分段线性系统分析与综合
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摘要
本文主要研究了分段线性系统(Piecewise Linear Systems)的输出反馈镇定、H_∞控制、保性能控制问题,并把分段线性系统理论应用于一类非线性系统的保性能控制设计中,取得了一些新的结论。主要创新如下:
一、给出了具有α稳定裕度约束的不确定分段线性系统的输出反馈控制器设计方法。对闭环增广系统构造了增广的分段二次Lyapunov函数,将极大化α性能的控制器设计问题转化成一个受双线性矩阵不等式组(Bilinear MatrixInequalities,BMIs)约束的非凸优化问题,动态输出反馈控制器便可以通过求解该BMIs问题得到。
二、给出了求解BMIs问题的一种新的基于遗传算法和内点法的混合算法。给出的新算法避免了传统算法的容易陷入局部极小、对大规模系统不再有效的缺点。用遗传算法对BMIs的双线性项中的一些变量进行随机搜索,使问题转化成一个受一组线性矩阵不等式(Linear Matrix Inequalities,LMIs)约束的半正定规划(Semidefinite Programming,SDP)问题,从而可利用基于内点法的相关软件进行求解。所设计算法对BMIs问题具有更强的全局最优的搜索能力,且实现方便。
三、给出了不确定分段线性系统的输出反馈H_∞控制器设计方法。通过对闭环增广系统设计的分段二次Lyapunov函数,将不确定分段线性系统的动态输出反馈H_∞控制器设计问题转化成一组BMIs的可行性问题,而最优H_∞控制器可以通过求解一个受BMIs约束的非凸优化问题得到。
四、分别给出了不确定分段线性系统的状态反馈和输出反馈保性能控制器设计方法。利用分段二次Lyapunov函数及Hamilton-Jacobi-Bellman(H-J-B)不等式方法,得到了分段二次型性能函数的下界和上界。其中性能函数下界可通过求解SDP问题得到,而性能函数最优上界可通过求解BMIs问题得到,从而得到了最优的保性能控制器。
五、给出了基于分段线性系统理论的非线性系统保性能控制设计方法。对非线性系统,将其进行分段线性化处理,然后用分段线性系统保性能控制设计的方法研究了非线性系统的最优状态反馈保性能控制的问题,从而可以通过求解一个BMIs问题得到最优保性能控制器。
In this dissertation,the output feedback stabilization,H_∞control,guaranteed cost control of piecewise linear systems are investigated deeply.In addition, the guaranteed cost controller design fbr a class of nonlinear systems is studied with the corresponding results of piecewise linear systems.The main contributions are as fbllows.
Firstly,the approach to the output feedback control fbr uncertain piecewise linear systems withα-stability constraint is presented.To make use of piecewise quadratic Lyapunov functions technique for the perfbrmance analysis of closed-loop augmented systems,the augmented piecewise-continuous quadratic Lyapunov functions are reconstructed.It is shown that the output feedback controller design procedure of uncertain piecewise linear systems withα-stability constraint can be cast as solving a set of bilinear matrix inequalities(BMIs).
Secondly,a new mixed algorithm that combines genetic algorithm(GA) and interior-point method is designed fbr solving the BMIs problem which is an NP hard problem.Specifically,some of the variables in BMIs are set to be searched by GA,then the corresponding non-convex problem reduces to the semidefinite programming(SDP)involving LMIs,which is convex and can be solved numerically with available software based on interior-point method.The proposed algorithm can be easy to carry out,and overcomes the shortcomings of the existent algorithms which are hard to converge to the global optimum or are not computationally tractable fbr large scale problems.
Thirdly,the design of robust H_∞output feedback controller fbr uncertain piecewise linear systems is presented.By constructing piecewise continuous Lyapunov functions fbr the closed-loop augmented systems,the H_∞output feedback controller design is cast as the feasibility problem of a set of BMIs,and the optimal H_∞controller can be obtained by solving a non-convex optimization problem under the constraints of BMIs.
Fourthly,the guaranteed cost control fbr uncertain piecewise linear systems via state feedback and output feedback are investigated respectively based on the piecewise quadratic Lyapunov functions technique and Hamilton-Jacobi-Bellman (H-J-B)inequality method.It has been shown that both the state feedback opti- mal controller and output feedback optimal controller,which minimize the upper bounds on cost function,can be obtained respectively by solving a non-convex optimization problem under the BMIs constraints.The controller obtained can be judged by a lower bound on cost function which can be obtained by SDP.
Finally,the optimal state feedback guaranteed cost controller design for the nonlinear systems is presented.The nonlinear systems are described as a class of uncertain piecewise linear systems,then with the results of guaranteed cost controller design for piecewise linear systems,the optimal guaranteed cost controller for the nonlinear systems can be obtained by solving a non-convex optimization problem under the BMIs constraints.
一、给出了具有α稳定裕度约束的不确定分段线性系统的输出反馈控制器设计方法。对闭环增广系统构造了增广的分段二次Lyapunov函数,将极大化α性能的控制器设计问题转化成一个受双线性矩阵不等式组(Bilinear MatrixInequalities,BMIs)约束的非凸优化问题,动态输出反馈控制器便可以通过求解该BMIs问题得到。
二、给出了求解BMIs问题的一种新的基于遗传算法和内点法的混合算法。给出的新算法避免了传统算法的容易陷入局部极小、对大规模系统不再有效的缺点。用遗传算法对BMIs的双线性项中的一些变量进行随机搜索,使问题转化成一个受一组线性矩阵不等式(Linear Matrix Inequalities,LMIs)约束的半正定规划(Semidefinite Programming,SDP)问题,从而可利用基于内点法的相关软件进行求解。所设计算法对BMIs问题具有更强的全局最优的搜索能力,且实现方便。
三、给出了不确定分段线性系统的输出反馈H_∞控制器设计方法。通过对闭环增广系统设计的分段二次Lyapunov函数,将不确定分段线性系统的动态输出反馈H_∞控制器设计问题转化成一组BMIs的可行性问题,而最优H_∞控制器可以通过求解一个受BMIs约束的非凸优化问题得到。
四、分别给出了不确定分段线性系统的状态反馈和输出反馈保性能控制器设计方法。利用分段二次Lyapunov函数及Hamilton-Jacobi-Bellman(H-J-B)不等式方法,得到了分段二次型性能函数的下界和上界。其中性能函数下界可通过求解SDP问题得到,而性能函数最优上界可通过求解BMIs问题得到,从而得到了最优的保性能控制器。
五、给出了基于分段线性系统理论的非线性系统保性能控制设计方法。对非线性系统,将其进行分段线性化处理,然后用分段线性系统保性能控制设计的方法研究了非线性系统的最优状态反馈保性能控制的问题,从而可以通过求解一个BMIs问题得到最优保性能控制器。
In this dissertation,the output feedback stabilization,H_∞control,guaranteed cost control of piecewise linear systems are investigated deeply.In addition, the guaranteed cost controller design fbr a class of nonlinear systems is studied with the corresponding results of piecewise linear systems.The main contributions are as fbllows.
Firstly,the approach to the output feedback control fbr uncertain piecewise linear systems withα-stability constraint is presented.To make use of piecewise quadratic Lyapunov functions technique for the perfbrmance analysis of closed-loop augmented systems,the augmented piecewise-continuous quadratic Lyapunov functions are reconstructed.It is shown that the output feedback controller design procedure of uncertain piecewise linear systems withα-stability constraint can be cast as solving a set of bilinear matrix inequalities(BMIs).
Secondly,a new mixed algorithm that combines genetic algorithm(GA) and interior-point method is designed fbr solving the BMIs problem which is an NP hard problem.Specifically,some of the variables in BMIs are set to be searched by GA,then the corresponding non-convex problem reduces to the semidefinite programming(SDP)involving LMIs,which is convex and can be solved numerically with available software based on interior-point method.The proposed algorithm can be easy to carry out,and overcomes the shortcomings of the existent algorithms which are hard to converge to the global optimum or are not computationally tractable fbr large scale problems.
Thirdly,the design of robust H_∞output feedback controller fbr uncertain piecewise linear systems is presented.By constructing piecewise continuous Lyapunov functions fbr the closed-loop augmented systems,the H_∞output feedback controller design is cast as the feasibility problem of a set of BMIs,and the optimal H_∞controller can be obtained by solving a non-convex optimization problem under the constraints of BMIs.
Fourthly,the guaranteed cost control fbr uncertain piecewise linear systems via state feedback and output feedback are investigated respectively based on the piecewise quadratic Lyapunov functions technique and Hamilton-Jacobi-Bellman (H-J-B)inequality method.It has been shown that both the state feedback opti- mal controller and output feedback optimal controller,which minimize the upper bounds on cost function,can be obtained respectively by solving a non-convex optimization problem under the BMIs constraints.The controller obtained can be judged by a lower bound on cost function which can be obtained by SDP.
Finally,the optimal state feedback guaranteed cost controller design for the nonlinear systems is presented.The nonlinear systems are described as a class of uncertain piecewise linear systems,then with the results of guaranteed cost controller design for piecewise linear systems,the optimal guaranteed cost controller for the nonlinear systems can be obtained by solving a non-convex optimization problem under the BMIs constraints.
引文
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