Stability Analysis of Adaptive Model Predictive Control for Discrete-Time Lur'e Systems

Conceptos Básicos

This paper analyzes the closed-loop stability of predictive cost adaptive control (PCAC) for output-feedback of a discrete-time Lur'e system using absolute stability criteria, namely the circle criterion and the Tsypkin test.

Resumen

The paper considers a discrete-time Lur'e (DTL) system, which consists of a linear system G connected in a feedback loop with a memoryless nonlinearity γ. The goal is to evaluate the effectiveness of PCAC for stabilizing the DTL system.
Key highlights:

PCAC performs online closed-loop linear system identification using recursive least squares (RLS) with variable-rate forgetting. The identified model is then used as the basis for receding-horizon optimization.
The closed-loop Lur'e system, comprising the positive feedback interconnection of the DTL system and the PCAC-based controller, is derived. This allows the application of absolute stability theory for analyzing the stability of the closed-loop system.
The discrete-time circle and Tsypkin criteria are used to evaluate the absolute stability of the closed-loop Lur'e system, where the adaptive controller is viewed as instantaneously linear time-invariant.
A numerical example demonstrates that, under additional excitation, the circle and Tsypkin criteria are satisfied, suggesting that PCAC globally asymptotically stabilizes the DTL system.
The effectiveness of PCAC in stabilizing the nonlinear system, despite the modeling mismatch between the linearized system and the actual self-oscillating behavior, is an interesting observation that warrants further research.

Estadísticas

None.

Citas

None.

Ideas clave extraídas de

Stability Analysis of Adaptive Model Predictive Control Using the Circle and Tsypkin Criteria

by Juan A. Pare... a las arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.04170.pdf

Consultas más profundas

How can the stability analysis be extended to account for the time-varying nature of the closed-loop Lur'e system dynamics, rather than the heuristic application of the absolute stability criteria?

To account for the time-varying nature of the closed-loop Lur'e system dynamics in a more rigorous manner, one approach could involve the development of time-varying Lyapunov functions. These Lyapunov functions would need to capture the changing dynamics of the system over time and provide a systematic way to analyze the stability properties. By incorporating time-varying Lyapunov functions into the stability analysis, it would be possible to establish stability guarantees that are more directly linked to the evolving dynamics of the system. This approach would require a deeper understanding of the time-varying behavior of the system and the formulation of Lyapunov functions that can account for these variations.

What are the potential limitations of the PCAC approach in stabilizing Lur'e systems with more complex nonlinearities or dynamics?

While PCAC (Predictive Cost Adaptive Control) has shown effectiveness in stabilizing Lur'e systems with certain types of nonlinearities, it may face limitations when dealing with more complex nonlinearities or dynamics. Some potential limitations of the PCAC approach in such scenarios include:

Model Mismatch: PCAC relies on accurate system identification for its predictive control law. In the presence of complex nonlinearities, the model used for prediction may not accurately capture the system dynamics, leading to suboptimal control performance.

Curse of Dimensionality: As the complexity of the system increases, the computational burden of solving the optimization problems associated with PCAC may become prohibitive. This can limit the real-time applicability of the approach in systems with high-dimensional state spaces.

Convergence Issues: In systems with highly nonlinear dynamics, the convergence of the adaptive controller may be slower or less stable. The adaptive nature of PCAC may struggle to adapt quickly to rapidly changing dynamics or disturbances.

Robustness Concerns: PCAC may lack robustness in the face of uncertainties or disturbances in systems with complex nonlinearities. The adaptive nature of the controller may not be able to handle uncertainties effectively, leading to performance degradation.

How can the insights from this stability analysis be leveraged to develop more robust and adaptive control strategies for a broader class of nonlinear systems?

The insights gained from the stability analysis of the closed-loop Lur'e system using absolute stability criteria can be leveraged to develop more robust and adaptive control strategies for a broader class of nonlinear systems in the following ways:

Lyapunov-Based Control Design: By extending the stability analysis to incorporate time-varying Lyapunov functions, control strategies can be designed based on Lyapunov stability theory. This approach provides a systematic framework for ensuring stability and convergence in nonlinear systems.

Adaptive Control Techniques: Building on the adaptive nature of PCAC, more advanced adaptive control techniques such as model reference adaptive control or adaptive sliding mode control can be explored. These techniques offer robustness to uncertainties and disturbances in nonlinear systems.

Nonlinear Control Strategies: Leveraging the understanding of nonlinear dynamics gained from the stability analysis, nonlinear control strategies such as feedback linearization, sliding mode control, or backstepping control can be employed. These strategies are well-suited for handling complex nonlinearities in a wide range of systems.

Data-Driven Approaches: Integrating data-driven approaches such as machine learning or reinforcement learning with the stability analysis insights can lead to the development of adaptive control strategies that learn from system data and adapt in real-time to changing dynamics.

By combining these approaches and tailoring them to the specific characteristics of the nonlinear systems at hand, more robust and adaptive control strategies can be developed to address a broader class of nonlinear systems effectively.