Contemporary Engineering Sciences, Vol. 11, 2018, no. 99, 4913-4920 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.810539 PID Controller Design for DC Motor Juan Pablo Trujillo Lemus Department of Physics and BIOIF Universidad Tecnológica de Pereira Pereira, Colombia Germán Correa Vélez Department of Mathematics and GIMAE Universidad Tecnológica de Pereira Pereira, Colombia Nancy Janet Castillo Rodríguez Department of Physics and DICOPED Universidad Tecnológica de Pereira Pereira, Colombia Copyright c 2018 Juan Pablo Trujillo Lemus. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The analysis of a DC machine (DC motor) is presented starting from a third order model of said system. The parameters of a commercial machine are also used with which the stability of the model is studied initially and then the design of a PID controller is presented to control the speed ω of the motor rotor. Keywords: DC machine, PID, controller design, Ziegler-Nichols criterion 1 Introduction The importance in the design of the control systems has grown during the last years to a great extent, taking a considerable participation in the industry.
4914 Juan Pablo Trujillo Lemus et al. Currently control devices are designed for different elements and processes of companies. Electric machines then play an important role in such systems and the study of stability and control of these also do so, because the processes often use DC motors to drive their workloads. These engines have quite simple functional and constructive models [1]. There are several well-known methods for controlling DC motors, such as: PI, PID, biposition, etc. These can be easily implemented using analog electronics. This article aims to present a speed control technique of a DC motor based on systems theory concepts. The proportional-integral-derivative controller, or PID controller, can be understood as a generic control device that has the great advantage of being able to be tuned to control a certain system or process [2]. 2 DC Machine Modeling Figure 1 shows the equivalent electric circuit of a DC motor. From this circuit we obtain the system of equations defined later by using the Kirchhoff law of mesh currents for the armature and field circuits and Newton s law of summation of torques for the rotor of the machine. Figure 1: DC motor. v f = R f i f + L f di f dt v a = c 1 i f ω + L a di a dt + R ai a J dω dt = c 2i f i a + c 3 ω Taking x 1 = i f, x 2 = i a and x 3 = ω. For a separately driven DC motor, the voltages v a and v f are independent control inputs. The system of appropri-
PID controller design for DC motor 4915 ate state equations is chosen, obtaining the model representation of the DC machine shown below [3]: ẋ 1 = R f L f x 1 + v f L f ẋ 2 = R a x 2 c 1 x 1 x 3 + v a L a L a L a ẋ 3 = c 3 J x 3 + c 2 J x 1x 2 The first equation is for the field circuit with v f, i f, R f and L f being its voltage, current and inductance respectively. The variables v a, i a, R a and L a correspond to the variables for the reinforcement circuit described by the second equation. The third equation refers to the torque equation for the axis, with J as inertia and B as the damping coefficient. The term c 1 i f ω is the e.m.f induced in the armature circuit, and c 2 i f i a is the torque produced by the interaction of the armature current with the flow of the field circuit [4]. To reduce and simplify the previous model, take the voltage v f as a constant, which makes the first equation disappear by taking I f = v f R f. ẋ 2 = R a L a x 2 c 1I f L a x 3 + u L a ẋ 3 = B J x 3 + c 2I f J x 2 Table 1 shows the parameters of a commercial DC motor, with which the corresponding analysis concerning this document will be developed. The constants Parameters Value Units R a 5.3 Ω L a 5.8 10 4 H J 1.4 10 6 kgm 2 k b 2.2 10 2 V s/rad k a 2.2 10 2 Nm/A B 2.01 10 6 Nms Table 1: DC motor parameters. k b and k a are given by: k b = c 1 I f, k a = c 2 I f The damping parameter B, also known as the viscous friction coefficient, is obtained by using the equation of the mechanical time constant t a given by: B = J t a k bk a R a,
4916 Juan Pablo Trujillo Lemus et al. where t a = 1.5s. Finally, the equations that will be analyzed are presented below: ẋ 2 = R a L a x 2 k b L a x 3 + u L a ẋ 3 = B J x 3 + k a J x 2 By replacing the parameters of table 1 in the previous system of equations, the following model is obtained in matrix form. ] [ ] [ ] [ ] [ẋ2 37.931 9137.9 x2 1724.1 = + u 1.4375 15714 0 ẋ 3 It is desired to design a PID controller For the system shown above [5], for this, initially the transfer function that relates the input to the output is presented as shown below. Applying the Laplace transform to the system of equations we have Sx 2 (s) = 37.931x 3 (s) 9137.9x 2 (s) + 1724.1U (s) x 3 Sx 3 (s) = 1.4375x 3 (s) + 15714x 2 (s) Relating the previous equations, we have to: [ ] S 2 + 9092.53S + 609458 x 2 (s) = U(s) 27093596.96 For this case, the output Y (s) is taken as the state variable x 3 (angular velocity ω of the rotor), which corresponds to the variable to be controlled. The transfer function of the plant is defined as. H (s) = Y (s) U (s) = 27093596.96 S 2 + 9092.53S + 609458 The stability of the system can be defined by the poles of the system, which, for the system to be stable, the roots of the denominator of the transfer function (poles), must be in the complex left half, as shown below. S 1,2 = 9092.53 ± 9092.53 2 4(1)(609458) 2(1) Therefore, we get S 1 = 67.53 S 2 = 9025
PID controller design for DC motor 4917 Figure 2: Response to input (a) step (b) boost. Figure 3: Geometric place of the system poles. Which indicates that the system is stable and not oscillatory (with imaginary part null). Figure 2 shows the response presented by the system previously shown by a unit step input. The response of the system to the unit step input is obtained by applying the inverse Laplace transform to the transfer function that is related to the output of the system as shown below. Y (s) = Applying Laplace inverse transform: 27093596.96 (S + 9025)(S + 67.53) U (s) y (t) = 44.455 + 0.335e 9025t 44.79e 67.53t
4918 Juan Pablo Trujillo Lemus et al. Now we want to design a PID controller to control the system before a unit step input. Figure 4: Closed loop of the system. 3 Criteria Ziegler-Nichols It is desired to tune a PID controller using the Ziegler-Nichols criterion, which consists of finding a gain k c with which the response of the system to a unit step input for the closed loop system is oscillatory and periodic. From this gain the proportional constant of the controller and the oscillation period P c are determined, with which the time constants used to find the integral gains (k i ) and derivative (k d ) are defined. T ype C(s) C(s) k p T i T d P k p ] 0.5k c P I k p [1 + 1 P T is 0.45k n c 1.2 P ID k p [1 + 1 T is + st d ] P 0.6k n P n c 2 8 Table 2: Obtaining driver parameters. For the system that has been studied, it has been found that K c is 1.2 with an oscillation period P n of 0.04s. From this information, it is also possible to determine the value of the time constants T i and T d and the gains of the PID controller using the equations presented in table 2, as shown below. T i = 0.02, T d = 0.005 k p = 0.7 k i = 36 k d = 0.0035 The figure shows the response of the system with the PID controller implemented. The comparison has been made with two parameters of k c different and with this the good response that the system has with the controller parameters found has been verified.
PID controller design for DC motor 4919 Figure 5: System response with PID. Figure 6: System response. 4 Conclusion The basic term is the proportional term, P, which generates a corrective control action proportional to the error. The integral term, I, generates a correction proportional to the integral of the error. This assures us that, if we apply a sufficient control effort, the tracking error is reduced to zero. The derivative term, D, generates a control action proportional to the error range change. This tends to have a stabilizing effect, but usually generates large control actions. It is extremely important to analyze in the first instance the type of system with which you are working, the degree of it will determine how narrow the values of the tuning gains can reach. It is also important to determine from the beginning the real stability of the system, since it does not make sense to implement a control process to a system that is naturally unstable, in which case it will be necessary to modify the plant to maintain the poles in the negative real half-axis, guaranteeing so its stability. Acknowledgements. We would like to thank the referee for his valuable suggestions that improved the presentation of this paper and our gratitude to the Department of Mathematics of the Universidad Tecnológica de Pereira (Colombia) and the GIMAE (Grupo de Investigación en Matemática Aplicada y Educación).
4920 Juan Pablo Trujillo Lemus et al. References [1] H. Khalil, Nonlinear Systems, Prentice-Hall, Jersey, Vol. 2, 1996. [2] N. Matsui, M. Shigyo, Brushless DC motor without position and speed sensors, IEEE Transactions on Industry Applications, 28 (1992), no. 1, 120-127. https://doi.org/10.1109/28.120220 [3] D. Xue, Chunna Zhao, YangQuan Chen, Fractional order PID control of a DC-motor with elastic shaft: a case study, American Control Conferen IEEEce, (2006). https://doi.org/10.1109/acc.2006.1657207 [4] D. Atherton, PID controller tuning, Computing and Control Engineering Journal, 10 (1999), no. 2, 44-50. https://doi.org/10.1049/cce:19990202 [5] K. Narendra, Control of Nonlinear Dynamical Systems Using Neural Networks: Controllability and Stabilization, IEEE Transactions on Neural Networks, 4 (1993), no. 2, 192-206. https://doi.org/10.1109/72.207608 Received: October 17, 2018; Published: November 25, 2018