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Optimal Control of Switched Systems Governed by Logical Switching Dynamics

Xiao Zhang, Min Meng, Changxi Li, Ka-Fai Cedric Yiu

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.

Optimal Control of Switched Systems Governed by Logical Switching Dynamics

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.
Paper Structure (15 sections, 7 theorems, 62 equations, 3 figures, 2 algorithms)

This paper contains 15 sections, 7 theorems, 62 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Mathematical Preliminaries
  4. System Model and Control Objective
  5. Deterministic Case
  6. Stochastic Case
  7. Control Objective
  8. Main Results
  9. Co-Design under Deterministic Logical Dynamics
  10. Co-Design under Stochastic Logical Dynamics
  11. An Illustrative Example
  12. Conclusion
  13. Auxiliary Lemmas
  14. Proof of Theorem \ref{['t5.1']}
  15. Proof of Theorem \ref{['t6.1']}

Key Result

Lemma 2.2

Figures (3)

  • Figure 1: Overall architecture of the proposed co-design framework. This diagram illustrates the coupling captured algebraically by \ref{['eq5.1']}--\ref{['eq5.2']} (presented in Section \ref{['S2']}). The logical control input $\gamma_t$ drives the logical dynamical system $f$, generating the switching signal $\theta_t$ that selects the active continuous subsystem $(A_{\iota(\theta_t)}, B_{\iota(\theta_t)})$. The continuous control input $u_t$ regulates the selected subsystem.
  • Figure 2: Monte Carlo simulation of the performance ratio $J/J^*$.
  • Figure 3: Continuous state trajectories over all Monte Carlo trials.

Theorems & Definitions (17)

  • Definition 2.1: che12
  • Lemma 2.2: che12
  • Lemma 2.3: che12
  • Example 2.4
  • Remark 2.5
  • Remark 2.6: Controllability Condition
  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3: Relation to Free-Switching Formulations
  • Remark 3.4: Interpretation and Implementation
  • ...and 7 more