Optimal Control of Switched Systems Governed by Logical Switching Dynamics

TL;DR

This paper presents a unified algebraic approach using semi-tensor products to optimally co-design logical and continuous controls for switched linear systems with internal logical switching dynamics, enabling tractable solutions.

Contribution

It introduces a novel semi-tensor product-based algebraic framework that embeds logical and continuous dynamics, deriving Riccati recursions for optimal control and decision rules.

Findings

Effective unified algebraic representation of logical and continuous dynamics.

Derivation of Riccati-type recursions for optimal control laws.

Hierarchical algorithm reduces computational complexity.

Abstract

This paper investigates the optimal co-design of logical and continuous controls for switched linear systems governed by controlled logical switching dynamics. Unlike traditional switched systems with arbitrary or state-dependent switching, the switching signals here are generated by an internal logical dynamical system and explicitly integrated into the control synthesis. By leveraging the semi-tensor product (STP) of matrices, we embed the coupled logical and continuous dynamics into a unified algebraic state-space representation, transforming the co-design problem into a tractable linear-quadratic framework. We derive Riccati-type backward recursions for both deterministic and stochastic logical dynamics, which yield optimal state-feedback laws for continuous control alongside value-function-based, state-dependent decision rules for logical switching. To mitigate the combinatorial explosion inherent in logical decision-making, a hierarchical algorithm is developed to decouple offline precomputation from efficient online execution. Numerical simulations demonstrate the efficacy of the proposed framework.