Certifying Set Attractivity for Discrete-Time Uncertain Nonlinear Switched Systems

Technical Summary: Certifying Set Attractivity for Discrete-Time Uncertain Nonlinear Switched Systems

Problem Statement
The paper addresses the challenge of certifying the robust local attractivity of sets for uncertain nonlinear switched systems in discrete time, where the switching law acts as a decision variable. While extensive literature exists on the stability analysis of linear and uncertainty-free switched systems (both continuous and discrete time), and some work covers uncertain linear systems, the problem remains a significant open issue for uncertain nonlinear switched systems. Existing Lyapunov-based methods are often nontrivial to apply to nonlinear systems, and set-theoretic approaches based on invariant sets have not yet provided general results for attractivity in the presence of uncertainties for this specific class of systems.

Methodology
The authors propose a framework centered on a new class of functions termed Attractivity Guarantee (AG)-functions. The methodology proceeds through the following logical steps:

1. Definition of AG-functions: An AG-function is a relaxation of the control Lyapunov function concept. It is a non-negative function $L: X \to \mathbb{R}_{\geq 0}$ that vanishes on a target set $\Omega$ and satisfies a specific decrease condition. Specifically, for states within a certain level set, there must exist a switching mode such that the supremum of the function's value over all possible uncertainty realizations decreases by a margin dependent on the distance to the set $\Omega$.
2. Theoretical Link to Attractivity: The paper proves (Proposition 2) that the existence of an AG-function associated with a set $\Omega$ guarantees the robust local attractivity of that set.
3. Constructive Approach via Contractive Sets: To address the difficulty of finding AG-functions directly for nonlinear systems, the authors propose a constructive method based on Robust Control Contractive Sets (RCCS). An RCCS is a set where, for every state, there exists a switching mode that drives the system into the interior of the set (accounting for all uncertainties).
4. Existence Proof: The authors demonstrate (Proposition 5) that if a compact RCCS exists, one can explicitly construct a piecewise-constant AG-function based on the "distance" of a state from the target set, measured by the number of steps required to reach the set via controllable sets.
5. Algorithmic Implementation:
   • For linear systems with additive uncertainty, the authors adapt an existing algorithm (Fiacchini and Jungers, 2014) to recursively compute robust controllable sets using Minkowski differences. This allows for the offline computation of a stabilizing switching law.
   • For nonlinear systems, a numerical procedure is proposed to approximate the controllable sets and verify the contractivity condition ($\Omega \subset \text{int}_X(C(\Omega))$) using a grid-based approach.

Key Contributions

• Introduction of AG-functions: The paper defines AG-functions as a tool to certify set attractivity with weaker requirements than traditional Lyapunov functions, specifically accommodating discontinuities arising from switching.
• Theoretical Guarantee: It establishes a rigorous logical chain: the existence of a Robust Control Contractive Set (RCCS) $\implies$ existence of an AG-function $\implies$ robust local attractivity of the set.
• Constructive Method: The paper provides a concrete method to generate AG-functions from RCCSs, defining the function based on the step-index of the robust controllable sets ($C_k(\Omega)$).
• Switching Law Synthesis: An explicit stabilizing switching law is derived (Equation 16) that selects the mode minimizing the step-index to the target set, ensuring the system trajectory remains within the domain of attraction and converges to the set.
• Linear and Nonlinear Validation: The framework is validated on linear switched systems with additive uncertainty and applied to a nonlinear case study.

Results

• Linear Case: The proposed algorithm successfully identifies RCCSs and stabilizing switching laws for linear systems with bounded uncertainties. In one example, the method identified a stabilizing law where uncertainty-free analysis suggested stability, but the presence of uncertainty prevented the identification of a contractive set, highlighting the necessity of the robust approach.
• Nonlinear Case Study (Antimicrobial Resistance): The framework was applied to a discrete-time model of bacterial population dynamics (susceptible and resistant strains) under immune response and antibiotic treatment.
  • The authors identified RCCSs for specific initial bacterial loads (e.g., total load $b_0 \leq 10,000$).
  • Numerical simulations demonstrated that for these loads, the system possesses an attractive set, meaning the bacterial population can be driven to a low-load state despite uncertainties in discretization and model parameters.
  • For higher initial loads ($b_0 = 120,000$), the contractivity condition failed, indicating the limits of the therapy protocol for high bacterial burdens.

Significance and Claims
The paper claims to provide the first general result for certifying set attractivity in uncertain nonlinear switched systems. Its significance lies in:

• Bridging a Gap: It extends set-theoretic control methods from linear/uncertainty-free systems to the more complex domain of uncertain nonlinear systems.
• Practical Relevance: The application to antimicrobial resistance (AMR) demonstrates the approach's utility in biological systems, specifically in assessing the fundamental limitations of antibiotic therapy protocols based on initial bacterial load.
• Robustness: The methodology explicitly accounts for uncertainties, providing a geometric certificate of asymptotic stability and constraint satisfaction that is robust to model variations and discretization errors.

The authors note that the approach is currently limited to local attractivity and that the construction of the switching law can be computationally intensive for general nonlinear systems due to the need to compute controllable sets offline. They suggest future work should focus on less conservative conditions for AG-function existence and performance-oriented switching strategies.