具有参数不确定性和外干扰系统的鲁棒H-状态反馈控制

本文考虑具有参数不确定性和外干扰系统的鲁棒Ｈ∞状态反馈控制问题。这个问题的解只需求解一个代数Ｒｉｃｃａｔｉ方程就可得到其状态反馈阵，且这个状态反馈阵仍保持和ＬＱ最优设计相同的形式。运用这样的状态反馈控制。即能保证具有参数不确定性系统是稳定的，又能达到Ｈ∞最优干扰抑制效果。     

Linear matrix inequalities, Riccati equations, and indefinitestochastic linear quadratic controls

This paper deals with an optimal stochastic linear-quadratic (LQ) control problem in infinite time horizon, where the diffusion term in dynamics depends on both the state and the control variables. In contrast to the deterministic case, we allow the control and state weighting matrices in the cost functional to be indefinite. This leads to an indefinite LQ problem, which may still be well posed due to the deep nature of uncertainty involved. The problem gives rise to a stochastic algebraic Riccati equation (SARE), which is, however, fundamentally different from the classical algebraic Riccati equation as a result of the indefinite nature of the LQ problem. To analyze the SARE, we introduce linear matrix inequalities (LMIs) whose feasibility is shown to be equivalent to the solvability of the SARE. Moreover, we develop a computational approach to the SARE via a semi-definite programming associated with the LMIs. Finally, numerical experiments are reported to illustrate the proposed approach     

Continuous-time singular linear–quadratic control: Necessary and sufficient conditions for the existence of regular solutions

The purpose of this paper is to provide a full understanding of the role that the constrained generalized continuous algebraic Riccati equation plays in singular linear–quadratic (LQ) optimal control. Indeed, in spite of the vast literature on LQ problems, only recently a sufficient condition for the existence of a non-impulsive optimal control has for the first time connected this equation with the singular LQ optimal control problem. In this paper, we establish four equivalent conditions providing a complete picture that connects the singular LQ problem with the constrained generalized continuous algebraic Riccati equation and with the geometric properties of the underlying system.     

Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon

This paper is concerned with a stochastic linear-quadratic (LQ) problem in an infinite time horizon with multiplicative noises both in the state and the control. A distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite. A new type of algebraic Riccati equation – called a generalized algebraic Riccati equation (GARE) – is introduced which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints. It is then shown that the well-posedness of the indefinite LQ problem is equivalent to a linear matrix inequality (LMI) condition, whereas the attainability of the LQ problem is equivalent to the existence of a “stabilizing solution” to the GARE. Moreover, all possible optimal controls are identified via the solution to the GARE. Finally, it is proved that the solution to the GARE can be obtained via solving a convex optimization problem called semidefinite programming.     

Infinite horizon indefinite stochastic linear quadratic control for discrete-time systems

This paper discusses discrete-time stochastic linear quadratic (LQ) problem in the infinite horizon with state and control dependent noise, where the weighting matrices in the cost function are assumed to be indefinite. The problem gives rise to a generalized algebraic Riccati equation (GARE) that involves equality and inequality constraints. The well-posedness of the indefinite LQ problem is shown to be equivalent to the feasibility of a linear matrix inequality (LMI). Moreover, the existence of a stabilizing solution to the GARE is equivalent to the attainability of the LQ problem. All the optimal controls are obtained in terms of the solution to the GARE. Finally, we give an LMI -based approach to solve the GARE via a semidefinite programming.     

The set of positive semidefinite solutions of the algebraic Riccatiequation of discrete-time optimal control

In this paper the algebraic Riccati equation (ARE) of the discrete-time linear-quadratic (LQ) optimal control problem and its set of positive semidefinite solutions is studied under the most general assumption which is output stabilizability. With respect to an appropriate basis, the discrete-time algebraic Riccati equation (DARE) decomposes into a Lyapunov equation and an irreducible Riccati equation. The focus is on the Riccati part which amounts to studying a DARE where all unimodular modes are controllable. A bijection between positive semidefinite solutions and certain well-defined sets of F-invariant subspaces is established which, together with its inverse, is order reversing. As an application, issues concerning positive definite or strong solutions are clarified. Analogous results for negative semidefinite solutions are valid only under an additional assumption on the unobservable subspace     

具有指定闭环特征值的离散时间最优调节器的设计

本文研究了LQ最优调节器的逆问题。在控制变量加权矩阵R给定的条件下,通过引入一组自由变量,给出了满足闭环系统特征值要求的状态加权矩阵Q的一种参数化表示结果。基于这一结果,文中研究了一类系统的LQ最优调节器之逆问题的解析解法。通过所求得的自由变量解,就可以直接确定系统的最优状态反馈控制器。     

LQ控制区段混合能矩阵的微分方程及其应用

本文根据计算结构力学与线性二次控制的对应关系,提出了连续时间有限区段的混合能 分块子矩阵Q2,G2及Φ2.推导出适用于LQ控制非定常课题的二区段连接的凝聚消元公式及 这些子矩阵的微分方程,可用级数展开求解这些方程.当△t很小时,这些分块子矩阵的高次 近似可以大大加速里卡提代数方程算法的收敛性.     

LQ最优控制系统中加权阵的确定

本文研究了LQ最优调节器的逆问题.在控制变量加权矩阵R给定的条件下,通过引入 一组自由变量,给出了满足闭环系统特征值要求的状态加权矩阵Q的一种参数化表示结果.基 于这种结果,研究了LQ逆问题的矩阵变换解法和一类系统的LQ逆问题的解法.此外,文中 还给出了不求解代数矩阵Riccati方程确定系统的最优状态反馈系数矩阵K的方法.     

线性二次闭环微分对策次优对策的双线性矩阵不等式方法

The suboptimal control program via memoryless state feedback strategies for LQ differential games with multiple players is studied in this paper. Sufficient conditions for the existence of the suboptimal strategies for LQ differential games are presented. It is shown that the suboptimal strategies of LQ differential games are associated with a coupled algebraic Riccati inequality. Furthermore, the problem of designing suboptimal strategies is considered. A non-convex optimization problem with BMI constrains is formulated to design the suboptimal strategies which minimizes the performance indices of the closed-loop LQ differential games and can be solved by using LMI Toolbox of MATLAB. An example is given to illustrate the proposed results.     

Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: the discrete-time case

In this paper, a new methodology is developed for the closed-form solution of a generalized version of the finite-horizon linear-quadratic regulator problem for LTI discrete-time systems. The problem considered herein encompasses the classical version of the LQ problem with assigned initial state and weighted terminal state, as well as the so-called fixed-end point version, in which both the initial and the terminal states are sharply assigned. The present approach is based on a parametrization of all the solutions of the extended symplectic system. In this way, closed-form expressions for the optimal state trajectory and control law may be determined in terms of the boundary conditions. By taking advantage of standard software routines for the solution of the algebraic Riccati and Stein equations, our results lead to a simple and computationally attractive approach for the solution of the considered optimal control problem without the need of iterating the Riccati difference equation.     

Suboptimal Strategies of Linear Quadratic Closed-loop Differential Games: An BMI Approach

The suboptimal control program via memoryless state feedback strategies for LQ differential games with multiple players is studied in this paper. Sufficient conditions for the existence of the suboptimal strategies for LQ differential games are presented. It is shown that the suboptimal strategies of LQ differential games are associated with a coupled algebraic Riccati inequality. Furthermore, the problem of designing suboptimal strategies is considered. A non-convex optimization problem with BMI constrains is formulated to design the suboptimal strategies which minimizes the performance indices of the closed-loop LQ differential games and can be solved by using LMI Toolbox of MATLAB, An example is given to illustrate the proposed results.     

奇异控制正则化及刚度移置法

基于结构力学中子结构链与 LQ 离散时间控制的模拟关系,本文给出了代数里卡提方程的两种等价表达形式,利用其中的势能列式提出了刚度移置法.采用刚度移置法及适当的变换导出了奇异控制问题的里卡提方程.     

The generalised discrete algebraic Riccati equation in linear-quadratic optimal control

This paper investigates the properties of the solutions of the generalised discrete algebraic Riccati equation arising from the classic infinite-horizon linear quadratic (LQ) control problem. In particular, a geometric analysis is used to study the relationship existing between the solutions of the generalised Riccati equation and the output-nulling subspaces of the underlying system and the corresponding reachability subspaces. This analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem, which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil. In establishing the main results of this paper, several ancillary problems on the discrete Lyapunov equation and spectral factorisation are also addressed and solved.     

Optimal control of coupled parabolic–hyperbolic non-autonomous PDEs: infinite-dimensional state-space approach

This paper deals with the design of an optimal state-feedback linear-quadratic (LQ) controller for a system of coupled parabolic–hypebolic non-autonomous partial differential equations (PDEs). The infinite-dimensional state space representation and the corresponding operator Riccati differential equation are used to solve the control problem. Dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the LQ-optimal control problem and also to guarantee the exponential stability of the closed-loop system. Thanks to the eigenvalues and eigenfunctions of the parabolic operator and also the fact that the hyperbolic-associated operator Riccati differential equation can be converted to a scalar Riccati PDE, an algorithm to solve the LQ control problem has been presented. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ optimal controller designed in the early portion of the paper is implemented for the original non-linear model. Numerical simulations are performed to show the controller performances.     

An equivalence result in linear-quadratic theory

We consider the zero-endpoint infinite-horizon LQ problem. We show that the existence of an optimal policy in the class of feedback controls is a sufficient condition for the existence of a stabilizing solution to the algebraic Riccati equation. This result is shown without assuming positive definiteness of the state weighting matrix. The feedback formulation of the optimization problem is natural in the context of differential games and we provide a characterization of feedback Nash equilibria both in a deterministic and stochastic context.     

Infinite time horizon nonzero-sum linear quadratic stochastic differential games with state and control-dependent noise

This paper discusses the infinite time horizon nonzero-sum linear quadratic （LQ） differential games of stochastic systems governed by Itoe＇s equation with state and control-dependent noise. First, the nonzero-sum LQ differential games are formulated by applying the results of stochastic LQ problems. Second, under the assumption of mean-square stabilizability of stochastic systems, necessary and sufficient conditions for the existence of the Nash strategy are presented by means of four coupled stochastic algebraic Riccati equations. Moreover, in order to demonstrate the usefulness of the obtained results, the stochastic H-two/H-infinity control with state, control and external disturbance-dependent noise is discussed as an immediate application.     

LQ逆问题解的一种有效算法

本文研究了LQ最优控制逆问题解的参数化表示结果和基于这一参数化表示结果的矩阵变换解法。研究的对象是线性时不变离散时间系统。此外,文中还给出了不求解代数矩阵Riccati方程确定系统的最优状态反馈系数矩阵K的方法。     

Infinite-dimensional LQ optimal control of a dimethyl ether (DME) catalytic distillation column

This contribution addresses the development of a linear quadratic (LQ) regulator in order to control the concentration profiles along a catalytic distillation column, which is modelled by a set of coupled hyperbolic partial differential and algebraic equations (PDAEs). The proposed method is based on an infinite-dimensional state-space representation of the PDAE system which is generated by a transport operator. The presence of the algebraic equations, makes the velocity matrix in the transport operator, spatially varying, non-diagonal, and not necessarily negative through of the domain. The optimal control problem is treated using operator Riccati equation (ORE) approach. The existence and uniqueness of the non-negative solution to the ORE are shown and the ORE is converted into a matrix Riccati differential equation which allows the use of a numerical scheme to solve the control problem. The result is then extended to design an optimal proportional plus integral controller which can reject the effect of load losses. The performance of the designed control policy is assessed through a numerical study.