# American Institute of Mathematical Sciences

May  2020, 3(2): 81-99. doi: 10.3934/mfc.2020007

## A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertainty

 College of Engineering, Department of Electrical Engineering, TAIF University KSA, TAIF, KSA

* Corresponding author: NasimUllah

Received  December 2019 Revised  January 2020 Published  May 2020

This research work proposes a novel triple mode sliding mode controller for a nonlinear system with measurement noise and uncertainty. The proposed control has the following goals (1) it ensures the transient and steady state robustness of the system in closed loop (2) it reduces chattering in the control signal with measurement noise. Fuzzy system is used to tune the appropriate order of the fractional operators for the proposed control system. Depending on the tuned range of the fractional operators, the proposed controller can operate effectively in the following three modes (1) classical sliding mode (SMC) (2) fractional order sliding mode (FSMC) (3) Integral sliding mode control (ISMC). With the noisy feedback, the performance of the classical SMC and SMC with boundary layer degrades significantly while ISMC shows better performance. However ISMC exhibits large transient overshoots.The proposed control method optimally selects the appropriate mode of the controller to ensure performance(transient and steady state) and suppresses the effect of noisy feedback. The proposed scheme is derived for the permanent magnet synchronous motor, s (PMSM) speed regulation problem which is subject to uncertainties, measurement noise and un-modeled dynamics as a case study. The effectiveness of proposed scheme is verified using numerical simulations.

Citation: Nasim Ullah, Ahmad Aziz Al-Ahmadi. A triple mode robust sliding mode controller for a nonlinear system with measurement noise and uncertainty. Mathematical Foundations of Computing, 2020, 3 (2) : 81-99. doi: 10.3934/mfc.2020007
##### References:

show all references

##### References:
(a): variation of $\alpha$ with $|S_{3} |$ (b): Variation of $\beta$ with $|S_{3} |$
(a): Fuzzy sets of input variable $|S_{3} |$(b): Fuzzy sets of out variable $\alpha$(c): Fuzzy sets of out variable $\beta$(d): Variation of $\beta$ with $\alpha$
(a) speed regulation, (b) control signal, (c) sliding surface
Enlarged view of (a) speed regulation, (b) control signal, (c) sliding surface
(a) speed error, (b) control signal, (c) sliding surface with $D(X, u, t)$
Enlarged view of (a) speed error, (b) control signal, (c) sliding surface with $D(X, u, t)$
(a) speed error, (b) control signal, (c) sliding surface with $D(X, u, t)$ and measurement noise
Enlarged view of (a) speed error, (b) control signal, (c) sliding surface with $D(X, u, t)$ and measurement noise
Speed regulation (b) Speed error (c) Control signal (d) Sliding surface with with $D(X, u, t)$ and measurement noise
(a) Speed error (b) Control signal (c) Sliding surface with with $D(X, u, t)$ and measurement noise
(a) Adaptation of $\alpha$ (b) Adaptation of $\beta$
 [1] Yuan Li, Ruxia Zhang, Yi Zhang, Bo Yang. Sliding mode control for uncertain T-S fuzzy systems with input and state delays. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 345-354. doi: 10.3934/naco.2020006 [2] Cecilia Cavaterra, Denis Enăchescu, Gabriela Marinoschi. Sliding mode control of the Hodgkin–Huxley mathematical model. Evolution Equations & Control Theory, 2019, 8 (4) : 883-902. doi: 10.3934/eect.2019043 [3] Hao Sun, Shihua Li, Xuming Wang. Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1447-1464. doi: 10.3934/dcdss.2020375 [4] Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020051 [5] Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004 [6] Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619 [7] Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. Impulsive control of a symmetric ball rolling without sliding or spinning. Journal of Geometric Mechanics, 2010, 2 (4) : 321-342. doi: 10.3934/jgm.2010.2.321 [8] Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058 [9] George A. Anastassiou. Fractional Ostrowski-Sugeno Fuzzy univariate inequalities. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3305-3317. doi: 10.3934/dcdss.2020111 [10] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 [11] Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861 [12] Mehar Chand, Jyotindra C. Prajapati, Ebenezer Bonyah, Jatinder Kumar Bansal. Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 539-560. doi: 10.3934/dcdss.2020030 [13] Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417 [14] Jacky Cresson, Fernando Jiménez, Sina Ober-Blöbaum. Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021012 [15] Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183 [16] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3529-3539. doi: 10.3934/dcdss.2020432 [17] Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259 [18] Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457 [19] Peng Cheng, Yanqing Liu, Yanyan Yin, Song Wang, Feng Pan. Fuzzy event-triggered disturbance rejection control of nonlinear systems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020119 [20] Aliki D. Muradova, Georgios K. Tairidis, Georgios E. Stavroulakis. Adaptive Neuro-Fuzzy vibration control of a smart plate. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 251-271. doi: 10.3934/naco.2017017

Impact Factor:

## Tools

Article outline

Figures and Tables