PID speed control of DC motor using meta-heuristic algorithms

ABSTRACT


INTRODUCTION
DC motors are actuators that produce angular rotation when supplied with electrical energy. They have significant importance in various electrical systems employed in domestic and industrial applications such as electrical vehicles, industrial mills and cranes, robots, and multiple home appliances [1], [2]. This importance is due to their advantageous characteristics like precision, convenience, and continuous control [3]. In order to drive the DC motor at appropriate speed or torque, it is necessary to have a proper control scheme.
PID controller is one of such control schemes employed in numerous industrial applications [4]. The term PID is an abbreviation for "proportional integral derivative" and a PID controller is a control system incorporating these three components. The integrator mitigates the controlled system's error, and the derivative provides improved output, adding to other advantageous reasons as to why PID controller has been preferred for more than eight decades [5]. The parameters of proportional, integrator and derivation gains, denoted respectively as , , , are tuned to obtain desired output from the controlled process [6]. There are several classical approaches to tune the PID controller namely Ziegler-Nichols [7], Cohen-Coon [8], Chien-Hrones-Reswick [9], Astrom and Hagglund [10]. However, these conventional methods typically consume a great deal of time as tuning of parameters must be done iteratively until optimal solution is obtained [11] and results in undesirable overshoot [12]. To overcome these disadvantages, number of PID tuning methods have been proposed in the literature. One of such approaches is the usage of metaheuristic techniques. Metaheuristic optimization techniques are stochastic techniques that provides sufficiently acceptable solution(s) iterating the candidate solution(s) improving a certain metric, often referred to as the fitness value. Metaheuristic algorithms can effectively overcome the problem of getting stuck in local optima while exploration in the feasible solution domain and provide effective optimization in problems with complexities of time or dimensions [13], [14]. Control of DC motor has been a popular area where several meta-heuristic algorithms find application [15], [16].
In this paper, two metaheuristic algorithms are presented as tuning methods to tune parameters of speed-controlled DC motor, namely, archimedes optimization algorithm (AOA) and dispersive flies optimization (DFO). The paper is set in the following order: Section 2 outlines the methodology employed in the study with a brief description of meta-heuristic algorithms, Section 3 illustrates results and relevant discussions, and Section 4 concludes the study.

METHOD 2.1. Modelling of DC motor
An externally excited DC motor is employed in this study. The schematic of armature-controlled DC motor is illustrated in Figure 1. The voltage ( ) is employed to regulate the angular velocity ( ) of the motor.
Where is the back EMF constant. The governing mathematical model for armature loop is Where is the armature current, is the inductance of armature winding, and is the armature resistance.

PID controller
This study assumes to achieve a disturbance rejection controller by using a step input as reference. The controller efficacy is evaluated with regards to overshoot. rise time, peak time, and settling time of the closed-loop step response. The transfer function of the PID controller is where , and represent proportional gain, integral gain, and derivative gain, respectively. Likewise, and represent the integral and derivative time constant. Also, = / , and = . The schematic diagram of the proposed controller for speed control of DC motor is illustrated in Figure 2. Finally, for no-load condition with PID speed controller, the closed-loop transfer function is given by (8).

Ziegler-Nichols (ZN) method
The ZN method [17] to find , , and is developed on the transient response of the system to be controlled. In this study, step response (open loop) method is employed. The open loop method involves locating the inflection point in the response curve where the slope of the response curve starts decreasing. The procedure is as, a) ensure that the response curve looks like an S-shaped curve as shown in Figure 3, for the open loop step response, b) draw a line tangent to the inflection point and measure the delay time ( ) and time constant ( ), c) measure the steady state gain of the plant ( ), and d) finally, compute the controller parameters from Table 1.

Meta-heuristic algorithms 2.4.1. Archimedes optimization algorithm
Archimedes optimization algorithm, in short AOA, is a physics-inspired metaheuristic technique proposed in 2020 [18]. It is based on the Archimedes' principle which states that for an object, submerged fully or partially in a fluid, buoyancy force acting on the object equates the displaced portion of the fluid's weight. In AOA, objects refer to the individuals of the population. The objects have physical properties like acceleration, volume, and density. AOA tries to converge to an optimum where these individuals are in equilibrium. In other words, resultant force acting on the object is zero and the object floats on the fluid. In initial stage of AOA, each object has random position in fluid. With iteration, AOA updates each object's density and volume. Iterations continue until termination criteria is met. The algorithm's implementation in optimization problem is illustrated by the pseudo-code.

Dispersive flies optimization
Dispersive flies optimization, introduced in 2014 [19], is inspired from two behaviours of flies: their swarming behaviour when they find a food source and their retreating and dispersing behaviour when encountered a threat. It has been employed in several discrete and continuous search spaces problems in the domain of medical imaging [20], training of deep neural network [21], optimization of machine learning algorithms [22]. DFO's implementation in optimization problem is illustrated by the pseudo-code.

Particle swarm optimization
Kennedy and Eberhart [23] suggested PSO which has its motivation in the collective behaviour of fauna which commute in groups. Each member in swarm is referred as a "particle" which moves around in the solution space. Their movements are governed by pre-defined rules. Each of these members, or particles, is assigned, a velocity value and a position value. The change in position is brought up by adjustment in velocity, which in turn depends on each member's best position and entire population's best position until Where =iteration number, =particle number, =velocity, =position, 1 , 2 =acceleration constants, ( , )=personal best position of ℎ particle, and ( )=global best position in population.

Performance index and response criteria
ITAE is a common performance index used in the design of a PID control. This index was selected to be our objective function because integral of square error and integral of absolute error, ISE and IAE respectively, weigh all error equally resulting in longer settling time. ITAE overcomes this limitation [27]. ITAE is evaluated using the (18).

= ∫ | ( )| ( )
Three response characteristics, particularly, the settling time, rise time and the overshoot of the plant introduced with the step input were observed. Then the response of the suggested algorithms, ZN method and PSO were compared. The data obtained are compared with that of PSO as it is the most used algorithm for synonymous task in literature.

Algorithm parameters
The  The dimension of the problem to be optimized by the algorithms is three, referring to the three gain values of PID controller: , , . The range of these gains used is.

RESULTS AND DISCUSSION
To investigate the efficacy of AOA and DFO, their performance in transient response were compared with ZN and PSO. The chosen algorithms for performance comparison are AOA-PID-ITAE, DFO-PID-ITAE and PSO-PID-ITAE. Transient response criteria mainly include percentage overshoot ( ), rise time ( ), settling time ( ), and peak time ( ). Table 3

Ziegler-Nichols method
As per the procedure described in Section 2.3., the parameters for computing the PID gains is obtained from Figure 3. The obtained parameters are = 1, = 0.40476 , and = 0.64285 . The PID gain parameters computed using these values with the corresponding transient response criteria are incorporated in Table 4. The closed-loop response of the motor using PID gain parameters obtained from Ziegler-Nichols method has rise time of 0.7768 , settling time of 1.2518 , peak time of 5.1184 , and no overshoot. Hence, the Ziegler-Nichols method seems to have improved the system's transient response by removing the disturbance and reducing rise time and peak time. Although Ziegler-Nichols removed the disturbance form the transient response, peak time increased from 2.5570 to 5.1184 .  Table 4 illustrates the best performance of the algorithms to produce optimal PID controller gains.    Figure 4 shows the closed-loop step response of the system for all these controllers. Although all the meta-heuristic algorithms can reduce the disturbance in comparison to open loop response, small percentage of overshoot is still prevalent in the system. Figure 4 illustrates the closed-loop step response of the system for all these controllers.  Table 5 provides the minimum value of the objective function these metaheuristic algorithms converge to after 100 iterations. This helps one conclude that the proposed tuning methods provide PID controller parameters with comparatively lower ITAE value which is a desired feature. DFO-PID controller, evidently, has the lowest ITAE with lowest standard deviation for 10 independent runs. Hence, the DFO-PID controller is the most accurate meta-heuristic algorithm based on ITAE fitness function. Likewise, DFO-PID is evident to show minimal variance in the result for different runs illustrating the high repeatability of the algorithm. AOA-PID ranks second after DFO-PID in terms of fitness function value as well.  Figure 5 illustrates the convergence of the meta-heuristic algorithm for the best simulation run. It is observed that the suggested methods take less iteration to converge to provide an optimal PID controller. It is observable that the DFO-PID converges faster than AOA-PID and PSO-PID. While DFO-PID took only 18 iterations to converge to the ITAE of 0.002484, AOA-PID took about 50 iterations to achieve approximately the same. On the contrary, PSO-PID could only converge to ITAE of 0.003153 even with 100 iterations. AOA-PID and DFO-PID outperform ZN method and PSO-method in transient response criteria as well as in minimizing ITAE value. Furthermore, these two proposed methods take comparatively less iterations to converge to optimal ITAE value than PSO-PID. Figure 5. Convergence plot of meta-heuristic algorithms

CONCLUSION
In this study, two new approaches are presented to obtain optimum gain parameters of PID controller to regulate a DC motor's rotational speed. In controller design process, meta-heuristic algorithms are utilized to minimize the ITAE fitness function. Transient response characteristics of DC motor speed control system were employed to evaluate the efficacy of meta-heuristic algorithms. In this study, AOA, DFO, and PSO algorithms are considered for performance comparison. The numerical figures and graphical simulation results conclude that the proposed techniques outperform the classical ZN method and the popular PSO method. Hence, the proposed technique can be employed to ensure optimum performance of PID controller in large electrical systems, process industry and automation sector, among others.