Parameterized linear matrix inequality techniques in fuzzy controlsystem design

This paper proposes different parameterized linear matrix inequality (PLMI) characterizations for fuzzy control systems. These PLMI characterizations are, in turn, relaxed into pure LMI programs, which provides tractable and effective techniques for the design of suboptimal fuzzy control systems. The advantages of the proposed methods over earlier ones are then discussed and illustrated through numerical examples and simulations     

State-feedback stabilisation for fuzzy bilinear uncertain system with disturbance via fuzzy control approach

This article presents a fuzzy control scheme for a class of Takagi–Sugeno (T–S) fuzzy uncertain bilinear system with disturbance (FUBSD). First, the T–S FUBSD is established and based on the parallel distributed compensation method, the overall robust H _∞ fuzzy controller is proposed to globally stabilise the T–S FUBSD. Then, some sufficient conditions are derived to guarantee the robust stabilisability of the overall fuzzy control system via linear matrix inequalities. Finally, a numerical example is utilised to demonstrate the validity and effectiveness of the proposed control scheme.     

Stability analysis and synthesis for an affine fuzzy system via LMIand ILMI: discrete case

This paper develops a stability analysis and controller synthesis methodology for a discrete affine fuzzy system based on the convex optimization techniques. In analysis, the stability condition under which the affine fuzzy system is quadratically stable is derived. Then, the condition Is recast in the formulation of Linear Matrix Inequalities (LMI) and numerically addressed. The emphasis of this paper, however, is on the synthesis of fuzzy controller based on the derived stability condition. In synthesis, the stabilizability condition turns out to be in the formulation of nonconvex matrix inequalities and is solved numerically in an iterative manner. Discrete iterative LMI (ILMI) approach is proposed to obtain the feasible solution for the synthesis of the affine fuzzy system. Finally, the applicability of the suggested methodology is demonstrated via some examples and computer simulations.     

Parameterized bilinear matrix inequality techniques for gain‐scheduling proportional integral derivative control design

Proportional‐integral‐derivative (PID) structured controller is the most popular class of industrial control but still could not be appropriately exploited in gain‐scheduling control systems. To gain the practicability and tractability of gain‐scheduling control systems, this paper addresses the gain‐scheduling PID control. The design of such a controller is based on parameterized bilinear matrix inequalities, which are then solved via a bilinear matrix inequality optimization problem of nonconvex optimization. Several computational procedures are developed for its computation. The merit of the developed algorithms is shown through the benchmark examples.     

Output tracking design of affine nonlinear plant via variablestructure system

An output tracking design for affine nonlinear systems via a variable structure system is proposed. The structure algorithm is used to determine if the considered system is left invertible. It is shown that if such a condition is satisfied, then a sliding mode controller can be designed to achieve output tracking. The decouplability condition in conventional approaches is now replaced by the weaker left-invertibility condition. Stable tracking is guaranteed if the internal mode is bounded-input-bounded-state stable. Simulation results are included to demonstrate the design procedure     

Non-linear control system design via fuzzy modelling and LMIs

The method of fuzzy-model-based control has emerged as an alternative approach to the solution of analysis and synthesis problems associated with plants that exhibit complex non-linear behaviour. At present, the literature in this field has addressed the control design problem related to the stabilization of state-space fuzzy models. In practical situations, however, where perturbations exist in the state-space model, the problem becomes one of robust stabilization that has yet to be posed and solved. The present paper contributes in this direction through the development of a framework that exploits the distinctive property of the fuzzy model as the convex hull of linear system matrices. Using such a quasi-linear model structure, the robust stabilization of complex non-linear systems, against modelling error and parametric uncertainty, based on static state or dynamic output feedback, is reduced to a linear matrix inequality (LMI) problem.     

A heuristic approach to solving a class of bilinear matrix inequality problems

Many canonical and modern control problems can be recast into the problem of a group of matrix inequalities. Some of them are in the form of linear matrix inequalities (LMIs), which can be solved very efficiently by the powerful LMI toolbox in Matlab, but some others are in the form of bilinear matrix inequalities. The characteristic of this latter class of problems is that when the so called “communicating variables” are fixed, the overall problem will be reduced to the problem in LMIs. Thus, how to find the communicating variables is the key to solve the whole problem. In this paper, an optimal estimate for the communicating variables is presented. We will illustrate our method by completely solving the problems of overshoot bound control and reachable set analysis for uncertain systems. Numerical examples are provided to show the effectiveness of the proposed method.     

Linear matrix inequality tests for frequency domain inequalities with affine multipliers

This paper revisits an alternative formulation of the Kalman-Yakubovich-Popov (KYP) Lemma, relating an infinite dimensional Frequency Domain Inequality (FDI) to a pair of finite dimensional Linear Matrix Inequalities (LMI). It is shown that this alternative formulation allows a certain class of the coefficient matrix of the FDI to be frequency dependent without introducing conservatism. The construction provided in the present paper is helpful in other problems where system augmentation is commonly used, such as those involving rational or polynomial multipliers for stability and performance analysis.     

Linear matrix inequality conditions for robustness and control design

In this paper, well‐known stability conditions for linear, time‐invariant systems subject to structured uncertainty are extended to encompass matrix valued and affine integral constraints on the system inputs and outputs. Necessary and sufficient analysis conditions are derived in terms of linear matrix inequalities. For a large class of these problems, the synthesis question can also be expressed as a linear matrix inequality. Copyright © 2001 John Wiley & Sons, Ltd.     

基于LMI的仿射型模糊控制系统的稳定性研究

基于凸优化方法，分析了仿射型模糊控制系统的稳定性与镇定问题。首先以LMI的形式给出了仿射型模糊系统稳定的充分条件；然后基于这一条件讨论了闭环系统的镇定问题。得出的镇定条件是BMI，使用轮换极小法将这一BMI问题转化为2个LMI问题，从而可以用MATLAB中的LMI优化工具箱进行求解。仿真例子说明了所提方法的有效性。     

Non-fragile control of fuzzy affine dynamic systems via piecewise Lyapunov functions

This paper addresses the robust H_∞ static output feedback (SOF) controller design problem for a class of uncertain fuzzy affine systems that are robust against both the plant parameter perturbations and controller gain variations. More specifically, the purpose is to synthesize a non-fragile piecewise affine SOF controller guaranteeing the stability of the resulting closed-loop fuzzy affine dynamic system with certainH_∞ performance index. Based on piecewise quadratic Lyapunov functions and applying some convexification procedures, two different approaches are proposed to solve the robust and non-fragile piecewise affine SOF controller synthesis problem. It is shown that the piecewise affine controller gains can be obtained by solving a set of linear matrix inequalities (LMIs). Finally, simulation examples are given to illustrate the effectiveness of the proposed methods.     

模糊不确定时滞系统的静态输出的反馈镇定:迭代线性矩阵不等式方法

对一类不确定非线性时滞系统利用模糊T＿S模型进行建模,研究了静态输出反馈镇定问题.用矩阵不等式的形式给出了模糊T＿S不确定时滞系统可通过静态输出反馈镇定的充分条件.并将矩阵不等式的条件转化为迭代线性矩阵不等式(ILMI),并给出相应的算法.     

Robust control of uncertain multi-inventory systems via linear matrix inequality

We consider a continuous time linear multi-inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. We address the problem of ε-stabilising the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which ε-stabilisability is possible through a saturated linear state feedback control. All the results are based on a linear matrix inequalities approach and on some recent techniques for the modelling and analysis of polytopic systems with saturations. Numerical simulations are provided.     

Analysis of bilinear systems via single-term Walsh series

This paper presents a new method of analysis of bilinear systems via the single-term Walsh series (STWS). The STWS method has definite advantages over the Walsh function (WF) method and block-pulse function (BPF) method. Both the discrete values and block-pulse values of the results are obtained for any length of time and selecting any positive integer as the number of intervals. The recursive relations are simple and save both computer time and storage. Illustrative examples are presented.     

Robust structure from motion with affine camera via low-rank matrix recovery

We present a novel approach to structure from motion that can deal with missing data and outliers with an affine camera. We model the corruptions as sparse error. Therefore the structure from motion problem is reduced to the problem of recovering a low-rank matrix from corrupted observations. We first decompose the matrix of trajectories of features into low-rank and sparse components by nuclear-norm and l1-norm minimization, and then obtain the motion and structure from the low-rank components by the classical factorization method. Unlike pervious methods, which have some drawbacks such as depending on the initial value selection and being sensitive to the large magnitude errors, our method uses a convex optimization technique that is guaranteed to recover the low-rank matrix from highly corrupted and incomplete observations. Experimental results demonstrate that the proposed approach is more efficient and robust to large-scale outliers.     

Calculating fuzzy inverse matrix using fuzzy linear equation system

Linear equation systems play a very important role in engineering, mathematics, statistics and other disciplines. Fuzzifying either parameters or variables or both in these systems has been one of the research areas in the fuzzy literature since these kinds of systems are encountered in many applications. These systems are generally called fuzzy linear equations. Various types of these models have been examined for a decade. The solution procedures of these systems depend on different methods such as extension principle and interval arithmetic. Also, the method which is often used in computing inverse of a matrix in real case could be extended to fuzzy case, which employs linear equation system and identity matrix. For this purpose, we propose a new method which includes some new definitions which are fuzzy zero number, fuzzy one number and fuzzy identity matrix. Based on these definitions, direct computation of fuzzy inverse matrix is done using fuzzy arithmetic and fuzzy equation system. Actually, this simply extends the notion used in real case to fuzzy case. Calculation is realized with two different settings. While the first one is called direct numerical solution, the other is obtained by choice of decision maker. It is noted that the uniqueness of the calculated fuzzy inverse matrix is not guaranteed.     

Analysis and design of fuzzy controller and fuzzy observer

This paper addresses the analysis and design of a fuzzy controller and a fuzzy observer on the basis of the Takagi-Sugeno (T-S) fuzzy model. The main contribution of the paper is the development of the separation property; that is, the fuzzy controller and the fuzzy observer can be independently designed. A numerical simulation and an experiment on an inverted pendulum system are described to illustrate the performance of the fuzzy controller and the fuzzy observer     

Asymptotic stabilization of the bilinear time-invariant system via piecewise-constant feedback

Sufficient conditions for the asymptotic stabilization in a preassigned domain of the nonhomogeneous bilinear system (BLS) via piecewise-constant feedback are given, based on those for an auxillary BLS with additional control in the drift term. Application to the swing equation is also discussed.     

Feedback stabilization of fuzzy systems via linear matrix inequalities

This paper considers the problem of designing robust feedback stabilization controller for a class of fuzzy systems. Based on piecewise Lyapunov functions, new design techniques for state feedback and output feedback controllers are proposed. The resultant controller is robust against measurement and modelling perturbations. The design conditions are non-conservative, and the design process is easy to construct by using commercially supported linear matrix inequalities software packages.     

Analysis of linear time-varying systems and bilinear systems via Taylor series

Linear time-varying systems and bilinear systems are each analysed via Taylor series. Using the operational matrix for integration and the product operational matrix, the dynamical equation of a linear time-varying system (or a bilinear system) is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the Taylor series expansions can be determined recursively by the algorithm derived. The algorithm proposed here is similar to those already developed for orthogonal functions; however, owing to the simplicity of the operational matrix of integration and the product operational matrix, Taylor series present considerable computational advantages compared with the other polynomial series, provided that both the input and the output signals are analytic functions of t.