Characterization of Modified PID Controllers to Improve the Response Time of Angular Velocity to the Voltage Step in DC Motors by Means of a Mathematical Model Diagram Using SIMULINK Software

The objective of this research was the characterization of modified PID controllers to improve the response time of direct current (DC) motors. The Degem System EB-109 experimentation module (includes DC motor and six-pole pair synchronous motor), Tektronix TDS 2012 oscilloscope, Fluke 175 multimeter and a simulation environment for block diagrams with Simulink software were used. The procedure began by making voltage and current measurements in the armature of the DC motor in order to calculate its mechanical-electrical constants and obtain its mathematical model. Then, using block diagrams, a closed-loop control system was designed for the DC motor, using the PID as a control block in a first phase, and then in a second phase, the modified PID. The system was analyzed with simulation software and the results of both phases were compared. With the modified PID controller, a better speed and stability response time was obtained, thus eliminating transient peaks.


Introduction
Technological innovation in any energy parameter is of vital importance to achieve better performance in the equipment and machines of the production processes [1]; [2]; [3]. Direct current (DC) motors, due to their high starting torque, are used as electric drivers in a variety of activities both in the industrial and residential sectors, having the ability to control and regulate their speed with precision [4]; [5]; [6]. The industrial sector has been growing sustainably through time and automatic control techniques, complying with the requirement and functionality, improving safety, production costs and obtaining efficient results [7]; [8]; [9].
Electric motors are used for a wide variety of residential, commercial and industrial operations, for this reason analysis of the operating parameters of DC motors is performed to improve starting speed response [10].
DC motor feedback systems require electromechanical sensors in control system configurations such as PID, PI and others in order to determine the position of the rotor and calculate an important parameter such as speed to achieve an improvement in performance and efficiency [11]; [12]; [13]. The correct selection and adjustment of the PI and PID controllers allows to obtain a higher performance as a function of the proportional gain parameters Kp and the integral time (Ti) [14]. The classic PID control is an accessible option due to its possibility of regulation of the position and speed immediately and as the most common alternative [15]; [16]. In addition, its analog implementation is simple and does not require large costs in components, while digital controllers integrate hardwaresoftware that can be private or "open source", they are also very useful in the automation of industrial processes [17] cited by [18]; [19].
Three forms of control intervene in the PID configuration: Proportional control (P) that calculates a value proportional to the current error (difference between the reference value and the feedback value); the integral control (I) that takes the data of the past error and integrates it in time, and the derivative (D) that calculates the derivative of the error in the current time by drawing a projection of the future error. Consequently, in the implementation of a PID controller it will be required to use regulation constants Kp, Ki and Kd for the control action and obtain a system output with a fast rise time and a fast settling time cited by [20]. Likewise, to determine the stability of the system, the closed-loop equivalent transfer function is determined using the Routh-Hurwitz criterion [21]; [22]; [23] and [24]. The existence of other method methods such as Tyreus-Luyben also allow the test of the closed loop method of Ziegler-Nichols, relatively changing the calculation of the parameters, but which are based on a gain of the period and the sustained oscillation [25]. The configuration that can be established in a PID can be slightly modified to improve its performance depending on the response to be obtained, as proposed by Eitelberg, who incorporated prior gain blocks to each controller, cited by [26] Figure 1 shows the diagram of a closed loop PID control with unity feedback where x(t) represents the reference signal or set-point, y(t) the output variable to be controlled, the PID control block is made up of Kp the proportional gain of the proportional control, Ki the integral gain of the integral control, Kd the derivative gain of the derivative control, e(t) is the error signal, u(t) the output of the PID control block that performs the action of control, F is the transfer function of the plant, Q represents the disturbance. The block to be optimized is the classic PID that will be replaced by a modified PID indicated in figure 6. Equation (1) is the formula for the classic PID controller, the formula for the modified PID controller is indicated in the equation (39). The present work aimed to compare the speed response time of the DC motor model in a classic closed-loop PID control system with a modified PID control system.

DC motor parameters
The DC motor parameters were calculated using the EB-109 DEGEM SYSTEM module, figure 2 shows a diagram that is part of the module and was used to measure the rotational speed of the DC motor that is fed by a variable voltage Va, the speed of rotation is transmitted to a synchronous generator of 6 pairs of poles using pulleys and delivering, the generator, an alternating voltage whose magnitude is proportional to its speed of rotation, this alternating voltage is rectified and filtered, obtaining a direct voltage VDC proportional to the rotational speed of the DC motor and also at variable voltage V a .  Figure 2. Angular speed measuring motor-generator diagram of the EB-109 module [27].
The ratio of the diameters of the synchronous generator pulleys (dAC) and the DC motor (dDC) is dAC/dDC= 5.3, also the relationship of the angular speeds of the DC motor (ND) and the synchronous generator (NS) is shown in equation To calculate the constant C, the direct voltage VDC and the wave period (T) of the alternating signal of the generator were measured, the Fluke 175 multimeter and the Tektronix TDS oscilloscope were used respectively, measuring the time periods on their screen and through cursors between every 2 wave peaks. It should be noted that these measurements are not exact for the same reason that the instruments have tolerances to their precision and on the oscilloscope screen with the use of the cursors there are also measurement errors. These measurements will allow us to calculate the rotational speed of the DC motor in rpm (ND), which by relating it to VDC we will obtain the constant C whose value will be approximate due to the tolerances in the precision of the measurements.
The first two columns of Table 1 are the VDC voltage measurements and the period T of the alternating waves of the generator. The VDC measurements were obtained by varying the supply voltage (Va) of the DC motor; the next three columns are the calculated values: frequency (F), generator rotational speed (NS) and ND. For the frequency the inverse of the period T was used, for NS the equation (5) formula to obtain the speed of the generator in rpm where F is the frequency generated in Hertz, p is the number of pairs of poles and 60 are the seconds that equals one minute. Equation (2) was used to calculate ND.  Figure 3 shows the graph of the ND and VDC points from table 1, estimating the linear trend equation (3) whose slope is the constant C=1191.7 rpm /V. It should be noted that the ideal is that it should not have the constant term (-32,557) but it must be considered that the measurements carried out also have a tolerance in their precision, The armature resistance of the motor DC (Ra) was obtained as the average of the following measurements made with the rotor locked using the Fluke 175 multimeter in the resistance function: 49.5 Ω, 48.5 Ω, 48.3 Ω, 48.1 Ω, 48.1 Ω, then Ra = 48.5 Ω.

Obtaining Kv, B
To calculate the parameter Kv (voltage constant (Vs / rad)) we are based on equation (9) therefore we need to calculate the counter electromotive force generated (Eg (volt)) equation (8) where Va is the continuous supply voltage of the DC motor, Ia its armature current and also calculate the angular velocity of the DC motor (wD(rad/s)) with equation (7) Figure 4 shows the graph of points Eg and wD from table 2 estimating the linear trend equation (11) whose slope corresponds to the sought parameter Kv = 0.0232 Vs/rad, as in the previous case of the linear estimation To obtain the parameter C, in this case the same thing also happens, that the appearance of the independent term (+0.0451) is due to the low precision in the measurements made with the three multimeters and it gives us to understand in the equation that with zero angular velocity a counter electromotive force is already being generated.  To calculate the parameter B we take the average value of its values given in the last column of table 2, resulting in B = 5.49 x10 -6 Nms/rad.

Obtaining the armature inductance (La), moment of inertia (J) and torque constant (kt).
To calculate the armature inductance of the DC motor (La) in Henries, we are based on equation (12), therefore it is required to measure the electrical time constant (a) of the coil La. For this case and with the rotor locked of the DC motor, it was applied to the input Va a step-type voltage of 4.48 volts and simultaneously measured with the oscilloscope at the ends of an external resistance of 56 Ω in series with the armature coil La. The time it took to reach 63% of the maximum voltage is the electrical time constant, its value being a=26 µs. Finally, using equation (12) we calculate the inductance of the armature, resulting in La=2.717 mH.
= + = To calculate the moment of inertia (J (Nms 2 )) we use equation (13)  This constant is estimated as 1/3 of the time that elapses between disconnecting the motor power supply and stopping it, resulting in a value m=371.6 ms, For the torque constant (Kt (Nm/A)) its numerical value is the same as that of the voltage constant (Kv), then Kv 0.0233 Nm/A.

DC motor model
In table 3 left column we have the equations for the DC motor model where vF(t), iF(t) are the field voltage and current, TM and TL the motor and load torque, va(t), ia(t) are the armature voltage and current, in the right column are the Laplace transformations of the respective instantaneous equations.   Figure 5 represents the block diagram of the DC motor model obtained in Simulink, with an input Va as the armature supply voltage of the DC motor and the outputs w(t) angular velocity. ia(t) armature current, theta position angle, TM motor torque.
To obtain the transfer function of the motor speed W(s) with respect to the supply voltage Va(s) we carry out the following steps: from equation (23) we solve for Ia(s), we take Eg(s) from the equation (20), then in equation (19) we replace Ia(s) Eg(s) obtaining equation (24): From equation (22) we solve for TM(s) and replace in equation (24) Table 4 in the first column shows the gains of the P, PD PI and PID control blocks and in the right column the closed loop transfer functions that have been obtained for each type of PID control In table 5 we apply the Routh-Hurwitz stability criterion for each of the PID controllers to obtain the gain of the controllers P (Kp proportional gain) and PD (Kp Kd proportional and derivative gain). In Table 6 we apply the Routh-Hurwitz stability criterion for each of the PID controllers to obtain the gain of the PI controllers (Kp Ki gain proportional and integral) and PID (Kp Ki Kd proportional, integral and derivative gain).

Modified PID control of closed loop DC motor
As mentioned in the introduction, the configuration in a PID can be slightly modified to improve its performance depending on the response to be obtained, as proposed by Eitelberg, who incorporated prior gain blocks to each controller, cited by [26]. They have added three gain blocks Fp, Fi and Fd, these blocks do not modify the regulation properties of the system since the disturbance signal does not circulate through these blocks. What is affected is the closed-loop transfer function, which will not be the same as that established by the classic PID controller.   Its transfer function in Laplace terms is given by equation (37), ordering as a fraction we have equation (38): Considering the PID (s) of equation (38), the closed-loop transfer function with modified PID control, of figure 6, is indicated in equation (39). Table 7 shows the summary of the values that Kp, Ki and Kd must take for the Routh-Hurwitz stability criterion to be fulfilled is shown. For the modified PID, the proportional gains Kp=1, Ki= 1, Kd=0 were used. The values of the three gain blocks with value 1 reduce the structure to the classic PID, we will see the effect of only increasing the gain Fp in hundredths and thousandth in speed response, these values are shown in table 8.