# Vietnamese - French conference in applied mathematics

Jul 9 – 11, 2018
Ho Chi Minh City University of Science
Asia/Ho_Chi_Minh timezone

## Robust controllers for parabolic systems using the Galerkin approximation

Jul 9, 2018, 2:45 PM
30m
Ho Chi Minh City University of Science

#### Ho Chi Minh City University of Science

227 Nguyễn Văn Cừ, Phường 4, T.P. Hồ Chí Minh

### 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)

### Presentation materials

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