线性离散奇异系统的不定二次最优控制问题: Krein空间方法

The finite time horizon indefinite linear quadratic(LQ) optimal control problem for singular linear discrete time-varying systems is discussed. Indefinite LQ optimal control problem for singular systems can be transformed to that for standard state-space systems under a reasonable assumption. It is shown that the indefinite LQ optimal control problem is dual to that of projection for backward stochastic systems. Thus, the optimal LQ controller can be obtained by computing the gain matrices of Kalman filter. Necessary and sufficient conditions guaranteeing a unique solution for the indefinite LQ problem are given. An explicit solution for the problem is obtained in terms of the solution of Riccati difference equations.     

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.     

Optimistic Value Model of Indefinite LQ Optimal Control for Discrete‐Time Uncertain Systems

Uncertainty theory is a branch of mathematics which provides a new tool to deal with the human uncertainty. Based on uncertainty theory, this paper proposes an optimistic value model of discrete‐time linear quadratic (LQ) optimal control, whereas the state and control weighting matrices in the cost function are indefinite, the system dynamics are disturbed by uncertain noises. With the aid of the Bellman's principle of optimality in dynamic programming, we first present a recurrence equation. Then, a necessary condition for the state feedback control of the indefinite LQ problem is derived by using the recurrence equation. Moreover, a sufficient condition of well‐posedness for the indefinite LQ optimal control is given. Finally, a numerical example is presented by using the obtained results.     

Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems

A new approach to study the indefinite stochastic linear quadratic (LQ) optimal control problems, which we called the “equivalent cost functional method”, is introduced by Yu (2013) in the setup of Hamiltonian system. On the other hand, another important issue along this research direction, is the possible state feedback representation of optimal control and the solvability of associated indefinite stochastic Riccati equations. As the response, this paper continues to develop the equivalent cost functional method by extending it to the Riccati equation setup. Our analysis is featured by its introduction of some equivalent cost functionals which enable us to have the bridge between the indefinite and positive-definite stochastic LQ problems. With such bridge, some solvability relation between the indefinite and positive-definite Riccati equations is further characterized. It is remarkable the solvability of the former is rather complicated than the latter, hence our relation provides some alternative but useful viewpoint. Consequently, the corresponding indefinite linear quadratic problem is discussed for which the unique optimal control is derived in terms of state feedback via the solution of the Riccati equation. In addition, some example is studied using our theoretical results.     

离散系统多步观测时滞的H∞输出反馈控制

针对一类多观测时滞离散系统,本文提出了基于Krein空间理论解决H∞输出反馈问题的新方法．利用H∞输出反馈控制问题与不定二次型之间的关系,时滞系统的H∞控制问题分解为LQ问题和时滞的H∞估计问题．通过重组观测,时滞的H∞估计问题可转化为无时滞的H∞估计问题,从而给出由两个Riccati方程决定的H∞控制器存在的充要条件．本文的方法不需要增广系统．     

离散系统多步观测时滞的H∞输出反馈控制

针对一类多观测时滞离散系统, 本文提出了基于Krein空间理论解决H_∞输出反馈问题的新方法. 利用H_∞输出反馈控制问题与不定二次型之间的关系, 时滞系统的H-infinity控制问题分解为LQ问题和时滞的H_∞估计问题. 通过重组观测, 时滞的H_∞估计问题可转化为无时滞的H_∞估计问题, 从而给出由两个Riccati方程决定的H_∞控制器存在的充要条件. 本文的方法不需要增广系统.     

Optimal control of discrete-time linear fractional-order systems with multiplicative noise

A finite horizon linear quadratic (LQ) optimal control problem is studied for a class of discrete-time linear fractional systems (LFSs) affected by multiplicative, independent random perturbations. Based on the dynamic programming technique, two methods are proposed for solving this problem. The first one seems to be new and uses a linear, expanded-state model of the LFS. The LQ optimal control problem reduces to a similar one for stochastic linear systems and the solution is obtained by solving Riccati equations. The second method appeals to the principle of optimality and provides an algorithm for the computation of the optimal control and cost by using directly the fractional system. As expected, in both cases, the optimal control is a linear function in the state and can be computed by a computer program. A numerical example and comparative simulations of the optimal trajectory prove the effectiveness of the two methods. Some other simulations are obtained for different values of the fractional order.     

Characterizing all optimal controls for an indefinite stochasticlinear quadratic control problem

This paper is concerned with a stochastic linear quadratic (LQ) control problem in the infinite-time horizon, with indefinite state and control weighting matrices in the cost function. It is shown that the solvability of this problem is equivalent to the existence of a so-called static stabilizing solution to a generalized algebraic Riccati equation. Moreover, another algebraic Riccati equation is introduced and all the possible optimal controls, including the ones in state feedback form, of the underlying LQ problem are explicitly obtained in terms of the two Riccati equations     

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     

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.     

一类双线性随机系统M量测不定LQ控制

研究了带有乘性噪声和受扰动观测的离散时间随机系统不定线性二次(Linear quadratic, LQ) 最优输出反馈控制问题. 对此类问题而言,二次成本函数的加权矩阵不定号,并且最优控制具有对偶效果.为在最优性和计算复杂度间 进行折衷,本文采用了一种M量测反馈控制设计方法.基于动态规划方法,将未来的测量结合到当前控制 计算当中的M量测反馈控制可以通过倒向求解一类与原系统维数相同的广义差分Riccati方程(Generalized difference Riccati equation, GDRE)得到.仿真结果 表明本文提出的算法与目前普遍采用的确定等价性方法相比具有优越性.     

Indefinite LQ optimal control for stochastic Takagi–Sugeno fuzzy system under sensor data scheduling: Finite-horizon case

The present paper considers the finite-horizon indefinite linear quadratic (LQ) control problem for stochastic Takagi–Sugeno (T-S) fuzzy systems with input delay. In this paper, we consider the presence of sensor data scheduling, which imposes a communication energy constraint and necessitates optimal state estimation for measurements. Then, by utilizing dynamic programming principles, the stochastic LQ problem under consideration can be solved, while the optimal control policy is developed in terms of the unique solutions to a set of coupled difference Riccati equations (CDREs). Specifically, for simple delay-free case, the linear matrix inequalities based conditions are also proposed, whose feasibility is shown to be equivalent to the well-posedness of the indefinite LQ control under consideration. As an application, our theoretic analysis is extended to study the intermittent observation model caused by random denial-of-service attack.     

奇异系统的不定号二次型指标最优控制问题

讨论奇异系统的不定号LQ问题 (二次型指标中的权矩阵含有负特征值的最优控制问题). 首先指出问题的可解性, 并给出了问题等价转化为奇异系统的奇异LQ问题的充要条件. 然后基于等价的奇异系统奇异LQ问题, 给出问题存在唯一最优控制—轨线对的充分条件. 最后用一个算例说明结论的正确性.     

多步观测时滞的控制

Based on an innovation analysis method in the Krein space, a sufficient and necessary condition is given for the existence of the solution of H1 control problem for a linear continuous-time system with multiple delays. By introducing a re-organized innovation sequence, the H1 control problem with delayed measurements is converted into a linear quadratic (LQ) problem and a delay-free H2 estimation problem in the Krein space. The controller is given in terms of two forward Riccati equations and a backward Riccati equation.     

时变广义系统线性二次最优控制

研究时变广义系统线性二次最优控制问题.通过引进时变广义系统脉冲能控性及脉冲能 观性等概念,建立了这类问题与标准状态空间系统二次指标问题的等价性.进而证明了解的 存在唯一性,给出了解的表示和最优反馈综合.