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Sufficient Conditions for Detectability of Approximately Discretized Nonlinear Systems

Seth Siriya, Julian D. Schiller, Victor G. Lopez, Matthias A. Müller

TL;DR

This work tackles the challenge of guaranteeing detectability for observers applied to approximately discretized nonlinear systems by linking continuous-time i-IOSS LMIs to discrete-time LMIs for a whole family of ADT models. The authors prove that, under a Lipschitz CT system and a consistency condition between the CT and ADT linearizations, the CT LMIs imply the DT LMIs for sufficiently small sampling period τ, enabling an i-IOSS Lyapunov function for the ADT family. They explicitly derive how the discrete-time LMIs scale with τ and show that Euler and RK2 discretizations satisfy the required linearization consistency, with RK2 offering explicit bounds under additional assumptions. An illustrative batch-reactor example demonstrates practical computational advantages, showing that verifying CT LMIs is substantially cheaper than verifying the corresponding DT LMIs on the ADT models. Collectively, the results provide a rigorous, broadly applicable pathway to ensure robust, sample-based state estimation in nonlinear monitoring and control tasks.

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.

Sufficient Conditions for Detectability of Approximately Discretized Nonlinear Systems

TL;DR

This work tackles the challenge of guaranteeing detectability for observers applied to approximately discretized nonlinear systems by linking continuous-time i-IOSS LMIs to discrete-time LMIs for a whole family of ADT models. The authors prove that, under a Lipschitz CT system and a consistency condition between the CT and ADT linearizations, the CT LMIs imply the DT LMIs for sufficiently small sampling period τ, enabling an i-IOSS Lyapunov function for the ADT family. They explicitly derive how the discrete-time LMIs scale with τ and show that Euler and RK2 discretizations satisfy the required linearization consistency, with RK2 offering explicit bounds under additional assumptions. An illustrative batch-reactor example demonstrates practical computational advantages, showing that verifying CT LMIs is substantially cheaper than verifying the corresponding DT LMIs on the ADT models. Collectively, the results provide a rigorous, broadly applicable pathway to ensure robust, sample-based state estimation in nonlinear monitoring and control tasks.

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.

Paper Structure

This paper contains 8 sections, 6 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Results
  4. Preservation of LMI conditions after approximate discretization
  5. i-IOSS Lyapunov functions for approximately discretized systems
  6. Consistency of linearizations for Euler- and RK2-discretized models
  7. Example
  8. Conclusion

Key Result

Theorem 1

Consider a set $\mathcal{X}\times\mathcal{U}\times\mathcal{D}\subseteq\mathbb{R}^n\times\mathbb{R}^q\times\mathbb{R}^m$, the CT system eqn:system-dynamics-eqn:output-equation, and the family of associated approximate DT systems eqn:approx-system-dynamics-eqn:approx-system-output-equation under Ass. Then, the family of ADT systems eqn:approx-system-dynamics-eqn:approx-system-output-equation with s

Theorems & Definitions (22)

  • Remark 1
  • Definition 1: CT LMI conditions schiller2024robust
  • Remark 2
  • Definition 2: DT LMI conditions schiller2023lyapunov
  • Definition 3: Consistency of linearizations
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Definition 4: i-IOSS Lyapunov function schiller2024robust
  • ...and 12 more