H2 guaranteed cost fuzzy control design for discrete-time nonlinear systems with parameter uncertainty

This paper presents a design method of H₂ guaranteed cost (GC) fuzzy controllers for discrete-time nonlinear systems with parameter uncertainties. The Takagi and Sugeno (T-S) fuzzy model with parameter uncertainties is employed to represent an uncertain discrete-time nonlinear system. A sufficient condition for the existence of H₂ GC fuzzy controllers is presented in terms of linear matrix inequalities (LMIs). The resulting fuzzy controllers not only guarantee that the closed-loop fuzzy system is quadratically stable, but also provide a guaranteed cost on the H₂ performance index. Furthermore, an optimal H₂ GC fuzzy controller in the sense of minimizing a bound on the guaranteed cost is provided by means of an LMI optimization procedure. Finally, it is also demonstrated, through numerical simulations on the backing up control of a truck-trailer, that the proposed design method is effective.     

A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems

This paper presents a systematic approach for decreasing conservativeness in stability analysis and control design for Takagi-Sugeno (TS) systems. This approach is based on the idea of multiple Lyapunov functions together with simple techniques for introducing slack matrices. Unlike some previous approaches based on multiple Lyapunov functions, both the stability and the stabilization conditions are written as linear matrix inequality (LMI) problems. The proposed approach reduces the number of inequalities and guarantees extra degrees of freedom to the LMI problems. Numeric examples illustrate the effectiveness of this method.     

Stabilization of a class of discrete-time switched systems via observer-based output feedback

In this paper, observer-based static output feedback control problem for discrete-time uncertain switched systems is investigated under an arbitrary switching rule. The main method used in this note is combining switched Lyapunov function (SLF) method with Finsler’s Lemma. Based on linear matrix inequality (LMI) a less conservative stability condition is established and this condition allows extra degree of freedom for stability analysis. Finally, a simulation example is given to illustrate the efficiency of the result.     

Static output-feedback stabilization of discrete-time Markovian jump linear systems: A system augmentation approach

This paper studies the static output-feedback (SOF) stabilization problem for discrete-time Markovian jump systems from a novel perspective. The closed-loop system is represented in a system augmentation form, in which input and gain-output matrices are separated. By virtue of the system augmentation, a novel necessary and sufficient condition for the existence of desired controllers is established in terms of a set of nonlinear matrix inequalities, which possess a monotonic structure for a linearized computation, and a convergent iteration algorithm is given to solve such inequalities. In addition, a special property of the feasible solutions enables one to further improve the solvability via a simple D-K type optimization on the initial values. An extension to mode-independent SOF stabilization is provided as well. Compared with some existing approaches to SOF synthesis, the proposed one has several advantages that make it specific for Markovian jump systems. The effectiveness and merit of the theoretical results are shown through some numerical examples.     

A new LMI condition for robust stability of discrete-time uncertain systems

In this paper, a new robust stability condition is derived for uncertain discrete-time linear systems with convex polytopic uncertainties. The condition is expressed in terms of a set of linear matrix inequalities (LMIs) involving only the vertices of the polytope domain. It enables us to determine robust stability of uncertain systems easily by solving some LMIs. A rigorous proof is given to show that an interesting result appeared recently is a special case of the proposed condition. Numerical examples also demonstrate the merit of the present condition in the aspect of conservativeness over other results in the literature.     

Relaxed stabilization conditions for continuous-time Takagi-Sugeno fuzzy control systems

This paper deals with the stabilization of continuous-time Takagi-Sugeno (T-S) fuzzy control systems. Based on fuzzy Lyapunov functions and nonparallel distributed compensation (non-PDC) control laws, new stabilization conditions are represented in the form of linear matrix inequalities (LMIs). The theoretical proof shows that the proposed conditions can provide less conservatism than the existing results in the literature. Moreover, in order to demonstrate the effectiveness of the non-PDC control laws, the problem of H_∞ controller design for T-S fuzzy systems is also studied. Simulation examples are given to illustrate the merits of the proposed methods.     

An improved robust stability and robust stabilization method for linear discrete-time uncertain systems

This paper addresses the problems of the robust stability and robust stabilization of a discrete-time system with polytopic uncertainties. A new and simple method is presented to directly decouple the Lyapunov matrix and the system dynamic matrix. Combining this method with the parameter-dependent Lyapunov function approach yields new criteria that include some existing ones as special cases. A numerical example illustrates the improvement over the existing ones.     

LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno's form

This paper presents the stabilization analysis for a class of nonlinear systems that are represented by a Takagi and Sugeno (TS) discrete fuzzy model (Takagi and Sugeno IEEE Trans. Systems Man Cybern. 15(1)(1985)116). The main result given here concerns their stabilization using new control laws and new nonquadratic Lyapunov functions. New relaxed conditions and linear matrix inequality-based design are proposed that allow outperforming previous results found in the literature. Two examples are also provided to demonstrate the efficiency of the approaches.     

Robust stabilization of linear discrete-time systems with time-varying input delay

This paper presents a sufficient condition to stabilize linear discrete-time systems with time-varying input delay and model uncertainties. The key idea consists of applying Artstein’s reduction method and the scaled bounded real lemma; the robust stabilization problem is cast as a set of linear matrix inequalities on a transformed delay-free system. Hence, a novel convex characterization of the problem, an alternative to bilinear matrix inequalities in prior literature, is provided. Predictor-based controllers and robust state-feedback stabilization are particular cases of the proposal. Furthermore, numerical examples show that this proposal may improve tolerance against delay variations when compared to other methods available in literature.     

LMI conditions for quadratic and robust D-stability of interval systems

Sufficient conditions for the quadratic D-stability and further robust D-stability of interval systems are presented in this paper. This robust D-stability condition is based on a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities （LMIs） defined at a series of partial-vertex-based interval matrices other than the total vertex matrices as in previous results. The results contain the usual quadratic and robust stability of continuous-time and discrete-time interval systems as particular cases. The illustrative example shows that this method is effective and less conservative for checking the quadratic and robust D-stability of interval systems.     

Reducing conservativeness in recent stability conditions of TS fuzzy systems

In this correspondence a new simple strategy for reducing the conservativeness in stability analysis of continuous-time Takagi-Sugeno fuzzy systems based on fuzzy Lyapunov functions is proposed. This new strategy generalizes previous results.     

Special decentralized control problems in discrete-time interconnected systems composed of two subsystems

In this paper, some special decentralized control problems are addressed for discrete-time interconnected systems. First it is pointed out that some subsystems must be unstable to ensure stability of the overall system in some special cases. Then a special kind of decentralized control problem is studied. This kind of problem can be viewed as harmonic control among independent subsystems. Research results show that two unstable systems can generate a stable system through some effective cooperations. In addition, a decentralized controller design method based on linear matrix inequality is also given by using parameter-dependent Lyapunov function method developed for the study of robust stability. A special linear star coupled dynamical network is also considered. The central subsystem must be unstable to stabilize the whole network under a special coupling. Several examples are given to illustrate the results.     

A less conservative LMI condition for the robust stability of discrete-time uncertain systems

In this paper, a less conservative condition for the robust stability of uncertain discrete-time linear systems is proposed. The uncertain parameters, assumed to be time-invariant, are supposed to belong to convex bounded domains (polytope type uncertainty). The stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain. A simple and numerically efficient feasibility test provides a set of Lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain. Examples illustrate the results.     

Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.     

Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno's form

This paper further studies stabilization of nonlinear systems represented by a Takagi-Sugeno (T-S) discrete fuzzy model. By extending a nonquadratic Lyapunov function and applying a nonparallel distributed compensation (non-PDC) law, a new stabilization conclusion is presented that is a generalization of some previous results in the literature. The new conclusion under a special situation is also suitable for a PDC law, which presents to be more relaxed than some existing results. An example is provided to demonstrate the effectiveness of the new conclusions.     

Stabilization for state/input delay systems via static and integral output feedback

This paper addresses the static and integral output feedback stabilization problems of continuous-time linear systems with an unknown state/input delay. By combining an augmentation approach and the delay partitioning technique, criteria for static and integral output feedback stabilizability are proposed in terms of nonlinear matrix inequalities with a free parameter matrix introduced. These new characterizations possess a special structure, which leads to linearized iterative computation. The effectiveness and merits of the proposed approach are shown through numerical examples.     

New results on stabilization of Markovian jump systems with time delay

This paper studies the problem of stochastic stabilization for a class of Markovian jump systems with time delay. A new delay-dependent stochastic stability criterion on the stochastic stability of the system is derived based on a novel Lyapunov-Krasovskii functional (LKF) approach. The equivalence and superiority to existing results are demonstrated. Then a state feedback controller, which guarantees the stochastic stability of the closed-loop system, is designed. Illustrative examples are provided to show the reduced conservatism and effectiveness of the proposed techniques.     

Dwell-time approach to input-output stability properties for a class of discrete-time dynamical systems

We consider a class of discrete-time dynamical systems that consist of switching of many linear time-invariant systems, and it can be used to cover a sampled-data version of Switched Linear Systems, which have been widely studied in the literature. For this class of dynamical systems, we establish the LMI (Linear Matrix Inequality) formulation of analyzing their input-output stability properties, including both ?-2 stability and passivity, by constraining switching signals via the concept of dwell time. The LMI formulation of ?-2 stability is applied to evaluating the closed-loop performance of an industrial refrigeration process that is regulated by several proportional-integral (PI) controllers in a switching structure.     

基于分段模糊Lyapunov函数的模糊系统稳定性分析和保性能设计

针对一类 Takagi-Sugeno (T-S) 连续模糊系统, 在分析模糊系统前提规则结构信息的基础上, 研究了其稳定性和保性能设计问题. 通过将模糊 Lyapunov 函数 (FLF) 和分段二次 Lyapunov 函数 (PQLF) 结合, 构造出分段模糊 Lyapunov 函数 (PFLF), 并提出了一种新的并行分配补偿 (PDC) 控制器. 基于 PFLF 方法, 得到了线性矩阵不等式 (LMI) 形式的模糊系统分析与设计的求解方法. 该方法继承了 FLF 与 PQLF 的优点. 仿真实例表明: 该方法所得稳定性判据更为宽松, 具有更好的保性能控制效果.     

Stability and stabilization of continuous-time singular hybrid systems

In this paper, the problems of stability and state feedback stabilization for a class of continuous-time singular hybrid systems are investigated. A new sufficient and necessary condition for a continuous-time singular hybrid system to be regular, impulse-free and stochastically stable is proposed in terms of a set of coupled strict linear matrix inequalities (LMIs). Moreover, a new sufficient and necessary condition is presented for the existence of the state feedback controller in terms of a set of coupled strict matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the obtained theoretical results.