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Non-fragile state observer design for neural networks with Markovian jumping parameters and time-delays
Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Non-fragile state observer design for neural networks with Markovian jumping parameters and time-delays V. Vembarasan a,b , P. Balasubramaniam a,∗ , Chee Seng Chan b a
Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Tamil Nadu, India
b
Centre of Image and Signal Processing, Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia
highlights • • • •
The non-fragile observer design for jumping neural networks with delays is investigated. Delay-dependent stability criteria are obtained in terms of LMIs. The observer gains are given from the LMI feasible solutions. The practical system of quadruple tank process is considered for the example.
article
info
Article history: Received 27 July 2013 Accepted 23 May 2014 Keywords: Linear matrix inequality Lyapunov–Krasovskii functional Markovian jumping parameters Neural networks Non-fragile observer design State estimation
abstract This paper investigates the non-fragile observer based design for neural networks with mixed time-varying delays and Markovian jumping parameters. By developing a reciprocal convex approach and based on the Lyapunov–Krasovskii functional, and stochastic stability theory, a delay-dependent stability criterion is obtained in terms of linear matrix inequalities (LMIs). The observer gains are given from the LMI feasible solutions. Finally, three numerical examples are given to illustrate the effectiveness of the derived theoretical results. Among them the third example deals the practical system of quadruple tank process. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Neural networks (NNs) can mimic the human brain, and they have been used for a wide variety of applications, for example, target tracking, machine learning, system identification and so on [1,2]. The analysis of dynamics for NNs in successful applications is preliminary. Time delays are unavoidably encountered both in biological and artificial neural systems, which may lead to oscillation and instability, hence in recent years increasing attention has been focused on stability analysis of NNs with time delays, see [3,4]. Many interesting results have been obtained, see [5–12], where [5,6] dealt the case of constant delays, authors in [9,10] studied the case of time-varying delays and the paper [11,12] considered the case of continuously distributed delays. Further, it is well known that delay-dependent results are generally less conservative than delay-independent results. Recently, delayed NNs have been focused on the stability analysis and a large amount of results have been available in the literature, see for example [13–17]. Further, most of the research on NNs has been restricted to simple cases of discrete delays. Since an NN usually has a spatial nature due to the presence of an amount of parallel
∗
Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail addresses: [email protected] (V. Vembarasan), [email protected] (P. Balasubramaniam), [email protected] (C.S. Chan).
http://dx.doi.org/10.1016/j.nahs.2014.05.006 1751-570X/© 2014 Elsevier Ltd. All rights reserved.
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V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
pathways of a variety of axon sizes and lengths, it is desired to model them by introducing distributed delays, for details see [16,17] and the references therein. In practice, sometimes a NN has finite state representations and modes may switch (or jump) from one to another at different times [18–21]. Recently, it has been revealed in [22] that, switching (or jumping) between different NNs modes can be governed by a Markovian chain. Specifically, the class of NNs with Markovian jumping parameters (MJPs) has two components in the state vector. The first one which varies continuously is referred to be the continuous state of the NNs and the second one which varies discretely is referred to be the mode of the NNs. For a specific mode, the dynamics of the NNs is continuous, but the parameter jumps among different modes may be seen as discrete events, see for example [23]. A state observer is used usually to reconstruct the states of a dynamic system and has very important applications in many aspects such as realization of feedback control, system supervision, gas-fired furnace system, and fault diagnosis. Naturally the neuron states are not often fully available in the network outputs in many applications, the neuron state estimation problem becomes important to utilize the estimated neuron state. Initially, Wang et al. [24] have investigated the state estimation problem for continuous-time NNs with time-varying delay through available output measurements and derived some sufficient conditions for the existence of the desired estimators. Therefore, delay-dependent state estimation problem has been studied widely for continuous-time NNs, see [25–34]. Recently authors in [32–34] applied sampled data control approach to study the delay-dependent state estimation problems for NNs. Further the state estimation problem for delayed genetic regulatory networks with randomly occurring uncertainties has been dealt in [35]. Very recently authors in [36–43] have studied state estimation for recurrent NNs with MJPs and mixed time delays. Sampled data approach for state estimation for NNs with MJPs has been considered in [42,43]. Meantime, the delay-dependent state estimation problem was widely studied for continuous-time delayed NNs in [29–31,41] by employing delay decomposition methods, free weighting matrix method, triple-integral approaches and the sufficient conditions are derived using linear matrix inequalities (LMIs). However, most results of the state estimation problems have concerned with the uncertainties appearing in system matrices only. Up to now, the design of state estimator to cope with the non-fragile observers have not been thoroughly studied yet. This brings up a ‘‘fragility’’ problem that has been of great interest in the signal processing area for some time now, but has been considered only recently by the control community [44]. The problem is that of determining the level of computer accuracy required to insure closed-loop specifications. Although it is generally known that many feedback systems require very accurate controller implementation, there have been few studies on the level of accuracy required to met given specifications, and on the design of controllers that require the least possible amount of accuracy, that is, non-fragile controllers. However up to date only limited works have been done with respect to the design of non-fragile controllers [45–49] for the system of differential equations with time-delays. Non-fragile controllers minimize the cost of implementation, and allow for on-line tuning of control parameters that are essential in practical applications. Note that very little research efforts have been made to design the state estimator coping with the time-varying delays and non-fragile controllers in the observer gains. The lack of the existing works are probably due to the fragility problem of controllers. Therefore, the aim of this paper is to shorten such a gap by designing the state estimator for NNs with MJPs and non-fragile observers and mixed time-varying delays. On the other hand, the reduction of the conservatism in delay-dependent criteria means that the feasible region is enlarged, which can increase the application region such as filtering, controller design, synchronization, and other important issues in control society. Therefore, the study of increasing the feasible region in delay-dependent stability criteria is an important and essential work in applying various control methods in dynamic systems with time-delays. In [50], authors have discussed reciprocally convex approach to derive stability results. In this paper, a similar type of stability criterion is applied to derive the sufficient conditions for the problem of state estimation for NNs with MJPs and non-fragile observers. To the best of authors’ knowledge, the non-fragile observer based state estimation of NNs with mixed time-varying delays and MJPs using reciprocal convex method has not been investigated yet. Suitable Lyapunov–Krasovskii functionals (LKF), reciprocal convex method and some analysis techniques are employed to derive sufficient conditions for delay-dependent stability criteria for the considered NNs with MJPs in terms of LMIs, which can be easily calculated by MATLAB LMI Control Toolbox. Numerical examples are given to illustrate the effectiveness of the proposed method. Motivated by the above discussions, the aim of this paper is to study the non-fragile observer based state estimation problem for a class of NNs with MJPs, and mixed time-varying delays. By constructing a suitable LKF including the available information of time-varying delays, sufficient conditions are established such that the resulting estimation error dynamics is asymptotically stable in the mean-square sense. The main contributions of this paper lie in the following two aspects: (i) the phenomena of non-fragile control is considered at the first time for the state estimation problem of NNs with MJPs; and (ii) the reciprocally convex combination method has been involved in that account for the improvement of the feasible region. Finally, numerical examples are provided to illustrate the developed state estimation scheme. Notations. Throughout this paper, Rn and Rn×m denote respectively the n-dimensional Euclidean space and the set of all n × m real matrices. The superscript T denotes the transposition and the notation X ≥ Y (respectively, X > Y ), where X and Y are symmetric matrices, meaning that X − Y is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimensions. | · | is the Euclidean norm in Rn . Moreover, let (Ω , F , P ) be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions. That is the filtration contains all P -null sets and is right continuous. n Denote by LP F0 [−δ, 0] the family of all F0 -measurable C ([−δ, 0]; R )-valued random variables ς = {ς (θ ) : −δ ≤ θ ≤ 0}
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
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such that sup−δ≤θ≤0 E ∥ς (θ )∥P < ∞, where E stands for the mathematical expectation operator with respect to the given probability measure P . The notation ∗ always denotes the symmetric block in one symmetric matrix.
2. Problem description and preliminaries Let {r (t ), t ≥ 0} be a right-continuous Markov process on the probability space which takes values in the finite space
S = {1, 2, . . . , N } with generator Γ = (γij ), (i, j ∈ S ) given by
P (r (t + △) = j|r (t ) = i) = o(∆)
where ∆ > 0, lim1t →0 (
γii = −
N
j=1,j̸=i
∆
γij ∆ + o(∆), 1 + γii ∆ + o(∆),
i ̸= j, i = j,
(1)
) = 0 and γij is transition rate from mode i to mode j satisfying γij ≥ 0 for i ̸= j with
γij , i, j ∈ S .
Consider the following NNs of mixed time-varying delays with MJPs and the network measurements described by x˙ (t ) = −A(r (t ))x(t ) + B(r (t ))g (x(t )) + B1 (r (t ))g (x(t − τ (t ))) + E (r (t ))
t
t −σ (t )
g (x(s))ds + J (r (t )),
y(t ) = C (r (t ))x(t ) + D(r (t ))f (t , x(t )),
(2)
where x(·) = [x1 (·), x2 (·), . . . , xn (·)]T ∈ Rn is neuron state vector, x(t ) = ϖ (t ), t ≤ 0; ϖ (t ) is the initial condition. A = diag{a1 , . . . , an } is a diagonal matrix with ai > 0, i = 1, . . . , n, the matrices B, B1 and E represent the connection weight matrix, the discretely delayed connection weight matrix, and distributively delayed connection weight matrix respectively. The matrices C ∈ Rm×n and D ∈ Rm×m are the constant matrices. g (x(·)) = [g1 (x1 (·)), . . . , gn (xn (·))]T ∈ Rn denotes the neuron activation function, J = [J1 , . . . , Jn ]T ∈ Rn is a constant input vector and f : R × Rn → Rm is the neuron-dependent nonlinear disturbances on the network outputs which are assumed to satisfy the underlying Assumption (H). τ (t ) and σ (t ) denote the discrete time-varying delays and the distributed time-varying delays, respectively, and satisfy 0 < τm ≤ τ (t ) ≤ τM , τ˙ (t ) ≤ µ and 0 ≤ σ (t ) ≤ σM , where τm , τM , µ and σM are constants. For the NNs (2), we construct the full-order state estimator as follows: x˙ˆ (t ) = −A(r (t ))ˆx(t ) + B(r (t ))g (ˆx(t )) + B1 (r (t ))g (ˆx(t − τ (t )))
+ E (r (t ))
t
t −σ (t )
g (ˆx(s))ds + J (r (t )) + K (r (t ))(y(t ) − yˆ (t )),
yˆ (t ) = C (r (t ))ˆx(t ) + D(r (t ))f (t , xˆ (t )),
(3)
where xˆ (t ) is the estimation of the neuron state, and K (r (t )) ∈ R is the estimator gain matrix to be designed. Notice that the Markov process {r (t ), t ≥ 0} takes values in the finite space S = {1, 2, . . . , N }. For notation simplicity, we denote A(i) := Ai , B(i) := Bi , B1 (i) := B1i , E (i) := Ei , J (i) := Ji , K (i) := Ki , C (i) := Ci , D(i) := Di . Non-fragile observer-based control for the practical implementation is considered in the following form: n×m
x˙ˆ (t ) = −Ai xˆ (t ) + Bi g (ˆx(t )) + B1i g (ˆx(t − τ (t ))) + Ei
t t −σ (t )
g (ˆx(s))ds + Ji + (Ki + 1Ki )(y(t ) − yˆ (t )),
yˆ (t ) = Ci xˆ (t ) + Di f (t , xˆ (t )),
(4)
where the real-valued matrices 1Ki (t ) represent possible controller gain fluctuations. It is assumed that 1Ki (t ) have the structure Hi Fi (t )GKi , in which Hi and GKi are known real constant matrices of appropriate dimensions with an unknown time-varying matrix satisfying FiT (t )Fi (t ) ≤ I. Define the error e(t ) := x(t ) − xˆ (t ), and φ(t ) := g (x(t )) − g (ˆx(t )), ψ(t ) := f (t , x(t )) − f (t , xˆ (t )). Thus, the error-state system is given by
e˙ (t ) = − Ai + (Ki + 1Ki )Ci e(t ) + Bi φ(t ) + B1i φ(t − τ (t )) + Ei
t t −σ (t )
φ(s)ds − (Ki + 1Ki )Di ψ(t ).
(5)
Obviously, e(s) is bounded and continuously differential on [−δ, 0], where δ = max{τM , σM }. Traditionally, the activation functions are assumed to be continuous, differentiable, monotonically increasing, and bounded, such as the sigmoid-type of function. However, in many electronic circuits, the input–output functions of amplifiers may be neither monotonically increasing nor continuously differentiable, hence, non monotonic functions can be more appropriate to describe the neuron activation in designing and implementing artificial NNs. In this paper, we make the following assumptions for the neuron activation functions.
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Assumption (H) ([51]). The neuron activation function g (·) and the neuron-dependent nonlinear disturbances f (·) in (2) are bounded and there exist four constant matrices φ − = diag{φ1− , φ2− , . . . , φn− }, φ + = diag{φ1+ , φ2+ , . . . , φn+ }, ψ − = diag{ψ1− , ψ2− , . . . , ψn− }, and ψ + = diag{ψ1+ , ψ2+ , . . . , ψn+ }, such that
φk−′ ≤
gk′ (α) − gk′ (β) 1
ψk−′ ≤
1
α−β
1
≤ φk+′ ,
(6)
1
fk′ (α, x(α)) − fk′ (β, x(β)) 2
2
x(α) − x(β)
2
≤ ψk+′ ,
(7)
2
for all α, β ∈ R, α ̸= β, k′1 = 1, 2, . . . , n and k′2 = 1, 2, . . . , m. By recalling definitions of φ(t ) and ψ(t ), from Assumption (H), we obtain the following inequality readily:
[φk′1 (t ) − φk−′ ek′1 (t )]T [φk′1 (t ) − φk+′ ek′1 (t )] ≤ 0,
k′1 = 1, 2, . . . , n,
1
1
[ψk′2 (t ) − ψk−′ ek′2 (t )]T [ψk′2 (t ) − ψk+′ ek′2 (t )] ≤ 0, 2
2
k′2 = 1, 2, . . . , m.
Let e(t ; ς ) be the trajectory of system (5) under the initial condition e(θ ) = ς (θ ) on −δ ≤ θ ≤ 0 in L2F0 ([−δ, 0], Rn ). It is obvious that the system (5) admits a trivial solution e(t ; 0) = 0. Before ending this section, a stability definition and a useful lemma are recalled. Definition 2.1 ([36]). For the error-state system (5) and every ς ∈ L2F0 ([−δ, 0], Rn ), the trivial solution is asymptotically stable in the mean square if, for every network mode lim E |e(t ; ς )|2 = 0.
t −→∞
Lemma 2.2 ([52] Schur Complement Lemma). Given constant matrices Ω1 , Ω2 and Ω3 withappropriate dimensions, where
Ω1T = Ω1 and Ω2T = Ω2 > 0, then Ω1 + Ω3T Ω2−1 Ω3 < 0, if and only if
Ω1 ∗
Ω3T − Ω2
< 0, or
−Ω2 ∗
Ω3 Ω1
< 0.
Lemma 2.3 ([53]). Let M = M T , H and G are real matrices of appropriate dimensions F T (t )F (t ) ≤ I, then M + HF (t )G + GT F T (t )H T < 0, if and only if there exists a positive scalar ϵ > 0 such that M + 1ϵ HH T + ϵ GT G < 0. 3. Main results In this section, we shall establish the main criterion based on the LMI approach. For presentation convenience, we introduce the following notations:
φn− + φn+ φ1− + φ1+ φ2− + φ2+ , Σ2 = diag , ,...,
Σ1 = diag{φ1 φ1 , φ2 φ2 , . . . , φn φn , }, − +
− +
− +
2
2
2
Σ3 = diag{ψ1 ψ1 , ψ2 ψ2 , . . . , ψn ψn , }, −
+
−
+
−
+
ψ1− + ψ1+ ψ2− + ψ2+ ψn− + ψn+ Σ4 = diag , ,..., . 2
2
2
Theorem 3.1. For given positive scalars τm , τM , σM and µ, the equilibrium solution of NNs with MJPs (5) is globally robustly asymptotically stable in the mean square if there existsymmetric positive definite matrices Pi , Q1 , Q2 , Q3 , R1 , R2 , S1 , S2 , further
for any matrices T with compatible dimensions, and scalars ϵ1i , ϵ2i , ϵ3i such that the following LMI holds:
Πi ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ξi Π9,9,i ∗ ∗ ∗ ∗ ∗ ∗
R2
T
TT
R2
≥ 0, diagonal matrices U > 0, V > 0, W > 0, and positive
Υ1
ϵ1i Υ2
Υ3
ϵ2i Υ4
Υ5
ϵ3i Υ6
0
0 0
0 0 0
0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
−ϵ1i I ∗ ∗ ∗ ∗ ∗
−ϵ1i I ∗ ∗ ∗ ∗
−ϵ2i I ∗ ∗ ∗
−ϵ2i I ∗ ∗
−ϵ3i I ∗
−ϵ3i I
< 0,
(8)
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
65
where
Π1,1,i = −Pi Ai − ATi Pi − Li Ci − CiT LTi +
N
γij Pj + Q1 + Q2 + Q3 − R1 − Σ1 U − Σ3 W ,
i =1
Π1,3,i = R1 ,
Π1,5,i = Pi Bi + Σ2 U ,
Π2,2,i = −(1 − µ)Q3 − R2 − Π2,6,i = Σ2 V ,
RT2
Π1,6,i = Pi B1i ,
+ T + T − Σ1 V , T
Π3,3,i = −Q1 − R1 − R2 ,
Π5,5,i = S1 + σ
2 M S2
− U,
Π1,7,i = Pi Ei ,
Π1,8,i = −Li Di + Σ4 W ,
Π2,3,i = R2 − T ,
Π2,4,i = R2 − T ,
T
Π3,4,i = T ,
Π4,4,i = −Q2 − R2 ,
T
Π6,6,i = −(1 − µ)S1 − V ,
Π7,7,i = −S2 , Π8,8,i = −W , Π9,9,i = −2Pi + N , T T T T T T Ξi = −ATi Pi − CiT LTi 0 · · · 0 B P B P E P − D L , i i i i 1i i i i 3
Υ3 = Υ1 ,
Υ2 = [GTKi CiT 0 · · 0 GTKi DTi ], ·
Υ1 = [−Hi 0 · · 0], · 7
Υ4 = [0 · · 0 GTKi CiT ], ·
6
Υ5 = [0 · · 0 − Hi ], ·
Υ6 = [0 · · 0 ·
7
GTKi DTi
],
7
N =τ
2 m R1
+ (τM − τm ) R2 . 2
7
Moreover, the state estimator gain matrix is given by Ki = Pi−1 Li . Proof. Choose the LKF for error-state system (5) as V (et , t , i) = V1 (et , t , i) + V2 (et , t , i) + V3 (et , t , i) + V4 (et , t , i),
(9)
where V1 (et , t , i) = eT (t )Pi e(t ), V2 (et , t , i) =
t
e (s)Q1 e(s)ds + T
t −τm
V3 (et , t , i) = τm
V4 (et , t , i) =
e (s)Q2 e(s)ds + T
t −τM 0
t
−τm
t
e˙ T (s)R1 e˙ (s)dsdθ + (τM − τm )
−τM
t t −τ (t )
t −τ (t )
−τm
t +θ
t
φ T (s)S1 φ(s)ds + σM
0
−σM
eT (s)Q3 x(s)ds, t
e˙ T (s)R2 e˙ (s)dsdθ , t +θ
t
φ T (s)S2 φ(s)dsdθ . t +θ
Let L be the weak infinitesimal generator of the random process {et , t ≥ 0}. Then, for each {r (t ) = i, i ∈ S }, it can be shown that
LV1 (et , t , i) = 2eT (t )Pi − Ai + (Ki + 1Ki )Ci e(t ) + Bi φ(t ) + B1i φ(t − τ (t ))
+ Ei
t t −σ (t )
N φ(s)ds − (Ki + 1Ki )Di ψ(t ) + γij eT (t )Pj e(t ),
(10)
j =1
LV2 (et , t , i) ≤ eT (t )(Q1 + Q2 + Q3 )e(t ) − eT (t − τm )Q1 e(t − τm ) + eT (t − τM )Q2 e(t − τM )
− (1 − µ)eT (t − τ (t ))Q3 e(t − τ (t )), t LV3 (et , t , i) = τm2 e˙ T (t )R1 e˙ (t ) − τm e˙ T (s)R1 e˙ (s)ds t −τm + (τM − τm )2 e˙ T (t )R2 e˙ (t ) − (τM − τm )
(11)
t −τm
e˙ T (s)R2 e˙ (s)ds,
(12)
t −τM
LV4 (et , t , i) ≤ φ T (t )S1 φ(t ) − (1 − µ)φ T (t − τ (t ))S1 φ(t − τ (t ))
+ σ φ (t )S2 φ(t ) − 2 M
T
t
φ (s) S2
T
t −σ (t )
t −σ (t )
φ(s)ds .
(13)
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V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
Now by applying Jensen’s Inequality Lemma [53] and Theorem 1 in [50], one can get t
e˙ T (s)R1 e˙ (s)ds ≤ −[e(t ) − e(t − τm )]T R1 [e(t ) − e(t − τm )],
−τm
(14)
t −τm
−(τM − τm )
t −τm t −τ (t )
e˙ T (s)R2 e˙ (s)ds − (τM − τm )
t −τ (t )
e˙ T (s)R2 e˙ (s)ds
t −τM
(τM − τm ) [e(t − τm ) − e(t − τ (t ))]T R2 [e(t − τm ) − e(t − τ (t ))] τ (t ) − τm (τM − τm ) − [e(t − τ (t )) − e(t − τM )]T R2 [e(t − τ (t )) − e(t − τM )] τM − τ (t ) T R2 T e(t − τm ) − e(t − τ (t )) e(t − τm ) − e(t − τ (t )) ≤− . T e(t − τ (t )) − e(t − τM ) e(t − τ (t )) − e(t − τM ) T R2 ≤−
(15)
Further for positive diagonal matrices U > 0, V > 0 and W > 0, we can get from Assumption (H) that e(t ) φ(t )
T
Σ1 U −Σ 2 U
e(t − τ (t )) φ(t − τ (t ))
−Σ 2 U U
T
Σ1 V −Σ 2 V
T
e(t ) ψ(t )
e(t ) ≤ 0, φ(t )
Σ3 W −Σ 4 W
−Σ 2 V V
−Σ 4 W
(16)
e(t − τ (t )) ≤ 0, φ(t − τ (t ))
(17)
e(t ) ≤ 0. ψ(t )
W
(18)
Now adding (10)–(18), we get
LV (et , t , i) ≤ ξ T (t ) Ωi + ΘiT N Θi + 1Πi (t ) ξ (t ),
(19)
where
ξ (t ) = e (t ) e (t − τ (t )) e (t − τm ) e (t − τM ) φ (t ) φ (t − τ (t )) T
T
T
T
(Ωi )9×9 = (Ωj,k,i ) = (Πj,k,i ) = (Πi )9×9 ,
T
T
T
t
t −σ (t )
T
φ(s)ds
Ψ (t ) , T
j, k varies between 1 and 9; except that
Ω1,1,i = −Pi Ai − ATi Pi − Pi Ki Ci − CiT KiT Pi +
N
γij Pj + Q1 + Q2 + Q3 − R1 − Σ1 U − Σ3 W ,
i=1
Ω1,8,i = −Pi Ki Di + Σ4 W , 1Πi (t ) = Υ1 F (t )Υ2 + Υ2T F T (t )Υ1T + Υ3 F (t )Υ4 + Υ4T F T (t )Υ3T + Υ5 F (t )Υ6 + Υ6T F T (t )Υ5T , T T T T T T B B E − D K . Θi = −ATi − CiT KiT 0 · · · 0 i 1i i i i 3
Also by Lemma 2.3, we can have
1Πi (t ) ≤ ϵ1i−1 Υ1 Υ1T + ϵ1i Υ2 Υ2T + ϵ2i−1 Υ3 Υ3T + ϵ2i Υ4 Υ4T + ϵ3i−1 Υ5 Υ5T + ϵ3i Υ6 Υ6T .
(20)
Substitute the inequality (20) in (19), one can get the following inequality:
Ωi + Θi N ΘiT + ϵ1i−1 Υ1 Υ1T + ϵ1i Υ2 Υ2T + ϵ2i−1 Υ3 Υ3T + ϵ2i Υ4 Υ4T + ϵ3i−1 Υ5 Υ5T + ϵ3i Υ6 Υ6T < 0.
(21)
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
67
In order to get the expected LMI condition of the error-state system (5), we will need to make some standard manipulations on the relation (21). By using the well-known Schur Complement Lemma (Lemma 2.2), the inequality (21) is equivalent to the following matrix inequality:
Ωi ∗ ∗ ∗ ∗ ∗ ∗ ∗
Θi N −N ∗ ∗ ∗ ∗ ∗ ∗
Υ1
ϵ1i Υ2
Υ3
ϵ2i Υ4
Υ5
ϵ3i Υ6
0
0 0
0 0 0
0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
−ϵ1i I ∗ ∗ ∗ ∗ ∗
−ϵ1i I ∗ ∗ ∗ ∗
−ϵ2i I ∗ ∗ ∗
−ϵ2i I ∗ ∗
−ϵ3i I ∗
< 0.
(22)
−ϵ3i I
By noting Ki = Pi−1 Li , pre- and post- multiplying (22), by diag{I , . . . , I , Pi N −1 , I , . . . , I } and diag{I , . . . , I , N −1 Pi , I , . . . , I },
8
6
8
6
respectively, and from view of the inequality −Pi N −1 Pi ≥ 2Pi − N resulting from (Pi − N )T N −1 (Pi − N ) ≥ 0, it is easy to see that the LMI condition (8) can guarantee that (22) holds. Hence, (21) is implied by the LMI condition (8). In other words, the LMI condition (8) and by Definition 2.1, implies that the error-state system (5) is globally robustly asymptotically stable in the mean square, and this completes the proof. Remark 3.2. The delay-dependent state estimation problem has been studied widely for continuous-time NNs, see [25–34]. Recently authors in [32–34] have applied sampled data control approach to study the delay-dependent state estimation problems for NNs. Further the sampled data approach for state estimation for NNs with MJPs has been considered in [42,43]. However up to now, there are no results employing non-fragile observer based state estimation for NNs with MJPs, which motivated our present study. Consider the following observer based non-fragile control error-state system NNs (5) without MJPs
e˙ (t ) = − A + (K + 1K )C e(t ) + Bφ(t ) + B1 φ(t − τ (t )) + E
t
t −σ (t )
φ(s)ds − (K + 1K )Dψ(t ).
(23)
Corollary 3.3. For given positive scalars τm , τM , σM and µ, the equilibrium solution of NNs without MJPs (23) is globally robustly asymptotically stable in the mean square if there exist symmetric positive definite matrices P , Q1 , Q2 , Q3 , R1 , R2 , S1 , S2 , further for any matrices T with compatible dimensions, and
scalars ϵ1 , ϵ2 , ϵ3 such that the following LMI holds
Π ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ξ 9,9 Π ∗ ∗ ∗ ∗ ∗ ∗
R2
TT
1 Υ
2 ϵ1 Υ
3 Υ
4 ϵ2 Υ
5 Υ
0
0 0
0 0 0
0 0 0 0
0 0 0 0 0
−ϵ1 I ∗ ∗ ∗ ∗ ∗
−ϵ1 I ∗ ∗ ∗ ∗
−ϵ2 I ∗ ∗ ∗
−ϵ2 I ∗ ∗
−ϵ3 I ∗
T
≥ 0, diagonal matrices U > 0, V > 0, W > 0, and positive
R2
6 ϵ3 Υ 0 0 0 < 0, 0 0 0
−ϵ3 I
where
1,1 = −PA − AT P − LC − C T LT + Q1 + Q2 + Q3 − R1 − Σ1 U − Σ3 W , Π 1,3 = R1 , 1,5 = PB + Σ2 U , 1,6 = PB1 , 1,7 = PE , 1,8 = −LD + Σ4 W , Π Π Π Π Π T T T 2,2 = −(1 − µ)Q3 − R2 − R2 + T + T − Σ1 V , 2,3 = R2 − T , 2,4 = R2 − T , Π Π Π 2,6 = Σ2 V , 3,3 = −Q1 − R1 − R2 , 3,4 = T T , Π Π Π 2 5,5 = S1 + σM S2 − U , 6,6 = −(1 − µ)S1 − V , Π Π 7,7 = −S2 , 8,8 = −W , 9,9 = −2P + N , Π Π Π T = −AT P − C T LT 0 · · · 0 BT P BT1 P E T P − DT LT , Ξ 3
1 = [−H 0 · · · 0], Υ 7
2 = [GTK C T 0 · · · 0 GTK DT ], Υ 6
4,4 = −Q2 − R2 , Π
(24)
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V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
3 = Υ 1 , Υ
4 = [0 · · · 0 GTK C T ], Υ
5 = [0 · · · 0 − H ], Υ
7
N =τ
6 = [0 · · · 0 GTK DT ], Υ
7
2 m R1
+ (τM − τm ) R2 . 2
7
Further, the state estimator gain matrix is given by K = P −1 L, and other terms are defined as same in Theorem 3.1.
Remark 3.4. Very recently authors in [36–43] have studied state estimation problem for recurrent NNs with MJPs and mixed time delays. Meantime, the delay-dependent state estimation problem was widely studied for continuous-time delayed NNs with MJPs by employing delay decomposition methods, free weighting matrix method, triple-integral approaches and the sufficient conditions are derived using linear matrix inequalities (LMIs). However, the result derived in this paper by using reciprocal convex approach [50] is new which is not discussed in any of the previous literature [36–43]. Moreover in the absence of the MJPs, the results derived in this paper are still new, the corresponding results are given in Corollary 3.3. Remark 3.5. The traditional assumption on the derivatives of the time-varying delays are less than 1 is no longer required in this analysis. Therefore theorems and corollaries in this paper are applicable to more general than previously investigated papers. In many cases, the information of the derivative of time delays is unknown because it is a very difficult task to obtain the precise values (even their boundedness or the boundedness of their derivatives) of time delay systems. Regarding this circumstance, rate-independent criteria for delay τ (t ) satisfying the conditions 0 < τm ≤ τ (t ) ≤ τM are derived by choosing Q3 and S1 to be zero in theorems and corollaries of this paper and then the corresponding results can be developed. Consider the following non-fragile control error-state system NNs (5) without MJPs and distributed time delays:
e˙ (t ) = − A + (K + 1K )C e(t ) + Bφ(t ) + B1 φ(t − τ (t )).
(25)
Corollary 3.6. For given positive scalars τm , τM , σM and µ, the equilibrium solution of NNs without MJPs (25) is globally robustly asymptotically stable in the mean square if there exist symmetric positive definite matrices P , Q1 , Q2 , Q3 , R1 , R2 , S1 , further for any matrices T with compatible dimensions, and
R2
TT
T
R2
≥ 0, diagonal matrices U > 0, V > 0, and positive scalars ϵ1 , ϵ2 such
that the following LMI holds:
ˆ Π ∗ ∗ ∗ ∗ ∗
Ξˆ ˆ Π7,7 ∗ ∗ ∗ ∗
Υˆ 1
ϵ1 Υˆ 2
Υˆ 3
ϵ2 Υˆ 4
0
0 0
0 0 0
0 0 0 0
−ϵ1 I ∗ ∗ ∗
−ϵ1 I ∗ ∗
−ϵ2 I ∗
< 0,
(26)
−ϵ2 I
where
ˆ 1,1 = −PA − AT P − LC − C T LT + Q1 + Q2 + Q3 − R1 − Σ1 U , Π ˆ 1 ,3 = R 1 , ˆ 1,5 = PB + Σ2 U , ˆ 1,6 = PB1 , Π Π Π T ˆ 2,2 = −(1 − µ)Q3 − R2 − R2 + T + T T − Σ1 V , Π ˆ 2 ,3 = R 2 − T T , ˆ 2,4 = R2 − T , Π Π ˆ 2 , 6 = Σ2 V , ˆ 3,3 = −Q1 − R1 − R2 , ˆ 3,4 = T T , ˆ 4,4 = −Q2 − R2 , Π Π Π Π ˆ 5 ,5 = S 1 − U , ˆ 6,6 = −(1 − µ)S1 − V , ˆ 7,7 = −2P + N , Π Π Π T ˆ = −AT P − C T LT 0 · · · 0 BT P BT1 P , Ξ 3
Υˆ 1 = [−H 0 · · 0], ·
Υˆ 2 = [GTK C T 0 · · 0], ·
5
Υˆ 3 = Υˆ 1 ,
5
Υˆ 4 = [0 · · 0 GTK C T ], ·
N = τm2 R1 + (τM − τm )2 R2 .
5
Further, the state estimator gain matrix is given by K = P −1 L, and other terms are defined as same in Theorem 3.1.
Remark 3.7. The study of randomly occurring incomplete information has become the active research topic. Very recently Lakshmanan et al. [35] studied the state estimation problem for delayed genetic regulatory networks with randomly
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
69
occurring uncertainties. Motivated by this in future, we can extend the present results to more general, that is, non-fragile robust state estimation problem for NNs with MJPs and randomly occurring uncertainties. This result will be investigated in the near future. Remark 3.8. When an NN is designed for a practical application, it is often required to be stable. The advantages of the newly-established stability criteria are that (i) less conservative conditions are formulated by means of LMIs, which can be facilitated readily by standard algorithms. According to [52], the LMI-based conditions can be solved by polynomial time algorithms with complexity proportional to MN 3 , where M is the total row size of the LMIs and N is the total number of scalar decision variables. Therefore, for NNs with hundreds or thousands of neurons, the LMIs-based conditions can be efficiently checked in practice; and (ii) the upper bound of the delay can be easily found such that the designed NNs can be clearly described. Therefore, it is expected that the results can provide a foundation to the design of this kind of NNs, and play an essential role to the potential applications in the areas such as adaptive control, combinatorial optimization, and signal processing. The engineering applications of the results obtained in this study will be one of our future research topics. 4. Numerical examples Example 4.1. Consider a three-neuron two-mode NNs with MJPs (2) with non-fragile control parameters as follows: 2 0 0
A1 =
0 3 0
0.3 1.1 −0.2
4 0 0
0 0 , 2
A2 =
0 3 0
1.3 0.7 −0.1
E2 =
0.5 −0.6 0.2
1.2 0.5 −0.4
E1 =
0 0 , 4
0.2 1.7 −0.3
−0.4 0.3 0.8
B1 =
−0.4 1.5 , 1.0
C1 =
0.2 −0.5 0.6
B2 =
−0.5 1.2 , 1.5
1 0 0
0.2 0.6 , −0.3
B11 =
0 0 0
2 cos(t ) + 0.03t 2 J1 = J2 = 2 sin(t ) − 0.03t 2 , 2 cos(t ) + 0.03t 2
0 0 , 1
0.8 0.2 0.4
−0.3 0.4 , −0.4
C2 =
1 0 0
0 1 0
0 0 , 0
0.1 −0.5 0.1
0.4 0.1 −0.7
0.2 0.4 , 0.3
B12 =
0.1 −0.4 0.2
0.8 0.1 −0.5
D1 = D2 = I ,
0.6 0.3 , 0.4
Γ =
−3 2
3 . −2
The activation function satisfies Assumption (H) with φ − = −0.5I , φ + = 0.4I, and the nonlinear disturbance also satisfies Assumption (H) with ψ − = 0.2I , ψ + = 0.4I. Thus, we can get the parameters as Σ1 = −0.2I , Σ2 = −0.05I , Σ3 = 0.08I , Σ4 = 0.3I. The activation function g (x(t )) =
|x(t ) + 1| − |x(t ) − 1| and nonlinear disturbance is of the form
1 5
t
f (t , x(t )) = 0.1 cos(x(t )) + 0.3. The discrete time-varying delay τ (t ) = ete+1 , and the distributed time-varying delay ρ(t ) = 0.5 sin(t ). Noticing that τ (t ) is a monotone increasing function with respect to t, we can easily obtain that τm = 0.5 et and τM = 1. In addition, ρM = 0.5. Moreover, it can be calculated that τ˙ (t ) = (et + ≤ 0.25, which implies µ = 0.25. 1)2 Using the MATLAB LMI solvers for the LMIs in Theorem 3.1, we obtain the corresponding non-fragile observer gain matrices as follows: −1
K1 = P 1
−0.2948 L1 = 0.0216 −0.1397
0.0650 −0.9827 −0.0686
0.0269 −0.0645 , −1.6613
−1
K2 = P 2
−1.7075 L2 = −0.0503 −0.0408
0.0433 −0.8756 0.0844
−0.1461 0.0487 . −0.2702
From the structures of 1Ki (t ), we have H1 = H2 = 0.3I , GK 1 = GK 2 = 0.1I and F (t ) = sin(t )I. The initial conditions of system (2) and the estimator (4) are taken as x(t ) = [4; 3; −2]T and xˆ (t ) = [2; −2; 1]T respectively for t ∈ [−1, 0]. The response of the state and estimation for the NNs are given in Fig. 1. In Fig. 2, NNs with time delays (5) are given which converge to zero asymptotically. Simulation of system modes of Example 4.1 is displayed in Fig. 3. Therefore, it follows from Theorem 3.1, that the error-state system NNs with MJPs (5) is globally asymptotically stable in the mean square. Remark 4.1. It is important to note that very limited works have been done on observer based non-fragile controllers for system of differential equations with time-delays [45–49]. But there are no related works on observer based non-fragile controller design for NNs. In order to fill such a gap, in this paper we aimed to design the non-fragile observer based design for NNs with MJPs and mixed time-varying delays. Hence we could not provide any comparison results. Example 4.2. Consider a three-neuron single-mode NNs without MJPs (2) with non-fragile control parameters as follows: 3 0 0
A=
0 3 0
0 0 , 3
0.4 −0.2 0.7
B=
−0.6 0.4 0.1
0.2 0.6 , −0.9
B1 =
0.5 −0.5 0.5
0.5 0.3 −0.1
0.4 0.4 , 1.2
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V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
a
b
c
Fig. 1. The response of the state, estimation trajectories in Example 4.1.
Fig. 2. The response of the error trajectories in Example 4.1.
Fig. 3. The simulation of system mode r (t ) = {1, 2} in Example 4.1.
1.9 0.2 −0.5
E=
0.8 0.5 −0.7
−0.5 0.9 , 1.8
1 0 0
C =
0 0 0
0 0 , 0
2 cos(t ) + 0.03t 2 J = 2 sin(t ) − 0.03t 2 . 2 cos(t ) + 0.03t 2
D = I,
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
71
The activation functions and the time-varying delays are as same in Example 4.1. From the structure of 1K (t ), we have H = GK = 0.5I and F (t ) = sin(t )I. Using the MATLAB LMI solvers for the LMIs in Corollary 3.3, we obtain the corresponding non-fragile observer gain matrices as follows:
−0.8232 L = 0.1951 −0.2809
−0.1316 −2.5119 −0.3244
K =P
−1
0.1340 −0.1302 . −2.9733
Therefore, it follows from Corollary 3.3, that the error-state system NNs without MJPs (23) are globally asymptotically stable. Example 4.3. Originally, NNs embody the characteristics of real biological neurons that are connected or functionally related in a nervous system. On the other hand, NNs can represent not only biological neurons but also other practical systems. One of them is the quadruple-tank process, which consists of four interconnected water tanks and two pumps. The inputs are the voltages to the two pumps and the outputs are the water levels of Tanks 1 and 2. This quadruple-tank process can be expressed clearly using the NNs model. Recently Lee et al., [34] proposed the state-space equation of the quadruple-tank process which can be rewritten to the form of system (25) and the parameters are given as follows: 0.0021 0 A= 0 0
0
0 B1 = −0.0143 −0.0119 H = 0.5I ,
−0.0424
0 0.0021 0 0
0 0.0424 0
0 0 −0.0114 −0.0131 0.5 0
GK =
0 0 −0.0165 −0.0059 0 0.5
0 0
−0.0060 −0.0063 B= 0
0 −0.0424 , 0 0.0424
T
0 0
−0.0065 −0.0051
−0.0029 −0.0073
0 0
0 0
0
,
0
0 0 , −0.0056 −0.0154
−0.0077 −0.0025 , 0
0
1 C = 0
0 1
0 0
0 , 0
0 J = , 0 0
F (t ) = sin(t )I .
The activation function satisfies Assumption (H) with φ − = −0.05I , φ + = 0.05I, and thus, we can get the parameters as
Σ1 = −0.0025I , Σ2 = 0. The activation function g (x(t )) = 0.01 |x(t ) + 1| − |x(t ) − 1| . The discrete time-varying delay
τ (t ) = 1 + 0.98sin( π2 t ) and µ = 0.99. Using the MATLAB LMI solvers to solve the LMIs in Corollary 3.6, we obtain the corresponding non-fragile observer gain matrices as follows: 0.7277 0.0002 −1 K =P L= 0.0936 0.0002
0.0002 0.7277 . 0.0002 0.0936
Originally, the water levels of the two lower tanks are the only accessible and usable information in the quadruple-tank process, whereas the water levels of the two upper tanks are not accessible. Although the water in the upper tanks overflows, no action can be taken. Therefore, there is a strong need to know the water level of the upper tanks, which is one of methods for obtaining the information to design the state estimator. Until now, the research of the quadruple-tank process focused on designing the stabilizing controller. Therefore, in the sense of practical applications, Lee et al., [34] have applied the sampled data control in their estimation method. However due to the fragility problem of the controllers, in this paper we have considered the non-fragile observer based design method to estimate the upper tank levels of the quadruple-tank process. The initial conditions of system (2) and the estimator (4) are taken as x(t ) = [−4; 4; −6; 5]T and xˆ (t ) = [10; −10; −10; 10]T respectively. The response of the state and estimation for the NNs are given in Fig. 4. In Fig. 5, NNs with time delays (25) are given which converge to zero asymptotically. Therefore, it follows from Corollary 3.6, that the error-state system NNs with MJPs (25) are globally asymptotically stable in the mean square. 5. Conclusion In this paper, we have dealt with the problem of non-fragile observer design for NNs with MJPs and time delays, which is new in the literature. By developing a reciprocal convex approach and defining appropriate LKFs, new LMI-based stability conditions are derived such that the estimation error system is stochastically globally asymptotically stable in the mean square. Numerical examples are demonstrated to show the merit of the derived results. By utilizing the proposed idea of this paper, future works will focus on stabilization and synchronization for various dynamic systems with time-delays such as NNs with MJPs [40], genetic regulatory networks [35], and multi-agent systems [21] with randomly occurring uncertainties.
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V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
a
b
c
d
Fig. 4. The response of the state, estimation trajectories in Example 4.3.
Fig. 5. The response of the error trajectories in Example 4.3.
Acknowledgments The authors would like to express their sincere gratitude to the Editor-in-Chief, Associate Editor, and Anonymous Reviewers for their valuable comments and suggestions to improve the quality of the manuscript. This research work is supported by DST INSPIRE Fellowship Grant DST/INSPIRE Fellowship/2011/278 dated 21.12.2011 and 03.10.2012 from the Department of Science and Technology, Ministry of Science and Technology, Government of India; Also this research work is supported by the High Impact Research MoE Grant UM.C/625/1/HIR/MoHE/FCSIT/08, H-22001-00-B0008 from the Ministry of Higher Education Malaysia. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Non-fragile state observer design for neural networks with Markovian jumping parameters and time-delays V. Vembarasan a,b , P. Balasubramaniam a,∗ , Chee Seng Chan b a
Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram - 624 302, Tamil Nadu, India
b
Centre of Image and Signal Processing, Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia
highlights • • • •
The non-fragile observer design for jumping neural networks with delays is investigated. Delay-dependent stability criteria are obtained in terms of LMIs. The observer gains are given from the LMI feasible solutions. The practical system of quadruple tank process is considered for the example.
article
info
Article history: Received 27 July 2013 Accepted 23 May 2014 Keywords: Linear matrix inequality Lyapunov–Krasovskii functional Markovian jumping parameters Neural networks Non-fragile observer design State estimation
abstract This paper investigates the non-fragile observer based design for neural networks with mixed time-varying delays and Markovian jumping parameters. By developing a reciprocal convex approach and based on the Lyapunov–Krasovskii functional, and stochastic stability theory, a delay-dependent stability criterion is obtained in terms of linear matrix inequalities (LMIs). The observer gains are given from the LMI feasible solutions. Finally, three numerical examples are given to illustrate the effectiveness of the derived theoretical results. Among them the third example deals the practical system of quadruple tank process. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Neural networks (NNs) can mimic the human brain, and they have been used for a wide variety of applications, for example, target tracking, machine learning, system identification and so on [1,2]. The analysis of dynamics for NNs in successful applications is preliminary. Time delays are unavoidably encountered both in biological and artificial neural systems, which may lead to oscillation and instability, hence in recent years increasing attention has been focused on stability analysis of NNs with time delays, see [3,4]. Many interesting results have been obtained, see [5–12], where [5,6] dealt the case of constant delays, authors in [9,10] studied the case of time-varying delays and the paper [11,12] considered the case of continuously distributed delays. Further, it is well known that delay-dependent results are generally less conservative than delay-independent results. Recently, delayed NNs have been focused on the stability analysis and a large amount of results have been available in the literature, see for example [13–17]. Further, most of the research on NNs has been restricted to simple cases of discrete delays. Since an NN usually has a spatial nature due to the presence of an amount of parallel
∗
Corresponding author. Tel.: +91 451 2452371; fax: +91 451 2453071. E-mail addresses: [email protected] (V. Vembarasan), [email protected] (P. Balasubramaniam), [email protected] (C.S. Chan).
http://dx.doi.org/10.1016/j.nahs.2014.05.006 1751-570X/© 2014 Elsevier Ltd. All rights reserved.
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V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
pathways of a variety of axon sizes and lengths, it is desired to model them by introducing distributed delays, for details see [16,17] and the references therein. In practice, sometimes a NN has finite state representations and modes may switch (or jump) from one to another at different times [18–21]. Recently, it has been revealed in [22] that, switching (or jumping) between different NNs modes can be governed by a Markovian chain. Specifically, the class of NNs with Markovian jumping parameters (MJPs) has two components in the state vector. The first one which varies continuously is referred to be the continuous state of the NNs and the second one which varies discretely is referred to be the mode of the NNs. For a specific mode, the dynamics of the NNs is continuous, but the parameter jumps among different modes may be seen as discrete events, see for example [23]. A state observer is used usually to reconstruct the states of a dynamic system and has very important applications in many aspects such as realization of feedback control, system supervision, gas-fired furnace system, and fault diagnosis. Naturally the neuron states are not often fully available in the network outputs in many applications, the neuron state estimation problem becomes important to utilize the estimated neuron state. Initially, Wang et al. [24] have investigated the state estimation problem for continuous-time NNs with time-varying delay through available output measurements and derived some sufficient conditions for the existence of the desired estimators. Therefore, delay-dependent state estimation problem has been studied widely for continuous-time NNs, see [25–34]. Recently authors in [32–34] applied sampled data control approach to study the delay-dependent state estimation problems for NNs. Further the state estimation problem for delayed genetic regulatory networks with randomly occurring uncertainties has been dealt in [35]. Very recently authors in [36–43] have studied state estimation for recurrent NNs with MJPs and mixed time delays. Sampled data approach for state estimation for NNs with MJPs has been considered in [42,43]. Meantime, the delay-dependent state estimation problem was widely studied for continuous-time delayed NNs in [29–31,41] by employing delay decomposition methods, free weighting matrix method, triple-integral approaches and the sufficient conditions are derived using linear matrix inequalities (LMIs). However, most results of the state estimation problems have concerned with the uncertainties appearing in system matrices only. Up to now, the design of state estimator to cope with the non-fragile observers have not been thoroughly studied yet. This brings up a ‘‘fragility’’ problem that has been of great interest in the signal processing area for some time now, but has been considered only recently by the control community [44]. The problem is that of determining the level of computer accuracy required to insure closed-loop specifications. Although it is generally known that many feedback systems require very accurate controller implementation, there have been few studies on the level of accuracy required to met given specifications, and on the design of controllers that require the least possible amount of accuracy, that is, non-fragile controllers. However up to date only limited works have been done with respect to the design of non-fragile controllers [45–49] for the system of differential equations with time-delays. Non-fragile controllers minimize the cost of implementation, and allow for on-line tuning of control parameters that are essential in practical applications. Note that very little research efforts have been made to design the state estimator coping with the time-varying delays and non-fragile controllers in the observer gains. The lack of the existing works are probably due to the fragility problem of controllers. Therefore, the aim of this paper is to shorten such a gap by designing the state estimator for NNs with MJPs and non-fragile observers and mixed time-varying delays. On the other hand, the reduction of the conservatism in delay-dependent criteria means that the feasible region is enlarged, which can increase the application region such as filtering, controller design, synchronization, and other important issues in control society. Therefore, the study of increasing the feasible region in delay-dependent stability criteria is an important and essential work in applying various control methods in dynamic systems with time-delays. In [50], authors have discussed reciprocally convex approach to derive stability results. In this paper, a similar type of stability criterion is applied to derive the sufficient conditions for the problem of state estimation for NNs with MJPs and non-fragile observers. To the best of authors’ knowledge, the non-fragile observer based state estimation of NNs with mixed time-varying delays and MJPs using reciprocal convex method has not been investigated yet. Suitable Lyapunov–Krasovskii functionals (LKF), reciprocal convex method and some analysis techniques are employed to derive sufficient conditions for delay-dependent stability criteria for the considered NNs with MJPs in terms of LMIs, which can be easily calculated by MATLAB LMI Control Toolbox. Numerical examples are given to illustrate the effectiveness of the proposed method. Motivated by the above discussions, the aim of this paper is to study the non-fragile observer based state estimation problem for a class of NNs with MJPs, and mixed time-varying delays. By constructing a suitable LKF including the available information of time-varying delays, sufficient conditions are established such that the resulting estimation error dynamics is asymptotically stable in the mean-square sense. The main contributions of this paper lie in the following two aspects: (i) the phenomena of non-fragile control is considered at the first time for the state estimation problem of NNs with MJPs; and (ii) the reciprocally convex combination method has been involved in that account for the improvement of the feasible region. Finally, numerical examples are provided to illustrate the developed state estimation scheme. Notations. Throughout this paper, Rn and Rn×m denote respectively the n-dimensional Euclidean space and the set of all n × m real matrices. The superscript T denotes the transposition and the notation X ≥ Y (respectively, X > Y ), where X and Y are symmetric matrices, meaning that X − Y is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimensions. | · | is the Euclidean norm in Rn . Moreover, let (Ω , F , P ) be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions. That is the filtration contains all P -null sets and is right continuous. n Denote by LP F0 [−δ, 0] the family of all F0 -measurable C ([−δ, 0]; R )-valued random variables ς = {ς (θ ) : −δ ≤ θ ≤ 0}
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
63
such that sup−δ≤θ≤0 E ∥ς (θ )∥P < ∞, where E stands for the mathematical expectation operator with respect to the given probability measure P . The notation ∗ always denotes the symmetric block in one symmetric matrix.
2. Problem description and preliminaries Let {r (t ), t ≥ 0} be a right-continuous Markov process on the probability space which takes values in the finite space
S = {1, 2, . . . , N } with generator Γ = (γij ), (i, j ∈ S ) given by
P (r (t + △) = j|r (t ) = i) = o(∆)
where ∆ > 0, lim1t →0 (
γii = −
N
j=1,j̸=i
∆
γij ∆ + o(∆), 1 + γii ∆ + o(∆),
i ̸= j, i = j,
(1)
) = 0 and γij is transition rate from mode i to mode j satisfying γij ≥ 0 for i ̸= j with
γij , i, j ∈ S .
Consider the following NNs of mixed time-varying delays with MJPs and the network measurements described by x˙ (t ) = −A(r (t ))x(t ) + B(r (t ))g (x(t )) + B1 (r (t ))g (x(t − τ (t ))) + E (r (t ))
t
t −σ (t )
g (x(s))ds + J (r (t )),
y(t ) = C (r (t ))x(t ) + D(r (t ))f (t , x(t )),
(2)
where x(·) = [x1 (·), x2 (·), . . . , xn (·)]T ∈ Rn is neuron state vector, x(t ) = ϖ (t ), t ≤ 0; ϖ (t ) is the initial condition. A = diag{a1 , . . . , an } is a diagonal matrix with ai > 0, i = 1, . . . , n, the matrices B, B1 and E represent the connection weight matrix, the discretely delayed connection weight matrix, and distributively delayed connection weight matrix respectively. The matrices C ∈ Rm×n and D ∈ Rm×m are the constant matrices. g (x(·)) = [g1 (x1 (·)), . . . , gn (xn (·))]T ∈ Rn denotes the neuron activation function, J = [J1 , . . . , Jn ]T ∈ Rn is a constant input vector and f : R × Rn → Rm is the neuron-dependent nonlinear disturbances on the network outputs which are assumed to satisfy the underlying Assumption (H). τ (t ) and σ (t ) denote the discrete time-varying delays and the distributed time-varying delays, respectively, and satisfy 0 < τm ≤ τ (t ) ≤ τM , τ˙ (t ) ≤ µ and 0 ≤ σ (t ) ≤ σM , where τm , τM , µ and σM are constants. For the NNs (2), we construct the full-order state estimator as follows: x˙ˆ (t ) = −A(r (t ))ˆx(t ) + B(r (t ))g (ˆx(t )) + B1 (r (t ))g (ˆx(t − τ (t )))
+ E (r (t ))
t
t −σ (t )
g (ˆx(s))ds + J (r (t )) + K (r (t ))(y(t ) − yˆ (t )),
yˆ (t ) = C (r (t ))ˆx(t ) + D(r (t ))f (t , xˆ (t )),
(3)
where xˆ (t ) is the estimation of the neuron state, and K (r (t )) ∈ R is the estimator gain matrix to be designed. Notice that the Markov process {r (t ), t ≥ 0} takes values in the finite space S = {1, 2, . . . , N }. For notation simplicity, we denote A(i) := Ai , B(i) := Bi , B1 (i) := B1i , E (i) := Ei , J (i) := Ji , K (i) := Ki , C (i) := Ci , D(i) := Di . Non-fragile observer-based control for the practical implementation is considered in the following form: n×m
x˙ˆ (t ) = −Ai xˆ (t ) + Bi g (ˆx(t )) + B1i g (ˆx(t − τ (t ))) + Ei
t t −σ (t )
g (ˆx(s))ds + Ji + (Ki + 1Ki )(y(t ) − yˆ (t )),
yˆ (t ) = Ci xˆ (t ) + Di f (t , xˆ (t )),
(4)
where the real-valued matrices 1Ki (t ) represent possible controller gain fluctuations. It is assumed that 1Ki (t ) have the structure Hi Fi (t )GKi , in which Hi and GKi are known real constant matrices of appropriate dimensions with an unknown time-varying matrix satisfying FiT (t )Fi (t ) ≤ I. Define the error e(t ) := x(t ) − xˆ (t ), and φ(t ) := g (x(t )) − g (ˆx(t )), ψ(t ) := f (t , x(t )) − f (t , xˆ (t )). Thus, the error-state system is given by
e˙ (t ) = − Ai + (Ki + 1Ki )Ci e(t ) + Bi φ(t ) + B1i φ(t − τ (t )) + Ei
t t −σ (t )
φ(s)ds − (Ki + 1Ki )Di ψ(t ).
(5)
Obviously, e(s) is bounded and continuously differential on [−δ, 0], where δ = max{τM , σM }. Traditionally, the activation functions are assumed to be continuous, differentiable, monotonically increasing, and bounded, such as the sigmoid-type of function. However, in many electronic circuits, the input–output functions of amplifiers may be neither monotonically increasing nor continuously differentiable, hence, non monotonic functions can be more appropriate to describe the neuron activation in designing and implementing artificial NNs. In this paper, we make the following assumptions for the neuron activation functions.
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Assumption (H) ([51]). The neuron activation function g (·) and the neuron-dependent nonlinear disturbances f (·) in (2) are bounded and there exist four constant matrices φ − = diag{φ1− , φ2− , . . . , φn− }, φ + = diag{φ1+ , φ2+ , . . . , φn+ }, ψ − = diag{ψ1− , ψ2− , . . . , ψn− }, and ψ + = diag{ψ1+ , ψ2+ , . . . , ψn+ }, such that
φk−′ ≤
gk′ (α) − gk′ (β) 1
ψk−′ ≤
1
α−β
1
≤ φk+′ ,
(6)
1
fk′ (α, x(α)) − fk′ (β, x(β)) 2
2
x(α) − x(β)
2
≤ ψk+′ ,
(7)
2
for all α, β ∈ R, α ̸= β, k′1 = 1, 2, . . . , n and k′2 = 1, 2, . . . , m. By recalling definitions of φ(t ) and ψ(t ), from Assumption (H), we obtain the following inequality readily:
[φk′1 (t ) − φk−′ ek′1 (t )]T [φk′1 (t ) − φk+′ ek′1 (t )] ≤ 0,
k′1 = 1, 2, . . . , n,
1
1
[ψk′2 (t ) − ψk−′ ek′2 (t )]T [ψk′2 (t ) − ψk+′ ek′2 (t )] ≤ 0, 2
2
k′2 = 1, 2, . . . , m.
Let e(t ; ς ) be the trajectory of system (5) under the initial condition e(θ ) = ς (θ ) on −δ ≤ θ ≤ 0 in L2F0 ([−δ, 0], Rn ). It is obvious that the system (5) admits a trivial solution e(t ; 0) = 0. Before ending this section, a stability definition and a useful lemma are recalled. Definition 2.1 ([36]). For the error-state system (5) and every ς ∈ L2F0 ([−δ, 0], Rn ), the trivial solution is asymptotically stable in the mean square if, for every network mode lim E |e(t ; ς )|2 = 0.
t −→∞
Lemma 2.2 ([52] Schur Complement Lemma). Given constant matrices Ω1 , Ω2 and Ω3 withappropriate dimensions, where
Ω1T = Ω1 and Ω2T = Ω2 > 0, then Ω1 + Ω3T Ω2−1 Ω3 < 0, if and only if
Ω1 ∗
Ω3T − Ω2
< 0, or
−Ω2 ∗
Ω3 Ω1
< 0.
Lemma 2.3 ([53]). Let M = M T , H and G are real matrices of appropriate dimensions F T (t )F (t ) ≤ I, then M + HF (t )G + GT F T (t )H T < 0, if and only if there exists a positive scalar ϵ > 0 such that M + 1ϵ HH T + ϵ GT G < 0. 3. Main results In this section, we shall establish the main criterion based on the LMI approach. For presentation convenience, we introduce the following notations:
φn− + φn+ φ1− + φ1+ φ2− + φ2+ , Σ2 = diag , ,...,
Σ1 = diag{φ1 φ1 , φ2 φ2 , . . . , φn φn , }, − +
− +
− +
2
2
2
Σ3 = diag{ψ1 ψ1 , ψ2 ψ2 , . . . , ψn ψn , }, −
+
−
+
−
+
ψ1− + ψ1+ ψ2− + ψ2+ ψn− + ψn+ Σ4 = diag , ,..., . 2
2
2
Theorem 3.1. For given positive scalars τm , τM , σM and µ, the equilibrium solution of NNs with MJPs (5) is globally robustly asymptotically stable in the mean square if there existsymmetric positive definite matrices Pi , Q1 , Q2 , Q3 , R1 , R2 , S1 , S2 , further
for any matrices T with compatible dimensions, and scalars ϵ1i , ϵ2i , ϵ3i such that the following LMI holds:
Πi ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ξi Π9,9,i ∗ ∗ ∗ ∗ ∗ ∗
R2
T
TT
R2
≥ 0, diagonal matrices U > 0, V > 0, W > 0, and positive
Υ1
ϵ1i Υ2
Υ3
ϵ2i Υ4
Υ5
ϵ3i Υ6
0
0 0
0 0 0
0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
−ϵ1i I ∗ ∗ ∗ ∗ ∗
−ϵ1i I ∗ ∗ ∗ ∗
−ϵ2i I ∗ ∗ ∗
−ϵ2i I ∗ ∗
−ϵ3i I ∗
−ϵ3i I
< 0,
(8)
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
65
where
Π1,1,i = −Pi Ai − ATi Pi − Li Ci − CiT LTi +
N
γij Pj + Q1 + Q2 + Q3 − R1 − Σ1 U − Σ3 W ,
i =1
Π1,3,i = R1 ,
Π1,5,i = Pi Bi + Σ2 U ,
Π2,2,i = −(1 − µ)Q3 − R2 − Π2,6,i = Σ2 V ,
RT2
Π1,6,i = Pi B1i ,
+ T + T − Σ1 V , T
Π3,3,i = −Q1 − R1 − R2 ,
Π5,5,i = S1 + σ
2 M S2
− U,
Π1,7,i = Pi Ei ,
Π1,8,i = −Li Di + Σ4 W ,
Π2,3,i = R2 − T ,
Π2,4,i = R2 − T ,
T
Π3,4,i = T ,
Π4,4,i = −Q2 − R2 ,
T
Π6,6,i = −(1 − µ)S1 − V ,
Π7,7,i = −S2 , Π8,8,i = −W , Π9,9,i = −2Pi + N , T T T T T T Ξi = −ATi Pi − CiT LTi 0 · · · 0 B P B P E P − D L , i i i i 1i i i i 3
Υ3 = Υ1 ,
Υ2 = [GTKi CiT 0 · · 0 GTKi DTi ], ·
Υ1 = [−Hi 0 · · 0], · 7
Υ4 = [0 · · 0 GTKi CiT ], ·
6
Υ5 = [0 · · 0 − Hi ], ·
Υ6 = [0 · · 0 ·
7
GTKi DTi
],
7
N =τ
2 m R1
+ (τM − τm ) R2 . 2
7
Moreover, the state estimator gain matrix is given by Ki = Pi−1 Li . Proof. Choose the LKF for error-state system (5) as V (et , t , i) = V1 (et , t , i) + V2 (et , t , i) + V3 (et , t , i) + V4 (et , t , i),
(9)
where V1 (et , t , i) = eT (t )Pi e(t ), V2 (et , t , i) =
t
e (s)Q1 e(s)ds + T
t −τm
V3 (et , t , i) = τm
V4 (et , t , i) =
e (s)Q2 e(s)ds + T
t −τM 0
t
−τm
t
e˙ T (s)R1 e˙ (s)dsdθ + (τM − τm )
−τM
t t −τ (t )
t −τ (t )
−τm
t +θ
t
φ T (s)S1 φ(s)ds + σM
0
−σM
eT (s)Q3 x(s)ds, t
e˙ T (s)R2 e˙ (s)dsdθ , t +θ
t
φ T (s)S2 φ(s)dsdθ . t +θ
Let L be the weak infinitesimal generator of the random process {et , t ≥ 0}. Then, for each {r (t ) = i, i ∈ S }, it can be shown that
LV1 (et , t , i) = 2eT (t )Pi − Ai + (Ki + 1Ki )Ci e(t ) + Bi φ(t ) + B1i φ(t − τ (t ))
+ Ei
t t −σ (t )
N φ(s)ds − (Ki + 1Ki )Di ψ(t ) + γij eT (t )Pj e(t ),
(10)
j =1
LV2 (et , t , i) ≤ eT (t )(Q1 + Q2 + Q3 )e(t ) − eT (t − τm )Q1 e(t − τm ) + eT (t − τM )Q2 e(t − τM )
− (1 − µ)eT (t − τ (t ))Q3 e(t − τ (t )), t LV3 (et , t , i) = τm2 e˙ T (t )R1 e˙ (t ) − τm e˙ T (s)R1 e˙ (s)ds t −τm + (τM − τm )2 e˙ T (t )R2 e˙ (t ) − (τM − τm )
(11)
t −τm
e˙ T (s)R2 e˙ (s)ds,
(12)
t −τM
LV4 (et , t , i) ≤ φ T (t )S1 φ(t ) − (1 − µ)φ T (t − τ (t ))S1 φ(t − τ (t ))
+ σ φ (t )S2 φ(t ) − 2 M
T
t
φ (s) S2
T
t −σ (t )
t −σ (t )
φ(s)ds .
(13)
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V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
Now by applying Jensen’s Inequality Lemma [53] and Theorem 1 in [50], one can get t
e˙ T (s)R1 e˙ (s)ds ≤ −[e(t ) − e(t − τm )]T R1 [e(t ) − e(t − τm )],
−τm
(14)
t −τm
−(τM − τm )
t −τm t −τ (t )
e˙ T (s)R2 e˙ (s)ds − (τM − τm )
t −τ (t )
e˙ T (s)R2 e˙ (s)ds
t −τM
(τM − τm ) [e(t − τm ) − e(t − τ (t ))]T R2 [e(t − τm ) − e(t − τ (t ))] τ (t ) − τm (τM − τm ) − [e(t − τ (t )) − e(t − τM )]T R2 [e(t − τ (t )) − e(t − τM )] τM − τ (t ) T R2 T e(t − τm ) − e(t − τ (t )) e(t − τm ) − e(t − τ (t )) ≤− . T e(t − τ (t )) − e(t − τM ) e(t − τ (t )) − e(t − τM ) T R2 ≤−
(15)
Further for positive diagonal matrices U > 0, V > 0 and W > 0, we can get from Assumption (H) that e(t ) φ(t )
T
Σ1 U −Σ 2 U
e(t − τ (t )) φ(t − τ (t ))
−Σ 2 U U
T
Σ1 V −Σ 2 V
T
e(t ) ψ(t )
e(t ) ≤ 0, φ(t )
Σ3 W −Σ 4 W
−Σ 2 V V
−Σ 4 W
(16)
e(t − τ (t )) ≤ 0, φ(t − τ (t ))
(17)
e(t ) ≤ 0. ψ(t )
W
(18)
Now adding (10)–(18), we get
LV (et , t , i) ≤ ξ T (t ) Ωi + ΘiT N Θi + 1Πi (t ) ξ (t ),
(19)
where
ξ (t ) = e (t ) e (t − τ (t )) e (t − τm ) e (t − τM ) φ (t ) φ (t − τ (t )) T
T
T
T
(Ωi )9×9 = (Ωj,k,i ) = (Πj,k,i ) = (Πi )9×9 ,
T
T
T
t
t −σ (t )
T
φ(s)ds
Ψ (t ) , T
j, k varies between 1 and 9; except that
Ω1,1,i = −Pi Ai − ATi Pi − Pi Ki Ci − CiT KiT Pi +
N
γij Pj + Q1 + Q2 + Q3 − R1 − Σ1 U − Σ3 W ,
i=1
Ω1,8,i = −Pi Ki Di + Σ4 W , 1Πi (t ) = Υ1 F (t )Υ2 + Υ2T F T (t )Υ1T + Υ3 F (t )Υ4 + Υ4T F T (t )Υ3T + Υ5 F (t )Υ6 + Υ6T F T (t )Υ5T , T T T T T T B B E − D K . Θi = −ATi − CiT KiT 0 · · · 0 i 1i i i i 3
Also by Lemma 2.3, we can have
1Πi (t ) ≤ ϵ1i−1 Υ1 Υ1T + ϵ1i Υ2 Υ2T + ϵ2i−1 Υ3 Υ3T + ϵ2i Υ4 Υ4T + ϵ3i−1 Υ5 Υ5T + ϵ3i Υ6 Υ6T .
(20)
Substitute the inequality (20) in (19), one can get the following inequality:
Ωi + Θi N ΘiT + ϵ1i−1 Υ1 Υ1T + ϵ1i Υ2 Υ2T + ϵ2i−1 Υ3 Υ3T + ϵ2i Υ4 Υ4T + ϵ3i−1 Υ5 Υ5T + ϵ3i Υ6 Υ6T < 0.
(21)
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
67
In order to get the expected LMI condition of the error-state system (5), we will need to make some standard manipulations on the relation (21). By using the well-known Schur Complement Lemma (Lemma 2.2), the inequality (21) is equivalent to the following matrix inequality:
Ωi ∗ ∗ ∗ ∗ ∗ ∗ ∗
Θi N −N ∗ ∗ ∗ ∗ ∗ ∗
Υ1
ϵ1i Υ2
Υ3
ϵ2i Υ4
Υ5
ϵ3i Υ6
0
0 0
0 0 0
0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
−ϵ1i I ∗ ∗ ∗ ∗ ∗
−ϵ1i I ∗ ∗ ∗ ∗
−ϵ2i I ∗ ∗ ∗
−ϵ2i I ∗ ∗
−ϵ3i I ∗
< 0.
(22)
−ϵ3i I
By noting Ki = Pi−1 Li , pre- and post- multiplying (22), by diag{I , . . . , I , Pi N −1 , I , . . . , I } and diag{I , . . . , I , N −1 Pi , I , . . . , I },
8
6
8
6
respectively, and from view of the inequality −Pi N −1 Pi ≥ 2Pi − N resulting from (Pi − N )T N −1 (Pi − N ) ≥ 0, it is easy to see that the LMI condition (8) can guarantee that (22) holds. Hence, (21) is implied by the LMI condition (8). In other words, the LMI condition (8) and by Definition 2.1, implies that the error-state system (5) is globally robustly asymptotically stable in the mean square, and this completes the proof. Remark 3.2. The delay-dependent state estimation problem has been studied widely for continuous-time NNs, see [25–34]. Recently authors in [32–34] have applied sampled data control approach to study the delay-dependent state estimation problems for NNs. Further the sampled data approach for state estimation for NNs with MJPs has been considered in [42,43]. However up to now, there are no results employing non-fragile observer based state estimation for NNs with MJPs, which motivated our present study. Consider the following observer based non-fragile control error-state system NNs (5) without MJPs
e˙ (t ) = − A + (K + 1K )C e(t ) + Bφ(t ) + B1 φ(t − τ (t )) + E
t
t −σ (t )
φ(s)ds − (K + 1K )Dψ(t ).
(23)
Corollary 3.3. For given positive scalars τm , τM , σM and µ, the equilibrium solution of NNs without MJPs (23) is globally robustly asymptotically stable in the mean square if there exist symmetric positive definite matrices P , Q1 , Q2 , Q3 , R1 , R2 , S1 , S2 , further for any matrices T with compatible dimensions, and
scalars ϵ1 , ϵ2 , ϵ3 such that the following LMI holds
Π ∗ ∗ ∗ ∗ ∗ ∗ ∗
Ξ 9,9 Π ∗ ∗ ∗ ∗ ∗ ∗
R2
TT
1 Υ
2 ϵ1 Υ
3 Υ
4 ϵ2 Υ
5 Υ
0
0 0
0 0 0
0 0 0 0
0 0 0 0 0
−ϵ1 I ∗ ∗ ∗ ∗ ∗
−ϵ1 I ∗ ∗ ∗ ∗
−ϵ2 I ∗ ∗ ∗
−ϵ2 I ∗ ∗
−ϵ3 I ∗
T
≥ 0, diagonal matrices U > 0, V > 0, W > 0, and positive
R2
6 ϵ3 Υ 0 0 0 < 0, 0 0 0
−ϵ3 I
where
1,1 = −PA − AT P − LC − C T LT + Q1 + Q2 + Q3 − R1 − Σ1 U − Σ3 W , Π 1,3 = R1 , 1,5 = PB + Σ2 U , 1,6 = PB1 , 1,7 = PE , 1,8 = −LD + Σ4 W , Π Π Π Π Π T T T 2,2 = −(1 − µ)Q3 − R2 − R2 + T + T − Σ1 V , 2,3 = R2 − T , 2,4 = R2 − T , Π Π Π 2,6 = Σ2 V , 3,3 = −Q1 − R1 − R2 , 3,4 = T T , Π Π Π 2 5,5 = S1 + σM S2 − U , 6,6 = −(1 − µ)S1 − V , Π Π 7,7 = −S2 , 8,8 = −W , 9,9 = −2P + N , Π Π Π T = −AT P − C T LT 0 · · · 0 BT P BT1 P E T P − DT LT , Ξ 3
1 = [−H 0 · · · 0], Υ 7
2 = [GTK C T 0 · · · 0 GTK DT ], Υ 6
4,4 = −Q2 − R2 , Π
(24)
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3 = Υ 1 , Υ
4 = [0 · · · 0 GTK C T ], Υ
5 = [0 · · · 0 − H ], Υ
7
N =τ
6 = [0 · · · 0 GTK DT ], Υ
7
2 m R1
+ (τM − τm ) R2 . 2
7
Further, the state estimator gain matrix is given by K = P −1 L, and other terms are defined as same in Theorem 3.1.
Remark 3.4. Very recently authors in [36–43] have studied state estimation problem for recurrent NNs with MJPs and mixed time delays. Meantime, the delay-dependent state estimation problem was widely studied for continuous-time delayed NNs with MJPs by employing delay decomposition methods, free weighting matrix method, triple-integral approaches and the sufficient conditions are derived using linear matrix inequalities (LMIs). However, the result derived in this paper by using reciprocal convex approach [50] is new which is not discussed in any of the previous literature [36–43]. Moreover in the absence of the MJPs, the results derived in this paper are still new, the corresponding results are given in Corollary 3.3. Remark 3.5. The traditional assumption on the derivatives of the time-varying delays are less than 1 is no longer required in this analysis. Therefore theorems and corollaries in this paper are applicable to more general than previously investigated papers. In many cases, the information of the derivative of time delays is unknown because it is a very difficult task to obtain the precise values (even their boundedness or the boundedness of their derivatives) of time delay systems. Regarding this circumstance, rate-independent criteria for delay τ (t ) satisfying the conditions 0 < τm ≤ τ (t ) ≤ τM are derived by choosing Q3 and S1 to be zero in theorems and corollaries of this paper and then the corresponding results can be developed. Consider the following non-fragile control error-state system NNs (5) without MJPs and distributed time delays:
e˙ (t ) = − A + (K + 1K )C e(t ) + Bφ(t ) + B1 φ(t − τ (t )).
(25)
Corollary 3.6. For given positive scalars τm , τM , σM and µ, the equilibrium solution of NNs without MJPs (25) is globally robustly asymptotically stable in the mean square if there exist symmetric positive definite matrices P , Q1 , Q2 , Q3 , R1 , R2 , S1 , further for any matrices T with compatible dimensions, and
R2
TT
T
R2
≥ 0, diagonal matrices U > 0, V > 0, and positive scalars ϵ1 , ϵ2 such
that the following LMI holds:
ˆ Π ∗ ∗ ∗ ∗ ∗
Ξˆ ˆ Π7,7 ∗ ∗ ∗ ∗
Υˆ 1
ϵ1 Υˆ 2
Υˆ 3
ϵ2 Υˆ 4
0
0 0
0 0 0
0 0 0 0
−ϵ1 I ∗ ∗ ∗
−ϵ1 I ∗ ∗
−ϵ2 I ∗
< 0,
(26)
−ϵ2 I
where
ˆ 1,1 = −PA − AT P − LC − C T LT + Q1 + Q2 + Q3 − R1 − Σ1 U , Π ˆ 1 ,3 = R 1 , ˆ 1,5 = PB + Σ2 U , ˆ 1,6 = PB1 , Π Π Π T ˆ 2,2 = −(1 − µ)Q3 − R2 − R2 + T + T T − Σ1 V , Π ˆ 2 ,3 = R 2 − T T , ˆ 2,4 = R2 − T , Π Π ˆ 2 , 6 = Σ2 V , ˆ 3,3 = −Q1 − R1 − R2 , ˆ 3,4 = T T , ˆ 4,4 = −Q2 − R2 , Π Π Π Π ˆ 5 ,5 = S 1 − U , ˆ 6,6 = −(1 − µ)S1 − V , ˆ 7,7 = −2P + N , Π Π Π T ˆ = −AT P − C T LT 0 · · · 0 BT P BT1 P , Ξ 3
Υˆ 1 = [−H 0 · · 0], ·
Υˆ 2 = [GTK C T 0 · · 0], ·
5
Υˆ 3 = Υˆ 1 ,
5
Υˆ 4 = [0 · · 0 GTK C T ], ·
N = τm2 R1 + (τM − τm )2 R2 .
5
Further, the state estimator gain matrix is given by K = P −1 L, and other terms are defined as same in Theorem 3.1.
Remark 3.7. The study of randomly occurring incomplete information has become the active research topic. Very recently Lakshmanan et al. [35] studied the state estimation problem for delayed genetic regulatory networks with randomly
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
69
occurring uncertainties. Motivated by this in future, we can extend the present results to more general, that is, non-fragile robust state estimation problem for NNs with MJPs and randomly occurring uncertainties. This result will be investigated in the near future. Remark 3.8. When an NN is designed for a practical application, it is often required to be stable. The advantages of the newly-established stability criteria are that (i) less conservative conditions are formulated by means of LMIs, which can be facilitated readily by standard algorithms. According to [52], the LMI-based conditions can be solved by polynomial time algorithms with complexity proportional to MN 3 , where M is the total row size of the LMIs and N is the total number of scalar decision variables. Therefore, for NNs with hundreds or thousands of neurons, the LMIs-based conditions can be efficiently checked in practice; and (ii) the upper bound of the delay can be easily found such that the designed NNs can be clearly described. Therefore, it is expected that the results can provide a foundation to the design of this kind of NNs, and play an essential role to the potential applications in the areas such as adaptive control, combinatorial optimization, and signal processing. The engineering applications of the results obtained in this study will be one of our future research topics. 4. Numerical examples Example 4.1. Consider a three-neuron two-mode NNs with MJPs (2) with non-fragile control parameters as follows: 2 0 0
A1 =
0 3 0
0.3 1.1 −0.2
4 0 0
0 0 , 2
A2 =
0 3 0
1.3 0.7 −0.1
E2 =
0.5 −0.6 0.2
1.2 0.5 −0.4
E1 =
0 0 , 4
0.2 1.7 −0.3
−0.4 0.3 0.8
B1 =
−0.4 1.5 , 1.0
C1 =
0.2 −0.5 0.6
B2 =
−0.5 1.2 , 1.5
1 0 0
0.2 0.6 , −0.3
B11 =
0 0 0
2 cos(t ) + 0.03t 2 J1 = J2 = 2 sin(t ) − 0.03t 2 , 2 cos(t ) + 0.03t 2
0 0 , 1
0.8 0.2 0.4
−0.3 0.4 , −0.4
C2 =
1 0 0
0 1 0
0 0 , 0
0.1 −0.5 0.1
0.4 0.1 −0.7
0.2 0.4 , 0.3
B12 =
0.1 −0.4 0.2
0.8 0.1 −0.5
D1 = D2 = I ,
0.6 0.3 , 0.4
Γ =
−3 2
3 . −2
The activation function satisfies Assumption (H) with φ − = −0.5I , φ + = 0.4I, and the nonlinear disturbance also satisfies Assumption (H) with ψ − = 0.2I , ψ + = 0.4I. Thus, we can get the parameters as Σ1 = −0.2I , Σ2 = −0.05I , Σ3 = 0.08I , Σ4 = 0.3I. The activation function g (x(t )) =
|x(t ) + 1| − |x(t ) − 1| and nonlinear disturbance is of the form
1 5
t
f (t , x(t )) = 0.1 cos(x(t )) + 0.3. The discrete time-varying delay τ (t ) = ete+1 , and the distributed time-varying delay ρ(t ) = 0.5 sin(t ). Noticing that τ (t ) is a monotone increasing function with respect to t, we can easily obtain that τm = 0.5 et and τM = 1. In addition, ρM = 0.5. Moreover, it can be calculated that τ˙ (t ) = (et + ≤ 0.25, which implies µ = 0.25. 1)2 Using the MATLAB LMI solvers for the LMIs in Theorem 3.1, we obtain the corresponding non-fragile observer gain matrices as follows: −1
K1 = P 1
−0.2948 L1 = 0.0216 −0.1397
0.0650 −0.9827 −0.0686
0.0269 −0.0645 , −1.6613
−1
K2 = P 2
−1.7075 L2 = −0.0503 −0.0408
0.0433 −0.8756 0.0844
−0.1461 0.0487 . −0.2702
From the structures of 1Ki (t ), we have H1 = H2 = 0.3I , GK 1 = GK 2 = 0.1I and F (t ) = sin(t )I. The initial conditions of system (2) and the estimator (4) are taken as x(t ) = [4; 3; −2]T and xˆ (t ) = [2; −2; 1]T respectively for t ∈ [−1, 0]. The response of the state and estimation for the NNs are given in Fig. 1. In Fig. 2, NNs with time delays (5) are given which converge to zero asymptotically. Simulation of system modes of Example 4.1 is displayed in Fig. 3. Therefore, it follows from Theorem 3.1, that the error-state system NNs with MJPs (5) is globally asymptotically stable in the mean square. Remark 4.1. It is important to note that very limited works have been done on observer based non-fragile controllers for system of differential equations with time-delays [45–49]. But there are no related works on observer based non-fragile controller design for NNs. In order to fill such a gap, in this paper we aimed to design the non-fragile observer based design for NNs with MJPs and mixed time-varying delays. Hence we could not provide any comparison results. Example 4.2. Consider a three-neuron single-mode NNs without MJPs (2) with non-fragile control parameters as follows: 3 0 0
A=
0 3 0
0 0 , 3
0.4 −0.2 0.7
B=
−0.6 0.4 0.1
0.2 0.6 , −0.9
B1 =
0.5 −0.5 0.5
0.5 0.3 −0.1
0.4 0.4 , 1.2
70
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
a
b
c
Fig. 1. The response of the state, estimation trajectories in Example 4.1.
Fig. 2. The response of the error trajectories in Example 4.1.
Fig. 3. The simulation of system mode r (t ) = {1, 2} in Example 4.1.
1.9 0.2 −0.5
E=
0.8 0.5 −0.7
−0.5 0.9 , 1.8
1 0 0
C =
0 0 0
0 0 , 0
2 cos(t ) + 0.03t 2 J = 2 sin(t ) − 0.03t 2 . 2 cos(t ) + 0.03t 2
D = I,
V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
71
The activation functions and the time-varying delays are as same in Example 4.1. From the structure of 1K (t ), we have H = GK = 0.5I and F (t ) = sin(t )I. Using the MATLAB LMI solvers for the LMIs in Corollary 3.3, we obtain the corresponding non-fragile observer gain matrices as follows:
−0.8232 L = 0.1951 −0.2809
−0.1316 −2.5119 −0.3244
K =P
−1
0.1340 −0.1302 . −2.9733
Therefore, it follows from Corollary 3.3, that the error-state system NNs without MJPs (23) are globally asymptotically stable. Example 4.3. Originally, NNs embody the characteristics of real biological neurons that are connected or functionally related in a nervous system. On the other hand, NNs can represent not only biological neurons but also other practical systems. One of them is the quadruple-tank process, which consists of four interconnected water tanks and two pumps. The inputs are the voltages to the two pumps and the outputs are the water levels of Tanks 1 and 2. This quadruple-tank process can be expressed clearly using the NNs model. Recently Lee et al., [34] proposed the state-space equation of the quadruple-tank process which can be rewritten to the form of system (25) and the parameters are given as follows: 0.0021 0 A= 0 0
0
0 B1 = −0.0143 −0.0119 H = 0.5I ,
−0.0424
0 0.0021 0 0
0 0.0424 0
0 0 −0.0114 −0.0131 0.5 0
GK =
0 0 −0.0165 −0.0059 0 0.5
0 0
−0.0060 −0.0063 B= 0
0 −0.0424 , 0 0.0424
T
0 0
−0.0065 −0.0051
−0.0029 −0.0073
0 0
0 0
0
,
0
0 0 , −0.0056 −0.0154
−0.0077 −0.0025 , 0
0
1 C = 0
0 1
0 0
0 , 0
0 J = , 0 0
F (t ) = sin(t )I .
The activation function satisfies Assumption (H) with φ − = −0.05I , φ + = 0.05I, and thus, we can get the parameters as
Σ1 = −0.0025I , Σ2 = 0. The activation function g (x(t )) = 0.01 |x(t ) + 1| − |x(t ) − 1| . The discrete time-varying delay
τ (t ) = 1 + 0.98sin( π2 t ) and µ = 0.99. Using the MATLAB LMI solvers to solve the LMIs in Corollary 3.6, we obtain the corresponding non-fragile observer gain matrices as follows: 0.7277 0.0002 −1 K =P L= 0.0936 0.0002
0.0002 0.7277 . 0.0002 0.0936
Originally, the water levels of the two lower tanks are the only accessible and usable information in the quadruple-tank process, whereas the water levels of the two upper tanks are not accessible. Although the water in the upper tanks overflows, no action can be taken. Therefore, there is a strong need to know the water level of the upper tanks, which is one of methods for obtaining the information to design the state estimator. Until now, the research of the quadruple-tank process focused on designing the stabilizing controller. Therefore, in the sense of practical applications, Lee et al., [34] have applied the sampled data control in their estimation method. However due to the fragility problem of the controllers, in this paper we have considered the non-fragile observer based design method to estimate the upper tank levels of the quadruple-tank process. The initial conditions of system (2) and the estimator (4) are taken as x(t ) = [−4; 4; −6; 5]T and xˆ (t ) = [10; −10; −10; 10]T respectively. The response of the state and estimation for the NNs are given in Fig. 4. In Fig. 5, NNs with time delays (25) are given which converge to zero asymptotically. Therefore, it follows from Corollary 3.6, that the error-state system NNs with MJPs (25) are globally asymptotically stable in the mean square. 5. Conclusion In this paper, we have dealt with the problem of non-fragile observer design for NNs with MJPs and time delays, which is new in the literature. By developing a reciprocal convex approach and defining appropriate LKFs, new LMI-based stability conditions are derived such that the estimation error system is stochastically globally asymptotically stable in the mean square. Numerical examples are demonstrated to show the merit of the derived results. By utilizing the proposed idea of this paper, future works will focus on stabilization and synchronization for various dynamic systems with time-delays such as NNs with MJPs [40], genetic regulatory networks [35], and multi-agent systems [21] with randomly occurring uncertainties.
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V. Vembarasan et al. / Nonlinear Analysis: Hybrid Systems 14 (2014) 61–73
a
b
c
d
Fig. 4. The response of the state, estimation trajectories in Example 4.3.
Fig. 5. The response of the error trajectories in Example 4.3.
Acknowledgments The authors would like to express their sincere gratitude to the Editor-in-Chief, Associate Editor, and Anonymous Reviewers for their valuable comments and suggestions to improve the quality of the manuscript. This research work is supported by DST INSPIRE Fellowship Grant DST/INSPIRE Fellowship/2011/278 dated 21.12.2011 and 03.10.2012 from the Department of Science and Technology, Ministry of Science and Technology, Government of India; Also this research work is supported by the High Impact Research MoE Grant UM.C/625/1/HIR/MoHE/FCSIT/08, H-22001-00-B0008 from the Ministry of Higher Education Malaysia. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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