Low-Gain Integral Control of Well-Posed Linear Infinite-Dimensional Systems with Input and Output Nonlinearities

Abstract

Time-varying low-gain integral control strategies are presented for asymptotic tracking of constant reference signals in the context of exponentially stable, wellposed, linear, infinite-dimensional, single-input-single-output, systems-subject to globally Lipschitz, nondecreasing input and output nonlinearities. It is shown that applying error feedback using an integral controller ensures that the tracking error is small in a certain sense, provided that (a) the steady-state gain of the linear part of the system is positive, (b) the reference value r is feasible in an entirely natural sense, and (c) the positive gain function t → k t is ultimately sufficiently small and not of class L 1 . Under a weak restriction on the initial data it is shown that (a), (b), and (c) ensure asymptotic tracking. If, additionally, the impulse response of the linear part of the system is a finite signed Borel measure, the global Lipschitz assumption on the output nonlinearity may be considerably relaxed.

Stability Results of Popov-Type for Infinite-Dimensional Systems with Applications to Integral Control

“…If additionally, lim t 3 1 gt 0, it follows from (4.10) that r wG0J ± u 1 P RG; J; w. In fact, it has been shown in [10] that if w is continuous and monotone, then r P RG; J; w is close to being a necessary condition for asymptotic tracking insofar as, if asymptotic tracking of r is achievable, whilst maintaining boundedness of J ± u together with ultimate continuity and ultimate boundedness of w g GJ ± u, then r P RG; J; w.…”

“…The following lemma, the proof of which can be found in the appendix of [10], shows that under certain conditions the initial-value problem (4.9) has a unique solution de®ned on R . Lemma 4.3.…”

Absolute stability and integral control for infinite-dimensional discrete-time systems

“…Remark 2.5), then results similar to Theorems 3.2 and 3.4 can be proved (see [15] for details). Finally, Theorems 3.2 and 3.4 constitute the discrete-time counterpart of the continuous-time low-gain integral control theory developed in [6,12]. Related discrete-time integral control results can be found in [16,17]: however, whilst similar in spirit, the relevant results in [16,17] are less general than Theorems 3.2 and 3.4 (for example, the choice of integrator gain is more restricted, the underlying linear system is assumed to be regular, output nonlinearities are not included).…”

“…In Section 3, we develop generalizations of this result to linear systems subject to input and/or output nonlinearities. The main results in Section 3 (Theorems 3.2 and 3.4) constitute the discrete-time counterpart of the continuous-time low-gain integral control theory developed in [6,12]. We close the paper with Section 4 which is devoted to applications of the input-output results in Sections 2 and 3 to infinite-dimensional state-space systems.…”

Integral control of infinite-dimensional systems in the presence of hysteresis: an input-output approach

“…If, additionally, lim t→∞ g(t) = lim t→∞ h(t) = 0, it follows from (4.2) that ρ − ϑ ∈ R(G, Φ, ψ). In fact, it has been shown in [6] that in the case of static input nonlinearities, if ψ is continuous and monotone, then ρ − ϑ ∈ R(G, Φ, ψ) is close to being a necessary condition for asymptotic tracking. …”