Optimal Control of Switched Systems Governed by Logical Switching Dynamics

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.

💡 Research Summary

This paper addresses the optimal co‑design of discrete logical controls and continuous inputs for switched linear systems whose switching signals are generated by an internal logical dynamical system rather than being arbitrarily assigned or purely state‑dependent. The authors first map each finite‑valued logical variable to a canonical basis vector and employ the semi‑tensor product (STP) of matrices to convert logical update functions into linear algebraic forms. By assembling the structure matrices of all logical nodes, they obtain an Algebraic State‑Space Representation (ASSR) that captures the evolution of the global logical state θₜ as θₜ₊₁ = L ⊗ γₜ ⊗ θₜ, where L is a logical matrix, γₜ is the logical control vector, and ⊗ denotes the STP. The continuous subsystem for mode i is described by (A_i, B_i); the active mode index ι(θₜ) is uniquely determined by the logical state. Consequently, the hybrid system can be written as a single augmented linear‑quadratic (LQ) model:

 θₜ₊₁ = L ⊗ γₜ ⊗ θₜ, xₜ₊₁ = Ā ⊗ θₜ xₜ + Ḃ ⊗ θₜ uₜ,

where xₜ ∈ ℝⁿ is the continuous state, uₜ ∈ ℝᵐ the continuous input, and Ā, Ḃ are block‑concatenations of the mode‑dependent matrices.

With this unified representation, the authors formulate a finite‑horizon quadratic performance index J = Σ₀^{T‑1}(xₜᵀQxₜ + uₜᵀRuₜ) + x_TᵀQ_f x_T and apply dynamic programming. The resulting cost‑to‑go matrices Pₜ satisfy a Riccati‑type backward recursion that simultaneously yields:

1. Continuous optimal control: uₜ* = –Kₜ(θₜ) xₜ, where Kₜ(θₜ) = (R + B_{ι(θₜ)}ᵀP_{t+1}B_{ι(θₜ)})⁻¹ B_{ι(θₜ)}ᵀP_{t+1}A_{ι(θₜ)}.

2. Logical optimal control: For each admissible logical input γₜ, a scalar cost J_logic(γₜ) = δ_{γₜ}ᵀMₜ(θₜ)δ_{γₜ} is computed, where Mₜ is constructed from P_{t+1}, L, Ā, and Ḃ. The optimal logical decision γₜ* minimizes this value, providing a value‑function‑driven, state‑dependent switching rule.

The framework is extended to stochastic logical dynamics by introducing a Markov transition matrix Π. The Riccati recursion then incorporates expectations, and the optimal logical policy minimizes the expected value function. Deterministic free‑switching LQR and optimal control of Markov Jump Linear Systems (MJLS) appear as special cases when logical dynamics are exogenous.

A major practical challenge is the combinatorial explosion of possible logical states and inputs. To mitigate this, the authors propose a hierarchical algorithm:

• Offline phase: Pre‑compute and store Pₜ and Kₜ for all reachable logical states and inputs using the backward Riccati recursion.

• Online phase: At each time step, query the current logical state θₜ and continuous state xₜ, retrieve the corresponding gain Kₜ(θₜ) and evaluate the pre‑computed scalar costs for all γₜ, then select the minimizing γₜ*. This decouples heavy matrix operations from real‑time execution, enabling fast decision making even for systems with thousands of logical configurations.

Numerical experiments illustrate the approach on a three‑mode switched system coupled with a two‑bit logical network (AND/OR gates). Compared with a conventional free‑switching LQR, the proposed method reduces the total quadratic cost by roughly 12 % while respecting logical constraints such as dwell‑time and safety interlocks. A stochastic example demonstrates that the expected‑cost optimal policy matches the MJLS solution but additionally exploits controllable logical inputs to achieve further performance gains.

The paper’s contributions are threefold: (i) a unified STP‑based algebraic embedding that resolves the structural mismatch between logical and continuous dynamics; (ii) closed‑form Riccati‑type synthesis that yields both continuous feedback gains and logical switching rules; and (iii) a hierarchical pre‑computation/online execution scheme that makes globally optimal co‑design tractable for real‑time applications. Limitations include memory demands for very large logical state spaces and the restriction to discrete‑time linear continuous dynamics; future work is suggested on compression techniques, extensions to nonlinear or continuous‑time models, and hardware‑accelerated implementations. Overall, the work provides a rigorous, scalable, and practically viable solution to optimal control of hybrid systems where logical switching is itself a controllable dynamical process.