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Observer-based interval type-2 fuzzy friction modeling and compensation control for steer-by-wire system

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Abstract

This paper studies the tracking control of the SbW system with unknown nonlinear friction torque and the unmeasured angular velocity. An observer-based adaptive interval type-2 fuzzy logic system controller is proposed to eliminate the adverse influence of the friction torque on the SbW system. Firstly, the angular velocity of the front wheels is estimated via the observer, such that the system sensitivity to measurement noise, the hardware cost, and the structural complexity are reduced. Then, an interval type-2 fuzzy logic system (IT2 FLS) is used to model the friction torque, in which the model and parameters are not effectively identified. IT2 FLS has a more exceptional ability to deal with uncertainties than the traditional type-1 fuzzy logic system (T1 FLS), so the friction modeling based on IT2 FLS has more satisfactory effect in practical application. Finally, an adaptive interval type-2 fuzzy logic system controller is proposed to achieve excellent tracking performance. The tracking error can be guaranteed to converge asymptotically to zero by the Lyapunov stability theory. The numerical simulations and hardware-in-loop (HIL) experiments verify the effectiveness and superiority of the proposed friction modeling method and control strategy.

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Abbreviations

\(\delta _{sm}\) :

Steering motor assembly rotational angle

\(\delta _{f}\) :

Front wheels’ steering angle

\(B_{sm}\) :

Viscous friction coefficient of steering motor assembly

\(B_{e}\) :

Viscous friction coefficient of equivalent system

\(J_{sm}\) :

Rotational inertia of steering motor assembly

\(J_{f}\) :

Rotational inertia of front wheels

\(J_{e}\) :

Rotational inertia of equivalent system

\(\tau _{sm}\) :

Control torque of steering motor assembly

\(\tau _{12}\) :

Load torque of steering motor assembly

\(\tau _{f}\) :

Driving torque of front wheels

\(\tau _{e}\) :

Self-aligning torque of front wheels

\(\tau _{fric}\) :

Friction torque of front wheels

k :

Steering motor assembly angle/front wheels angle

\(F_{c}\) :

Coulomb friction torque of front wheels

\(F_{s}\) :

Static friction torque of front wheels

\(v_{s}\) :

Stribeck velocity of front wheels

\(\sigma _{0}\) :

Stiffness coefficient of bristles

\(\sigma _{1}\) :

Damping coefficient of bristles

\(\sigma _{2}\) :

Viscous friction coefficient of front wheels

\(g(\dot{\delta _{f}})\) :

Stribeck effect of front wheels

SbW:

Steer-by-wire

T1 FS:

Type-1 fuzzy set

T2 FS:

Type-2 fuzzy set

IT2 FS:

Interval type-2 fuzzy set

IT2 FSs:

Interval type-2 fuzzy sets

KM:

Karnik-Mendel

SPR:

Strict positive real

FOU:

Footprint of uncertainty

UMF:

Upper membership function

LMF:

Lower membership function

MAE:

Maximum absolute error

RMSE:

Root mean square error

SD:

Standard deviation

GMS:

Generalized Maxwell-slip

RBFN:

Radial basis function network

FLS:

Fuzzy logic system

HIL:

hardware-in-loop

T1 FLS:

Type-1 fuzzy logic system

T1 FLSs:

Type-1 fuzzy logic systems

T2 FLSs:

Type-2 fuzzy logic systems

IT2 FLS:

Interval type-2 fuzzy logic system

IT2 FLSs:

Interval type-2 fuzzy logic systems

GT2 FLSs:

Generalized type-2 fuzzy logic systems

IT2-TSK-FLS:

Interval type-2 Takagi-Sugeno-Kang fuzzy logic system

AT1FLSC:

Adaptive Type-1 fuzzy logic system controller

AIT2FLSC:

Adaptive Interval type-2 fuzzy logic system controller NASTSM Nested Adaptive Super-twisting Sliding Mode

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 51775103, in part by the State Key Lab of Digital Manufacturing Equipment & Technology under Grant DMETKF2020015.

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, 110819, Shenyang, Liaoning, China

    Gang Luo, Bingxin Ma & Yongfu Wang

  2. School of Automation Science and Electrical Engineering, Beihang University, 100191, Beijing, China

    Zezheng Wang

  3. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, 430074, Wuhan, China

    Jianfeng Xu

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  1. Gang Luo
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Correspondence to Yongfu Wang.

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Luo, G., Wang, Z., Ma, B. et al. Observer-based interval type-2 fuzzy friction modeling and compensation control for steer-by-wire system. Neural Comput & Applic 33, 10429–10448 (2021). https://doi.org/10.1007/s00521-021-05801-5

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