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\(H_\infty \) tracking adaptive fuzzy integral sliding mode control for a train of self-balancing vehicles

  • Mansour KarkoubEmail author
  • Chien-Chih Weng
  • Tzu-Sung Wu
  • Wen-Shyong Yu
  • Ming-Guo Her
Article
  • 65 Downloads

Abstract

A series of self-balancing vehicles is stabilized and controlled in this research work. The system is composed of several Segway \(\frac{{\mathrm{TM}}}{}\) type platforms interconnected through flexible links. The system is inherently highly nonlinear and underactuated which makes the tracking task very challenging. In this paper, \(H_\infty \) tracking adaptive fuzzy integral sliding mode control scheme is proposed for n-self-balancing interconnected vehicles system where uncertainties and disturbances are included. First, a nonlinear dynamic model with uncertainties for the train system with n-vehicles is derived using the Lagrangian method assuming the vehicles moving in tandem on a inclined path. Then, the dynamics of the train system with n-vehicles is formulated as an error dynamics according to a specified reference signal. A fuzzy technique with an on-line estimation scheme is developed to approximate the dynamics of the train system with n-vehicles. The advantage of employing an adaptive fuzzy system is the use of linear analytical results instead of estimating nonlinear uncertain functions in dynamics with an online update law. Using the concept of parallel distributed compensation, the adaptive fuzzy scheme combined with the integral sliding mode control scheme is synthesized to address the system uncertainties and the external disturbances such that \(H_\infty \) tracking performance is achieved. Simulation results for 2-self-balancing interconnected vehicles system are presented to show the effectiveness and performance of the proposed control scheme.

Keywords

Underactuated system Self-balancing vehicles Adaptive fuzzy control Integral sliding mode control \(H_\infty \) tracking performance Riccati-like equation Lyapunov stability theorem 

List of symbols

\(M_{w_{_{i}}}\)

Mass of the wheel of the ith self-balancing vehicle

\(M_{c_{_{i}}}\)

Mass of the body of the ith self-balancing vehicle

\(x_{w_{_{i}}}\)

Lateral displacement of the wheel of the ith self-balancing vehicle

\(x_{c_{_{i}}}\)

Lateral displacement of the cabin of the ith self-balancing vehicle

\(y_{w_{_{i}}}\)

Vertical displacement of the wheel of the ith self-balancing vehicle

\(y_{c_{_{i}}}\)

Vertical displacement of the cabin of the ith self-balancing vehicle

\(J_{w_{_{i}}}\)

Moment of inertia of the wheel of the ith self-balancing vehicle

\(J_{c_{_{i}}}\)

Moment of inertia of the cabin of the ith self-balancing vehicle

\(J_{m_{_{i}}}\)

Moment of inertia of the motor of the ith self-balancing vehicle

\(u_{_{i}}\)

Motor torque output of the ith self-balancing vehicle

\(\theta _{w_{_{i}}}\)

Angle of rotation of the wheel of the ith self-balancing vehicle

\(\delta _{c_{_{i}}}\)

The deviation angle of the cabin of the ith self-balancing vehicle

\(\psi _{c_{_{i}}}\)

Angle of rotation of the cabin of the ith self-balancing vehicle

\(r_{_{w}}\)

Outer radius of the wheel of the self-balancing vehicle

r

Inner radius of the wheel of the self-balancing vehicle

L

Distance of the c.g. from the axis of rotation

\(\alpha \)

Road inclination

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Mansour Karkoub
    • 1
    Email author
  • Chien-Chih Weng
    • 1
  • Tzu-Sung Wu
    • 1
  • Wen-Shyong Yu
    • 2
  • Ming-Guo Her
    • 3
  1. 1.Texas A & M UniversityDohaQatar
  2. 2.Department of Electrical EngineeringTatung UniversityTaipeiTaiwan
  3. 3.Department of Mechanical EngineeringTatung UniversityTaipeiTaiwan

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