be computed asa J / M , ) = ( l / T ) C f = k - T + I e ( i ) ( a e ( i ) / M , ) . Thequantity ( d e i ) d e j ) can be computed in a dynamic fash-

ion using the method discussed in Section IV and used inthe following rule to update 8:

where 17 is the step size in the gradient procedure .Fig. 18(a) shows the outputs of the plant and the iden-tification model when the weights in the neural networkwere adjusted after an interval of 10 time steps using astep size of 7 = 0.01. The input to the plant (and themodel) was u ( k ) = sin (2 rk /2 5) . In Fig . 18(b) , thef u n c t i o n f ( u ) = ( U - 0 . 8 ) u ( u + 0 . 5 ) , as well as thefunction realized by the three layer neural network after50 000 steps for U E [ - , 11, are shown. As seen fromthe figure, the neural network approximates the givenfunction quite accurately.VI. CONTROLF DYNAMICALYSTEMS

As mentioned in Section 11, for the sake of mathemat-ical tractability most of the effort during the past two dec-ades in the model reference adaptive control theory hasbeen directed towa rds the control of linear time-invariantplants with unknown parameters. Muc h of the theoreticalwork in the late 1970's was aimed at determining adaptive


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laws for the adjustment of the control parameter vector8 ( k )which would result in s table overall systems. In 1980[30]-[33], it was conclusively shown that for both dis-crete-time and continuous-time systems, such stableadaptive laws could be determined provided that someprior information concerning the plant transfer functionwas available. Since that time much of the research in thearea has been directed towards determining conditionswhich assure the robustness of the overall system underdifferent types of perturbations.In contrast to the above, very little work has been re-ported on the global adaptive control of plants describedby nonlinear difference or differential equations. It is inthe control of such systems that we are primarily inter-ested in this section. Since, in most problems, very littletheory exists to guide the analysis, one of the aims is toindicate precisely how the nonlinear control problem dif-fers from the linear one and the nature of the theoreticalquestions that have to be answered.Algebraic and Analytic Parts ofAdaptive Control Prob-lems: In conventional adaptive control theory, two stagesare generally distinguished in the adaptive process. In thefirst, referred to as the algebraic part, it is first shownthat the controller has the necessary degrees of freedomto achieve the desired objective. More precisely, if someprior information regarding the plant is given, it is shownthat a controller parameter vector O* exists for every valueof the plant parameter vector p, so that the output of thecontrolled plant together with the controller approachesthe output of the reference model asymptotically. T he an-alytic part of the problem is then to determine stableadaptive laws for adjusting 8 ( k ) so that limk,, O ( k ) =O and the output error tends to zero.Direct and Indirect Control: For over 20 years, twodistinct approaches have been used [ l ] to control a plantadaptively. These are 1) direct control and 2) indirectcontrol. In direct control, the param eters of the controllerare directly adjusted to reduce some norm of the outputerror. In indirect control, the parameters of the plant areestimated as the elements of a vector b ( k ) at any instantk and the parameter vector 0 ( k ) of the controller is chosenassuming that p ( k ) epresents the true value p of the plantparameter vector. Even when the plant is assumed to belinear and time invariant, both direct and indirect adap-tive control result in overall nonlinear systems. Figs. 19and 20 represent the structure of the overall adaptive sys-tem using the two methods for the adaptive control of alinear time-invariant plant [ 3.A. Adaptive Control of Nonlinear Systems Using NeuralNetworks

For a detailed treatment of direct and indirect controlsystems the reader is referred to [ 3. The same approacheswhich have proved successful for linear plants can also beattempted when nonlinear plants have to be adaptivelycontrolled. T he structure used fo r the identification modelas well as the controller are strongly motivated by those

IController U-

Fig. 19. Direct adaptive control.

I I

19

Referencemodel

U 1ontrollerl l - - II 1 I IFig. 20. Indirect adaptive control.

used in the linear case. However, in place of the lineargains, nonlinear neural networks are used.Methods for identifying nonlinear plants using delayedvalues of both plant input and output were discussed inthe previous section and Fig. 1 1 shows a general identi-fication model. Fig. 21shows a controller whose outputis the control input to the plant and whose inputs are thedelayed values of the plant input and output, respectively.1. Indirect Control: At present, methods for directlyadjusting the control parameters based on the output error(between the plant and the reference model outputs) arenot available. This is because the unknown nonlinear plantin Fig. 21lies between the con troller and the output errore , . Hence, until such methods are developed, adaptivecontrol of nonlinear plants has to be carried out using in-direct methods. This implies that the methods describedin Section V have to be first used on line to identify theinput-output behavior of the plant. Using the resultingidentification model, which contains neural networks andlinear dynamical elements as subsystems, the parametersof the controller are adjusted. This is shown in Fig. 22.It is this procedure of identification followed by controlthat is adopted in this section. Dynamic b ack propagationthrough a system consisting of only neural networks andlinear dynamic elements was discussed in Section IV todetermine the gradient of a performance index with re-spect to the adjustable parameters of a system. Since iden-tification of the unknown plant is carried out using onlyneural networks and tapped delay lines, the identificationmodel can be used to com pute the partial derivatives of aperformance index with respect to the controller parame-ters.

B. Simulation ResultsThe procedure adopted to adaptively control a nonlin-ear plant depends largely on the prior information avail-able regarding the unknown plant. This includes knowl-


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U Nonlinear h . .

Referencemodel

'

Referencemodel f/m

IPIFig. 21. Direct adaptive control of nonlinear plants using neural networks.d Neuralnetwork N TDL

Neuralnetwork N e

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Fig. 2 2 . Indirect adaptive control using neural networks.

edge of the number of equilibrium states of the unforcedsystem, their stability properties, as well as the amplitudeof the input for which the output is also bounded. Forexample, if the plant is known to have a bounded outputfor all inputs U belonging to some compact set U, thenthe plant can be identified off line using the methods out-lined in Section V. During identification, the weights inthe identification model can be adjusted at every instantof time ( I; = 1 or at discrete time intervals ( 7; > 1 .Once the plant has been identified to the desired level ofaccurac y, control action can be initiated so that the outputof the plant follows the output of a stable reference model.It must be emphasized that even if the plant has boundedoutputs for bounded inputs, feedback control may resultin unbounded solutions. Hen ce, for on-line control, iden-tification and control must proceed simultaneously. Thetime intervals Ti nd T,, respectively, over which theidentification and control parameters are to be updatedhave to be judiciou sly chos en in such a case.Five exa mples , in which nonlinear plants are adaptivelycontrolled , are included below an d illustrate the ideas dis-cussed earlier. As in the previous section, each exampleis chosen to emphasize a specific point.I. Example 7: We consider here the problem of con-trolling the plant discussed in example 2 which is de-scribed by th e difference equationYp(k + 1) = f [ Y p ( k ) , Y p ( k - 1>3+ U ( k )

where the functionf[ Yp (k ) , Yp(k - 01

is assumed to be unknown. A reference model is de-scribed by the second-order difference equationy m ( k + 1 ) = 0.6ym(k) + 0.2ym(k - 1) + r ( k )

where r ( k ) is a bounded reference input. If the outputerror e, ( k ) is defined as e, ( k ) = yp k )- ym k), he aimof control is to determ ine a bounded control input u ( k )such that limk, e , ( k ) = 0. If the function f [ . in (19)is known, it follows directly that at stage k , k ) can becomputed from a knowledge of y p (k ) and its past valuesas

U(k) = -f[Yp(k), Yp(k - I ) ] + 0.6Yp(k)+ 0.2yp(k - 1 ) + r ( k ) ( 2 0 )resulting in the error difference equation e, ( k + 1 =0.6eC k)+ 0 .2e C(k - 1). Since the reference model isasymptotically stable, it follows that limk,, e , ( k ) = 0for arbitrary initial condition s. Howev er, sincef [ . is un-known, it is estimated on line asfas discussed in example2 using a neural network N and the series-parallel method.The control input to the plant at any instant k is com-puted using N [ in place off as

U@) = - N [ Yp(kL Yp(k - I ) ] + 0.6Yp(k)+ 0.2yp(k - 1 ) + r ( k ) . 21)This results in the nonlinear difference equation

Y,(k + 1)= f [Y p( k ) , Yp(k - I ) ] - N[Yp(kL Yp(k - 113

+ 0.6yp(k) + 0.2yp(k - 1) + r ( k ) ( 2 2 )governing the behavior of the plant. The structure of theoverall system is shown in Fig. 23.In the first stage, the unknown plant was identified offline using random inputs as described in example 2. Fol-lowing this, (21) was used to generate the control input.The response of the controlled system with a referenceinput r ( k ) = sin ( 2 ~ k / 2 5 ) s shown in Fig. 24(b).In the second s tage, both identification and control wereimplemented simultaneously using different values of Tiand T,. The asymptotic response of the system when iden-tification and control start at k = 0 with T, = T, = 1 isshown in Fig. 25(a). Since it is desirable to adjust thecontrol parameters a t a slowe r rate than the identificationparameters, the experiment was repeated with 7;. = 1 andT, = 3 and is shown in Fig. 25(b). Sin ce the identificationprocess is not complete for small values of k, the controlcan be theoretically unstable. However, this was not ob-served in the sim ulations . If the control is initiated at time


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t(a) (b)

Fig. 24. Example 7: (a) Response for no control action. (b) Response forr = sin 2 i r k / 2 5 ) with control.

Fig. 2 3 . Example 7: Structure of the overall system.

20 4l 80 80 100 0 20 U 80 80 100

IWO 0020 OMO 0e80 sa80 iwco oooo 0020 om0 0880 OSBO iwoo(a) (b)

Fig. 25. Example 7 : (a) Response when control is initiated at k = 0 withT, = T, = I . (b) Response when control is initiated at k = 0 and T, =1 and T, = 3 .

k = 0 using nominal values of the parameters of the neuralnetwork with Ti = T, = 10, the output of the plant wasseen to increase in an unbounded fashion a s shown in Fig.26.

The simulations reported above indicate that for stableand efficient on-line con trol, the identification must b e suf-ficiently accurate before control action is initiated andhence T, and T, should be chosen with care.


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' Response with feedback control 1.0 ,, . , . , . , . , .Response n t h feedback control

1.0-20 40 SO 80 100 0 20 40 80 so 100(C) (d)Fig. 31. Example 10: (a) , (b) Outputs of the reference model and the plantwhen no control action is taken. ( c) , (d) Outputs of the reference modeland the plant with feedback control.the form

which was identified successfully in exam ple 3 . Choosingthe reference model asy m ( k + 1) = 0 .6ym(k ) + r ( k )

t h e aim once again is t o c hoose u (k ) so t ha t l imk+ m= u3, the control input in this case is chosen asJ yp (k ) -Y m (k)l I ~ ~ [ Y , I~ p / ( l ~ ; ) a n d g [ u l

( 2 5 )U@) = g - [ -f[Y,(k)I + 0.6Yp(k) + r k ) ]w h e r e f a n d g-I are th,e estimates off and g-I, respec-tively. The estimates f and t are obtained as describedearlier using neural networks Nf nd N g . S in ce [ U ] hasbeen realized as the output of a neural network N,, theweights of a neural network N , E I (shown in Fig.3 2 ) can be adjusted so that N R N , ( r ) = r as r ( k ) variesover the interval [ -4 ,4 ] . The r ange [ -4 ,4 ] w a s c hose nfor r ( k ) since this assures that the input to the identifi-cation mogel varies over the same range for which theest imatesfand t are valid. In Fig . 3 3N , [ N , r ) ] s plot-ted against r and is seen to be unity ov er the entire range.

The determination of N,. was carried out over 25 000 timesteps using a random input uniformly distributed in theinterval [ -4, 41 and a step size of = 0.01. Since theplant nonlinearities f and g as well as g-' have been es-timated using neural networks Nfi g , and N,., respec-tively, the control input to the plant can be determinedusing ( 2 5 ) . The output of the plant to a reference inputr ( k ) = sin ( 2 n k / 2 5 ) + sin ( 2 n k / 10 ) is shown in Fig.


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Fig. 33. Example 1 1 : Plot of the function NR[iVc(r)].

0 20 40 , 60 BO 100 0 20 40 80 BO 100(a) (b)

Fig. 34. Example 1 1 : (a) Outputs of the reference model and plant withouta feedback controller. (b) Outputs of the reference model and plant witha feedback controller.

34(a) when a feedback c ontrolle r is not used; the responsewith a con troller is shown in Fig. 34(b). Th e response inFig. 34(b) is identical to that of the reference model andis almost indistinguishable from it. Hence, from this ex-ample we co nclude that i t may b e possible in some casesto generate a control input to an unknown plant so thatalmost perfect model following is achieved.

VII. COMMENTSND CONCLUSIONSIn this paper models for the identification and controlof nonlinear dynamic systems are suggested. Thesemodels, which include multilayer neural networks as wellas linear dynamics, can be viewed as generalized neuralnetworks. In the specific models given , the delayed valuesof relevant signals in the sy stem are used as inputs to mu l-tilayer neural networks. Methods for the adjustment ofparameters in generalized neural networks a re treated andthe concept of dy namic back propagation is introduced inthis context to gen erate partial derivatives of a perfor-

mance index w ith respect to adjustable parameters on line.However, in many identifiers and controllers it is shownthat by using a series-parallel model, the gradient can beobtained with the simpler static back-propagation method.

The simulation studies on low order nonlinear dynamicsystems reveal that identification and control using themethods suggested can b e very effective. T here is everyreason to believe that the same methods can also be usedsuccessfully for the identification an d control of multivar-iable system s of higher dimen sions. Hen ce, they shouldfind wide application in many areas of applied science.Several assumptions were made concerning the plantcharacteristics in the simulation studies to achieve satis-factory identification and control. For example, in allcases the plant was assumed to have bounded outputs forthe class of inputs specified. An obvious and importantextension of the methods in the future will be to the con-trol of unstable systems in some compact domain in thestate space. All the plants were also assumed to be of rel-ative degree unity (i.e., input at k affects the output atk + l ) , minimum phase (i.e., no unbounded input lies inthe null space of the operator representing the plant) andModels I1 and I11 used in control problems assumed thatinverses of operators existed and could be approximated.Future work will at tempt to relax some or al l of theseassumptions. Further, in al l cases the gradient m ethod isused exclusively for the adjustment of the parameters ofthe plant. Since it is well known that such methods can


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to 1987 and currently he is the Director of the Center for Systems Scienceat Yale. He is the author of numerous technical publications in the area ofsystems theory. He is the author of the books Frequency Domain Criteriafor Absolute Stabili ty (coauthor J. H . Taylor) published by Academic Pressin 1973; Stable Adaptive Systems (coauthor A. M . Annaswamy); andLearning Automata-An Introduction (coauthor M . A. L. Thathachar) pub-lished by Prentice-Hall in 1989. He is also the editor of the books Appli-cations ofAdap tive Conrrol (coeditor R.V. Monopoli) published by Aca-demic Press in 1980 and Adaprive and Learning Systems (Plenum Press,1986). He is currently editing a reprint volume entitled Recent Advancesin Adaptive Control (coeditors R. Orteg a and P. Dorato) which will bepublished by the IEEE Press. His research interests are in the areas ofstability theory, adaptive control, learning automata and the control ofcompl ex systems using neural networks. H e has served on various nationaland international technical committees as well as several editorial boardsof technical journals.Dr. Narendra is a member of Sigma Xi and the American MathematicalSociety and a Fellow of the American Association for the Advancement ofScience and the IEE (U.K .) . He was the recipient of the 1972 Franklin V .

Taylor Award of the IEEE Systems, Man, and Cybernetics Society and theGeorge S. Axelby Best Paper Award of the IEE E Control Systems Societyin 1988.*

Kannan Parthasarathy was born in India on Au-gust 31, 1 964. He received the B.Tech. deg ree inelectrical engineering from the Indian Institute ofTechnology, Madras, in 1986 and the M.S. de-gree and M.Phil. degree in electrical engineeringfrom Yale University in 19 87 and 1 988, respec-tively. At p resent he is a doctoral candidate at YaleUniversity. His research interests include adap-tive and learning systems, neural networks andtheir application to adaptive control problems.Mr. Parthasarathv is the reciDient of the Con-necticut High Technology Graduate Scholarship for the year 1989-1990.

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Identification and Control of Dynamical Systems Using Neural Networks