On Finite-Gain Stabilizability of Linear Systems Subject to Input Saturation

Abstract

This paper deals with (global) finite-gain input/output stabilization of linear systems with saturated controls. For neutrally stable systems, it is shown that the linear feedback law suggested by the passivity approach indeed provides stability, with respect to every L p-norm. Explicit bounds on closed-loop gains are obtained, and they are related to the norms for the respective systems without saturation. These results do not extend to the class of systems for which the state matrix has eigenvalues on the imaginary axis with nonsimple (size > 1) Jordan blocks, contradicting what may be expected from the fact that such systems are globally asymptotically stabilizable in the state-space sense; this is shown in particular for the double integrator.

Input-to-state stability of Lur’e systems

“…Results which are (vaguely) related to Proposition 3.4 can be found in [20], where it is shown that, under suitable assumptions, a "small" signal ISS property holds for Lur'e systems with nonlinearities of "saturation" type.…”

“…Moreover, a further corollary (Corollary 3.11) shows that the conditions of the usual textbook version of the circle criterion for global asymptotic stability (see [7,16,27]) are actually sufficient for ISS. Finally, we mention that if A is not Hurwitz and f is bounded (for example, if f is of "saturation" type), then the nonlinearity is not "powerful" enough to counteract large (but bounded) inputs (at least if im B ⊂ im B e ) and the Lur'e system (1.1) is not ISS (see [20] and Proposition 3.4 in the current paper). Correspondingly, it is not difficult to show that if A is not Hurwitz, f is bounded and every complex output feedback gain in the ball {F : F − K < r } is stabilizing, then there does not exist α ∈ K ∞ such that (1.2) holds (see Proposition 3.4).…”

Control Lyapunov functions for homogeneous “Jurdjevic-Quinn” systems

“…The same discussion holds replacing V λ with W λ satisfying (18). Reshaping V 0 via the flow of a vector field G has the interesting advantage that it replaces the above difficult condition on V λ given by equation (20) by a simpler condition on G as shown by the following lemma: Lemma 4.1.…”

“…In [18] (proof of Lem. 2), the case where f 0 is a linear vector field and where the control vector fields depends on the control (systems are non affine in the control but it has a particular structure) is studied.…”

“…Let us however mention two occurences in the litterature where such a step was taken in more particular cases, and as part of a proof in [18] (proof of Lem. 2) for linear systems with saturations and in [19] (proof of Cor.…”