Uncertain nonlinear system control using hybrid fuzzy LQRsliding mode technique optimized with evolutionary algorithm

Uncertain nonlinear system

control using hybrid fuzzy LQR￾sliding mode technique optimized

with evolutionary algorithm

Nguyen Ngoc Son

Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam

Cao Van Kien

Faculty of Electronics Technology, Industrial University of Ho Chi Minh City,

Ho Chi Minh City, Vietnam, and

Ho Pham Huy Anh

Faculty of Electrical-Electronics Engineering (FEEE),

Ho Chi Minh City University of Technology, VNU-HCM, Vietnam

Abstract

Purpose – This paper aims to propose an advanced tracking control of the uncertain nonlinear dynamic

system using a novel hybrid fuzzy linear quadratic regulator (LQR)-proportional-integral-derivative (PID)

sliding mode control (SMC) optimized by differential evolution (DE) algorithm.

Design/methodology/approach – First, a swing-up and balancing control is presented for an

experimental uncertain nonlinear Pendubot system perturbed with friction. The DE-based optimal SMC

scheme is used to optimally swing up the Pendubot system to the top equilibrium position. Then the novel

hybrid fuzzy-based on LQR fusion function and PID controller optimized by DE algorithm is innovatively

applied for balancing and control the position of the first link of the Pendubot in the down-right position with

tracking sinusoidal signal reference.

Findings – Experimental results demonstrate the robustness and effectiveness of the proposed approach in

balancing control for an uncertain nonlinear Pendubot system perturbed with internal friction.

Originality/value – This manuscript is an original research paper and has never been submitted to any

other journal.

Keywords Differential evolution (DE) algorithm, Optimal hybrid PID-fuzzy sliding mode control,

Uncertain nonlinear pendubot system perturbed with friction

Paper type Research paper

1. Introduction

The Pendubot system has fewer actuators than the degrees of freedom to be controlled

(Spong and Block, 1995). The Pendubot system is under-actuated as the angular acceleration

of the second link cannot be controlled directly. The study of Pendubot will facilitate further

research for more complicated under-actuated systems such as space robots, walking robots

This research is totally funded by National Foundation for Science and Technology Development

(NAFOSTED) under grant 107.01-2018.10, Vietnam.

Conflicts of Interest. The authors declare no conflict of interest.

Hybrid fuzzy

LQR-sliding

mode

technique

1893

Received 9 August 2018

Revised 23 March 2019

Accepted 25 March 2019

Engineering Computations

Vol. 36 No. 6, 2019

pp. 1893-1912

© Emerald Publishing Limited

0264-4401

DOI 10.1108/EC-08-2018-0356

The current issue and full text archive of this journal is available on Emerald Insight at:

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and underwater robots. Under-actuated mechanical systems have their own difficulties

within the control criterion. The control problems related to high nonlinearity, model

inaccuracy and uncertainties.

The principal control task of the Pendubot concerns the problem of swing-up two links

approximately into the balancing location and then keeping the Pendubot system stabilized

at the upright position. Paper (Mahindrakar and Banavar, 2006) introduced the control law

to swing-up the Pendubot from an initial domain to a neighborhood of the vertically upward

equilibrium position and then keeping the Pendubot system stabilized at the upright

position using a linear feedback controller based on linear quadratic regulator (LQR)

controller. Paper (Fantoni et al., 2000) proposed a controller for swinging the first joint of

Pendubot to a near position of the unstable balancing point. The balancing control is based

on an energy approach and the passivity properties of the system. In paper (Prasad et al.,

2014), the authors presented the modeling and simulation for optimal control design of

nonlinear inverted pendulum dynamic system using PID controller and LQR method, which

has been presented for both cases of without and with disturbance input. Freidovich et al.

(2008) introduced the novel virtual holonomic method, as to produce rhythmic motions for

the Pendubot system. In paper (Albahkali et al., 2009), linearization of the dynamics of the

Pendubot about this equilibrium results in a completely controllable system and allows a

linear controller to be designed for local asymptotic stability. A swing-up control was

implemented based on a series of rest-to-rest maneuvers of the first link about its vertically

upright configuration. Papers (Yoo, 2013; Xiong et al., 2016) introduced a sliding mode

control method to stabilize a class of underactuated systems. In paper (Xin and Juuri, 2012),

the authors showed the design of a strongly stabilizing controller for the upright equilibrium

point of the Pendubot by designing a specific output and using the pole-zero relation of the

linearized model around the equilibrium point. Nevertheless, the main drawback from these

papers above-mentioned is that almost is related to the Pendubot stability and control

performance meanwhile it needs to further investigate the Pendubot system subjected to

external noise and disturbances. Moreover, the control performance of Pendubot is strongly

depending on the highly nonlinearity and un-modeled dynamics of the system.

To solve this problem, the intelligent control has been considered as an effective strategy

for the control of the Pendubot system concerning external noise and uncertainties. Recently,

fuzzy control has become an alternative to conventional control algorithms to deal with

complex processes and efficiently combine the advantages of classical controllers. Paper

(Begovich et al., 2002) presented a combining the linear regulator approach with Takagi–

Sugeno (T–S) fuzzy methodology to achieve trajectory tracking for the Pendubot system.

Papers (Li et al., 2004; Meda-Campaña et al., 2015) designed a fuzzy controller for keeping the

first link swinging periodically while the second link maintains standing vertically. The

controller was applied on an under-actuated system known as “Pendubot” and the results are

compared with a stabilizer designed by linear matrix inequalitys. In paper (Xia et al., 2014),

the energy-based controller incorporated with fuzzy neural network compensator (ECFNNC)

was designed. Simulations and experimental results of the Pendubot actuated by a DC servo

motor were given to test the ECFNNC controller. Azimi and Koofigar (2015) proposed an

adaptive fuzzy back-stepping controller to control of the under-actuated systems, ensuring

the robustness against uncertainties and disturbances. However, a suitable choice of control

variables was considered in the fuzzy control design of these papers. While, a trial-and-error

method often used in building a fuzzy rule for controlling a nonlinear system and cannot

guarantee that the proposed fuzzy control system obtained a good performance.

To avoid the trial-and-error method, many researchers have successfully used evolutionary

algorithms to build the fuzzy rule. For examples, papers (Gorzałczany and Rudzinski, 2016
;

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