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A robust training algorithm of discrete-time MIMO RNN and application in fault tolerant control of robotic system

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Abstract

In this paper, a novel robust training algorithm of multi-input multi-output recurrent neural network and its application in the fault tolerant control of a robotic system are investigated. The proposed scheme optimizes the gradient type training on basis of three new adaptive parameters, namely, dead-zone learning rate, hybrid learning rate, and normalization factor. The adaptive dead-zone learning rate is employed to improve the steady state response. The normalization factor is used to maximize the gradient depth in the training, so as to improve the transient response. The hybrid learning rate switches the training between the back-propagation and the real-time recurrent learning mode, such that the training is robust stable. The weight convergence and L 2 stability of the algorithm are proved via Lyapunov function and the Cluett’s law, respectively. Based upon the theoretical results, we carry out simulation studies of a two-link robot arm position tracking control system. A computed torque controller is designed to provide a specified closed-loop performance in a fault-free condition, and then the RNN compensator and the robust training algorithm are employed to recover the performance in case that fault occurs. Comparisons are given to demonstrate the advantages of the control method and the proposed training algorithm.

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Abbreviations

\({\hat{V}}(k), {\hat{W}}(k)\) :

Estimated weights of RNN’s output layer and hidden layer, respectively

V*(k), W*(k):

Optimal weights of RNN’s output layer and hidden layer, respectively

\({\hat{y}}(k), {\hat{x}}(k)\) :

Output and state vector of RNN, respectively

\(\Upphi(\cdot), f(e(k))\) :

Activation function and cost function of RNN, respectively

αv(k), αw(k):

Learning rates of the robust adaptive gradient training, respectively

βv(k), βw(k):

Hybrid learning rates of the robust adaptive gradient training, respectively

ρv(k), ρw(k):

Normalization factors of the robust adaptive gradient training, respectively

e(k), ev(k), ew(k):

Training and estimation errors of RNN, respectively

e r (k):

Tracking error of the robot control system

A(k), B(k):

Jacobian matrix of the error gradient of RNN

H1, H2:

Feedforward and feedback operator in the closed-loop systems, respectively

θ d (k), θ(k):

Command and actual joint angle of the robot, respectively

K p , K v :

Diagonal feedback gain of joint angle positions and velocities, respectively

τ0(k):

Computed torque

τ c (k):

Compensation torque contributed by the RNN

M(θ(k)):

Inertia matrix of the robot

C(θ(k − 1), θ(k − 2)):

Coriolis/centripetal torque matrix of the robot

G(θ(k)):

Gravity vector of the robot

D(θ(k), θ(k − 1)):

Disturbance including the noise, static, and dynamic frictions

F(θ(k), θ(k − 1)):

Failure function of the robot induced by model uncertainty

k :

Sampling index

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Acknowledgements

The author would like to appreciate the valuable comments of the anonymous reviewer, which helped to improve the quality of this paper greatly.

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Authors and Affiliations

  1. School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798, Singapore

    Yilei Wu, Jinchuan Zheng & Qing Song

  2. Department of Computer Science and Technology, State Key Laboratory of Intelligent Technology and Systems, Tsinghua University, 100084, Beijing, China

    Fuchun Sun

Authors
  1. Yilei Wu
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  2. Fuchun Sun
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  3. Jinchuan Zheng
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  4. Qing Song
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Corresponding author

Correspondence to Yilei Wu.

Additional information

This work was supported by the National Science Fund for Distinguished Young Scholars (Grant No: 60625304).

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Wu, Y., Sun, F., Zheng, J. et al. A robust training algorithm of discrete-time MIMO RNN and application in fault tolerant control of robotic system. Neural Comput & Applic 19, 1013–1027 (2010). https://doi.org/10.1007/s00521-010-0343-2

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