IN the past decades, neural networks have been studied

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1 Proceedings of the International MultiConference of Engineers and Computer Scientists 18 Vol II IMECS 18 March Hong Kong Mixed H-infinity and Passivity Analysis for Neural Networks with Mixed Time-Varying Delays via Feedback Control Sunisa Luemsai Thongchai Botmart Abstract The problem of mixed H and passivity analysis for neural networks with mixed time-varying delays which consist of discrete and distributed time-varying delays via feedback control is investigated. Based on designing feedback controller and constructing a Lyapunov-Krasovskii functional (LKF) comprising novel integral terms the neural networks system is exponentially stable satisfying mixed H and passivity performance. Moreover improved Jensen inequalities and convex combination idea are utilized to derive sufficient conditions in terms of linear matrix inequalities (LMIs). A numerical example is employed to demonstrate the effectiveness of the proposed method. Index Terms neural networks mixed H and passivity analysis exponential stability Lyapunov-Krasovskii functional discrete and distributed time-varying delays I. INTRODUCTION IN the past decades neural networks have been studied by abundant researchers due to their fruitful applications in many areas such as pattern recognition signal processing optimization problems automatic control engineering parallel computation and so on 1] ] 8] ] 7]. Meanwhile it is well known that time delay is a usual phenomenon that occurs in neural networks and existence of time delay which usually causes a source of poor performance oscillation divergence and even instability of the system ]. Moreover time delay is often encountered in a neural networks model due to communication time and the finite switching speed of the amplifiers in hardware implementation 8]. Thus stability analysis for neural networks with constant discrete timevarying and distributed time-varying delays have received a great deal of attention 4] 7] 11] 1] 5]. On the other hand the passivity play important roles of stability of neural networks with time delay and is more attractive to attention. The main key of passivity theory is that the passive properties can keep the system internally stable and represent the property of energy consumption. Particularly it is effective tool related to the circuit analysis Manuscript received December 7 17; revised January The first author was supported by Science Achievement Scholarship of Thailand (SAST) and Department of Mathematics Faculty of Science Khon Kaen University. The second author was supported by the National Research Council of Thailand and Faculty of Science Khon Kaen University 18 and the Thailand Research Fund (TRF) the Office of the Higher Education Commission (OHEC) (grant number : MRG839). Sunisa Luemsai is with the Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen 4 Thailand. luemsai15@gmail.com. Thongchai Botmart is with the Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen 4 Thailand. thongbo@kku.ac.th. Correspondence should be addressed to Thongchai Botmart thongbo@kku.ac.th. and has received much attention from the control areas. In addition it has also been extensively applied in many physical systems such as fuzzy control signal processing network control ] and sliding mode control 3]. Hence the passivity of neural networks with time delay has studied in 1] 19] 4] ]. Also since H control design expresses the control problem as a mathematical optimization problem to finding the controller solution much attention of the H approach becomes theoretical and practical importance 3] 18]. Furthermore the H approach has the merit over classical control techniques in that they are readily applicable to problems involving multivariate systems with cross-coupling between channels 1]. However most recently combined H and passivity are more interested attention in the study and this problem with various systems becomes increasing interest among the researchers and it was first presented in 13] 14]. Particularly the problems of mixed H and passive synchronization for complex dynamical networks with sampled-data control have been studied in 17] 1]. Moreover the problem of memristive neural networks analysis with mixed H and passivity state estimation was investigated in 15]. Motivated by above discussions in this paper we studied the problem of mixed H and passivity analysis for neural networks with discrete and distributed time-varying delays via feedback control which researchers have not studied yet. The purpose is to focus on the exponential stability satisfying mixed H and passivity performance for the neural networks such that a Lyapunov-Krasovskii functional comprising double triple and quadruple integral terms and designing feedback controller are utilized. Moreover improved Jensen inequalities and convex combination idea are employed to derive sufficient conditions in terms of linear matrix inequalities (LMIs). Finally a numerical example is provided to demonstrate the effectiveness of the proposed method. Notation: The following notations will be used in this paper R and R n denote the set of real numbers and the n-dimentionals real spaces respectively. P > or P < denotes that the matrix P is symmetric and positive definite or negative definite matrix. The notation P T and P 1 denote the transpose and the inverse of P respectively. I denotes the identity matrix with appropriate dimensions. The symbol denotes the symmetric block in a symmetric matrix. diag{...} exhibits block diagonal matrix composed of elements in the bracket. e i denotes the unit column vector having one element on its ith row and zeros elsewhere. For x R n the norm of ISBN: ISSN: (Print); ISSN: 78-9 (Online) IMECS 18

2 Proceedings of the International MultiConference of Engineers and Computer Scientists 18 Vol II IMECS 18 March Hong Kong x denoted by x is defined by x = ( n i=1 x i ) 1/ ; x(t + ϵ) cl = max{sup max{h k } ϵ x(t + ϵ) sup max{h k } ϵ ẋ(t + ϵ) }. II. PRELIMINARIES Consider the following neural network model with both discrete time-varying delay and distributed time-varying delay: ẋ(t) = Ax(t) + Bf(x(t)) + Cg(x(t h(t))) +D k1 (t) t k (t) j(x(s)) ds + Eω(t) + u(t) z(t) = C 1 x(t) + C x(t h(t)) (1) k1 (t) +C 3 j(x(s)) ds + C 4 ω(t) t k (t) x(t) = ϕ(t) t ϱ ] where n denotes the number of neurons in the network x(t) = x 1 (t) x (t)... x n (t)] T R n is the neuron state vector z(t) R n is the output vector ω(t) R n is the deterministic disturbance input which belongs to L ) u(t) R n is the control input f(x(t)) g(x(t)) j(x(t)) R n are the neuron activation functions A = diag{a 1 a... a n } is a positive diagonal matrix B C D are interconnection weight matrices E C 1 C C 3 C 4 are given real matrices ϕ(t) is the initial function. The variable h(t) and k i (t) (i = 1 ) represent the discrete and distributed time-varying delays that satisfy h 1 h(t) h k 1 k 1 (t) k (t) k where h 1 h k 1 k ϱ = max{h k } are known real constant scalars. The following assumptions are made for later use. (H1) The activation function f is continuous and there exist constants F i and F + i such that F i f i(α 1 ) f i (α ) α 1 α F + i for all α 1 α and let F i = max{ F i F + i } where f = f 1 f... f n ] T and for any i {1... n} f i () =. (H) The activation function g is continuous and there exist constants G i and G + i such that G i g i(α 1 ) g i (α ) α 1 α G + i for all α 1 α and let G i = max{ G i G+ i } where g = g 1 g... g n ] T and for any i {1... n} g i () =. (H3) The activation function j is continuous and there exist constants J i and J + i such that J i j i(α 1 ) j i (α ) α 1 α J + i for all α 1 α and let Ji = max{ J i J + i } where j = j 1 j... j n ] T and for any i {1... n} j i () =. The state feedback controller takes the following form: u(t) = Kx(t). () By substituting equation () into equation (1) we get ẋ(t) = (K A)x(t) + Bf(x(t)) + Cg(x(t h(t))) +D k1 (t) t k (t) j(x(s)) ds + Eω(t) z(t) = C 1 x(t) + C x(t h(t)) (3) k1 (t) +C 3 j(x(s)) ds + C 4 ω(t) t k (t) x(t) = ϕ(t) t ϱ ]. To derive the main results we will introduce the following definitions and lemmas. Definition 1. 1] The system (3) with ω(t) = is exponentially stable if there exist two constants η > and ϖ > such that x(t) ηe ϖt x(ϵ) cl. (4) Definition. 1] For given scalar σ 1] the system (3) is exponentially stable and satisfies a mixed H /passivity performance index δ if the following two conditions can be guaranteed simultaneously: (1) the system (3) is exponentially stable in view of Definition 1. () under zero original condition there exists a scalar δ > such that the following inequality is satisfied: σz T (t)z(t) + (1 σ)δz T (t)ω(t) ] dt δ ω T (t)ω(t) ] dt (5) for any T p and any non-zero ω(t) L ). Lemma 3. 5] 9] Suppose η 1 < η and x(t) R n for any positive definite matrix P the following inequalities holds: η1 (η η 1 ) (η η1) η3 t η η1 t η η1 η η1 η t+ x T (s)p x(s) ds x T (s) dsp η t+λ P t+ η t+λ η η1 t η x(s) ds x T (s)p x(s) ds d x T (s) ds dp η1 η x T (s)p x(s) ds dλ d x T (s) ds dλ d t+λ x(s) ds dλ d. t+ x(s) ds d Lemma 4. 17] For a positive definite matrix R > any a differentiable function {x(u) u a b]} the following ISBN: ISSN: (Print); ISSN: 78-9 (Online) IMECS 18

3 Proceedings of the International MultiConference of Engineers and Computer Scientists 18 Vol II IMECS 18 March Hong Kong inequality hold: b ẋ T (α)rẋ(α) 1 a b a x(b) x(a)]t R x(b) x(a)] + 3 x(b) + x(a) ] T b x(α) dα b a b a a R x(b) + x(a) ] b x(α) dα. b a a III. MAIN RESULTS In this section the sufficient conditions which ensure the neural networks system (3) to be exponentially stable satisfying mixed H and passivity performance index δ are given. For the convenience of presentation we denote F 1 =diag{f1 F 1 + F F +... F n F n + } { F F =diag 1 + F 1 + F + F +... F n + F n + G 1 =diag{g 1 G+ 1 G G+... G n G + n } { G G =diag 1 + G + 1 G + G+... G n + G + n J 1 =diag{j1 J 1 + J J +... J n J n + } { J J =diag 1 + J 1 + J + J +... J n + J n + } } } ξ T (t) = x T (t) ẋ T (t) x T (t h 1 ) x T (t h ) x T (t h(t)) f T (x(t)) g T (x(t h(t))) j T (x(t)) 1 t h 1 t h 1 x T (s) ds 1 t h t h x T 1 t h1 (s) ds h(t) h 1 t h(t) xt (s) ds 1 t h(t) h h(t) t h h1 h(t) x T (s) ds k 1 (t) t k (t) jt (x(s)) ds t+ xt (s) ds d h(t) h t+ xt (s) ds d ω T (t) Theorem 5. For given scalars h 1 h k 1 k 1 δ and σ 1] if there exist eleven n n matrices P > Q 1 > Q > R 1 > R > U > L > X 1 > X > N > Z and three n n positive diagonal matrices Y 1 > Y > Y 3 > such that the following LMI holds: wherein with: π (ij) = π (ji) ]. Ξ + Ξ 1 < () Ξ + Ξ < (7) Ξ 1 = e 15 X 1 e T 15 Ξ = e 14 X 1 e T 14 Ξ = π (ij) ]1 1 π (11) = Q 1 + Q 4R 1 4R + σc T 1 C 1 F 1 Y 1 J 1 Y Z 1 N T A + (h h 1 ) 4 X 1 (h h 1 ) 4 X π (1) = P 1 N T + Z T N T A π (13) = R 1 π (14) = R π (15) = σc1 T C π (1) = F Y N T B π (17) = 1 N T C π (18) = J Y 3 π (19) = R 1 π (11) = R π (113) = σc1 T C N T D π (114) = h h 1 X π (115) = h h 1 X π (11) = σc1 T C 4 (1 σ)δc1 T + 1 N T E π () = h 1R 1 + h R + (h h 1 ) U N T + (h3 h3 1 ) 3 X π () = N T B π (7) = N T C π (13) = N T D π (1) = N T E π (33) = Q 1 4R 1 4U π (35) = U π (39) = R 1 π (311) = U π (44) = Q 4R 4U π (45) = U π (41) = R π (41) = U π (55) = 8U G 1 Y + σc T C π (57) = G Y π (511) = U π (51) = U π (513) = σc T C 3 π (51) = σc T C 4 (1 σ)δc T π () = Y 1 π (77) = Y π (88) = (k k 1 ) L Y 3 π (99) = 1R 1 π (11) = 1R π (1111) = 1U π (11) = 1U π (1313) = L + σc3 T C 3 π (131) = σc3 T C 4 (1 σ)δc3 T π (1414) = X 1 X π (1415) = X π (1515) = X 1 X π (11) = σc4 T C 4 (1 σ)δc4 T δ I another terms are then the system (3) is exponentially stable and satisfies a mixed H /passivity performance index δ. Furthermore the desired controller gains can be given as: K = N 1 Z. (8) Proof. We consider the following Lyapunov-Krasovskii functional candidate for the system (3) as 9 V (t) = V i (t) (9) i=1 ISBN: ISSN: (Print); ISSN: 78-9 (Online) IMECS 18

4 Proceedings of the International MultiConference of Engineers and Computer Scientists 18 Vol II IMECS 18 March Hong Kong where V 1 (t) = x T (t)p x(t) V (t) = V 3 (t) = t h 1 x T (s)q 1 x(s) ds V 4 (t) = h 1 V 5 (t) = h t h x T (s)q x(s) ds h 1 t+s h V (t) = (h h 1 ) V 7 (t) = (k k 1 ) V 8 (t) = (h h 1) V 9 (t) = (h3 h 3 1) ẋ T (τ)r 1 ẋ(τ) dτ ds ẋ T (τ)r ẋ(τ) dτ ds (1) t+s h1 h t+s k1 k t+s h1 h t+λ h1 h X ẋ(s) ds dφ dλ d. ẋ T (τ)uẋ(τ) dτ ds j T (x(τ))lj(x(τ)) dτ ds λ x T (s)x 1 x(s) ds dλ d t+φ ẋ T (s) Time derivatives of V i (t) i = along the trajectories of (3) are as follow: 1 ẋ(α) dα (14) ẋ(α) dα (15) 1 (t) = x T (t)p ẋ(t) + ẋ T (t)p x(t) (11) (t) = x T (t)q 1 x(t) x T (t h 1 )Q 1 x(t h 1 ) (1) 3 (t) = x T (t)q x(t) x T (t h )Q x(t h ) (13) 4 (t) = h 1ẋ T (t)r 1 ẋ(t) h 1 ẋ T (α)r t h 1 5 (t) = h ẋ T (t)r ẋ(t) h ẋ T (α)r t h (t) = (h h 1 ) ẋ T (t)uẋ(t) h1 (h h 1 ) ẋ T (α)uẋ(α) dα (1) t h 7 (t) = (k k 1 ) j T (x(t))lj(x(t)) (k k 1 ) k1 t k j T (x(α))lj(x(α)) dα (k k 1 ) j T (x(t))lj(x(t)) (k (t) k 1 (t)) k1(t) t k (t) j T (x(α))lj(x(α)) dα (17) 8 (t) = (h h 1) x T (t)x 1 x(t) (h h 1) 4 h1 h t+ x T (s)x 1 x(s) ds d (18) 9 (t) = (h3 h 3 1) ẋ T (t)x ẋ(t) (h3 h 3 1) 3 h1 h t+λ ẋ T (s)x ẋ(s) ds dλ d. (19) Utilizing Lemma 4. the following relations are easily obtained: h 1 t h 1 ẋ T (α)r 1 ẋ(α) dα ξ T (t) e 1 e 3 ] R 1 e 1 e 3 ] T ξ(t) 3ξ T (t) e 1 + e 3 e 9 ] R 1 e 1 + e 3 e 9 ] T ξ(t) () h t h ẋ T (α)r ẋ(α) dα ξ T (t) e 1 e 4 ] R e 1 e 4 ] T ξ(t) 3ξ T (t) e 1 + e 4 e 1 ] R e 1 + e 4 e 1 ] T ξ(t) (1) h1 (h h 1 ) ẋ T (α)uẋ(α) dα t h ξ T (t) e 5 e 4 ] U e 5 e 4 ] T ξ(t) 3ξ T (t) e 5 + e 4 e 1 ] U e 5 + e 4 e 1 ] T ξ(t) ξ T (t) e 3 e 5 ] U e 3 e 5 ] T ξ(t) 3ξ T (t) e 3 + e 5 e 11 ] U e 3 + e 5 e 11 ] T ξ(t). () On the other hand we have the following relation from Lemma 3. (k (t) k 1 (t)) (h h 1) k1 (t) t k (t) j T (x(α))lj(x(α)) dα ξ T (t)e 13 Le T 13ξ(t) (3) h1 h t+ x T (s)x 1 x(s) ds d ξ T (t)e 15 X 1 e T 15ξ(t) εξ T (t)e 15 X 1 e T 15ξ(t) (1 ε)ξ T (t)e 14 X 1 e T 14ξ(t) ξ T (t)e 14 X 1 e T 14ξ(t) (4) where ε = h (t) h 1 h h 1 (h3 h 3 1) ξ T (t) h1 h t+λ ẋ T (s)x ẋ(s) ds dλ d h h ] 1 e 1 e 15 e 14 X h h ] T 1 e 1 e 15 e 14 ξ(t). (5) Define Y 1 = diag{y 1 y... y n } > Y = diag{ỹ 1 ỹ... ỹ n } > and Y 3 = diag{ŷ 1 ŷ... ŷ n } > we get from assumptions (H1) (H) and (H3) respectively that ] T ] ] x(t) F1 Y 1 F Y 1 x(t) () f(x(t)) F Y 1 Y 1 f(x(t)) ] T ] ] x(t h(t)) G1 Y G Y x(t h(t)) g(x(t h(t))) G Y Y g(x(t h(t))) (7) ] T ] ] x(t) J1 Y 3 J Y 3 x(t). (8) j(x(t)) J Y 3 Y 3 j(x(t)) ISBN: ISSN: (Print); ISSN: 78-9 (Online) IMECS 18

5 Proceedings of the International MultiConference of Engineers and Computer Scientists 18 Vol II IMECS 18 March Hong Kong We introduce some auxiliary equality as follows = x T (t) 1 N T + ẋ T (t) N T ] ẋ(t) + (N 1 Z A)x(t) + Bf(x(t)) + Cg(x(t h(t))) + D k1 (t) t k (t) ] j(x(s)) ds + Eω(t). (9) Adding the right-hand sides of (9) to V (t) we can get from Eq. (11)-(8) that V (t)+σz T (t)z(t) (1 σ)δz T (t)ω(t) δ ω T (t)ω(t) ( ξ T (t) εξ (1) + (1 ε)ξ ()) ξ(t) (3) where Ξ (i) = Ξ + Ξ i (i = 1 ) with Ξ and Ξ i are defined in () (7). Since ε 1 the term εξ (1) + (1 ε)ξ () is a convex combination of Ξ (1) and Ξ (). These combinations are negative definite only if the following conditions hold simultaneously: Ξ (1) < (31) Ξ () < (3) therefore (31)(3) is equivalent to () and (7). Thus according to Eq. () (7) we have V (t) + σz T (t)z(t) (1 σ)δz T (t)ω(t) δ ω T (t)ω(t) <. (33) Then under the zero original condition it can be inferred that for any T p σz T (t)z(t) (1 σ)δz T (t)ω(t) δ ω T (t)ω(t) dt V (t) + σz T (t)z(t) (1 σ)δz T (t)ω(t) δ ω T (t)ω(t) dt < which indicates that σz T (t)z(t) (1 σ)δz T (t)ω(t) dt δ ω T (t)ω(t) dt. In this case the condition (5) is assured for any non-zero ω(t) L ). If ω(t) = in view of Eq. (33) we have V (t) < σz T (t)z(t) (34) Applying the same method as in 11] we can find that the system (3) is exponentially stable. Therefore according to Definition. the system (3) is exponentially stable and satisfies a mixed H /passivity performance index δ. This completes the proof. IV. NUMERICAL EXAMPLE In this section a numerical example is presented to illustrate the effectiveness of our results through the maximum allowable delay bound which is defined as the maximum delay value that retains the stability of the system. Example. Consider the system (3) with the following parameters : k 1 = k =. σ =.1 δ = 1 1 =.9 =. ] ] ] A = B = C = ].5.5].3.4 ] D = E = C = ] ].4] C = C.1 3 = C = ] ].3..3 F 1 = F. = ].1 ]..3 G 1 = G.1 = ] ].1 ]..3 1 J 1 = J. = and I =..1 1 LMIs of () (7) in Theorem 5. are solved and the corresponding maximum allowable values of h for different values of h 1. The maximum allowable values of h are shown in TABLE I. TABLE I THE MAXIMUM ALLOWABLE VALUES OF h FOR DIFFERENT VALUES OF h 1 Method h 1 = h 1 =.5 h 1 = 1 h 1 = h 1 = 3 Theorem V. CONCLUSION In this paper a new criterion has been derived for the mixed H and passivity analysis of neural networks with discrete and distributed time-varying delays via feedback control. By designing feedback controller and a Lyapunov- Krasovskii functional comprising double triple and quadruple integral terms have been utilized the sufficient conditions guaranteed exponential stability satisfying mixed H and passivity performance for the neural networks have been obtained. Eventually a numerical example has been presented to demonstrate the effectiveness of the proposed method. ACKNOWLEDGMENT The authors thank anonymous reviewers for their valuable comments and suggestions. REFERENCES 1] L. O. Chua and L. Yang Cellular Neural Networks: Applications IEEE Transaction on Circuits and System vol. 35 no. 1 pp ] M. A. Cohen and S. Grossberg Absolute Stability of Global Pattern Formation and Parallel Memory Storage by Competitive Neural Networks IEEE Transaction on System Man and Cybernetics vol. 13 no. 5 pp ] Y. Du X. Liu and S. Zhong Robust Reliable H Control for Neural Networks with Mixed Time Delays Chaos Solitons and Fractals vol. 91 pp ] A. Farnam R. M. Esfanjani and A. Ahmadi Delay-Dependent Criterion for Exponential Stability Analysis of Neural Networks with Time- Varying Delays IFAC-PapersOnLine vol. 49 no. 1 pp ] A. Farnam and R. M. Esfanjani Improved Linear Matrix Inequality Approach to Stability Analysis of Linear Systems with Interval Time- Varying Delays Journal of Computational and Applied Mathematics vol. 94 pp ] H. Gao T. Chen and T. Chai Passivity and Passification for Networked Control Systems SIAM Journal on Control and Optimization vol. 4 no. 4 pp ISBN: ISSN: (Print); ISSN: 78-9 (Online) IMECS 18

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