On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations

Abstract

Abstract.A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results by Oostveen and Curtain [Automatica 34 (1998) 953-967]. All the results are illustrated in detail by an electrical transmission line example of the distortionless loaded RLCG-type. The paper uses extensively the philosophy of reciprocal systems with bounded generating operators as recently studied and used by Curtain in (2003) [Syst.

Corrigendum to: “On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations”

“…Via the feedback law equation u = −μ 0 y this implies that for any x 0 : u ∈ L 2 (0, ∞). Now [2], Lemma 2.11, p. 177, implies that for every initial condition x 0 the first equation of (2.5) has a unique weak solution, whence, by Ball's theorem [1], p. 371 (see also [4], p. 259), the operator A 0 generates a C 0 -semigroup {S 0 (t)} t≥0 on H which is AS. Now, for every x 0 ∈ D(A 0 ) and each t ≥ 0, (2.4) yields…”

“…and it corresponds to the Lur'e control system of [2], Figure 1.1, p. 170, with f (y) = μ 0 y. Since c # is admissible andĝ ∈ H ∞ (C + ), for L 2 (0, ∞)-controls the output is given by y = P x 0 + Fu where P and F stand for the extended observability map and the extended input-output operator, both associated with (2.5).…”

“…This counterexample demonstrates that the assumptions of [2], Theorem 4.1, p. 186, are not enough to ensure non-negativity of H.…”

“…The authors are deeply indebted to Hartmut Logemann, Department of Mathematics, University of Bath, UK for pointing out a counterexample, repeated below, showing that the statement of [2], Theorem 4.1, p. 186, is wrong.…”

A general transfer function representation for a class of hyperbolic distributed parameter systems

“…The above definition of the transfer function can be extended to a more general class of systems, where some unboundedness of the input and output operators is possible, as shown by Salamon (1987), Curtain et al (1992), Callier and Winkin (1993), Grabowski and Callier (2001a;2001b), or Cheng and Morris (2003). Such general systems are interesting since they allow, e.g., the exact boundary control as well as the pointwise observation, as considered in Section 2.4.…”