# American Institute of Mathematical Sciences

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doi: 10.3934/mcrf.2021001

## Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities

 1 Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK 2 School of Engineering & the Built Environment, Edinburgh Napier University, Merchiston Campus, 10 Colinton Road, Edinburgh EH10 5DT, UK

* Corresponding author: Chris Guiver

Received  June 2020 Revised  November 2020 Published  January 2021

A low-gain integral controller with anti-windup component is presented for exponentially stable, linear, discrete-time, infinite-dimensional control systems subject to input nonlinearities and external disturbances. We derive a disturbance-to-state stability result which, in particular, guarantees that the tracking error converges to zero in the absence of disturbances. The discrete-time result is then used in the context of sampled-data low-gain integral control of stable well-posed linear infinite-dimensional systems with input nonlinearities. The sampled-date control scheme is applied to two examples (including sampled-data control of a heat equation on a square) which are discussed in some detail.

Citation: Max E. Gilmore, Chris Guiver, Hartmut Logemann. Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021001
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##### References:
Quantization function $q_\delta$
Square domain $\Omega \subseteq \mathbb R^2$
Feasible sets of reference vectors $r = {\left( {\begin{array}{*{20}{l}} {{r_1}}&{{r_2}} \end{array}} \right)^T}$ for $\phi_1$ (darker grey) and $\phi_2$ (lighter gray) regions. The reference $r$ in (4.8) is marked with a cross
Model data as in (4.5), (4.6), (4.8) and (4.9). (a) Initial temperature profile $z^0$. (b) Temperature profile of solution $z$ at time $t = 20$. (c) Outputs. (d) Held inputs. In panels (c) and (d), the solid and dashed lines correspond to $\tau = 0.25$ and $\tau = 0.5$, respectively. The dotted lines in panel (c) are the components of the reference
Model data as in (4.5), (4.6), (4.8), (4.10) and (4.11). (a) Outputs. (b) Held inputs. In panels (a) and (b), the solid and dashed lines correspond to the external forcing $v$ and $3 v$, respectively. The dotted lines in panel (a) are the components of the reference
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