Discretization of Linear Systems using the Matrix Exponential
Steven Dahdah, James Richard Forbes
TL;DR
The paper addresses discretizing continuous-time linear systems with process and measurement noise by introducing a single matrix-exponential construction that yields $A^{\mathrm{d}}$, $B^{\mathrm{d}}$, and $Q^{\mathrm{d}}$ directly from an augmented block matrix. It shows that $A^{\mathrm{d}}=\exp(A\Delta t)$, $B^{\mathrm{d}}=\int_{0}^{\Delta t} \exp(A(\Delta t-s)) B \, ds$, and $Q^{\mathrm{d}}=\int_{0}^{\Delta t} \exp(A(\Delta t-s)) L Q L^{\mathsf{T}} \exp(A^{\mathsf{T}}(\Delta t-s)) ds$, all of which can be obtained from a single exponential $\boldsymbol{\Upsilon}=\exp(\boldsymbol{\Xi}\Delta t)$. The method builds on and reconciles ideas from Van Loan and Farrell, providing a consistent and compact discretization procedure for state, input, and noise covariances. Practically, it simplifies the discretization workflow and can improve the accuracy and efficiency of estimation and control in linear stochastic systems.
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.