Exponential Stability of Parametric Optimization-Based Controllers via Lur'e Contractivity

Główne pojęcia

Investigating the exponential stability of LTI systems with parametric optimization-based controllers using a novel contractivity result for Lur’e systems.

Streszczenie

The article explores the stability of LTI systems driven by controllers derived from parametric optimization problems. It focuses on parametric projection controllers, establishing sufficient conditions for exponential stability. The study includes results for single-integrator systems and applications to state-dependent saturated control systems and control barrier function-based control. The analysis is based on contraction theory and contractivity of Lur’e systems, providing a tutorial review of the virtual system method. The main contributions include necessary and sufficient conditions for contractivity of Lur’e systems, convergence results for single integrators, and numerical simulations demonstrating global exponential stability.

Statystyki

Regularity results are studied in [5, Chapter 5].
Stability properties of CLF-CBF QP controllers are relaxed in [15].
A variant of the CLF-CBF QP controller is studied in [15].
The condition in [2] corresponds to inequality (7) with λ = 1.
Lipschitzness property is crucial for continuity.

Cytaty

"Controllers derived from optimization problems are ubiquitous in systems and control."
"One large class of optimization-based controllers solves optimal control problems offline."
"Parametric optimization is essential in model predictive control and CBF-based control."
"Our analysis provides novel sufficient conditions for exponential stability."

Kluczowe wnioski z

Exponential Stability of Parametric Optimization-Based Controllers via Lur'e Contractivity

by Alexander Da... o arxiv.org 03-14-2024

https://arxiv.org/pdf/2403.08159.pdf

Głębsze pytania

How can the assumptions be relaxed to apply the analysis to a broader class of LTI systems

To apply the analysis to a broader class of LTI systems, the assumptions can be relaxed in several ways. One approach is to consider non-convex optimization problems for controllers instead of strictly convex ones. This relaxation allows for a wider range of controller designs and applications. Additionally, relaxing the assumption of linearity for the nominal controller opens up possibilities for nonlinear control strategies. By considering more general forms of parametric programs and allowing for nonlinearity in both the system dynamics and controller design, we can extend the applicability of the analysis to a broader class of LTI systems.

What are the implications if the controller only approximately solves the parametric program

If the controller only approximately solves the parametric program, it may lead to suboptimal performance or stability issues in practice. The level of approximation could impact how well constraints are satisfied or how effectively objectives are achieved. In some cases, small deviations from optimal solutions may not significantly affect system behavior, while in other cases they could lead to instability or poor performance. It is essential to analyze these effects carefully when dealing with approximate solutions to ensure that overall system behavior remains within acceptable bounds.

How does the virtual system method compare to classical stability analysis techniques

The virtual system method offers a unique perspective on stability analysis compared to classical techniques. While traditional methods often rely on direct Lyapunov functions or input-to-state stability properties, the virtual system approach provides an elegant way to establish exponential convergence by transforming complex systems into simpler time-varying structures amenable to contraction theory principles. This method leverages contractivity results for Lur'e systems and enables rigorous analysis even when dealing with state-dependent controllers derived from parametric optimization problems.