Identification and Adaptive Control of Markov Jump Systems: Sample Complexity and Regret Bounds

Abstract

Learning how to effectively control unknown dynamical systems is crucial for intelligent autonomous systems. This task becomes a significant challenge when the underlying dynamics are changing with time. Motivated by this challenge, this paper considers the problem of controlling an unknown Markov jump linear system (MJS) to optimize a quadratic objective. By taking a model-based perspective, we consider identification-based adaptive control of MJSs. We first provide a system identification algorithm for MJS to learn the dynamics in each mode as well as the Markov transition matrix, underlying the evolution of the mode switches, from a single trajectory of the system states, inputs, and modes. Through martingale-based arguments, sample complexity of this algorithm is shown to be $\mathcal{O}(1/\sqrt{T})$. We then propose an adaptive control scheme that performs system identification together with certainty equivalent control to adapt the controllers in an episodic fashion. Combining our sample complexity results with recent perturbation results for certainty equivalent control, we prove that when the episode lengths are appropriately chosen, the proposed adaptive control scheme achieves $\mathcal{O}(\sqrt{T})$ regret, which can be improved to $\mathcal{O}(polylog(T))$ with partial knowledge of the system. Our proof strategy introduces innovations to handle Markovian jumps and a weaker notion of stability common in MJSs. Our analysis provides insights into system theoretic quantities that affect learning accuracy and control performance. Numerical simulations are presented to further reinforce these insights.

Improved results using Martingale-based arguments

References in corpus (10)

• Global Convergence of Policy Gradient Methods for the Linear Quadratic Regulator
• Near optimal finite time identification of arbitrary linear dynamical systems
• Non-Asymptotic Analysis of Robust Control from Coarse-Grained Identification
• Logarithmic Regret Bound in Partially Observable Linear Dynamical Systems
• Model-Free Linear Quadratic Control via Reduction to Expert Prediction
• Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems
• Efficient Optimistic Exploration in Linear-Quadratic Regulators via Lagrangian Relaxation
• Online Learning and Optimization of Markov Jump Affine Models
• Certainty Equivalent Quadratic Control for Markov Jump Systems
• Mode Clustering for Markov Jump Systems