Riccati differential equation in optimal filtering of periodic non-stabilizable systems

This paper discusses the periodic solutions of the matrix Riccati differential equation in the optimal filtering of periodic systems. Special emphasis is given to non-stabilizable systems and the question addressed is the existence and uniqueness of a steady-state periodic non-negative definite solution of the periodic Riccati differential equation which gives rise to an asymptotically stable steady-state filter. The results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution. The convergence of the general solution of the periodic Riccati differential equation to a periodic equilibrium solution is also investigated. The results are extensions of existing time-invariant systems results to the case of periodic systems     

Periodic strong solution for the optimal filtering problem oflinear discrete-time periodic systems

The periodic Riccati difference equation (PRDE) for the optimal filtering problem of linear periodic discrete-time systems is addressed. Specifically, the author provides a number of results on the existence, uniqueness, and stability properties of symmetric periodic nonnegative-definite solutions of the periodic Riccati difference equation in the case of nonreversible and nonstabilizable periodic systems. The convergence of symmetric periodic nonnegative-definite solutions of the periodic Riccati difference equation is also analyzed. The results have been established under weaker assumptions and include both necessary and sufficient conditions. The existence and properties of symmetric periodic nonnegative-definite solutions of the PRDE are established directly from the PRDE     

Cyclomonotonicity and stabilizability properties of solutions ofthe difference periodic Riccati equation

The Kalman filter associated with a discrete-time linear T-periodic system is tested. The problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first T values of the Kalman filter gain is stabilizing. Sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic Riccati equation. Potential applications are found in filter design, quasi-linearization techniques for the periodic Riccati equation, and the design of receding-horizon control strategies for periodic and multirate systems. When specialized to time-invariant systems, the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time-invariant Riccati equation and the existence of periodic stabilizing feedback     

Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.This work has been partially supported by Spanish DGICYT Grant No. PB91-O535.     

Numerical solution of the discrete-time periodic Riccati equation

In this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic Riccati equation. This method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the Euler-Lagrange difference equations. In doing so, it is possible to extract a basis for the stable deflating subspace of the extended pencil, from which the Riccati solution is obtained. This algorithm is an extension of the standard QZ algorithm and retains its attractive features, such as quadratic convergence and small relative backward error. A method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with     

Boundary problems and periodic Riccati equations

We consider the resolution problem of a periodic Riccati equation from the point of view of a generalized boundary problem that lets us reduce the differential problem to an algebraic equation of Riccati type. Sufficient conditions for the existence and calculus of special types of solutions are given.     

Controllability, stabilizability, and matrix Riccati equations for periodic systems

In this note, controllability, stabilizahility, and related concepts for periodically time-varying systems are discussed. Especially, it is proved that the definition of controllability employed in connection with the existence of periodic solutions of periodic matrix Riccati equations is equivalent to Kalman's original definition.     

Periodic solutions of periodic Riccati equations

For periodically time-varying matrix Riccati equations, controllability and observability (in the usual sense) are shown to be sufficient for the existence of a unique positive definite periodic solution.     

The difference periodic Ricati equation for the periodic predictionproblem

Gives a comprehensive treatment of several important aspects of the discrete-time periodic Riccati equation (DPRE) arising from the prediction problem for linear discrete-time periodic systems. The authors analyze the symmetric periodic positive semidefinite (SPPS) solution of the DPRE under appropriate assumptions of stabilizability and detectability of the periodic system. Among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the SPPS solution and the stability of the resulting closed-loop system. Some of these results can be seen as extensions of the corresponding results for the time-invariant case; however, a number of them contain contributions to the time-invariant case as well. The paper also gives a numerical algorithm based on an iterative linearization procedure for computing the SPPS solution. The algorithm is a periodic version of Kleinman's algorithm for the time-invariant case     

On the periodic solutions of the Matrix Riccati equation

We investigate the structure of the periodic orbits of timeinvariant matrix Riccati equations. Matrix Riccati equations are of critical importance in control, estimation, differential games, scattering theory, and in several other applications. It is therefore important to understand the principal features of the phase portraits of Riccati equations, such as the existence and structure of periodic solutions.     

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

周期黎卡提微分方程正定解的存在性

讨论了标准的周期黎卡提微分方程,给出了其存在埃尔米特周期正定（HPPD）解的一个完整的充分必要条件。准确地说,在经过一个适当的状态空间基底变换后该条件通过能稳性和能检测性概念表述。结果表明,当HP－PD解存在时,它或者是唯一的,或者有无限多个。这一结果可以看作是Richardson和Kwong的结果对周期时变情况的扩展。     

On positive definite solutions to the algebraic Riccati equation

Complete necessary and sufficient conditions for the existence of a positive definite solution to the algebraic Riccati equation are given. It is also shown that when a positive definite solution exists, it is either unique, or else there are uncountably many such solutions.     

Robust stabilization of linear systems with norm-bounded time-varying uncertainty

In this paper, robust stabilization of a class of linear systems with norm-bounded time-varying uncertainties is considered. It is shown that for this class of uncertain systems quadratic stabilizability via linear control is equivalent to the existence of a positive definite symmetric matrix solution to a (parameter-dependent) Riccati equation. Also, a construction for the stabilizing feedback law is given in terms of the solution to the Riccati equation.     

Remarks on the periodic orbits of the matrix Riccati equation

We investigate certain questions concerning the periodic structure of the matrix Riccati differential equation with constant coefficients. A closed-form expression for the periodic solutions is obtained for both the cases involving distinct or repeated eigenvalues in the associated linear hamiltonian system. Previous results are extended by establishing that periodic solutions are bounded if and only if the span of their range does not intersect the orthogonal complement of the controllable subspace of the associated linear system.     

Fault detection of linear discrete-time periodic systems

In this note, an approach to the design of optimal fault detection systems for linear discrete-time periodic systems is proposed, which leads to an optimized compromise between robustness to unknown disturbances and sensitivity to faults. The needed computation mainly consists in solving a difference periodic Riccati system. The proposed approach is finally illustrated by a numerical example.     

在镇定理论中控制系统与具有滞后的控制系统的等价性

本文由勒卡提矩阵方程与勒卡提矩阵微分方程的正定对称解构造了正定二次型函数,给出了在镇定理论中定常及时变线性控制系统与具有滞后的定常及时变线性控制系统的等价性。同时给出了滞后界限的估计公式。     

A miscellany of results of an equation of Count J. F. Riccati

A collection of results on the Riccati equation is presented. The questions addressed are the existence of strong solutions of the algebraic Riccati equation and the convergence of solutions of the Riccati difference equation to those of the algebraic equation. The results derived utilize detestability conditions only.     

Stabilizability and detectability of linear periodic systems

In this paper, four characterizations of stabilizability and detectability of linear periodic systems are considered. Two of them look as natural extensions of the classical definitions given for time-invariant systems. The remaining two are modal characterizations which turn out to be useful in the analysis of the periodic Lyapunov and Riccati equations. It is shown that all these notions of stabilizability (and detectability) are in fact equivalent to each other.One of the various definitions calls for the existence of the Kalman canonical decomposition of periodic systems. This issue is addressed in the Appendix.     

线性时滞系统的无源控制

研究一类线性时滞系统通过线性无记忆状态反馈控制律的无源控制问题。通过某个Ｒｉｃｃａｔｉ矩阵方程对称正定解的存在性,给出了使得闭环系统严格无源的控制器存在条件。进而,利用这个方程的正定解给出了无源化控制器的一个构造方法。