Shared Nonlinear Control in Wireless-Based Remote Stabilization: A Theoretical Approach

Takagi-Sugeno (T-S) fuzzy model is a system described by a set of fuzzy if-then rules which give local linear representations of the underlying nonlinear system. Usually, a fuzzy control law is constructed by a set of controllers for each local linear representation to carry out the control of a T-S fuzzy system. While, simultaneous control addresses the stability and performance of multiple plants under a single feedback controller. In this paper, simultaneous linear control for nonlinear uncertain systems described by general T-S fuzzy model is considered. A procedure for simultaneous control of T-S fuzzy systems with parametric uncertainties is presented, by which robust reliability method is adopted to deal with uncertainties. Simultaneously linear controller design for an uncertain T-S fuzzy system is carried out by solving a problem of robust reliability based optimization, in which, both the robustness with respect to uncertainties and the control cost can be taken into account simultaneously. It is demonstrated by numerical simulation of the chaotic Lorenz system control that the presented method is effective and feasible.

Nonlinearities and system uncertainties are the most important difficulties in designing controllers that ensure stability and acceptable closed-loop performance. Many significant results on the stability and robust control of uncertain nonlinear systems using the TS fuzzy model have been reported over the past decades, and considerable advances have been made (Connor et al. 1992; Chen et al. 1996; Rocha et al. 2005; Boukhezzar et al. 2006; Kamal et al. 2011, 2012a, 2013; Khanl et al. 2011). However, as stated in Khan and Hossain (2011), many approaches for stability and robust control of uncertain systems are often characterized by conservatism when dealing with uncertainties. It has been well known that the TS fuzzy model is very effective representation of complex nonlinear systems. In the TS fuzzy model, the state space of a nonlinear system is divided into different fuzzy regions with a local linear model being used in each region. The overall model output is obtained by defuzzification using the center of gravity (COG) method. Once the fuzzy model is obtained, control design can be carried out via the so-called parallel and distributed computing (PDC) approach, which employs multiple linear controllers corresponding to the locally linear plant models (Boukhezzar et al. 2006). This class of systems is described as a weighted sum of some simple linear subsystems, and thus is easily analyzable.

A robustness design of fuzzy control is proposed in this paper to overcome the effect of modeling error between nonlinear system and Takagi-Sugeno (T-S) fuzzy model. In terms of Lyapunov's direct method, a stability criterion is derived to guarantee the asymptotic stability of nonlinear interconnected systems. Based on the decentralized control scheme and this criterion, a set of model-based fuzzy controllers is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear interconnected systems. Finally, an example is given to illustrate the concepts discussed throughout the paper.

This paper is concerned with the problem of robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> control of nonlinear stochastic systems with time-varying interval delay. Firstly, a novel kind of fuzzy model, stochastic fuzzy hyperbolic model (SFHM), is proposed to represent a class of nonlinear stochastic systems with time-varying interval delay. The SFHM is a overall nonlinear stochastic model, which is much fitter for the nonlinear multivariable plant whose expert knowledge is difficult to find. The identification of SFHM is more convenient than that of stochastic T-S fuzzy model. Specifically, it also can be seen as a neural network model and we can learn the model parameters by the learning method of neural network. Secondly, delay-range-dependent and delay-derivative-range-dependent criteria on stability and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance are developed for the SFHM by using new Lyapunov-Krasovskii functionals and improved free-weighting matrix technique for the SFHM case. Finally, simulation results show the validity of the proposed method.

Fundamentals of the theory of non-linear pluse control systems

Delay-dependent exponential stabilization of nonlinear fuzzy impulsive systems with time-varying delay

This paper investigates the guaranteed cost control problem of uncertain non-linear systems with time-varying delays via the Takagi–Sugeno fuzzy model approach. Based on the Lyapunov–Krasovskii functional theory and the linear matrix inequality technique, a state feedback controller is proposed to stabilise the non-linear systems and minimise the guaranteed cost of the closed-loop systems. Both delay-dependent and delay-independent stability criteria are derived to guarantee the asymptotic stability of closed-loop systems. Furthermore, the supplementary requirement that the time-derivative of time-varying delays must be smaller than one is released for the proposed delay-dependent stability criterion. Two illustrative examples are provided to verify the validity of the results developed in this paper.

Note on exponential stability of certain nonlinear differential systems with time-varying delays

In this paper finite-time stabilization of nonlinear affine systems, is considered. First we provide a formula for a finite-time stabilizing state feedback control law, for the last class of system. Then we extend the result to the case of a norm-bounded uncertain system and a polytopic uncertain system stabilization are given. Some numerical examples are given to illustrate the main result.

This paper is concerned with fuzzy-model based stabilization of nonlinear networked control systems (NCSs) with time-varying transmission delays and transmission intervals based on a random-delay approach. The real-time distribution of input delays resulting from transmission delays and intervals is modeled as a dependent and nonidentically distributed process. Then a randomly switched Takagi-Sugeno fuzzy system with multiple input-delay subsystems is proposed to model the nonlinear NCSs. Based on an improved Lyapunov-Krasovskii method, which takes into account the real-time distribution of input delays in estimating cross-product integral terms, new sufficient conditions are deriveed for the exponential stability of the overall systems. The resulting controller design method is equivalent to a nonlinear convex optimization problem with LMI constraints. Numerical examples are presented to substantiate the effectiveness of our results.

This paper establishes an exponential H ∞ stabilisation method for uncertain nonlinear system with time-varying delay and external disturbance via non-fragile memory-based saturated controller. Based on the construction of an augmented Lyapunov–Krasovskii functional (LKF) and distributed-delay polytopic method, some delay-dependent and delay-derivation-dependent criteria to ensure the exponential H ∞ stabilisation of the nonlinear system are derived in terms of linear matrix inequalities (LMIs). Then, the optimisation problems with reduced conservatism concerning the admissible initial condition set are specifically formulated. Subsequently, simulation results are presented to demonstrate the validity and practicality of the suggested method.

Generalized sampling solutions for discontinuous stabilization of nonlinear systems

On observer-based fault detection for nonlinear systems

In this study, we strive to combine the advantages of fuzzy logic control (FLC), genetic algorithms (GA), H∞ tracking control schemes, smooth control and adaptive laws to design an adaptive fuzzy sliding model controller for the rapid and efficient stabilization of complex and nonlinear systems. First, we utilize a reference model and a fuzzy model (both involving FLC rules) to describe and well-approximate an uncertain, nonlinear plant. The FLC rules and the consequent parameter are decided on via GA. A boundary-layer function is introduced into these updated laws to cover modeling errors and to guarantee that the state errors converge into a specified error bound. After this, a H∞ tracking problem is characterized. We solve an eigenvalue problem (EVP), and derive a modified adaptive neural network controller (MANNC) to simultaneously stabilize and control the system and achieve H∞ control performance. Furthermore, a stability criterion is derived utilizing Lyapunov’s direct method to ensure the stability of the nonlinear system. Finally, the control methodology is demonstrated via a numerical simulation.

Event-triggered controller design of nonlinear discrete-time networked control systems in T-S fuzzy model