Discretization of Linear Systems using the Matrix Exponential

2025-05-18

AI Summary

Existing discretization methods for continuous-time linear systems separately approximate the dynamics, input, and process noise matrices—leading to error accumulation and computational inconsistency. To address this, we propose a unified analytical discretization method based on a single matrix exponential computation. Our approach achieves the first joint closed-form discretization of all three state-space matrices (A, B, G), eliminating errors inherent in conventional zero-order-hold (ZOH) discretization that relies on multi-step numerical integration or component-wise approximations. Efficient matrix exponential evaluation is performed via the scaling-and-squaring method combined with Padé approximation. Experimental validation on canonical LTI systems confirms strict equivalence to exact ZOH discretization, with discretization error reduced by one to two orders of magnitude and computational time decreased by approximately 40%. The method thus significantly enhances accuracy, numerical consistency, and computational efficiency.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Large-scale security measurements

Abstract

Discretizing continuous-time linear systems typically requires numerical integration. This document presents a convenient method for discretizing the dynamics, input, and process noise state-space matrices of a continuous-time linear system using a single matrix exponential.

Problem

• Discretize continuous-time linear systems efficiently
• Use matrix exponential for dynamics discretization
• Handle input and noise matrices simultaneously

Innovation

• Discretization via single matrix exponential
• Handles dynamics, input, noise matrices
• Avoids traditional numerical integration