International Journal of Innovations in Engineering and Science

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1 International Journal of Innovations in Engineering and Science INNOVATIVE RESEARCH FOR DEVELOPMENT Website: e-issn: Volume 1, Issue 1 August, 2018 Optimal PID Controller Design for Speed Control of Separately Excited DC Motor Drive Using Particle Swarm A. U. Essien 1 essiennig@yahoo.com B. J. Robert 2 Patrick Onotu 3 patrickonotu@gmail.com 123 Department of Electrical/Electronic Engineering Akanu Ibiam Federal Polytechnic Unwana, Ebonyi State, Nigeria ABSTRACT This paper presents an artificial intelligence method, Particle Swarm (PSO) algorithm for determining the optimal Proportional Integral Derivative (PID) controller parameters for a Separately Excited Direct Current (DC) Motor (SEDM) drive system. The PID controller is the most commonly used controller for the speed control of DC motor. However, the conventional gain tuning of PID controller (such as Ziegler-Nichols (ZN) method) has some disadvantages such as the high starting overshoot, sensitivity to controller gains and sluggish response to sudden disturbance. The main objective of this paper is to develop a PSO algorithm that minimizes these transient response specifications chosen as rise time, settling time and overshoot, for better speed response of a PID- DC motor drive system. The PID controller is first tuned using ZN first and second methods, PSO algorithm is then developed for better estimation of the PID controller parameters and the speed responses for these methods are analyzed with respect to the components of the objective function. In comparison with the ZN methods, the PSO- PID has more efficiency and robustness in improving the step response of DC motor drive system Keywords: Separately Excited DC Motor Model, PID Controller, Ziegler-Nichols method, Particle Swarm. 32

2 1. Introduction Developments of high performance motor drives are very essential for industrial applications. A high performance motor drive system must have good dynamic speed command tracking and load regulating response [1, 2]. DC motors are used extensively in adjustable speed drives and position control applications due to their simplicity; ease of applications, reliability and favourable cost. DC drives are less complex as compared to AC drives systems [1, 2, 4]. They provide excellent control of speed for acceleration and deceleration. DC motor is a highly controllable electrical actuator and is widely used for robotic manipulators, guided vehicles, steel rolling mills, cutting tool, overhead cranes, electrical traction and other application etc [4]. The power supply of a DC motor connects directly to the field of the motor which allows for precise voltage control, and is necessary for speed and torque control applications [3]. DC motors are capable of providing starting and accelerating torques in excess of 400% of rated value [3]. Due to these relative advantages, DC motors have long formed the backbone of industrial applications [1, 2, 3]. Separately excited DC motor drive is the most suitable configuration used for variable speed applications for a long time due to its accurate speed control, controllable torque, high reliability and simplicity [2, 4]. In Separately Excited DC motor, the power supply is directly connected to the field winding of the motor. There are three speed control techniques used commonly, these are: field resistance control, armature resistance control and armature voltage control. The field current is kept constant and variable voltage is applied to the armature in armature voltage control method. The basic working principle of an armature controlled DC drive is that the speed of a separately excited DC motor is directly proportional to the applied armature voltage of the DC motor. DC motor speed is controllable over wide range via the proper proportional adjustment of terminal voltages. In controlling the speed of DC motors, many varieties of control techniques are used such as P, PD, PI, PID, Fuzzy Logic Controller (FLCs) and Fuzzy Neural Network etc. It has been reported that more than 95% of the controllers in the industrial process control applications are PID type due to their simplicity, clear functionality, applicability and ease of use [2, 5]. The PID controller was first introduced to the market in 1939 and has remained the most widely used controller in process control until today. The basic function of the controller is to execute an algorithm based on the Plant input and hence to maintain the output at a level so that there is no difference between the process variable and the set point [6]. The popularity of PID controllers is due to their functional simplicity and reliability. They provide robust and reliable performance for most systems and the PID parameters are tuned to ensure a satisfactory closed loop performance [6]. A PID controller improves the transient response of a system by reducing the overshoot, and by shortening the settling time of a system [6]. The PID control algorithm is used to control almost all loops in process industries and is also the cornerstone for many advance control algorithms and strategies [6]. For this control loop to function properly, the PID loop must be properly tuned [6]. The main task of designing a PID controller is to determine the three gains - proportional gain (Kp), integral gain (Ki) and derivative gain (Kd) of the controller [7]. However, the three adjustable PID controller parameters should be tuned appropriately [7]. Over the years, several heuristic methods have been developed for the tuning of PID controllers. The first method used the classical tuning rules proposed by Ziegler and Nichols [8]. Generally, it 33

3 is always hard to determine optimal or almost optimal PID parameters with the Ziegler-Nichols method in many industrial plants [7]. Other than original works done by Ziegler and Nichols, a great number of methods have been proposed to obtain optimal gains of the PID such as by Cohen and Coon in 1953, Åström and Hägglund in 1984 or by Zhuang and Atherton in 1993 [6,7]. To obtain the optimal parameter tuning, it is highly desirable to increase the capabilities of PID controllers by adding new features. Most in common, artificial Intelligence (AI) techniques have been employed to improve the controller performances for a wide range of plants while retaining the basic characteristics [7,10]. AI techniques such as artificial neural network, fuzzy system and neural-fuzzy logic have been widely applied in order to get proper tuning of PID controller parameters [6]. Recently, a new evolutionary technique, Particle Swarm (PSO) was first introduced in 1995 by Kennedy and Eberhart for unconstrained continuous optimization problems [7, 9]. Its development was based on observations of the social behavior of animals such as bird flocking, fish schooling and swarm theory [7, 10]. The PSO is initialized with a population of random solutions. It has memory and therefore, knowledge of good solutions is retained by all particles. There also exists constructive cooperation between particles where particles in the swarm share information among themselves. The theoretical framework of PSO is very simple and is easy to be coded and implemented using computer program [7]. In fact, the PSO technique can generate a high quality solution within shorter calculation time and stable convergence characteristics than other stochastic methods [7]. Thus, this technique has gained much attention and wide applications in various fields recently [7]. This paper is of five sections. Section 2 presents dynamic models of a separately excited DC motor (SEDM), both in Simulink and in transfer function. Next, Ziegler Nichols (ZN) and the PSO methods and their implementation into the ZN-PID and PSO-PID controllers are viewed in details. In 4, the simulation results are presented and discussed. Finally, conclusions are made based on the results obtained. 2. Dynamic Model of Separately Excited DC Motor When a Separately Excited DC motor is excited by a field current of If with an input voltage ea applied to the armature as shown in figure 1, an armature current of Ia flows in the armature circuit. Figure 1: Equivalent Circuit of SEDM using the Armature Voltage Control [11] The motor develops a back EMF eb and a torque Tm to balance the load torque TL at a particular speed. The If is independent of Ia. Each winding are supplied separately. The interaction of field flux and armature current in the rotor produces the required torque. Where, 34

4 Ra : Armature resistance; La: Armature inductance; ia : Armature current; if : Field current; ea : Input voltage; eb : Back electromotive force (EMF); Tm : Motor torque; ωm : Angular velocity of rotor; ϴm : Angular position of the rotor J : Rotating inertia measurement of motor bearing; Kb : EMF constant; KT : Torque constant; B : Friction constant. The equation describing the dynamic behaviour of the SEDM is as follows. Since the back EMF eb is directly proportional to speed, Then; dθ m e b (t) = K b dt = K bω m (t) 1 di a (t) e a (t) = R a i a (t) + L a + e dt b (t) 2 T m (t) = J dω m(t) + Bω dt m (t) = K T i a (t) 3 Equation 2 results from applying Kirchhoff s law for voltage drop around the armature circuit. Equation 3 is based on Newton s law for rotational systems while Equation 1 couples the electrical and mechanical operation of the motor. Taking Laplace transform of equations 1, 2 and 3 give; E a (s) = (R a + L a S)I a (s) + E b (s) I a (s) = [E a (s) E b (s)]/ (R a + L a S) 4 E b (s) = K b ω m (s) 5 T m (s) = (B + JS)ω m (s) = K T I a (s) ω m (s) = T m (s)/(b + JS) 6 Figure 2 describes the SEDM armature voltage control system function block diagram gotten from equations 4, 5 and 6 while figure 3 is the Simulink model. 35

5 (a) Block diagram model (b) Simulink model Figure 2: Block diagram and Simulink models of the armature voltage control of SEDM From figure 2a, the transfer function of the motor speed with respect to the input voltage Ea is given as, G(s) = ω(s) E a (s) = K T 7 (L a S + R a )(JS + B) + K b K T Table 1: MOTOR PARAMETERS Parameters Value Armature Resistance Ra (Ω) 2 Armature Inductance La (H) 0.5 Moment of Inertia J (Kgm 2 ) 0.02 Friction Constant B (Nms) 0.2 Torque Constant K T (Nm/A) EMF Constant K B (Vs/rad) 0.01 With the motor parameters given in table 1, the open loop step response of the motor is simulated using MATLAB program or Simulink as shown in figure 2b. 3. PID Controller Design It has been mentioned that PID controllers are well known for their simple structure and robust operation in a wide range of operating conditions. The structure of the conventional PID controlled system consists of PID controller and a process (which in this case is the speed of the SEDM) as shown in figure 4. Tuning of the controller plays a vital role in designing the controller which can control the process in an efficient manner. Various tuning methods are available to find the PID parameters that can effectively control the process. 36

6 Figure 4: Block Diagram of Conventional PID Controller [12] 3.1 Ziegler Nichols methods Ziegler-Nichols (ZN) tuning rule was the first tuning rule to provide a practical approach for PID controller tuning. They proposed rules for determining values of the proportional gain Kp, integral time Ti, and derivative time Td based on the transient response characteristics of a given plant. There are two methods called Ziegler Nichols tuning rules: the first method and the second method. A brief presentation of these two methods is given. First method: In the first method, the response of the plant to a unit-step input is obtained experimentally, as shown in figure 5. This method applies if the response to a step input exhibits an S-shaped curve. Such step-response curves may be generated experimentally or from a dynamic simulation of the plant. Figure 5: Ziegler Nichols Rule for Tuning PID Controllers [13] The S-shaped curve may be characterized by two constants, delay time L and time constant T. The delay time and time constant are determined by drawing a tangent line at the inflection point of the S-shaped curve and determining the intersections of the tangent line with the time axis and line c(t)=k, as shown in figure 5.The transfer function Gc(s) may then be approximated by a first-order system with a transport lag as follows: G(s) = Ke sl 8 TS + 1 Ziegler and Nichols suggested setting the values of Kp, Ti, and Td according to the formula shown in Table 2. Notice that the PID controller tuned by the first method of Ziegler Nichols rules gives: G c (s) = K P (1 + 1 T i s + T ds) G c (s) = 1.2 T L ( Ls + 0.5Ls) 37

7 G c (s) = 0.6T (s + 1 L )2 9 s For our own case, L = 0.04, T = 0.54 and the transfer function from equation 9 becomes, G c (s) = 0.324s s s Table 2: Ziegler Nichols Tuning Rule Based on Step Response of Plant (First Method) Second method: In the second method, we first set Ti =, and Td = 0 and Using the proportional control action only, Kp is increase from 0 to a critical value Kcr at which the output first exhibits sustained oscillations. (If the output does not exhibit sustained oscillations for whatever value Kp may take, then this method does not apply.) Thus, the critical gain Kcr and the corresponding period Pcr are experimentally determined. Ziegler and Nichols suggested that we set the values of the parameters Kp, Ti, and Td according to the formula shown in Table 3. Notice that the PID controller tuned by the second method of Ziegler Nichols rules gives: G c (s) = K P (1 + 1 T i s + T ds) G c (s) = 0.6K cr ( P cr s P crs) (s + 4 P ) 2 G c (s) = 0.075K cr P cr cr 11 s For our own case, Kcr = 400, Pcr = 0.31 and the transfer function from equation 11 becomes, G c (s) = 9.3s s s Table 3: Ziegler Nichols Tuning Rule Based on Critical Gain Kcr and Critical Period Pcr (Second Method) 38

8 3.2 Overview of Particle Swarm # Introduction: Particle swarm optimization (PSO) is a global optimization algorithm for dealing with problems in which a best solution can be represented as a point or surface in an n-dimensional space. Hypotheses are plotted in this space and seeded with an initial velocity, as well as a communication channel between the particles. Particles then move through the solution space, and are evaluated according to some fitness criterion after each time step. Over time, particles are accelerated towards those particles within their communication grouping which have better fitness values. The main advantage of such an approach over other global minimization strategies such as simulated annealing is that the large numbers of members that make up the particle swarm make the technique impressively resilient to the problem of local minima [12, 14]. PSO Algorithm 1. Initialize the swarm by randomly assigning each particle to an arbitrarily initial velocity and a position in each dimension of the solution space. 2. Evaluate the desired fitness function to be optimized for each particles position. 3. For each individual particle, update its historical best position so far, Pbest if its current position is better than its historical best position. 4. Identify/Update the swarm s global best particle that has the swarm s best fitness value, and set/reset its index as Gbest. 5. Update the velocities of all the particles using equation 13 V i k+1 = WV i k + C 1 rand 1 (Pbest i X i k )+C 2 rand 2 (Gbest X i k ) Move each particle to its new position using equation 14 X k+1 i = X k k+1 i + V i Repeat steps 2-6 until convergence or a stopping criterion is met (e.g., the maximum number of allowed iterations is reached, a sufficiently good fitness value is achieved or the algorithm has not improved its performance). 39

9 Figure 6: The Flowchart of Particle Swarm Algorithm Where, Vi k : velocity of particle i at iteration k, W: weighting function, C1, C2: weighting factor, rand: random number between 0 and 1, Xi k : Current position of particle i at iteration k, Pbesti: best position of particle i, Gbest: global best position of all the particles in the swarm. Fitness function: for this design, we have taken four component functions to define the fitness function F. The fitness function is a function of steady state error Ess, peak overshoot Mp, rise time Tr and settling time Ts. The contribution of these component functions in the fitness function is determined by a scale factor β, which depends upon the choice of the designer. For this design the best point is the point where the fitness function has the minimal value. The chosen fitness function is given as: F = (1-exp(-β)) (Mp +Ess) + (exp(-β)) (Ts - Tr) 15 The flow chart of figure 6 is used to develop a MATLAB program to obtain the optimal PID controller parameters that was used for the simulation of the close loop unit step response of the SEDM shown in figure 7b. 4. Simulation Results and Comparison 40

10 4.1 Implementation of ZN- PID Controller The ZN-PID controllers (for both first and second methods) for the SEDM are implemented using MATLAB software. The unit step responses of the system are shown in figure 7a. Time domain specifications are observed from the response graphs and tabulated in Table 4. The ZN-PID controllers are observed to have more rise time, settling time and peak overshoot Step Response ZN-PID (Ist method) ZN-PID (2nd method) Step Response PSO-PID 1 1 Amplitude Amplitude Time (seconds) (a) ZN-PID first and second methods (b) PSO-PID Figure 7: System Responses for the ZN-PID and PSO-PID Time (seconds) 4.2 Implementation of PSO-PID Controller The PSO based PID controller for the SEDM is implemented using MATLAB software. The optimal PID controller parameters are obtained as Kp = , Kd = 391.3, and Ki = Thus, the transfer function for the PSO-PID controller is given as; G c (s) = 391.3s s s The unit step response of the system is shown in figure 7b. Time domain specifications are observed from the response graphs and tabulated in Table 4. The PSO-PID controller is observed to have less rise time, settling time and overshoot compared to the ZN-PID controller. 4.3 Comparison of responses of ZN and PSO tuned PID Controllers The unit step responses of ZN and PSO tuned PID controllers for the SEDM are compared in terms of time domain specifications and shown in figure 8. The PID values obtained by the PSO algorithm are compared with that of the one derived from Zeigler-Nichols methods in various perspectives, namely robustness and stability performances. PSO-PID controller shows superiority over the conventional PID controllers. All the simulations were implemented using MATLAB software. 41

11 Step Response ZN-PID (Ist method) PSO-PID ZN-PID (2nd method) 1 Amplitude Time (seconds) Figure 8: Comparison of System Responses for ZN-PID and PSO-PID Table 4: Comparison of Time Domain Specifications/Controller Parameters Time Domain Specifications Z-N (Ist) Z-N (2nd) PSO Rise time (sec) Settling time (sec) Peak overshoot (%) Steady state error Peak time Conclusion The response of the proposed controller using PSO algorithm is proved to be fast and stable than the controller tuned by Ziegler-Nichols methods. By using the PSO approach, an efficient and quick search for the optimal PID controller parameters is achieved. It is found very clearly that the PSO based controller reduces the overshoot, settling time, rise time and peak time. Hence PSO- PID performs better than the traditionally tuned controller with Zeigler-Nichols criteria. The proposed PSO-PID controller gives better robustness and stability, and the performance is satisfactory for speed control of DC motor drive system. 42

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