Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S−1/2B*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H∞ for positive real transfer functions of the form D+S−1/2B*(author−A)−1,B

Analytic and algebraic properties of Riccati equations:A survey

AbstractThis is a survey of recent results on the classical problems of the analytic properties of Riccati equations and algebraic properties of Riccati equations and applications to spatially distributed systems

Analytic system problems and J-lossless coprime factorization for infinite-dimensional linear systems

AbstractThis paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an H∞-norm bound to a class of systems with irrational transfer matrices. Using the coprime factorization description, the H∞-control problem can be reduced to two stable analytic system problems. Such problems have solutions if and only if a certain J-lossless factorization exists. The full H∞-synthesis problem is shown to be equivalent to the solution of two nested J-lossless factorizations. If the irrational transfer matrix has a state-space realization, then the known state-space formulas for the H∞-control problem may be recovered using the relationship between J-lossless factorizations and solutions of Riccati equations. However, the results derived here are valid for a larger class of infinite-dimensional systems

STRONG STABILIZATION OF (ALMOST) IMPEDANCE PASSIVE SYSTEMS BY STATIC OUTPUT FEEDBACK

The plant to be stabilized is a system node E with generating triple (A, B, C) and transfer function G, where A generates a contraction semigroup on the Hilbert space X. The control and observation operators B and C may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator E such that, if we replace G(s) by G(s) + E, the new system Sigma(E) becomes impedance passive. An easier case is when G is already impedance passive and a special case is when Sigma has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback u = -kappa y + v, where u is the input of the plant and kappa > 0, stabilizes Sigma, strongly or even exponentially. Here, y is the output of Sigma and v is the new input. Our main result is that if for some E is an element of L(U), Sigma(E) is impedance passive, and Sigma is approximately observable or approximately controllable in infinite time, then for sufficiently small kappa the closed-loop system is weakly stable. If, moreover, sigma(A)boolean AND iR is countable, then the closed-loop semigroup and its dual are both strongly stable

A Comparison of finite-dimensional controller designs for distributed parameter systems

This paper compares five different approaches to the design of finite-dimensional controllers for linear infinite-dimensional systems. The approaches are varied and include state and frequency domain methods, exact controller designs, controller designs by approximation and robust controller designs

Robust Controllers for Dead-time Systems

It is shown that the classical controllers for dead-time systems, known as Smith predictors, provide a suitable basis for the design of optimal robust stabilizing controllers. Only in case of an unstable plant a modification of the Smith predictor is necessary. A simple first order dead-time system serves as illustrative example

Analytic Solutions of Matrix Riccati Equations with Analytic Coefficients

For matrix Riccati equations of platoon-type systems and of systems arising from PDEs, assuming the coefficients are analytic or rational functions in a suitable domain, analyticity of the stabilizing solution is proved under various hypotheses. General results on analytic behavior of stabilizing solutions are developed as well

Coprime factorization and robust stabilization for discrete-time infinite-dimensional systems

We solve the problem of robust stabilization with respect to right-coprime factor perturbations for irrational discrete-time transfer functions. The key condition is that the associated dynamical system and its dual should satisfy a finite-cost condition so that two optimal cost operators exist. We obtain explicit state space formulas for a robustly stabilizing controller in terms of these optimal cost operators and the generating operators of the realization. Along the way we also obtain state space formulas for Bezout factors