A new robust control framework for heat transfer equations through $ L_1 $ robust stability analysis and the design of a 2-DoF robust LQI controller

Taewan Kim; Jung Hoon Kim; Jihyun Park; Taewan Kim; Jung Hoon Kim; Jihyun Park

doi:10.3934/math.2025859

AIMS Mathematics

2025, Volume 10, Issue 8: 19217-19239. doi: 10.3934/math.2025859

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A new robust control framework for heat transfer equations through $ L_1 $ robust stability analysis and the design of a 2-DoF robust LQI controller

1.
Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea
2.
Smart Mobility R&D Division, Korea Institute of Robotics & Technology Convergence (KIRO), Pohang 37666, Republic of Korea

Received: 12 June 2025 Revised: 06 August 2025 Accepted: 20 August 2025 Published: 25 August 2025
MSC : 93D09

This paper developed a new robust control framework for uncertain heat transfer systems with a rapid thermal annealing (RTA) processes. We first employed the spatial discretization and Taylor linearization schemes in the nonlinear partial differential equation describing the dynamic behavior of heat transfer systems, by which a linear time-invariant (LTI) system with model uncertainties was derived. More precisely, the model uncertainties were decomposed into feedforward and feedback components, and they were shown to be bounded in terms of the $ L_\infty $(-induced) norm. This representation allowed us to establish an $ L_1 $ robust stability condition for uncertain heat transfer systems. We next designed a 2-degree-of-freedom (2-DoF) robust linear quadratic integral (LQI) controller to achieve the $ L_1 $ robust stability condition and reduce the effects of the model uncertainties on the associated tracking accuracy. Finally, some simulations were given to verify the effectiveness of the developed method.
- heat transfer equations,
- $ L_1 $ robust stability,
- 2-DoF control,
- small-gain theorem
Citation: Taewan Kim, Jung Hoon Kim, Jihyun Park. A new robust control framework for heat transfer equations through $ L_1 $ robust stability analysis and the design of a 2-DoF robust LQI controller[J]. AIMS Mathematics, 2025, 10(8): 19217-19239. doi: 10.3934/math.2025859

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Abstract

This paper developed a new robust control framework for uncertain heat transfer systems with a rapid thermal annealing (RTA) processes. We first employed the spatial discretization and Taylor linearization schemes in the nonlinear partial differential equation describing the dynamic behavior of heat transfer systems, by which a linear time-invariant (LTI) system with model uncertainties was derived. More precisely, the model uncertainties were decomposed into feedforward and feedback components, and they were shown to be bounded in terms of the $ L_\infty $(-induced) norm. This representation allowed us to establish an $ L_1 $ robust stability condition for uncertain heat transfer systems. We next designed a 2-degree-of-freedom (2-DoF) robust linear quadratic integral (LQI) controller to achieve the $ L_1 $ robust stability condition and reduce the effects of the model uncertainties on the associated tracking accuracy. Finally, some simulations were given to verify the effectiveness of the developed method.

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