[Paper Review] To sample or not to sample: Self-triggered control for nonlinear systems

Adolfo Anta, Paulo Tabuada|ArXiv.org|Jun 4, 2008

Adaptive Control of Nonlinear Systems18 references4 citations

TL;DR

This paper proposes self-triggered control for nonlinear systems, enabling aperiodic controller updates based on real-time state measurements to reduce resource usage. It develops self-trigger conditions for state-dependent homogeneous and polynomial systems, proving that inter-execution times scale predictably under homogeneity, enabling efficient, stable control without periodic sampling or dedicated hardware monitoring.

ABSTRACT

Feedback control laws have been traditionally implemented in a periodic fashion on digital hardware. Although periodicity simplifies the analysis of the mismatch between the control design and its digital implementation, it also leads to conservative usage of resources such as CPU utilization in the case of embedded control. We present a novel technique that abandons the periodicity assumption by using the current state of the plant to decide the next time instant in which the state should be measured, the control law computed, and the actuators updated. This technique, termed self-triggered control, is developed for two classes of nonlinear control systems, namely, state-dependent homogeneous systems and polynomial systems. The wide applicability of the proposed results is illustrated in two well known physical examples: a jet engine compressor and the rigid body.

Motivation & Objective

• To address the inefficiency of periodic control in digital implementations, which wastes CPU, bandwidth, and energy by updating regardless of system state.
• To develop aperiodic control strategies that trigger updates only when necessary, improving resource utilization in embedded and networked control systems.
• To extend self-triggered control—where the controller computes its next update time—to nonlinear systems, specifically homogeneous and polynomial systems.
• To establish theoretical foundations for self-triggered control under information constraints, linking system dynamics, performance, and resource allocation.
• To demonstrate applicability through physical examples: a jet engine compressor and rigid body dynamics.

Proposed method

• Proposes a self-triggered control framework where the controller uses current state measurements to compute the next update time, eliminating periodic sampling.
• Develops self-trigger conditions based on Lyapunov-like functions and homogeneity properties for state-dependent homogeneous systems.
• Introduces an auxiliary extended system to analyze inter-execution times, leveraging homogeneity of degree l−1 to derive scaling laws.
• Uses flow commutativity and φ-related vector fields to prove that inter-execution times are preserved under system transformations.
• Applies dilation invariance to relate inter-execution times across different scaling factors, enabling scalable computation.
• Employs a state-dependent execution rule defined by ηZ ∘ z(t,z) = c, where c > 0 is a performance threshold.

Experimental results

Research questions

• RQ1 How can self-triggered control be designed for nonlinear systems to avoid periodic sampling while ensuring stability?
• RQ2 What theoretical conditions ensure that self-triggered updates maintain system performance and stability in nonlinear systems?
• RQ3 How do inter-execution times scale under system homogeneity, and can this be exploited for efficient computation?
• RQ4 Can self-triggered control be applied to polynomial and homogeneous systems without requiring continuous monitoring?
• RQ5 What is the relationship between the original system and an extended auxiliary system that enables analysis of self-triggering intervals?

Key findings

• Self-triggered control for nonlinear systems achieves stable performance with reduced resource usage by scheduling updates based on current state, avoiding periodic execution.
• For state-dependent homogeneous systems, inter-execution times scale with λ^(l−1) under dilation, enabling scalable computation of update times.
• The inter-execution time τ(x) for the original system equals the inter-execution time τ̃(λx,λ) for the extended system, up to scaling by λ^(l−1).
• Invariant sets in the original system induce corresponding invariant sets in the extended system, preserving stability properties under scaling.
• The proposed method avoids dedicated hardware for continuous monitoring by using state measurements to compute the next update time.
• The framework is validated on two physical systems—jet engine compressor and rigid body—demonstrating broad applicability.