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[15] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1996. [16] J. J. Hopfield, “Neural networks and physical systems with emergent collective behavior,” Proc. Nat. Acad. Sci. USA, vol. 79, pp. 2554–2558, 1982. [17] S. Amari, “Dynamics of pattern formation in lateral-inhibition type neural fields,” Biol. Cybern., vol. 27, pp. 77–87, 1977. [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. 213–245, Dec. 1995. [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. 23–35, 1997. [20] 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. Cambridge, MA: MIT Press, 1994, pp. 282–290. , “PERAC: Aneural architecture to control artificial animals,” [21] Robot. Autonom. Syst., vol. 16, no. 2–4, pp. 291–320, 1995. [22] S. Leprêtre, P. Gaussier, and J. P. Cocquerez, “From navigation to active object recognition,” in SAB 2000, Paris, France, Sept. 2000. [23] S. Moga and P. Gaussier, “A neuronal structure of learning by imitation,” in Proc. Eur. Conf. Artificial Life, vol. 1674, F. Mondada, D. Floreano, and J. D. Nicoud, Eds., Lausanne, France, Sept. 1999, pp. 314–318. [24] P. Andry, P. Gaussier, S. Moga, J. P. Banquet, and J. Nadel, “The dynamics of imitation processes: from temporal sequence learning to implicit reward communication,” IEEE Trans. Man, Syst., Cybern. A, vol. 31, pp. 431–442, Sept. 2001. [25] P. Gaussier, C. Joulain, J. P. Banquet, S. Leprêtre, and A. Revel, “The visual homing problem: an example of robotics/biology cross fertilization,” Robot. Auton. Syst., vol. 30, pp. 155–180, 2000. [26] P. Gaussier, S. Leprêtre, C. Joulain, A. Revel, M. Quoy, and J. P. Banquet, “Animal and robot learning: experiments and models about visual navigation,” in Proc. 7th Eur. Workshop Learning Robots, Edinburgh, U.K., 1998, pp. 46–55. [27] P. Gaussier, A. Revel, J. P. Banquet, and V. Babeau, “From view cells and place cells to cognitive map learning: processing stages of the hippocampal system,” Biol. Cybern., vol. 86, pp. 15–28, 2002. [28] E. C. Tolman, “Cognitive maps in rats and men,” Psychol. Rev., vol. 55, no. 4, 1948. [29] C. Thinus-Blanc, Animal Spatial Navigation. Singapore: World Scientific, 1996. [30] A. Revel, P. Gaussier, S. Leprêtre, and J. P. Banquet, “Planification versus sensory-motor conditioning: what are the issues,” in From Animals to Animats: Simulation of Adaptive Behavior, 1998, p. 423. [31] M. A. Arbib and I. Lieblich, “Motivational learning of spatial behavior,” in Syst. Neuro., J. Metzler, Ed. New York: Academic, 1977, pp. 221–239. [32] N. A. Schmajuk and A. D. Thieme, “Purposive behavior and cognitive mapping: a neural network model,” Biol. Cybern., vol. 67, pp. 165–174, 1992. [33] I. A. Bachelder and A. M. Waxman, “Mobile robot visual mapping and localization: A view-based neurocomputational architecture that emulates hippocampal place learning,” Neural Networks, vol. 7, no. 6/7, pp. 1083–1099, 1994. [34] B. Schölkopf and H. A. Mallot, “View-based cognitive mapping and path-finding,” Adapt. Beh., vol. 3, pp. 311–348, 1995. [35] G. Bugmann, J. G. Taylor, and M. J. Denham, “Route finding by neural nets,” in Neural Networks, J. G. Taylor, Ed. Henley-on-Thames, U.K.: Alfred Waller Ltd., 1995, pp. 217–230. [36] O. Trullier, S. I. Wiener, A. Berthoz, and J. A. Meyer, “Biologically based artificial navigation systems: review and prospects,” Progr. Neurobiol., vol. 51, pp. 483–544, 1997. [37] M. O. Franz, B. Schölkopf, H. A. Mallot, and H. H. Büthoff, “Learning view graphs for robot navigation,” Auton. Robots, vol. 5, pp. 111–125, 1998. [38] P. Gaussier, S. Leprêtre, M. Quoy, A. Revel, C. Joulain, and J. P. Banquet, Interdisciplinary Approaches to Robot Learning. Singapore: World Scientific, 2000, vol. 24, pp. 53–94. [39] N. Burgess, M. Recce, and J. O’Keefe, “A model hippocampal function,” Neural Networks, vol. 7, no. 6/7, pp. 1065–1081, 1994. [40] A. Revel, P. Gaussier, and J. P. Banquet, “Taking inspiration from the hippocampus can help solving robotics problems,” in Proc. IEEE Eur. Symp. Artificial Neural Networks, Bruges, Belgium, Apr. 1999.
[41] R. E. Bellman, “On a routing problem,” Quart. Appl. Math., vol. 16, pp. 87–90, 1958. [42] M. Quoy, P. Gaussier, S. Leprêtre, A. Revel, C. Joulain, and J. P. Banquet, “A neural model for the visual navigation and planning of a mobile robot,” in Proc. Eur. Conf. Artificial life, Lausanne, France, Sept. 1999. [43] N. A. Schmajuk and H. T. Blair, “Place learning and the dynamics of spatial navigation: a neural network approach,” Adapt. Beh., vol. 1, pp. 353–385, 1992. [44] J. Droulez and A. Berthoz, “A neural network model of sensory topic maps with predictive short term memory,” Proc. Nat. Acad. Sci., vol. 88, pp. 9653–9657, 1991. [45] J. A. Scott Kelso, Dynamic Patterns: The Self-Organization of Brain and Behavior. Cambridge, MA: MIT Press, 1995. [46] P. Gaussier, S. Moga, J. P. Banquet, and M. Quoy, “From perceptionaction loops to imitation processes,” Appl. Artif. Intell., vol. 1, no. 7, pp. 701–727, 1998. [47] E. Daucé, M. Quoy, and B. Doyon, “Resonant spatio-temporal learning in large random neural networks,” Biol. Cybern., vol. 87, pp. 185–198, 2002.
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 1980s [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, 2001; revised July 2, 2002. This work was supported by CONACYT, Mexico, under Project 32059A 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. 45091, Guadalajara, Mexico (e-mail:
[email protected]). J. P. Perez is with CINVESTAV, Unidad Guadalajara, Guadalajara, Jalisco, C.P. 45091, Mexico, on leave from the School of Mathematics and Physics of Universidad Autonoma de Nuevo Leon (UANL), Monterrey, Mexico. Digital Object Identifier 10.1109/TSMCA.2003.811509
1083-4427/03$17.00 © 2003 IEEE
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 33, NO. 4, JULY 2003
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 [10]; digressing from previous publications, [10] 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 [10], 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 ( )ur
(1)
8 t m 2 [0; +1), (t) 2 = 0:
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 H 1 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) + gp (x) d (6) r where d 2 < is the disturbance and f (0; t) = 0. 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(t0 ) and for any d continuous on [0; 1), the solution exits for all t 0 and satisfies
j (t)j (j (t0)j; t 0 t0 ) + sup jd(t)j 0t
t
(7)
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for all t0 and t such that 0 t0 t. Consider the system which, in addition to the disturbance d, also has a control input ur
_ = f ( ) + gp ( ) d + g ( )ur : (8) The time dependence for f , gp , and g are omitted for simplicity.
Definition 3: The system (8) is input-to-state stabilizable if there exists a control law ur = (x) continuous everywhere with (0) = 0 (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 : (jdj) ) u2< inf Lf V + Lg V + Lg V u < 0
IV. NEURAL NETWORK STABILIZATION
where Lg 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.
In this section, we establish and prove our main result as the following theorem. Theorem 1: Under Assumption 1, the input defined as
u = 0(W > W + I )f (x)
III. NEURAL NETWORK DESCRIPTION We consider a Hopfield-type neural network described by the system of differential equations in the form
(9) x_ = 0Ax + W f (x) + u 8 t n2 [0; +1), x(t) 2 0, W 2 : f (x):
(f (x))
(10)
For kL > 1, the change of variable y = (1=kL )x allows to obtain (10), without losing generality. It is worth noting that (p1 0 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 [10], conditions for ISS stability of (9) are stated as follows
kW k2 < k02 k3PkPk k L kP k < 3 ; P = P > > 0; A> P + P A = 0 I where k 1 k stands for any matrix norm.
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) 0
Equation (13) is a Lure type candidate Lyapunov function candidate. The time derivative of V , along the trajectories of (9), is given by
V_
> 0Ax + W f (x) + u) > > > = 0(f (x)) Ax + (f (x)) Wf (x) + (f (x)) u > > > = 0(f (x)) Ax + 12 2(f (x)) Wf (x) + (f (x)) u: = (f (x)) (
(14)
From (14), it is ready to obtain
Lf V = 0(f (x))> Ax + 12 2(f (x))> Wf (x) ; Lg V = (f (x))> (15) where Lf V and Lg V are the Lie derivatives of V [6] with respect to f0 and g , defined in (11). Equation (14) can also be written as n V_ = 0 ai fi (xi )xi + 12 2(f (x))> Wf (x) + (f (x))> u: (16) i=1 From (16), and considering (12), we obtain
V_
0(f (x))>x + 12
> Wf (x)
2(f (x))
> u:
+ (f (x))
Taking into account the following property [1] for 8 a;
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_ = f0 (x) + g(x)u f0 (x) = 0Ax + W f (x) g(x) = I n2n :
(12)
(17)
b 2 a + b> b 2a> b and defining a = f (x) and b = W f (x), it follows > > > > (f (x)) f (x) + (f (x)) W W f (x) 2(f (x)) Wf (x):
(18) Substituting (18) in (17), we obtain
V_ (11)
0(f (x))>x + 12 (f (x))>f (x)
> > > + 12 (f (x)) W W f (x) + (f (x)) u:
(19)
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Fig. 1.
Fig. 2.
N.N. phase plane.
At this stage, we propose the following input
u = 0(W > W + I )f (x) 01 > (20) = 0 R(x) (Lg V ) 0 1 where is a positive constant and R(x) is a function of x in general, but here it is chosen to be
R(x)01 =
W >W
1
+I
:
V_
0f (x)>f (x) + 12 (f (x))>f (x) 0 12 (f (x))>W > W f (x) 0 (f (x))>f (x) 0(f (x))> 0 12 I + W > W f (x):
lim t!1
2 V (x) +
with
1 02 L V l(x) = f
t
0
> R(x)u) d
(l(x) + u
l(x) = 02
(21)
(22)
+
0 R(x)01(Lg V )>
01 (Lg V )> > W > W + I f (x): = (f (x)) = Lg V R(x)
(26)
l(x) + u> R(x)u > > = 2 (f (x)) Ax 0 2 (f (x)) W f (x)
> W > W f (x) + (f (x))> f (x) > W > W + I f (x) + (f (x)) > > = 2 (f (x)) Ax 0 2 (f (x)) W f (x) > W > W f (x) + 2 (f (x))> f (x): + 2 (f (x))
(27)
Now, we substitute (20) in (14) to obtain
0(f (x))>Ax + (f (x))>W f (x) > 0W > W + I f (x) + (f (x)) V_ = 0(f (x))> Ax + (f (x))> W f (x) 0 (f (x))> W > W f (x) 0 (f (x))>f (x): Multiplying both sides of (28) by 02 , we obtain 02 V_ = 2 (f (x))>Ax 0 2 (f (x))>W f (x) V_
=
> W > W f (x) + 2 (f (x))> f (x):
+ 2 (f (x))
(28)
(29) (30)
To this end, substituting (30) in (22), we obtain
J (u) = (24)
Replacing inequality (18) in (24), then
l(x) 2(f (x))> Ax 0 (f (x))> f (x)
lim t!1
2 V (x(t)) +
f
= lim 2 V (x(t)) t!1 = 2 V (x(0)):
t
0
02 V_ d
0 2 V (x(t)) + 2 V (x(0))g
Thus, the minimum of the cost functional is giving by J (u) = for the optimal input (20). In summary, the optimal and input-to-state stabilizing input is finally given as
2 V (x(0)),
(25)
In (25), it is ready to verify that l(x) > 0 for all x 6= 0 and 1. Therefore l(x) is radially unbounded [6]. This is a requirement to apply the inverse optimal control technique [7].
limkxk!1 l(x) =
R(x)
l(x) + u> R(x)u = 02V_ :
+ (f (x))
0 (f (x))> W > W f (x) > > > + (f (x)) W W f (x) + (f (x)) f (x) > l(x) 2 (f (x)) Ax:
0 Lg V R(x)01
Comparing (29) with (27), we establish
0(f (x))>Ax + (f (x))>W f (x)
> W > W + I f (x) > > = 2 (f (x)) Ax 0 2 (f (x)) W f (x) > > > + (f (x)) W W f (x) + (f (x)) f (x):
u> R(x)u =
+ (f (x))
2 Lg V R(x)01 (Lg V )> : (23) Next, we substitute the definitions of Lf , Lg V and R(x)01 in (23) to obtain
Next, we consider the term u> R(x)u, which can be written as
formulated as
Under Assumption 1, the proposed control law (20) 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 (20) and define a cost functional as follows
J (u) =
ISS-N.N. phase plane.
Taking into account (24) and (26), the term l(x) + u> R(x)u can be
The motivation for this choice of the proposed input will be seen from the optimization discussed below. Substituting (20) in (19), we obtain
V_
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u = 0 W >W
If (9) has as input (31) then
x_ = 0Ax + W
+I
f (x):
0 W >W 0 I
(31)
f (x):
(32)
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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 x _1 x _2
=
0
2
0
x1
0
2
x2
+
1:0
2:0
tanh(x1 )
:
4:0
tanh(x2 )
03 0
+
u1 u2
with u1 = u2 = u. As discussed in [10], this example does not fulfill the conditions to be ISS. Fig. 1 portraits its phase pane for u = 0. 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.
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, 1976. [2] C. T. Chen, Linear System Theory and Design. New York: Saunders, 1984. [3] M. Forti et al., “Necessary and sufficient conditions for absolute stability of neural networks,” IEEE Trans. Circuits Syst. I, vol. 41, pp. 491–494, July 1994. [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. 3088–3092. [5] E. Kaszkurewics and A. Bhaya, “On a class of globally stable neural circuits,” IEEE Trans. Circuits Syst. I, vol. 41, pp. 171–174, Feb. 1994. [6] H. Khalil, Nonlinear System Analysis, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [7] M. Krstic and H. Deng, Stabilization of Nonlinear Uncertain Systems. New York: Springer-Verlag, 1998. [8] K. Matsouka, “Stability conditions for nonlinear continuous neural networks with asymmetric connections weights,” Neural Networks, vol. 5, pp. 495–500, 1992. [9] A. N. Michel, J. A. Farrell, and W. Porod, “Qualitative analysis of neural networks,” IEEE Trans. Circuits Syst., vol. 36, pp. 229–243, Feb. 1989. [10] E. N. Sanchez and J. P. Perez, “Input-to-state stability analysis for dynamic neural networks,” IEEE Trans. Circuits Syst. I, vol. 46, pp. 1395–1398, Nov. 1999. [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. 857–863, 2001. [12] E. N. Sanchez, J. P. Perez, L. J. Ricalde, and G. Chen, “Chaos synchronization via adaptive recurrent neural control,” in Proc. 40th IEEE CDC, Orlando, FL, Dec. 2001, pp. 3536–3539. [13] S. Sastry, Nonlinear Systems. New York: Springer-Verlag, 1999. [14] R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. New York: Springer-Verlag, 1997.