Speaker
Description
We want to compute a $T$-periodic symmetric solution $X(t)$ of a $T$-periodic differential matrix Riccati equation
$-\dot{X}(t)=X(t)A(t)+A^T(t)X(t)-X(t)B(t)R^{-1}(t)B^T(t)X(t)+Q(t)$
such that all solutions of the feedback system $\dot{x} = [A(t)-B(t)R^{-1}(t)B^T(t)X(t)]x(t)$ are asymptotically stable, i.e. $\lim_{t\to\infty}x(t)=0$. The standard solvability condition for the matrix Riccati equation requires that the $T$-periodic Hamiltonian matrix
$H(t) = \begin{bmatrix}A(t) & -B(t)R^{-1}(t)B^T(t) \\ -Q(t) & - A^T(t)\end{bmatrix}$
has no characteristic multipliers on the unit circle $|z|=1$.
The transition matrices $\Phi_k=\Phi_H(t_{k+1},t_k)$ for the periodic system with the matrix $H(t)$ are calculated using the Runge-Kutta method for $k=0,1,\ldots,P-1$ on an equidistant grid $t_k=kh$ with step $h=T/P$. The monodromy matrices for $H(t)$ sampled on the grid $t_k$ are matrix products $\Psi_k=\Phi_{k+P-1}\ldots\Phi_{k+1}\Phi_k$, where the indices are taken modulo $P$. The sampled solution $X(t_k)$ is determined by the block eigenvalue problem
$\Psi_k\begin{bmatrix}I\\X(t_k)\end{bmatrix}= \begin{bmatrix}I\\X(t_k)\end{bmatrix}\Lambda_k$,
where the discrete-stable matrices $\Lambda_k$ are the monodromy matrices of the aforementioned feedback system.
In paper [1], an orthogonal elimination method was proposed for accurate computation of $\Psi_k$. In paper [2], an $O(P\log P)$ algorithm was developed for computing all monodromy matrices $\Psi_k$. In this talk, I present a faster $O(P)$ method for computing the entire solution $X(t_k)$, $k=0,1,\ldots,P-1$. Alternative algorithms are discussed in paper [3].
References
[1] A.N. Malyshev, Siberian Math J., 30(4):559-567, 1989.
[2] P. Benner et al, Parallel Algorithms for LQ Optimal Control of Discrete-Time Periodic Linear Systems, J. Parallel Distrib. Comp, 62:306-325, 2002.
[3] S. Gusev et al, A numerical evaluation of solvers for the periodic Riccati differential equation, BIT Numer Math, 50:301-329, 2010.