Off-policy reinforcement learning for $ H_\infty $ control design (1311.6107v3)

Abstract: The $H_\infty$ control design problem is considered for nonlinear systems with unknown internal system model. It is known that the nonlinear $ H_\infty $ control problem can be transformed into solving the so-called Hamilton-Jacobi-Isaacs (HJI) equation, which is a nonlinear partial differential equation that is generally impossible to be solved analytically. Even worse, model-based approaches cannot be used for approximately solving HJI equation, when the accurate system model is unavailable or costly to obtain in practice. To overcome these difficulties, an off-policy reinforcement leaning (RL) method is introduced to learn the solution of HJI equation from real system data instead of mathematical system model, and its convergence is proved. In the off-policy RL method, the system data can be generated with arbitrary policies rather than the evaluating policy, which is extremely important and promising for practical systems. For implementation purpose, a neural network (NN) based actor-critic structure is employed and a least-square NN weight update algorithm is derived based on the method of weighted residuals. Finally, the developed NN-based off-policy RL method is tested on a linear F16 aircraft plant, and further applied to a rotational/translational actuator system.

• The paper proposes an off-policy RL method that reformulates the nonlinear H∞ control problem by learning the Hamilton-Jacobi-Isaacs equation directly from system data.
• It leverages a neural network actor-critic structure with least-square updates, simplifying complex nonlinear dynamics into tractable computation for linear systems.
• Simulation results on an F16 aircraft plant and a nonlinear actuator system validate robust convergence and effective L2-gain performance in diverse control scenarios.

Off-policy Reinforcement Learning for $H_\infty$ Control Design: A Comprehensive Overview

The paper "Off-policy Reinforcement Learning for $H_\infty$ Control Design" presents a pivotal exploration of using reinforcement learning (RL) techniques to address the complex problem of $H_\infty$ control design in nonlinear systems where internal system models are unknown. The authors, Biao Luo, Huai-Ning Wu, and Tingwen Huang, develop an innovative approach that leverages off-policy reinforcement learning, propelling the $H_\infty$ control design forward amidst traditional challenges in model acquisition and computational solutions of associated equations.

Central to the paper is the transformation of the nonlinear $H_\infty$ control problem, which typically requires solving the challenging Hamilton-Jacobi-Isaacs (HJI) equation, into a data-driven RL framework. The authors propose an off-policy RL method that learns the solution of the HJI equation directly from real-time data obtained from system interactions, bypassing the need for an explicit system model. The major innovation here is the deployment of an off-policy strategy, which allows for the utilization of arbitrary policies for data generation, a significant advantage over on-policy methods typically constrained to data from evaluating policies.

Theoretical and Methodological Contributions

1. Transformation and Learning Methodology: The paper details the transformation of the $H_\infty$ control problem into an RL framework, presenting a novel off-policy learning method. The methodology hinges on constructing a surrogate learning algorithm where the optimal control policy and disturbance rejection levels are learned from data rather than derived through traditional model awareness strategies.
2. Neural Network-based Implementation: It employs a neural network (NN) based actor-critic structure. By doing so, the authors utilize a least-square NN weight update algorithm derived from the method of weighted residuals, facilitating the learning process.
3. Linear System Simplification: For linear systems, the authors simplify the complex nonlinear dynamic equations into algebraic Riccati equations (ARE), drastically reducing computational overhead and facilitating easier implementation in a real-world setting.
4. Convergence Analysis: A rigorous convergence analysis is presented, proving that the proposed method is mathematically equivalent to a Newton's method for finding solutions and ensuring that the approach reliably converges to stable, optimal solutions of the HJI equation over iterations.

Numerical Insights and Results

The authors validate their methodology through simulation studies on a linear F16 aircraft plant and a highly challenging nonlinear rotational/translational actuator system. Their results demonstrate robust convergence of the NN parameters and establish the method's practical applicability in achieving prescribed $L_2$-gain performance levels in both linear and nonlinear settings.

Implications and Future Directions

The implications of this research are notable. From a theoretical perspective, the paper bridges a gap in control theory by integrating off-policy RL within $H_\infty$ control, thereby expanding the application scope to situations where obtaining an accurate model of the system is unfeasible. Practically, the framework provides a blueprint for real-time control system applications wherein model error can be significant, offering a way to mitigate disturbance without explicit model dependency.

Looking forward, the principles established in this paper may be extended to more complex systems, potentially incorporating multi-agent dynamics or adaptive mechanisms for decentralized control architectures. Furthermore, with the prevalence of data-rich environments, this approach could prove instrumental in areas like autonomous navigation, robotic control systems, and smart grid management, where model uncertainties are a persistent challenge.

In conclusion, this paper presents a robust, computationally feasible approach to $H_\infty$ control design using RL techniques, possibly redefining how control systems can be designed and optimized in model-deficient environments. Its comprehensive submission establishes a significant benchmark and sets forth avenues for subsequent exploration in reinforcement learning applications across complex system domains.