### Speaker

Dr
Duc Duy PHAN
(Tampere University of Technology)

### Description

This is a joint work with Lassi Paunonen and Petteri Laakkonen, Tampere University of Technology.
We consider the robust output tracking problem on state space $H$
$$ \dot{x} (t) = A x(t) + B u(t),$$
$$ y(t) = C x(t) + D u(t),$$
where $x$ is the state, $u$ is the input (control), and $y$ is the output (observation). Our goal is to design a dynamic feedback controller of the form
$$\dot{z}(t) = \mathcal{G}_1 z(t) + \mathcal{G}_2 e(t),$$
$$u(t) = K z(t),$$
where $e(t) = y(t)- y_{ref}(t)$ is the regulation error in such a way that the output $y(t)$ of the system converges asymptotically to a given reference signal $y_{ref}(t)$. We propose a new way of designing finite-dimensional robust controllers based on Galerkin approximations of infinite-dimensional controllers presented before in [Pau16]. For a class of sesquilinear form $A$ and assumptions of approximation schemes proposed in [BI88,BI97,Mor94], we prove that the finite dimensional controllers solve the Robust Output Regulation Problem. \\
[BI88] H. T. Banks and K. Ito.
A unified framework for approximation in inverse problems for distributed parameter systems.
Control Theory Adv. Tech., 1988.
[BI97] H. T. Banks and K. Ito. Approximation in LQR Problems for Infinite Dimensional Systems With Unbounded Input Operators.
J. Math. Systems Estim. Control, 1997.
[Mor94] K. A. Morris. Design of finite-dimensional controllers for infinite-dimensional systems by approximation.
J. Math. Systems Estim. Control, 4(2):30, 1994.
[Pau16] L. Paunonen. Controller Design for Robust Output Regulation of Regular Linear Systems.
IEEE Transactions on Automatic Control}, 61(10):2974--2986, Oct 2016.

### Primary author

Dr
Duc Duy PHAN
(Tampere University of Technology)