elhawwary2013reduction

Reduction theorems for stability of closed sets with application to backstepping control design

Authors: Mohamed I. El-Hawwary, Manfredi Maggiore · Year: 2013 · Venue: Automatica 49(1):214–222 Raw: md · pdf

Summary

Solves a reduction problem for asymptotic stability of closed sets in nonlinear systems: given nested closed, positively-invariant sets ${\Gamma }_1\subset {\Gamma }_2$ with ${\Gamma }_1$ asymptotically stable relative to ${\Gamma }_2$, find conditions under which ${\Gamma }_1$ is asymptotically stable in the full state space. Analogous reduction results are given for plain stability and for attractivity. The payoff is a reduction-based backstepping technique for upper-triangular systems that needs no Lyapunov function and mitigates the controller-complexity blow-up of classical backstepping.

Key Claims

• Reduction principle (sets). Relative asymptotic stability of ${\Gamma }_1$ w.r.t. ${\Gamma }_2$ plus asymptotic stability of ${\Gamma }_2$ (under the stated conditions) lifts to asymptotic stability of ${\Gamma }_1$ — the set-stability analogue of the cascade reduction lemma.
• Cascade application. The theorems specialize to stability of sets for cascade-connected systems and to hierarchical control design.
• Backstepping without a Lyapunov function. For upper-triangular control systems, a reduction-based backstepping design avoids constructing an explicit Lyapunov function and curbs controller-complexity growth.

Method

Set-stability analysis on a domain $X\subset {\mathbb{R}}^n$ with locally-Lipschitz $f$; nested positively-invariant closed sets ${\Gamma }_1\subset {\Gamma }_2$; conditions phrased on the flow $\varphi (t,x_0)$ and relative stability rather than on a single Lyapunov function.

Regime note. A general nonlinear-systems / set-stability paper — no spacecraft or manipulator model — so the free-flying vs free-floating distinction does not arise. Relevance is as a tool for certifying stability of cascade / hierarchical designs.

Relevance to thesis

The closed-loop of a free-flying space manipulator under a nominal guidance/control law is naturally hierarchical and often decomposes into cascades whose stability is most cleanly stated as stability of a set (e.g. a tracking manifold) rather than a point. This paper’s set-reduction machinery is a candidate tool for certifying such designs without building a single monolithic Lyapunov function, complementing the point-cascade results of panteley2001growth.

Connections

Topics: Cascaded Systems · Lyapunov Stability Sources: panteley2001growth · khalil2002nonlinear

Key Equations / Quotes

From the abstract:

“Given two closed, positively invariant subsets of the state space of a nonlinear system, ${\Gamma }_1\subset {\Gamma }_2$, assuming that ${\Gamma }_1$ is asymptotically stable relative to ${\Gamma }_2$, find conditions under which ${\Gamma }_1$ is asymptotically stable.”

“For upper triangular control systems, we present a reduction-based backstepping technique that does not require the knowledge of a Lyapunov function, and mitigates the problem of controller complexity arising in classical backstepping design.”

Open Questions

• How do the set-reduction conditions map onto the specific tracking-manifold / cascade structure of the free-flying manipulator closed loop?