Dynamic State-Feedback Control for LPV Systems: Ensuring Stability and LQR Performance

TL;DR

This work addresses stabilization and optimal control for polytopic LPV systems with a constant input matrix by introducing a dynamic state-feedback controller that evolves via a projected policy gradient flow to minimize the LQR cost. The authors establish quadratic stability through finite LMIs on a polytopic augmentation, guarantee boundedness of the controller trajectories, and prove convergence of the adaptive gain to the time-varying LQR optimum for constant parameter trajectories. They show that, under reasonable assumptions, frozen-time closed-loop systems are asymptotically stable and that stability is preserved even for arbitrarily fast parameter variations, provided the parameter-variation rate is managed. Simulations demonstrate stability retention and significant transient performance improvements over static gains, highlighting practical viability for rapidly varying LPV dynamics.

Abstract

In this paper, we propose a novel dynamic state-feedback controller for polytopic linear parameter-varying (LPV) systems with constant input matrix. The controller employs a projected gradient flow method to continuously improve its control law and, under established conditions, converges to the optimal feedback gain of the corresponding linear quadratic regulator (LQR) problem associated with constant parameter trajectories. We derive conditions for quadratic stability, which can be verified via convex optimization, to ensure exponential stability of the LPV system even under arbitrarily fast parameter variations. Additionally, we provide sufficient conditions to guarantee the boundedness of the trajectories of the dynamic controller for any parameter trajectory and the convergence of its feedback gains to the optimal LQR gains for constant parameter trajectories. Furthermore, we show that the closed-loop system is asymptotically stable for constant parameter trajectories under these conditions. Simulation results demonstrate that the controller maintains stability and improves transient performance.

Figures (4)

• Figure 1: Structured overview of the logical dependencies among the contributions, assumptions, propositions, and theorems of Section \ref{['sec:stability']}. The subsections are separated by Assumptions \ref{['ass:bound']} and \ref{['ass:opt']}.
• Figure 2: Stability regions $\mathcal{K}_\rho$, optimal feedbacks $K_\rho^*$ and stability boundaries $f_1=f_2 =0$ for $\rho=0.5, \rho=1$ and $\rho=2$.
• Figure 3: Visualization of the stability region $\mathcal{K}_{\mathcal{P}}$, the set $\mathcal{C}$ and trajectories of gradient flow and projected gradient flow for $\rho=0.5 \to \rho = 2$.
• Figure 4: Trajectories of the states $x(t)$ and the feedback gains $K(t)$ for the fast-varying parameter trajectory $\rho(t)$.

Theorems & Definitions (35)

• Proposition 1
• Theorem 1
• Proposition 2
• Definition 1: Dynamic state-feedback controller
• Remark 1
• Definition 2: Closed-loop System
• Remark 2
• Proposition 3
• proof
• Lemma 1