Monotonicity of maximal solutions of algebraic Riccati equations

A monotonicity result for the maximal solution of the equation XBB^*X − A^*X − XA − Q = 0, Q = Q^*, (A, B) stabilizable, is proved.     

Strong solutions of the discrete-time algebraic Riccati equation

Conditions are given under which a solution of the DARE is positive semidefinite if and only if all the eigenvalues of its associated closed-loop matrix are in the closed unit disc.     

The limiting form of the continuous algebraic Riccati equation

In this paper solution-preserving transformations of algebraic Riccati equations are examined. As an illustrative example we deal with the limiting cost of the linear-quadratic cheap control problem.     

On parameter dependence of solutions of algebraic riccati equations

We study the behavior of Hermitian solutions, especially the maximal ones, of algebraic Riccati equations whose coefficients depend on real parameters. The cases of analytic dependence on one parameter andC′ dependence (0≤r≤∞) on many parameters are considered. The basic assumption made is stabilizability. Partially supported by an NSF grant.     

Curvature properties of the algebraic Riccati equation

We prove that the solution to the algebraic Ricatti equation (ARE) is concave with respect to a nonnegative-definite symmetric state weighting matrix Q when the input weighting matrix R = R^T > 0. We also prove that the solution to the ARE is concave with respect to a positive-definite diagonal input weighting matrix R when Q = Q^T ≥ 0.     

A class of marked invariant subspaces with an application to algebraic Riccati equations

Invariant subspaces of a matrix A are considered which are obtained by truncation of a Jordan basis of a generalized eigenspace of A. We characterize those subspaces which are independent of the choice of the Jordan basis. An application to Hamilton matrices and algebraic Riccati equations is given.     

The algebraic Riccati equation — A polynomial approach

Polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic Riccati equation.     

Existence and uniqueness of unmixed solutions of the discrete-time algebraic Riccati equation

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 .     

On the numerical solution of a structured nonsymmetric algebraic Riccati equation

A class of nonsymmetric algebraic Riccati equation, where one of the two linear coefficients is block diagonal, is studied. These equations arise in the modeling of an adaptive MMAP[K]/PH[K]/1$MMAP\left[K\right]/PH\left[K\right]/1$ queue. Some theoretical results are proved, and two new algorithms are introduced that exploit the diagonal structure of the linear coefficient.     

On upper bounds for the unified algebraic Riccati equation

In recent years, several eigenvalues, norms and determinants bounds have been investigated separately for the solutions of continuous and discrete Riccati equations. In this paper, an upper bound for solution of the unified Riccati equation is presented. In the limiting cases, the result reduces to a new upper bound for the solution of continuous and discrete Riccati equation.     

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.     

Intervals of solutions of the discrete-time algebraic Riccati equation

If two solutions YZ of the DARE are given then the set of solutions X with YXZ can be parametrized by invariant subspaces of the closed loop matrix corresponding to Y. The paper extends the geometric theory of Willems from the continuous-time to the discrete-time ARE making the weakest possible assumptions.     

离散时间代数Riccati方程解矩阵的特征值分析

针对离散时间代数Riccati方程DTARE的唯一对称正定解X的特征值,通过矩阵的恒等变形,给出了一种新的分析方法.最后获得解X的极值特征值的上界和下界,以及解X的特征值的和———迹的一个下界.     

Upper solution bounds of the continuous coupled algebraic Riccati matrix equation

In this article, by using some matrix identities, we construct the equivalent form of the continuous coupled algebraic Riccati equation (CCARE). Further, with the aid of the eigenvalue inequalities of matrix's product, by solving the linear inequalities utilising the properties of M-matrix and its inverse matrix, new upper matrix bounds for the solutions of the CCARE are established, which improve and extend some of the recent results. Finally, a corresponding numerical example is proposed to illustrate the effectiveness of the derived results.     

Monotonicity and stabilizability- properties of solutions of the Riccati difference equation: Propositions, lemmas, theorems, fallacious conjectures and counterexamples

The problem considered is that of selecting an initial covariance matrix for the Kalman filter to ensure that the closed-loop filter at every subsequent time instant is exponentially asymptotically stable as a time-invariant filter. Sufficient conditions are derived based on monotonicity properties of the solution of the Riccati difference equation. The results have application in observer design, and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included.     

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.     

The dual algebraic Riccati equations and the set of all solutions of the discrete-time Riccati equation

In this paper, two new pairs of dual continuous-time algebraic Riccati equations (CAREs) and dual discrete-time algebraic Riccati equations (DAREs) are proposed. The dual DAREs are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix. Then, in the case of singular matrices, a generalised inverse is introduced to deal with the dual DARE problem. These dual AREs can easily lead us to an iterative procedure for finding the anti-stabilising solutions, especially to DARE, by means of that for the stabilising solutions. Furthermore, we provide the counterpart results on the set of all solutions to DARE inspired by the results for CARE. Two examples are presented to illustrate the theoretical results.     

Stability radius for structured perturbations and the algebraic Riccati equation

In this paper we generalize the notion of stability radius introduced in [1] to allow for structured perturbations. We then relate the stability radius to the existence of Hermitian solutions of an algebraic Riccati equation and give some applications of this result.     

On the existence of solutions of the Riccati differential equation

We generalize a technique given in C. Martin [1], to obtain a characterization of finite escape times for time-varying Riccati equations which also apply to the non-definite case.