Discrete-time Fuzzy Systems Research Articles

Static output feedback control for discrete-time fuzzy bilinear system

The problem of designing fuzzy static output feedback controller for T-S discrete-time fuzzy bilinear system (DFBS) is presented. Based on parallel distribution compensation method, some sufficient conditions are derived to guarantee the stability of the overall fuzzy system. The stabilization conditions are further formulated into linear matrix inequality (LMI) so that the desired controller can be easily obtained by using the Matlab LMI toolbox. In comparison with the existing results, the drawbacks, such as coordinate transformation, same output matrices, have been eliminated. Finally, a simulation example shows that the approach is effective.

Linear-like discrete-time fuzzy control in the regulation of irrigation canals

A linear-like discrete-time fuzzy controller was designed to control and stabilize a single-pool irrigation canal. Saint Venant equations for open-channel flow were linearized using the Taylor series and a finite-difference approximation of the original nonlinear partial differential equations. Using the linear optimal control theory, a traditional linear quadratic regulator (LQR) was first developed for an irrigation canal with a single-pool, and the results were observed. Then a linear-like global system representation of a discrete-time fuzzy system was proposed by viewing a discrete-time fuzzy system in a global concept and unifying the individual matrices into synthetic matrices. This linear-like representation aided development of a design scheme for a global optimal fuzzy controller in the way of the general linear quadratic approach. Based on this kind of system representation, a discrete-time optimal fuzzy control law that can achieve global minimum effect was developed. An example problem with a single-pool was considered for evaluating the performance of the discrete-time optimal fuzzy controller in the control of irrigation canals. The results obtained with the optimal fuzzy controller were compared to the results obtained with a traditional linear quadratic regulator. The discrete-time fuzzy controller was the best for the operation of the canal system, reaching the optimal performance index under unknown demands.

Stability Analysis and Controller Design for Discrete-Time Fuzzy Systems With Time-Varying Delay

Abstract Convex conditions for stability analysis and control synthesis for discrete-time fuzzy systems with time-varying delay are proposed in this paper. The conditions depend on the variation rate of the time-delay and are obtained by considering a parameter dependent Lyapunov-Krasovskii functional. To reduce the conservatism, besides the standard techniques to include extra matrix variables, some extra equations related to the control signal are also added. It is shown that this approach leads to a less expensive convex formulation that the standard way. In all cases, the conditions are formulated as an LMI (linear matrix inequality) feasibility test, that can be efficiently solved in polynomial time by specialized numerical solvers. Numerical examples are presented to compare and illustrate the efficiency of the proposed conditions.

Relaxed ${\cal H}_{\infty }$ Stabilization Conditions for Discrete-Time Fuzzy Systems With Interval Time-Varying Delays

To derive less-conservative delay- and range-dependent <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> stabilization conditions for discrete-time Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delays, the use of a fuzzy-weighting-dependent Lyapunov-Krasovskii functional (FWLKF), in which all variables are set to be affinely or quadratically dependent on fuzzy weighting functions, is proposed. Subsequently, parameterized linear matrix inequality (PLMI)-based <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sub> stabilization conditions are derived by following the free-weighting matrix approach. To fully exploit the convexity of fuzzy weighting functions, the derived PLMIs are sequentially replaced by a finite set of LMIs by considering all possible conditions associated with fuzzy weighting functions.

Relaxed delay-dependent stabilization criteria for discrete-time fuzzy systems based on a switching fuzzy model and piecewise Lyapunov functions

This paper presents delay-dependent stability analysis and controller synthesis methods for discrete-time Takagi-Sugeno (T-S) fuzzy systems with time delays. The T-S fuzzy system is transformed to an equivalent switching fuzzy system. Consequently, the delay-dependent stabilization criteria are derived for the switching fuzzy system based on the piecewise Lyapunov function. The proposed conditions are given in terms of linear matrix inequalities (LMIs). The interactions among the fuzzy subsystems are considered in each subregion, and accordingly the proposed conditions are less conservative than the previous results. Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable. Finally, a design example is given to show the validity of the proposed method.

New LMI Approach to Fuzzy $H_{\infty}$ Filter Designs

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This brief is concerned with the <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> filtering problem for a class of discrete-time fuzzy systems. The objective is to find a new way to design a stable filter guaranteeing the asymptotic stability and a prescribed <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex> </formula> performance of the filtering error system. By using the fuzzy Lyapunov function approach and adding slack matrix variables, a new condition for <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> performance analysis is proposed. Then, a linear matrix inequality (LMI)-based approach is developed for designing the fuzzy <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> filter, which can be obtained from the solution of convex optimization problems. When these LMIs are feasible, an explicit expression of a desired <formula formulatype="inline"><tex Notation="TeX">$H_{\infty}$</tex></formula> fuzzy filter is given. Finally, a numerical example is provided to illustrate the effectiveness and less conservatism of the proposed approach. </para>

Robust generalised ℋ 2 and ℋ ∞ static output feedback control for uncertain discrete-time fuzzy systems

This study is concerned with the generalised ℋ2 and the ℋ∞ static control problems for discrete-time fuzzy systems with time-varying norm-bounded parametric uncertainties. Based on the solutions to certain linear matrix inequalities LMIs, static output feedback controllers are designed, which can ensure the asymptotic stability and the generalised ℋ2 and ℋ∞ performance level, respectively. Differenting from the existing results, the design conditions in this study are derived in terms of LMIs directly, and the local output matrices are allowed to be different. Three illustrative examples are provided to demonstrate the effectiveness and the reduced conservatism of the proposed methods.

Delay-Dependent $\hbox{H}_{\infty }$ Filter Design for Discrete-Time Fuzzy Systems With Time-Varying Delays

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper investigates delay-dependent <formula formulatype="inline"><tex Notation="TeX">$\hbox{H}_{\bm\infty }$</tex></formula> filter design problems for discrete-time fuzzy systems with time-varying delays. First, a novel delay-dependent piecewise Lyapunov–Krasovskii functional (DDPLKF) is proposed in which both the upper bound of delays and the delay interval are considered. Based on this DDPLKF, the delay-dependent stability criteria for discrete-time systems with constant or time-varying delays are obtained, respectively. Then, delay-dependent full-order and reduced-order <formula formulatype="inline"><tex Notation="TeX">$\hbox{H}_{\bm\infty }$</tex></formula> filter design approaches are proposed. The filter parameters can be obtained by solving a set of linear matrix inequalities (LMIs). Simulation examples are also given to illustrate the performance of the proposed approaches. It is shown that our approaches are less conservative and that the corresponding <formula formulatype="inline"><tex Notation="TeX">$\hbox{H}_{\bm\infty }$</tex></formula> filters can achieve better performance than the existing approaches. </para>

Stability Analysis and Stabilization for Discrete-Time Fuzzy Systems With Time-Varying Delay

This paper is concerned with the problems of stability analysis and stabilization for discrete-time Takagi-Sugeno fuzzy systems with time-varying state delay. By constructing a new fuzzy Lyapunov function and by making use of novel techniques, an improved delay-dependent stability condition is obtained, which is dependent on the lower and upper delay bounds. The merit of the proposed stability condition lies in its reduced conservatism, which is achieved by avoiding the utilization of some bounding inequalities for the cross products between two vectors. Then, a delay-dependent stabilization approach based on a parallel distributed compensation scheme is developed for both state feedback and observer-based output feedback cases. The proposed stability and stabilization conditions are formulated in terms of linear matrix inequalities, which can be solved efficiently by using existing optimization techniques. Two illustrative examples are provided to demonstrate the effectiveness of the results proposed in this paper.

Piecewise $H_{\infty}$ Controller Design of Uncertain Discrete-Time Fuzzy Systems With Time Delays

This paper considers the robust H infin control of uncertain discrete-time fuzzy systems with time delays based on piecewise Lyapunov--Krasovskii functionals. It is shown that the stability with H infin disturbance attenuation performance can be established for the closed-loop fuzzy control systems if there exists a piecewise Lyapunov--Krasovskii functional, and moreover, the functional and the corresponding controller can be obtained by solving a set of linear matrix inequalities that are numerically feasible. A numerical example is given to demonstrate the efficiency and the advantage of the proposed method.

H ∞ Fuzzy Filtering for Discrete-Time Nonlinear Systems via Fuzzy Lyapunov Function Approach

This paper is concerned with the H ? fuzzy filtering problem for a class of discrete-time fuzzy systems. The objective is to design a stable filter guaranteeing the asymptotic stability and a prescribed H ? performance of the filtering error system. Motivated by the parallel distributed compensation (PDC) technique, a new filter model is proposed in this paper. Both full-order and reduced-order filters are established, and they can be obtained from the solution of convex optimization problems in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired H ? fuzzy filter is given. A numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach.

Stability Analysis and $H_{\infty}$ Controller Design of Discrete-Time Fuzzy Large-Scale Systems Based on Piecewise Lyapunov Functions

This paper is concerned with stability analysis and H(infinity) decentralized control of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. The fuzzy large-scale systems consist of J interconnected discrete-time Takagi-Sugeno (T-S) fuzzy subsystems, and the stability analysis is based on Lyapunov functions that are piecewise quadratic. It is shown that the stability of the discrete-time fuzzy large-scale systems can be established if a piecewise quadratic Lyapunov function can be constructed, and moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. The H(infinity) controllers are also designed by solving a set of LMIs based on these powerful piecewise quadratic Lyapunov functions. It is demonstrated via numerical examples that the stability and controller synthesis results based on the piecewise quadratic Lyapunov functions are less conservative than those based on the common quadratic Lyapunov functions.

Stability Analysis - Multiconvexity Approach

A new stability condition in terms of LMIs is studied in this paper, continuous- and discrete-time fuzzy systems treated in a unified manner. Based on a premise-dependent Lyapunov function and multiconvexity, we release the conservatism that commonly exists in the common P approach.

Fuzzy Constrained Min-Max Model Predictive Control Based on Piecewise Lyapunov Functions

This paper proposes two novel stable fuzzy model predictive controllers based on piecewise Lyapunov functions and the min-max optimization of a quasi-worst case infinite horizon objective function. The main idea is to design state feedback control laws that minimize the worst case objective function based on fuzzy model prediction, and thus to obtain the optimal transient control performance, which is of great importance in industrial process control. Moreover, in both of these predictive controllers, piecewise Lyapunov functions have been used in order to reduce the conservatism of those existent predictive controllers based on common Lyapunov functions. It is shown that the asymptotic stability of the resulting closed-loop discrete-time fuzzy predictive control systems can be established by solving a set of linear matrix inequalities. Moreover, the controller designs of the closed-loop control systems with desired decay rate and input constraints are also considered. Simulations on a numerical example and a highly nonlinear benchmark system are presented to demonstrate the performance of the proposed fuzzy predictive controllers.

Output Feedback Stabilization of Discrete-Time Fuzzy Systems with Immeasurable Premise Variables

Output Feedback Stabilization of Discrete-Time Fuzzy Systems with Immeasurable Premise Variables

Dynamic output feedback H ∞ control of discrete-time fuzzy systems: a fuzzy-basis-dependent Lyapunov function approach

This article deals with an output feedback H ∞ control problem for a class of discrete-time fuzzy dynamic systems. A full-order dynamic output feedback H ∞ control design approach is developed by combining a fuzzy-basis-dependent Lyapunov function and a transformation on the controller parameters, which leads to sufficient conditions in the form of strict linear matrix inequalities (LMIs). The fuzzy-basis-dependent results are less conservative due to the generality of the fuzzy-basis-dependent Lyapunov function used which includes the fuzzy-basis-independent one as a special case. It has been shown that the underling full-order dynamic output feedback H ∞ control problem can be solved as LMI optimization problems that can be computed numerically very efficiently. Finally, two numerical examples, concerning the control of a discrete-time chaotic Lorenz system and an inverted pendulum, are given to demonstrate the applicability of the proposed approach.

Decentralized robust stabilization of discrete-time fuzzy large-scale systems with parametric uncertainties: a LMI method

Decentralized robust stabilization problem of discrete-time fuzzy large-scale systems with parametric uncertainties is considered. This uncertain fuzzy large-scale system consists of N interconnected T-S fuzzy subsystems, and the parametric uncertainties are unknown but norm-bounded. Based on Lyapunov stability theory and decentralized control theory of large-scale system, the design schema of decentralized parallel distributed compensation (DPDC) fuzzy controllers to ensure the asymptotic stability of the whole fuzzy large-scale system is proposed. The existence conditions for these controllers take the forms of LMIs. Finally a numerical simulation example is given to show the utility of the method proposed.

Formula omitted] filtering of discrete-time fuzzy systems via basis-dependent Lyapunov function approach

This paper is concerned with the H ∞ filtering problem for a class of discrete-time fuzzy systems. Attention is focused on the design of a stable filter guaranteeing a prescribed noise attenuation level in the H ∞ sense. By using basis-dependent Lyapunov functions, sufficient conditions for the solvability of this problem are obtained. It has been shown that the H ∞ filtering problem can be solved as a linear matrix inequality (LMI) optimization problem. Two examples are provided to demonstrate the applicability of the proposed approach.

Delay-dependent stability analysis and stabilization for discrete-time fuzzy systems with state delay: a fuzzy Lyapunov-krasovskii functional approach

This correspondence studies stability analysis and stabilization for discrete-time Takagi and Sugeno fuzzy systems with state delay. First, a new fuzzy Lyapunov-Krasovskii functional (LKF) is constructed to derive a delay-dependent stability condition for open-loop fuzzy systems. Then, a delay-dependent stabilization approach based on a nonparallel distributed compensation scheme is provided for closed-loop fuzzy systems. Both state feedback and observer-based control cases are considered. The proposed stability and stabilization conditions are represented in terms of linear matrix inequalities (LMIs), which can be solved efficiently by using existing LMI optimization techniques. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Control Performance of Discrete-Time Fuzzy Systems Improved by Neural Networks

A new control scheme is proposed to improve the system performance for discrete-time fuzzy systems by tuning control grade functions using neural networks. According to a systematic method of constructing the exact Takagi-Sugeno (T-S) fuzzy model, the system uncertainty is considered to affect the membership functions. Then, the grade functions, resulting from the membership functions of the control rules, are tuned by a back-propagation network. On the other hand, the feedback gains of the control rules are determined by solving a set of LMIs which satisfy sufficient conditions of the closed-loop stability. As a result, both stability guarantee and better performance are concluded. The scheme applied to a truck-trailer system is verified by satisfactory simulation results.