Effective digital implementation of fuzzy control systems based on approximate discrete-time models

This paper addresses an effective digital implementation of fuzzy control systems via an intelligent digital redesign (IDR) approach. The purpose of IDR is to effectively convert an existing continuous-time fuzzy controller to an equivalent sampled-data fuzzy controller in the sense of the state-matching. The authors show that, under reasonable assumptions, the IDR based on the exact discrete-time models can be reduced to the IDR based on the approximate discrete-time models. The state-matching error between the closed-loop trajectories is carefully analyzed using the integral quadratic functional approach. The estimation of the state-matching error is presented using the linear matrix inequality (LMI) techniques. The problem of designing the sampled-data fuzzy controller to minimize the estimation as well as to guarantee the stability is formulated and solved as the convex optimization problem with LMI constraints. It is also shown that the resulting sampled-data fuzzy controller recovers the performance of the continuous-time fuzzy controller as the sampling period approaches zero. A numerical example is used to demonstrate the effectiveness of the proposed design technique.     

Fuzzy filter for nonlinear sampled-data systems: Intelligent digital redesign approach

This paper presents a fuzzy filter design method for nonlinear sampled-data systems using an intelligent digital redesign (IDR) technique. Based on a Takagi–Sugeno (T–S) fuzzy model, discretized closed-loop systems with pre-designed analog fuzzy and digital fuzzy filters are presented. An IDR problem is given to guarantee both state-matching condition and asymptotic stability. Sufficient conditions for solving the IDR problem are proposed and are derived in terms of linear matrix inequalities (LMIs). Finally, a simulation example is given to show the effectiveness of the proposed method.     

Robust guaranteed cost control of uncertain fuzzy systems under time-varying sampling

In practice, the system is often modeled as a continuous-time fuzzy system, while the control input is applied only at discrete instants. This system is called a sampled-data control system. In this paper, robust guaranteed cost control for uncertain sampled-data fuzzy systems is discussed. A guaranteed cost control where a quadratic cost function is bounded by a certain scalar, not only stabilizes a system but also considers a control performance. A typical sampled-data control is the zero-order input, which can be represented as a piecewise-continuous delay. Here we take a delay system approach to the sampled-data guaranteed cost control problem. The closed-loop system with a sampled-data state feedback controller becomes a system with time-varying delay. First, guaranteed cost control performance conditions for the closed-loop system are given in terms of linear matrix inequalities (LMIs). Such conditions are derived by using Leibniz–Newton formula and free weighting matrix method for fuzzy systems under the assumption that sampling time is not greater than some prescribed scalar. Then, a design method of robust guaranteed cost state feedback controller for uncertain sampled-data fuzzy systems is proposed. Examples are given to illustrate our robust sampled-data guaranteed cost control design.     

<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> digital redesign for LTI systems

The so-called digital redesign (DR) is a sampled-data (SD) controller design method where an analogue controller is designed firstly, and then transformed to an approximately equivalent digital controller in the sense of state-matching. In this approach, the SD controller is designed by reducing the discrepancy between the discrete-time (DT) counterpart of the closed-loop SD control system and the continuous-time (CT) closed-loop system. In this paper, we develop a DR strategy for CT linear time-invariant systems. More specifically, H _∞ norm of the error dynamic system between the CT and DT plants is minimized for the optimal state-matching performance at every sampling point. The design problem is formulated as linear matrix inequalities which can be efficiently solved by using convex optimization techniques. Finally, an example is given to illustrate the effectiveness of the proposed method.     

TS-based sampled-data model predictive controller for continuous-time nonlinear systems

This paper proposes a new fuzzy model predictive control approach for continuous-time nonlinear systems in terms of linear matrix inequalities (LMIs). The proposed approach is based on the Takagi–Sugeno fuzzy modeling, a quadratic Lyapunov function, and a sampled-data parallel distributed compensation controller with constant sampling time. The goal is designing the sampled-data controller such that at each sampling time, the stability of the closed-loop system is guaranteed and an infinite horizon cost function is minimised. The main advantage of the proposed approach is to eliminate the approximations induced from discretizing the original system and cost function upper bound minimisation. Consequently, a lower bound of the cost function is obtained and the performance of the proposed model predictive controller is improved compared to the recently published papers in the same field of interest. In addition, the Euclidean norm constraint of the control input vector is derived in terms of LMIs. To illustrate the merits of the proposed approach, the proposed technique is applied to a continuous stirred tank reactor system.     

Decentralised control for large-scale sampled-data systems: digital redesign approach

In this paper, digital redesign (DR) techniques are presented for a decentralised controller of large-scale sampled-data systems. To improve the performance of the previous DR technique, the state-matching error is defined and directly minimised by using the state-matching error cost function. Also, the discretisation error of the interconnection term is eliminated through an exact discrete-time design approach. Sufficient conditions of the proposed DR techniques are obtained in the Lyapunov sense and are converted into optimal problems with linear matrix inequalities. Finally, two numerical examples are provided to demonstrate the performance improvement of the proposed techniques.     

Digitalizing a Fuzzy Observer-Based Output-Feedback Control: Intelligent Digital Redesign Approach

This paper concerns an intelligent digital redesign (IDR) technique for a Takagi-Sugeno fuzzy observer-based output-feedback control (FOBOFC) system. The term IDR involves converting an existing analog control into an equivalent digital counterpart in the sense of state-matching. The IDR problem is herein viewed as a minimization problem of the norm distances between nonlinearly interpolated linear operators to be matched. Its constructive condition with global rather than local state-matching is formulated in terms of bilinear matrix inequalities. The main features of the proposed method are that the state estimation error in the plant dynamics is considered in the IDR condition that plays a crucial role in the performance improvement; the stability property is preserved by the proposed IDR algorithm; the separation principle is shown when the premise variables are measurable; finally, the IDR condition for a more general FOBOFC-an estimated premise variable case is conducted. A numerical example is demonstrated to visualize the feasibility of the developed methodology     

Decentralised sampled-data control for large-scale systems with nonlinear interconnections

This paper presents a decentralised sampled-data control technique for a class of large-scale systems, which are considered to consist of linear subsystems and nonlinear interconnections. The decentralised sampled-data controller design problem is established using a closed-loop subsystem. Based on the controller design problem, the stability condition is derived for a closed-loop large-scale system, and the maximum interconnection bound is guaranteed to satisfy the stability condition. Also, its sufficient condition is formulated in terms of linear matrix inequalities. Finally, the effectiveness of the proposed technique is verified by using an example of the multi-machine power system.     

Controlling linear systems with nonlinear perturbations under sampled-data output feedback: digital redesign approach

This article develops a digital redesign (DR) technique for sampled-data observer-based output-feedback control of a continuous-time linear system with nonlinear perturbation. It is assumed that the nonlinear perturbation is a locally Lipschitz function. To deal with the discrete-time modelling error in nonlinear systems, as opposed to the previous approach, the DR problem is configured as a stabilisation one for error dynamics between the closed-loop system of nominal linear model under an analogue state-feedback controller and that of the linear system with the nonlinear perturbation under a sampled-data output-feedback controller. A constructive DR condition is formulated in the format of linear matrix inequalities. The stability of the actual sampled-data control system is guaranteed within the DR procedure. The effectiveness of the proposed DR methodology is demonstrated through a numerical simulation.     

Observer-based sampled-data control for nonlinear systems: Robust intelligent digital redesign approach

This paper suggests a solution for a robust digital implementation of the output-feedback control system by utilizing intelligent digital redesign (IDR). The general IDR method involves designing a digital fuzzy controller which is comparable to a pre-designed analog one via state-matching. However, under containing the parametric uncertainties, the new problem about the structural property of the uncertainties is inevitably occurred in the procedure of discretization. For solving that, we use the bilinear and inverse-bilinear approximation method and the concerned IDR is viewed as a convex minimization problem of the norm distances between linear operators to be matched. Also, we consider the state estimation error in the plant dynamics so that we attempt to improve the control performance. The robust stability property is preserved by the proposed IDR method and its condition is represented in terms of linear matrix inequalities (LMIs). The simulation results for permanent magnet synchronous motor (PMSM) system are demonstrated to visualize the feasibility of the proposed method.     

Fuzzy guaranteed cost control for nonlinear systems with time-varying delay

This paper focuses on the problem of guaranteed cost control for Takagi-Sugeno (T-S) fuzzy systems with time-varying delayed state. A linear quadratic cost function is considered as a performance index of the closed-loop fuzzy system. Then, the guaranteed cost control of the closed-loop fuzzy system is discussed, and the sufficient conditions are provided for the construction of a guaranteed cost controller via state feedback and observer-based output feedback. When these conditions, which are given in terms of the feasibility of linear matrix inequalities (LMIs), are satisfied, the designed state feedback controller and observer-based controller gain matrices can be obtained via a convex optimization problem. Two illustrative examples are provided to demonstrate the effectiveness of the approaches proposed in this paper.     

A new intelligent digital redesign for T-S fuzzy systems: global approach

This paper proposes a novel and efficient global intelligent digital redesign technique for a Takagi-Sugeno (T-S) fuzzy system. The term of intelligent digital redesign involves converting an existing analog fuzzy-model-based controller into an equivalent digital counterpart in the sense of state-matching. The proposed method should be notably discriminated from the previous works in that it allows us to globally match the states of the overall closed-loop T-S fuzzy system with the predesigned analog fuzzy-model-based controller and those with the digitally redesigned fuzzy-model-based controller, and further to examine the stabilizability by the redesigned controller in the sense of Lyapunov. The key idea is that the global intelligent digital redesign problem is viewed as a convex minimization problem of the norm distance between nonlinearly interpolated linear operators to be matched. Sufficient conditions for the global state-matching and the stability of the digitally controlled system are formulated in terms of linear matrix inequalities (LMIs). A complex nonlinear system, Duffing-like chaotic oscillator is simulated and demonstrated to validate the feasibility and effectiveness of the proposed digital redesign technique, which implies the safe applicability to the digital control system.     

Multi-rate digital redesign of cascaded and dynamic output feedback systems

In this paper, a new indirect digital redesign method is presented for multi-rate sampled-data control systems with cascaded and dynamic output feedback controllers. These analogue controllers are often pre-designed based on desirable frequency specifications, such as bandwidth, natural angular frequency, etc. To take advantage of the digital controller over the analogue controller, digital implementation of these analogue controllers are often desirable. As only measured input-output signals are available, an ideal state reconstructing algorithm is utilised to obtain the multi-rate discrete-time states of the original continuous-time system. Based on the Chebyshev quadrature method, the gains of the multi-rate cascaded and the output feedback digital controllers are determined from their continuous-time counterparts according to the different sampling rates employed in the different parts of the closed-loop system. As a result, the respective analogue controllers with the high-frequency and low-frequency characteristics can be implemented using the respective fast-rate sampling and slow-rate sampling digital controllers. Unlike the classical direct bilinear transform method, which is an open-loop direct digital redesign method, the proposed digital controllers take into account the state-matching of the original continuous-time closed-loop system and the digitally redesigned sampled-data closed-loop system. To further improve the state-matching performance, an improved digital redesign approach is also developed to construct the multi-rate cascaded and dynamic output feedback digital controllers. Illustrative examples are given to demonstrate the effectiveness of the developed methods.     

基于T-S模型的非线性时滞系统非脆弱保性能模糊控制

针对一类用T-S模糊模型描述的非线性时滞系统,采用状态反馈的并行分布补偿方法,研究其保守性较小的非脆弱保性能模糊控制问题,使闭环系统在控制器存在加性摄动的情况下,其闭环性能指标值低于确定的上界.利用线性矩阵不等式处理方法,导出了非脆弱保性能模糊控制律的存在条件,通过建立和求解一个线性矩阵不等式问题,给出了非脆弱保性能模糊控制律的设计方法.仿真结果表明了所提出方法的有效性.     

Delay-dependent guaranteed cost control for T-S fuzzy systems with time delays

This study introduces a guaranteed cost control method for nonlinear systems with time-delays which can be represented by Takagi-Sugeno (T-S) fuzzy models with time-delays. The state feedback and generalized dynamic output feedback approaches are considered. The generalized dynamic output feedback controller is presented by a new fuzzy controller architecture which is of dual indexed rule base. It considers both the dynamic part and the output part of T-S fuzzy model which guarantees that the controller without any delay information can stabilize time-delay T-S fuzzy systems. Based on delay-dependent Lyapunov functional approach, some sufficient conditions for the existence of state feedback controller are provided via parallel distributed compensation (PDC) first. Second, the corresponding conditions are extended into the generalized dynamic output feedback closed-loop system via so-called generalized PDC technique. The upper bound of time-delay can be obtained using convex optimization such that the system can be stabilized for all time-delays whose sizes are not larger than the bound. The minimizing method is also proposed to search the suboptimal upper bound of guaranteed cost function. The effectiveness of the proposed method can be shown by the simulation examples.     

Robust digital implementation of fuzzy control for uncertain systems and its application to active magnetic bearing system

In this paper, we propose the robust digital control for active magnetic bearing (AMB) systems. For achieving the robust stability, we deal with the uncertainties of the given system based on the Takagi-Sugeno (T-S) fuzzy model. Also, in order to solve the digital implementation for real plants, this paper presents a robust intelligent digital redesign (IDR) method. The term IDR involves converting an analog controller into an equivalent digital one in the sense of state-matching. The uncertainties in the plant dynamics is shown in the IDR condition by virtue of the pade and inverse-pade approximation method. Also, the robust stability property is preserved by the proposed method. The sufficient conditions for robust controller are obtained in terms of solutions to linear matrix inequalities (LMIs). Finally, simulation results for two AMB systems are demonstrated to visualize the feasibility of the proposed method.     

Intelligent Digital Redesign for Sampled-data Fuzzy Control Systems Based on State-matching Error Cost Function Approach

International Journal of Control, Automation and Systems - This paper presents a novel intelligent digital redesign (IDR) technique for a sampled-data fuzzy controller of fuzzy systems which can be...     

非线性时变时延系统的模糊采样最优控制

针对非线性时变时延系统,采用输入时延和自由权重矩阵方法研究模糊采样最优控制问题.应用T-S模糊系统表征非线性系统,控制器是零阶保持采样信号.由线性矩阵不等式给出最优控制准则,所设计的模糊采样控制器在闭环系统渐近稳定意义下可保证期望最优控制性能.最后,通过卡车拖车系统实验验证模糊采样控制设计方案的可行性.     

Robust nonfragile guaranteed cost control for uncertain T-S fuzzy Markov jump systems with mode-dependent average dwell time and input constraint

This paper aims to investigate the problem of robust nonfragile guaranteed cost control for uncertain Takagi-Sugeno fuzzy systems with Markov jump parameters, time-varying delay and input constraint. A nonfragile mode-dependent fuzzy controller with mode-dependent average dwell time (MDADT) is designed with input constraint. A sufficient condition is developed to ensure that the resulting closed-loop system is robust almost surely asymptotically stable with guaranteed cost index not exceeding the specified upper bound. Subsequently, the controller gain and upper bound of the guaranteed cost index can be obtained by solving a set of linear matrix inequalities. Finally, numerical and practical examples are provided to demonstrate the performance of the proposed approach.