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Multi-Objective Optimal Feedback Controls for Under-Actuated Dynamical System

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Abstract

This paper presents a study of optimal control design for a single-inverted pendulum (SIP) system with the multi-objective particle swarm optimization (MOPSO) algorithm. The proportional derivative (PD) control algorithm is utilized to control the system. Since the SIP system is nonlinear and the output (the pendulum angle) cannot be directly controlled (it is under-actuated), the PD control gains are not tuned with classical approaches. In this work, the MOPSO method is used to obtain the best PD gains. The use of multi-objective optimization algorithm allows the control design of the system without the need of linearization, which is not provided by using classical methods. The multi-objective optimal control design of the nonlinear system involves four design parameters (PD gains) and six objective functions (time-domain performance indices). The Hausdorff distances of consecutive Pareto sets, obtained in the MOPSO iterations, are computed to check the convergence of the MOPSO algorithm. The MOPSO algorithm finds the Pareto set and the Pareto front efficiently. Numerical simulations and experiments of the rotary inverted pendulum system are done to verify this design technique. Numerical and experimental results show that the multi-objective optimal controls offer a wide range of choices including the ones that have comparable performances to the linear quadratic regulator (LQR) control.

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Abbreviations

B arm :

Viscous damping coefficient of arm

B p :

Viscous damping coefficient of pendulum

g :

Gravity constant

l arm :

Rotary arm length from pivot to center of mass

l p :

Distance from pivot to center of mass

L p :

Full length of pendulum

J arm :

Rotary arm moment of inertia about pivot

J p :

Pendulum moment of inertia about pivot

K g :

Total gearbox ratio

K m :

Back electromotive force constant

K t :

Motor torque constant

marm,mp :

Rotary arm mass, pendulum mass

r :

Full length of rotary arm

R m :

Motor armature resistance

V m :

Motor armature voltage

α :

Angle of the pendulum

ζf :

Damping ratio of the digital differentiator

ηg,ηm :

Gearbox efficiency, motor efficiency

θ :

Angle of the rotary arm

ω cf :

Cutoff frequency of the digital differentiator

ω max :

Motor maximum speed

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Authors and Affiliations

  1. School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo, 255000, Shandong, China

    Zhichang Qin  (秦志昌)

  2. School of Mechanical Engineering, Tianjin University of Technology, Tianjin, 300384, China

    Ying Xin  (辛 英)

  3. Tianjin Key Laboratory for Advanced Mechatronic System Design and Intelligent Control, Tianjin University of Technology, Tianjin, 300384, China

    Ying Xin  (辛 英)

  4. School of Engineering, University of California, Merced, CA, 95343, USA

    Jianqiao Sun  (孙建桥)

Authors
  1. Zhichang Qin  (秦志昌)
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  2. Ying Xin  (辛 英)
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  3. Jianqiao Sun  (孙建桥)
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Corresponding author

Correspondence to Zhichang Qin  (秦志昌).

Additional information

Foundation item: the National Natural Science Foundation of China (Nos. 11572215 and 11702162), and the Natural Science Foundation of Shandong Province (No. ZR2018LA009)

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Qin, Z., Xin, Y. & Sun, J. Multi-Objective Optimal Feedback Controls for Under-Actuated Dynamical System. J. Shanghai Jiaotong Univ. (Sci.) 25, 545–552 (2020). https://doi.org/10.1007/s12204-020-2211-2

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  • DOI: https://doi.org/10.1007/s12204-020-2211-2

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