Duality and H8-Optimal Control of Coupled ODE-PDE Systems

Sachin Shivakumar; Amritam Das; Siep Weiland; Matthew M. Peet

doi:10.1109/CDC42340.2020.9303989

Duality and H8-Optimal Control of Coupled ODE-PDE Systems

Sachin Shivakumar, Amritam Das, Siep Weiland, Matthew M. Peet

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

15 Citations (Scopus)

Abstract

In this paper, we present a convex formulation of H8 -optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE) system and show that stability and H8 performance of the PIE system implies that of the ODE-PDE system. We then construct a dual PIE system and show that asymptotic stability and H8 performance of the dual system is equivalent to that of the primal PIE system. Next, we pose a convex dual formulation of the stability and H8 -performance problems using the Linear PI Inequality (LPI) framework. Next, we use our duality results to formulate the stabilization and H8 - optimal state-feedback control problems as LPIs. LPIs are a generalization of LMIs to Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB toolbox. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples.

Original language	English
Title of host publication	59th IEEE Conference on Decision and Control (CDC 2020)
Publisher	Institute of Electrical and Electronics Engineers
Pages	5689-5696
Number of pages	8
ISBN (Electronic)	978-1-7281-7447-1
DOIs	https://doi.org/10.1109/CDC42340.2020.9303989
Publication status	Published - 11 Jan 2021
Event	59th IEEE Conference on Decision and Control, CDC 2020 - Virtual/Online, Virtual, Jeju Island, Korea, Republic of Duration: 14 Dec 2020 → 18 Dec 2020 Conference number: 59 https://cdc2020.ieeecss.org/

Conference

Conference	59th IEEE Conference on Decision and Control, CDC 2020
Abbreviated title	CDC
Country/Territory	Korea, Republic of
City	Virtual, Jeju Island
Period	14/12/20 → 18/12/20
Internet address	https://cdc2020.ieeecss.org/

Funding

of the ODE-PDE are inherited from the PIE. Our duality results allow can be used with LPIs to find stabilizing and H∞-optimal state-feedback controllers for PIEs and these controllers can then be used to regulate the associated ODE-PDEs. We have demonstrated the accuracy and scalability of the resulting algorithms by applying the results to several illustrative examples. While the scope of the paper is limited to inputs entering through the ODE or in-domain, we believe the results can be extended to inputs at the boundary. ACKNOWLEDGMENT This work was supported by Office of Naval Research Award N00014-17-1-2117, and National Science Foundation grants CMMI-1935453 and CNS-1739990. REFERENCES

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10.1109/CDC42340.2020.9303989

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title = "Duality and H8-Optimal Control of Coupled ODE-PDE Systems",

abstract = "In this paper, we present a convex formulation of H8 -optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE) system and show that stability and H8 performance of the PIE system implies that of the ODE-PDE system. We then construct a dual PIE system and show that asymptotic stability and H8 performance of the dual system is equivalent to that of the primal PIE system. Next, we pose a convex dual formulation of the stability and H8 -performance problems using the Linear PI Inequality (LPI) framework. Next, we use our duality results to formulate the stabilization and H8 - optimal state-feedback control problems as LPIs. LPIs are a generalization of LMIs to Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB toolbox. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples.",

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Shivakumar, S, Das, A , Weiland, S & Peet, MM 2021, Duality and H8-Optimal Control of Coupled ODE-PDE Systems. in 59th IEEE Conference on Decision and Control (CDC 2020)., 9303989, Institute of Electrical and Electronics Engineers, pp. 5689-5696, 59th IEEE Conference on Decision and Control, CDC 2020, Virtual, Jeju Island, Korea, Republic of, 14/12/20. https://doi.org/10.1109/CDC42340.2020.9303989

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AU - Shivakumar, Sachin

AU - Das, Amritam

AU - Weiland, Siep

AU - Peet, Matthew M.

N1 - Conference code: 59

PY - 2021/1/11

Y1 - 2021/1/11

N2 - In this paper, we present a convex formulation of H8 -optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE) system and show that stability and H8 performance of the PIE system implies that of the ODE-PDE system. We then construct a dual PIE system and show that asymptotic stability and H8 performance of the dual system is equivalent to that of the primal PIE system. Next, we pose a convex dual formulation of the stability and H8 -performance problems using the Linear PI Inequality (LPI) framework. Next, we use our duality results to formulate the stabilization and H8 - optimal state-feedback control problems as LPIs. LPIs are a generalization of LMIs to Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB toolbox. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples.

AB - In this paper, we present a convex formulation of H8 -optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE) system and show that stability and H8 performance of the PIE system implies that of the ODE-PDE system. We then construct a dual PIE system and show that asymptotic stability and H8 performance of the dual system is equivalent to that of the primal PIE system. Next, we pose a convex dual formulation of the stability and H8 -performance problems using the Linear PI Inequality (LPI) framework. Next, we use our duality results to formulate the stabilization and H8 - optimal state-feedback control problems as LPIs. LPIs are a generalization of LMIs to Partial Integral (PI) operators and can be solved using PIETOOLS, a MATLAB toolbox. Finally, we illustrate the accuracy and scalability of the algorithms by constructing controllers for several numerical examples.

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M3 - Conference contribution

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BT - 59th IEEE Conference on Decision and Control (CDC 2020)

PB - Institute of Electrical and Electronics Engineers

T2 - 59th IEEE Conference on Decision and Control, CDC 2020

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ER -

Duality and H8-Optimal Control of Coupled ODE-PDE Systems

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