A review on old and new results on robust regulation of DPS with infinite-dimensional exosystems

Abstract

In this paper the robust regulation problem for infinite-dimensional systems \begin{equation*} \dot x = Ax + Bu + F_s v, \qquad y = Cx + Du + F_m v, \end{equation*} with infinite-dimensional exosystems $\dot v = S v$ is discussed. The feedback controller is of the form \begin{equation*} \dot z = \mathcal{G}_1 z + \mathcal{G}_2 e, \qquad u = K z. \end{equation*} All the spaces involved are infinite-dimensional. The purpose of the feedback controller is to stabilize the closed loop system and to asymptotically track the reference and perturbation signal generated by the exosystem. The first key idea is the existence of a dynamic (bounded) steady state operator, which gives the asymptotic state of the stabilized closed loop system as time goes to infinity. This operator satisfies an operator Sylvester equation. The controller $(\mathcal{G}_1, \mathcal{G}_2)$ is robustly regulating if the Sylvester equation decomposes so that a regulation constraint will be satisfied. In the presentation various definitions of Internal Model Principle, including an infinite-dimensional one, guaranteeing robust regulation, will be discussed. The second important step is in stabilizing the closed loop system by a proper choice of the controller. As the robust controller contains a $p$-copy of the exosystem, the closed loop system cannot be stabilized exponentially; instead strong or weak stabilization must be used. A necessary condition for the existence of bounded dynamic steady state operator is the nonexistence of system zeros on the spectrum of the exosystem. In the infinite-dimensional case, the behaviour of the system transfer function at infinity also plays an important role. The presentation reviews and combines the recent results of T. Hämäläinen, L. Paunonen. P. Laakkonen and S. Pohjolainen.