Reduced-order Riccati Equations Research Articles

Silverman algorithm and the structure of discrete-time stochastic systems

It is a well-known, yet poorly understood fact that, contrary to the continuous-time case, the same discrete-time process y can be represented by minimal linear models (see (1.1a), (1.1b) below) which may either have a non-singular or a singular D matrix. In fact, models with D=0 have been commonly used in the statistical literature. On the other hand, for models with a singular D matrix the Riccati difference equation of Kalman filtering involves in general the pseudo-inversion of a singular matrix. This “cheap filtering” problem, dual to the better known “cheap control” problem, has been studied for several decades in connection with the so-called “invariant directions” of the Riccati equation. For a singular D, a reduction of the order of the Riccati equation is in general possible. The reasons for such a reduction do not seem to be completely clear either. In this paper we provide an explanation of this phenomenon from the classical point of view of “zero flipping” among minimal spectral factors. Changing D's occurs whenever zeros are “flipped” from z=∞ to their reciprocals at z=0. It is well known that for finite zeros, the zero-flipping process takes place by multiplication of the underlying spectral factor by a suitable rational all-pass matrix function. For infinite zeros, zero flipping is implemented by a dual version of the Silverman structure algorithm. Using this interpretation, we derive a new algorithm for filtering of non-regular processes, based on a reduced-order Riccati equation. We also obtain a precise characterization of the reduction of the order of the Riccati equation which is afforded by zeros either at z=∞ or at the origin. This order reduction has traditionally been associated with the study of invariant directions, a point of view which, as we show, does not capture the essence of the phenomenon.

Complete solution of the 4-blockH? control problem with infinite and finite j?-axis zeros

This paper discusses the 4-block H∞ control problem with infinite and finite jω-axis invariant zeros in the state-space realizations of the transfer functions from the control input to the controlled output and from the disturbance input to the measurement output, where these realizations are induced from a stabilizable and detectable realization of the generalized plant. This paper extends the DGKF approach to the H∞ control problem but permitting infinite and finite jω-axis invariant zeros by using the eigenstructures related to these zeros. Necessary and sufficient conditions are presented for checking solvability through checking the stabilizing solutions of two reduced-order Riccati equations and examining matrix norm conditions related to the jω-axis zeros. The parameterization of all suitable controllers is given in terms of a linear fractional transformation involving a certain fixed transfer function matrix and together with a stable transfer function matrix with gain less than 1 which is free apart from satisfying certain interpolation conditions. Copyright © 2000 John Wiley & Sons, Ltd.

Robust Stabilization of Singularly Perturbed Systems with Uncertainties

This paper considers the robust stabilization of singularly perturbed systems with time-varying unknowbut-bounded uncertainties. The H∞ control method are used to establish the stability of the closed-loop system. The construction of the stabilizing controller involves solving a certain algebraic Riccati equation with small parameter e > 0. In order to overcome the computation difficulties caused by small parameter e, we propose the e-less quadratically stabilizing controller by using the results of the solution for above a algebraic Riccati equation. It is shown that if the reduced order Riccati equations have a positive definite stabilizing solution then the given uncertain linear system with the proposed controller independ of e is quadraticaly stabilizable. To show the effectiveness of the proposed algorithm, numerical examples are included.

The general nonstrict algebraic Riccati inequality

If ( A, B) is stabilizable, it is well known how to verify the solvability or the existence of positive definite solutions of the nonstrict algebraic Riccati inequality (ARI) A∗X + XA − XBB∗X + Q ≥0. This paper serves to demonstrate how far it is presently possible to generalize these results if the uncontrollable modes of ( A, B) are not restricted. We prove a new reduction principle which may be formulated as follows: The ARI has a solution iff a certain reduced-order Riccati equation (for a stabilizable system), a linear equation, and a reduced-order nonstrict Lyapunov inequality (corresponding to the uncontrollable modes on the imaginary axis) have solutions. Basically, the quadratic Riccati inequality is hence reduced to a linear Lyapunov inequality. We discuss how this principle may be applied to actually check the solvability of the ARI. As the main motivation for our work we briefly explain the consequences of these results for the general state-feedback H ∞-optimization problem.