Reference Output Tracking in Boolean Control Networks

In this paper, the problem of tracking a given reference output trajectory is investigated for the class of Boolean control networks, by resorting to their algebraic representation. First, the case of a finite-length reference trajectory is addressed, and the analysis and algorithm first proposed in [17] are extended to be able to deal with arbitrary initial conditions and to identify all possible solutions. The approach developed for the finite-length case is then adjusted to cope with periodic reference output trajectories. The results of the paper are illustrated through an example.

💡 Research Summary

This paper addresses the output‑tracking problem for Boolean control networks (BCNs), a class of discrete‑time logical systems whose states, inputs, and outputs are binary (0/1). While previous works have largely focused on constant‑output regulation or on tracking under a fixed initial condition, this study extends the analysis to arbitrary initial states and to both finite‑length and periodic reference output trajectories.

The authors begin by recalling the algebraic representation of a BCN via the semi‑tensor product (STP):
 x(t + 1) = L ⊗ u(t) ⊗ x(t), y(t) = H x(t),
where x ∈ L_N, u ∈ L_M, y ∈ L_P, L ∈ L_{N×N·M}, and H ∈ L_{P×N}. The matrix L_tot = L₁ ∨ … ∨ L_M aggregates all possible state transitions irrespective of the input.

To handle an arbitrary initial condition, the paper introduces the one‑step indistinguishability class C_i for each possible output δ_i^P, represented compactly by the vector v_i = Hᵀ δ_i^P. For a given reference output sequence {y_r(t)}_{t=1}^T, the vectors v(t) are assembled, and a recursive sequence α(t) is defined as
 α(1) = v(1), α(t) = v(t) ⊙ (L_tot α(t‑1)) for t ≥ 2,
where ⊙ denotes element‑wise multiplication. The non‑zero entries of α(t) identify all states that can produce the required output at time t while being reachable from some state active in α(t‑1). Consequently, α(T) ≠ 0 guarantees the existence of at least one compatible state trajectory.

However, global trackability (i.e., from every possible initial state) requires an additional condition: the set X₁ of admissible first states of compatible trajectories must be reachable in a single step from any state in the network. The authors extract X₁ by back‑propagating from the non‑zero entries of α(T) using the transpose of L_tot. The necessary and sufficient condition for universal trackability is then: (i) X₁ ≠ ∅ and (ii) ∀ x₀ ∈ L_N, ∃ u ∈ L_M such that L ⊗ u ⊗ x₀ ∈ X₁. This is formalized in Theorem 2.

Based on these insights, the paper proposes a modified algorithm that (1) computes α(t) for t = 1,…,T, (2) reconstructs X₁, (3) checks the one‑step reachability of X₁ from every state, and (4) if the condition holds, enumerates all feasible control sequences that achieve exact tracking. The algorithm runs in O(T·N·M) time for the forward propagation and O(N²) for the reachability test, which is substantially more efficient than applying the original