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Modeling and iterative learning control of spatially distributed parameter systems with sensing and actuation over a selected area of the domain

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Abstract

This paper gives new contributions to the development of iterative learning control for distributed parameter systems, based on using finite difference schemes to construct a finite-dimensional approximate model of the dynamics for control law design. To form a basis for the new results, systems whose dynamics are described by a fourth-order partial differential equation are considered together with the associated accuracy and numerical stability checks. Some previous control law designs use only a spatial variable as the control input, which can be a serious obstacle to practical implementation since many actuators and sensors must be deployed. This paper’s new design is based on spatially homogeneous sensing and excitation over a selected sub-area of the domain considered. Supporting numerical case studies are given to support the analysis.

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Funding

The work was partially supported by the National Science Centre in Poland, grant no. 2020/37/B/ST7/03280.

Author information

Authors and Affiliations

  1. Municipal Department of Transport, 65-713, Zielona Gora, Poland

    Blazej Cichy

  2. Institute of Information Theory and Automation, The Czech Academy of Sciences, Prague, Czech Republic

    Petr Augusta

  3. Institute of Automation, Electronic and Electrical Engineering, University of Zielona Gora, Zielona Gora, Poland

    Krzysztof Galkowski

  4. School of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK

    Eric Rogers

Authors
  1. Blazej Cichy
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  2. Petr Augusta
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  3. Krzysztof Galkowski
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  4. Eric Rogers
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Correspondence to Blazej Cichy.

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Appendix A Notation

Appendix A Notation

$$\begin{aligned}&A=\left[ \begin{array}{lllllllllllll}\!\! A_1^{(\frac{n+1}{2})} &{} A_2 &{} A_3 &{} O &{} \cdots &{} \cdots &{}\cdots &{}\cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots &{} O \\ A_2^\mathrm {T} &{} A_1^{(\frac{n+3}{2})} &{} A_2 &{} A_3 &{} O &{} \ddots &{}\ddots &{}\ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ A_3^\mathrm {T} &{} A_2^\mathrm {T} &{} A_1^{(\frac{n+5}{2})} &{} A_2 &{} A_3 &{} O &{} \ddots &{}\ddots &{} \ddots &{} \ddots &{}\ddots &{} \ddots &{} \vdots \\ O &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{}\ddots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{}\ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} A_3&{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{}\ddots &{} \ddots &{} \ddots &{} \ddots &{} A_1^{(n-1)} &{} A_2 &{} A_4&{} \ddots &{} \ddots &{} \ddots &{} \ddots &{}\vdots \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} A_3^\mathrm {T} &{} A_2^\mathrm {T} &{} A_1^{(n)} &{} A_2 &{} A_3 &{} \ddots &{} \ddots &{} \ddots &{}\vdots \\ \vdots &{} \ddots &{} \ddots &{}\ddots &{} \ddots &{} A_4^\mathrm {T} &{} A_2^\mathrm {T} &{} A_1^{(n-1)} &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{} \ddots &{} \ddots &{}\ddots &{} \ddots &{} \ddots &{} A_3^\mathrm {T} &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} O \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} O &{} A_3^\mathrm {T} &{} A_2^\mathrm {T} &{} A_1^{\frac{(n+5}{2})} &{} A_2 &{} A_3 \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} O &{} A_3^\mathrm {T} &{} A_2^\mathrm {T} &{} A_1^{(\frac{n+3}{2})} &{} A_2 \\ O &{} \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots &{} \cdots &{} O &{} A_3^\mathrm {T} &{} A_2^\mathrm {T} &{} A_1^{(\frac{n+1}{2})} \!\!\end{array}\right] ,\nonumber \\&A_1^{(X)}={\begin{bmatrix} S &{} Q &{} 0 &{} \cdots &{} 0 \\ Q &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ 0 &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} Q \\ 0 &{} \cdots &{} 0 &{} Q &{} S \\ \end{bmatrix}}, \, A_2={\begin{bmatrix} P &{} P &{} 0 &{} \cdots &{} \cdots &{} 0 \\ 0 &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ 0 &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ 0 &{} \cdots &{} \cdots &{} 0 &{} P &{} P \\ \end{bmatrix}}, \, A_3={\begin{bmatrix} 0 &{} R &{} 0 &{} \cdots &{} \cdots &{} 0 \\ 0 &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ 0 &{} \cdots &{} \cdots &{} 0 &{} R &{} 0 \\ \end{bmatrix}}. \end{aligned}$$
(51)

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Cichy, B., Augusta, P., Galkowski, K. et al. Modeling and iterative learning control of spatially distributed parameter systems with sensing and actuation over a selected area of the domain. Multidim Syst Sign Process 32, 1237–1258 (2021). https://doi.org/10.1007/s11045-021-00780-1

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