Stabilizability and positiveness of solutions of the jump linear quadratic problem and the coupled algebraic Riccati equation

This note addresses the jump linear quadratic problem of Markov jump linear systems and the associated algebraic Riccati equation. Necessary and sufficient conditions for stability of the optimal control and positiveness of Riccati solutions are developed. We show that the concept of weak detectability is not only a sufficient condition for the finiteness of cost functional to imply stability of the associated trajectory, but also a necessary one. This, together with a characterization developed here for the kernel of the Riccati solution, allows us to show that the control solution stabilizes the system if and only if the system is weakly detectable, and that the Riccati solution is positive-definite if and only if the system is weakly observable. The connection between the algebraic Riccati equation and the control problem is made, as far as the minimal positive-semidefinite solution for the algebraic Riccati equation is identified with the optimal solution of the linear quadratic problem. Illustrative numerical examples and comparisons are included.     

Optimal reduced-order state estimation for unstable plants

The problem of optimal reduced-order steady-state state estimation is considered for the case in which the plant has unstable poles. In contrast to the standard full-order estimation problem involving a single algebraic Riccati equation, the solution to the reduced-order problem involves one modified Riccati equation and one Lyapunov equation coupled by a projection matrix. This projection is completely distinct from the projection obtained by Bernstein and Hyland (1985) for stable plants.     

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     

Solution of the singular discrete regulator problem using eigenvector methods

The discrete regulator problem with a singular state matrix cannot be solved using either the eigenvectors or the sign function of the associated compound matrix, since then, the implicit equation defining the feedback matrix cannot be transformed into the regular discrete Riccati equation. In this paper, it is shown that the solution of a continuous Riccati equation, whose matrix parameters depend on the matrices of both discrete system and cost functional, is the solution of the implicit equation. Since the continuous Riccati equation has symmetric spectrum with respect to the imaginary axis, it has no singularity problem and therefore the matrix sign function method can be used for its solution.     

On the linear quadratic minimum-time problem

A nontraditional minimum-time problem that includes quadratic-state and control-weighting terms in the performance index is investigated. This formulation provides a convenient solution to the problem that uses the solution of the Riccati equation to compute the optimal feedback gain and the optimal time. In some cases the latter is simply found using the derivative of the Riccati equation solution     

On the discrete and continuous time infinite-dimensional algebraic Riccati equations

The standard state space solution of the finite-dimensional continuous time quadratic cost minimization problem has a straightforward extension to infinite-dimensional problems with bounded or moderately unbounded control and observation operators. However, if these operators are allowed to be sufficiently unbounded, then a strange change takes place in one of the coefficients of the algebraic Riccati equation, and the continuous time Riccati equation begins to resemble the discrete time Riccati equation. To explain why this phenomenon must occur we discuss a particular hyperbolic PDE in one space dimension with boundary control and observation (a transmission line) that can be formulated both as a discrete time system and as a continuous time system, and show that in this example the continuous time Riccati equation can be recovered from the discrete time Riccati equation. A particular feature of this example is that the Riccati operator does not map the domain of the generator into the domain of the adjoint generator, as it does in the standard case.     

Solvability and asymptotic behavior of generalized Riccatiequations arising in indefinite stochastic LQ controls

The optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential Riccati equation. In this paper, we provide general necessary and sufficient conditions for the solvability of this generalized differential Riccati equation. Furthermore, its asymptotic behavior is investigated along with its connection to the generalized algebraic Riccati equation associated with the linear quadratic control problem in finite time horizon. Examples are presented to illustrate the results established     

On the solution of linear boundary value problems by extended invariant imbedding

Unstable linear boundary value problems can be solved by the method of Invariant Imbedding in a stable manner. Instead of integration of the system equations this method requires the integration of a matrix Riccati equation, which depends on the boundary values of the problem. The dimension of the Riccati equation is determined by a suitable decoupling of the system equations. Invariant Imbedding now fails, if this decoupling does not correspond with the boundary condition. In addition, the Riccati equation has to be solved once more for each new boundary condition. An extension algorithm is defined, which maps the boundary value problem into a problem of double dimension. This “extended” boundary value is solved by a modified Invariant Imbedding. The resulting “Extended (Dual) Invariant Imbedding” is always applicable and does not depend on the boundary conditions. The corresponding “extended” Riccati equation has to be integrated only once “offline”. If the boundary condition is changed, only systems of linear equations have to be solved “online”.     

Criterion for the convergence of the solution of the Riccati differential equation

The optimal control problem for a linear system with a quadratic cost function leads to the matrix Riccati differential equation. The convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix. A necessary and sufficient condition for convergence is derived for stabilizable systems, even if the output in the cost function is not detectable. An algorithm is developed to determine the limiting value of the solution, which is one of the symmetric positive semidefinite solutions of the algebraic Riccati equation. Examples for convergence and nonconvergence are given. A discussion is also included of the convergence properties of the solution of the Riccati differential equation to any real symmetric (not necessarily positive semidefinite) solution of the algebraic Riccati equation.     

Existence conditions and properties for the maximal periodic solution of periodic Riccati difference equations

The existence and properties of the maximal symmetric periodic solution of the periodic Riccati difference equation, is analysed for the optimal filtering problem of linear periodic discrete-time systems. Special emphasis is given to systems not necessarily reversible and subject only to a detectability assumption. Necessary and sufficient conditions for the existence and uniqueness of periodic non-negative definite solutions of the periodic Riccati difference equation which gives rise to a stable filter are also established. Furthermore, the convergence of non-negative definite solutions of the Riccati equation is investigated.     

Sequential decomposition of the matrix Riccati equation and its application to the linear quadratic regulator problem

The differential matrix Riccati equation for the multi-input-multi-output linear quadratic optimal regulator problem is considered. Two methods are presented, successive and parallel, that decompose this equation into a set of Riccati equations that correspond to optimal regulator problems of possibly reduced dimensions. The additivity of the solutions to the equations obtained by the successive sequential decomposition method (SDM) and the additivity of the inverses of the solutions to the equations obtained by the parallel SDM are established. Some duality relations between the successive and the parallel methods are presented via the use of the adjoint Riccati equation. The theory developed is extended to the algebraic matrix Riccati equation as a limiting case. The application of the SDMs in the infinite-time linear quadratic regulator problem is investigated. Special attention is paid to the partially ‘cheap’ problem where the cost of some of the regulator controls or of their combinations tends asymptotically to zero. Explicit expressions for the asymptotic optimal cost are derived and the behaviour of the asymptotic optimal root loci is investigated.     

一类广义Riccati矩阵方程对称解的双迭代算法

研究了一类广义系统控制理论导出的Riccati矩阵方程对称解的数值计算方法．运用牛顿算法将Riccati矩阵方程的对称解问题转化为线性矩阵方程的对称解或者对称最小二乘解问题,采用修正共轭梯度法解决导出的线性矩阵方程的对称解问题,可建立求Riccati矩阵方程对称解的双迭代算法．数值算例表明,双迭代算法是有效的．     

A Nonrecursive Solution Method for the Linear-Quadratic Optimal Control Problem with a Singular Transition Matrix

In the optimal linear regulator problem the control vector is usually determined by solving the algebraic matrix Riccati equation using successive substitutions. This, however, can be rather inefficient from a computational point of view. A nonrecursive method which requires that the transition matrix is nonsingular has been proposed by Vaughan (1970). In the present paper we present a nonrecursive solution to the matrix Riccati equation for the case that the transition matrix may be singular. We show that this procedure leads to the same numerical results as the standard iteration of the matrix Riccati equation.     

Correlated noise filtering and invariant directions for the Riccati equation

The Riccati equation associated with a class of discrete-time correlated noise problems is examined, and the concept of invariant directions for this equation is introduced. For single-output systems the set of such directions is completely characterized. Deletion of these directions by an appropriate transformation of the Riccati equation results in a minimal order equation for computation. This transformation also reveals the underlying structure of the optimal filter for the correlated noise problem.     

Optimum H designs under sampled state measurements

We present the complete solution to the H^∞-optimal control problem when only sampled values of the state are available. For linear time-varying systems the optimum controller is characterized in terms of the solution of a particular generalized Riccati-differential equation, with the optimum performance determined by the conjugate point conditions associated with a family of generalized Riccati differential equations. For the infinite-horizon time-invariant problem, however, the optimum controller is characterized in terms of the solution of a particular generalized algebraic Riccati equation, and the performance is determined in terms of the conjugate-point conditions of a single generalized Riccati equation, defined on the longest sampling interval. If the distribution of the sampling times is also taken as part of the general design, uniform sampling turns out to be optimal for the infinite horizon case, while for the finite horizon problem a nonuniform sampling generally leads to a better performance.     

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     

Quadratic optimal control of stable systems through spectral factorization

We consider the infinite horizon quadratic cost minimization problem for a linear system with finitely many inputs and outputs. A common approach to treat a problem of this type is to construct a semigroup in an abstract state space, and to use infinite-dimensional control theory. However, this approach is less appealing in the case where there are discrete time delays in the impulse response, because such time delays force both the control operator and the observation operator to be unbounded at the same time. In order to be able to include this case we take an alternative approach. We work in an input-output framework, and reduce the problem to a symmetric Wiener-Hopf problem, that can be solved by means of a canonical factorization of the symbol. In a standard shift semigroup realization this amounts to factorizations of the Riccati operator and the feedback operator into convolution operators and projections. Our approach leads to a new significant discovery: in the case where the impulse response of the system contains discrete time delays, the standard Riccati equation is incorrect; to get the correct Riccati equation the feed-through matrix of the system must be partially replaced by the feed-through matrix of the spectral factor. This means that, before it is even possible to write down the correct Riccati equation, a spectral factorization problem must first be solved to find one of the weighting matrices in this equation.     

On the design of optimum distributed parameter system with boundary control function

The problem of optimum control of a distributed parameter system with boundary control is studied. The distributed parameter system considered is described by theN-dimensional wave equation. The error measure is quadratic. The control function is unconstrained. The Riccati equation for optimum boundary control is derived. Methods for solving the Riccati equation and calculating optimum control are discussed. The resulting control is closed loop.     

H∞滤波问题数值求解的精细积分算法

有限时间H∞滤波的Riccati方程和滤波方程分别为非线性矩阵微分方程和线性变系数微分方程,而且Riccati微分方程解的存在性还依赖于参数 γ-2,因此求这些方程的数值解一般比较困难.按照结构力学与最优控制的模拟关系,Riccati方程解存在的临界参数 γ-2cr对应于广义Rayleigh商的一阶本征值.因此可以用精细积分法结合扩展的Wittrick-Williams(W-W)算法计算 γ-2cr .并求解Ricclati方程,滤波微分方程的解也可以由精细积分法计算.     

Some computational results on size reduction inH∞ design

The authors explore the properties of the algebraic Riccati equation. New results on the rank of the solution to the algebraic Riccati equation are obtained and lead to an efficient algorithm for reducing the size of the antistable transfer function matrix in the model-matching problem. In the 1-block problem, the method is taken further to obtain explicit formulas for a minimal realization of the matrix. The approach is direct and numerically reliable as it is based almost entirely on orthogonal transformations