System Identiﬁcation Using Adjustable RBF Neural Network with Stable Learning Algorithms Wen Yu1 and Xiaoou Li2 1

2

Departamento de Control Autom´ atico, CINVESTAV-IPN A.P. 14-740, Av.IPN 2508, M´exico D.F., 07360, M´exico [email protected] Secci´ on de Computaci´ on, Departamento de Ingenier´ıa El´ectrica, CINVESTAV-IPN A.P. 14-740, Av.IPN 2508, M´exico D.F., 07360, M´exico

Abstract. In general, RBF neural network cannot match nonlinear systems exactly. Unmodeled dynamic leads parameters drift and even instability problem. According to system identiﬁcation theory, robust modiﬁcation terms must be included in order to guarantee Lyapunov stability. This paper suggests new learning laws for normal and adjustable RBF neural networks based on Input-to-State Stability (ISS) approach. The new learning schemes employ a time-varying learning rate that is determined from input-output data and model structure. The calculation of the learning rate does not need any prior information such as estimation of the modeling error bounds.

1

Introduction

Resent results show that RBF neural network seems to be very eﬀective to identify a broad category of complex nonlinear systems when we do not have complete model information [10]. It is well known that normal identiﬁcation algorithms are stable for ideal plants [4]. In the presence of disturbance or unmodeled dynamics, these adaptive procedures can go to instability easily. The lack of robustness in parameters identiﬁcation was demonstrated in [2] and became a hot issue in 1980s. Several robust modiﬁcation techniques were proposed in [4]. The weight adjusting algorithms of neural networks is a type of parameters identiﬁcation, the normal gradient algorithm is stable when neural network model can match the nonlinear plant exactly [11]. Generally, we have to make some modiﬁcations to the normal gradient algorithm or backpropagation such that the learning process is stable. For example, in [6] some hard restrictions were added in the learning law, in [13] the dynamic backpropagation was modiﬁed with NLq stability constraints. Another generalized method is to use robust modiﬁcation techniques of robust adaptive control [4]. [8] applied σ−modiﬁcation, [5] used modiﬁed δ−rule, and [15] used dead-zone in the weight tuning algorithms. The motivation of this paper is to prove that the normal gradient law and backpropagation-like algorithm without robust modiﬁcations are L∞ stable for identiﬁcation error. F. Yin, J. Wang, and C. Guo (Eds.): ISNN 2004, LNCS 3174, pp. 212–217, 2004. c Springer-Verlag Berlin Heidelberg 2004

System Identiﬁcation Using Adjustable RBF Neural Network

213

Input-to-state stability (ISS) is an elegant approach to analyze stability besides Lyapunov method. It can lead to general conclusions on the stability by using input and state characteristics. We will use input-to-state stability approach to obtain some new learning laws that do not need robust modiﬁcations. A simple simulation gives the eﬀectiveness of the suggested algorithm. To the best of our knowledge, ISS approach for RBF neural network was not still applied in the literature. The adjustable RBF neural network is referred to the activation function of RBF neural network can be updated by a learning algorithm. In this ISS approach is applied to system identiﬁcation via RBF neural network. Two cases are considered: (1) For normal RBF neural network, the activation functions are assumed to be known, and learning is carried on the weights, (2) Learning algorithm concerns both the activation functions and the weights. The new stable algorithms with time-varying learning rates are applied.

2

Identiﬁcation Using Normal RBF Neural Network

Consider following discrete-time nonlinear system in NARMA form y(k) = f [y (k − 1) , y (k − 2) , · · · u (k − 1) , u (k − 2) , · · · ] = f [X (k)]

(1)

T

where X (k) = [y (k − 1) , y (k − 2) , · · · u (k − d) , u (k − d − 1) , · · · ] ∈ n , f (·) is an unknown nonlinear diﬀerence equation representing the plant dynamics, u (k) and y (k) are measurable scalar input and output, d is time delay. A normal RBF neural network can be expressed as y (k) = W (k) Φ [X (k)]

(2)

where the weights W (k) = [w1 · · · wn ] , the activation Φ [X (k)] = function 2 T i , ci and σi are [φ1 · · · φn ] . The Gaussian function is φi = exp − xiσ−c i the center and width parameters of the activation function of φi . When we have some prior information of the identiﬁed plant, we can construct the activation function φ1 · · · φn . In this section we assume φ1 · · · φn are given by prior knowledge. The object of RBF neural modeling is to ﬁnd the weights W (k), such that the output y (k) of RBF neural networks (2) can follow the output y (k) of nonlinear plant (1). Let us deﬁne identiﬁcation error as e (k) = y (k) − y (k) . We will use the modeling error e (k) to train the RBF neural networks (2) online such that y (k) can approximate y(k). According to function approximation theories of RBF neural networks [3], the identiﬁed nonlinear process (1) can be represented as y (k) = W ∗ Φ [X (k)] − µ1 (k)

(3)

where W ∗ is unknown weights which can minimize the unmodeled dynamic µ1 (k) . The identiﬁcation error can be represented by (2) and (3) (k) Φ [X (k)] + µ1 (k) e (k) = W

(4)

214

W. Yu and X. Li

(k) = W (k) − W ∗ . In this paper we are only interested in open-loop where W identiﬁcation, we assume that the plant (1) is bounded-input and boundedoutput (BIBO) stable, i.e., y(k) and u(k) in (1) are bounded. By the bound of the activation function Φ, µ1 (k) in (3) is bounded. The following theorem gives a stable gradient descent algorithm for RBF neural modeling. Theorem 1. If we use the RBF neural networks (2) to identify nonlinear plant (1), the following gradient descent algorithm with a time-varying learning rate can make identiﬁcation error e (k) bounded W (k + 1) = W (k) − ηk e (k) ΦT [X (k)]

(5)

η

2 , 0 < η ≤ 1. The normalized identiﬁcation 1 + Φ [X (k)] e(k) satisﬁes the following average performance error eN (k) = 1+max Φ[X(k)] 2 ( ) k

where the scalar ηk =

lim sup T →∞

T 1 2 eN (k) ≤ µ1 T

(6)

k=1

2 where µ1 = max µ1 (k) . k

Proof. We selected a positive deﬁned scalar Lk as

2

Lk = W (k)

(7)

By the updating law (5), we have (k + 1) = W (k) − ηk e (k) ΦT [X (k)] W Using the inequalities a − b ≥ a − b , 2 ab ≤ a2 + b2 for any a and b. By using (4) and 0 ≤ ηk ≤ η ≤ 1, we have

2

2

(k) − ηk e (k) ΦT (X) − W (k)

∆Lk = Lk+1 − Lk = W 2

2

= ηk2 e (k) Φ [X (k)] − 2ηk e (k) [e (k) − µ1 (k)] 2 2 2 ≤ ηk2 e (k) Φ[X (k)] − 2ηk e(k) + 2ηk e (k) µ1 (k)

2 2 2 ≤ −ηk e (k) 1 − ηk ΦT (X) + ηk µ1 (k) Since ηk =

η 1 + Φ [X (k)]

2,

2

ηk 1 − ηk Φ [X (k)] ≥

ηk 1+max(Φ[X(k)]2 ) k

So

(8)

≥

= ηk

1−

1 + Φ [X (k)]

η 2 1+max(Φ[X(k)]2 ) k

η

2

2

Φ [X (k)]

System Identiﬁcation Using Adjustable RBF Neural Network 2

2

∆Lk ≤ −π e (k) + η µ1 (k) where π is deﬁned as π =

η 2 , 1+max(Φ[X(k)]2 )

215

(9)

2

Because n min w i

≤ Lk ≤

2

2 k

2 2 n max w i , where n min w i and n max w i are K∞ -functions, and π e (k) 2 is an K∞ -function, η µ (k) is a K-function. So Lk admits a ISS-Lyapunov function. By [7], the dynamic of the identiﬁcation error is input-to-state stable. From (4) and (7) we know Lk is the function of e (k) and µ1 (k) . The ”INPUT” corresponds to the second term of (9), i.e., the modeling error µ1 (k). The ”STATE” corresponds to the ﬁrst term of (8), i.e., the identiﬁcation error e (k) . Because the ”INPUT” µ1 (k) is bounded and the dynamic is ISS, the ”STATE” e (k) is bounded. (8) can be rewritten as 2 e (k) (10) ∆Lk ≤ −η 2 + η1 µ 2 1 + max Φ [X (k)] k

Summarizing (10) from 1 up to T , and by using LT > 0 and L1 is a constant, we obtain T 2 η eN (k) ≤ L1 − LT + T η1 µ ≤ L1 + T η1 µ k=1

(6) is established.

3

Identiﬁcation Using Adjustable RBF Neural Network

When we regard the plant as a black-box, neither the weight nor the activation function are known. Now the object of the RBF neural modeling is to ﬁnd the weights, as well as the activation functions φ1 · · · φn , such that the RBF neural networks (2) can follow the nonlinear plant (1). Similar as (3), (1) can be represented as (11) y (k) = W ∗ Φ∗ [X (k)] − µ2 (k) 2 x −c∗ T where Φ [X (k)] = [φ∗1 · · · φ∗n ] , φ∗i = exp − iσ∗ i . In the case of three i

independent variables, a smooth function f has Taylor formula as l−1 ∂ ∂ ∂ k

1 0 0 0 x1 − x1 f (x1 , x2 , x3 ) = + x2 − x2 + x3 −x3 f +Rl k! ∂x1 ∂x2 ∂x3 0 k=0

where Rl is the remainder of the Taylor formula. If we let x1 , x2 , x3 correspond wi , ci and σi , x01 , x02 , x03 correspond wi∗ , c∗i and σi∗ ,

+

n i=1

∂(wi φi ) ∂ci

(ci − c∗i ) +

n

(wi − wi∗ ) φi i=1 n ∂(wi φi ) (σi − σi∗ ) ∂σi i=1

y (k) = y (k) + µ2 +

(12) +R

216

W. Yu and X. Li

where R is second order approximation error of the Taylor series. Using the chain rule, we get ∂(wi φi ) ∂ci ∂(wi φi ) ∂σi

= =

∂(wi φi ) ∂φi ∂(wi φi ) ∂φi

∂φi ∂ci ∂φi ∂σi

i = wi 2φi xiσ−c 2 i

i) = wi 2φi (xi −c σ3

2

i

The identiﬁcation error is Φ + 2W D1 Φ1 C + 2W D2 Φ1 Ω + ζk ek = W T ∗ ∗ T = W (k)−W , Φ = [φ1 · · · φn ] , ζk = R + µ2 , C W = [(c1 − c1 ) , · · · (cn − cn )] , ∗

1 n Φ1 = diag [φ1 · · · φn ] , D1 = diag x1σ−c , , · · · xnσ−c 2 2 n 1 2 2 (x1 −c1 ) (xn −cn ) ∗ = [(σ1 − σ ) , · · · (σn − σ ∗ )]T D2 = diag ,Ω ,··· n 1 σ3 σ3 n

1

(13) Theorem 2. If we use the adjustable RBF neural network (2) to identify nonlinear plant (1), the following backpropagation algorithm makes identiﬁcation error e (k) bounded W (k + 1) = W (k) − ek ηk Φ C (k + 1) = C (k) − 2ek ηk W (k) D1 Φ1 Ω (k + 1) = Ω (k) − 2ek ηk W (k) D2 Φ1 T

T

where C = [c1 , · · · cn ] , Ω = [σ1 , · · · σn ] , ηk = 2

2

η 1+Ψk ,

(14) 2

Ψk = Φ +

4 W D1 Φ1 + 4 W D2 Φ1 , 0 < η ≤ 1. The average of the identiﬁcation error satisﬁes J = lim sup T →∞

where π =

η (1+Ψk )2

> 0, ζ = max ζk2

T 1 2 η ek ≤ ζ π T

(15)

k=1

k

Proof. We selected a positive deﬁned scalar Lk as

2

2

2

(k) + C (k) + Ω (k)

Lk = W

(16)

So we have

2

2

2

∆Lk = W (k) − ηk ek Φ − W (k) + C (k) − 2ηk ek W D1 Φ1

2

2

2

− C (k) + Ω (k) − 2ηk ek W D2 Φ1 − Ω (k)

2 (k) Φ + 4η 2 W D1 Φ1 2 e2 = ηk2 Φ e2k − 2ηk ek W k k (k) + 4η 2 W D2 Φ1 2 e2 − 4ηk ek W D2 Φ1 Ω (k) W D1 Φ1 C −4ηk ek k k 2 2 2 2 2 = ηk ek Φ + 4 W D1 Φ1 + 4 W D2 Φ1 Φ + 2W D1 ΦC + 2W D2 ΦΩ −2ηk ek W

The remaining parts are similar to the proof of Theorem 1.

(17)

System Identiﬁcation Using Adjustable RBF Neural Network

4

217

Conclusion

This paper applies input-to-state stability approach to adjustable RBF neural networks and proposes robust learning algorithms which can guarantee the stability of training process. The proposed algorithms are eﬀective. The main contributions are: (1) By using ISS approach, we conclude that the commonlyused robustifying techniques in discrete-time neural modeling, such as projection and dead-zone, are not necessary. (2) New algorithms with time-varying learning rates are proposed, which are robust to any bounded uncertainty.

References 1. Cybenko, G.: Approximation by Superposition of Sigmoidal Activation Function. Math.Control Sig. Syst., 2 (1989) 303-314 2. Egardt, B.: Stability of Adaptive Controllers. Lecture Notes in Control and Information Sciences, Vol.20, Springer-Verlag, Berlin (1979) 3. Haykin,S.: Neural Networks- A Comprehensive Foundation. Macmillan College Publ. Co., New York (1994) 4. Ioannou, P.A., Sun, J.: Robust Adaptive Control. Prentice-Hall, Inc, Upper Saddle River: NJ (1996) 5. Jagannathan, S., Lewis, F.L.: Identiﬁcation of Nonlinear Dynamical Systems Using Multilayered Neural Networks. Automatica, 32 (1996) 1707-1712 6. Jin, L., Gupta, M.M.: Stable Dynamic Backpropagation Learning in Recurrent Neural Networks. IEEE Trans. Neural Networks, 10 (1999) 1321-1334 7. Jiang, Z.P., Wang, Y.: Input-to-State Stability for Discrete-Time Nonlinear Systems. Automatica, 37 (2001) 857-869 8. Kosmatopoulos, E.B., Polycarpou, M.M., Christodoulou, M.A., Ioannou, P.A.: High-Order Neural Network Structures for Identiﬁcation of Dynamical Systems. IEEE Trans. on Neural Networks, 6 (1995) 442-431 9. Mandic, D.P., Hanna, A.I., Razaz, M.A.: Normalized Gradient Descent Algorithm for Nonlinear Adaptive Filters Using a Gradient Adaptive Step Size. IEEE Signal Processing Letters, 8 (2001) 295-297 10. Peng,H., Ozaki,T., Ozaki, V.H., Toyoda, Y.: A Parameter Optimization Method for Radial Basis Function Type Models. IEEE Trans. Neural Networks, 14 (2003) 432-438 11. Polycarpou, M.M., Ioannou, P.A.: Learning and Convergence Analysis of NeuralType Structured Networks. IEEE Trans. Neural Networks, 3 (1992) 39-50 12. Song, Q., Xiao, J., Soh, Y.C.: Robust Backpropagation Training Algorithm for Multilayered Neural Tracking Controller. IEEE Trans. Neural Networks, 10 (1999) 1133-1141 13. Suykens, J.A.K., Vandewalle, J., De Moor, B.: NLq Theory: Checking and Imposing Stability of Recurrent Neural Networks for Nonlinear Modelling. IEEE Transactions on Signal Processing, 45 (1997) 2682-2691 14. Yu, W., Li, X.: Some New Results on System Identiﬁcation with Dynamic Neural Networks. IEEE Trans. Neural Networks, 12 (2001) 412-417 15. Yu, W., Poznyak , A.S., Li, X.: Multilayer Dynamic Neural Networks for Nonlinear System On-line Identiﬁcation. International Journal of Control, 74 (2001) 18581864

2

Departamento de Control Autom´ atico, CINVESTAV-IPN A.P. 14-740, Av.IPN 2508, M´exico D.F., 07360, M´exico [email protected] Secci´ on de Computaci´ on, Departamento de Ingenier´ıa El´ectrica, CINVESTAV-IPN A.P. 14-740, Av.IPN 2508, M´exico D.F., 07360, M´exico

Abstract. In general, RBF neural network cannot match nonlinear systems exactly. Unmodeled dynamic leads parameters drift and even instability problem. According to system identiﬁcation theory, robust modiﬁcation terms must be included in order to guarantee Lyapunov stability. This paper suggests new learning laws for normal and adjustable RBF neural networks based on Input-to-State Stability (ISS) approach. The new learning schemes employ a time-varying learning rate that is determined from input-output data and model structure. The calculation of the learning rate does not need any prior information such as estimation of the modeling error bounds.

1

Introduction

Resent results show that RBF neural network seems to be very eﬀective to identify a broad category of complex nonlinear systems when we do not have complete model information [10]. It is well known that normal identiﬁcation algorithms are stable for ideal plants [4]. In the presence of disturbance or unmodeled dynamics, these adaptive procedures can go to instability easily. The lack of robustness in parameters identiﬁcation was demonstrated in [2] and became a hot issue in 1980s. Several robust modiﬁcation techniques were proposed in [4]. The weight adjusting algorithms of neural networks is a type of parameters identiﬁcation, the normal gradient algorithm is stable when neural network model can match the nonlinear plant exactly [11]. Generally, we have to make some modiﬁcations to the normal gradient algorithm or backpropagation such that the learning process is stable. For example, in [6] some hard restrictions were added in the learning law, in [13] the dynamic backpropagation was modiﬁed with NLq stability constraints. Another generalized method is to use robust modiﬁcation techniques of robust adaptive control [4]. [8] applied σ−modiﬁcation, [5] used modiﬁed δ−rule, and [15] used dead-zone in the weight tuning algorithms. The motivation of this paper is to prove that the normal gradient law and backpropagation-like algorithm without robust modiﬁcations are L∞ stable for identiﬁcation error. F. Yin, J. Wang, and C. Guo (Eds.): ISNN 2004, LNCS 3174, pp. 212–217, 2004. c Springer-Verlag Berlin Heidelberg 2004

System Identiﬁcation Using Adjustable RBF Neural Network

213

Input-to-state stability (ISS) is an elegant approach to analyze stability besides Lyapunov method. It can lead to general conclusions on the stability by using input and state characteristics. We will use input-to-state stability approach to obtain some new learning laws that do not need robust modiﬁcations. A simple simulation gives the eﬀectiveness of the suggested algorithm. To the best of our knowledge, ISS approach for RBF neural network was not still applied in the literature. The adjustable RBF neural network is referred to the activation function of RBF neural network can be updated by a learning algorithm. In this ISS approach is applied to system identiﬁcation via RBF neural network. Two cases are considered: (1) For normal RBF neural network, the activation functions are assumed to be known, and learning is carried on the weights, (2) Learning algorithm concerns both the activation functions and the weights. The new stable algorithms with time-varying learning rates are applied.

2

Identiﬁcation Using Normal RBF Neural Network

Consider following discrete-time nonlinear system in NARMA form y(k) = f [y (k − 1) , y (k − 2) , · · · u (k − 1) , u (k − 2) , · · · ] = f [X (k)]

(1)

T

where X (k) = [y (k − 1) , y (k − 2) , · · · u (k − d) , u (k − d − 1) , · · · ] ∈ n , f (·) is an unknown nonlinear diﬀerence equation representing the plant dynamics, u (k) and y (k) are measurable scalar input and output, d is time delay. A normal RBF neural network can be expressed as y (k) = W (k) Φ [X (k)]

(2)

where the weights W (k) = [w1 · · · wn ] , the activation Φ [X (k)] = function 2 T i , ci and σi are [φ1 · · · φn ] . The Gaussian function is φi = exp − xiσ−c i the center and width parameters of the activation function of φi . When we have some prior information of the identiﬁed plant, we can construct the activation function φ1 · · · φn . In this section we assume φ1 · · · φn are given by prior knowledge. The object of RBF neural modeling is to ﬁnd the weights W (k), such that the output y (k) of RBF neural networks (2) can follow the output y (k) of nonlinear plant (1). Let us deﬁne identiﬁcation error as e (k) = y (k) − y (k) . We will use the modeling error e (k) to train the RBF neural networks (2) online such that y (k) can approximate y(k). According to function approximation theories of RBF neural networks [3], the identiﬁed nonlinear process (1) can be represented as y (k) = W ∗ Φ [X (k)] − µ1 (k)

(3)

where W ∗ is unknown weights which can minimize the unmodeled dynamic µ1 (k) . The identiﬁcation error can be represented by (2) and (3) (k) Φ [X (k)] + µ1 (k) e (k) = W

(4)

214

W. Yu and X. Li

(k) = W (k) − W ∗ . In this paper we are only interested in open-loop where W identiﬁcation, we assume that the plant (1) is bounded-input and boundedoutput (BIBO) stable, i.e., y(k) and u(k) in (1) are bounded. By the bound of the activation function Φ, µ1 (k) in (3) is bounded. The following theorem gives a stable gradient descent algorithm for RBF neural modeling. Theorem 1. If we use the RBF neural networks (2) to identify nonlinear plant (1), the following gradient descent algorithm with a time-varying learning rate can make identiﬁcation error e (k) bounded W (k + 1) = W (k) − ηk e (k) ΦT [X (k)]

(5)

η

2 , 0 < η ≤ 1. The normalized identiﬁcation 1 + Φ [X (k)] e(k) satisﬁes the following average performance error eN (k) = 1+max Φ[X(k)] 2 ( ) k

where the scalar ηk =

lim sup T →∞

T 1 2 eN (k) ≤ µ1 T

(6)

k=1

2 where µ1 = max µ1 (k) . k

Proof. We selected a positive deﬁned scalar Lk as

2

Lk = W (k)

(7)

By the updating law (5), we have (k + 1) = W (k) − ηk e (k) ΦT [X (k)] W Using the inequalities a − b ≥ a − b , 2 ab ≤ a2 + b2 for any a and b. By using (4) and 0 ≤ ηk ≤ η ≤ 1, we have

2

2

(k) − ηk e (k) ΦT (X) − W (k)

∆Lk = Lk+1 − Lk = W 2

2

= ηk2 e (k) Φ [X (k)] − 2ηk e (k) [e (k) − µ1 (k)] 2 2 2 ≤ ηk2 e (k) Φ[X (k)] − 2ηk e(k) + 2ηk e (k) µ1 (k)

2 2 2 ≤ −ηk e (k) 1 − ηk ΦT (X) + ηk µ1 (k) Since ηk =

η 1 + Φ [X (k)]

2,

2

ηk 1 − ηk Φ [X (k)] ≥

ηk 1+max(Φ[X(k)]2 ) k

So

(8)

≥

= ηk

1−

1 + Φ [X (k)]

η 2 1+max(Φ[X(k)]2 ) k

η

2

2

Φ [X (k)]

System Identiﬁcation Using Adjustable RBF Neural Network 2

2

∆Lk ≤ −π e (k) + η µ1 (k) where π is deﬁned as π =

η 2 , 1+max(Φ[X(k)]2 )

215

(9)

2

Because n min w i

≤ Lk ≤

2

2 k

2 2 n max w i , where n min w i and n max w i are K∞ -functions, and π e (k) 2 is an K∞ -function, η µ (k) is a K-function. So Lk admits a ISS-Lyapunov function. By [7], the dynamic of the identiﬁcation error is input-to-state stable. From (4) and (7) we know Lk is the function of e (k) and µ1 (k) . The ”INPUT” corresponds to the second term of (9), i.e., the modeling error µ1 (k). The ”STATE” corresponds to the ﬁrst term of (8), i.e., the identiﬁcation error e (k) . Because the ”INPUT” µ1 (k) is bounded and the dynamic is ISS, the ”STATE” e (k) is bounded. (8) can be rewritten as 2 e (k) (10) ∆Lk ≤ −η 2 + η1 µ 2 1 + max Φ [X (k)] k

Summarizing (10) from 1 up to T , and by using LT > 0 and L1 is a constant, we obtain T 2 η eN (k) ≤ L1 − LT + T η1 µ ≤ L1 + T η1 µ k=1

(6) is established.

3

Identiﬁcation Using Adjustable RBF Neural Network

When we regard the plant as a black-box, neither the weight nor the activation function are known. Now the object of the RBF neural modeling is to ﬁnd the weights, as well as the activation functions φ1 · · · φn , such that the RBF neural networks (2) can follow the nonlinear plant (1). Similar as (3), (1) can be represented as (11) y (k) = W ∗ Φ∗ [X (k)] − µ2 (k) 2 x −c∗ T where Φ [X (k)] = [φ∗1 · · · φ∗n ] , φ∗i = exp − iσ∗ i . In the case of three i

independent variables, a smooth function f has Taylor formula as l−1 ∂ ∂ ∂ k

1 0 0 0 x1 − x1 f (x1 , x2 , x3 ) = + x2 − x2 + x3 −x3 f +Rl k! ∂x1 ∂x2 ∂x3 0 k=0

where Rl is the remainder of the Taylor formula. If we let x1 , x2 , x3 correspond wi , ci and σi , x01 , x02 , x03 correspond wi∗ , c∗i and σi∗ ,

+

n i=1

∂(wi φi ) ∂ci

(ci − c∗i ) +

n

(wi − wi∗ ) φi i=1 n ∂(wi φi ) (σi − σi∗ ) ∂σi i=1

y (k) = y (k) + µ2 +

(12) +R

216

W. Yu and X. Li

where R is second order approximation error of the Taylor series. Using the chain rule, we get ∂(wi φi ) ∂ci ∂(wi φi ) ∂σi

= =

∂(wi φi ) ∂φi ∂(wi φi ) ∂φi

∂φi ∂ci ∂φi ∂σi

i = wi 2φi xiσ−c 2 i

i) = wi 2φi (xi −c σ3

2

i

The identiﬁcation error is Φ + 2W D1 Φ1 C + 2W D2 Φ1 Ω + ζk ek = W T ∗ ∗ T = W (k)−W , Φ = [φ1 · · · φn ] , ζk = R + µ2 , C W = [(c1 − c1 ) , · · · (cn − cn )] , ∗

1 n Φ1 = diag [φ1 · · · φn ] , D1 = diag x1σ−c , , · · · xnσ−c 2 2 n 1 2 2 (x1 −c1 ) (xn −cn ) ∗ = [(σ1 − σ ) , · · · (σn − σ ∗ )]T D2 = diag ,Ω ,··· n 1 σ3 σ3 n

1

(13) Theorem 2. If we use the adjustable RBF neural network (2) to identify nonlinear plant (1), the following backpropagation algorithm makes identiﬁcation error e (k) bounded W (k + 1) = W (k) − ek ηk Φ C (k + 1) = C (k) − 2ek ηk W (k) D1 Φ1 Ω (k + 1) = Ω (k) − 2ek ηk W (k) D2 Φ1 T

T

where C = [c1 , · · · cn ] , Ω = [σ1 , · · · σn ] , ηk = 2

2

η 1+Ψk ,

(14) 2

Ψk = Φ +

4 W D1 Φ1 + 4 W D2 Φ1 , 0 < η ≤ 1. The average of the identiﬁcation error satisﬁes J = lim sup T →∞

where π =

η (1+Ψk )2

> 0, ζ = max ζk2

T 1 2 η ek ≤ ζ π T

(15)

k=1

k

Proof. We selected a positive deﬁned scalar Lk as

2

2

2

(k) + C (k) + Ω (k)

Lk = W

(16)

So we have

2

2

2

∆Lk = W (k) − ηk ek Φ − W (k) + C (k) − 2ηk ek W D1 Φ1

2

2

2

− C (k) + Ω (k) − 2ηk ek W D2 Φ1 − Ω (k)

2 (k) Φ + 4η 2 W D1 Φ1 2 e2 = ηk2 Φ e2k − 2ηk ek W k k (k) + 4η 2 W D2 Φ1 2 e2 − 4ηk ek W D2 Φ1 Ω (k) W D1 Φ1 C −4ηk ek k k 2 2 2 2 2 = ηk ek Φ + 4 W D1 Φ1 + 4 W D2 Φ1 Φ + 2W D1 ΦC + 2W D2 ΦΩ −2ηk ek W

The remaining parts are similar to the proof of Theorem 1.

(17)

System Identiﬁcation Using Adjustable RBF Neural Network

4

217

Conclusion

This paper applies input-to-state stability approach to adjustable RBF neural networks and proposes robust learning algorithms which can guarantee the stability of training process. The proposed algorithms are eﬀective. The main contributions are: (1) By using ISS approach, we conclude that the commonlyused robustifying techniques in discrete-time neural modeling, such as projection and dead-zone, are not necessary. (2) New algorithms with time-varying learning rates are proposed, which are robust to any bounded uncertainty.

References 1. Cybenko, G.: Approximation by Superposition of Sigmoidal Activation Function. Math.Control Sig. Syst., 2 (1989) 303-314 2. Egardt, B.: Stability of Adaptive Controllers. Lecture Notes in Control and Information Sciences, Vol.20, Springer-Verlag, Berlin (1979) 3. Haykin,S.: Neural Networks- A Comprehensive Foundation. Macmillan College Publ. Co., New York (1994) 4. Ioannou, P.A., Sun, J.: Robust Adaptive Control. Prentice-Hall, Inc, Upper Saddle River: NJ (1996) 5. Jagannathan, S., Lewis, F.L.: Identiﬁcation of Nonlinear Dynamical Systems Using Multilayered Neural Networks. Automatica, 32 (1996) 1707-1712 6. Jin, L., Gupta, M.M.: Stable Dynamic Backpropagation Learning in Recurrent Neural Networks. IEEE Trans. Neural Networks, 10 (1999) 1321-1334 7. Jiang, Z.P., Wang, Y.: Input-to-State Stability for Discrete-Time Nonlinear Systems. Automatica, 37 (2001) 857-869 8. Kosmatopoulos, E.B., Polycarpou, M.M., Christodoulou, M.A., Ioannou, P.A.: High-Order Neural Network Structures for Identiﬁcation of Dynamical Systems. IEEE Trans. on Neural Networks, 6 (1995) 442-431 9. Mandic, D.P., Hanna, A.I., Razaz, M.A.: Normalized Gradient Descent Algorithm for Nonlinear Adaptive Filters Using a Gradient Adaptive Step Size. IEEE Signal Processing Letters, 8 (2001) 295-297 10. Peng,H., Ozaki,T., Ozaki, V.H., Toyoda, Y.: A Parameter Optimization Method for Radial Basis Function Type Models. IEEE Trans. Neural Networks, 14 (2003) 432-438 11. Polycarpou, M.M., Ioannou, P.A.: Learning and Convergence Analysis of NeuralType Structured Networks. IEEE Trans. Neural Networks, 3 (1992) 39-50 12. Song, Q., Xiao, J., Soh, Y.C.: Robust Backpropagation Training Algorithm for Multilayered Neural Tracking Controller. IEEE Trans. Neural Networks, 10 (1999) 1133-1141 13. Suykens, J.A.K., Vandewalle, J., De Moor, B.: NLq Theory: Checking and Imposing Stability of Recurrent Neural Networks for Nonlinear Modelling. IEEE Transactions on Signal Processing, 45 (1997) 2682-2691 14. Yu, W., Li, X.: Some New Results on System Identiﬁcation with Dynamic Neural Networks. IEEE Trans. Neural Networks, 12 (2001) 412-417 15. Yu, W., Poznyak , A.S., Li, X.: Multilayer Dynamic Neural Networks for Nonlinear System On-line Identiﬁcation. International Journal of Control, 74 (2001) 18581864