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Dissipativity-Guaranteed Distributed Model Predictive Controller for Reconfigurable Large-Scale System

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Abstract

This paper presents a distributed model predictive control strategy for reconfigurable large-scale systems to maintain global stability when there are subsystems joining in or leaving the system during closed-loop operation, which is also called the plug-and-play control. Dissipativity is an input–output property, and the overall dissipativity property of a large-scale system can be described by the ‘linear’ combination of the dissipativity properties of its subsystems based on their interconnection topology. The capacity of capturing interaction effects among subsystems on overall stability makes dissipative system theory a useful tool to design control system for reconfigurable large-scale system. The redesign is restricted to subsystems whose input–output property will be directly influenced by the topology change in order to allow quick operation. The overall stability condition can still be guaranteed if the dissipativity property of the redesigned subsystems does not spoil the overall dissipativity condition. Simulation results of a power network system with subsystems joining in and leaving the network show the effectiveness of the proposed strategy.

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Acknowledgements

This work was supported by the National Nature Science Foundation of China (61590924, 61233004, 61673273).

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  1. Department of Automation, Shanghai Jiao Tong University, Shanghai, 200240, China

    Ye He & Shaoyuan Li

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  1. Ye He
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  2. Shaoyuan Li
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Correspondence to Shaoyuan Li.

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Appendices

Appendix A Dissipativity Property of Area 2 in Scenario 1

Here, the supply rate coefficient matrix of area 2 in scenario 1 before and after reconfiguration is given for example.

$$\begin{aligned} Q_{c2}= & {} \left( \begin{matrix} 0.5192&{}\quad 0&{}\quad 0.0057&{}\quad -0.0143\\ 0&{}\quad -2.5055&{}\quad 0.0018&{}\quad -0.0038\\ 0.0057&{}\quad 0.0018&{}\quad -1.4550&{}\quad 0.8702\\ -0.0143&{}\quad -0.0038&{}\quad 0.8702&{}\quad -1.555 \end{matrix}\right) \end{aligned}$$
(34)
$$\begin{aligned} S_{c2}= & {} \left( \begin{matrix} -0.0057&{}\quad 0.0143&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ -0.0018&{}\quad 0.0038&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0.3906&{}\quad 0.1775&{}\quad 0.0347&{}\quad 0.0068&{}\quad -0.0433&{}\quad 0.0185\\ 0.1685&{}\quad 0.3504&{}\quad 0.0780&{}\quad 0.0416&{}\quad 0.0780&{}\quad -0.0314 \end{matrix}\right) \end{aligned}$$
(35)
$$\begin{aligned} R_{c2}= & {} \left( \begin{matrix} 0.8277&{}\quad -1.2163&{}\quad -0.0851&{}\quad -0.0929&{}\quad 0.0675&{}\quad -0.0600\\ -1.2163&{}\quad 0.8546&{}\quad -0.0780&{}\quad -0.0416&{}\quad -0.0780&{}\quad 0.0314\\ -0.0851&{}\quad -0.0780&{}\quad 0.3452&{}\quad -0.2930&{}\quad -0.0119&{}\quad 0.0101\\ -0.0929&{}\quad -0.0416&{}\quad -0.2930&{}\quad 0.3760&{}\quad 0.0004&{}\quad 0.0027\\ 0.0675&{}\quad -0.0780&{}\quad -0.0119&{}\quad 0.0004&{}\quad 0.1967&{}\quad -0.2095\\ -0.0600&{}\quad 0.0314&{}\quad 0.0101&{}\quad 0.0027&{}\quad -0.2095&{}\quad 0.2382 \end{matrix}\right) \nonumber \\ \end{aligned}$$
(36)
$$\begin{aligned} \tilde{Q}_{c2}= & {} \left( \begin{matrix} 91.2645&{}\quad 0&{}\quad 0.0400&{}\quad -0.0444\\ 0&{}\quad -807.1916&{}\quad 0.0002&{}\quad -0.0488\\ 0.0400&{}\quad 0.0002&{}\quad -32.3273&{}\quad 1.1519\\ -0.0444&{}\quad -0.0488&{}\quad 1.1519&{}\quad -32.8557 \end{matrix}\right) \end{aligned}$$
(37)
$$\begin{aligned} \tilde{S}_{c2}= & {} \left( \begin{matrix} -0.0399&{}\quad 0.0444&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ -0.0002&{}\quad 0.0488&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 28.8476&{}\quad 2.4783&{}\quad -0.0942&{}\quad -0.0906&{}\quad 0.1009&{}\quad -0.0119&{}\quad 0.1351&{}\quad -0.1898\\ 2.5986&{}\quad 28.5571&{}\quad 0.2968&{}\quad 0.1031&{}\quad -0.0162&{}\quad 0.0221&{}\quad -0.1981&{}\quad 0.3405 \end{matrix}\right) \nonumber \\\end{aligned}$$
(38)
$$\begin{aligned} \tilde{R}_{c2}= & {} \left( \begin{matrix} -25.1409&{}\quad -6.2300&{}\quad 0.0335&{}\quad 0.0662&{}\quad -0.1451&{}\quad -0.0030&{}\quad 0.0067&{}\quad -0.0336\\ -6.2300&{}\quad -24.2576&{}\quad -0.2975&{}\quad -0.1047&{}\quad 0.0183&{}\quad -0.0216&{}\quad 0.1984&{}\quad -0.3407\\ 0.0335&{}\quad -0.2975&{}\quad 0.4424&{}\quad -0.3727&{}\quad 0.0090&{}\quad 0.0148&{}\quad 0.0483&{}\quad -0.0805\\ 0.0662&{}\quad -0.1047&{}\quad -0.3727&{}\quad 0.4998&{}\quad -0.0276&{}\quad -0.0573&{}\quad 0.0214&{}\quad -0.0376\\ -0.1451&{}\quad 0.0183&{}\quad 0.0090&{}\quad -0.0276&{}\quad 0.4095&{}\quad -0.3692&{}\quad -0.0320&{}\quad 0.0525\\ -0.0030&{}\quad -0.0216&{}\quad 0.0148&{}\quad -0.0573&{}\quad -0.3692&{}\quad 0.4168&{}\quad 0.0277&{}\quad -0.0183\\ 0.0067&{}\quad 0.1984&{}\quad 0.0483&{}\quad 0.0214&{}\quad -0.0320&{}\quad 0.0277&{}\quad 0.6104&{}\quad -0.8179\\ -0.0336&{}\quad -0.3407&{}\quad -0.0805&{}\quad -0.0376&{}\quad 0.0525&{}\quad -0.0183&{}\quad -0.8179&{}\quad 1.1254 \end{matrix}\right) \nonumber \\ \end{aligned}$$
(39)

Appendix B Dissipativity Property of Area 5 in Scenario 2

And the supply rate coefficient matrix of area 5 in scenario 2 before reconfiguration is provided as follows.

$$\begin{aligned} Q_{c5}= & {} \left( \begin{matrix} 81.2242&{}\quad 0&{}\quad 0.0366&{}\quad -0.0567\\ 0&{}\quad -1562.5154&{}\quad 0.0686&{}\quad -0.0906\\ 0.0366&{}\quad 0.0686&{}\quad -31.6405&{}\quad -0.2095\\ -0.0567&{}\quad -0.0906&{}\quad -0.2095&{}\quad -31.1488 \end{matrix}\right) \end{aligned}$$
(40)
$$\begin{aligned} S_{c5}= & {} \left( \begin{matrix} -0.0365&{}\quad 0.0567&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ -0.0686&{}\quad 0.0906&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 29.8175&{}\quad 2.5966&{}\quad -0.2418&{}\quad -0.3851&{}\quad 0.2968&{}\quad -0.2340\\ 1.9295&{}\quad 28.8412&{}\quad -0.0313&{}\quad 0.6099&{}\quad -0.4103&{}\quad 0.4193 \end{matrix}\right) \quad \end{aligned}$$
(41)
$$\begin{aligned} R_{c5}= & {} \left( \begin{matrix} -27.9627&{}\quad -4.3190&{}\quad 0.2788&{}\quad 0.3260&{}\quad -0.3291&{}\quad 0.2589\\ -4.3190&{}\quad -26.5317&{}\quad 0.0321&{}\quad -0.6116&{}\quad 0.4107&{}\quad -0.4183\\ 0.2788&{}\quad 0.0321&{}\quad 2.1178&{}\quad -2.0372&{}\quad -0.1396&{}\quad 0.0212\\ 0.3260&{}\quad -0.6116&{}\quad -2.0372&{}\quad 2.0157&{}\quad 0.1803&{}\quad -0.0981\\ -0.3291&{}\quad 0.4107&{}\quad -0.1396&{}\quad 0.1803&{}\quad 0.2081&{}\quad -0.2169\\ 0.2589&{}\quad -0.4183&{}\quad 0.0212&{}\quad -0.0981&{}\quad -0.2169&{}\quad 0.2679 \end{matrix}\right) \nonumber \\ \end{aligned}$$
(42)

When the area 4 is removed from the system, for area 5, the process input from area 4 is removed as well. No offline dissipativity design is needed, the supply rate coefficient matrix of area 5 after reconfiguration can be obtained by deleting the corresponding rows and columns.

$$\begin{aligned} \tilde{Q}_{c5}= & {} Q_{c5} \end{aligned}$$
(43)
$$\begin{aligned} \tilde{S}_{c5}= & {} \left( \begin{matrix} -0.0365&{}\quad 0.0567&{}\quad 0&{}\quad 0\\ -0.0686&{}\quad 0.0906&{}\quad 0&{}\quad 0\\ 29.8175&{}\quad 2.5966&{}\quad -0.2418&{}\quad 0.2968\\ 1.9295&{}\quad 28.8412&{}\quad -0.0313&{}\quad -0.4103 \end{matrix}\right) \end{aligned}$$
(44)
$$\begin{aligned} \tilde{R}_{c5}= & {} \left( \begin{matrix} -27.9627&{}\quad -4.3190&{}\quad 0.2788&{}\quad -0.3291\\ -4.3190&{}\quad -26.5317&{}\quad 0.0321&{}\quad 0.4107\\ 0.2788&{}\quad 0.0321&{}\quad 2.1178&{}\quad -0.1396\\ -0.3291&{}\quad 0.4107&{}\quad -0.1396&{}\quad 0.2081 \end{matrix}\right) \end{aligned}$$
(45)

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He, Y., Li, S. Dissipativity-Guaranteed Distributed Model Predictive Controller for Reconfigurable Large-Scale System. Circuits Syst Signal Process 39, 1873–1895 (2020). https://doi.org/10.1007/s00034-019-01257-0

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