Mixed H2/H∞ stabilization design for memristive neural networks

doi:10.1016/j.neucom.2019.07.002

Neurocomputing

Volume 361, 7 October 2019, Pages 92-99

https://doi.org/10.1016/j.neucom.2019.07.002 Get rights and content

Abstract

This study considers the multiobjective stabilization design problem for memristive neural networks (MNNs). Initially, by using a set of logical switched functions, the original MNN is transformed into another model which is easy to be dealt with. Then, based on the transformed model and using the Lyapunov direct method, a mixed H₂/H_∞ control design is developed in the form of linear matrix inequalities (LMIs), such that the closed-loop MNN is exponentially stable and an H₂ performance bound is given while providing a prescribed H_∞ performance of disturbance attenuation. Furthermore, via the existing LMI optimization technique, a suboptimal mixed H₂/H_∞ controller can be constructed in the sense of making the H₂ performance bound as small as possible. Finally, numerical simulations exhibit the feasibility and validity of the proposed design method.

Introduction

As the fourth basic element of electrical circuits, memristor was postulated in [1] and fabricated by the Hewlett-Packard Company [2]. The unparalleled properties of the memristor such as nonvolatility characteristic, “pinched hysteresis loop” in voltage-current plane, and nanometer dimension, make this device have the exclusive capability to remember the past quantity of electric charge and be treated as the promising circuit device for imitating human brain [3], [4], [5], [6]. Therefore, by replacing the resistors, memrisors were applied to formulate the memristive neural networks (MNNs) [7], [8], [9].

Because of the strong information storage capacity and computation power, the MNNs were applied in many fields, such as associative memory [10], [11], image processing [12], robotic manipulator control implementation [13], and secure communication [14], [15]. Over the past decade, the dynamical behavior analysis and synthesis for MNNs have been studied extensively, which are dramatically related to the successful applications [16], [17], [18], [19], [20], [21], [22], [23], [24]. Moreover, a necessary condition for the proper operation of a neural network is that it is stable within the dynamic range of prescribed inputs. For example, in neural networks of associative memories, locally stable equilibria store information and form distributed memory structures [25]. The stabilization problem is closely related to the stability issue and the medical treatment of neuropathy patients in biological neural networks is a typical application for stabilization of neural networks [25]. Therefore, there has been much work investigating the stability and stabilization for MNNs. In [26], the exponential stability analysis of MNNs with time-varying delays was studied in the sense of Filippov. Utilizing the differential inclusion theory, the exponential stabilization design for MNNs was firstly investigated while the guaranteed quadratic performance was also considered [27]. Afterwards, the chaotic MNNs were stabilized by designing the intermittent controller [28] and the finite-time stability or stabilization problem for MNNs was considered via the finite-time stability theorem [29], [30], [31]. In [32] and [33], the stabilization for Takagi-Sugeno fuzzy MNNs and the exponential stability of MNNs with mixed time-varying delays were investigated, respectively. Moreover, the characteristic function and convex combination methods were employed to handle the exponential stability and stabilization of MNNs [34] as well as the robust analysis technique was used to address the exponential stabilization issue for MNNs under saturation sampled-data control [35]. Recently, the global asymptotic stability and stabilization of MNNs with communication delays have been studied by means of event-triggered sampling control [36].

On the other hand, the external disturbance for MNNs such as electromagnetic interference is ubiquitous and inevitable in the stabilization design. When the external disturbance exists, it usually degrades the performance and may even cause instability of MNNs. Then, it is very necessary to suppress the disturbance effect in MNNs. More recently, the constrained state feedback stabilization design and the constrained observer-based stabilization design for MNNs with H_∞ performance have been studied in [37] and [38], respectively. In addition, little attention has been paid to the guaranteed H₂ performance of MNNs with exception [27]. However, in practice, the designed controller needs to guarantee that a system is stable while having both H_∞ and H₂ performance [39], [40], [41], [42]. As we know, despite the previous efforts, such a multiobjective stabilization design of MNNs has not been dealt with yet, which motivates the present study.

In this article, a multiobjective stabilization design approach will be proposed for MNNs. Based on a tractable model for the MNN, a mixed H₂/H_∞ control design is developed in the form of linear matrix inequalities (LMIs), such that the closed-loop MNN is exponentially stable and an H₂ performance bound is given while providing a prescribed H_∞ disturbance attenuation performance. Further, a suboptimal controller can also be constructed by making the H₂ performance bound as small as possible utilizing the LMI optimization technique [43], [44]. Finally, one numerical example is provided to illustrate the feasibility and validity of the proposed method.

The main contribution of this paper is that the multiobjective stabilization design problem for MNNs including H_∞ and H₂ performance is considered for the first time. For the sake of specifying the mixed performance objectives of closed-loop MNN with a less conservatism, two different Lyapunov functions and the improved formulation in Lemma 2.3 are adopted.

Notations: $R^{m \times n},$ $R^{n}$ and $R$ represent the set of all m × n real matrices, the n-dimensional Euclidean space and the set of real numbers, respectively. ‖ · ‖ denotes the standard Euclidean norm and ${∥ x ∥}_{M}^{2}$ stands for x^TMx. The symbol * in some matrix indicates an ellipsis for symmetry terms and the superscript T is the transpose of a matrix or a vector. λ_max( · ) and λ_min( · ) denote the maximal and minimal eigenvalues of a matrix, respectively. ⊗ denotes the Kronecker product. For a matrix $Q = Q^{T},$ Q < ( ≤ ,  > ,  ≥ ) 0 indicates it is negative definite (negative semidefinite, positive definite, positive semidefinite, respectively).

Section snippets

Problem formulation and preliminaries

This study considers the following MNN: $\dot{x} (t) = - D x (t) + A (x (t)) g (x (t)) + E u (t) + F w (t)$ where $x (t) \in R^{n}$ is the state; $g (x (t)) ≜ {[g_{1} (x_{1} (t)) g_{2} (x_{2} (t)) \dots g_{n} (x_{n} (t))]}^{T} \in R^{n}$ indicates the neuron activation function; $D ≜ diag {d_{1}, d_{2}, \dots, d_{n}} > 0$ and A(x(t))≜(a_ij(x_i(t)))_n × n denote the connection weight matrix; $u (t) \in R^{m}$ is the control input; $w (t) \in R^{s}$ is the exogenous disturbance satisfying $\int_{0}^{\infty} {∥ w (t) ∥}^{2} d t < \infty$ ; $E \in R^{n \times m}$ and $F \in R^{n \times s}$ are the known constant matrices. For each $j \in S ≜ {1, 2, \dots, n},$ the neuron activation function g_j( · ) with $g_{j} (0) = 0$

Mixed H₂/H_∞ stabilization design

In this section, for the MNN (1), via introducing the two different Lyapunov function candidates, an LMI-based mixed H₂/H_∞ stabilization design method is proposed and a suboptimal controller is also constructed by means of addressing an LMI optimization problem.

In the first place, given some control law (6), a sufficient condition is given to achieve the H_∞ performance (7) for the MNN (1).

Theorem 3.1

For some given scalar γ > 0, if there exists a matrix $X_{1} > 0 \in R^{n \times n}$ and diagonal matrices ${\bar{Λ}}_{1} > 0 \in R^{n \times n},$ ${\bar{M}}_{1} > 0 \in R^{n^{2} \times}$

Numerical simulations

In the following, one will consider the three-neuron MNN (1) with $\begin{matrix} D & = & 1.5 I_{3}, E = [\begin{matrix} 1 & 0 \\ - 0.5 & 1 \\ 1.5 & 2 \end{matrix}], \\ F & = & [\begin{matrix} 0.1 \\ 0.2 \\ 0.1 \end{matrix}], g_{1} (s) = g_{2} (s) = g_{3} (s) = \tanh (s) \end{matrix}$ and $\begin{matrix} A (x (t)) = [\begin{matrix} a_{11} (x_{1} (t)) & a_{12} (x_{1} (t)) & 0 \\ a_{21} (x_{2} (t)) & a_{22} (x_{2} (t)) & a_{23} (x_{2} (t)) \\ 0 & a_{32} (x_{3} (t)) & 1.3 \end{matrix}] \end{matrix}$ where $\begin{matrix} a_{11} (x_{1} (t)) & = & {\begin{matrix} 1, | x_{1} (t) | ⩽ 1 \\ 1.5, | x_{1} (t) | > 1 \end{matrix} \\ a_{12} (x_{1} (t)) & = & {\begin{matrix} 1.4, | x_{1} (t) | ⩽ 1 \\ 1.7, | x_{1} (t) | > 1 \end{matrix} \\ a_{21} (x_{2} (t)) & = & {\begin{matrix} - 3, | x_{2} (t) | ⩽ 1 \\ - 3.2, | x_{2} (t) | > 1 \end{matrix} \\ a_{22} (x_{2} (t)) & = & {\begin{matrix} 2, | x_{2} (t) | ⩽ 1 \\ 1.6, | x_{2} (t) | > 1 \end{matrix} \\ a_{23} (x_{2} (t)) & = & {\begin{matrix} - 1.3, | x_{2} (t) | ⩽ 1 \\ - 2.2, | x_{2} (t) | > 1 \end{matrix} \\ a_{32} (x_{3} (t)) & = & {\begin{matrix} 1.8, | x_{3} (t) | ⩽ 1 \\ 1.4, | x_{3} (t) | > 1 . \end{matrix} \end{matrix}$ Hence, it is clear that $L = I_{3}$ and $\begin{matrix} A_{0} = [\begin{matrix} 1.25 & 1.55 & 0 \\ - 3.1 & 1.8 & - 1.75 \\ 0 & 1.6 \end{matrix} \end{matrix}$

Conclusions

This study has dealt with the multiobjective stabilization design problem for MNNs. Initially, by introducing a set of logical switched functions, the original MNN is transformed into another model which is easy to be handled. Based on the transformed model, via the two different Lyapunov function candidates, a mixed H₂/H_∞ stabilization design is then developed in the form of LMIs, where an H₂ performance bound and a prescribed H_∞ disturbance attenuation performance are both provided. Finally,

Declarations of interest

The authors declare that they have no conflict of interest.

Acknowledgments

This work was supported in part by the National Natural Science Foundation for Distinguished Young Scholars of China under Grant 61625302, and in part by the National Natural Science Foundation of China under Grants 61473011 and 61721091. The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions which have improved the presentation of this paper.

Xiao-Wei Zhang received the B.S. degree in mathematics and applied mathematics from Handan College, Handan, China and the M.S. degree in operations research and cybernetics from Yanshan University, Qinhuangdao, China, in 2012 and 2015, respectively. He is currently pursuing the Ph.D. degree in control theory and control engineering from Beihang University (Beijing University of Aeronautics and Astronautics), Beijing, China.

His current research interests include stability of neural networks,

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His current research interests include stability of neural networks, distributed parameter systems, and fuzzy modeling and control.

Huai-Ning Wu received the B.E. degree in automation from Shandong Institute of Building Materials Industry, Jinan, China and the Ph.D. degree in control theory and control engineering from Xi’an Jiaotong University, Xi’an, China, in 1992 and 1997, respectively.From August 1997 to July 1999, he was a Post-doctoral Research Fellow with the Department of Electronic Engineering at Beijing Institute of Technology, Beijing, China. Since August 1999, he has been with the School of Automation Science and Electrical Engineering, Beihang University (formerly Beijing University of Aeronautics and Astronautics), Beijing. From December 2005 to May 2006, he was a Senior Research Associate with the City University of Hong Kong (CityU), Kowloon, Hong Kong. From October to December during 2006–2008 and from July to August in 2010, 2011 and 2013, he was a Research Fellow with CityU. He is currently a Professor with Beihang University and a Distinguished Professor of Yangtze River Scholar with the Ministry of Education of China. His current research interests include robust control, fault-tolerant control, distributed parameter systems, and fuzzy/neural modeling and control.

Dr. Wu was a recipient of the China National Funds for Distinguished Young Scientists. He serves as an Associate Editor of the IEEE Transactions on Fuzzy Systems, and the IEEE Transactions on Systems, Man & Cybernetics: Systems.

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Mixed H2/H∞ stabilization design for memristive neural networks

Abstract

Introduction

Section snippets

Problem formulation and preliminaries