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Robust adaptive sliding mode control design for uncertain stochastic systems with time-varying delay

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Abstract

Based on a least square wavelet kernel support vector machine (LS-SVM) and a hybrid wavelet kernel least square SVM, two adaptive integral sliding mode control schemes are presented for uncertain stochastic systems with time-varying delay. Adaptive parameter is chosen based on estimation of uncertain function and Swarm Optimization algorithm. First, an integral sliding surface is constructed. Using the linear matrix inequalities, a sufficient condition for the existence of sliding surface is then derived. This condition guarantees the global stochastic stability of stochastic dynamics in the specified switching surface. Next, LS-SVM is used to establish an adaptive controller. Also, an on-line learning rule for the weight vector and bias term are derived. Simulation results show that the proposed controllers can achieve a satisfactory performance.

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Abbreviations

\(x(t)\in R^{n}\) :

State vector

\(u(t)\in R^{m}\) :

Control input

w(t):

One-dimensional Brownian motion

\(\Delta A, \Delta A_d , \Delta C, \Delta C_d \) :

Unknown time-varying matrices representing system parameter uncertainties and uncertainties of stochastic perturbation

\(M_1 , M_2 , L_1 , L_2 , L_3 , L_4 \) :

Known real constant matrices with appropriate dimensions

\(F_1 (t), F_2 (t)\) :

Unknown real time-varying matrices with Lebesgue measurable elements

\(\varphi (t)\) :

Continuous vector valued initial condition

\(f\left( {x(t),t} \right) \in R^{n}\) :

An unknown nonlinear function

\(\gamma >0\) :

The known constant

\(\mu >0\) :

Regularization item

\(\alpha _i \) :

Lagrange multipliers

\(K(x_i ,x_j )\) :

Kernel function of SVM

\(\psi (z)\) :

Sinc wavelet

a :

Dilation factor

\(z, a\in R, t\) :

Translation factor

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Authors and Affiliations

  1. Department of Control Engineering, Faculty of Electrical and Computer Engineering, University of Tabriz, P.O. Box 5166753389, Tabriz, Iran

    Bahar Ahmadi & Hamid Nourisola

  2. Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, University of Sistan and Baluchestan, Zahedan, Iran

    Saeed Tavakoli

Authors
  1. Bahar Ahmadi
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  2. Hamid Nourisola
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  3. Saeed Tavakoli
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Correspondence to Saeed Tavakoli.

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Ahmadi, B., Nourisola, H. & Tavakoli, S. Robust adaptive sliding mode control design for uncertain stochastic systems with time-varying delay. Int. J. Dynam. Control 6, 413–424 (2018). https://doi.org/10.1007/s40435-016-0298-y

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  • DOI: https://doi.org/10.1007/s40435-016-0298-y

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