Position and Speed Tracking of DC Motor Based on Experimental Analysis in LabVIEW ^†

Abstract

DC motors are widely used in industries to provide mechanical power in speed and torque. The position and speed control of DC motors is receiving interest from the scientific community in robotics, especially in robotic arms, a flexible joint manipulator. The current research work is based on the position control of DC motors using experimental investigations in LabVIEW. The linear control strategy is applied to track the position and speed of the DC motor with comparative analysis in the LabVIEW platform and simulation analysis in MATLAB. The tracking error in the hardware setup based on LabVIEW programming is slightly greater than the simulation analysis in MATLAB due to the inertial load of the motor during steady-state conditions. The controller output shows the input voltage applied to the DC motor varies between 0 and 8 V to ensure minimal steady error while tracking the position and speed of the DC motor.

1. Introduction

In the current technological era, the design and development of high-performance motor drives is increasing exponentially due to their applications in electric trains, robotics, DC appliances, biomedical equipment and other industrial applications. Generally, the high-performance motor drive system is based on proficient speed command tracking and a load regulating system to perform any task in industry [1,2,3]. DC drives are normally less expensive and provide excellent control of speed for acceleration and deceleration compared to AC motors. The most common emerging applications of DC motors are robotic and electrical equipment. Therefore, the speed and position control of DC motors has been investigated in the last few decades. One challenging aspect of the DC motor when designing a control system is the uncertain and nonlinear characteristic which degrades the performance of controllers [4,5]. In addition to industrial applications, DC motors are extensively used in various serve devices like flight simulators, optoelectronic tracking platforms and missile electromechanical actuators. However, various uncertainties exist in motor serve systems including modeling errors, torque disturbances and parametric perturbations hindering its performance and efficacy [6,7,8]. The disturbance torques can be quite large for cheap DC motors and can cause performance issues. The conventional proportional integral derivative (PID) controller may not be very effective for eliminating the speed ripples of motors with small inertia due to the periodic nature of the disturbance torque [9,10,11,12,13].

The robust control method based on the disturbance observer shows a strong inhibitory effect against various external disturbances and parametric variations by improving the control performance of the system. The high-precision serve system relies on such robust control strategies. The sliding mode control variable structure control (SMVSC) is a nonlinear control strategy exhibiting strong robustness against external disturbances and parametric variations [14,15,16,17,18,19,20]. Besides this, the global sliding mode control reflects the system trajectory on the sliding surface by constructing nonlinear switching function and later eliminates the initial reaching function [21,22,23]. The current research is based on the position and speed control of the DC motor for undergraduate students to investigate and compare the performance of DC motors using hardware setups based on LabVIEW programming and simulation analysis in MATLAB. The mathematical model of the DC motor is derived using physics and electromagnetism laws. The PID controller is designed to track the position of the DC motor. The performance of the PID controller is investigated while tracking the position and speed during steady-state conditions. It is observed that when using hardware based on experimental setup, the tracking error is slightly greater compared to the simulation analysis in MATLAB due to the inertial load of the DC motor during the switching phase of the pulse generator signal. The controller gains and specification of the input reference signal are the same for both analyses. The rest of the paper is organized as follows: Section 2 describes the mathematical model DC motor; Section 3 works on the control scheme design using a PID controller; Section 4 discusses the simulation results in MATLB and the hardware implementation results in LabVIEW; Section 5 provides the conclusions.

2. Mathematical Model of DC Motor

The DC motor is an electromechanical device that provides displacement at the output side for various voltage levels at the input. The mathematical model of an armature-controlled DC motor was derived by Mablekos in 1980. Figure 1 shows the permanent magnets that generate the magnetic field referred to as the fixed field. The armature is a rotating part through the current; $i_A(t)$ is the flow, and the experienced force at $90°$ can be stated as $F=ILB\sin \theta$ for $\theta =90°, F=ILB$, while $L$ is the length of the conductor, and $B$ is the magnetic field strength. The resulting torque occurs due to the force experienced by the armature, which causes the armature to rotate. The $e_{indcued}=vBL$, $v$ is the velocity of the conductor, $e_{indcued}$ is the induced voltage that is proportional to the angular speed of the armature in the magnetic field. The induced voltage can be stated as $V_b=K_b\frac{d{\theta }_m(t)}{dt}=K_{b}s{\theta }_m\left(s\right).$ Applying $KVL$ to the closed-loop circuit is shown in Figure 1 as

$$E\left(t\right)=i_a\left(t\right)R_a+L_a\frac{d{i}_a}{dt}+V_b\left(t\right)$$

(1)

Applying the Laplace transform to Equation (1) as in Equation (2),

$$E\left(s\right)=I_a\left(s\right){(R}_a+L_{a}s{I}_a\left(s\right)+V_b\left(s\right)$$

(2)

Putting the value of $V_b=K_b\frac{d{\theta }_m(t)}{dt}=K_{b}s{\theta }_m\left(s\right)$ in Equation (2) as in Equation (3),

$$E\left(s\right)=I_a\left(s\right){(R}_a+L_{a}s)+K_{b}s{\theta }_m(s)$$

(3)

The torque ${\mathsf{{\rm T}}}_{\mathrm{m}}(\mathrm{s})$ developed by the motor is proportional to the armature current $I_a\left(s\right)$, which can be expressed as ${\mathsf{{\rm T}}}_{\mathrm{m}}(\mathrm{s})={\mathrm{K}}_{\mathrm{t}}{I}_a\left(s\right)$ or $I_a\left(s\right)=\frac{{\mathsf{{\rm T}}}_{\mathrm{m}}(\mathrm{s})}{{\mathrm{K}}_{\mathrm{t}}}$. Then, the value of $I_a\left(s\right)$ is put in Equation (3) as in Equation (4):

$$E\left(s\right)=\frac{{\mathsf{{\rm T}}}_{\mathrm{m}}(\mathrm{s})}{{\mathrm{K}}_{\mathrm{t}}}{(R}_a+L_{a}s)+K_{b}s{\theta }_m(s)$$

(4)

Expressing the torque in terms of ${\theta }_m(s)$, $J_m$ is the moment of inertia of the armature $J_a$ and load inertia for $J_{load}$ for the QENT DC motor $J_m=J_a+J_{load}$. This can be expressed as ${\mathsf{{\rm T}}}_{\mathrm{m}}\left(\mathrm{s}\right)={(\mathrm{J}}_{\mathrm{m}}{\mathrm{s}}^2+{\mathrm{D}}_{\mathrm{m}}\mathrm{s}){\theta }_m(s)$ and its value is incorporated into Equation (4) as in Equation (5):

$$E\left(s\right)=\frac{{(\mathrm{J}}_{\mathrm{m}}{\mathrm{s}}^2+{\mathrm{D}}_{\mathrm{m}}\mathrm{s}){\theta }_m(s)}{{\mathrm{K}}_{\mathrm{t}}}(R_a+L_{a}s)+K_{b}s{\theta }_m\left(s\right)$$

(5)

Assuming that the armature $R_a>>L_a$ as $L_a$ can be neglected in Equation (5) as in Equation (6),

$$E\left(s\right)=\frac{{(\mathrm{J}}_{\mathrm{m}}{\mathrm{s}}^2+{\mathrm{D}}_{\mathrm{m}}\mathrm{s}){\theta }_m(s)}{{\mathrm{K}}_{\mathrm{t}}}\times R_a+K_{b}s{\theta }_m\left(s\right)$$

(6)

Simplifying Equation (6) as in Equation (7),

$$\frac{{\theta }_m\left(s\right)}{E\left(s\right)}=\frac{\raisebox{1ex}{$K_t$}\!\left/ \!\raisebox{-1ex}{$R_a$}\right.}{s(J_{m}s+\frac{K_t{K}_b}{R_a})}$$

(7)

The physical parameters of DC motor is shown in

Table 1. Physical parameter of the DC motor.

Parameter Name	Description	Unit
$R_a$	Terminal resistance	$8.4 \mathsf{\Omega }$
$K_t$	Torque constant	0.042 Nm/A
$K_b$	Motor back-emf constant	0.042 V/(rad/s)
$J_m$	Rotor inertia	4.0 × 10−6 kg m2
$J_l$	Load inertia	0.6 × 10−6 kg m2
$L_m$	Rotor inductance	1.16 mH
$m_l$	Estimated damping of the pivot	0.016 kg
$m_d$	Disk mass	0.053 kg

with their description and unit values.

3. PID Controller Design for Position Tracking of DC Motor

Figure 2 shows the cascaded control strategy based on a PID system to reflect the ${\theta }_m$ to ${\theta }_{ref}$ in a closed-loop scheme.

The controller input based on a PID controller can be defined as in Equation (8) [24,25].

$$E(s)=k_p{(\theta }_{ref}-{\theta }_m(t))+k_i\int {(\theta }_{ref}-{\theta }_m(t))dt+k_v\frac{d}{dt}({\theta }_{ref}-{\theta }_m(t))$$

(8)

Simplifying and taking the Laplace transform $\mathcal{L}$ of Equation (8) as in Equation (9),

$$E(s)=k_p{\theta }_{ref}(s)-k_p{\theta }_m\left(s\right)+\frac{k_i}{s}{\theta }_{ref}(s)-\frac{k_i}{s}{\theta }_m(s)-k_{v}s{\theta }_m(s)$$

(9)

By substituting the value of $E\left(s\right)$ as the input to the plant from Equation (9) in Equation (10),

$$\frac{{\theta }_m\left(s\right)}{1086.96}\left(s^2+45.65s\right)=k_p{\theta }_{ref}\left(s\right)-k_p{\theta }_m\left(s\right)+\frac{k_i}{s}{\theta }_{ref}\left(s\right)-\frac{k_i}{s}{\theta }_m\left(s\right)-k_{v}s{\theta }_m\left(s\right)$$

(10)

Simplifying and rearranging Equation (10) as in Equation (11),

$$\frac{{\theta }_m\left(s\right)}{{\theta }_{ref}}=\frac{1086.96(k_{p}s+k_i)}{s^3+s^2\left(45.65+{1086.96k}_v\right)+{1086.96k}_{p}s+{1086.96k}_i}$$

(11)

The third-order characteristic polynomial equation can be defined as in Equation (12).

$${(s}^2+2\zeta {\omega }_{n}s+{\omega }_n^2)\left(s+p_o\right)=s^3+\left(2\zeta {\omega }_n{+p}_o\right)s^2+{(\omega }_n^2+2\zeta {\omega }_n{p}_o)s+{\omega }_n^2{p}_o$$

(12)

Then, compare the denominator of Equations (11) and (12) to compute the $k_p$, $k_i$ and $k_v$ gains of the PID controller for $\zeta =1$ and ${\omega }_n=30 \mathrm{rad}/\sec$ and $p_o$ = 1 as $0.883\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$deg$}\right., 0.0132\raisebox{1ex}{$deg$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \mathrm{a}\mathrm{n}\mathrm{d} 0.827\frac{A}{\raisebox{1ex}{$deg$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}$, respectively.

4. Simulations and Experimental Hardware Results in MATLAB and LabVIEW

The investigated results achieved based on the simulation results in MATLAB and the experimental analysis using LabVIEW are identified in Section 4. Figure 3a shows the position control of the DC motor based on a PID controller while using the gains values $k_p$, $k_v$ and $k_i$ shown in Equation (12). The actual position ${\theta }_m\left(s\right)$ accurately tracks the ${\theta }_{ref}$ with a tracking error of less than 5% in the MATLAB simulations. The input to the DC motor is shown in Figure 3b as the input voltage level $E(s)$ that varies from $-4 \mathrm{V} \mathrm{to} 4 \mathrm{V}$. Figure 3a,b shows that, as the position of the DC motor is varies, the input voltage level also varies between two peaks. The Figure 3a shows that tracking error is quite negligible using MATLAB simulation based on PID controller.

Figure 4a shows the position control of the DC motor based on the hardware platform using LabVIEW. The ${\theta }_m\left(s\right)$ shows the measured position of the DC motor tracking the ${\theta }_{ref}$ with a significant steady-state error due to the inertial load of the motor during the switching of the pulse generator signal as a reference position level. Figure 4b shows that the controller output $E\left(s\right)$ is voltage applied as the input to the DC motor. The controller output $E\left(s\right)$ varies from $-4 \mathrm{V} \mathrm{to} 4 \mathrm{V}$ as the position of the DC motor changes.

Figure 5a shows the speed control of the DC motor based on a PID controller. The ${\omega }_m$ is the actual speed of the DC motor as it tracks the ${\omega }_{ref}$ with minor deviations in tracking error during the switching of the square wave or pulse generator signal. Figure 5b shows the input voltage level being provided to the DC motor as the controller output signal. The amplitude of the input voltage $E\left(s\right)$ to the DC motor has a variation of $\left(0-8\right) \mathrm{V}$.

5. Conclusions

This paper investigated the position and speed control of the DC motor based on the PID controller and achieved a nominal performance with a steady-state error of less than 5% in a simulation based on MATLAB and experimental analysis using the LabVIEW platform. It is observed that the inertial load impacts the tracking performance of the linear controllers. The speed control of the DC motor has an eminent flexible joint manipulator that can be further extended while handling various perturbations including torque disturbances as well.