Quadratic optimal control of stable well-posed linear systems

Abstract

Abstract. We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

Absolute-stability results in infinite dimensions

“…In this section we use the results of § 3 to derive absolute-stability results for wellposed state-space systems. There are a number of equivalent definitions of well-posed systems (see Curtain & Weiss 1989;Salamon 1987Salamon , 1989Staffans 1997Staffans , 2001Staffans , 2004Staffans & Weiss 2002;Weiss 1989Weiss , 1994). We will be brief in the following and refer the reader to the above references for more details.…”

“…Section 4 is devoted to applications of the input-output results in § 3 to the class of well-posed state-space systems which are documented, for example, in Curtain & Weiss (1989), Salamon (1987Salamon ( , 1989, Staffans (1997Staffans ( , 2001Staffans ( , 2004, Staffans & Weiss (2002) and Weiss (1989Weiss ( , 1994. We remark that the class of well-posed, linear, infinitedimensional systems allows for considerable unboundedness in the control and observation operators and is therefore rather general: it includes most distributed parameter systems and time-delay systems (retarded and neutral) which are of interest in applications.…”

Strong stabilization of (almost) impedance passive systems by static output feedback

“…We denote the generating triple of Σ s by (A s , B s , C s ) and its transfer function by G s . Note from (38) and Figure 3 that there is no feedback loop involved in the transformation between the inputs and outputs of the scattering passive system Σ s and of Σ κ . So the latter is in fact a well-posed system and from (38) we deduce that its generating triple (A κ , B κ , C κ ) and G κ are given by…”

“…Necessary and sufficient conditions for well-posedness were given in Curtain and Weiss [10]. For alternative definitions, background and examples we refer to Salamon [33], Staffans [38], [42], [43], Weiss [50], [51], Weiss and Rebarber [54] and Weiss, Staffans and Tucsnak [55]. All well-posed systems are compatible, see Staffans and Weiss [43].…”

A Convex Sum-of-Squares Approach to Analysis, State Feedback and Output Feedback Control of Parabolic PDEs

“…The problem of point actuation with full-state feedback was considered in [21] (and related work) and extended in [20] to output feedback controller synthesis through the use of a Luenberger observer. An alternative Riccati-based approach for static output feedback of a certain class of well-posed operators can be found in [35], [36], [42]. A limitation of these Riccati-based methods, however, is that they rely on finitedimensional numerical methods for obtaining the operatorvalued solution.…”