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

TL;DR

This paper demonstrates that almost impedance passive systems can be strongly stabilized using static output feedback, especially when they are approximately observable or controllable, extending previous results to more general systems.

Contribution

The paper introduces a stabilization method for impedance passive systems with unbounded operators, using static output feedback under approximate controllability or observability.

Findings

Weak stability achieved for small feedback gain rac14;

Strong stability if the spectrum on the imaginary axis is countable

Applicable to boundary control systems like wave, beam, and heat equations

Abstract

The plant to be stabilized is a system node $\Sigma$ with generating triple $(A,B,C)$ and transfer function $\bf G$, where $A$ generates a contraction semigroup on the Hilbert space $X$. The control and observation operators $B$ and $C$ may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator $E$ such that, if we replace ${\bf G}(s)$ by ${\bf G}(s)+E$, the new system $\Sigma_E$ becomes impedance passive. An easier case is when $\bf G$ is already impedance passive and a special case is when \mm $\Sigma$ has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback $u=-\kappa y+v$, where $u$ is the input of the plant and $\kappa>0$, stabilizes $\Sigma$, strongly or even exponentially. Here, $y$ is the output of \m $\Sigma$ and $v$ is the new input. Our main result is that if for some $E\in{\mathcal L}(U)$, $\Sigma_E$ is impedance passive, and \m $\Sigma$ is approximately observable or approximately controllable in infinite time, then for sufficiently small $\kappa$ the closed-loop system is weakly stable. If, moreover, $\sigma(A)\cap i{\mathbb R}$ is countable, then the closed-loop semigroup and its dual are both strongly stable.