[Paper Review] Symbolic Models for Nonlinear Time-Varying Time-Delay Systems via Alternating Approximate Bisimulation

Giordano Pola, Pierdomenico Pepe+1 more|arXiv (Cornell University)|Nov 26, 2010

Stability and Control of Uncertain Systems29 references3 citations

TL;DR

This paper proposes symbolic models for nonlinear time-varying time-delay systems using alternating approximate bisimulation, introducing incremental input-delay-to-state stability (δ-IDSS) characterized via Lyapunov-Krasovskii functionals. The key contribution is proving that such systems admit computable symbolic models that are alternating approximately bisimilar to the original system with adjustable precision.

ABSTRACT

Time-delay systems are an important class of dynamical systems that provide a solid mathematical framework to deal with many application domains of interest. In this paper we focus on nonlinear control systems with unknown and time-varying delay signals and we propose one approach to the control design of such systems, which is based on the construction of symbolic models. Symbolic models are abstract descriptions of dynamical systems where one symbolic state and one symbolic input correspond to an aggregate of states and an aggregate of inputs. We first introduce the notion of incremental input-delay-to-state stability and characterize it by means of Lyapunov-Krasovskii functionals. We then derive sufficient conditions for the existence of symbolic models that are shown to be alternating approximately bisimilar to the original system. Further results are also derived which prove the computability of the proposed symbolic models in a finite number of steps.

Motivation & Objective

• Address the challenge of controlling nonlinear systems with unknown and time-varying delays, which are common in engineering and biological systems.
• Extend symbolic model-based control design from known/constant delays to time-varying and unknown delay signals.
• Provide a formal framework for verifying complex specifications (e.g., safety, liveness) in continuous-time delay systems using discrete abstractions.
• Ensure the symbolic models are computable in finite time under boundedness assumptions on states, inputs, and delays.
• Establish a link between control-theoretic stability and formal methods through a novel stability notion: δ-IDSS.

Proposed method

• Introduce the novel notion of incremental input-delay-to-state stability (δ-IDSS), which quantifies trajectory mismatch under different initial conditions, inputs, and delay signals.
• Characterize δ-IDSS using Lyapunov-Krasovskii functionals that depend on the difference between two system trajectories.
• Construct symbolic models where each symbolic state and input represent aggregates of continuous states and inputs, with time-delay signals also abstracted.
• Establish alternating approximate bisimulation between the original system and its symbolic model, ensuring behavioral approximation within a user-defined precision.
• Propose a finite-step computation procedure for the symbolic model under boundedness assumptions on state, input, and delay sets.
• Adapt algorithms from prior work to improve computational efficiency, validated via a numerical case study with a 2D time-delay system.

Experimental results

Research questions

• RQ1 Can symbolic models be constructed for nonlinear time-varying time-delay systems with unknown delays?
• RQ2 What stability property is necessary and sufficient to ensure that symbolic models are alternating approximately bisimilar to the original system?
• RQ3 How can the precision of the symbolic abstraction be controlled and made arbitrarily small?
• RQ4 Is the symbolic model computable in finite time under practical boundedness assumptions?
• RQ5 Can the proposed framework support complex control specifications beyond stabilization, such as synchronization or safety?

Key findings

• The proposed δ-IDSS condition is characterized via a Lyapunov-Krasovskii functional that captures the sensitivity of trajectories to differences in initial conditions, inputs, and time-delay signals.
• For the numerical example, a δ-IDSS Lyapunov-Krasovskii functional was constructed with β(ω,t) = 4.3580e^(-1.0870t)ω, γ_U(ω) = 13.5647ω, and γ_D(ω) = 194.1666ω, confirming δ-IDSS for the system.
• With ε = 0.12, the symbolic model was constructed using τ = 2, λ_X = 0.02, λ_U = 5×10⁻⁴, and λ_D = 1.4×10⁻⁴, satisfying the required bisimulation inequality.
• The symbolic model was computed in approximately 203,215 seconds using standard algorithms, but optimized algorithms reduced the solution time to 7,692 seconds on a standard laptop.
• The symbolic control strategy successfully achieved synchronization in the original system under time-varying delay Δ(t) = 1.5 + 0.5sin(0.01t), as verified by simulation.
• The framework ensures that the symbolic model is alternating approximately bisimilar to the original system, enabling formal verification and synthesis of complex control specifications.