Discrete-time delayed standard neural network model and its application

The research on the theory and application of artificial neural networks has achieved a great success over the past two decades. Recently, increasing attention has been paid to recurrent neural networks, which are rich in dynamics, highly parallelizable, and easily implementable with VLSI. Due to these attractive features, RNNs have widely been applied to system identification, control, optimization and associative memories[1]. Stability analysis, which is critical to any applications of R…     

时滞标准神经网络模型及其应用

提出一种新的神经网络模型---时滞标准神经网络模型(DSNNM),它由线性动力学系统和有界静态时滞非线性算子连接而成.利用不同的Lyapunov泛函和S方法推导出DSNNM全局渐近稳定性和全局指数稳定性的充分条件,这些条件可表示为线性不等式(LMI)形式.大多数时滞(或非时滞)动态神经网络(DANN)稳定性分析或神经网络控制系统都可以转化为DSNNM,以便用统一的方法进行稳定性分析或镇定控制.从DSNNM应用于时滞联想记忆(BAM)神经网络的稳定性分析以及PH中和过程神经控制器的综合实例,可以看出,得到的稳定性判据扩展并改进了以往文献中的稳定性定理,而且可将稳定性分析推广到非线性控制系统的综合.     

Stability analysis of discrete-time recurrent neural networks based on standard neural network models

In order to conveniently analyze the stability of various discrete-time recurrent neural networks (RNNs), including bidirectional associative memory, Hopfield, cellular neural network, Cohen-Grossberg neural network, and recurrent multiplayer perceptrons, etc., the novel neural network model, named standard neural network model (SNNM) is advanced to describe this class of discrete-time RNNs. The SNNM is the interconnection of a linear dynamic system and a bounded static nonlinear operator. By combining Lyapunov functional with S-Procedure, some useful criteria of global asymptotic stability for the discrete-time SNNMs are derived, whose conditions are formulated as linear matrix inequalities. Most delayed (or non-delayed) RNNs can be transformed into the SNNMs to be stability analyzed in a unified way. Some application examples of the SNNMs to the stability analysis of the discrete-time RNNs shows that the SNNMs make the stability conditions of the RNNs easily verified.     

Unified stabilizing controller synthesis approach for discrete-time intelligent systems with time delays by dynamic output feedback

A novel model, termed the standard neural network model (SNNM), is advanced to describe some delayed (or non-delayed) discrete-time intelligent systems com- posed of neural networks and Takagi and Sugeno (T-S) fuzzy models. The SNNM is composed of a discrete-time linear dynamic system and a bounded static nonlinear operator. Based on the global asymptotic stability analysis of the SNNMs, linear and nonlinear dynamic output feedback controllers are designed for the SNNMs to stabilize the closed-loop systems, respectively. The control design equations are shown to be a set of linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms to determine the control signals. Most neural-network-based (or fuzzy) discrete-time intelligent systems with time delays or without time delays can be transformed into the SNNMs for controller synthesis in a unified way. Three application examples show that the SNNMs not only make controller synthesis of neural-network-based (or fuzzy) discrete-time intelligent systems much easier, but also provide a new approach to the synthesis of the controllers for the other type of nonlinear systems.     

LMI designmethod for networked-based PID control

In this paper, we propose a methodology for the design of networked PID controllers for second-order delayed processes using linear matrix inequalities. The proposed procedure takes into account time-varying delay on the plant, time-varying delays induced by the network and packed dropouts. The design is carried on entirely using a continuous-time model of the closed-loop system where time-varying delays are used to represent sampling and holding occurring in a discrete-time digital PID controller.     

A New Method for Stabilization of Networked Control Systems With Random Delays

We consider the stabilization problem for a class of networked control systems in the discrete-time domain with random delays. The sensor-to-controller and controller-to-actuator delays are modeled as two Markov chains, and the resulting closed-loop systems are jump linear systems with two modes. The necessary and sufficient conditions on the existence of stabilizing controllers are established. It is shown that state-feedback gains are mode-dependent. An iterative linear matrix inequality (LMI) approach is employed to calculate the state-feedback gains.     

Controller synthesis for networked control systems

This paper presents a discrete-time model for networked control systems (NCSs) that incorporates all network phenomena: time-varying sampling intervals, packet dropouts and time-varying delays that may be both smaller and larger than the sampling interval. Based on this model, constructive LMI conditions for controller synthesis are derived, such that stabilizing state-feedback controllers can be designed. Besides the proposed controller synthesis conditions a comparison is made between the use of parameter-dependent Lyapunov functions and Lyapunov-Krasovskii functions for stability analysis. Several examples illustrate the effectiveness of the developed theory.     

Hybrid state-space fuzzy model-based controller with dual-ratesampling for digital control of chaotic systems

We develop a hybrid state-space fuzzy model-based controller with dual-rate sampling for digital control of chaotic systems. A Takagi-Sugeno (TS) fuzzy model is used to model the chaotic dynamic system and the extended parallel-distributed compensation technique is proposed and formulated for designing the fuzzy model-based controller under stability conditions. The optimal regional-pole assignment technique is also adopted in the design of the local feedback controllers for the multiple TS linear state-space models. The proposed design procedure is as follows: an equivalent fast-rate discrete-time state-space model of the continuous-time system is first constructed by using fuzzy inference systems. To obtain the continuous-time optimal state-feedback gains, the constructed discrete-time fuzzy system is then converted into a continuous-time system. The developed optimal continuous-time control law is finally converted into an equivalent slow-rate digital control law using the proposed intelligent digital redesign method. The main contribution of the paper is the development of a systematic and effective framework for fuzzy model-based controller design with dual-rate sampling for digital control of complex such as chaotic systems. The effectiveness and the feasibility of the proposed controller design method is demonstrated through numerical simulations on the chaotic Chua circuit     

Robust predictor feedback for discrete-time systems with input delays

This work studies the design problem of feedback stabilisers for discrete-time systems with input delays. A backstepping procedure is proposed for disturbance-free discrete-time systems. The feedback law designed by using backstepping coincides with the predictor-based feedback law used in continuous-time systems with input delays. However, simple examples demonstrate that the sensitivity of the closed-loop system with respect to modelling errors increases as the value of the delay increases. The paper proposes a Lyapunov redesign procedure that can minimise the effect of the uncertainty. Specific results are provided for linear single-input discrete-time systems with multiplicative uncertainty. The feedback law that guarantees robust global exponential stability is a nonlinear, homogeneous of degree 1 feedback law.     

Stability Analysis of Discrete-Time Recurrent Neural Networks With Stochastic Delay

This paper is concerned with the stability analysis of discrete-time recurrent neural networks (RNNs) with time delays as random variables drawn from some probability distribution. By introducing the variation probability of the time delay, a common delayed discrete-time RNN system is transformed into one with stochastic parameters. Improved conditions for the mean square stability of these systems are obtained by employing new Lyapunov functions and novel techniques are used to achieve delay dependence. The merit of the proposed conditions lies in its reduced conservatism, which is made possible by considering not only the range of the time delays, but also the variation probability distribution. A numerical example is provided to show the advantages of the proposed conditions.      

Stable adaptive neurocontrol for nonlinear discrete-time systems

This paper presents a novel approach in designing neural network based adaptive controllers for a class of nonlinear discrete-time systems. This type of controllers has its simplicity in parallelism to linear generalized minimum variance (GMV) controller design and efficiency to deal with complex nonlinear dynamics. A recurrent neural network is introduced as a bridge to compensation simplify controller design procedure and efficiently to deal with nonlinearity. The network weight adaptation law is derived from Lyapunov stability analysis and the connection between convergence of the network weight and the reconstruction error of the network is established. A theorem is presented for the conditions of the stability of the closed-loop systems. Two simulation examples are provided to demonstrate the efficiency of the approach.     

Fuzzy combination of fuzzy and switching state-feedback controllers for nonlinear systems subject to parameter uncertainties.

This paper presents a fuzzy controller, which involves a fuzzy combination of local fuzzy and global switching state-feedback controllers, for nonlinear systems subject to parameter uncertainties with known bounds. The nonlinear system is represented by a fuzzy combined Takagi-Sugeno-Kang model, which is a fuzzy combination of the global and local fuzzy plant models. By combining the local fuzzy and global switching state-feedback controllers using fuzzy logic techniques, the advantages of both controllers can be retained and the undesirable chattering effect introduced by the global switching state-feedback controller can be eliminated. The steady-state error introduced by the global switching state-feedback controller when a saturation function is used can also be removed. Stability conditions, which are related to the system matrices of the local and global closed-loop systems, are derived to guarantee the closed-loop system stability. An application example will be given to demonstrate the merits of the proposed approach.     

Infinite horizon LQR for systems with multiple delays in a single input channel

This paper is concerned with the linear quadratic regulation (LQR) problem for both linear discrete-time systems and linear continuous-time systems with multiple delays in a single input channel. Our solution is given in terms of the solution to a two-dimensional Riccati difference equation for the discrete-time case and a Riccati partial differential equation for the continuous-time case. The conditions for convergence and stability are provided.     

A new neural network-based approach for self-tuning control of nonlinear SISO discrete-time systems

This article presents a new neural network-based approach for self-tuning control of nonlinear single-input single-output (SISO) discrete-time dynamic systems. According to the approach, a neural network ARMAX (NN-ARMAX) model of the system is identified and continuously updated, using an online training algorithm. Control design is accomplished by solving an optimal discrete-time linear quadratic tracking problem using an observer-type linear state-space Kalman innovation model, which is built from the parameters of a local linear version of the NN-ARMAX model. The state-feedback control law is implemented using the Kalman state, which is calculated without estimating the noise covariance properties. The proposed control approach is shown to be very effective and outperforms the self-tuning control approach based on a linear ARMAX model on two simulation examples.     

Linear functional observers for systems with delays in State variables: the discrete-time case

This note extends to the discrete-time case the design of linear functional state observers, recently developed for continuous-time delay systems. Sufficient conditions for the stability dependent of delays and stability independent of delays are derived using linear and bilinear matrix inequalities [(LMIs) and (BMIs)] formulations.     

H2-optimal sampled-data control

The H₂-optimal control of continuous-time linear time-invariant systems by sampled-data controllers is discussed. Two different solutions, state space and operator theoretic, are given. In both cases, the H₂ sampled-data problem is shown to be equivalent to a certain discrete-time H₂ problem. Other topics discussed include input-output stability of sampled-data systems, performance recovery in digital implementation of analog controllers, and sampled-data control of systems with the possibility of multiple-time delays     

Matrosov theorem for parameterized families of discrete-time systems

A version of Matrosov's theorem for parameterized discrete-time time-varying systems is presented. The theorem is a discrete-time version of the continuous-time result in Loria et al., 2002 (δ-persistency of excitation: a necessary and sufficient condition for uniform attractivity, 2002, submitted for publication). Our result facilitates controller design for sampled-data nonlinear systems via their approximate discrete-time models. An application of the theorem to establishing uniform asymptotic stability of systems controlled by model reference adaptive controllers designed via approximate discrete-time plant models is presented.     

Stability analysis of discrete-time recurrent neural networks

We address the problem of global Lyapunov stability of discrete-time recurrent neural networks (RNNs) in the unforced (unperturbed) setting. It is assumed that network weights are fixed to some values, for example, those attained after training. Based on classical results of the theory of absolute stability, we propose a new approach for the stability analysis of RNNs with sector-type monotone nonlinearities and nonzero biases. We devise a simple state-space transformation to convert the original RNN equations to a form suitable for our stability analysis. We then present appropriate linear matrix inequalities (LMIs) to be solved to determine whether the system under study is globally exponentially stable. Unlike previous treatments, our approach readily permits one to account for non-zero biases usually present in RNNs for improved approximation capabilities. We show how recent results of others on the stability analysis of RNNs can be interpreted as special cases within our approach. We illustrate how to use our approach with examples. Though illustrated on the stability analysis of recurrent multilayer perceptrons, the approach proposed can also be applied to other forms of time-lagged RNNs.     

Robust output-feedback control of linear discrete-time systems

The problem of designing H_∞ dynamic output-feedback controllers for linear discrete-time systems with polytopic type parameter uncertainties is considered. Given a transfer function matrix of a system with uncertain real parameters that reside in some known ranges, an appropriate, not necessarily minimal, state-space model of the system is described which permits reconstruction of all its states via the delayed inputs and outputs of the plant. The resulting model incorporates the uncertain parameters of the transfer function matrix in the state-space matrices. A recently developed linear parameter-dependent LMI approach to state-feedback H_∞ control of uncertain polytopic systems is then used to design a robust output-feedback controllers that are of order comparable to the one of the plant. These controllers ensure the stability and guarantee a prescribed performance level within the uncertainty polytope.     

Multiple models switching control based on recurrent neural networks

This paper presents a novel approach in designing adaptive controller to improve the transient performance for a class of nonlinear discrete-time systems under different operating modes. The proposed scheme consists of generalized minimum variance (GMV) controllers and a compensating controller. GMV controllers are based on the known nominal linear multiple models, while the compensating controller is based upon a recurrent neural network. The adaptation law of network weight is derived from Lyapunov stability theory. A suitable switching control strategy is applied to choose the best controller by the performance indices at every sampling instant. Simulations are discussed in order to illustrate the merits of the proposed method.