Theory and Numerical Solution of Differential and Algebraic Riccati Equations

Abstract

Since Kalman's seminal work on linear-quadratic control and estimation problems in the early 1960s, Riccati equations have been playing a central role in many computational methods for solving problems in systems and control theory, like controller design, Kalman filtering, model reduction, and many more. We will review some basic theoretical facts as well as computational methods to solve them, with a special emphasis on the many contributions Volker Mehrmann had regarding these subjects. IntroductionThe algebraic and differential Riccati equations (AREs/DREs) play a fundamental role in the solution of problems in systems and control theory. They have found widespread applications in applied mathematics and engineering, many of which can be found in the monographs [1,36,68,86]. In this chapter, we focus on Riccati equations associated to control problems, as these have always inspired Volker Mehrmann's work, and he has mainly focused on the resulting symmetric Riccati equations -symmetric in the sense that the associated Riccati operators map symmetric (Hermitian) matrices onto symmetric (Hermitian) matrices. Hence, also the solutions to the Riccati equations we will consider are expected to be symmetric (Hermitian). A class of nonsymmetric AREs that arises, e.g., in queuing theory, certain fluid flow problems, and transport theory (see, e.g., [36]) is of importance as well, but for conciseness, we will omit these AREs even though Volker has also contributed to this area [81].In most of the literature on AREs and DREs, the motivation is taken from the classical linear-quadratic regulator (LQR) problem. This was the topic of Volker's habilitation thesis [74], where, building upon earlier work by Bender and Laub [6,7], he extended the LQR theory to so-called descriptor systems, i.e., systems with

RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations

“…(b) The first part follows as in (a). If Ξ ≥ 0 andX is a stabilizing solution to (10), then X = Ξ +X ≥ 0 and A − GX =Ã − GX is stable, which makes X the stabilizing solution to (1).…”

“…(b) Conversely, if X is a solution to (10), then X = Ξ + X is a solution to the original Riccati equation (1). Moreover, if Ξ ≥ 0 and X is a stabilizing solution to (10), then X = Ξ + X is the stabilizing solution to (1).…”

“…(a) This is a straightforward computation which follows by inserting X = X − Ξ and the formula for the residual of Ξ into (10), see also [10,23].…”

“…One such approach is motivated by Theorem 1 and the discussion about shift selection in [2]. The Hamiltonian matrix H = Ã G C * C − Ã * is associated to the residual equation (10), where Ξ = X k is the approximation after k steps of the RADI iteration. If (λ, [ r q ]) is a stable eigenpair of H, and σ k+1 = λ is used as the shift, then…”