Continuous-time Algebraic Riccati Equations Research Articles

Generalized continuous-time Riccati theory

A Riccati-like equation which incorporates as special cases the standard continuous-time algebraic Riccati equation and the constrained continuous-time algebraic Riccati equation is introduced and studied. A characterization of the conditions under which an equation of general form has a stabilizing solution are investigated in terms of the so-called proper deflating subspace of the extended Hamiltonian pencil.

Isolated semidefinite solutions of the continuous-time algebraic riccati equation

The set of all negative-semidefinite solutions of the CAREA*X+XA+XBB*X−C*C=0 is homeomorphic to a well defined set ofA-invariant subspaces provided that the purely imaginary eigenvalues ofA are controllable. Based on that homeomorphism isolated n.s.d. solutions of the CARE are characterized by properties of their kernels.

Properties of a quadratic matrix equation and the solution of the continuous-time algebraic Riccati equation

We discuss some properties of a quadratic matrix equation with some restrictions, then use these results on the algebraic Riccati equation to get a new algorithm. The algorithm sufficiently takes account of the structure of the associated matrix; hence it is very effective.

Parallel Algorithms for Solving the Algebraic Riccati Equation via the Matrix Sign Function

In this paper parallel algorithms for solving the continuous-time algebraic Riccati equation both on shared memory multiprocessors and distributed memory multiprocessors are presented. The algorithms are based on the matrix Sign function of the Hamiltonian matrix associated with the equation. In the case of shared memory multiprocessors, the use of LAPACK and BLAS-3 subroutines allows the generation of reliable, portable and efficient subroutines for solving the algebraic Riccati equation. In the case of distributed memory multiprocessors, the distribution of the data wrapped by blocks of columns among the processors and the use of LAPACK and BLAS-3 subroutines improves the efficiency of the algorithms as reported in the experimental results.

Lattice properties of sets of semidefinite solutions of continuous-time algebraic riccati equations

The set of negative-semidefinite solutions of the algebraic Riccati equation A∗X + XA + XBB∗X − C∗C = 0 is studied under the most general hypothesis, namely that (A, B) is stabilizable modulo the undetectable subspace of (A, C). If is a lattice, then is isomorphic to 0 ⊕ S where T0 is the direct sum of copies of the set of nonnegative real numbers and is a closure of which is order isomorphic to a system of well-defined A-invariant subspaces. Necessary and sufficient conditions are given such that (or) is a lattice.

Decomposition and Parametrization of Semidefinite Solutions of the Continuous-Time Algebraic Riccati Equation

Negative-semidefinite solutions of the $\operatorname{ARE}\mathcal{R}(X) = A^ * X + XA + XBB^ * X - C^ * C = 0$ are studied. With respect to an appropriate basis the ARE breaks up into a Lyapunov equation $A_0^ * X_0 + X_0 A_0 = 0$, where $A_0 $ has only purely imaginary eigenvalues, and an indecomposable Riccati equation $\mathcal{R}_r (X_r ) = A_r^ * X_r + X_r A_r + X_r B_r B_r^ * X_r - C_r^ * C_r = 0$ such that each solution $x \leq 0$ is of the form $X = {\operatorname {diag}}(X_0 ,X_r )$ . The focus is on the solutions $\mathcal{S} = \{ X| X . = {\operatorname {diag}}(0,X_r ),\mathcal{R}_r (X_r ) = 0,X_r \} \leq 0$. The set $\mathcal{S}$ has as an order-isomorphic image a well-defined set $\mathcal{N}$ of A-invariant subspaces. The characterization of $\mathcal{N}$ involves the stabilizable and the uncontrollable subspace of $(A,B,C)$.

A Galois correspondence between sets of semidefinite solutions of continuous-time algebraic Riccati equations

The sets of negative semidefinite solutions T i of two algebraic Riccati equations R i(X)=A ∗ i X+XA i+XB iB ∗ i−C ∗ iC i=(I, X)H i(I, X) ∗, i=1, 2, are compared under the hypothesis that H 1⩽ H 2. If X + 1 and X + 2 are the greatest solutions in T 1 and T 2 respectively, then X + 1⩽ X + 2. A more general result will be proved which allows the comparison of other solutions of T 1 and T 2 besides the extremal ones and which in the case of stabilizability leads to a Galois connection between T 1 and T 2. The comparison results are based on one hand on a decomposition of the equations R i(X)=0 into Lyapunov matrix equations and genuine Riccati equations which induce a corresponding decomposition of the solutions in T i , and the other hand on a parametrization of the Riccati components by A i -invariant subspaces.

183 The constrained continuous time algebraic Riccati equation

183 The constrained continuous time algebraic Riccati equation

Krylov Subspace Methods for Solving Large Lyapunov Equations

This paper considers several methods for calculating low-rank approximate solutions to large-scale Lyapunov equations of the form $AP + PA' + BB' = 0$. The interest in this problem stems from model reduction where the task is to approximate high-dimensional models by ones of lower order. The two recently developed Krylov subspace methods exploited in this paper are the Arnoldi method [Saad, Math. Comput., 37 (1981), pp. 105–126] and the Generalised Minimum Residual method (GMRES) [Saad and Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869]. Exact expressions for the approximation errors incurred are derived in both cases. The numerical solution of the low-dimensional linear matrix equation arising from the GMRES method is discussed and an algorithm for its solution is proposed. Low rank solutions of discrete time Lyapunov equations and continuous time algebraic Riccati equations are also considered. Throughout this paper, the authors tackle problems in which B has more than one column with the use of block Krylov schemes.

The Constrained Continuous Time Algebraic Riccati Equation

A generalization of the well-known Riccati equation, suitable for singular linear quadratic control and the singular H∞-control problem, is studied. A necessary and sufficient condition for the existence of the stabilizing solution is given in terms of a possibly singular matrix pencil. The result justifies a Schur-like algorithm for computing this stabilizing solution.

Computational Experience in Solving Algebraic Riccati Equations Using Gaussian Symplectic Transformations

An efficient algorithm for solving continuous-time algebraic Riccati equations (CAREs) is presented. The algorithm employs condition-controlled Gaussian symplectic transformations, a symmetric updating scheme for computing the stabilizing solution of CARE, and accurate approximations of eigenvalues of the Hamiltonian matrix involved in the optimal problem. The Hamiltonian structure is preserved throughout the computations. Numerical results show that the algorithm can be used safely, even when large condition numbers of the transformation matrices are allowed.

Linear Quadratic Problems with Indefinite Cost for Discrete Time Systems

This paper deals with the discrete-time, infinite-horizon linear quadratic problem with indefinite cost criterion. Given a discrete-time linear system, an indefinite cost-functional and a linear subspace of the state space, the problem of minimizing the cost-functional over all inputs that force the state trajectory to converge to the given subspace is considered. A geometric characterization of the set of all Hermitian solutions of the discrete-time algebraic Riccati equation is given. This characterization forms the discrete-time counterpart of the well-known geometric characterization of the set of all real symmetric solutions of the continuous-time algebraic Riccati equation as developed by Willems [IEEE Trans. Automat. Control, 16 (1971), pp. 621–634] and Coppel [Bull. Austral. Math. Soc., 10 (1974), pp. 377–401]. In the set of all Hermitian solutions of the Riccati equation the solution that leads to the optimal cost for the above-mentioned linear quadratic problem is identified. Finally, necessary and sufficient conditions for the existence of optimal controls are given.

Perturbation Analysis of Matrix Quadratic Equations

Nonlocal perturbation analysis of a general type matrix quadratic equation is presented using the technique of contractive operators. The continuous-time algebraic matrix Riccati equation arising in linear-quadratic optimization is considered as a particular case.

Doubling algorithm for continuous-time algebraic Riccati equation

A second-order convergent algorithm is presented which gives an approximation to the unique positive definite solution of the continuous-time algebraic Riccati equation (CARE). First the CARE is transformed into the discrete-time algebraic Riccati equation (DARE) using the transformation given by Hitz and Anderson (1972). Then the discrete-time doubling algorithm, whose initial values are expressed in forms suitable for computation, is applied to the DARE. Next, it is shown that this algorithm is convergent under the condition that the CARE has the unique positive definite solution. Finally an ‘inverse’ of the Hitz and Anderson (1972) transformation is presented, which transforms the DARE into the CARE. It is proved that the ‘inverse’ transformation preserves the stabilizability, controllability, detectability and observability of the DARE.