Controlled Invariance in Fully Actuated Max-plus Linear Systems with Precedence Semimodules

Given a max-plus linear system and a semimodule, the problem of computing the maximal controlled invariant subsemimodule is still open to this day. In this paper, we consider this problem for the specific class of fully actuated systems and constraints in the form of precedence semimodules. The assumption of full actuation corresponds to the existence of an input for each component of the system state. A precedence semimodule is the set of solutions of inequalities typically used to represent time-window constraints. We prove that, in this setting, it is possible to (i) compute the maximal controlled invariant subsemimodule and (ii) decide the convergence of a fixed-point algorithm introduced by R.D. Katz in strongly polynomial time.

💡 Research Summary

The paper tackles the longstanding open problem of computing the maximal controlled invariant subsemimodule for max‑plus linear systems, focusing on the special case of fully actuated systems with precedence‑type constraints. A fully actuated system is one where the input matrix B is the max‑plus identity, meaning each state component can be directly influenced by an independent control input. Precedence semimodules are defined by inequalities of the form x ≥ A⊗x, which model time‑window constraints common in discrete event systems such as manufacturing lines or railway networks.

The authors first review max‑plus algebra and the representation of precedence constraints through weighted directed graphs. They recall that a solution to x ≥ A⊗x exists iff the Kleene star A* converges, which is equivalent to the absence of infinite‑weight paths in the graph G(A). In finite graphs this reduces to checking for positive‑weight circuits; in infinite graphs more subtle infinite‑weight paths may appear without any positive circuit.

The core contribution is a polynomial‑time method to decide whether a trajectory satisfying both the system dynamics and the precedence constraints exists, and to compute the maximal (A,B)-invariant subsemimodule when it does. By exploiting full actuation, the dynamics x(k+1)=A⊗x(k)⊕u(k) can be rewritten as x(k+1)≥A⊗x(k) with u(k) chosen arbitrarily large, allowing the control input to be eliminated from the feasibility analysis. The constraints are expressed as three inequalities involving matrices L, C, and \tilde R, which together with the dynamics form a block‑structured precedence matrix M_K. The infinite‑horizon version M_∞ captures the entire trajectory.

A recursive sequence of matrices Π_k is introduced: Π₀ = C*, Π_{k+1} = (L Π_k R ⊕ C)*, where R = A ⊕ \tilde R. The authors prove that the system is consistent (i.e., admits an infinite trajectory satisfying all constraints) iff Π_{n²+1}=Π_{n²} and Π_{n²} has only finite entries. This condition can be checked in O(n⁵) time. Weak consistency (existence of finite‑length feasible trajectories for any horizon) holds iff every Π_k has only finite entries, which can be verified in O(n⁹) time. These results rely on the equivalence between the absence of infinite‑weight paths (or positive circuits) in the graph of M_∞ and the algebraic stabilization of the Π‑sequence.

The paper then connects these findings to the geometric approach to controlled invariance. Katz (2007) defined a fixed‑point operator ϕ on subsets of Rⁿ_max, where ϕ(X)=X∩A^{-1}(X⊖Im B). The maximal (A,B)-invariant subsemimodule K* is the limit of iterating ϕ on a given semimodule K, provided the iteration stabilizes in finite steps. In the general max‑plus setting, convergence is not guaranteed. However, under the fully actuated and precedence‑semimodule assumptions, the authors show that convergence of ϕ coincides with the stabilization of the Π‑sequence, thus enabling a strongly polynomial algorithm to compute K* exactly.

An illustrative example demonstrates a system where weak consistency holds (no positive‑weight circuits) but full consistency fails due to an infinite‑weight path, leading to the non‑existence of a feasible infinite trajectory. The authors also present case studies from robotic job‑shops and railway networks, showing how the theory can be applied to real‑world discrete‑event systems.

In conclusion, the paper provides a significant breakthrough: it identifies a tractable subclass of max‑plus linear systems where the maximal controlled invariant subsemimodule can be computed efficiently, and it supplies concrete polynomial‑time algorithms for both feasibility checking and invariant computation. Future work is suggested on extending the results to partially actuated systems, time‑varying or non‑linear constraints, and distributed implementations for large‑scale networks.