Numerical methods for large-scale differential matrix equations in control theory

20 Apr 2017
8:30-9:20
Amphi 3

Numerical methods for large-scale differential matrix equations in control theory

In this talk, we consider large-scale continuous-time differential Lyapunov and differential Riccati equations having low rank right-hand sides:

(1)   \begin{equation*} %\left\{\begin{aligned} \dot X(t) =A^T\,X(t)+X(t)\,A-X(t)\,B\,B^T\,X(t)+C^T\,C,\;\; (DRE) \\ %X(0)&=X_0 \\ %\end{aligned} %\right. \end{equation*}

or

(2)   \begin{equation*} %\left\{\begin{aligned} \dot X(t) =A^T\,X(t)+X(t)\,A+C^T\,C,\;\; (DLE) \\ %X(0)=X_0 \\ %\end{aligned} %\right. \end{equation*}

These equations appear in many problems such in control theory for finite horison or in model reduction for large scale time-dependent dynamical systems.

(DRE) and (DLE) are generally solved by Backward Differentiation Formula (BDF) (or Rosenbrock) methods leading to large scale algebraic Lyapunov or Riccati equation which has to be solved for each timestep. However, these techniques are not effective for large problems because, at each timestep, one has to solve large scale algebraic Lyapunov or Riccati equations which could be very expensive. Here, we propose new approaches based on projection on small subspaces. For differential Lyapunov equations, we construct approximate solutions from the exponential expression of the exact solution using Krylov subspace methods to approximate exponential of a matrix times a block of vectors. For differential Riccati equations, we project the problem onto a small block Krylov or extended block Krylov subspace and then and obtain a low-dimentional differential algebraic Riccati equation. The latter matrix differential problem is solved by Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. We give some theoretical results and simple expressions of the residual norms allowing the implementation of a stop test in order to limit the dimension of the projection spaces. Uppers bounds for the norm of the errors are also given. The proposed numerical experiments show the effectiveness of our approaches.