Robust nonlinear fractional order fuzzy PD plus fuzzy I controller applied to robotic manipulator

Abstract

The aim of this article is to utilize fractional calculus for performance enhancement of nonlinear fuzzy PD + I controller. A fractional order fuzzy PD + I controller (FOFPD + I) is designed and implemented to control complex, uncertain and nonlinear robotic manipulator. FOFPD + I controller is derived from fractional order PD and fractional order I controller. The proposed control strategy has an adaptive capability due to its nonlinear gains and preserves the linear structure of fractional order PD + I controller. Further, integer-order fuzzy PD + I controller (FPD + I) and conventional PID controllers are also designed for comparative analysis. The optimum parameter values of FOFPD + I, FPD + I and PID controllers are obtained using non-dominated sorting genetic algorithm-II. The effectiveness of proposed controller is examined for reference tracking and disturbance rejection problems of robotic manipulator. The designed controllers are also validated experimentally on DC servomotor. Simulation and experimental results prove the superiority of FOFPD + I controller as compared to its integer-order equivalent and conventional PID controllers for control of robotic manipulator.

Appendix

$\tau_{1}$	Torque for link-1	$T$	Sampling time	$o.p$	Output positive
$\tau_{2}$	Torque for link-2	$K_{\text{P}}$	Proportional gain for FOFPD controller	$o.n$	Output negative
$\theta_{1}$	Actuator angle measured from $X - a$x is to AB	$K_{\text{D}}$	Derivative gain for FOFPD controller	$o.z$	Output zero
$\theta_{2}$	Actuator angle measured from extended line of $AB$ to the line $BC$	$K_{\text{I}}$	Integral gain for FOFI controller	$w_{1}$	Objective-1
$l_{1}$	Length of link-1, $AB$	$\grave{u}_{\text{PD}} \left( {nT} \right)$	FOFPD controller output	$w_{2}$	Objective-2
$l_{2}$	Length of link-2, $BC$	$\Delta u_{\text{I}} \left( {nT} \right)$	FOFI controller output	$R_{\text{T}}$	Reference trajectory
$m_{1}$	Mass of link-1	$D^{\lambda }$	Fractional operator	$K_{{u{\text{PD}}}}$	Output gain for FOFPD controller
$m_{2}$	Mass of link-2	$r.p$	Fractional rate of error positive	$K_{\text{uI}}$	Output gain for FOFI controller
$K_{\text{p}}^{\text{PD}}$	Proportional gain for conventional FOPD controller	$r.n$	Fractional rate of error negative	$\lambda$	Fractional order
$K_{\text{D}}^{\text{PD}}$	Derivative gain for conventional FOPD controller	$e.p$	Error positive	$g$	Acceleration due to gravity
$K_{\text{I}}^{\text{I}}$	Integral gain for conventional FOI controller	$e.n$	Error negative