Robust PID‐PSD Controller Design: BMI Approach

A new robust proportional‐integral‐derivative (PID)–proportional‐sum‐derivative (PSD) controller design method based on linear (bilinear) matrix inequalities (LMI, BMI) is proposed for uncertain affine linear system. The design procedure guarantees the parameter dependent quadratic stability, and guaranteed cost control with a new quadratic cost function (LQRS) including the derivative term for the state vector as a tool to influence the overshoot and response rate. The second approach to the PSD controller design procedure is based on a Lyapunov function with a special term corresponding to the time‐delay part of the control algorithm. The results obtained are illustrated on three examples to show the robust PID, PSD control design procedure and the influence of the choice of matrix S in the extended cost function.     

H_∞ controller synthesis of piecewise discrete time linear systems

This paper presents an H∞ controller design method for pieccwise discrete time linear systems based on a piecewise quadratic Lyapunov function. It is shown that the resulting closed loop system is globally stable with guaranteed H∞ perfomiance and the controller can be obtained by solving a set of bilinear lnatrLx inequalities. It has been shown that piecewise quadratic Lyapunov functions are less conservative than the global qnadnmc Lyapunov functions. A simulation example is also given to illustrate the advantage of the proposed approach.     

跳变双线性随机系统饱和执行器的镇定

对于一类具有Markov跳变参数的双线性离散随机系统,研究其饱和执行器问题.分别采用一般二次型Lyapunov函数、饱和关联Lyapunov函数进行系统随机稳定性分析,以椭圆不变集构造随机稳定域,提出两种依赖于模态跳变率的饱和状态控制器设计方法,两种方法均以线性矩阵不等式的形式给出.     

Robust stabilization of the Takagi-Sugeno fuzzy model via bilinearmatrix inequalities

Quadratic stability has enabled, mainly via the linear matrix inequality framework, the analysis and design of a nonlinear control system from the local matrices of the system's Takagi-Sugeno (T-S) fuzzy model. It is well known, however, that there exist stable differential inclusions, hence T-S fuzzy models whose stability is unprovable by a globally quadratic Lyapunov function. At present, literature in the broader area of stability analysis suggests piecewise-quadratic stability as a means to avoid such conservatism. This paper generalizes the idea and proposes a framework that supports less conservative sufficient conditions for the stability of the T-S model by using piecewise-quadratic generalized Lyapunov functions. The advocated approach results in the formulation of the controller synthesis, which, herein, aims for robust stabilization, as a problem of bilinear rather than linear matrix inequalities. Simulation studies, which include an algorithm for solution of bilinear matrix inequalities, demonstrate the proposed method     

Output-feedback sampled-data polynomial controller for nonlinear systems

This paper presents the stability analysis and control synthesis for a sampled-data control system which consists of a nonlinear plant and an output-feedback sampled-data polynomial controller connected in a closed loop. The output-feedback sampled-data polynomial controller, which can be implemented by a microcontroller or a digital computer, is proposed to stabilize the nonlinear plant. Based on the Lyapunov stability theory, stability conditions in terms of sum of squares are obtained to guarantee the stability and to aid the design of a polynomial controller. A simulation example is given to demonstrate the effectiveness of the proposed control approach.     

一类多项式非线性系统鲁棒H∞控制

针对一类具有多项式向量场的仿射型不确定非线性系统,给出一种基于多项式平方和(sum of squares,SOS)技术的鲁棒H∞状态反馈控制器设计方法.该方法的优点在于控制器的设计避开了直接求解复杂的哈密尔顿-雅可比不等式(Hamilton Jacobi inequality,HJI)和构造Lyapunov函数带来的困难.将鲁棒稳定性分析和控制器设计问题转化为求解以Lyapunov函数为参数的矩阵不等式,该类不等式可利用SOS技术直接求解.此外,在前文基础上研究了基于SOS规划理论与S-procedure技术的局部稳定鲁棒H∞控制器设计方法.最后以非线性质量弹簧阻尼系统作为仿真算例验证该方法的有效性.     

Optimal fuzzy controller based on non-monotonic Lyapunov function with a case study on laboratory helicopter

This paper presents a new approach to design an observer-based optimal fuzzy state feedback controller for discrete-time Takagi–Sugeno fuzzy systems via LQR based on the non-monotonic Lyapunov function. Non-monotonic Lyapunov stability theorem proposed less conservative conditions rather than common quadratic method. To compare with optimal fuzzy feedback controller design based on common quadratic Lyapunov function, this paper proceeds reformulation of the observer-based optimal fuzzy state feedback controller based on common quadratic Lyapunov function. Also in both methodologies, the dependence of optimisation problem on initial conditions is omitted. As a practical case study, the controllers are implemented on a laboratory twin-rotor helicopter to compare the controllers' performance.     

不确定广义双线性系统的鲁棒镇定

针对带有不确定参数的广义双线性系统的鲁棒镇定问题进行研究, 其中不确定参数是时变且范数有界的. 本文的目的是设计状态反馈控制器, 使得对所有满足条件的不确定参数闭环系统都是渐近稳定的. 通过引入广义二次稳定的概念、采用Lyapunov方法, 分别给出了不确定广义双线性系统在两种不同情形下可镇定的充分条件; 此外, 两个数值例子分别说明了两种设计方法的有效性和合理性.     

Guaranteed Cost Control of Polynomial Fuzzy Systems via a Sum of Squares Approach

This paper presents the guaranteed cost control of polynomial fuzzy systems via a sum of squares (SOS) approach. First, we present a polynomial fuzzy model and controller that are more general representations of the well-known Takagi–Sugeno (T-S) fuzzy model and controller, respectively. Second, we derive a guaranteed cost control design condition based on polynomial Lyapunov functions. Hence, the design approach discussed in this paper is more general than the existing LMI approaches (to T-S fuzzy control system designs) based on quadratic Lyapunov functions. The design condition realizes a guaranteed cost control by minimizing the upper bound of a given performance function. In addition, the design condition in the proposed approach can be represented in terms of SOS and is numerically (partially symbolically) solved via the recent developed SOSTOOLS. To illustrate the validity of the design approach, two design examples are provided. The first example deals with a complicated nonlinear system. The second example presents micro helicopter control. Both the examples show that our approach provides more extensive design results for the existing LMI approach.      

A Sum-of-Squares Approach to Modeling and Control of Nonlinear Dynamical Systems With Polynomial Fuzzy Systems

This paper presents a sum of squares (SOS) approach for modeling and control of nonlinear dynamical systems using polynomial fuzzy systems. The proposed SOS-based framework provides a number of innovations and improvements over the existing linear matrix inequality (LMI)-based approaches to Takagi--Sugeno (T--S) fuzzy modeling and control. First, we propose a polynomial fuzzy modeling and control framework that is more general and effective than the well-known T--S fuzzy modeling and control. Secondly, we obtain stability and stabilizability conditions of the polynomial fuzzy systems based on polynomial Lyapunov functions that contain quadratic Lyapunov functions as a special case. Hence, the stability and stabilizability conditions presented in this paper are more general and relaxed than those of the existing LMI-based approaches to T--S fuzzy modeling and control. Moreover, the derived stability and stabilizability conditions are represented in terms of SOS and can be numerically (partially symbolically) solved via the recently developed SOSTOOLS. To illustrate the validity and applicability of the proposed approach, a number of analysis and design examples are provided. The first example shows that the SOS approach renders more relaxed stability results than those of both the LMI-based approaches and a polynomial system approach. The second example presents an extensive application of the SOS approach in comparison with a piecewise Lyapunov function approach. The last example is a design exercise that demonstrates the viability of the SOS-based approach to synthesizing a stabilizing controller.      

Quantized robust ℋ∞ control of discrete-time systems with random communication delays

The article considers stability and robust ?_∞ controller design of discrete-time systems with random communication delays and state quantization. A finite state Markov process is used to model communication delays between sensors and controllers. Measurements are assumed to be quantized by a logarithmic quantizer, and the effect of quantization errors are incorporated into the controller design. Based on a Lyapunov–Krasovskii approach, novel methodologies for analysing stability and designing a time-delay mode-dependent quantized state feedback controller are proposed. The controller is obtained through solving bilinear matrix inequalities (BMIs) using the cone complementarity linearisation algorithm.     

Analysis of partial stability problems using sum of squares techniques

A methodology for algorithmic construction of Lyapunov functions for problems concerning the stability of an equilibrium with respect to part of the system variables is proposed. This methodology utilizes the previously developed sum of squares technique to determine Lyapunov certificates. Conditions for stability with respect to part of the variables are developed that allow for Lyapunov functions to be determined in terms of a sum of squares. Asymptotic stability conditions in terms of sum of squares polynomials are developed for autonomous and non-autonomous systems. An example is presented which demonstrates the methodology and gives insight into the new stability conditions.     

Output feedback controller design for uncertain piecewise linear systems

This paper proposes output feedback controller design methods for uncertain piecewise linear systems based on piecewise quadratic Lyapunov function. The α-stability of closed-loop systems is also considered. It is shown that the output feedback controller design procedure of uncertain piecewise linear systems with α-stability constraint can be cast as solving a set of bilinear matrix inequalities （BMIs）. The BMIs problem in this paper can be solved iteratively as a set of two convex optimization problems involving linear matrix inequalities （LMIs） which can be solved numerically efficiently. A numerical example shows the effectiveness of the proposed methods.     

广义双线性系统的最优控制

最优控制是自动控制理论的重要研究分支,本文首次对广义双线性系统的最优控制问题进行研究.利用李雅普诺夫稳定性理论和广义李雅普诺夫方程的解来设计最优控制器,使得闭环系统全局渐近稳定且使广义二次性能指标最小.此外,还给出最优化控制器的设计方法,整个设计过程简单,具有较少的保守性,例子表明设计方法的有效性和合理性.     

H/sub /spl infin// controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and bilinear matrix inequalities

This work presents an H/sub /spl infin// controller design method for fuzzy dynamic systems based on techniques of piecewise smooth Lyapunov functions and bilinear matrix inequalities. It is shown that a piecewise continuous Lyapunov function can be used to establish the global stability with H/sub /spl infin// performance of the resulting closed-loop fuzzy control systems and the control laws can be obtained by solving a set of bilinear matrix inequalities (BMIs). Two examples are given to illustrate the application of the proposed methods.     

An improved sum‐of‐squares based approach to fuzzy tracking control design of nonlinear systems

In this paper, using a more general Lyapunov function, less conservative sum‐of‐squares (SOS) stability conditions for polynomial‐fuzzy‐model‐based tracking control systems are derived. In tracking control problems the objective is to drive the system states of a nonlinear plant to follow the system states of a given reference model. A state feedback polynomial fuzzy controller is employed to achieve this goal. The tracking control design is formulated as an SOS optimization problem. Here, unlike previous SOS‐based tracking control approaches, a full‐state‐dependent Lyapunov matrix is used, which reduces the conservatism of the stability criteria. Furthermore, the SOS conditions are derived to guarantee the system stability subject to a given H_∞ performance. The proposed method is applied to the pitch‐axis autopilot design problem of a high‐agile tail‐controlled pursuit and another numerical example to demonstrate the effectiveness and benefits of the proposed method.     

A sum of squares approach to backstepping controller synthesis for piecewise affine and polynomial systems

This paper develops a backstepping controller synthesis methodology for piecewise polynomial (PWP) systems in strict form. The main contribution of the paper is to formulate sufficient conditions for controller design for PWP systems in strict form as a sum of squares feasibility problem under the assumption that an initial control Lyapunov function exists to start the iterative backstepping procedure. This problem can then be translated into a convex SDP problem and solved by available software packages. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares polynomial control Lyapunov function for PWP systems with discontinuous vector fields. The second case addresses the construction of a PWP control Lyapunov function for PWP systems with continuous vector fields. One major advantage of the proposed method is the fact that it can handle systems with discontinuous vector fields and sliding modes. The new synthesis method is applied to several numerical examples. One of these examples offers the first convex optimization solution to piecewise affine (PWA) control of a benchmark circuit system addressed before in the literature using non‐convex PWA control solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Novel Approach to Switched Controller Design for Linear Continuous‐Time Systems

In this paper, we study a novel approach to the design of an output feedback switched controller with an arbitrary switching algorithm for continuous‐time invariant systems that is described by a novel plant model as a gain‐scheduled plant using the multiple quadratic stability and quadratic stability approaches. In the proposed design procedure, there is no need to use the notion of the "dwell time". The obtained results are in the form of bilinear matrix inequalities (BMI). Numerical examples show that, in the proposed method, the design procedure is less conservative and gives more possibilities than that described in the papers published previously.     

Output feedback optimal guaranteed cost control of uncertain piecewise linear systems

This paper proposes the output feedback optimal guaranteed cost controller design method for uncertain piecewise linear systems based on the piecewise quadratic Lyapunov functions technique. By constructing piecewise quadratic Lyapunov functions for the closed‐loop augmented systems, the existence of the guaranteed cost controller for closed‐loop uncertain piecewise linear systems is cast as the feasibility of a set of bilinear matrix inequalities (BMIs). Some of the variables in BMIs are set to be searched by genetic algorithm (GA), then for a given chromosome corresponding to the variables in BMIs, the BMIs turn to be linear matrix inequalities (LMIs), and the corresponding non‐convex optimization problem, which minimizes the upper bound on cost function, reduces to a semidefinite programming (SDP) which is convex and can be solved numerically efficiently with the available software. Thus, the output feedback optimal guaranteed cost controller can be obtained by solving the non‐convex optimization problem using the mixed algorithm that combines GA and SDP. Numerical examples show the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

A new calculation method of feedback controller gain for bilinear paper-making process with disturbance

This paper presents a new method to calculate the feedback control gain for a class of multivariable bilinear system, and also applied this method on the control of two sections of paper-making process with disturbance. The robust H∞ control problem is to design a state feedback controller such that the robust stability and a prescribed H∞ performance of the resulting closed-loop system are ensured. The controller turns out to be robust with respect to the disturbance in the plant. Utilizing the Schur complement and some variable transformations, the stability conditions of the multivariable bilinear systems are formulated in terms of Lyapunov function via the form of linear matrix inequality (LMI). The gain of controller will be calculated via LMI. Finally, the application examples of a headbox section and a dryer section of paper-making process are used to illustrate the applicability of the proposed method.