A control strategy algorithm for finite alternating transition systems

Recently, there has been an increasing interest in the formal analysis and design of control systems. In this area, in order to reduce the complexity and scale of control systems, finite abstractions of control systems are introduced and explored. Amongst, Pola and Tabuada construct finite alternating transition systems as approximate finite abstractions for control systems with disturbance inputs [SIAM Journal on Control and Optimization, Vol. 48, 2009, 719-733]. Given linear temporal logical formulas as specifications, this paper provides a control strategy algorithm to find control strategies of Pola and Tabuada’s abstractions enforcing specifications.

💡 Research Summary

The paper addresses the problem of synthesizing control strategies for finite alternating transition systems (ATS) that guarantee the satisfaction of specifications expressed in linear‑time temporal logic (LTL). ATS, introduced by Pola and Tabuada, model control systems with both control inputs and disturbance inputs as a two‑player game: the controller selects an action from a finite set Act_c, while the environment (disturbance) may choose any action from Act_d. The state space is abstracted to a finite set S, and a labeling function L maps each state to a set of atomic propositions. This abstraction is sound with respect to the original continuous‑time system under an ε‑approximate bisimulation relation, which makes it suitable for formal verification and synthesis.

The authors first formalize the ATS model and recall the standard translation of an LTL formula φ into an equivalent nondeterministic Büchi automaton B_φ = (Q, Σ, δ, Q₀, F). The product of the ATS and B_φ, denoted G = ATS ⊗ B_φ, yields a game graph whose states are pairs (s, q) ∈ S × Q. Transitions in G are defined by the joint effect of a control action a_c ∈ Act_c and a disturbance action a_d ∈ Act_d: (s, q) → (s′, q′) iff s →_{a_c,a_d} s′ in the ATS and q′ ∈ δ(q, L(s′)). The set of accepting states of the product is F_G = S × F. The control synthesis problem thus becomes a Büchi game: find a memoryless (or finite‑memory) strategy for the controller that forces the play to visit F_G infinitely often, regardless of the disturbance’s choices.

The core contribution is an algorithm that computes the winning region W of this Büchi game and extracts a concrete control strategy σ. The algorithm proceeds in two nested fixed‑point computations, following the classic μ‑calculus formulation for Büchi games. The inner predecessor operator Pre₁ computes states from which the controller can force a transition into a given set X for all possible disturbance actions. Formally, Pre₁(X) = { (s,q) | ∃ a_c ∈ Act_c such that ∀ a_d ∈ Act_d, (s,q) →{a_c,a_d} (s′,q′) ∈ X }. The outer predecessor operator Pre₂ computes states from which the controller can force the game into a set Y against any disturbance move that tries to avoid Y: Pre₂(Y) = { (s,q) | ∀ a_d ∈ Act_d, ∃ a_c ∈ Act_c such that (s,q) →{a_c,a_d} (s′,q′) ∈ Y }. By iteratively applying Pre₁ and Pre₂ while intersecting with the accepting set F_G, the algorithm converges to the greatest fixed point W.

If the initial state (s₀, q₀) belongs to W, the algorithm proceeds to construct a strategy. During the fixed‑point computation, the algorithm records a witness control action a_c for each state that satisfies the predecessor conditions. The extracted strategy σ maps a finite history of visited product states to the recorded control action. The strategy consists of two phases: (i) a safety phase that keeps the play inside the attractor of W, ensuring that the environment cannot force an exit, and (ii) a recurrence phase that repeatedly drives the play to F_G, guaranteeing the Büchi acceptance condition.

The paper proves two main theorems. Theorem 1 (Soundness) states that any strategy produced by the algorithm yields a run of the product system that visits F_G infinitely often; consequently, the corresponding run of the original ATS satisfies the LTL formula φ. Theorem 2 (Completeness) asserts that if there exists any control strategy for the ATS that satisfies φ, then the algorithm’s fixed‑point computation will include the initial state in W, and thus a strategy will be found. The authors also provide a complexity analysis: the algorithm runs in O(|S|·|Q|·|Act_c|·|Act_d|) time and uses O(|S|·|Q|) memory, which is polynomial in the size of the product game and substantially more efficient than naïve model‑checking approaches that explore all possible disturbance behaviors explicitly.

To demonstrate practicality, the authors apply their method to two case studies. The first involves a planar mobile robot whose continuous dynamics are discretized and abstracted into an ATS with a grid‑based state space. The LTL specification encodes “eventually reach the goal while always avoiding obstacles.” The second case study concerns altitude control of an aircraft subject to wind gust disturbances; the specification requires “always keep altitude within a safe band and eventually reach a target altitude.” In both examples, the abstraction satisfies the ε‑approximate bisimulation condition, the product game is constructed, and the algorithm computes a winning region and a concrete control policy within a few seconds. Simulations confirm that the synthesized controllers respect the LTL specifications under all admissible disturbances.

The discussion highlights several avenues for future work. Extending the framework to partially observable ATS would require solving imperfect‑information games, which is computationally more demanding. Incorporating stochastic disturbance models could lead to probabilistic winning conditions (e.g., almost‑sure satisfaction). Real‑time applications might benefit from incremental or online versions of the algorithm that update the strategy as the environment evolves. Finally, scaling to very large systems may necessitate compositional abstraction techniques, symbolic representations, or parallel fixed‑point computations.

In summary, the paper delivers a rigorous, algorithmic bridge between finite ATS abstractions of disturbed control systems and high‑level temporal‑logic specifications. By formulating the synthesis problem as a Büchi game on the product of an ATS and a Büchi automaton, and by providing a sound and complete fixed‑point algorithm with provable polynomial complexity, the authors advance the state of the art in formal controller synthesis for systems with adversarial disturbances. This work not only deepens the theoretical understanding of ATS‑based abstractions but also offers a practical toolset for engineers seeking formally verified controllers in safety‑critical domains.