Adaptive sliding-mode observer for second order discrete-time MIMO nonlinear systems based on recurrent neural-networks

  • Iván SalgadoEmail author
  • Hafiz Ahmed
  • Oscar Camacho
  • Isaac Chairez
Original Article


This manuscript introduces a novel methodology to solve the state estimation of discrete-time multi-input multi-output (MIMO) nonlinear systems with uncertain dynamics. The mathematical model of the nonlinear systems considered in this paper satisfies the usual Lagrangian structure that characterizes many mechanical, electrical or electromechanical systems. A recurrent neural network (RNN) estimates the uncertain dynamics of the MIMO system with an updating law based on a particular variant of the discrete-time version of the super-twisting algorithm (DSTA). A Lyapunov stability analysis defines the convergence zone for the state estimation error throughout the solution of a matrix inequality. The convergence zone for the estimation is smaller when the DSTA and the RNN work together in an observer. Numerical examples demonstrate how the adaptive observer reduces the zone of convergence and the oscillations in the steady state compared with a discrete version of the STA with additional linear correcting terms. An experimental implementation shows how the observer estimates the unknown states of a Van Der Pol Oscillator. A comparison against some variations of the DSTA justifies the advantages of the mixed DSTA-RNN observer.


State estimation Lyapunov theory Sliding modes Recurrent neural networks Discrete-time super twisting algorithm Second order systems 



The authors thank the economical support provided by Instituto Politécnico Nacional through the Research Grants labeled SIP-20181009, SIP-2018043 and SIP-20180330.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.UPIBI-Instituto Politécnico NacionalMexico CityMexico
  2. 2.CIDETEC-Instituto Politécnico NacionalMexico CityMexico
  3. 3.Coventry UniversityCoventryUK

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