# Delay Systems

## By Sébastien Boisgérault, MINES ParisTech

### August 18, 2017

# Contents

# About

Systems of delay-differential algebraic equations (DDAE) – or *delay systems* for the sake of brevity – combine differential and algebraic equations with delayed variables in the right-hand side. A typical example^{1} would be: \[
\begin{array}{rll}
\dot{x}(t) &=& E x(t) - FG y(t) \\
y(t) &=& \displaystyle e^{T E} x(t-T) -
\int_{-T}^0 e^{-\theta E} FG y(t+\theta) d\theta
\end{array}
\]

# Presentations

# Papers

# Notes

this example describes the interaction between the system \[\dot{x}(t) = E x(t) + Fu(t)\] with a

*deadtime*– \(x(t)\) is unknown at time \(t\), only the value \(x(t-T)\) is available for some delay \(T>0\) – and a predictor-controller designed to stabilize it (with a finite-spectrum assignment for example). Think of it as an improvement of the classic Smith predictor.↩MAREVA is the Applied Mathematics Minor of MINES ParisTech “Master’s in Science and Executive Engineering” degree.↩