On Sampling Time and Invariance

TL;DR

The paper addresses safety guarantees for adaptive-sampling control by introducing $M$-step hold invariance, a generalization of traditional control invariance that enforces constraints at all samples while allowing controlled sampling-rate changes. It defines both nominal and robust variants, $\mathcal{C}^M_{\infty,\cdot}$ and $\mathcal{RC}^M_{\infty,\cdot}$, and proves monotonicity properties $\mathcal{C}^{M+1}_{\infty,T_s}\subseteq\mathcal{C}^{M}_{\infty,T_s}$ and $\mathcal{RC}^{M+1}_{\infty,T_s}\subseteq\mathcal{RC}^{M}_{\infty,T_s}$, enabling safe switching between sampling rates under an $M$-step hold controller. The framework is instantiated for discrete LTI models with polytopic constraints, including explicit computation of precursor sets and invariant sets, and extended to a nonlinear example by bounding inter-sample discretization and modeling errors with a disturbance set. This work supplies a principled foundation for adaptive-sampling controllers that maintain inter-sample constraint satisfaction, bridging discrete-model invariance with real-time, continuous-time safety guarantees. The results have practical implications for safety-critical systems where switching sampling rates is advantageous for task complexity and resource use.

Abstract

Invariant sets define regions of the state space where system constraints are always satisfied. The majority of numerical techniques for computing invariant sets have been developed for discrete-time systems with a fixed sampling time. Understanding how invariant sets change with sampling time is critical for designing adaptive-sampling control schemes that ensure constraint satisfaction. We introduce M-step hold control invariance, a generalization of traditional control invariance, and show its practical use to assess the link between control sampling frequency and constraint satisfaction. We robustify M-step hold control invariance against model mismatches and discretization errors, paving the way for adaptive-sampling control strategies.

Figures (5)

• Figure 1: $\mathcal{C}_{\infty,T_s} \subset\mathcal{C}_{\infty,MT_s}$ for exact discretizations of a constrained double integrator with $T_s=0.5$ and $M=3$. Up-sampled optimal inputs calculated for $f_{MT_s}$ applied to $f_{T_s}$ and $f_c$ can violate constraints.
• Figure 2: $\mathcal{C}^M_{\infty,T_s}$ with larger $M$ are subsets of those with smaller $M$.
• Figure 3: Visualization of $\mathcal{W}[\text{mod}(k,M)]$ for a 1D system. The uncertainty ($\epsilon_c$ and $\epsilon_d$) resets with each measurement.
• Figure 4: $\mathcal{RC}^M_{\infty,T_s}$ with larger $M$ are subsets of those with smaller $M$.
• Figure 5: $\mathcal{RC}^M_{\infty,T_s}$ for the nonlinear example.

Theorems & Definitions (16)

• Definition 1
• Definition 2
• Remark 1
• Definition 3
• Definition 4
• Definition 5
• Lemma 1
• proof
• Theorem 1
• Definition 6