Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator (1805.09388v1)

Abstract: We consider adaptive control of the Linear Quadratic Regulator (LQR), where an unknown linear system is controlled subject to quadratic costs. Leveraging recent developments in the estimation of linear systems and in robust controller synthesis, we present the first provably polynomial time algorithm that provides high probability guarantees of sub-linear regret on this problem. We further study the interplay between regret minimization and parameter estimation by proving a lower bound on the expected regret in terms of the exploration schedule used by any algorithm. Finally, we conduct a numerical study comparing our robust adaptive algorithm to other methods from the adaptive LQR literature, and demonstrate the flexibility of our proposed method by extending it to a demand forecasting problem subject to state constraints.

• The paper introduces a polynomial-time robust adaptive control algorithm that achieves sub-linear regret in LQR systems.
• It employs semidefinite programming to ensure near-optimal stability with parameter estimation converging at a T^(-1/3) rate.
• The study highlights the trade-off between computational feasibility and regret minimization, setting the stage for future robust control research.

Analyzing Regret Bounds in Robust Adaptive Control for LQR Systems

The paper "Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator" explores the adaptive control mechanisms of Linear Quadratic Regulator (LQR) systems where the system models are partially unspecified, and the cost function follows a quadratic form. This paper introduces a polynomial-time algorithm designed to tackle the uncertainties inherent in adaptive LQR problems, presenting a noteworthy achievement as it provides high probability guarantees of sub-linear regret. The research also navigates the intricate relationship between regret minimization and parameter estimation, contributing a novel lower bound for expected regret linked with the exploration schedule of any algorithm.

Key Contributions and Methodology

The authors capitalize on advancements in estimating linear systems and robust controller synthesis to develop a robust adaptive controller algorithm. The proposed method guarantees stability, maintaining near-optimal performance consistently, with induced regret bound by $(T^{2/3})$ as temporal iterations progress. This algorithm employs finite-dimensional semidefinite programs with dimensions scaling logarithmically in terms of time, $T$. Additionally, the algorithm ensures the estimation of system parameters at a rate of $(T^{-1/3})$ in the operator norm context. These estimations are critical in practice, where accurate system modeling is often required even if optimal control is theoretically achievable without precise models.

The paper's notable contribution is the successful sub-linear regret bound for a polynomial-time adaptive control algorithm, achieved without resorting to unrealistic assumptions. The results showcase fundamental limitations in balancing parameter identification and regret minimization, indicating the theoretical sharpness of their analysis up to logarithmic factors.

Related Work and Comparative Analysis

The paper positions itself in the broader domain of LQR control of unknown dynamic systems, drawing on existing frameworks such as Optimism in the Face of Uncertainty (OFU) and Thompson Sampling (TS). While both OFU and TS offer paths to optimal regret $(\sqrt{T})$, they demand computational prescriptions that are often dense and challenging due to the non-convex nature of the tasks.

In contrast, the robust method presented circumvents these computational obstacles via convex optimization techniques aligned with robust controls, providing a practical and computationally feasible alternative. The distinct methodologies reveal a trade-off: OFU and TS can potentially achieve lower regret bounds, but with prohibitive computational complexity.

Practical and Theoretical Implications

Practically, the formulation contributes significantly to applications requiring robust adaptive controls, such as demand forecasting under restricted environmental constraints. The implications reach further into the theory of control systems, particularly in the adaptive context. From a theoretical standpoint, the authors extend the understanding of the trade-offs intrinsic to adaptive control with uncertainties, laying groundwork that influences future developments in robust control theory and algorithms.

Future Research Directions

This exploration uncovers multiple potential avenues, notably the quest for techniques capable of achieving $(\sqrt{T})$ regret bounds within similar computational constraints. Future work might examine the convergence of policy gradients or try applying the dynamic data learned from adaptive regime switches to enhance exploration inherent to stochastic environments. Addressing nonlinear dynamics and imposing safety constraints offer additional research challenges, harnessing adaptive mechanisms in even broader contexts.

Conclusion

In summary, the authors have rendered a substantial contribution to robust and adaptive control theories through their vigorous analysis and algorithmic development, effectively juxtaposing precision in parameter estimation against overall system performance. The comprehensive numerical evaluations reintegrate the method's viability and extend its applications to real-world dynamic systems where performance stability is pivotal. The paper is thus pivotal for researchers enhancing adaptive control mechanisms and those steering adaptive LQR systems from theoretical robustness to practice-oriented reliability.