Iterative Data-driven Control Design for Unknown Linear Systems: A System Theoretic Analysis

Core Concepts

This article analyzes the fundamental mechanisms and properties of indirect and direct data-driven policy iteration methods for solving the linear quadratic regulator (LQR) problem when the system dynamics are unknown. The analysis provides insights into the role of system identification in establishing convergence, sample complexity, and robustness.

Abstract

The paper investigates two iterative data-driven control design approaches for solving the linear quadratic regulator (LQR) problem when the system dynamics are unknown:

Indirect Policy Iteration (IPI):

Combines recursive least squares (RLS) for online system identification with a model-based policy iteration scheme.
Analyzes the closed-loop dynamics of the interconnected RLS and policy iteration algorithms.
Establishes convergence and robustness properties without requiring persistence of excitation in the data.

Direct Policy Iteration (DPI):

Builds on a recently proposed direct data-driven policy iteration method.
Extends the method to address potential identifiability issues in noise-free scenarios.
Compares the strengths and limitations of the indirect and direct approaches in terms of sample complexity, convergence, and excitation requirements.

The key insights from the analysis are:

IPI exhibits better sample complexity and robustness to lack of excitation compared to DPI.
The availability of an estimated model in IPI plays a crucial role in establishing these advantages.
The analysis provides a system-theoretic perspective on the fundamental differences between indirect and direct data-driven control.

Stats

"The estimation error of the recursive least squares initialized with ˆθ0 and H0 = aI, a > 0 is bounded by: ∥ˆθi −θ∥F ≤∆θUpper
i ≤f(∥ˆθ0 −θ∥F , i) + g(∥jnon∥∞)."
"For any initialization ˆA0, ˆB0 and stabilizable pair (A, B), given a locally persistent sequence {Di} with any lower bound α > 0 and persistency window N ∈Z++ and M ∈Z++, there always exists an index ist such that ∀i ≥ist, the estimate obtained from RLS ˆAi, ˆBi, is stabilizable."

Quotes

"If Assumption 1 and Assumption 2 are satisfied, then the coupled recursive least squares and policy iteration system formulated by (19)-(21) admits the equivalent dynamical system representation..."
"Then, for any positive integer τIPI ∈Z++ the estimate ˆPi and ˆθi satisfy the following relationships: ∥ˆPi −P∗∥F ≤β(∥ˆP0 −P∗∥F , i) + γ(∥Ω∥∞), ∥ˆθi −θ∥F ≤∆θUpper
i ≤f(∥ˆθ0 −θ∥F , i) + g(∥jnon∥∞)."

Key Insights Distilled From

The Role of Identification in Data-driven Policy Iteration: A System Theoretic Study

by Bowen Song,A... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2401.06721.pdf

Deeper Inquiries

How can the indirect and direct policy iteration approaches be extended to handle nonlinear or time-varying system dynamics

To extend the indirect and direct policy iteration approaches to handle nonlinear or time-varying system dynamics, several modifications and adaptations can be made:
Indirect Policy Iteration:

Nonlinear System Dynamics: For nonlinear systems, the model estimation step in the indirect policy iteration can be adjusted to incorporate nonlinear system identification techniques such as neural networks, Gaussian processes, or kernel methods. This would involve updating the model estimate using nonlinear regression methods to capture the system dynamics accurately.
Time-Varying System Dynamics: To handle time-varying dynamics, the recursive least squares algorithm can be modified to include adaptive mechanisms that can track changes in the system parameters over time. This would involve updating the system model estimates dynamically based on the evolving system behavior.

Direct Policy Iteration:

Nonlinear System Dynamics: In the direct policy iteration approach, handling nonlinear system dynamics would require using nonlinear control techniques such as model predictive control (MPC) or reinforcement learning algorithms that can directly optimize policies for nonlinear systems without the need for explicit model identification.
Time-Varying System Dynamics: For time-varying dynamics, the direct policy iteration approach can be extended by incorporating adaptive control strategies that can adapt the controller parameters in real-time to account for changes in the system dynamics.

Overall, extending the indirect and direct policy iteration approaches to nonlinear or time-varying system dynamics would involve integrating advanced modeling and control techniques that can capture the complexities of such systems and adapt to changes over time.

What are the implications of the analysis on the design of exploration strategies for data collection in the context of data-driven control

The analysis on the design of exploration strategies for data collection in the context of data-driven control has several implications:

Balancing Exploration and Exploitation: The analysis provides insights into how exploration strategies can be designed to balance the trade-off between exploring new regions of the state space to gather informative data and exploiting existing knowledge to improve control performance.
Optimizing Data Collection: By understanding the convergence properties and limitations of the data-driven control algorithms, exploration strategies can be optimized to focus on collecting data points that are most beneficial for improving the system identification and control performance.
Adaptive Exploration: The system-theoretic perspective developed in the analysis can guide the design of adaptive exploration strategies that can adjust the exploration intensity based on the system's response and the quality of the data collected.
Robustness to Lack of Excitation: The analysis highlights the importance of ensuring sufficient excitation in the data for accurate system identification. Exploration strategies can be designed to ensure robustness to the lack of excitation by actively seeking diverse data points to cover the system's operating range.

Overall, the analysis provides a framework for designing exploration strategies that can enhance the efficiency and effectiveness of data collection in data-driven control applications.

Can the system-theoretic perspective developed in this work be applied to analyze other classes of data-driven control algorithms beyond policy iteration

The system-theoretic perspective developed in this work can be applied to analyze other classes of data-driven control algorithms beyond policy iteration in the following ways:

Reinforcement Learning Algorithms: The system-theoretic analysis can be extended to analyze reinforcement learning algorithms such as deep Q-learning, actor-critic methods, and policy gradient algorithms. By modeling these algorithms as dynamical systems and studying their convergence properties, robustness, and performance guarantees, insights can be gained into their behavior and design principles.
Model Predictive Control (MPC): The system-theoretic perspective can be applied to analyze MPC algorithms in data-driven control settings. By examining the closed-loop dynamics of MPC systems, stability properties, and optimization convergence, the analysis can provide a deeper understanding of the interaction between the controller and the system dynamics.
Adaptive Control Strategies: The system-theoretic framework can be used to analyze adaptive control strategies that dynamically adjust controller parameters based on online data. By studying the convergence and robustness properties of adaptive algorithms, the analysis can guide the design of effective adaptive control schemes in data-driven settings.

By applying the system-theoretic perspective to a broader range of data-driven control algorithms, researchers can gain valuable insights into the underlying principles, limitations, and performance characteristics of these algorithms.