Model-based neural algorithms for parameter estimation

Model-based neural algorithms for parameter estimation

Intelligent Systems Model-Based Neural Algorithms for Parameter Estimation* FREDRIK P. SKANTZE and ANURADHA M. ANNASWAMY* Adaptive Control Laborator...

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Intelligent Systems

Model-Based Neural Algorithms for Parameter Estimation* FREDRIK P. SKANTZE and ANURADHA M. ANNASWAMY*

Adaptive Control Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

ABSTRACT Recently 0-adaptive neural networks (TANN) were developed for parameter estimation for systems with nonlinear parametrization [1]. This paper discusses the estimation problem when additional information is available about the model structure. In particular, algorithms are presented for recursive parameter estimation for systems with partial nonlinear parametrization and for systems where the nonlinearities appear in an additive manner in the regression equation. Training procedures are developed for the neural networks which ensure the stability of the algorithms. It is shown how the training procedures can be modified to ensure stability in the presence of a bounded disturbance. The complexity of the neural networks needed to perform the identification tasks is greatly reduced compared to the TANN algorithm proposed in [1]. Simulation results are presented which demonstrate the capabilities of the algorithms. ©Elsevier Science Inc. 1998

1.

INTRODUCTION

Q u e s t i o n s r e l a t e d to prediction, regulation, a n d fault diagnosis in complex e n g i n e e r i n g systems can be posed as p r o b l e m s of identification a n d control of n o n l i n e a r d y n a m i c systems. A g e n e r a l solution to these prob-

* This work was supported in part by the Electrical Power Research Institute under contract No. 8060-13 and in part by the National Science Foundation under grant No. ECS-9296070. *Corresponding author.

INFORMATION SCIENCES 104, 107-128 (1998) © Elsevier Science Inc. 1998 655 Avenue of the Americas, New York, NY 10010

0020-0255/98/$19.00 PII S0020-0255(97)00077-7

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F. P. SKANTZE AND A. M. ANNASWAMY

lems, especially in the presence of nonlinearities, is obviously a difficult task. While conventional analytical techniques are applicable in the context of linear systems, very few methods exist for identification and control of nonlinear systems. An additional source of complexity is the presence of uncertainties due to unforeseen changes in the environment and variations in system characteristics. Hence all the identification, control and fault diagnosis tasks have to be carried out amidst these uncertainties. A new approach, for carrying out these tasks, which was investigated recently by a number of researchers and enjoyed popular attention is the use of neural networks. Because of their capabilities as universal approximators which can learn by examples, neural networks were suggested for identifying and controllling nonlinear systems for which analytical solutions cannot be found, for example [1, 8, 10, 11, 3]. In most of these approaches, neural networks were used as a "black box" model of the system under consideration [8, 10, 11, 3]. If the problem is one of identification, the network models the relation between system input and output. In a control problem, the neural network is trained to aproximate the controller, which is the map between the tracking error and the control input. The results obtained in these investigations indicate that when very little prior information is available about the underlying systems, the approach used leads to significant improvement. However, a drawback of this method is that it provides little insight into the structure and dynamics of the actual system since the parameters of the neural network have no physical meaning. In many engineering systems, models that capture the dominant system behavior are available. The underlying physics can be quantified by conservation equations or constitutive relations. In such situations, where a "gray box" model is available, it is often infeasible to use conventional analytical techniques due to the presence of nonlinearities and uncertainties. Recently, a model-based approach was suggested in [1] for identification tasks in systems using neural networks. In this approach, the neural network is trained to learn the map between the system variables and the values of the physical parameters, 0, associated with the system. To distinguish these neural networks from those identifying the system input/output characteristics, they are referred to as 0-adaptive neural networks (TANN) since they adapt to the system parameters, 0. This approach is particularly attractive when the system parameters occur nonlinearly since few analytical tools exist for estimating 0 in this case. In this paper, we introduce new model-based neural algorithms for identifying 0 by making judicious use of the structure of the underlying physical model.

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In [1], two algorithms were presented for the identification of the parameter vector 0 of a dynamic system,

y,=f(,b1_l,0)

(1)

where y, is the output of the system at time t and ~bt_ 1 represents past values of measurable system variables. In the first approach, which is referred to as the block estimation method, the neural network is trained to approximate the implicit function between system input/output data and the parameters. The second approach is a recursive estimation method where the neural network updates the parameter estimates on-line based on new samples of the system response. In both cases, conditions under which the algorithms lead to stable estimation, training procedures for the networks, and illustrative examples which show the nature of the parameter convergence were described. The main property of a neural network that is exploited in these tasks is its ability as a universal approximator of a nonlinear function [9]. For a radial basis function network, the approximating error can be shown to be of the order of (l/N) 2/n [2], where N is the number of basis functions and n is the input dimension of the nonlinear function being approximated. At the same time it should be noted that the size of the neural network grows exponentially with the number of the variables for which it has to be trained. It is therefore of interest to determine identification algorithms that yield the smallest possible parameter error norm with the minimum number of inputs to the neural network. In this paper, we introduce new recursive algorithms which make use of model-based parametrizations. In particular we will consider two special cases of (1), T

Yt =f(q~N, t, ON) + dPL, , OL,

(2)

Yt = ~-"~fi(4)i, ,,0~),

(3)

and

i=1

f(•N,_I, 0N) and fi(~)i,_l, Oi), i= 1,2 ..... m, are nonlinear functions of their arguments. It would be desirable to exploit the linear substructure

where

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F. P. SKANTZE AND A. M. ANNASWAMY

of (2) and the additive property of the nonlinearities in (3) in ways which reduce the complexity of the neural networks needed to perform the identification tasks. Such algorithms are developed in this paper. In order to enhance readability we will adhere to the following notation throughout the paper. The estimate of a quantity x will be denoted by 2 and the estimation error of x will be denoted by ~, where $ = 2 - x . The change in x between time t - 1 and t is given by A x t = x t - x t _ r A function of several variables which are themselves functions of time is denoted as ft if its arguments are evaluated at time t and possibly at past instants t - 1, t - 2 . . . . . For example, f(Xt_l,Yt_2) will be denoted as f t - i and so on. ft corresponds to an estimate of ft with all of its arguments that are not measurable replaced by their estimates. Section 2 briefly reviews linear parameter estimation and provides insights that will be important for understanding the problem when the parameters occur nonlinearly. In Section 3, the TANN algorithm from [1] is reviewed. In Section 4, we introduce an algorithm for systems with partial nonlinear parametrization as in (2). The estimation problem for regression equations of the form of (3), where the nonlinearities appear additively, is discussed in Section 5. Some concluding remarks are offered in Section 6. 2.

LINEAR PARAMETRIZATION

Parameter identification in systems where the parameters occur linearly were studied extensively [7]. Several algorithms were proposed over the years and conditions under which accurate identification can be carried out are well known. In this section, we briefly discuss one such algorithm for the sake of completeness and comparison. In particular, we highlight the features of the algorithm that enable us to derive the neural algorithm that is applicable to the system in (1) with a combined parametrization, since we would like to make use of the fact that the parameter vector 0L in (2) occurs linearly by combining ideas from linear parameter estimation with the neural computational structures developed in [1]. Consider a dynamic system of the form,

y,= 4~L lo,

(4)

where Yt is the system output at time t, ~bt_ 1 is a vector which is a linear or nonlinear function of past values of the inputs and output of the system and 0 is the unknown parameter vector that needs to be estimated. The recursive estimation problem can be stated as follows. We want to find a

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function F such that the algorithm,

Ot=Ot-l +F(Y,,qbt-,,O,-,),

(5)

ensures that 0t ~ 0 as t ~ ~. For a system which is linearly parameterized such as (4), a number of possible choices for F exist depending on the criterion of best fit. If we choose F such that 0 lies on the hyperplane H = { 0 : y,=~br_10} and at the same time minimizes the cost function J=½lO1-Ot 112, then we obtain the well known projection algorithm (e.g., [4]), (6) where

Y, = q~L 10t 1--Yt,

kt-i

1 d+l~b,_ll 2'

and d is a small positive constant. A standard technique used to analyze the stability of algorithms of the form of (5) is to define a positive definite function of the parameter error. If the change of this function at each time instant is negative or zero, then the algorithm is guaranteed to be stable. With this in mind, we define a positive definite function Vt as

z,= 4Ta, The change in Vt is given by

A~=2AotTot_I + AOtTAot,

(7)

where k 0 , = 0 , - 0 , _ 1" The second term on the right-hand side of (7)will always be positive and thus has a destabilizing effect. In order for AG to be negative, the first term must be negative and of larger magnitude than the second term. If we substitute for A~, from (6) and make use of the fact that I~bt_ 1[2/(d + I~bt_ t l2) < 1, we obtain A< ~
(8)

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F . P . SKANTZE AND A. M. ANNASWAMY

Since the system in (4) is linear in the parameter vector 0, the following identity holds,

4'L1 gt l=-~t"

(9)

This leads to

AV~

d+14',_a1240,

(10)

which verifies the stability of the algorithm. The identity in (9) does not hold for systems which are nonlinearly parametrized. This makes it considerably more difficult to develop stable recursive estimation algorithms for these systems. We will address this issue in the next section. 3.

NONLINEAR PARAMETRIZATION

We consider in this section the parameter identification problem for a dynamic system of the form,

yt=f(4't 1,0),

(11)

where y, is the output of the system at time t, 4,t- 1 represents past values of measurable inputs and the output, and 0 is a vector of unknown system parameters in ~ n. 4' and 0 lie in the compact sets • and O, respectively. In particular, we are interested in systems where f is a nonlinear function of the parameter vector 0, since these systems cannot be put into the form of (4). Just as in the previous section, we want to find a function F which makes use of past values of the inputs and output of the system as well as the previous parameter estimates to ensure that the equilibrium point 0= 0 of the estimation algorithm in (5) is stable. However, when the parameter vector 0 is a nonlinear function of f as in (11), few analytical tools are available to derive such a function F. In [1] an algorithm is proposed where a neural network, N, is trained to perform the role of the function F. That is, the weights, W, of the neural network are chosen such that 0 t = 0 t , + vN(gt, 4,,_,, 0t_ 1; W),

(12)

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113

results in a stable estimation algorithm, u represents a dead-zone modification to be discussed later. To gain insight into the stability properties of this algorithm, it is useful to normalize N. This results in a new neural network, N*, given by

N, = -kt15tNt*,

(13)

where k, = 1/(d + [Art*12) and d > 0. Substituting (13) into (12) gives Ot ~- Ot 1 - k t 1 5 t N t *"

(14)

Accordingly, as in the previous section, we define a positive definite function Vt = 0~0t. Again the change in Vt at time t is given by AVt = 2Aotrot_l+AOtrAO t. Substituting for A0t from (14) followed by some algebraic manipulation gives

A Vt ..< k, ( - 215, N,* T(~, 1 --}-15t2),

(15)

which is of the same form as (8). In the previous section, the identity ~b~r , ~}t-1 =15, was used to establish that AV~~<0. It is evident from (15) that if we could train N* such that Nt*T0t_l =15t, then AVt~<0. However, for a nonlinear system of the form of (11), it is not possible to accomplish this, since Nt*r0t_l is a linear function of 0, whereas )7, =f(c~t_ 1, Ot 1)f(~b,-1, 0) is a nonlinear function of 0. We can however express the prediction error as 15t= N* T0t_ 1 + R(~bt_ 1, 0, 0t- 1), where R t _ l represents an error term. Intuitively, we can see that if R t_ 1 is small compared to 15t, then AVt will be less than or equal to zero. If we choose 6>12/ (1-a)sup4)~.,0~o,O~ o R(~,O,O), where 0 ~ < a < l , then we can ensure that I)St[> 2JR t 11by choosing the dead zone in (12) as v:{~

if 133'1> & otherwise.

The resulting change in Vt is given by

AVt<

aY!

d + [Nt.[2 ~<0,

(16)

when [fit[> 15 and AVt = 0 if [fit[~<6. Hence, it follows that there exists a neural network N such that the algorithm in (12) is stable.

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F . P . SKANTZE AND A. M. ANNASWAMY

In order to carry out parameter estimation, the neural network needs to be trained. For this purpose, the network requires a target output so that during training, the difference between the target and the actual system output of the network can be used to adjust its weights. However, since F cannot be expressed in closed form, the neural network in (12) has no direct target output that can be used for its training. We can, however, generate a distal target for N. Let the function A V d represent the desired value for AV in (16) and let A V e be the difference between the two so that

AVe = a V - aV .

(17)

We can then use A V d as a distal target and train the neural network by adjusting its weights in such a way that IAVeldecreases. The choice of the distal target, A V d, should be such that the neural network can make AVe small. If we compare (16) with (10), we see that the AV achieved in both cases are of the same form. Therefore it seems natural to choose A V d as given by (10), but with the variable ~bt_ 1 replaced by a quantity C,_ 1, which performs the corresponding function for the nonlinear system in (11) as does ~bt_ 1 for the linear system in (4). That is, we define a target AVd for AV as

AVd'

d + ]C,_ 1]2 "

(18)

To a first order approximation, the dynamics of (11) in the neighborhood of some parameter value 0 0 can be described by the linear equation y t = ( O f t _ l / O O ) l r o O . Thus, (cTft_l/C)O)loo plays a similar role for the nonlinear system as does ~b for the linear system. Noting that we are only interested in the magnitude of C and that a large C is more conservative since it leads to a smaller magnitude of AVd, we choose Ct- 1 = sup

°~ft- 1 .

0~O

Because it is known that belongs to compact set ~ , choosing 0j ~ ®, 0j ~ O, and and AVdj using (11) and (18), given by

0 and 0 belong to compact set ® and ~b we can form the jth training pattern by ~bj~ • and calculating the corresponding 15j respectively. The j t h training pattern is then

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By repeating this process for all variations of 0, 0, and 4), the complete training set is obtained as Ttrai,=

I,.J Tj,

I<<.j<~Q

where Q denotes the number of training patterns. Using the same procedure, a testing set Ttest can be formed for cross validation. The neural network can be trained by choosing its weights so as to minimize the cost function, Q

J(W) = Y'. max{0, AVe,} 2.

(19)

j=l

Because the neural network output, N, is only indirectly related to the cost function J(W) through (15), (17), and (19), this procedure for adjusting the weights of the neural network is referred to as training with a distal teacher [6]. It should be noted that since this is a nonlinear optimization problem, it might converge to a local minima. The reader is referred to [1] for further details on algorithms available for performing the minimization of J(W) and for a more elaborate discussion on the formation of the training set as well as of the calculation of the size of the dead zone. 4.

PARTIAL NONLINEAR PARAMETRIZATION

4.1. THE ALGORITHM The algorithms presented in Sections 2 and 3 can both be represented in the form of the update law,

For the linear system y, = ~b,r ] 0, the prediction error is given by Y,=~b,r-, ~, 1,

(20)

and it was shown that choosing F¢ as ~b,_ ~ resulted in a stable estimation algorithm. For a system of the form y, =f(~b,_ 1, 0), the prediction error is given by

yt=f(4)t_l,0t

1)--f(4)t_,,O).

(21)

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F. P. SKANTZE

AND A. M. ANNASWAMY

In this case we choose F, as the output, N,, of a neural network and train the neural network in such a way as to ensure the stability of the algorithm. In this section, we consider the dynamic system,

where y, is the output of the system at time t, +N,_,, and c#+_, represent past values of measurable inputs and the output and 0, and 13, are vectors of unknown system parameters in 8’~ and ‘%‘L. 4,,, and 8, lie in compact sets @.N and O,, respectively. The task is to estimate the parameter vectors & and 0,. The prediction error for this system is given by

(23) where j, is assumed to lie in compact set $. Our discussions in Sections 2 and 3 indicated that a linear algorithm suffices for a prediction error of the form of (20), whereas for a prediction error of the form of (20, a neural network can be used. This suggests that for the system in (22), since the prediction error is of the form of (23), a neural network is only needed to estimate the parameter vector e,, which occurs nonlinearly. We therefore propose the following algorithm,

(24) (25) where

k

1 ‘-I=

d+lC,_1~2+~~~,_,~2’

d>O,

(26)

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117

otherwise, 2 6>j 1--a

R( Cu, ON,ON),

sup

(27)

y E ~ , ~N E (~N, ONE ~N, ONE ~)N

R = f - - f --NTON, and 0 ~,9~nN such that the equilibrium point (ON : O, 0 L : O) of the parameter identification algorithm,

Ou,: Ou, ,-- ukt.gtF(.gt, &N, ,,ON, ,), o\, = oL, ,-

6L, ,.

is stable, where, kt =

1

2,

d + Iftl z + I~bL, ~1

u= { 1 0

d > 0,

(28)

if lytl> 6, otherwise,

2

sup

6>_- 1--a

R(CbN,ON,ON),

(29)

f;C ~II , ~N E ¢~N' ONE ~)U' ONE (~N

n =fi--f--FT"ON, and 0 ~ a < l . Proof. Let F t be a continuous function and choose 6 such that (29) is satisfied. Consider the positive definite function Vt = VN, + VL, ,

(30)

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F. P. SKANTZE AND A. M. A N N A S W A M Y

where VN, = 0urON, and VL, = 0[,0L,. After some algebraic manipulation, we obtain AVt = 2AO,rO,_ 1 + AO,rAOt • If I)3,1~< 6, then A~t = 0 so that Vt = Vt 1 which implies stability. When 1:9,1> 6 we have

AVt= - 2kt-~,( FtTON, ~+ 4aTL, ~OL, ,) + k2t y2(IFt12 +149L, ,I2) =k,5,2 [ _ 2F, rON,, + 4~[, lOLl 1 +

IF, I2+I~L, ,[2

)

d_t_lFtlm_t_l~t , 2


2F, rON, + C~L, r OL, " +R

f~

) 2R + Y, + 1

Inl

~
--

a'~2

<0.

d +-lF,12 +l~bL, ,I2 Thus we have established that either lY,I< a so that AVt --0, or lY,I> a in which case AVt < - a k , f~t2 <0. It therefore follows that 0 = 0 is a stable equilibrium point. []

4.2. TRAINING ^

The neural network in (24) only receives )~t, (~N, t' and ON, 1 as inputs whereas the original TANN algorithm, which does not take into account the linear relationship between 0L and Yt, requires all measurable system variables as inputs. Thus the algorithm presented in (24) and (25) has the potential to reduce the complexity of the neural network needed to perform the identification task. This, however, will not be the case if the training procedure outlined in Section 3 is used to train the neural network. The reason for this is that AV is a function of all the system variables. Thus, if we use AVd as a distal target and adjust the weights of the neural network such that IAV~L=JAV U AVdl decreases, the training set must contain variations of ~bN, q~L, ON, OL, ON, and 0L, which means that no reduction in complexity was achieved. In this section, we develop an alternative training procedure, which adjusts the weights of the neural network to achieve the same AVd by using distal targets which only

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119

depend on )3, ¢bN, iN, and ON. As a result, the training set only contains variations of these variables, which results in a drastic reduction in the complexity of the neural network in (24). Suppose in Theorem 1, we substitute N for F and let k be given by (26) instead of (28). The following two conditions must be satisfied for the algorithm to be stable, sgn(3~t) NtTON, >/sgn()3t) (./~_l --f,-1 ) -- - ~ l ) ~ t l ,

IN, I~IC,_ tl, where a is given by (27). Thus, for the two quantities, P = sgn( )~,) Nf0N, ,,

(31)

M=IN~I,

(32)

we can define two targets,

Pd = sgn(JT,) (j~ , -f,_ t) -- ~lJ~,l,

Ma=lC, 11. The jth training pattern is then given by

where ON and t~N belong to ON, and 4~Ns and )7s lie in ~N and ~t, respectiveJly. The tr'aining set is obtained as Ttrain =

U Tj, l <~j~
where Q denotes the number of training patterns. Ttest can be formed in a similar manner. If we let Pe = P 1- P a j and M e~ = M .3- M a 1, then we want to choose the weights, W, of the neural network such that Pe~ >t 0 and M~ ~ O. This can be done by minimizing the cost function,

j=l

O'p

O"M

'

120

F . P . SKANTZE A N D A. M. A N N A S W A M Y

where Q denotes the number of training patterns. Note that Pej only contributes to the cost if it is negative and M e only contributes if it is positive. O-p and trM are normahzatlon constants which are usually chosen as the standard deviations of Pd and Me, respectively• Since the neural network output, Nt, is only indirectly related to the cost function J through (31), (32), and (33), the process of updating the weights of the neural network can be viewed as training with a distal teacher, After the training is completed, 6 can be calculated by evaluating the neural network output, Nj, for each training pattern, ~. The dead zone is then given by •

.

2 6 >/T--z-d

J

,

sup R j,

I<~j<~Q

where Ri=fi-fi-NirOu, and 0~
6, since AVd, < -akt_ 1)72 < 0, AVt will also be negative as long as the approximating error, e, of the neural network is not too large• For a neural network which satisfies the property of a "universal approximator" [5, 9], e can be made arbitrarily small by choosing the appropriate network structure and weights. •

4.3.

.

J

R O B U S T N E S S TO A BOUNDED D I S T U R B A N C E

The algorithm described in Sections 4.1 and 4.2 was developed based on the assumption of a perfect system model• This assumption rarely holds in

MODEL-BASED NEURAL ALGORITHMS

121

practice, because of the inevitable presence of uncertainty due to disturbances, modeling errors and measurement noise. An approach to modeling these error terms is in the form of an additive disturbance signal, w r Equation (22) then takes on the form, Yt = f ( ¢N,.

,,

ON) + qbT, lOL +Wt"

(34)

If the training procedure described in Section 4.2 is used to estimate 0, then the presence of wt could cause instability. In particular, we can no longer guarantee that AVt < 0. The objective of this section is to modify the training procedure such that V~ retains its nonincreasing property in the presence of w,. It is assumed that there exists a known positive constant 1) such that sup Iw tl < ~ . The prediction error for the system in (34) is given by

)Tl=f( ~Nt ,,ONt_l)--f( ~Nt..l,ON) ~- ~tT lot t I--W, • If the parameter update laws are given by (24) and (25) as before, then the two conditions needed for AV to be nonpositive are sgn()7,) Nf0N, , >/sgn( )7,)(J~ 1-Jr,-, ) - - ~ l ) 7 t l - s g n (

IN, I < IC,_,l. Thus, for the two quantities, P = sgn()7,) NtV0N, ,, M = I N , I, we can define two targets pd = sgn()7,)(ft_,--f, Md =

IQ_,I.

,) - }--~--al)7,1+ sgn()7,) f~,

)7,)w,,

122

F. P. SKANTZE AND A. M. ANNASWAMY

The neural network can then be trained by minimizing the cost function given by (33). This will ensure that AVe<0 even in the presence of the disturbance w r It should be noted that since the targets for the neural network depend on l~, the dead zone, 3, will also depend on ~. 4.4.

SIMULATION RESULTS

We describe the behavior of the algorithm in (24) and (25) in the context of parameter estimation for the following system from [1],

Yt

ONYt 1 l+e-O,~y~l

+OLut

1.

ON and 0L are unknown parameters to be estimated. It is known a priori that the numerical values of the parameters are between 1 and 2 and that the input is in the range between 1 and 3. Given the bounds on the input and the parameters, y is confined to the interval [1.5, 8.5]. The neural network was trained for seven variations of )7t, Yt-~, 01, ~, and ON, respectively, resulting in 74-- 2401 training patterns, a was chosen to be 0.5, which resulted in a 3 of 0.07. A radial basis function network with 500 centers was trained using the procedure outlined in the previous sections. Simulations with noise-free and noisy measurements were performed. The input to the system was u t = 2 + sin(27rt/8). Figures 1 and 2 show simulation results for the estimation of the parameters with noise-free measurements. The dashed lines represent the actual parameter values and the solid lines are the estimates. Figures 3 and 4 show simulation results when measurement noise was added to y. The noise is modeled as a random signal uniformly distributed on the interval [ - 0 . 1 , 0.1], resulting in a ~q of 0.1. In both simulations the parameter estimates converge until the algorithms stop updating because of the dead-zone modification. The rate of convergence is slower when the measurements are noisy. In order to achieve comparable^ results with the original TANN algorithm the six variables 4~1, ~b2, 01, 02, ON, and 0 L would have to be varied seven times each, resulting in 76= 117649 training patterns. Thus the complexity of the estimation problem was reduced by a factor of 49. 5.

ADDITIVE NONLINEARITIES

In the previous section, we showed that the identification problem is simplified substantially if the nonlinear function f in (11) contains a linear substructure so that it can be written in the form f(4~t- l, 0) =f(4'N, ,, 0N)

MODEL-BASED

NEURAL

123

ALGORITHMS

2111

1.9 1.8 1.7

1,6 z 1.5

1.4 1.3

,2f 1.1

1 0

I

I

50

100

150

Iteration

Fig. 1. Estimates of

ON

(noise-free measurements).

+ ~bLr 0L, where,

4)N ¢=[4~L]

and

ON 0=[0L].

This raises the question if there are other special forms of f which will simplify the identification of 0. One such case is when f can be expressed as a sum of nonlinear functions f(~bt_ 1, 0) = y~m i= 1 fi( l~i, 1' Oi)' where,

,=

and

0Iil

F. P. S K A N T Z E AND A. M. A N N A S W A M Y

124 2

1.9 1.8 1.7

1.6

t

.u 1.5

¢D

1.4 1.3 1.2 1.1

1

0

I

I

50

100

150

Iteration

Fig. 2. Estimates of 0L (noise-free measurements).

(hi S a r e measurable functions of time, and 0 is an unknown parameter. The system is then described by the regression equation,

Y,= ~fi((hi, 1,Oi),

(35)

i=1

where (hi and 0i lie in compact sets qbi and ®i, respectively, i = 1..... m. For the system in (11), the impact of 0i on Yt depends on the values of (hi, 1,'",(hm , and 01 ..... Oi_l,0i+l,...,Om. For this reason, the neural network in ~12) must be trained for variations of all the system variables. However, for the system in (35), the impact of 0i on Yt only depends on the value of (hi,_~- In this section, we present an algorithm which makes full use of this by using a separate neural network to estimate each of the parameter vectors. The neural network, N/, used to estimate the parameter vector 0i only needs to be trained for variations of the signals related to the ith subsystem, namely .9, (hi t |, 0i, t' and Oi, resulting in a drastic

MODEL-BASED NEURAL ALGORITHMS

125

1.9 1.8 1.7

1.6 z 1.5

1.4 1.3 1.2 1.1

1

0

I

i

I

I

I

|

100

200

300

400

500

600

Iteration

Fig. 3. Estimatesof 0N (noisymeasurements). reduction in complexity. The algorithm is given by

01,=~1' 1-- ~-/kl

,)TtN,()Tt,~b,, ,,{}1, ,),

Omt=~rnt 1-- pkt lfltNm( fit's)m, l '~m, i)' where, kt_ l =

1 d + 2m=

p= / 1 0

l[Ci,

1]2'

if I~,1> a, otherwise,

d>O

700

126

F . P . SKANTZE AND A. M. ANNASWAMY 2

1.9 1.8

1.7 1.6 .a 1.5 ~D

1.4 1.3 1.2

1.1

0

I

I

100

200

I

i

300 400 Iteration

I

I

500

600

700

Fig. 4. Estimates of 0L (noisy measurements).

O f i,- 1 C~, =

sup

OiE ~)i

00~ '

and 0 ~

MODEL-BASED NEURAL ALGORITHMS

127

and their respective targets, l+a

Pd, = ---U-sgn(9,)()~,,-f~, ,), Mai:[Ci ' ,1. If we let Pe, = P i - Pal, and Me~= M i -Md,, then we can train the neural network by choosing the weights, Wi, so as to minimize the cost function,

gi(w) = E

[(min0ej/

j=l

O'M i

where Qi denotes the number of training patterns in the training set of the ith neural network and (Pe, Me, ) are the errors in the targets correspondi) 1 . . ing to the . jth .data . set of the. training set of the. tth neural network. P~i) . . . . only contributes to the cost If it Is negative and M e only contributes if It ~s . . . . ij . positive, o-e, and Crg, are normalization constants which are usually chosen as the standard deviations of Pd~ and Ma,, respectively. 6.

CONCLUDING REMARKS

Estimates of the physical parameters of a system provide valuable information for tasks such as control and fault detection. For systems which are nonlinearly parameterized, however, few analytical techniques are available for parameter estimation. In [1] a novel, model-based approach to neural network based system identification was proposed where the neural network is trained to provide an estimate of the system parameters at each instant of time. This algorithm thus has the potential to improve the performance of control and fault detection in dynamic systems in the presence of nonlinearities. However, the neural network has to be trained for variations in all the system variables and grows exponentially in size when the number of variables is increased. In this paper, we present algorithms which make use of special structures of the nonlinearities to reduce the complexity of the neural network. In particular, algorithms for nonlinear systems with a linear subsystem as well as systems where the nonlinearities appear additively, are developed. It is shown that the training procedures developed result in stable estimators in the presence of bounded disturbances and require a significantly smaller number of training patterns. This is further verified by simulation results.

128

F. P. S K A N T Z E A N D A. M. A N N A S W A M Y

The authors thank Dr. Ssu-Hsin Yu for his many helpful suggestions and comments. REFERENCES 1. A. M. Annaswamy and S. Yu, 0-adaptive neural networks: A new approach to parameter estimation, IEEE Trans. Neural Networks" 7:907-918 (1996). 2. A. R. Barron, Approximation and estimation bounds for artificial neural networks, in: Proceedings of the 4th Annual Workshop on Computational Learning Theory, 1991, pp. 243-249. 3. F.-C. Chen and H. K. Khalil, Adaptive control of a class of nonlinear discrete-time systems using neural networks, IEEE Trans. Automat. Contr. 40(5):791-801 (1995). 4. G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction, and Control, PrenticeHall, Englewood Cliffs, NJ, 1984. 5. K. Hornik, M. Stinchcombe, and H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2:359-366 (1989). 6. M. I. Jordan and D. E. Rumelhart, Forward models: Supervised learning with a distal teacher, Cognitive Sci. 16:307-354 (1992). 7. L. Ljung, System Identification: Theory for the User, Prentice-Hall, Englewood Cliffs, NJ, 1987. 8. K. S. Narendra and K. Parthasarathy, Identification and control of dynamical systems using neural networks, IEEE Trans. Neural Networks 1(1):4-26 (1990). 9. J. Park and I. W. Sandberg, Universal approximation using radial-basis function networks, Neural Comput. 3:246-257 (1991). 10. R. M. Sanner and J.-J. E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neural Networks 3(6):837-863 (1992). 11. M. A. Sartori and P. J. Antsaklis, Implementations of learning control systems using neural networks, IEEE Contr. Syst. Mag. 12(2):49-57 (1992).

Received 28 September 1995; revised 15 October 1996