Robust Quadratic-Optimal Control of TS-Fuzzy-Model-Based Dynamic Systems With Both Elemental Parametric Uncertainties and Norm-Bounded Approximation Error

This paper considers the design problem of the robust quadratic-optimal parallel-distributed-compensation (PDC) controllers for Takagi–Sugeno (TS) fuzzy-model-based control systems with both elemental parametric uncertainties and norm-bounded approximation error. By complementarily fusing the robust stabilizability condition, the orthogonal functions approach (OFA), and the hybrid Taguchi genetic algorithm (HTGA), an integrative method is presented in this paper to design the robust quadratic-optimal PDC controllers such that 1) the uncertain TS-fuzzy-model-based control systems can be robustly stabilized, and 2) a quadratic integral performance index for the nominal TS-fuzzy-model-based control systems can be minimized. In this paper, the robust stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). By using the OFA and the LMI-based robust stabilizability condition, the robust quadratic-optimal PDC control problem for the uncertain TS-fuzzy-model-based dynamic systems is transformed into a static constrained-optimization problem represented by the algebraic equations with constraint of LMI-based robust stabilizability condition, thus greatly simplifying the robust optimal PDC control design problem. Then, for the static constrained-optimization problem, the HTGA is employed to find the robust quadratic-optimal PDC controllers of the uncertain TS-fuzzy-model-based control systems. Two design examples of the robust quadratic-optimal PDC controllers for an uncertain inverted pendulum system and an uncertain nonlinear mass–spring–damper mechanical system are given to demonstrate the applicability of the proposed integrative approach.      

Optimal control of Takagi–Sugeno fuzzy-model-based systems representing dynamic ship positioning systems

Orthogonal function approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA) are used to solve quadratic finite-horizon optimal controller design problems in both a fuzzy parallel distributed compensation (PDC) controller and a non-PDC controller (linear state feedback controller) for Takagi–Sugeno (TS) fuzzy-model-based control systems for dynamic ship positioning systems (TS-DSPS). Based on the OFA, an algorithm requiring only algebraic computation is used to solve dynamic equations for TS-fuzzy-model-based feedback and is then integrated with HTGA to design quadratic finite-horizon optimal controllers for TS-DSPS under the criterion of minimizing a quadratic finite-horizon integral performance index, which is also converted to algebraic form by the OFA. Integration of OFA and HTGA in the proposed approach enables use of simple algebraic computation and is well adapted to the computer implementation. Therefore, it facilitates design tasks of quadratic finite-horizon optimal controllers for the TS-DSPS. The applicability of the proposed approach is demonstrated in the example of a moored tanker designed using quadratic finite-horizon optimal controllers.     

Design of Optimal Controllers for Takagi–Sugeno Fuzzy-Model-Based Systems

By the use of the elegant operational properties of the orthogonal functions, a direct computational algorithm for solving the Takagi-Sugeno (TS) fuzzy-model-based feedback dynamic equations is first developed in this paper. The basic idea is that the state variables are expressed in terms of the orthogonal functions. The new method simplifies the procedure of solving the TS fuzzy-model-based feedback dynamic equations into the successive solution of a system of recursive formulas taking only two terms of the expansion coefficients. Based on the presented recursive formulas, the developed computational algorithm only involves the straightforward algebraic computation. Then, the developed algorithm is integrated with the hybrid Taguchi-genetic algorithm (HTGA) to design both the quadratic optimal fuzzy parallel-distributed-compensation (PDC) controller and the quadratic-optimal non-PDC controller (quadratic optimal linear-state feedback controller) of the TS fuzzy-model-based control systems under the criterion of minimizing a quadratic integral performance index, where the quadratic integral performance index is also converted into the algebraic form by using the orthogonal-function approach (OFA). The proposed new approach, which integrates the OFA and the HTGA, is nondifferential, nonintegral, straightforward, and well adapted to the computer implementation. The computational complexity can, therefore, be reduced remarkably. Thus, this proposed approach facilitates the design tasks of the quadratic optimal controllers for the TS fuzzy-model-based control systems. A design example of the quadratic optimal controllers for the translational oscillator system with an eccentric rotational proof mass actuator is given to demonstrate the applicability of the proposed approach     

线性时滞系统的耗散控制

研究了一类线性时滞系统的二次耗散控制问题,基于线性矩阵不等式（LMI）方法导出了耗散控制器存在的充分条件,通过线性矩阵不等式的可行解构造出耗散态状态反馈和动态输出反馈控制律,相应的闭环系统是二次稳定和严格（Q,S,R）耗散的,本文的主要贡献是统一了线性时滞系统现有的H∞控制和无源控制结果。     

一类时滞组合系统的优化保成本容错控制

对一类时滞组合系统，研究了依赖时滞的分散状态反馈可靠保成本控制器的设计问题．采用线性矩阵不等式方法，导出了时滞依赖可靠保成本控制器的存在条件和参数化表示．进而，通过建立和求解一个线性矩阵不等式组约束的凸优化问题，给出了优化保成本容错控制器的设计方法．仿真实例验证了这个设计法的有效性．     

Design of robust-optimal output feedback controllers for linear uncertain systems using LMI-based approach and genetic algorithm

This paper considers the robust-optimal design problems of output feedback controllers for linear systems with both time-varying elemental (structured) and norm-bounded (unstructured) parameter uncertainties. Two new sufficient conditions are proposed in terms of linear-matrix-inequalities (LMIs) for ensuring that the linear output feedback systems with both time-varying elemental and norm-bounded parameter uncertainties are asymptotically stable, where the mixed quadratically-coupled parameter uncertainties are directly considered in the problem formulation. A numerical example is given to show that the presented sufficient conditions are less conservative than existing ones reported recently. Then, by integrating the hybrid Taguchi-genetic algorithm (HTGA) and the proposed LMI-based sufficient conditions, a new integrative approach is presented to find the output feedback controllers of the linear systems with both time-varying elemental and norm-bounded parameter uncertainties such that the control objective of minimizing a quadratic integral performance criterion subject to the stability robustness constraint is achieved. A design example of the robust-optimal output feedback controller for the AFTI/F-16 aircraft control system with the time-varying elemental parameter uncertainties is given to demonstrate the applicability of the proposed new integrative approach.     

不确定时滞系统鲁棒镇定新方法

针对一类同时具有状态和控制滞后的不确定时滞系统,提出了一种新的时滞依赖型鲁棒镇定设计方法.通过引入一种新的线性状态变换,分离出时滞依赖因子,采用Lyapunov-Krasovskii泛函方法,导出了不确定时滞系统可鲁棒镇定的时滞依赖型的新判据,所得结论以线性矩阵不等式组的解的形式表示.仿真结果表明所得结论较之已有结果更为简单,具有更小保守性,且有更广泛的应用范围,不仅可以解决小时滞问题,也可用于解决大时滞问题.     

不确定非线性系统的时滞依赖保性能控制

针对一类基于T-S模糊模型表示的具有时变状态时滞和范数有界不确定性非线性系统，研究了时滞依赖保性能模糊控制器设计问题。对于用T-S模糊模型表示的非线性时滞系统，已知系统的时变时滞本身及其变化率的上界，采用并行分散补偿技术，通过选取合适的Lyapunov函数，推导了依赖时滞上界及变化率上界的时滞保性能模糊控制器存在的充分条件，进而通过建立和求解LMI（线性矩阵不等式）约束的凸优化问题，给出了保性能控制律的设计方法。数值算例表明了该方法的有效性。     

Robust Controllability of T–S Fuzzy-Model-Based Control Systems With Parametric Uncertainties

The robust controllability problem for the Takagi–Sugeno (T–S) fuzzy-model-based control systems is studied in this paper. Under the assumption that the nominal T–S fuzzy-model-based control systems are locally controllable (i.e., each fuzzy rule of the nominal T–S fuzzy-model-based control systems has a full row rank for its controllability matrix), a sufficient condition is proposed to preserve the assumed property when the parameter uncertainties are added into the nominal T–S fuzzy-model-based control systems. The proposed sufficient condition can provide the explicit relationship of the bounds on parameter uncertainties to preserve the assumed property. Besides, a robustly global controllability condition and the related robustly global stabilizability condition of the uncertain T–S fuzzy-model-based control systems are also presented in this paper. A nonlinear mass–spring–damper mechanical system with parameter uncertainties is given as an example to illustrate the application of the proposed sufficient conditions.      

Synthesis of minimax optimal controllers for uncertain time-delay systems with structured uncertainty

This paper is concerned with the design of robust state feedback controllers for a class of uncertain time-delay systems. The uncertainty is assumed to satisfy a certain integral quadratic constraint. The controller proposed is a minimax optimal controller in the sense that it minimizes the maximum value of a corresponding linear quadratic cost function over all admissible uncertainties. The controller leads to an absolutely stable closed loop uncertain system and is constructed by solving a finite dimensional parameter-dependent algebraic Riccati equation.     

Robust stabilization of nonlinear multiple time-delay large-scale systems via decentralized fuzzy control

To overcome the effect of modeling errors between nonlinear multiple time-delay subsystems and Takagi-Sugeno (T-S) fuzzy models with multiple time delays, a robustness design of fuzzy control is proposed in This work. In terms of Lyapunov's direct method, a delay-dependent stability criterion is hence derived to guarantee the asymptotic stability of nonlinear multiple time-delay large-scale systems. Based on this criterion and the decentralized control scheme, a set of model-based fuzzy controllers is then synthesized via the technique of parallel distributed compensation (PDC) to stabilize the nonlinear multiple time-delay large-scale system. Finally, a numerical example with simulations is given to demonstrate the concepts discussed throughout This work.     

New Robust Stability Conditions and Design of Robust Stabilizing Controllers for Takagi–Sugeno Fuzzy Time-Delay Systems

In this paper, we consider robust stability and stabilization of uncertain Takagi-Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability of a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems and it thus leads to a conservative fuzzy controller design as well. We first consider rather relaxed robust stability conditions of uncertain fuzzy systems. To this end, we introduce an auxiliary system to the original fuzzy time-delay system to obtain generalized delay-dependent stability conditions. Such an auxiliary system has some arbitrary matrices that generalize not only the system representation but also delay-dependent stability conditions. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time-delay, and are given in terms of linear matrix inequalities (LMIs). Then, we compare our delay-dependent stability conditions with other conditions in the literature, and show that our conditions guarantee the stability of a wider class of systems than others. Next, we consider the robust stabilization problem with memoryless and delayed state feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give three examples that illustrate our results.     

Robust control for a class of T-S fuzzy systems with interval time-varying delay

The problem of delay-dependent stability analysis and controller design for a class of T-S fuzzy systems with interval state time-varying delay is considered. Based on Lyapunov stability theory, defining a new Lyapunov-Krasovskii functional and introducing some free-weighting matrices, a new delay-dependent criterion is given to ensure the systems asymptotically stable. The merit of the proposed conditions lies in the less conservativeness than the existing ones, which is achieved by considering the mean of time-delay interval and the introduction of the free variables. By the concept of parallel distributed compensation (PDC), a delay-dependent condition for the existence of a fuzzy state feedback control law with memory is proposed. Another merit is the consideration of the memory of the controller. All conditions are shown in terms of linear matrix inequalities (LMIs), which can be solved efficiently by using the LMI optimization techniques. Two numerical examples are given to illustrate the feasibility and validity of the proposed approach.     

离散线性时滞系统的次优控制:逐次逼近法

A successive approximation approach for designing optimal controllers is presented for discrete linear time-delay systems with a quadratic performance index. By using the successive approximation approach, the original optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems without time-delay and time-advance terms. The optimal control law obtained consists of an accurate feedback terms and a time-delay compensation term which is the limit of the solution sequence of the adjoint equations. By using a finite-step iteration of the time-delay compensation term of the optimal solution sequence, a suboptimal control law is obtained. Simulation examples are employed to test the validity of the proposed approach.     

Suboptimal Control for Discrete Linear Systems with Time-delay:A Successive Approximation Approach

A successive approximation approach for designing optimal controllers is presented for discrete linear time-delay systems with a quadratic performance index.By using the successive approximation approach,the original optimal,control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems without time-delay and time- advance terms.The optimal control law obtained consists of an accurate feedback terms and a time-delay compensation term which is the limit of the solution sequence of the adjoint equations. By using a finite-step iteration of the time-delay compensation term of the optimal solution sequence, a suboptimal control law is obtained.Simulation examples are employed to test the validity of the proposed approach.     

不确定时滞系统的保性能控制

基于时滞系统的一个新的指数稳定性定理,考虑了时不变不确定时滞系统的保性能控制问题.利用一种新的分析技术,通过解两个Riccati不等式和一个线性矩阵不等式得一保性能控制器,该控制器可保证闭环系统指数稳定且使系统满足一定的性能指标.与一般的不确定时滞系统保性能控制不同,所得的性能指标的上界中含有参数,可通过调整参数值的大小来使性能指标达到最优.所得Riccati不等式中含有时滞,因此该判据是时滞相关的.数值算例证明了该方法的适用性.     

线性时滞系统依赖于时滞的H∞状态反馈控制

对具有纯滞后输入的线性时滞系统,在系统的状态时滞与控制输入时滞不同时,研 究了依赖时滞的H∞状态反馈控制器设计问题,其控制器存在的充分条件由一个线性矩阵不 等式(LMI)的形式给出,并给出了相应的H∞控制器的综合设计方法.     

Some algebraic aspects of the strong stabilizability of time-delaylinear systems

This note considers some algebraic aspects of the output feedback strong stabilizability of time-delay linear systems. A sufficient condition for the existence of a stable finite-dimensional stabilizing compensator for a single-input-single-output (SISO) time-delay system is given. An illustrative example is also given, and two open problems are posed     

不确定时滞系统的鲁棒绝对稳定性研究

基于Lyapunov稳定性理论，采用线性矩阵不等式处理方法，研究具有参数不确定性的时滞系统鲁棒绝对稳定性问题．导出了用线性矩阵不等式系统可行性描述的系统鲁棒绝对稳定的滞后依赖型条件，据此可计算出最大的允许时滞界．通过求解一组线性矩阵不等式，给出使得闭环系统鲁棒绝对稳定的无记忆状态反馈控制律设计方法。     

不确定离散时滞系统经动态输出反馈的时滞依赖保性能控制

考虑了不确定离散时滞系统经动态输出反馈的保性能控制器设计问题．利用离散时滞系统的广义系统表示形式以及有关向量交叉乘积项的界的处理方法，以线性矩阵不等式的形式，给出了一个时滞依赖的保性能输出反馈控制器存在的充分条件．一个数值例子说明了所给方法的有效性．