Abstract
This paper deals with the H ∞ control problem of neural networks with time-varying delays. The system under consideration is subject to time-varying delays and various activation functions. Based on constructing some suitable Lyapunov–Krasovskii functionals, we establish new sufficient conditions for H ∞ control for two cases of time-varying delays: (1) the delays are differentiable and have an upper bound of the delay-derivatives and (2) the delays are bounded but not necessary to be differentiable. The derived conditions are formulated in terms of linear matrix inequalities, which allow simultaneous computation of two bounds that characterize the exponential stability rate of the solution. Numerical examples are given to illustrate the effectiveness of our results.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
1 Introduction
In the area of control, signal processing, pattern recognition and image processing, delayed neural networks have many useful applications. Some of these applications require that the equilibrium points of the designed network be stable. In both biological and artificial neural systems, time delays due to integration and communication are ubiquitous and often become a source of instability. The time delays in electronic neural networks are usually time-varying, and sometimes vary violently with respect to time due to the finite switching speed of amplifiers and faults in the electrical circuitry. Therefore, stability and control of delayed neural networks is a very important issue, and various stability criteria have been reported in the literature (see, for example, [1–6]). In conducting a periodicity or stability analysis of a neural network, the conditions to be imposed on the neural network are determined by the characteristics of various activation functions as well as network parameters. When neural networks are designed for problem solving, it is desirable for their activation functions to be general. To facilitate the design of neural networks, it is important that the neural networks with various activation functions and time-varying delays are studied. On the other hand, the problem of H ∞ control of dynamical time-delay systems are of practical and theoretical interest due to their useful applications in image processing, especially in classification of patterns, associative memories and other areas (see, for example, [7–14]). For the H ∞ control problem, appropriate methods for linear time-delay systems usually make use of the Lyapunov functional approach, whereby the H ∞ conditions are obtained via solving either matrix inequalities or algebraic Riccati-type equations [15–17]. Regarding H ∞ control of neural networks, the papers [18–20] proposed a state feedback H ∞ control law for the asymptotic stabilization of neural networks with constant time delays. To the best of our knowledge, the H ∞ control problem of neural networks with time-varying delays has not been fully studied yet, which are important in both theories and applications. This motivates our research.
In this paper, we investigate the H ∞ control with exponential stability for neural networks with time-varying delays. The novel features here are that the time-varying delay is present in the observation output with various activation functions, and the controllers to be designed must satisfy some exponential stability constraints on the closed-loop poles. Based on constructing a set of augmented Lyapunov–Krasovskii functionals, a H ∞ controller is designed to achieve exponential stabilization of the neural networks for two cases of time-varying delays: the delays are differentiable and have an upper bound of the delay-derivatives and the delays are bounded but not necessary to be differentiable. The conditions are obtained in terms of LMIs, which allow simultaneous computation of two bounds that characterize the exponential stability rate of the solution and can be easily determined by utilizing MATLABs LMI Control Toolbox.
This paper is organized as follows. Section 2 presents notations, definitions and auxiliary propositions required for the proof of the main results. H ∞ control design for delayed neural networks for two cases of time-varying delays: (1) the delays are differentiable and have an upper bound of the delay-derivatives and (2) the delays are bounded but not necessary to be differentiable, and numerical examples of the results are presented in Sects. 3 and 4, respectively.
2 Preliminaries
The following notations will be used throughout this paper. R + denotes the set of all real non-negative numbers; R n denotes the n-dimensional space with the scalar product \(\langle \cdot , \cdot \rangle\) and the vector norm \(\| \cdot \|;\; R^{n\times r}\) denotes the space of all matrices of (n × r)-dimension. Matrix A is symmetric if A = A T, where A T denotes the transpose of A. I n denotes the identity matrix in R n; λ(A) denotes the set of all eigenvalues of \(A; \lambda_{\rm max}(A) = \max\{\mathop{\rm Re}\lambda: \lambda \in \lambda(A)\}; \lambda_{\rm min}(A) = \min\{\mathop{\rm Re}\lambda: \lambda \in \lambda(A)\}; C([a, b], R^n)\) denotes the set of all R n-valued continuous functions on \([a, b]; L_2([0,\infty], R^r)\) denotes the set of all square-integrable R r-valued functions on \([0,\infty]\). Matrix A is semi-positive definite (A ≥ 0) if \(\langle Ax,x\rangle \geq 0\) for all \(x \in R^n; A\) is positive definite (A > 0) if \(\langle Ax,x\rangle > 0\) for all x ≠ 0; A ≥ B means A − B ≥ 0. Let us denote \(x_t := \{x(t+s), s\in [-h,0]\}\) the segment of the trajectory x(t) with the norm \(\|x_t\| =\sup\nolimits_{t\in[-h,0]}\|x(t+s)\|\).
Consider the following delayed neural networks with control input and observation output
where \(\tau = \max\{\tau_1,\tau_2\}, x(t) \in R^n\) is the state vector of the neural networks; \(u(t) \in L_2([0,s), R^m), s >0, m\leq n, \) is the control input; \(w(t)\in L_2([0,\infty), R^r), r\leq n, \) is the uncertain input of the neural networks; \(z(t)\in R^l, l\leq n,\) is the observation output; n is the number of neurals; \(f(x(t)) = [f_1(x_1(t)), f_2(x_2(t)),{\ldots}, f_n(x_n(t))]^T, g(x(t)) = [g_1(x_1(t)), g_2(x_2(t)), {\ldots}, g_n(x_n(t))]^T, h(x(t)) = [h_1(x_1(t)), h_2(x_2(t)), {\ldots}, h_n(x_n(t))]^T\) are the neural activation functions; the diagonal matrix \(A= \hbox {diag}(a_1, a_2, {\ldots}, a_n), a_i >0,\) represents the self-feedback term; the matrices \(W_0, W_1\in R^{n\times n},W_2\in R^{l\times n}\) denote, respectively, the connection weights; \(B\in R^{n\times m}, D\in R^{l\times m}\) denote the control input matrices; \(B_1\in R^{n\times r}\) denotes the uncertain/perturbation input matrix; \( C\in R^{l\times n} \) denotes the observation output matrix; the initial functions \(\phi(t)\in C([-\tau,0],R^n)\) with the uniform norm \(||\phi|| = \max\nolimits_{t\in [-\tau, 0]}\|\phi(t)\|;\) the time-varying delay functions τ1(t), τ2(t) satisfy either (H1) or (H2):
-
(H1)
\( 0\leq \tau_1(t) \leq \tau_1, \quad \dot \tau_1(t)\leq \delta_1 <1, \quad \forall t\in R^+; \)
\( 0\leq \tau_2(t) \leq \tau_2,\quad \dot \tau_2(t)\leq \delta_2 <1, \quad \forall t\in R^+,\)
-
(H2)
\(0\leq \tau_1(t) \leq \tau_1, \quad 0\leq \tau_2(t) \leq \tau_2, \quad \forall t\in R^+.\)
In this paper, we consider various activation functions f(x), g(x), h(x), f(0) = h(0) = g(0) = 0, which are globally Lipschitzian with the Lipschitz constants ξ i > 0, σ i , > 0, η i > 0 such that
Definition 2.1
Given β > 0. The system (2.1), where w(t) = 0, is β-stabilizable if there is a feedback control law u(t) = Kx(t) such that every solution x(t, ϕ) of the closed-loop system
satisfies
Definition 2.2
Let the numbers β > 0, γ > 0 be given. The H ∞ control problem for system (2.1) has a solution if there exists a memoryless state feedback controller u(t) = Kx(t) satisfying the following two requirements:
-
1.
The system (2.1) is β-stabilizable.
-
2.
There is a number c 0 > 0 such that
$$ \sup \frac{\int_0^\infty\|z(t)\|^2{\text{d}}t}{c_0\|\phi\|^2+ \int_0^\infty \|w(t)\|^2{\text{d}}t } \leq \gamma, $$(2.4)
where the supremum is taken over all \(\phi(t)\in C([-{\tau},0],R^n)\) and the nonzero uncertainty \(w(t)\in L_2([0,\infty),R^r).\) In this case, we say that the feedback H ∞ control u(t) = Kx(t) exponentially stabilizes the system.
The following lemmas are essential for the proofs in the subsequent section.
Proposition 2.1
Let P, Q be matrices of appropriate dimensions and Q is symmetric positive definite. Then
The proof of the above proposition is easily derived from completing the square:
Proposition 2.2
(Schur complement lemma [21]) Given constant symmetric matrices X, Y and Z, where Y > 0. Then X + Z T Y −1 Z < 0 if and only if
Proposition 2.3
(Razumikhin stability theorem [22]) Consider the time-delay system \(\dot x(t) = f(t,x_t), x(t) =\phi(t), t\in [-h,0].\) Assume that \(u, v, w: R^+\rightarrow R^+\) are nondecreasing, and u(s), v(s) are positive for s ≥ 0, v(0) = u(0) = 0, and q > 1. If there is a function \(V(t,x): R^+\times R^n\rightarrow R^+\) such that
-
1.
\(u(\|x\|) \leq V(t,x)\leq v(\|x\|), \; t\in R^+, x\in R^n\)
-
2.
\(\dot V(t,x(t)) \leq - w(\|x(t)\|) \hbox { if } V(t+s, x(t+s)) \leq qV(t,x(t)), \forall s\in [-h,0], t\in R^+,\)
then the zero solution of system is asymptotically stable.
3 Main results
Let us denote
Theorem 3.1
Assume the condition (H1). Given β > 0, the H ∞ control of system (2.1) has a solution if there exist a symmetric positive definite matrix \(P\in R^{n \times n},\) and two diagonal positive definite matrices \(D_i \in R^{n\times n}, i=1,2,\) such that the following LMIs hold:
The feedback H ∞ control law is defined by
Proof
Consider the following time-varying Lyapunov–Krasovskii functional for the closed-loop system (2.3):
where
It is easy to verify that
Taking the time derivative of V(·) in t along the solution we obtain
Using Proposition 2.1 for the estimation
we have
By adding and substituting into the right-hand side of the inequality (3.6) four items
and using the condition (2.2) and the diagonal matrices G > 0, H > 0, F > 0 for the following estimations
we have
Then, using the Schur complement lemma and Proposition 2.2, we obtain
Therefore, we obtain
where z(t) = [x(t), f(x(t))] and
Letting w(t) = 0, and noting that
and that N < 0 is, by Schur complement lemma, equivalent to \({\mathcal{N}} <0, \) where
we obtain from (3.2), (3.3) that
Therefore, from differential inequality (3.8), it follows that
Using the condition (3.5), we have
To complete the proof of the theorem, it remains to show the γ-optimal level condition (2.4). For this, we consider the relation
Since V(t, x t ) ≥ 0, t ≥ 0, we have
and hence
Observe that the value of \(\|z(t)\|^2\) is defined due to (2.1) and (3.1) as
Using Proposition 2.1, we have
then
Submitting the estimation of \(\dot V(t,x_t)\) and \(\|z(t)\|^2\) respectively defined from (3.7) and (3.10) into (3.9), we obtain
Applying Proposition 2.1 for the estimation
we have
equivalently,
Letting \(s\rightarrow \infty,\) and setting \(c_0 = \frac{\alpha_2}{\gamma} > 0,\) we obtain that
for all nonzero \(w(t)\in L_2([0,\infty),R^r),\) ϕ(t) ∈ C([ − h, 0], R n). This completes the proof of the theorem.
Remark 3.1
Theorem 3.1 provides sufficient conditions for solving the H ∞ control problem of the Hopfield delayed neural network (2.1) in terms of LMIs, which allows for an arbitrary prescribed stability degree β. The LMI feasibility will depend on parameters of the system under consideration as well as some upper limits for the Lipschitz constants and the time delays. The feasibility of the LMIs (3.2)–(3.3) can be tested by the reliable and efficient MATLABs LMI Control Toolbox [23].
In the sequel, the H ∞ control problem for the system (2.1) will be solved further with no restriction on the derivative of the time-varying delay function. For this, we set
Theorem 3.2
Assume the condition (H2). The H ∞ control of system (2.1) has a solution if there exist a symmetric positive definite matrix P and two diagonal positive definite matrices \(D_i \in R^{n\times n}, i=1,2,\) such that the following LMIs hold:
The feedback H ∞ control law is defined by
Proof
Let us set P τ = P + e −τ I n . We consider the following Lyapunov–Krasovskii functional
Taking the time derivative of V(·) in t along the solution and using the feedback control (3.13), we obtain
Using Proposition 2.1, we have
Then, we have
By adding and substituting into the right-hand side of the inequality (3.15) five items
and using the condition (2.2) and the diagonal matrices D 1 > 0, D 2 > 0, H > 0, F > 0 for the following estimations
we have
In the light of the Razumikhin theorem, Proposition 2.3, we assume that for any \(\epsilon > 0,\) such that
and using the condition (3.14), we have
Therefore, from (3.16) it follows that
Now letting \(\epsilon \rightarrow 0^+, \) and w(t) = 0 in (3.17), we obtain
where z(t) = [x(t), f(x(t))] and
Note that N < 0 is, by Schur complement lemma, equivalent to \({\mathcal{N}} <0,\) where
Since
the conditions (3.18) gives
and hence taking the conditions (3.11), (3.12) into account, there is α > 0 such that
Hence, the zero solution of the closed-loop system, by using the Razuminkhin-type stability theorem, Proposition 2.3, is asymptotically stable. The exponential estimation of the solution, as in the proof of Theorem 3.1, follows from the differential inequality
and hence
The condition (2.4) is proved by the same arguments used in Theorem 3.1. This completes the proof of the theorem.
Remark 3.2
Note that by using the Razumikhin stability theorem, only knowledge of the upper bound of the time-delay function is required in condition (H2) and no additional information of the delay is necessary, which is of particular interest for many practical processes. However, unlike the LMI conditions obtained in Theorem 3.1 that allow for an arbitrary prescribed stability degree β, the exponential rate of the system (2.1) obtained in Theorem 3.2 depends on the solution P of the LMIs (3.11) and (3.12) as well as on the time delay.
4 Numerical examples
Example 4.1
Consider the system (2.1) with the delay function \(\tau_1(t) =\sin^2(0.25t), \tau_2(t)=\cos^2(0.4t)\) and
Given β = 0.5, δ1 = 0.5, δ2 = 0.8, γ = 100, by using LMI toolbox of MATLAB, we have both the LMI (3.2), (3.3) feasible with
The feedback H ∞ control is defined by (3.4) as
and the solution x(t, ϕ) satisfies
Figure 1 shows the trajectories of solutions x 1(t) and x 2(t) of the closed-loop system (2.1) with the initial condition \(\phi(t) =(1, 0.2), t\in [1,0].\)
Example 4.2
Consider the system (2.1) with the time-delay functions
where \(\beta(t) = t, t\in [0, 0.5]; = -t+1, t\in (0.5,1]. \)
It is worth noting that the delay functions τ1(t),τ2(t) are bounded τ1 = 2, τ2 = 1, but non-differentiable, and therefore, the methods used in [11–14] are not applicable to this system. Given β = 0.5, γ = 1000, by using LMI toolbox of MATLAB, we have both the LMI (3.11), (3.12) feasible with
The feedback H ∞ control is defined by (3.13) as
Figure 2 shows the trajectories of solutions x 1(t) and x 2(t) of the closed-loop system (2.1) with the initial condition \(\phi(t) =(-1, 0.6), t\in [-2,0]. \)
5 Conclusion
The H ∞ control problem with exponential stability for neural networks with time-varying delays has been studied. Based on constructing a set of augmented Lyapunov–Krasovskii functionals, new sufficient conditions for H ∞ control have been established for two cases of time-varying delays: the delays are differentiable and have an upper bound of the delay-derivatives; and the delays are bounded but not necessary to be differentiable. The derived conditions are formulated in terms of LMIs. Upon the feasibility of the LMIs, all the control parameters can be easily computed and the design of a H ∞ controller can be accomplished.
References
Balasubramaniam P, Vembarasan V, Rakkiyappan R (2011) Global robust asymptotic stability analysis of uncertain switched Hopfield neural networks with time delay in the leakage term. Neural Comput Appl. doi:10.1007/s00521-011-0639-x
Hunt KJ, Sbarbaro D, Zbikowski R, Gawthrop PJ (1992) Neural networks for control systems: a survey. Automatica 28:1083–1112
Kao Y, Gao C, Han W (2010) Global exponential robust stability of reaction diffusion interval neural networks with continuously distributed delays. Neural Comput Appl 19:867–873
Singh V (2009) Modified criteriafor global robust stability of interval delay neural networks. Appl Math Comput 215:3124–3133
Liu F, Wu M, He Y, Yokoyama R (2011) Improved delay-dependent stability analysis for uncertain stochastic neural networks with time-varying delay. Neural Comput Appl 20:441–449
Phat VN, Trinh H (2010) Exponential stabilization of neural networks with various activation functions and mixed time-varying delays. IEEE Trans Neural Netw 21:1180–1185
Francis BA (1978) A course in H ∞ control theory. Springer, Berlin
Keulen van B (1993) H ∞ control for distributed parameter systems: a state-space approach. Birkhauser, Boston
Senthilkumar T, Balasubramaniam P (2011) Delay-dependent robust H ∞ control for uncertain stochastic T–S fuzzy systems with time-varying state and input delays. Int J Syst Sci 42:877–887
Senthilkumar T, Balasubramaniam P (2011) Robust H ∞ control for nonlinear uncertain stochastic T–S fuzzy systems with time-delays. Appl Math Lett 24:1986–1994
Senthilkumar T, Balasubramaniam P (2011) Delay-dependent robust stabilization and H ∞ control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays. J Optim Theory Appl 151:100–120
Park JH, Ji DH, Won SC, Lee SM (2008) H ∞ synchronization of time-delayed chaotic systems. Appl Math Comput 204:170–177
Park JH, Ji DH, Won SC, Lee SM, Choi SJ (2009) H ∞ control of Lur’e systems with sector and slope restricted nonlinearities. Phys Lett A 373:3734–3740
Lee SM, Ji DH, Kwon OM, Park JH (2011) Robust H ∞ filtering for a class of discrete-time nonlinear systems. Appl Math Comput 217:7991–7997
Niculescu SI (1998) H ∞ memoryless control with an α-stability constraint for time-delay systems: an LMI approach. IEEE Trans Autom Control 43:739–743
Phat VN, Vinh DQ, Bay NS (2008) L 2-stabilization and H ∞ control for linear non-autonomous time-delay systems in Hilbert spaces via Riccati equations. Adv Nonl Var Ineq 11:75–86
Souza de CE, Li X (1999) Delay-dependent robust H ∞ control of uncertain linear state-delayed systems. Automatica 35:1313–1321
Lin FJ, Lee TS, Lin CH (2001) Robust H ∞ controller design with recurrent neural network for linear synchronous motor drive. IEEE Trans Indus Electron 50:456–470
Liu M (2008) Robust H ∞ control for uncertain delayed nonlinear systems based on standard neural network models. Neurocomputing 71:3469–3492
Huang H, Feng G (2009) Delay-dependent H ∞ and generalized L 2 filtering for delayed neural networks. IEEE Trans Circ Syst I 56:846–857
Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in systems and control theory. SIAM, Philadenphia
Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New York
Gahinet P, Nemirovskii A, Laub AJ, Chilali M (1995) LMI control toolbox for use with MATLAB. The MathWorks, Inc, Natick
Acknowledgments
This work was supported by the National Foundation for Science and Technology Development, Vietnam and the Faculty Strategic Fund, Deakin University, Australia. The authors wish to thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Phat, V.N., Trinh, H. Design of H ∞ control of neural networks with time-varying delays. Neural Comput & Applic 22 (Suppl 1), 323–331 (2013). https://doi.org/10.1007/s00521-012-0820-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00521-012-0820-x