Input-to-State Stabilization of Dynamic Neural Networks

Transcription

1 532 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 4, JULY 23 [15] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice-Hall, [16] J. J. Hopfield, Neural networks and physical systems with emergent collective behavior, Proc. Nat. Acad. Sci. USA, vol. 79, pp , [17] S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol. Cybern., vol. 27, pp , [18] G. Schöner, M. Dose, and C. Engels, Dynamics of behavior: theory and applications for autonomous robot architectures, Robot. Auton. Syst., vol. 16, no. 2 4, pp , Dec [19] E. Bicho and G. Schöner, The dynamics approach to autonomous robotics demonstrated on a low-level vehicle platform, Robot. Auton. Syst., vol. 21, pp , [2] P. Gaussier and S. Zrehen, A topological map for on-line learning: emergence of obstacle avoidance in a mobile robot, in From Animals to Animats: SAB 94. 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Moga, J. P. Banquet, and M. Quoy, From perceptionaction loops to imitation processes, Appl. Artif. Intell., vol. 1, no. 7, pp , [47] E. Daucé, M. Quoy, and B. Doyon, Resonant spatio-temporal learning in large random neural networks, Biol. Cybern., vol. 87, pp , 22. Input-to-State Stabilization of Dynamic Neural Networks Edgar N. Sanchez and Jose P. Perez Abstract As a continuation of their previous published results, in this paper the authors propose a new methodology, for input-to-state stabilization of a dynamic neural network. This approach is developed on the basis of the recent introduced inverse optimal control technique for nonlinear control. An example illustrates the applicability of the proposed approach. Index Terms Dynamic neural networks, Lyapunov analysis, nonlinear systems, stability. I. INTRODUCTION Neural networks have became an important methodology to various scientific areas for solving difficult problems and for improving system performance. Among the different proposed neural network schemes, the Hopfield-type neural network [4] remains an important architecture due to successful applications in solving associative memory, pattern recognition, identification and control, and optimization problems as well as its easy VLSI implementation. Analysis of dynamic neural network stability has attracted a great deal of attention since the late 198s [9]. When a neural network is employed as an associative memory, the existence of many equilibrium points is a needed feature. However many engineering applications, such as identification and control, involve optimization problems, where it is required to have a well-defined solution for all possible initial conditions. From a mathematical perspective, this means that the neural network should have a unique equilibrium point, which is Manuscript received October 19, 21; revised July 2, 22. This work was supported by CONACYT, Mexico, under Project 3259A and also by the UANL School of Mathematics and Physics. This paper was recommended by Associate Editor H. Takagi. E. N. Sanchez is with CINVESTAV, Unidad Guadalajara, C.P. 4591, Guadalajara, Mexico ( sanchez@gdl.cinvestav.mx). J. P. Perez is with CINVESTAV, Unidad Guadalajara, Guadalajara, Jalisco, C.P. 4591, Mexico, on leave from the School of Mathematics and Physics of Universidad Autonoma de Nuevo Leon (UANL), Monterrey, Mexico. Digital Object Identifier 1.119/TSMCA /3$ IEEE

2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 4, JULY stable and globally attractive. Therefore, analysis of neural network global asymptotic stability has been intensively investigated lately; see, for instance [3], [5], and [8], and references therein. As an extension, input-to-state stability (ISS) analysis, for this kind of neural networks, is for the first time presented in [1]; digressing from previous publications, [1] does not require constant inputs. Even if conditions for global asymptotic stability are already well established, as far as we know, no publication presents results on stabilization. As a continuation of our previous research [1], in this paper, we present the input-to-sate stabilization for Hopfield-type neural networks. We develop this analysis on the basis of the so-called inverse optimal control, recently introduced [7], [14] in the field of nonlinear control. Using this technique, we propose a new methodology in order to achieve the mentioned stabilization. The applicability of the approach is illustrated by one example. II. MATHEMATICAL PRELIMINARIES Before proceeding with the discussion of the proposed methodology, we briefly present some useful concepts for nonlinear systems. A. Lure Type Candidate Lyapunov Functions Consider the nonlinear system _ = f () +g ()u r (1) where, 8 t 2 [; +1), (t) 2 < n is the system state, u r(t) 2< m is the input, f () = (f 1(); f 2() 111f n()) > : < n! < n is a nonlinear vector valued function, and g () = (g 1 (); g 2 () 111g n ()) 2 < n2m, with g i () a nonlinear vector valued function, 8 i =1111n. Taking into account that functions, which satisfy the sector condition, are quite important for the research reported in this paper, we include their definition. Definition 1[6]: A memoryless nonlinearity : [; 1)2< n!< n is said to satisfy a sector condition if [ (t; y) K > miny] [ (t; y) K maxy] ; 8 t 2 [; 1); 8 y 2 < n (2) for some matrices K max 2 < n2n ;K min 2 < n2n, where K = K max K min is a positive definite symmetric matrix and the interior of is connected and contains the origin. If =< n then (:; :) satisfies the sector condition globally, in which case it is said that (:; :) belongs to a sector [K min ;K max ]. If (2) holds with strict inequality then (:; :) is said to belong to a sector (K min ;K max ). For a vector valued function L, which belongs to a sector (; K max ), it is possible to define a positive definite Lure type function [6] as follows V (y) = y ( L(z)) > n dz = y Li(z i ) dz i : Let define a new nonlinear vector valued function f L() = (f L1 (); f L2 () 111f Ln ()) > : < n!< n, such that f Li ( j ): >; j > =; j = <; j < jf Li ( j )j <k j ; j 6= ; i; j =1111n where j1jis the absolute value. Then f L () belongs to the sector (; K), with K = k 2 I n2n, and k = max(k j ). This condition implies (f L()) > (f L() K) <, 8kk 6=. On the basis of the Lure problem [6], [13], for (1), it is possible to define the following Lure type candidate Lyapunov function V = (f L (z)) > n dz = f Li (z i ) dz i )8t 2 [; +1); V : < n!<: (3) Considering that f L (z) belongs to the sector (; K), then V is positive definite and radially unbounded, i.e.: V () > for all 6= and limkk!1 V () =1. B. Inverse Optimal Control This part closely follows [7], [14]. As stated in [14], optimal stabilization guarantees several desirable properties for the closed loop system, including stability margins. In a direct approach, we would have to solve the Hamilton Jacobi Bellman (HJB) equation, which in not an easy task. Besides, the robustness achieved is largely independent of the particular choice of the cost functional. These facts motivated the research community to pursue the development of design methods, which solve the inverse problem of optimal stabilization. For system (1), in the inverse optimal control approach, a stabilizing feedback is designed first and then shown to optimize a cost functional of the form 1 J = l()+u > r R()u r dt: The problem is inverse because the functions l() and R() are a posteriori determined by the stabilizing feedback, rather than a priori chosen by the design. A stabilizing control law u r() solves an inverse optimal problem for system (1), if it can be expressed as u r() = 1 2 k() = 1 2 R1 () L g V () > R() =R > () > (4) where V () is a positive semidefinite function, such that the negative semidefiniteness of V _ is achieved with the control u r (). That is _V = L f V () 1 Lg V ()k() : (5) 2 For (4) and (5), L f V and L g V are the Lie derivatives of V [6] with respect to f and g, defined in (1). When the function l() is selected as l() =L f V ()+ 1 2 L g V ()k() then V (x) is a solution of the HJB equation l()+l f V () 1 4 L g V () R 1 L g V (x) > =: On the basis of the inverse optimal control approach, disturbance attenuation is considered in [7]. Two types of disturbance attenuation are considered. 1) input-to-state stabilization and 2) differential games. They are demonstrated to be equivalent. These results can be seen as the solution of a nonlinear H1 problem without requiring the Hamilton Jacobi Isaacs (HJI) partial differential equation to be solved. In order to solve the disturbance attenuation problem, the following nonlinear system is considered _ = f (; t) +g p(x) d (6) where d 2< r is the disturbance and f (; t)=. In this section, from this point, all concepts are taken from [7]. Definition 2: The system (6) is said to be input-to-sate stable (ISS) if there exist a class KL function [6] and a class K function [6], such that for any x(t ) and for any d continuous on [; 1), the solution exits for all t and satisfies j(t)j (j(t )j; t t )+ sup jd(t)j (7) t t

3 534 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 4, JULY 23 for all t and t such that t t. Consider the system which, in addition to the disturbance d, also has a control input u r _ = f () +g p () d + g ()u r : (8) The time dependence for f, g p, and g are omitted for simplicity. Definition 3: The system (8) is input-to-state stabilizable if there exists a control law u r = (x) continuous everywhere with () = (small control property) such that the closed loop system is ISS with respect to d. Definition 4: A smooth positive defined radially unbounded function [6] V : < n!<is called an ISS-control Lyapunov function (iss-clf) for (8) if there exist a class K 1 [6] function such that the following implication hold for all 6= and all d 2< r jj >(jdj) ) inf L f V + L g V + L g Vu < u2< where L g V stand for the respective Lie derivative. The system (8) is input-to-state stabiliziable if and only if there exists an iss-clf with the small control property. III. NEURAL NETWORK DESCRIPTION We consider a Hopfield-type neural network described by the system of differential equations in the form _x = Ax + Wf(x) +u (9) where, 8 t 2 [; +1), x(t) 2 < n is the neural network state, u(t) 2 < n is the input, A 2 < n2n is a diagonal matrix with elements a i >, W 2 < n2n is the interconnection matrix, and f (x) = (f1(x1); f2(x2) 111f n (x n )) > : < n! < n is a nonlinear vector valued activation function. In this correspondence, similar to [5], we assume that for each function f i (x i ) the following properties hold: (p1) f i: <!(1; +1) is a globally Lipschitz-continuous map [6], with Lipschitz constant k L ;k L 2 (; 1), i.e., jf i (x i ) f i (y i )j k L jx i y i j, 8 x i 2<, 8 y i 2<, with j1jthe respective absolute value. (p2) f i (x i )x i >, 8 x i 6=and f i () =. (p3) f i is monotonically increasing. These properties imply: (i1) f belongs to the sector (; K), K = k L 2 I n2n. (i2) limkxk!1(f(x)) > x = 1. In fact, (p1; p2) ) (i1), and (p2; p3) ) (i2). Considering k L 1, from (p1; i1), it immediately follows (f(x)) > x (f(x)) > f(x): (1) For k L > 1, the change of variable y =(1=k L)x allows to obtain (1), without losing generality. It is worth noting that (p1 p3) are not restrictive and are usually assumed in the literature related to stability analysis of dynamic neural networks. Additionally, in this paper, digressing from other publications, no further restriction is imposed on the input. In [1], conditions for ISS stability of (9) are stated as follows kw k 2 < 3kP k k 2 kp k L kp k < 3 ; P = P > > ; A > P + PA = I where k1kstands for any matrix norm. In this paper, we propose a new approach such that, even if these conditions are not fulfilled, (9) is input-to-state stabilized. Equation (9) can also be expressed as _x = f(x)+ g(x)u f(x) =Ax + Wf(x) g(x) =I n2n : (11) It is clear that x =is an equilibrium point of (11), for u =. Now, let define as follows = min(a i) (12) where min(:) stands for the minimum value. Before to proceed with the analysis, without loss of generality, we consider Assumption 1: For the neural network (9), A is selected such that >1=2. This requirement is not restrictive. If (9) does not fulfill it, as explained in [2], a simple linear state feedback control law can be used to locate the eigenvalues of A in order to satisfy this assumption, without destroying the neural network structure of (9). IV. NEURAL NETWORK STABILIZATION In this section, we establish and prove our main result as the following theorem. Theorem 1: Under Assumption 1, the input defined as u = (W > W + I)f(x) input-to-state stabilizes the dynamic neural network (9). Besides, this stabilizing control law minimizes a meaningful cost functional, as established below. Proof: We first find a Lyapunov function candidate that satisfies all the requirements to be an input-to-state control Lypaunov function (iss-clf). Such a function is essential for the design of an input-to-state stabilizing input. We choose the following candidate x V = (f(z))> dz: (13) Equation (13) is a Lure type candidate Lyapunov function candidate. The time derivative of V, along the trajectories of (9), is given by _V =(f(x)) > (Ax + Wf(x)+u) = (f(x)) > Ax +(f(x)) > Wf(x) +(f(x)) > u = (f(x)) > Ax (f(x)) > Wf(x) +(f(x)) > u: (14) From (14), it is ready to obtain L f V = (f(x)) > Ax (f(x)) > Wf(x) ; L g V =(f(x)) > (15) where L f V and L g V are the Lie derivatives of V [6] with respect to f and g, defined in (11). Equation (14) can also be written as n _V = a i f i (x i )x i (f(x)) > Wf(x) +(f(x)) > u: (16) From (16), and considering (12), we obtain _V (f(x)) > x (f(x)) > Wf(x) +(f(x)) > u: (17) Taking into account the following property [1] for 8 a; b 2< n a > a + b > b 2a > b and defining a = f (x) and b = Wf(x), it follows (f(x)) > f(x)+(f(x)) > W > Wf(x) 2(f(x)) > Wf(x): (18) Substituting (18) in (17), we obtain _V (f(x)) > x (f(x))> f(x) (f(x))> W > Wf(x) +(f(x)) > u: (19)

4 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 4, JULY Fig. 1. N.N. phase plane. Fig. 2. ISS-N.N. phase plane. At this stage, we propose the following input u = (W > W + I)f (x) = R(x) 1 (L gv ) > (2) where is a positive constant and R(x) 1 is a function of x in general, but here it is chosen to be R(x) 1 = 1 W > W + I : The motivation for this choice of the proposed input will be seen from the optimization discussed below. Substituting (2) in (19), we obtain _V f (x) > f(x)+ 1 2 (f(x))> f(x) 1 2 (f(x))> W > Wf(x) (f(x)) > f(x) _V (f(x)) > 1 2 I + W > W f (x): (21) Under Assumption 1, the proposed control law (2) globally asymptotically stabilizes the neural network (9). Therefore, (13) is iss-clf [7]. Note that (9) is input-to-state stabilizable, because the iis-clf fulfills the property of small control [7]. Besides, the inverse optimal assignment problem, defined below, is solvable. For the purpose of assigning the control gain, following [7], we consider (2) and define a cost functional as follows t J(u) = lim t!1 2V (x) + (l(x)+u > R(x)u) d (22) with l(x) 1 = 2L f V + 2 L g V R(x) 1 (L g V ) > : (23) Next, we substitute the definitions of L f, L gv and R(x) 1 in (23) to obtain l(x) =2 (f(x)) > Ax +(f(x)) > Wf(x) + (f(x)) > W > W + I f (x) =2(f(x)) > Ax 2(f(x)) > Wf(x) + (f(x)) > W > Wf(x) +(f(x)) > f(x): (24) Replacing inequality (18) in (24), then l(x) 2(f(x)) > Ax (f(x)) > f(x) (f(x)) > W > W f (x) + (f(x)) > W > Wf(x) +(f(x)) > f(x) l(x) 2(f(x)) > Ax: (25) In (25), it is ready to verify that l(x) > for all x 6= and limkxk!1 l(x) =1. Therefore l(x) is radially unbounded [6]. This is a requirement to apply the inverse optimal control technique [7]. Next, we consider the term u > R(x)u, which can be written as u > R(x)u = L g VR(x) 1 R(x) R(x) 1 (L g V ) > = L g VR(x) 1 (L g V ) > = (f(x)) > W > W + I f (x): (26) Taking into account (24) and (26), the term l(x) +u > R(x)u can be formulated as l(x)+u > R(x)u =2(f(x)) > Ax 2(f(x)) > Wf(x) + (f(x)) > W > W f (x)+(f(x)) > f(x) + (f(x)) > W > W + I f (x) =2(f(x)) > Ax 2(f(x)) > Wf(x) +2(f(x)) > W > W f (x) +2(f(x)) > f(x): (27) Now, we substitute (2) in (14) to obtain _V = (f(x)) > Ax +(f(x)) > Wf(x) +(f(x)) > W > W + I f (x) _V = (f(x)) > Ax +(f(x)) > Wf(x) (f(x)) > W > W f (x) (f(x)) > f(x): (28) Multiplying both sides of (28) by 2, we obtain 2 V _ =2(f(x)) > Ax 2(f(x)) > Wf(x) +2(f(x)) > W > W f (x) +2(f(x)) > f(x): (29) Comparing (29) with (27), we establish l(x)+u > R(x)u = 2 V: _ (3) To this end, substituting (3) in (22), we obtain t J(u) = lim t!1 2V (x(t)) + 2 Vd _ = lim f2v (x(t)) 2V (x(t)) + 2V (x())g t!1 =2V (x()): Thus, the minimum of the cost functional is giving by J(u) = 2V (x()), for the optimal input (2). In summary, the optimal and input-to-state stabilizing input is finally given as If (9) has as input (31) then u = W > W + I f (x): (31) _x = Ax + W W > W I f (x): (32)

5 536 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 4, JULY 23 Comment 1: Equation (31) could be used for global asymptotic stabilization of any given dynamic neural network formulated as (9). This result ensures a global minimum, which is relevant for optimization. Additionally, (32) still preserves the structure of a neural network. Comment 2: The approach, on which the present paper is based, has already been used by the authors and collaborators to develop recurrent neural control schemes, applied to complex nonlinear systems such as chaotic ones [11], [12]. V. EXAMPLE To illustrate the applicability of the proposed methodology, we include the following example. The neural network is given as _x1 _x2 = 2 2 x1 x2 + 1: 2: 3: 4: tanh(x1) tanh(x2) + u1 with u1 = u2 = u. As discussed in [1], this example does not fulfill the conditions to be ISS. Fig. 1 portraits its phase pane for u =. Then, we apply the input given as (31) to this neural network; the resulting phase plane is presented in Fig. 2. As can be seen, the neural network is input-to-state stabilized. VI. CONCLUSIONS We have presented a novel approach for input-to-state stabilization of dynamic neural networks. The proposed input is developed based on the inverse optimal control approach. This result would help for the implementation of robust nonlinear control by dynamic neural networks. Research is being pursued along this line. u2 ACKNOWLEDGMENT The authors thank the anonymous reviewers, which helped to improve this paper. REFERENCES [1] R. G. Bartle, The Elements of Real Analysis. New York: Wiley, [2] C. T. Chen, Linear System Theory and Design. New York: Saunders, [3] M. Forti et al., Necessary and sufficient conditions for absolute stability of neural networks, IEEE Trans. Circuits Syst. I, vol. 41, pp , July [4] J. J. Hopfield, Neurons with graded reponse have collective computational propertites like those of two-state neurons, in Proc. Nat. Acad. Sci., vol. 81, May 1984, pp [5] E. Kaszkurewics and A. Bhaya, On a class of globally stable neural circuits, IEEE Trans. Circuits Syst. I, vol. 41, pp , Feb [6] H. Khalil, Nonlinear System Analysis, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, [7] M. Krstic and H. Deng, Stabilization of Nonlinear Uncertain Systems. New York: Springer-Verlag, [8] K. Matsouka, Stability conditions for nonlinear continuous neural networks with asymmetric connections weights, Neural Networks, vol. 5, pp , [9] A. N. Michel, J. A. Farrell, and W. Porod, Qualitative analysis of neural networks, IEEE Trans. Circuits Syst., vol. 36, pp , Feb [1] E. N. Sanchez and J. P. Perez, Input-to-state stability analysis for dynamic neural networks, IEEE Trans. Circuits Syst. I, vol. 46, pp , Nov [11] E. N. Sanchez, J. P. Perez, and G. Chen, Using dynamic neural networks to generate chaos: An inverse optimal control approach, Int. J. Bifurcation Chaos, vol. 11, pp , 21. [12] E. N. Sanchez, J. P. Perez, L. J. Ricalde, and G. Chen, Chaos synchronization via adaptive recurrent neural control, in Proc. 4th IEEE CDC, Orlando, FL, Dec. 21, pp [13] S. Sastry, Nonlinear Systems. New York: Springer-Verlag, [14] R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. New York: Springer-Verlag, 1997.