# Contents

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 example1 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

### Introduction to Delay Systems

A MAREVA2 course.

### Delay Equations – A Case for Algebro-Differential Systems.

Mines ParisTech Mathematics and Systems Seminar.

### Design of Algebraic Observers for Brass Instruments

ISMA 2014, with Brigitte d’Andrea-Novel.

# Papers

### A Core Theory of Delay Systems

Expository paper.
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### Design of Algebraic Observers for Brass Instruments

with Brigitte d’Andrea-Novel.
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# Notes

1. 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.

2. MAREVA is the Applied Mathematics Minor of MINES ParisTech “Master’s in Science and Executive Engineering” degree.