Robust Data-Driven Tube-Based Zonotopic Predictive Control with Closed-Loop Guarantees for Unknown Linear Systems

Kernkonzepte

A robust data-driven tube-based zonotopic predictive control (TZPC) approach is proposed for discrete-time linear systems with unknown dynamics and bounded noise, ensuring stability and recursive feasibility.

Zusammenfassung

The proposed TZPC approach consists of two phases:

Learning Phase:

• An over-approximation of all models consistent with past input and noisy state data is obtained using zonotope properties.
• A nominal system, feedback policy, and error dynamics are defined.

Control Phase:

• A data-driven predictive control problem is formulated, integrating terminal ingredients to guarantee recursive feasibility.
• The optimal control problem ensures robust satisfaction of state and input constraints.

The key contributions are:

1. Proof of recursive feasibility of the optimal control problem.
2. Guarantee of robust exponential stability for the closed-loop system.
3. Comparative analysis showing that TZPC significantly reduces execution time compared to previous data-driven predictive control methods.

The effectiveness and competitive performance of TZPC are demonstrated through numerical simulations, including a double integrator example and a simplified building thermal model.

Statistiken

The maximum execution time for the solver over 5 runs is 0.15 minutes for TZPC, compared to 40 minutes for ZPC and 78.26 minutes for TZDDPC.

Zitate

"The advantages of using TZPC include constraint satisfaction and shorter execution time. The latter is achieved by TZPC by circumventing the direct inclusion of reachable sets in the optimization constraints."

Wichtige Erkenntnisse aus

Robust Data-Driven Tube-Based Zonotopic Predictive Control with Closed-Loop Guarantees

by Mahsa Farjad... um arxiv.org 09-24-2024

https://arxiv.org/pdf/2409.14366.pdf

Tiefere Fragen

How can the proposed TZPC approach be extended to handle time-varying constraints or input-output data?

The proposed Tube-Based Zonotopic Predictive Control (TZPC) approach can be extended to handle time-varying constraints and input-output data by incorporating adaptive mechanisms that adjust the control strategy based on real-time data. This can be achieved through the following strategies:

Dynamic Constraint Adjustment: The constraints on the control inputs and states can be made time-dependent by defining them as functions of time or system states. For instance, the sets ( U(t) ) and ( X(t) ) can be updated at each time step based on the current operating conditions or external factors, such as environmental changes in building automation systems.

Online Learning: Implementing an online learning framework allows the TZPC to adapt to changing dynamics and constraints. By continuously updating the zonotope representations of the system based on new input-output data, the controller can refine its predictions and constraints dynamically. This could involve using techniques such as recursive least squares or Kalman filtering to estimate the system parameters in real-time.

Modified Optimization Problem: The optimization problem formulated in the control phase can be modified to include time-varying parameters. This involves redefining the cost function and constraints to reflect the current state of the system and its environment, ensuring that the controller remains effective under varying conditions.

Incorporation of Feedback Mechanisms: Feedback from the system can be utilized to adjust the control inputs and constraints in real-time. By monitoring the system's performance and adjusting the control strategy accordingly, the TZPC can maintain stability and performance even in the presence of time-varying conditions.

By integrating these strategies, the TZPC framework can effectively manage time-varying constraints and input-output data, enhancing its applicability in dynamic environments such as HVAC systems in buildings.

What are the potential challenges and limitations of applying TZPC to high-dimensional systems or nonlinear dynamics?

Applying the TZPC approach to high-dimensional systems or nonlinear dynamics presents several challenges and limitations:

Computational Complexity: As the dimensionality of the system increases, the computational burden associated with solving the optimization problem grows significantly. The complexity of managing zonotopes and ensuring recursive feasibility can lead to longer computation times, making real-time implementation challenging.

Curse of Dimensionality: In high-dimensional spaces, the volume of the state and input spaces increases exponentially, which can complicate the representation of reachable sets and the design of robust invariant sets. This can lead to difficulties in ensuring that the controller remains effective across the entire state space.

Nonlinear Dynamics: The TZPC framework is primarily designed for linear systems. Extending it to handle nonlinear dynamics requires additional considerations, such as the need for local linearization techniques or the use of more complex zonotopic representations that can capture the behavior of nonlinear systems. This can complicate the design and analysis of the controller.

Stability Guarantees: While the TZPC approach provides closed-loop stability guarantees for linear systems, ensuring similar guarantees for high-dimensional or nonlinear systems may require additional analysis and potentially more conservative designs. The robustness of the controller in the presence of model uncertainties and disturbances may also be compromised.

Data Availability and Quality: The effectiveness of the TZPC approach relies heavily on the availability and quality of input-output data. In high-dimensional systems, obtaining sufficient and accurate data for model learning can be challenging, potentially leading to suboptimal control performance.

Addressing these challenges requires ongoing research and development, including the exploration of advanced computational techniques, robust control strategies, and improved data-driven methodologies.

Can the TZPC framework be integrated with other data-driven techniques, such as those based on behavioral systems theory, to further enhance its capabilities?

Yes, the TZPC framework can be integrated with other data-driven techniques, including those based on behavioral systems theory, to enhance its capabilities in several ways:

Behavioral Modeling: By incorporating behavioral systems theory, the TZPC can leverage a broader range of modeling techniques that focus on the input-output behavior of systems rather than explicit state-space representations. This can be particularly beneficial for systems where the underlying dynamics are complex or poorly understood.

Enhanced Robustness: Integrating behavioral approaches can improve the robustness of the TZPC framework by allowing it to account for uncertainties and variations in system behavior more effectively. Behavioral models can provide insights into the system's response to disturbances, enabling the design of more resilient control strategies.

Improved Data Utilization: Behavioral systems theory emphasizes the use of data to inform control strategies. By combining TZPC with these techniques, the framework can better utilize available data, leading to more accurate predictions and improved performance in real-time applications.

Multi-Model Approaches: The integration can facilitate the development of multi-model control strategies, where different models are used to represent various operating conditions or system behaviors. This can enhance the adaptability of the TZPC framework, allowing it to switch between models based on real-time data.

Synergistic Algorithms: The combination of TZPC with behavioral systems theory can lead to the development of synergistic algorithms that incorporate the strengths of both approaches. For example, the predictive capabilities of TZPC can be enhanced by the behavioral insights, leading to more effective control strategies.

Overall, integrating the TZPC framework with other data-driven techniques, such as those based on behavioral systems theory, can significantly enhance its capabilities, making it more versatile and effective in managing complex control problems across various applications.