New perspectives on output feedback stabilization at an unobservable target

Abstract

We address the problem of dynamic output feedback stabilization at an unobservable target point. The challenge lies in according the antagonistic nature of the objective and the properties of the system: the system tends to be less observable as it approaches the target. We illustrate two main ideas: well chosen perturbations of a state feedback law can yield new observability properties of the closed-loop system, and embedding systems into bilinear systems admitting observers with dissipative error systems allows to mitigate the observability issues. We apply them on a case of systems with linear dynamics and nonlinear observation map and make use of an ad hoc finite-dimensional embedding. More generally, we introduce a new strategy based on infinite-dimensional unitary embeddings. To do so, we extend the usual definition of dynamic output feedback stabilization in order to allow infinite-dimensional observers fed by the output. We show how this technique, based on representation theory, may be applied to achieve output feedback stabilization at an unobservable target.

Output Feedback Stabilization of Non-uniformly Observable Systems

“…To do so, we need to obtain, from the estimation ẑ of z, an estimation x of x, the state of the original system. This is obtained thanks to a left-inverse of the embedding τ , whose existence follows from the next theorem proved in [8]. Indeed, noetherianity of analytic maps allows to show that finitely many bounded linear forms are sufficient to discriminate any two points of an embedded compact.…”

“…Indeed, the constant input u = 0 makes the system unobservable in any positive time, due to the skew-symmetry of A combined with the symmetry of the output. Actually, this system is not stabilizable by means of a dynamic output feedback when A is not invertible, as shown in [8]. In [12], time-varying stabilization strategies are considered to tackle this issue.…”

“…While the embedding obtained in Section 4.1 was highly dependent of this choice, the framework of Section 4.2 is sufficiently general to deal more output maps. If there exists a map h such that h(y) = Cτ (x), then the output feedback stabilization strategy of Theorem 4.3 still holds (see [8,Theorem 4.4]) when C * lies in the span of a finite number of elements of the canonical Hilbert basis of X. Due to Gelfand-Raikov theorem, output maps satisfying this condition are dense for the uniform convergence on compact sets of R 2 .…”

“…[8]). The closed-loop system (u(t))ẑ − αC * (C ẑ − J 0 (µ 2y)),u(t k ) = Kπ(ẑ(t − k )) + δN 2 (ẑ(t − k ) − 1) u(t) = u(t k ), t ∈ [t k , t k+1 )…”

Exponential Stabilizability and Observability at the Target Imply Semiglobal Exponential Stabilizability by Templated Output Feedback

“…Note that assumption o4) naturally excludes the case of systems that are unobservable at the target point (see e.g. [9]). Nevertheless, our observability condition is generic (i.e.…”