Linear quadratic regulation for discrete-time systems with state delays and multiplicative noise

In this paper, the linear quadratic regulation problem for discrete-time systems with state delays and multiplicative noise is considered. The necessary and sufficient condition for the problem admitting a unique solution is given. Under this condition, the optimal feedback control and the optimal cost are presented via a set of coupled difference equations. Our approach is based on the maximum principle. The key technique is to establish relations between the costate and the state.     

On a general optimal algorithm for multirate output feedbackcontrollers for linear stochastic periodic systems

A modified optimal algorithm for multirate output feedback controllers of linear stochastic periodic systems is developed. By combining the discrete-time linear quadratic regulation (LQR) control problem and the discrete-time stochastic linear quadratic regulation (SLQR) control problem to obtain an extended linear quadratic regulation (ELQR) control problem, one derives a general optimal algorithm to balance the advantages of the optimal transient response of the LQR control problem and the optimal steady-state regulation of the SLQR control problem. In general, the solution of this algorithm is obtained by solving a set of coupled matrix equations. Special cases for which the coupled matrix equations can be reduced to a discrete-time algebraic Riccati equation are discussed. A reducable case is the optimal algorithm derived by H.M. Al-Rahmani and G.F. Franklin (1990), where the system has complete state information and the discrete-time quadratic performance index is transformed from a continuous-time one     

Explicit Necessary and Sufficient Conditions for Quadratic Linearization

The existing necessary and sufficient conditions for quadratic linearization of control affine systems are in effect, constructive in nature. The exception is the classical result, which requires one to check involutive properties of distributions of the quadratic polynomials to be linearized. Nevertheless, the latter condition is difficult to verify. In this paper, we provide necessary and sufficient conditions for quadratic linearization that are not constructive but are based on checking a linear system of equations involving quadratic polynomial terms only. The system needs only to be put in controller normal form of the linear part which is known. Thus, the result is verifiable and explicit. Also, if required, once the quadratic linearization conditions are met, we show how to construct the coordinate and state feedback transformation to implement the linearization.     

约束随机线性二次最优控制的研究

本文研究线性终端状态约束下不定随机线性二次最优控制问题.首先利用Lagrange Multiplier 定理得到了存在最优线性状态反馈解的必要条件, 而在加强的条件下也得到了最优控制存在的充分条件. 从某种意义上讲, 以往关于无约束随机线性二次最优控制的一些结果可以看成本文主要定理的推论.     

Maximal and Stabilizing Hermitian Solutions for Discrete-Time Coupled Algebraic Riccati Equations

Discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control and H _∞-control of Markovian jump linear systems are considered. First, the equations that arise from the quadratic optimal control problem are studied. The matrix cost is only assumed to be hermitian. Conditions for the existence of the maximal hermitian solution are derived in terms of the concept of mean square stabilizability and a convex set not being empty. A connection with convex optimization is established, leading to a numerical algorithm. A necessary and sufficient condition for the existence of a stabilizing solution (in the mean square sense) is derived. Sufficient conditions in terms of the usual observability and detectability tests for linear systems are also obtained. Finally, the coupled algebraic Riccati equations that arise from the H _∞-control of discrete-time Markovian jump linear systems are analyzed. An algorithm for deriving a stabilizing solution, if it exists, is obtained. These results generalize and unify several previous ones presented in the literature of discrete-time coupled Riccati equations of Markovian jump linear systems. Date received: November 14, 1996. Date revised: January 12, 1999.     

Partial pole placement by LQ regulators: an inverse problemapproach

This paper gives a necessary and sufficient condition under which a state feedback control law places part of the closed-loop poles exactly at specified points and, at the same time, is linear quadratic optimal for some quadratic weightings. This is made possible by means of a solution to the inverse problem of optimal control. A design example is given to illustrate the result     

时变不确定系统鲁棒控制器设计的新方法

基于包含两个二次项的分段Lyapunov函数,研究了线性时变不确定系统的鲁棒控制器设计问题.所考虑的系统由两个矩阵的凸组合构成,通过引入一个附加矩阵,推导出鲁棒控制器存在的充分条件.该控制器的状态反馈增益的求解问题可以转化为一组带有两个比例参数的线性矩阵不等式的凸优化问题.最后的数值示例说明了该设计方法的可行性.     

Quadratic stabilizability of linear systems with structuralindependent time-varying uncertainties

The author investigates the problem of designing a linear state feedback control to stabilize a class of single-input uncertain linear dynamical systems. The systems under consideration contain time-varying uncertain parameters whose values are unknown but bounded in given compact sets. The method used to establish asymptotical stability of the closed-loop system (obtained when the feedback control is applied) involves the use of a quadratic Lyapunov function. The author first shows that to ensure a stabilizable system some entries of the system matrices must be sign invariant. He then derives necessary and sufficient conditions under which a system can be quadratically stabilized by a linear control for all admissible variations of uncertainties. The conditions show that all uncertainties can only enter the system matrices in such a way as to form a particular geometrical pattern called an antisymmetric stepwise configuration. For the systems satisfying the stabilizability conditions, a computational control design procedure is also provided and illustrated via an example     

Suboptimal control of linear time-varying multi-delay systems via shifted Legendre polynomials

A direct method based on using shifted Legendre polynomials is developed to obtain suboptimal control for linear time-varying systems with multiple state and control delays and quadratic performance index. In this method, both the control and state variables are first expanded into finite shifted Legendre series. The governing delay-differential equation is then converted to a set of linear algebraic equations through use of the operational matrices of integration and delay. The problem finally becomes the simple one of finding the unknown coefficients of the control variables alone, which minimizes the quadratic form of performance index.     

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．     

参数不确定广义大系统的保性能分散控制

对一类范数有界时不变参数不确定的连续广义大系统和一个二次型性能指标,研究了其保性能分散控制问题.目的是设计一状态反馈分散控制器,使得对所有容许的不确定性,闭环系统不仅是鲁棒稳定的,而且性能指标有一上界.应用线性矩阵不等式方法,给出了一个用线性矩阵不等式表达的保性能分散控制器存在的充分条件;在此条件可解时,给出了保性能分散控制律的表达式.最后,举例说明了该方法的应用.     

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.     

A decentralized team decision problem with an exponential cost criterion

A static decentralized team is represented by the nodes of a network working together to optimize the expected value of an exponential of a quadratic function of the state and control variables. The information consists of known linear functions of the normally distributed state corrupted by additive Gaussian noise. For certain ranges of the system parameters, the stationary condition for optimality is satisfied by a linear decision rule operating on the available information. These stationary conditions reduce to a set of algebraic matrix equations and a matrix inequality condition from which the values of the decision gains are determined. Although the stationary conditions are necessary for the linear control law to be minimizing in the class of nonlinear control laws, sufficiency is obtained for our linear controller to be minimizing in the class of linear control laws. Since the quadratic performance criterion produces the only previously known closed form decentralized decision rule, the exponential criterion is an important generalization.     

Linear control with incomplete state feedback and known initial-state statistics†

A method is proposed for the design of a linear, time-invariant state feedback control for a linear, time-invariant, finite-dimensional system with infinite duration of control (regulator problem), which makes use of only those state variables which are available for measurement, providing that these are sufficient to render the system stable. The expected value vector and the covariance matrix for the initial state are presumed to be known.

The cost function is quadratic and is expressed in terms of the initial state statistics and the cost-weighting matrix. The necessary conditions are derived for the minimization of the expected value of the cost. The minimization results in a set of m simultaneous polynomial equations in m unknowns where m is the product of the number of the available state variables and the number of control signals formed out of these. The theory is illustrated by a simple example of a third-order system.     

Robust stability and stabilization of discrete singular systems: an equivalent characterization

This note deals with the problems of robust stability and stabilization for uncertain discrete-time singular systems. The parameter uncertainties are assumed to be time-invariant and norm-bounded appearing in both the state and input matrices. A new necessary and sufficient condition for a discrete-time singular system to be regular, causal and stable is proposed in terms of a strict linear matrix inequality (LMI). Based on this, the concepts of generalized quadratic stability and generalized quadratic stabilization for uncertain discrete-time singular systems are introduced. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are obtained in terms of a strict LMI and a set of matrix inequalities, respectively. With these conditions, the problems of robust stability and robust stabilization are solved. An explicit expression of a desired state feedback controller is also given, which involves no matrix decomposition. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.     

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.     

Analytic solution for a class of linear quadratic open-loop Nash games

A method for solving the asymmetric coupled Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic games is presented. The class of games studied here is one in which the state weighting matrices in player's cost functionals are proportional to each other. By writing in a special order the necessary conditions for open-loop Nash strategy, a matrix with specific properties is derived. These properties are then exploited to solve the two-point boundary-value problem. Some special cases are discussed and a simple example is given to illustrate the solution procedure.