Delay Systems

By Sébastien Boisgérault, MINES ParisTech

August 18, 2017



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: \[ \begin{array}{rll} \dot{x}(t) &=& E x(t) - FG y(t) \\ y(t) &=& \displaystyle e^{T E} x(t-T) - \int_{-T}^0 e^{-\theta E} FG y(t+\theta) d\theta \end{array} \]




  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.