Transfer functions of infinite-dimensional systems: positive realness and stabilization

Abstract

We consider a general class of operator-valued irrational positive-real functions with an emphasis on their frequency-domain properties and the relation with stabilization by output feedback. Such functions arise naturally as the transfer functions of numerous infinite-dimensional control systems, including examples specified by PDEs. Our results include characterizations of positive realness in terms of imaginary axis conditions, as well as characterizations in terms of stabilizing output feedback, where both static and dynamic output feedback are considered. In particular, it is shown that stabilizability by all static output feedback operators belonging to a sector can be characterized in terms of a natural positive-real condition and, furthermore, we derive a characterization of positive realness in terms of a mixture of imaginary axis and stabilization conditions. Finally, we introduce concepts of strict and strong positive realness, prove results which relate these notions and analyse the relationship between the strong positive realness property and stabilization by feedback. The theory is illustrated by examples, some arising from controlled and observed partial differential equations.

Stability and convergence properties of forced infinite-dimensional discrete-time Lur'e systems

“…We next state three lemmas which underpin our development. The first lemma is a discrete-time version of [15,Proposition 5.6]. We omit the proof, since it is similar to that in [15].…”

The generalised singular perturbation approximation for bounded real and positive real control systems

“…The assumption that G is rational implies that G is analytic on C 0 \∆, and it is well-known (see [20,Proposition 3.3]) that analyticity and the positive realness condition (4.1) together imply that G in fact has no poles in C 0 , and hence G ∈ H(C 0 , C m×m ). Rational positive real functions may have simple imaginary axis poles, such as s → 1/s, and need not be proper, such as s → s.…”

“…They parallel the results in Section 3: the first contains state-space properties of the positive real GSPA and the second contains frequency domain properties and error bounds. Adopting the nomenclature convention used in [20], we say that the rational, C m×mvalued function G is strongly positive real if 3), let r ∈ n − 1 and ξ ∈ C with Re(ξ) ≥ 0 which is not a pole of G. Then there exists proper, rational, and positive real G ξ r ∈ H(C 0 , C m×m ) which has a state-space realisation of dimension r, such that (2.7) holds andδ…”

“…For ξ ∈ C 0 , and for statement (iii), we shall require (A, B, C, D) defined in (6.5), which is stable by statement (3) of Lemma 6.2. Calculations starting from (3.1) and (3.2) respectively show that The first equations in (6.17) and (6.18) may respectively be rewritten as 20) and…”

Infinite-dimensional Lur’e systems with almost periodic forcing

“…In fact, if H is positive real, then H is holomorphic on C 0[15, Proposition 3.3].The following result can be considered as an incremental version of the circle criterion. Let Σ, Z 1 and Z 2 be as in Theorem 3.4, let i, j ∈ {1, 2} and let…”