Robust exponential stabilization of positive uncertain switched neural networks with actuator saturation and sensor faults

Highlights

• A new yet general positive switched neural network model is established to fill the academic gap.

• The influence of sensor faults and actuator saturation on models is considered simultaneously. Besides, with interval uncertainties taken into consideration, a more comprehensive interval observer is constructed.

• The proposed method in this paper can be applied to the general nonlinear hybrid systems. Application of some novel techniques from the theory of nonlinear network cluster to positive uncertain switched systems yields a deep insight into the nonlinear dynamics under investigation.

Abstract

This article focuses on the robust exponential stabilization of positive uncertain switched neural networks subject to actuator saturation and sensor faults. Given the existence of interval uncertainty and the constraint concerning positivity of the original system, a new positive state-bounding observer is constructed to guarantee the coinstantaneous estimation of system state and sensor faults. To deal with actuator saturation, the convex hull scheme is employed. By designing the state-feedback controller and utilizing the multiple time-varying linear co-positive Lyapunov function, sufficient conditions for the robust exponential stability on the studied system are established under dwell-time switching. Furthermore, for optimizing the observer matrix, an iterative algorithm is developed. Eventually, a numerical example is exploited to illuminate the feasibility and effectiveness of both the deduced results and the proposed approaches.

Introduction

Positive systems that involve inherently nonnegative attribute variables such as the number of animal populations, absolute temperatures and substance concentrations are ubiquitous in the real world. Such systems whose states and outputs are always nonnegative in allusion to nonnegative initial conditions and inputs have numerous practical applications in domains like biology, communications and sociology [1], [2], [3]. When neural networks are devised to address multifarious issues concerning them, such as detection, estimation and optimization, relevant state and output vectors need to inherit positive constraints, which creates a class of positive neural networks. The adhibition of these networks exists in neural control, recognition, the implementation of monotone-regular behaviors and a variety of disciplines in the industries and academia [4], [5], [6]. Therefore, it is of realistic significance and value to investigate the stability of positive neural networks. Whereas, this issue has just attracted increasing research attention for the past few years, with still relatively few accomplishments reported in the literature [7], [8], [9]. At the same time, positive switched linear systems have caught widespread attention in many fields such as formation flight [10], [11], communication system [12], [13], turbofan engine model [14] and medical system [15]. What is notable is that although the primary characters of both positive neural networks and conventional linear switched systems have been well understood, to our best knowledge, no results on the stability of positive switched neural networks have yet been reported in the literature. This paper in these contexts will concentrate on studying the robust exponential stability of such neural networks for dwell-time switched signals.

On account of nonnegativity, the dynamic behavior on positive nominal neural networks presents remarkable characteristics, which can normally simplify performance analysis. There has been evidence that the $l_1$-gain is applicable to positive neural networks in terms of performance characterization and robustness [16]. However, on the one hand, different from the system defined in the whole space, the variables of positive neural networks are always limited within the first orthogonal cone. The design methods of traditional systems are often restricted in dealing with these positive systems, thereby bringing new challenges to the analysis and synthesis of positive neural networks. On the other hand, the system parameters in practice are usually uncertain due to some of inevitable external factors such as random fault, environmental noise and configuration aging. Spurred on by the above, some results have been reported [17], [18]. Taking [17] as example, an interval observer that can observe the upper and lower bounds of the system states is introduced, and the stabilization problem of the positive system is studied under the dynamic output feedback controller. In fact, it is the close association of both interval observers design and positive system theory that makes it vital and natural to design the observer gains for ensuring that the error dynamics between the observer and the original system is always positive. And until now, several methods have been developed for designing interval observers on uncertain positive systems, to mention a few, delay systems [18], time-varying systems [19], discrete-time systems [20] and nonlinear systems [21]. It is worth noting that the design of the interval observer (state-bounding observer) for the positive neural networks is still a blank in the academic circle.

In addition, both sensor faults caused by random interference in ground wire, signal line break or short line connection and actuator saturation generated by physical limitations or consideration of safety factors may inevitably lead to system performance degradation or closed-loop system instability in the actual control system. For one thing, in the existing literature [22], [23], [24], [25], [26], various fault estimation means are proposed by constructing iterative observers, robust observers, descriptor observers and sliding mode observers. Among all these schemes, the descriptor observer method is superior in multi-fault and/or interference estimation. The faults are decoupled from the system state by means of generalized augmented transformation, based on which descriptor observers are constructed with guaranteed performance to estimate the original system state and faults simultaneously [27], [28]. For another, the stability of positive switched systems with actuator saturation is studied in [14], [29] and [30]. Nevertheless, there is no literature dealing with both sensor faults and actuator saturation. Compared with the positive switched systems without both, it is difficult to find a workable framework for analysis. After all, due to the complex dynamic behavior of the positive switched systems, coupled with sensor faults and actuator saturation, the difficulty of analysis and control synthesis increases sharply. Therefore, theoretical challenges and technical shortcomings prompt us to explore the actual performance evaluation methods of positive switched systems with sensor faults and actuator saturation.

Motivated by the foregoing discussion, we investigate robust exponential stability for positive switched neural networks considering sensor faults and actuator saturation. The primary difficulties in designing positive observers and establishing such stabilization conditions rest with the inherent nonnegative constraint on state estimation and how to compelling “energy” of the overall network to decrease at switching moments, respectively. To cope with these difficulties, a positive state-bounding observer is constructed to concurrently estimate the system state and the faults, after which the multiple time-varying linear co-positive Lyapunov function is developed to acquire some sufficient conditions for the robust exponential stabilization under dwell-time switching. To conclude, the contributions of this article is mainly embodied in four aspects: (1) As an extension of [14], a new yet general positive switched neural network model is established to fill the academic gap. (2) For the first time, the influence of sensor faults and actuator saturation on models is considered simultaneously. Besides, with interval uncertainties taken into consideration, a more comprehensive interval observer is constructed. (3) Sufficient conditions for robust exponential stabilization are obtained with the aid of the multiple time-varying linear co-positive Lyapunov function under dwell-time switching, and the attraction domain of the system state is subsequently estimated. (4) The existence conditions of a desired observer are established in the form of linear programming which is computationally more efficient compared with linear matrix inequality techniques. Afterwards, the iterative optimization algorithm is performed to solve the optimized matrices of state-bounding observer.

The layout of the remaining sections is shown below: Section 2 furnishes notations and preparatory works about positive uncertain switched neural networks. Section 3 puts forward the theorems for robust exponential stabilization analysis, controller devise and positive state-bounding observer design. Moreover, an illustrative example is presents in Section 4. In the end, Section 5 draws a conclusion.