多时滞Hopfield神经网络的鲁棒稳定性分析及吸引域的估计

1、Vol.32,No.1ACTA AUTOMATICA SINICA January,2006Analysis for Robust Stability of Hopeld Neural Networks withMultiple Delays 1ZHANG Hua-Guang JI Ce ZHANG Tie-Yan(Key Laboratory of Process Industry Automation,Ministry of Education,Northeastern University,Shenyang 110004(E-mail:zorks,m ultiple dela ys,pa

2、rameter p erturbations,robust stabilit y ,nequalit yn of Hopeld neural net w orks with symmetric in terconne s and gained abundan t fruits 1,2in classication,parallel in solving some optimization problems 3.Ho w ever,it is im of in terconnecting structure due to the inuences of parame dela y occurs

3、inevitably and ma y bring oscillation durineural net w orks 4.Hence,it is imp ortan t to consider theNo.1ZHANG Hua-Guang et al.:Analysis for Robust Stability of Hopeld Neural Networks (85x i M i (2Considering the inuences of perturbations,then (1can be described asx (t =(C +C x (t +(T 0+T 0S (x (t +

4、Kk =1(T k +T k S (x (t k (3where C =diagc 1,c n n n and T k n n ,k =0,K are time-invariant matrices representing the norm-bounded uncertainties.Assumption 2.We assume that the norm of the perturbations C and T k ,k =0,K are bounded andC T 0T k =HF A B 0B K (4where F is an unknown matrix representing

5、 parametric perturbations which satisesF T F E (5where E is an identical matrix,and A,B 0,B K can be regarded as the known structural matrices of perturbations with appropriate dimensions.Denition.The equilibrium point of system (1is said to be globally robustly stable with respect to the uncertaint

6、ies C and T k ,k =0,K ,if the equilibrium point of system (3is globally asymptotically stable.Distinctly,the origin is an equilibrium point of (1and (3.Thus,in order to study the global robust stability of the zero solution of system (1with respect to parametric uncertainties C and T k ,k =0,K ,it s

7、uces to investigate the globally asymptotic stability of the zero solution ofsystem (3.Now,the interconnecting matrix T =T 0+T 0+Kk =1(T k +T k is nonsymmetric due tothe inuences of uncertainties T k ,k =0,K .Lemma 1.(8Michel,et al.For a functional dierential equation with time delay x(t =f (t,x t ,

8、if there exists a continuous functional V (t,such that there exist non-decreasing continuous functions u,v,w :+,which satisfy u (0=v (0=0,u ( (0 V (t, v (|and V(t, w ( (0 ,then the solution x =0of the functional dierential equation is asymptotically stable.In the above Lemma, denotes the Euclidean v

9、ector norm on n .x t (denotes the restriction of x (to the interval t K ,t translated to K ,0.For s K ,0,we have x t (s =x (t +s ,where t 0.For any C (K ,0,n ,we dene |=max (t :t K ,0.Lemma 29.If U,V and W are real matrices of appropriate dimensions with M satisfying M =M T ,thenM +UV W +W T V T U T

10、 0(6for all V T V E ,if and only if there exists a positive constant such thatM +1UU T +W T W 0(786ACTA AUTOMATICA SINICA Vol.32No.1ZHANG Hua-Guang et al.:Analysis for Robust Stability of Hopeld Neural Networks (8788ACTA AUTOMATICA SINICA Vol.32P HH T P 00000.000+A T A A T B K E MA T B 0E M E M B T

11、K A E M B T K B K EM E M B T K B 0E M .E M B T 0A E M B T 0B K E ME M B T 0B 0E M 0, k = 1, , K, the equilibrium point x = 0 of system (3 is asymptotically stable if K K 2 P0 C + k=0 k + ( M 2 P0 2 k=0 1 ( Tk + Tk 2 2 k (22 M where P0 = diag1/c1 , , 1/cn , M = maxi : 1 i n. Here, denotes the matrix

12、norm T induced by the Euclidean vector norm, i.e., P0 = max (P0 P0 . M The corollary follows from S 0, i = 1, , n, k = 0, , K, such that the following linear matrix inequality (LMI holds K T P P C + C T k +K+ A A 1 P TK E M T K +K E M BK BK E M 0 . . . 0 0 . . . . . . 0 0 . . . . 0 . . . 0 . . . . .

13、 P T0 E M 0 . . . 0 T 0 +0 E M B0 B0 E M 0 . . . . . . 0 PH . . . . . . 0 K+ I 1 0 . . . 0 . . . . . . . . . 0 . . . . . . . . . 0 . . . . 0 0 PH k=0 T E M TK P . . . . . . M T TP E 0 TP H . . . . . . TP H . . . . . . . . . . . . . . . . 0 . . 0 . . . . . 0 0I (23 The proof of the corollary and the

14、simulations are omitted due to space limitation. 4 Conclusion During the implementation process of Hopeld neural networks by electronic circuits, time delays and parameter perturbations are inevitable. This paper studies the robust stability of a class of Hopeld neural network models with multiple d

15、elays and parameter perturbations, and gives the sufcient conditions for the asymptotic stability of equilibrium point x = 0 for arbitrarily bounded delay k , k = 1, , K, which take the form of linear matrix inequality. Since the perturbation norms are bounded in general, we give a useful corollary

16、by means of the denition and properties of matrix norm. In applications, the bound of delays is frequently not very large and is usually known. Therefore, the next research work is to discuss further whether we can obtain the sucient conditions for the robust stability of equilibrium point, which de

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21、438443 8 Michel A N, Wang K, Hu B. Qualitative Theory of Dynamical SystemsThe Role of Stability Preserving Mappings. Second Edition. New York: Marcel Dekker, 2001 9 Xu S, Lam J, Lin Z and Galkowski K. Positive real control for uncertain two dimensional systems. IEEE Transactions on Circuits and Systems-I, 2002, 49(11: 16591666 10 Boyd S, El Ghaoui L, Feron E. Linear matrix inequalities in system and control theory. Studies in Applied Mathematics. Philadelphia: SIAM, 1994 ZHANG Hua-Guang Received his Ph. D. degree from Southeastern University in 1991. He entered Automatic Control Department,