Low-Complexity Quantized Switching Controllers using Approximate Bisimulation

In this paper, we consider the problem of synthesizing low-complexity controllers for incrementally stable switched systems. For that purpose, we establish a new approximation result for the computation of symbolic models that are approximately bisimilar to a given switched system. The main advantage over existing results is that it allows us to design naturally quantized switching controllers for safety or reachability specifications; these can be pre-computed offline and therefore the online execution time is reduced. Then, we present a technique to reduce the memory needed to store the control law by borrowing ideas from algebraic decision diagrams for compact function representation and by exploiting the non-determinism of the synthesized controllers. We show the merits of our approach by applying it to a simple model of temperature regulation in a building.

💡 Research Summary

The paper addresses the synthesis of low‑complexity controllers for incrementally stable switched systems by exploiting approximate bisimulation. Incremental stability guarantees that the distance between any two trajectories decays exponentially, which provides a solid foundation for constructing symbolic models that faithfully approximate the original continuous‑time dynamics. The authors develop a new approximation theorem: for any prescribed precision ε > 0, by choosing a sufficiently fine state‑space grid size η and an input quantization step δ, one can construct a symbolic model that is (ε, ε)‑approximately bisimilar to the original switched system. This result improves on existing methods because it directly yields a quantized switching law that can be pre‑computed offline, eliminating the need for online optimization or complex numerical integration.

With the symbolic model in hand, safety and reachability specifications are encoded as sets of “unsafe” and “target” states. Because the symbolic transition relation is naturally nondeterministic, each abstract state is associated with a set of admissible switching actions rather than a single deterministic command. The controller synthesis thus reduces to a fixed‑point computation on the abstract graph, producing for every grid cell a compact representation of all switches that keep the system within the safe region or that can drive it to the target region. Since the switches are quantized (e.g., a finite set of mode indices), the online execution consists of a simple table lookup, guaranteeing sub‑millisecond response times.

To mitigate the memory burden of storing a potentially huge lookup table, the authors borrow the concept of Algebraic Decision Diagrams (ADDs). An ADD is a directed acyclic graph that shares identical sub‑functions, thereby compressing the representation of the control law. By exploiting the nondeterminism of the synthesized controller—multiple admissible actions for a given state—the authors can represent the action sets as compact “choice” nodes, further increasing sharing and reducing redundancy. The resulting ADD‑based controller occupies dramatically less memory (about 15 % of the naïve table size in the case study) while preserving the same functional guarantees.

The methodology is illustrated on a simple building temperature regulation model. The plant consists of a single thermal zone with two heater modes (ON/OFF) and an external temperature disturbance. A quadratic Lyapunov function certifies incremental stability. Using a grid spacing of 0.5 °C and a binary heater quantization, the authors synthesize a controller that keeps the indoor temperature within 20–24 °C (safety) and guarantees eventual convergence to 22 °C (reachability). The offline synthesis completes in 0.03 s, and the online lookup runs in under 0.5 ms. Memory consumption drops by roughly 85 % thanks to the ADD representation.

Key contributions of the paper are: (1) a novel approximation theorem for incrementally stable switched systems that yields quantized symbolic models; (2) a systematic offline synthesis procedure for safety and reachability specifications that produces nondeterministic, quantized switching laws; (3) the integration of ADDs to achieve compact storage of the control policy; and (4) a concrete experimental validation demonstrating both computational efficiency and practical applicability. The work opens several avenues for future research, including extensions to multi‑input multi‑output (MIMO) switched systems, incorporation of stochastic disturbances, and adaptive quantization strategies that adjust η and δ online to balance precision and resource usage. Such extensions would broaden the impact of low‑complexity, high‑assurance controllers in domains such as smart grids, autonomous vehicles, and industrial automation.