Nonlinear Feedback Stability Analysis via Dissipativity with Dynamic Supply Rates

Concepts de base

Nonlinear feedback systems can be analyzed for Lyapunov and asymptotic stability using a general notion of dissipativity with dynamic supply rates, which extends classical dissipativity with static supply rates.

Résumé

The paper proposes a general notion of dissipativity with dynamic supply rates for nonlinear systems. This extends classical dissipativity with static supply rates and dynamic supply rates of miscellaneous quadratic forms.
The main results concern Lyapunov and asymptotic stability analysis for nonlinear feedback dissipative systems characterized by dissipation inequalities with respect to compatible dynamic supply rates, but involving possibly different and independent auxiliary systems. Dissipativity conditions guaranteeing stability of the state of the feedback systems, without concerns on the stability of the state of the auxiliary systems, are provided.
The key results specialize to a simple coupling test for the interconnection of two nonlinear systems described by dynamic (Ψ, Π, Υ, Ω)-dissipativity, and are shown to recover several existing results in the literature, including small-gain, passivity indices, static (Q, S, R)-dissipativity, dissipativity with terminal costs, etc. Comparison with the input-output approach to feedback stability analysis based on integral quadratic constraints is also made.

Stats

The paper does not contain any explicit numerical data or statistics. It focuses on the theoretical development of the dissipativity framework and its application to feedback stability analysis.

Citations

"Dissipativity theory abstracts the notion of energy and its dissipation in dynamical systems, and may be viewed as a generalisation of Lyapunov theory for autonomous systems to open systems with input and outputs."
"When it comes to robust closed-loop (asymptotic/exponential) stability analysis based on dissipativity, conservatism may be reduced with the aid of stable and stably invertible dynamical multipliers."
"Motivated in part by the prowess and utility of multipliers, the notion of dissipativity with dynamic supply rates has been considered for robust stability analysis in various contexts."

Idées clés tirées de

Feedback Stability Analysis via Dissipativity with Dynamic Supply Rates

by Sei Zhen Kho... à arxiv.org 04-30-2024

https://arxiv.org/pdf/2209.08322.pdf

Questions plus approfondies

How can the proposed dissipativity framework with dynamic supply rates be extended to address robust stability analysis in the presence of uncertainties or disturbances

The proposed dissipativity framework with dynamic supply rates can be extended to address robust stability analysis in the presence of uncertainties or disturbances by incorporating uncertainty models into the system dynamics and supply rates. One approach is to consider dynamic supply rates that capture the effects of uncertainties or disturbances on the system behavior. This can involve modifying the dynamic operators in the supply rates to account for uncertain parameters or disturbances that affect the system's dissipation properties. By formulating dissipativity conditions that account for these uncertainties, the stability analysis can be extended to robust stability analysis, ensuring system stability in the presence of uncertainties or disturbances.

What are the potential limitations or drawbacks of the dynamic supply rate approach compared to the input-output approach based on integral quadratic constraints

One potential limitation of the dynamic supply rate approach compared to the input-output approach based on integral quadratic constraints is the complexity of formulating and verifying dissipativity conditions with dynamic supply rates. Dynamic supply rates introduce additional dynamics and dependencies in the dissipativity framework, which may increase the computational complexity of stability analysis. In contrast, the input-output approach based on integral quadratic constraints offers a more structured and systematic framework for stability analysis, with well-established methods for formulating and verifying stability conditions. Additionally, the input-output approach may provide more insights into system behavior and performance through the analysis of integral quadratic constraints.

Can the dynamic dissipativity framework be applied to the analysis and design of interconnected systems beyond the feedback configuration considered in this paper

The dynamic dissipativity framework can be applied to the analysis and design of interconnected systems beyond the feedback configuration considered in the paper by extending the dissipativity conditions to accommodate more complex interconnections. This can involve considering multiple interconnected systems with dynamic supply rates and auxiliary systems, allowing for the analysis of larger-scale interconnected systems. The framework can be applied to networked systems, distributed systems, and multi-agent systems, where dissipativity properties play a crucial role in ensuring stability and performance. By adapting the dynamic dissipativity framework to various interconnection structures and system architectures, it can provide valuable insights into the stability and behavior of interconnected systems in diverse applications.