Nonsymmetric Algebraic Riccati Equations Research Articles

Decomposition of Optimal Control and Filtering Tasks for Linear Continuous Stochastic Systems

The algebraic regulator and filter Riccati equations are decomposed into reduced-order algebraic Riccati equations. That is, the exact solution of the global algebraic Riccati equation is found in terms of the reduced-order nonsymmetric algebraic Riccati equations. Using the separation principle the corresponding LQG optimal problem is obtained in terms of the local subsystems.

Local convergence analysis of conjugate gradient methods for solving algebraic Riccati equations

Necessary and sufficient conditions are given for local convergence of the conjugate gradient (CG) method for solving symmetric and nonsymmetric algebraic Riccati equations. For these problems, the Frobenius norm of the residual matrix is minimized via the CG method, and convergence in a neighborhood of the solution is predicated on the positive definiteness of the associated Hessian matrix. For the nonsymmetric case, the Hessian eigenvalues are determined by the squares of the singular values of the closed-loop Sylvester operator. In the symmetric case, the Hessian eigenvalues are closely related to the squares of the closed-loop Lyapunov singular values. In particular, the Hessian is positive definite if and only if the associated operator is nonsingular. The invertibility of these operators can be expressed as a noncancellation condition on the eigenvalues of the closed-loop matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Pole placement and order reduction in two-time-scale control systems through Riccati iteration

A transformation of variables taken from singular perturbations may be applied to two-time-scale linear systems in state space form to reduce the system to block-diagonal form with slow and fast modes decoupled. The transformation is easily computed by applying the new “Riccati Iteration.” The iteration yields a solution to the nonsymmetric algebraic Riccati equation obtained by partitioning the original system matrix A. The numerical procedure is initiated with the trivial iterate L 0 = 0, and is globally convergent to the desired unique time scale decoupling solution. After transformation, the decoupled system may be used in controller design to achieve exact closed-loop pole placement in the slow subsystem without altering the poles of the fast subsystem. The decoupled form may also be used to reduce system order by setting a small parameter to zero. Provided the fast subsystem is stable, the order reduction can be expected to yield a good approximation to the original system. These methods are demonstrated using the 16th order linear model of a turbofan engine.