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LMI-based linear parameter varying PID control design and its application to an aircraft control system

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Abstract

To bridge the gap between real-world applications and theoretical achievements, this paper proposes a new synthesis approach of gain-scheduled proportional–integral–derivative (PID) control for linear parameter-varying (LPV) systems. It is recognized that the synthesis problem of PID controllers for LPV systems should be formulated as nonconvex optimization problems. To avoid this situation, using a special matrix transformation, novel synthesis conditions with linear matrix inequality constraints are provided in this paper. The stability of the resulting closed-loop system is guaranteed theoretically based on a parameter-dependent Lyapunov function, and two types of robust performance (bounded \(L_2\) norm and induced \(L_2\) norm) are also achieved in the corresponding synthesis conditions. Then, the control system of aircraft is designed based on the proposed method, and the system responses are compared with the traditional LPV-PID control and the LPV dynamic output feedback control.

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Data availability

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

This research was sponsored by the National Natural Science Foundation of China under Grant #61473186.

Author information

Authors and Affiliations

  1. School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Bixuan Huang, Bei Lu & Qifu Li

  2. Department of Mechanical Engineering, The University of British Columbia, Vancouver, V6T 1Z4, Canada

    Ryozo Nagamune

Authors
  1. Bixuan Huang
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  2. Bei Lu
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  3. Ryozo Nagamune
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  4. Qifu Li
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Contributions

The authors confirm contribution to the paper as follows: BH developed the theoretical formalism, performed the numerical simulation, and drafted the manuscript; BL conceived the original idea, supervised the project, and finalized the manuscript; RN helped verify the theory; QL helped verify the numerical results. All authors reviewed the results and approved the final version of the manuscript.

Corresponding author

Correspondence to Bei Lu.

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The authors declare that they have no conflict of interest.

Ethical standard statement

This article does not contain any studies with human or animal subjects performed by any of the authors.

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Appendix

Appendix

The system parameters at \( h=1000\) ft and \( V=300\) ft/s are given as

$$\begin{aligned} \left[ \begin{array}{cc} A_\mathrm{{air}} &{} B_\mathrm{{air}} \\ C_\mathrm{{air}} &{} D_\mathrm{{air}} \end{array} \right] =\left[ \begin{array}{ccccc|cc} -0.0303 &{} -8.0728 &{} -2.8148 &{} -32.1739 &{} 0.3684 &{} 0 &{} 0.0057 \\ -0.0007 &{} -0.5918 &{} 0.9042 &{} -6.2315\times 10^{-15} &{} -0.0002 &{} 0 &{} -0.0013 \\ 0 &{} -1.1893 &{} -0.8983 &{} 0 &{} 0 &{} 0 &{} -0.0634 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 64.94 &{} 0 \\ \hline 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \end{array} \right] . \end{aligned}$$

The control gains of the induced \( L_2 \) norm LPV-PID control are given as

$$\begin{aligned} \left[ \begin{array}{ccc} K_{P}&K_{I}&K_{D} \end{array} \right] =\left[ \begin{array}{cccc|c|cc} -5370.5244 &{} -19.4187 &{} 1.0971 &{} 4335.5679 &{} -1465.9746 &{} -507.1957 &{} 225.3770 \\ 51604.6521 &{} -2.5148 &{} -11.0291 &{} -41042.0583 &{} 14696.6093 &{} 5404.5756 &{} 5404.5756 \end{array} \right] . \end{aligned}$$

The control gains of the \( L_2 \) norm LPV-PID control are given as

$$\begin{aligned} \left[ \begin{array}{ccc} K_{P}&K_{I}&K_{D} \end{array} \right] =\left[ \begin{array}{cccc|c|cc} -83.1068 &{} -0.4452 &{} 0.0006 &{} 83.0968 &{} -0.0010 &{} -1.1243 &{} 4.0368 \\ 1064.6304 &{} 0.0105 &{} -0.1167 &{} -1042.0765 &{} 0.0056 &{} 60.2930 &{} -51.3934 \end{array} \right] . \end{aligned}$$

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Huang, B., Lu, B., Nagamune, R. et al. LMI-based linear parameter varying PID control design and its application to an aircraft control system. AS 5, 309–321 (2022). https://doi.org/10.1007/s42401-021-00104-y

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  • DOI: https://doi.org/10.1007/s42401-021-00104-y

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