Sensing, Computing and Automation

Volume XX(SX)

Supplementary

August

2007

Dynamics of Continuous, Discrete & Impulsive Systems Series A: Mathematical Analysis

Editor-in-Chief Xinzhi Liu, University of Waterloo Special Issue on

Advances in Neural Networks--Theory and Applications

Part 1

DCDIS 14(S1) 1-502 (2007)

Watam Press x Waterloo

ISSN 1201-3390

Preface

During the past several decades, neural networks and their related fields such as evolutionary algorithms, fuzzy logic and expert systems have seen a considerable development both in theories and applications. Recently, the research on the basis of neural networks has been further enhanced especially in the continuous and discrete algorithms and their computational complexity analysis. The combination among those soft computing techniques has led to many hybrid and intelligent systems such as neuro-fuzzy systems and genetic neural systems. Moreover, many experimental products based on neural network techniques have been successfully moved from laboratories to real applications. This special issue aims to report some recent advances in the theory and applications of neural networks. Papers submitted to this special issue can be roughly classified into three groups, i.e., neural network theory, neural network models and algorithms, and neural network applications and hardware. All submitted papers have been refereed strictly by experts in the related fields based on the criteria of originality, significance, quality and clarity. This special issue contains 139 well-selected papers from a large volume of submissions, which cover the major topics of the fields and represent the current trend and research interests. Taking this opportunity, the guest editors of this special issue would like to deliver their sincere thanks to the reviewers for their careful and valuable comments on the submitted manuscripts and for their detailed and helpful suggestions which have improved the quality of the manuscripts. As well, we would like to thank the chief editor of this journal for his encouragement, support and guidance during the preparation of this special issue.

Guest editors: Yi Zhang, Bo Zhang and Shiji Song

i

DCDIS A Supplement, Advances in Neural Networks, Vol. XX(SX) XX--XX

Recurrent High-Order Neural Network Control for Discrete-Time Output Trajectory Tracking 1

Alma Y. Alanis1 , Edgar N. Sanchez2 , Alexander G. Loukianov1 and Guanrong Chen3 CINVESTAV, Unidad Guadalajara, Apartado Postal 31-438, Plaza La Luna, Guadalajara, Jalisco, C.P. 45091, Mexico, e-mail:[email protected] 2 CINVESTAV, Unidad Guadalajara, on sabbatical leave at CUCEI, Universidad de Guadalajara, Mexico. 3 Department of Electronic Engineering, City University of Hong Kong, China The best-known training approach for recurrent neural networks (RNN) is the back propagation through time learning [19]. However, it is merely a first-order gradient descent method and hence its learning speed is very slow. Recently, some Extended Kalman Filter (EKF) based algorithms have been introduced to the training of neural networks [18]. With an EKF-based algorithm, the learning convergence can be improved. Over the past decade, the EKF-based training of neural networks, both feedforward and recurrent ones, has proven to be reliable and practical for many applications [18], [20]. In this paper, we propose a scheme for trajectory tracking based on the block control technique [5], using a new neural observer for a class of MIMO discrete-time nonlinear systems. This observer is based on a discretetime recurrent high-order neural network (RHONN), which estimates the state vectors of the unknown plant dynamics. The learning algorithm for the RHONN is based on an EKF. Once the neural network structure is determined, the block control technique is used to develop the corresponding trajectory tracking controller. This paper also includes the respective stability analysis, from the Lyapunov approach, for the whole system. Finally, the applicability of the proposed design is illustrated by a meaningful example: output chaos synchronization of discrete-time systems.

Abstract – This paper presents the design of an adaptive controller based on the block control technique, and a new neural observer for a class of MIMO discrete-time nonlinear systems. The observer is based on a recurrent high-order neural network (RHONN), which estimates the state vectors of the unknown plant dynamics. The learning algorithm for the RHONN is based on an Extended Kalman Filter (EKF). This paper also includes the respective stability analysis, using the Lyapunov approach, for the whole system, which include the nonlinear plant, the neural observer trained with the EKF and the block controller. Simulation results are included to illustrate the applicability of the proposed scheme. Keywords – Recurrent high-order neural network, Extended Kalman filter, Nonlinear observer, Sliding mode, Discrete-time block control.

I. Introduction Nonlinear trajectory tracking is an important research subject ([1], [6], [8], [10], [12], and references cited therein; mostly for continuous-time systems). In the recent literature on adaptive and robust controls, numerous approaches have been proposed for nonlinear control system trajectory tracking, among which the block control strategy provides a well-suited design methodology [5]. In most nonlinear control designs, it is usually assumed that all the system states are measurable. In practice, however, only parts of these states are measured directly. For this reason, nonlinear state estimation remains an important topic for study in the nonlinear systems theory [13]. Numerous approaches have been proposed for the design of nonlinear observers, yielding many interesting results in dierent directions ([7], [13] and references therein). Most of these approaches in developing nonlinear observers require a nominal mathematical model of the plant dynamics, if not fully then at least partially [13]. Recurrent neuralnetwork observers have also been proposed, and they do not require a precise plant model. This technique is therefore attractive and actually has been successfully applied to state estimation [13], [15]. These works were developed mostly for continuous-time systems. Nonlinear discrete-time neural observers, on the other hand, have been seldom discussed [15].

II. Mathematical Preliminaries Let n denote the sampling step, n 5 0 ^ Z+ , |·| be the absolute value, and k·k be the Euclidian norm for vectors and an adequate norm for matrices. Following [6], consider an MIMO nonlinear system, { (n + 1) = I ({ (n) > x (n)) | (n) = k ({ (n))

(1) (2)

where { 5 · · · > q

(6)

where { bl (l = 1> 2> · · · > q) is the state of the lth neuron, Ol is the respective number of higher-order connections, {L1 > L2 > · · · > LOl } is a collection of non-ordered subsets of {1> 2> · · · > q}, q is the state dimension, zl (l = 1> 2> · · · > q) is the respective on-line adapted weight {(n)> x(n)) is given by vector, and }l (b 6 6 5 5 gl (1) m5L1 # lm m }l1 : : 9 9 .. : (7) }l ({(n)> x(n)) = 7 ... 8 = 9 . 8 7 glm (Ol ) }lOl m5LOl # lm with gml (n) being a nonnegative integers, 6 5 5 # l1 V(b {1 ) .. 9 : 9 .. 9 : 9 . . 9 : 9 9 #l : 9 V(b { q) 1 :=9 #l = 9 9 #l : 9 x1 q+1 : 9 9 9 : 9 .. .. 7 8 7 . . # lq+p

zl (n + 1) = zl (n) +  l Nl (n) h (n) > l = 1> · · · > q Nl (n) = Sl (n) Kl (n) Pl (n) (12) Sl (n + 1) = Sl (n)  Nl (n) Kl> (n) Sl (n) + Tl with £ ¤1 Pl (n) = Ul + Kl> (n) Sl (n) Kl (n) h (n) = | (n)  |b (n)

and 6 : : : : : : : : 8

where h (n) 5 ===> Ol and l = 1> ===> q. Usually, Sl and Tl are initialized as diagonal matrices, with entries Sl (0) and Tl (0), respectively. It is important to remark that Kl (n) > Nl (n) and Sl (n) for the EKF are bounded (for a detailed explanation, see [17]).

(10)

where {l is the lth plant state, }l is a bounded approximation error, which can be reduced by increasing the number of the adjustable weights [16]. Assume that there exists ideal weights vector zl such that k}l k can be minimized on a compact set }l 

x (n)) + g (n) | (n) = F{ (n)

(11)

(16)

where { 5 q | (n) = F{ (n) (17)

with Dl (n) = Sl (n)  Gl (n) + Tl Gl (n) = el (n) }l ({(n)> x(n)) + Nl (n) Kl> (n) Sl (n) and i (n) = z 0}l , (24) can be expressed as elW (n) Sl (n) z el (n) Yl (n) = z z elW (n) [El (n)] z el (n) +2 { eW (n) F W N W [Dl (n)] Nl (n) F { e (n) W W W W +i (n)  l i (n) + { e (n) F Ol  l Ol F { e (n) z elW (n) Sl (n) z el (n)  { eWl (n) l { el (n)

For system (17) > a neural Luenberger observer (RHONO) is used, with the following structure: ¤> £ bl (n) = = = { bq (n) { b1 (n) = = = { { b (n) = { bl (n + 1) = zl> }l (b {(n)> x(n)) + Ol h (n) |b (n) = F { b (n) > l = 1> · · · > q

 ke { (n)k2 kl Nl Fk2 kDl (n)k 2 2

 ke { (n)k l

(18)

¯ ¯2 2  kz el (n)k kEl (n)k + ¯0}l ¯  l ¯ ¯ +2 kz el (n)k k}l ({(n)> x(n))k ¯0}l ¯  l

with Ol 5 < , zl and }l as in (6). The weight vectors are updated on-line with a decoupled EKF (12)  (15) > the output error is defined by s

h (n) = | (n)  |b (n)

+ kz el (n)k2 k}l ({(n)> x(n))k2 l with El (n) = Gl (n)  Tl

(19)

2

{ e (n) = { (n)  { b (n)

(20)

l

where Hl (n) = l k l Nl Fk2l kDl (n)kkOl Fk2 kDl (n)k > 2 Il (n) = kEl (n)k  k}l ({(n)> ¯x(n))k l and Jl (n) = ¯  0 ¯ ¯ kzl  zl max k k}l ({(n)> x(n))k }l l = Then Yl (n)  0 when s¯ ¯ ¯0} ¯2 kDl (n)k + 2Jl (n) l  1 ke { (n)k A Hl (n)

Then the dynamic of (20) can be expressed as el (n) }l ({(n)> x(n)) { el (n + 1) = z +0}l  Ol Fe { (n)

(21)

with 0}l = }l + gl (n) = By the other hand the dynamic of (11) is el (n)  l Nl (n) h (n) z el (n + 1) = z

2

Yl (n)   ke { (n)k Hl (n)  kz el (n)k Il (n) ¯ 0 ¯2 ¯ ¯ + } l + 2Jl (n)

and the state estimation error is

(22)

or

By summarizing (12)-(20), we obtain the first main result as follows. Theorem 1 : For system (17), the nonlinear observer (18) trained with the EKF-based algorithm (12), ensures that the output error (14) and the estimation error (20) are semi-globally uniformly ultimately bounded. Proof: Consider first the Lyapunov function candidate Then for the candidate Lyapunov function

s¯ ¯ ¯0} ¯2 kDl (n)k + 2Jl (n) l  2 kz el (n)k A Il (n)

Therefore the solution of (21) and (22) is stable. Hence, the estimation error and the RHONO weights are SGUUB [9]. Considering (18) and (14) it is easy too see that the output error has an algebraic relation with { e (n) for that reason if { e (n) is bounded h (n) is bounded too.

el (n) Sl (n) z el (n) Yl (n) = z +e {l (n) l { el (n) (23) Yl (n) = Y (n + 1)  Y (n) = z el (n + 1) Sl (n + 1) z el (n + 1) +e {l (n + 1) l { el (n + 1) z el (n) Sl (n) z el (n)  { el (n)  l { el (n)

h (n) = F { e (n) kh (n)k = kFk ke { (n)k

V. Controller Design Consider the nonlinear system defined as

Using (11) and (12) in (23)

{ (n + 1) = i ({ (n)) + E ({ (n)) x (n) + g (n) | (n) = F{ (n) (25)

W

el (n)  l Nl (n) h (n)] [Dl (n)] Yl (n) = [z × [z el (n)   l Nl (n) h (n)] W

2

 ke { (n)k kOl Fk kDl (n)k

{ (n)]  l [i (n)  Ol Fe { (n)] + [i (n)  Ol Fe z el (n) Sl (n) z el (n)  { el (n)  l { el (n)

where { 5 u  1 (26) ¤> £ {1 (n) = = = {l (n) = = = {u (n) > where { (n) = ¤> £ g1 (n) = = = gl (n) = = = gu (n) and g (n) = £ ¤> {l (n) = {1 (n) = = = {l (n) > l = 1> · · · > u  1, and the set of numbers (q1 > · · · > qu ), which define the structure of system (26), satisfy q1  q2  · · ·  qu  p. Define the following transformation:

+{gu (n)  {u (n) Note that when kxht (n)k  x0 > the equivalent control is applied, yielding motion on the sliding manifold V (n) = 0. In the case of kxht (n)k A x0 > the proposed control x (n) strategy is x0 kxht , and the closed-loop system is ht (n)k V (n + 1) =

}1 (n) = {1 (n)  {g (n) }2 (n) = {2 (n)  [E1 ({1 (n))]1 (N1 }1 (n)) 1

+ [E1 ({1 (n))] .. . }u (n) = {u (n)  {gu (n)

(i1 ({1 (n)) + g1 (n))

¡ ¢ ¢ ¡ V (n) + i3 {1 (n)  {2g (n) + {2 (n) ¶ μ x0 × 1 kxht (n)k

Along any solution of the system, the Lyapunov candidate function Y (n) = V > (n) V (n) gives (27)

Y (n) = V > (n + 1) V (n + 1)  V > (n) V (n) ¶¸> · μ x0 = (V (n) + iv (n)) 1  kxht (n)k ¶ μ x0 × (V (n) + iv (n)) 1  kxht (n)k

with |g (n) = {g (n) as the desired trajectory for tracking. Using (27), system (26) can be rewritten as }1 (n + 1) = N1 }1 (n) + E1 }2 (n) .. . }u1 (n + 1) = Nu1 }u1 (n) + Eu1 }u (n) (28) }u (n + 1) = iu ({ (n)) + Eu ({ (n)) x (n) +gu (n)  {gu (n)

V > (n) V (n) Ã  2 kV (n)k Ã

To design the control law, we use the sliding mode block control technique. The surface is derived from the block control procedure, and a natural selection for the sliding surface V (n) = 0 is V (n) = }u (n) = 0. Thus, system (28) is represented as

+

! x0 ° 1 °  kiv (n)k °Eu ° !2

x0 ° 1 °  kiv (n)k °Eu °

where iv (n) = {u (n) ° + °{g (n) + iu ({ (n)) + gu (n)  {u>g (n + 1) > and if °Eu1 ° kiv (n)k  x0 holds, then Y (n)  0= The third main result of this paper is the following. Proposition 1. Given a desired output trajectory |g , a dynamic system with output |, and a neural network with output |b, the following inequality holds [5]:

}1 (n + 1) = N1 }1 (n) + E1 }2 (n) .. . }u1 (n + 1) = Nu1 }u1 (n) + Eu1 V (n) (29) V (n + 1) = iu ({ (n)) + Eu ({ (n)) x (n) +gu (n)  {gu (n)

|  |k + k|g  |bk k|g  |k  kb

Once the sliding surface is selected, the next step is to define x (n), as ( xht (n) for kxht (n)k  x0 x (n) = (30) x (n) x0 kxht for kxht (n)k A x0 ht (n)k

where |g  | is the system output tracking error, and |b  | is the output estimation error and |g  |b is the output tracking error of the nonlinear observer. Based on this proposition, it is possible to divide the tracking objective into two parts [5]: 1. Minimization of |b  |, which can be achieved by the proposed on-line nonlinear observer algorithm (12) as established in Theorem 1. 2. Minimization of |g  |b. For this, a tracking algorithm is developed on the basis of the nonlinear observer (6).

where the equivalent control is calculated from V (n + 1) = 0, as 1

xht (n) = [Eu ({ (n))] × (iu ({ (n)) + {u>g (n + 1)  gu (n))

4

This minimization is obtained by designing the control law (30), as stated in Theorem 2. It is possible to establish Proposition 1 due to the separation principle for discrete-time nonlinear systems [11] as follows. Theorem 3. (Separation Principle) [11] : The asymptotic stabilization problem of the system (1)-(2) is solvable via estimated state feedback, if and only if, the system (1)-(2) is asymptotically stabilizable and exponentially detectable. Corollary 1 [11]: There is an exponential observer for a Lyapunov stable discrete-time nonlinear system (1)(2) with x = 0 if, and only if, the linearized system of the system (1)-(2) is detectable.

1

x 10

4

Tracking

Time for state estimation 0.5

0

-0.5

x1

-1

-1.5

Estimation (- -)

-2

-2.5 Plant(-) Reference (-.-)

-3

-3.5

10

20

30

40

50 Time

60

70

80

90

100

Fig. 1. Time evolution of the state {1 (n) (solid line), its estimate { b1 (n) (dashed line) and its reference {1>g (n) (dash-dot line) 4

Estimation (- -) 2

0

x1

VI. An Example In this section, to illustrate the applicability of the proposed approach, we present the output synchronization of two representative chaotic systems, via simulations. Nonlinear systems have very rich dynamical behaviors including chaos, characterized by high sensitivity to parameter variation, external disturbance, and particularly tiny changes of initial conditions [2]. Chaotic attractors are globally bounded but locally unstable, which presents some important properties, which have technological applications [2]. Two typical discrete chaotic systems are used for simulation here, on their synchronization under the control using the proposed scheme.

-4

-6

-8 1

3

4

5

6

7

8

9

10

Fig. 2. Time evolution zoom for the state {1 (n) (solid line), and its estimate { b1 (n) (dashed line)

In order to estimate the state of (31), the following observer is used: {1 (n)) + z12 (n) { b2 (n) { b1 (n + 1) = z11 (n) V (b { b2 (n + 1) = z21 (n) V 2 (b {1 (n)) +z22 (n) V 2 (b {2 (n)) + z23 (n) x (n) |b (n) = { b1 (n)

The Hénon system is given by [1]

which is in the block canonical form, with the control law (30) and

(31)

xht (n) = [z23 (n)]1 ¡ ¢ × iu ({ (n)) + {g2 (n + 1)

where d and e are real parameters.

where iu ({ (n)) = z21 (n) V 2 ({1 (n)) + z22 (n) V 2 ({2 (n)) {g2 (n) = [z12 (n)]1 [z11 (n) V ({1 (n)) + n1 {1 (n)]

B. The Lozi System The Lozi system is given by [3] {1>g (n + 1) = s |{1>g (n)| + {2>g (n) + 1 {2>g (n + 1) = t{1>g (n) |1>g (n) = {1>g (n)

2

Time

A. The Hénon System

{1 (n + 1) = d{21 (n) + {2 (n) + 1 {2 (n + 1) = e{1 (n) | (n) = {1 (n)

Plant(-)

-2

The results are shown in Figs. 1—6, which verify the applicability of the proposed approach.

(32)

VII. Conclusions In this paper, we have discussed a discrete-time output trajectory tracking by means of a recurrent high-order neural network (RHONN) controller and its application to chaos synchronization. First, a nonlinear observer is designed based on a RHONN trained with an EKF-based algorithm, where the training of the nonlinear observer is performed on-line in a parallel configuration. Then, using the designed control mechanism, output synchronization of two chaotic systems is performed by means of

where s and t are real parameters. C. Simulation Results The Hénon chaotic attractor is generated by (31) with d = 1=4> e = 0=3> "1 (0) = 0=4> "2 (0) = 0=4. The goal is to force the chaotic Hénon attractor to synchronize to the Lozi attractor generated by (32) with s = 1=8> t = 1> "1 (0) = 5000> "2 (0) = 5000=

5

PLANT

4

2

2

Plant(-)

1.5

x 10

1

1 0

0.5 x1(k)

x2

-1

0

-2

-0.5 -3

-1 -4

-1.5 Estimation (- -) -2

2

4

6

8

10 Time

12

14

16

18

-5 -5

20

1

[7] x 1(k) 0

-0.5

[8]

-1

[9] -0.5

0

0.5

1

1.5

x1 (k+1)

[10]

Fig. 4. Original Hénon attractor

[11]

the sliding-mode block control technique. Simulation results illustrate the applicability of the proposed control scheme. Acknowledgement: The authors thank the support of CONACYT Mexico, through Projects 39866Y and 39811Y.

[12]

[13] [14]

References

[2] [3] [4] [5] [6]

G. Chen and J. L. Moiola, “An overview of bifurcation, chaos and nonlinear dynamics in control systems”, The Franklin Institute Journal, Vol. 331B, pp. 819-858, 1994. G. Chen and X. Yu (eds.), Chaos Control: Theory and Applications, Springer-Verlad, Berlin, 2003. G. Chen and X. Dong (eds.), From Chaos to Order: Methodologies, Perspectives and Applications, World Scientific Pub. Co., Singapore, 1998. C. K. Chui and G. Chen, Kalman Filtering with Real-Time Applciations, Springer-Verlag, New York, 1st ed., 1987; 3rd ed., 1999. R. A. Felix, E. N. Sanchez and A.G. Loukianov, “Avoding controller singularities in adaptive recurrent neural control”, Proceedings IFAC’05, Praga, July, 2005. S. S. Ge, J. Zhang and T. H. Lee,“Adaptive neural net-

[15]

[16] [17]

[18]

[19] REFERENCE

4

2

x 10

[20]

1

0

-1 x1d(k)

[1]

-2

-3

-4

-5 -5

-4

-3

-2

-1

0

1

2 4

x1d(k+1)

-1

0

1

2 x 10

Fig. 6. Synchronized attractor

0.5

-1

-2

4

ORIGINAL HÉNON

-1.5 -1.5

-3

x1(k+1)

Fig. 3. Time evolution zoom for the state {2 (n) (solid line), and its estimate { b2 (n) (dashed line) 1.5

-4

x 10

Fig. 5. Reference Lozi attractor

6

work control for a class of MIMO nonlinear systems with disturbances in discrete-time”, IEEE Transactions on Systems, Man and Cybernetics, Part B, vol. 34, No. 4, August, 2004. R. Grover and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering, 2nd ed., John Wiley and Sons, New York, USA, 1992. H. Khalil, Nonlinear Systems, 2nd ed., Prentice Hall, Upper Saddle River, N.J., USA, 1996. Y. H. Kim and F. L. Lewis, High-Level Feedback Control with Neural Networks, World Scientific, Singapore, 1998. M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley and Sons, New York, USA, 1995. W. Lin. and C. I. Byrnes,“Design of Discreh-Time Nonlinear Control Systems Via Smooth Feedback”, IEEE Transactions on Automatic Control, vol. 39, no. 11, pp. 2340-2346, 1994. A. G. Loukianov, B. Castillo-Toledo, and S. Dodds, “Robust stabilization of a class of uncertain system via block decomposition and VSC”, Int. J. Robust Nonlinear Control, vol. 12, pp. 1317—1338, 2002. A. S. Poznyak, E. N. Sanchez and W. Yu, Dierential Neural Networks for Robust Nonlinear Control, World Scientific, Singapore, 2001. E. N. Sanchez, A. Y. Alanis and G. Chen, “Recurrent neural networks trained with Kalman filtering for discrete chaos reconstruction”, Proceedings of Asian-Pacific Workshop on Chaos Control and Synchronization’04, Melbourne, Australia, July, 2004. L. J. Ricalde and E. N. Sanchez, “Inverse optimal nonlinear high order recurrent neural observer”, International Joint Conference on Neural Networks IJCNN 05, Montreal, Canada, August, 2005. G. A. Rovithakis and M. A. Chistodoulou, Adaptive Control with Recurrent High -Order Neural Networks, SpringerVerlag, New York, USA, 2000. Y. Song and J. W. Grizzle, “The extended Kalman Filter as Local Asymptotic Observer for Discrete-Time Nonlinear Systems”, Journal of Mathematical systems, Estimation and Control, vol. 5, no. 1, pp. 59-78, Birkhauser-Boston, 1995. S. Singhal and L. Wu, “Training multilayer perceptrons with the extended Kalman algorithm”, in D. S. Touretzky (ed.), Advances in Neural Information Processing Systems, vol. 1, pp. 133-140, Morgan Kaufmann, San Mateo, CA, USA, 1989. P. J. Werbos, “Backpropagation through time: what it does and how to do it”, Proceedings of the IEEE, vol. 78, no. 10, pp. 1550 - 1560, 1990. R. J. Williams and D. Zipser, “A learning algorithm for continually running fully recurrent neural networks”, Neural Computation, vol. 1, pp. 270—280, 1989.