The paper describes the set of solutions of the discrete-time algebraic Riccati equation. It is shown that each solution is a combination of a pair of opposite unmixed solutions. There is a one-to-one correspondence between solutions and invariant subspaces of the closed loop matrix of an unmixed solution. The results of the paper provide an extended counterpart of the parametrization theory of...

In this paper we propose a nonrecursive method for solving the general discrete-time algebraic Riccati equation related to the Hm control problem (Hm-DARE). We have achieved this by casting the problem of solving a given H,-DARE to the problem of solving an auxiliary continuous-time algebraic Riccati equation associated with the H , control problem (Hm-CARE) for which the well known nonrecursiv...

where A,B and C are complex matrices of dimensions n × n, n × p and q × n respectively. We focus on the set T = {X | R(X) = 0, X ≤ 0} of negative-semidefinite solutions and we shall characterize those elementsX of T which are isolated (in the topology which T inherits as a subset of the normed space C n×n). Such an isolated X has the property that for a sufficiently small there is no solution Y...

In this paper, asymmetric Algebraic Riccati Equations are analyzed. We derive a new parametrization of the set of solutions, which may be viewed as alternative to the classical one based on Lagrangian subspaces of a “pseudo-Hamiltonian” matrix. Generalizing the symmetric case, the proposed parametrization is given in terms of pairs of invariant subspaces of two related “feedback” matrices. Exte...

This paper deals with two interrelated issues. One is an invariant subspace approach to finding solutions for the algebraic Riccati equation for a class of infinite dimensional systems. The second is approximation of the solution of the algebraic Riccati equation by finite dimensional approximants. The theory of exponentially dichotomous operators and bisemigroups is instrumental in our approach.

A solution X of a discrete-time algebraic Riccati equation is called unmixed if the corresponding closed-loop matrix Φ(X) has the property that the common roots of det (sI−Φ(X)) and det (I− sΦ(X)∗) (if any) are on the unit circle. A necessary and sufficient condition is given for existence and uniqueness of an unmixed solution such that the eigenvalues of Φ(X) lie in a prescribed subset of C. A...

This paper develops an anti-windup scheme for systems with rate-limited actuators. The main results show how a full-order anti-windup compensator can be synthesised using an algebraic Riccati equation and several free parameters. A further result then shows how the free parameters may be chosen to influence, in an intuitive way, the local L2 gain of the system and the size of the region of attr...

In this note we present equations that uniquely determine the maximumpossible imaginaryvalue of the closed loop eigenvalues in an LQ-optimal system, irrespective of how the state weight matrix is chosen, provided a real symmetric solution of the algebraic Riccati equation exists. In addition, the corresponding state weight matrix and the solution to the algebraic Riccati equation are derived fo...