[Paper Review] Is Lipschitz Continuity Preserved under Sampled-Data Discretization?

Masoud Abbaszadeh|arXiv (Cornell University)|Dec 27, 2016

Control Systems and Identification23 references3 citations

TL;DR

This paper investigates whether Lipschitz continuity is preserved under zero-order hold (ZOH) based approximate discretization of nonlinear continuous-time systems. It derives analytical expressions for the discrete-time Lipschitz and one-sided Lipschitz constants in terms of the continuous-time constants, system matrix norms, sampling time, and Jacobian derivatives, showing that the property is preserved under bounded conditions with explicit error bounds.

ABSTRACT

Usually, given a continuous-time nonlinear model, a closed form solution for an exact discretization cannot be found explicitly, originating the need of approximating discrete-time models. This note studies the preservation of the Lipschitz continuity under approximate discretizations.

Motivation & Objective

• To analyze whether the Lipschitz continuity property of nonlinear continuous-time systems is preserved under ZOH-based approximate discretization.
• To extend the analysis to the one-sided Lipschitz condition, which is less restrictive than standard Lipschitz continuity.
• To derive explicit analytical expressions for the discrete-time Lipschitz constants based on continuous-time parameters.
• To provide quantitative bounds on the preservation of stability and performance guarantees in sampled-data control systems.
• To support the use of higher-order approximate models in control design by establishing continuity preservation under discretization.

Proposed method

• Uses Taylor-Lie series expansion of the continuous-time system solution within each sampling interval under ZOH assumption.
• Derives approximate discrete-time models up to third order using successive time derivatives of the system dynamics.
• Applies matrix norm and inner product properties to bound the Lipschitz condition in discrete time.
• Uses the induced norm of the system matrix A and the spectral norm of the Jacobian ∂f/∂x to quantify growth in state differences.
• Establishes bounds on the discrete-time Lipschitz constant γd as a function of T, γc, ∥A∥, and higher-order derivatives.
• Derives separate expressions for the one-sided Lipschitz constant ρd, which depends on both ρc and γc.

Experimental results

Research questions

• RQ1 Does the standard two-sided Lipschitz continuity condition persist in approximate discrete-time models derived via ZOH discretization?
• RQ2 Can the one-sided Lipschitz condition be preserved under the same discretization, and how does it relate to the two-sided case?
• RQ3 What is the analytical relationship between the continuous-time Lipschitz constant and the discrete-time counterpart under ZOH?
• RQ4 How do sampling time T and system matrix norms affect the preservation of Lipschitz continuity in discretized models?
• RQ5 What are the explicit expressions for the discrete-time Lipschitz constants in second- and third-order approximate models?

Key findings

• The discrete-time two-sided Lipschitz constant γd is bounded by Tγc + T²(∥A∥γc + γc²/2) for second-order approximation.
• For third-order models, γd includes additional terms involving the second derivative norm β and the function norm M, with explicit dependence on T³.
• The one-sided Lipschitz constant ρd is expressed as Tρc + (T²/2)∥A∥(ρc + γc + ρcγc), showing coupling between one-sided and two-sided constants.
• The discrete Lipschitz constant grows linearly with sampling time T, but higher-order terms depend on system matrix and Jacobian norms.
• The results confirm that Lipschitz continuity is preserved under ZOH discretization as long as the continuous system satisfies the condition and T is sufficiently small.
• The derived bounds provide a foundation for stability analysis and LMI-based controller design in sampled-data nonlinear systems.