Sufficient Conditions for Detectability of Approximately Discretized Nonlinear Systems

TL;DR

This paper establishes conditions under which discretized models of continuous nonlinear systems remain detectable, emphasizing the importance of sampling period and discretization schemes like Euler and Runge-Kutta.

Contribution

It provides LMI-based conditions ensuring detectability of discretized systems derived from continuous systems, applicable to common discretization methods.

Findings

Detectability is preserved under small sampling periods.

Euler and Runge-Kutta methods satisfy the conditions.

Practical example demonstrates applicability.

Abstract

In many sampled-data applications, observers are designed based on approximately discretized models of continuous-time systems, where usually only the discretized system is analyzed in terms of its detectability. In this paper, we show that if the continuous-time system satisfies certain linear matrix inequality (LMI) conditions, and the sampling period of the discretization scheme is sufficiently small, then the whole family of discretized systems (parameterized by the sampling period) satisfies analogous discrete-time LMI conditions that imply detectability. Our results are applicable to general discretization schemes, as long as they produce approximate models whose linearizations are in some sense consistent with the linearizations of the continuous-time ones. We explicitly show that the Euler and second-order Runge-Kutta methods satisfy this condition. A batch-reactor system example is provided to highlight the usefulness of our results from a practical perspective.