[Paper Review] Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases

Antoine Chaillet, Yacine Chitour+2 more|arXiv (Cornell University)|Feb 23, 2007

Control and Stability of Dynamical Systems10 references4 citations

TL;DR

This paper establishes uniform stabilization for linear systems with persistently exciting (PE) control signals, proving that a fixed linear time-invariant state feedback $ K $ can globally asymptotically stabilize systems when $ A $ is neutrally stable or when the system is a double integrator. The key result is that such a stabilizing gain $ K $ exists and depends only on $ (A,B) $, $ \mu $, and $ T $, even when the control signal $ \alpha(t) $ vanishes over intervals, provided the PE condition holds.

ABSTRACT

Consider the controlled system $dx/dt = Ax + α(t)Bu$ where the pair $(A,B)$ is stabilizable and $α(t)$ takes values in $[0,1]$ and is persistently exciting, i.e., there exist two positive constants $μ,T$ such that, for every $t\geq 0$, $\int_t^{t+T}α(s)ds\geq μ$. In particular, when $α(t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback $u=Kx$, with $K$ only depending on $(A,B)$ and possibly on $μ,T$, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator.

Motivation & Objective

• To determine whether a fixed linear time-invariant (LTI) state feedback $ K $ can stabilize a linear system under persistently exciting (PE) control signals, even when the control is intermittently zero.
• To address the challenge of stabilizing systems where the control input $ \alpha(t)Bu $ becomes inactive over intervals, making the system uncontrollable at times.
• To provide a constructive solution for stabilization using only the system matrices $ (A,B) $, and the PE parameters $ \mu, T $, without requiring knowledge of the exact switching times of $ \alpha(t) $.
• To extend the notion of averaging stabilization to non-scalar, non-averagable systems by proving the existence of a universal stabilizing gain $ K $ under PE conditions.

Proposed method

• The authors analyze the system $ \dot{x} = Ax + \alpha(t)Bu $, where $ \alpha(t) \in [0,1] $ is a PE signal satisfying $ \int_t^{t+T} \alpha(s)ds \geq \mu $ for all $ t \geq 0 $, with $ \mu, T > 0 $.
• For the neutrally stable case, they use Lyapunov-based arguments and the structure of the system's dynamics to show that a fixed $ K $ stabilizes the system uniformly across all such $ \alpha(t) $.
• For the double integrator, they exploit the nilpotent structure of $ A $ and use a transformation to Brunovsky normal form to reduce the problem to a scalar-like system.
• They apply weak-$ \star $ convergence arguments in $ L^\infty $ and use Gronwall's inequality to prove continuity of solutions with respect to $ \alpha $, ensuring stability under perturbations of the PE signal.
• The proof for the full-rank $ B $ case uses a change of coordinates to reduce the system to $ \dot{x} = Ax + \alpha u $, where $ u = -kx $, and shows that the transformed system satisfies $ \dot{y} = -k\alpha y $, leading to exponential decay.

Experimental results

Research questions

• RQ1 Can a fixed linear time-invariant state feedback $ K $ stabilize a linear system when the control signal $ \alpha(t) $ is persistently exciting but may vanish over intervals?
• RQ2 Is it possible to achieve global asymptotic stabilization without relying on real-time knowledge of $ \alpha(t) $'s switching pattern, using only $ (A,B) $, $ \mu $, and $ T $?
• RQ3 Does the persistency of excitation condition suffice to ensure stabilization via a single, fixed $ K $, even when the system becomes uncontrollable during parts of the trajectory?
• RQ4 Can the averaging intuition from scalar systems be extended to multi-dimensional systems with non-averagable dynamics, such as the double integrator?
• RQ5 What conditions on $ A $ and $ B $ allow for a universal stabilizing gain $ K $ under PE control signals?

Key findings

• For neutrally stable $ A $, there exists a fixed $ K $ such that the closed-loop system is globally asymptotically stable for all $ \alpha(t) \in \mathcal{G}(T,\mu) $, regardless of the signal's switching behavior.
• For the double integrator system, a fixed $ K $ exists that stabilizes the system uniformly under any PE signal $ \alpha(t) $, even when $ \alpha(t) $ vanishes on intervals.
• When $ \mathrm{rank}(B) = 2 $, the system $ \dot{x} = Ax + \alpha Bu $ is uniformly stabilized by a fixed $ K $, with the proof relying on transformation to a scalar-like system via coordinate change.
• The stabilizing gain $ K $ depends only on $ (A,B) $, $ \mu $, and $ T $, and not on the specific form of $ \alpha(t) $, making it robust to signal irregularities.
• The proof establishes uniform convergence of solutions under weak-$ \star $ convergence of $ \alpha $, ensuring stability is preserved under perturbations of the PE signal.
• The paper shows that the naive averaging intuition from scalar systems does not extend directly to higher-dimensional systems, but a fixed $ K $ still ensures stabilization under PE conditions.