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Optimization of BLDC Cascaded Position and Speed PI Controllers using PSOMohammed Issa Al-Busaidi, Student: s121510Abstract—Particle Swarm Optimization (PSO) is one of the optimization techniques and a kind of evolutionary computation technique. The method has been found to be robust in solving problems featuring nonlinearity and nondifferentiability, multiple optima, and high dimensionality through adaptation, which is derived from the social-psychological theory. The objective of this project is to implement PSO technique in tuning a cascaded position and speed PI controllers of Brushless DC (BLDC) Motor in order to minimize the output response steady-state error, rising time, settling time and overshoot percentage. The optimization process along with the BLDC model will be executed using Matlab and SIMULINK software.INTRODUCTIONDC motor drive are very common in applications where adjustable speed, good speed control and frequent starting braking and reversing are required due to their design simplicity, robustness and ease of control. BLDC motor in specific are unique since its flux is provided through a permanent magnet compared to a conventional DC motor.BLDC motors combine the advantages conventional DC motor un terms of better velocity capability and mechanical commutator with the advantages of AC motors such as simple structure, higher reliability and maintenance free. Additionally, BLDC motor has more advantages, for example: smaller volume, high force, simple system structure and high performance compared to volume [4].Fig. 1. BLDC motor block diagram with its Transfer Function. From the control point of view, dc motor exhibit excellent control characteristics because of the decoupled nature of the field and armature mmf’s [1]. PID control with its three term functionality covering treatment to both transient and steady-states response, offers the simplest and yet most efficient solution to many real world control problems. In spite of the simple structure and robustness of this method, optimally tuning gains of PID controllers have been quite difficult [1]. Many techniques have been implemented to tune the parameters of PID controller. Those parameters directly affect both the transient and steady-state response and can cause undesirable performance if not calculated properly.In this project, implementation of PSO will be experimented to tune a cascaded position and speed PI controllers. the operation of PSO will be discussed as well as the development of the objective function and lastly the results of the optimization will be illustrated.BLDC MotorThe stator of BLDC motor is the coil, and the rotor is the permanent magnet. The stator develops the magnetic field to rotate the rotor. Hall effect sensor detects the rotor position as the commutating signals. Therefore, BLDC motors use permanent magnets instead of coil in the armature and so do not need brushes [4].The dynamic characteristics of BLDC motors are similar to permanent magnet DC motors. The characteristic equations of BLDC motors can be represented as [1]: where νapp (t) is the applied voltage, ω(t)is the motor speed, L is the inductance of the stator, νemf(t) it the back electromotive force, T is the torque of the motor, D is the viscous coefficient, J is the moment of inertia, Kt is the motor torque constant, and Kb is the back electromotive force constant. Fig. 1 shows the block diagram of the BLDC motor with its transfer function.Fig. 2 Flow chart of PSO algorithms TABLE IBLDC Motor ParametersParametersValues and unitsR2.7 ΩKb0.8790 Vs rad-1Kt0.8811 Kg-m/AD0.0015 Kg-m s/radL13.9 mHJ0.0043 Kgm s2/radThe parameters of the motor used for simulation are shown in Table. 1: Particle Swarm OptimizationParticle swarm optimization is attributed to Kennedy, Eberhart and Shi [3]. It is an iterative gradient-free search algorithm inspired by biological swarming such as bird flocking. The search starts with a random set (here called swarm) of solutions (here called particles). The particles travel through a search space and are rated according to a user defined objective function. Their movements are a function of the individual experience and information acquired from other particles. However, the velocity vectors are not deterministic. The instantaneous strength of social and individual behavior varies randomly for each particle in each iteration. In a basic version of the algorithm the only piece of information shared among the particles is the global best solution found so far. Each particle stores also its best solution found so far. The velocity and position update rules are as follows [3]: (5) (6)where: j is the particle identification number, i denotes the iteration number, vj and pj are speed and position of the j-th particle, ppbest stores the best solution proposed jj so far by the j-th particle (pbest), pgbest denotes the best solution found so far by the swarm (gbest), c1, c2 and c3 are the explorative factor (inertia weight), the individuality factor and the social factor, respectively. The search path is not deterministic because of the last two terms in (5) that include multiplication by the random numbers rand() generated for each particle in each iteration. The random numbers are uniformly distributed in the unit interval. Fig. 2 illustrates the algorithm flow chart of PSO for BLDC motor.The Objective Function [3]The performance measure to be minimized contains the following objectives of the PI controller, Minimize the steady state error, the difference between the input and output of the system in the limit as time goes to infinity, i.e. when the transient response reaches a steady state. With no overshoot the steady state error is eliminated when the steady state velocity of the vehicle reaches the desired velocity, Minimize the rise time, time required for system response to rise from: 10% to 90% (Over damped); 5% to 95%; 0% to 100% (Under damped) of the final steady state value of the desired response, Minimize the maximum overshoot, Maximum Overshoot is the maximum peak value of the response curve measured from the desired response of the system, and Minimize the settling time, Time required for response to reach and stay within 2% of final value.The developed objective function has to account for all previous measures discussed. There are some commonly used performance measures as the integral of squared error (ISE):Fig. 3 Response of BLDC PI controller when using ISE as fitness function. (7)where e(t) denotes control error, or the generalized ISE, (8)where is a subjective weighting factor, or the integral of squared error and derivative of control effort, (9)where u(t) denotes control effort (control signal) and is again a subjective weighting factor. Additionally, it is common practice to add terms that take into account overshooting or crossing acceptable levels for control signal. The nature of the PSO enables the designer to work with any form of performance index. However, our main goal is to keep this stage simple without sacrificing the performance of the resulting system. All proposed here performance measures have the form of (10)where the function f1BLDC introduces penalty for position control error after the time equal to a theoretical time needed to travel a test angle under assumption that the only physical limits are these related to maximal absolute values of electromagnetic torque and angular speed, the function f2BLDC switches off the penalty for reference torque variations during transients forced by the test reference speed and test disturbance torque, and the function f3 BLDC penalizes for an overshoot. Fig. 4 Response of BLDC PI controller when using customized ISEDCE as fitness function. For the discussed PI controllers tuning task, each particle is a vector of can- didate settings for both controllers. Thes