Stable and Quadratic Optimal Control for TS Fuzzy-Model-Based Time-Delay Control Systems

For the finite-horizon optimal control problem of the Takagi-Sugeno (TS) fuzzy-model-based time-delay control systems, by integrating the delay-dependent stabilizability condition, the shifted-Chebyshev-series approach (SCSA), and the hybrid Taguchi-genetic algorithm (HTGA), an integrative method is presented to design the stable and quadratic optimal parallel distributed compensation (PDC) controllers. In this paper, the delay-dependent stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). Based on the SCSA, an algebraic algorithm only involving the algebraic computation is derived in this paper for solving the TS fuzzy-model-based time-delay feedback dynamic equations. In addition, by using the SCSA, the stable and quadratic optimal PDC control problem for the TS fuzzy-model-based time-delay control systems is replaced by a static parameter optimization problem represented by the algebraic equations with constraint of the LMI-based stabilizability condition, thus greatly simplifying the stable and optimal PDC control design problem. The computational complexity for both differential and integral in the stable and optimal PDC control design of the original dynamic systems may therefore be reduced considerably. Then, for the static constrained optimization problem, the HTGA is employed to find the stable and quadratic optimal PDC controllers of the TS fuzzy-model-based time-delay control systems. A design example of the stable and quadratic optimal PDC controllers for the continuous-stirred-tank-reactor system is given to demonstrate the applicability of the proposed integrative approach.     

方块脉冲函数用于线性时变系统的分析和最优控制

本文给出了方块脉冲函数的一些运算性质.利用这些性质求解线性时变系统的状态方程和基于二次型性能指标的最优控制规律,得出了便于应用的均匀分段恒定解答.较沃尔什函数逼近法容易导出形式简明的递推算法,且子区间的分段数可选取任意整数,因而节省计算机内存和机时,有助于提高计算和控制的精度.     

Analysis and Synthesis of Robust Control Systems Using Linear Parameter Dependent Lyapunov Functions

This note provides sufficient robust stability conditions for continuous time polytopic systems. They are obtained from the Frobenius-Perron Theorem applied to the time derivative of a linear parameter dependent Lyapunov function and are expressed in terms of linear matrix inequalities (LMI). They contain as special cases, various sufficient stability conditions available in the literature to date. As a natural generalization, the determination of a guaranteed H₂ cost is addressed. A new gain parametrization is introduced in order to make possible the state feedback robust control synthesis using parameter dependent Lyapunov functions through linear matrix inequalities. Numerical examples are included for illustration     

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.      

分段线性函数应用于线性时变系统的最优控制

本文给出了分段线性函数的一些运算性质,利用这些性质求解线性时变系统基于二次型性能指标的最优反馈控制律,推导出了形式简明的求解算法,该算法较之于方块脉冲函数算法具有更高的计算精度。     

分段线性函数应用于延时系统的最优控制

本文把分段线性函数引入到求解延时系统的最优控制中，推得延时线性系统二次型最优控制问题的分段控制解答及其状态轨线估计，算例结果表明，该算法比方块脉冲函数算法具有更高的计算精度。     

Fine Tuning Of Membership Functions For Fuzzy Neural Systems

This paper presents a new method for fine‐tuning the Gaussian membership functions of a fuzzy neural network ( FNN ) to improve approximation accuracy. This method results in special shape membership functions without the convex property. We first recall that any continuous function can be represented by a linear combination of Gaussian functions with any standard deviation. Therefore, the Gaussian membership function in the second layer of the FNN can be replaced by several small Gaussian functions; the weighting vectors of this new network (called FNN₅ ) can then be updated using the backpropagation algorithm. The proposed method can adapt proper membership functions for any nonlinear input/output mapping to achieve highly accurate approximation. Convergence analysis shows that the weighting vectors of the FNN₅ eventually converge to the optimal values. Simulation results indicate that (a) this approach improves approximation accuracy, and (b) that the number of rules can be reduced for any given level of accuracy. For the purpose of illustrating the proposed method, the FNN₅ is also applied to tune PI controllers such that gain and phase margins of the closed‐loop system achieve the desired specifications.     

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     

Stability Analysis and Performance Design for Fuzzy-Model-Based Control System Under Imperfect Premise Matching

This paper investigates the stability analysis and performance design of nonlinear systems. To facilitate the stability analysis, the Takagi–Sugeno (T–S) fuzzy model is employed to represent the nonlinear plant. Under the imperfect premise matching in which T–S fuzzy model and fuzzy controller do not share the same membership functions, a fuzzy controller with enhanced design flexibility and robustness property is proposed to control the nonlinear plant. However, the nice characteristic given by the perfect premise matching, leading to conservative stability conditions, vanishes. In this paper, under the imperfect premise matching, information of membership functions of the fuzzy model and controller are considered in stability analysis. With the introduction of slack matrices, relaxed linear matrix inequality (LMI)-based stability conditions are derived using Lyapunov-based approach. Furthermore, LMI-based performance conditions are provided to guarantee system performance. Simulation examples are given to illustrate the effectiveness of the proposed approach.      

Stability Conditions for LMI-Based Fuzzy Control From Viewpoint of Membership Functions

In this paper, we investigate the stability conditions for linear matrix inequality (LMI)-based fuzzy control design. Especially, we focus on the dependence of the stability upon membership functions. In general, the membership functions in the rule bases of Takagi-Sugeno (T-S) fuzzy model and controllers are the same and restricted between 0 and 1. In contrast to this setting, we obtain some new results when different membership functions are considered and their values lying outside the interval of [0,1] are allowed. Applying Lyapunov equation and a convex hull of fuzzy subsystems, we first establish a relationship between the stable interval characteristic polynomial and a set of feasible LMIs. Then Kharitonov's theorem gives an insight for the solvability of stabilization problems using LMI-based design and, this leads that the membership functions have an influence on stability. On the other hand, the LMI condition leads to the well-known results for LMI-based fuzzy control design. We further indicate that the different LMI conditions arise due to the same or different membership functions and find their own applications on adaptive fuzzy control. Finally, if the unit interval constraint is removed, an LMI condition for global stability is obtained     

Decentralized Control of Autonomous Swarm Systems Using Artificial Potential Functions: Analytical Design Guidelines

This paper presents a framework for decentralized control of self-organizing swarm systems based on the artificial potential functions (APFs). In this scheme, multiple agents in a swarm self-organize to flock and achieve formation control through attractive and repulsive forces among themselves using APFs. In particular, this paper presents a set of analytical guidelines for designing potential functions to avoid local minima for a number of representative scenarios. Specifically the following cases are addressed: 1) A non-reachable goal problem (a case that the potential of the goal is overwhelmed by the potential of an obstacle, 2) an obstacle collision problem (a case that the potential of the obstacle is overwhelmed by the potential of the goal), 3) an obstacle collision problem in swarm (a case that the potential of the obstacle is overwhelmed by potential of other robots in a group formation) and 4) an inter-robot collision problem (a case that the potential of the robot in a formation is overwhelmed by potential of the goal). The simulation results showed that the proposed scheme can effectively construct a self-organized swarm system with the capability of group formation, navigation and migration in the presence of obstacles.Category (5) – Intelligent Systems/Intelligent Control/Fuzzy Control/Prosthetics/Robot Motion Planning     

方块脉冲函数用于非线性系统的分析以及最优控制的综合

本文用方块脉冲函数方法讨论了解非线性微分方程系统的收敛性和稳定性.在分析和综合非线性系统最优控制规律中,得到了分段恒定解答的递推算法,算法证明简单,除对基于二次型性能指标的线性时变系统有效外,也可用于求解线性最优控制系统的Riccati方程.     

非线性控制系统中的Лурье函数

本文在已有文献基础上,提出最佳和次佳函数概念,并给出寻找方法。这些结果既可以用来作理论分析,又可以用来实际确定绝对稳定角域中的K值,与传统的频率方法相比,优点是能较方便地给出相应的型V函数。     

非线性控制系统中的Лypbe函数

本文在已有文献基础上，提出最佳和次佳Лypbe函数概念，并给出寻找方法。这些结果既可以用来作理论分析，又可以用来实际确定绝对稳定角域中的K值，与传统的频率方法相比，优点是能较方便地给出相应的 Лypbe型V函数。     

采用三角型隶属度函数的模糊系统的插值特性

研究了采用三角型隶属度函数的模糊系统的插值特性.针对常见的Mamdani和TSK两类模糊系统,研究了模糊系统与分段插值函数在函数逼近上的等价性.在此基础上,分别给出了两类模糊系统在插值方面的逼近误差估计.这一结果为模糊系统的结构分析提供了新的研究方法和理论依据.     

A Genetic Approach for Simultaneous Design of Membership Functions and Fuzzy Control Rules

Based on the genetic algorithm (GA), an approach is proposed for simultaneous design of membership functions and fuzzy control rules since these two components are interdependent in designing a fuzzy logic controller (FLC). With triangular membership functions, the left and right widths of these functions, the locations of their peaks, and the fuzzy control rules corresponding to every possible combination of input linguistic variables are chosen as parameters to be optimized. By using a proportional scaling method, these parameters are then transformed into real-coded chromosomes, over which the offspring are generated by rank-based reproduction, convex crossover, and nonuniform mutation. Meanwhile, the concept of enlarged sampling space is used to expedite the convergence of the evolutionary process. To show the feasibility and validity of the proposed method, a cart-centering example will be given. The simulation results will show that the designed FLC can drive the cart system from any given initial state to the desired final state even when the cart mass varies within a wide range.     

Asymptotic Stability Analysis for Switched Stochastic Nonlinear Systems Using Mode-dependent Uniformly Stable Functions

International Journal of Control, Automation and Systems - In this paper, we intend to investigate uniform global asymptotic stability in probability (UGAS-P) for a class of time-varying switched...     

Homogeneous Feedback Control of Nonlinear Systems Based on Control Lyapunov Functions

This paper is concerned with homogeneous feedback control for a class of nonlinear systems. By using control Lyapunov functions, homogeneous controllers of homogeneous and nonhomogeneous systems are respectively constructed, under which the stabilization of the systems under considerations are guaranteed. Then, the design method is extended for uncertain systems by means of homogeneous domination theory. Compared with the traditional design method based on robust control Lyapunov functions, the present design reduces the difficulty of constructing controllers. Finally, several simulation examples are given to illustrate the correctness of the design.     

采用正交函数辨识电液控制系统

本文采用正交函数BPF将含有未知参数的系统线性微分方程转化为代数向量方程进行求解而完成辨识过程,获取表征系统特性的传递函数或微分方程参数.仿真辨识及对实际电液控制系统的辨识结果均表明,BPF辨识法具有原理简洁、实现方便及辨识精度高等优点.任何可积函数均可作为辨识输入信号,辨识过程可由递推算法实现.所以BPF辨识法能方便有效地用于动态过程的系统实时在线辨识.