Design of Speed Controller for Permananent Magnet Synchronous Motor Drive Using Genetic Algorithm Based Lower Order System Mo delling

Problem statement: In this study, a Model Order Reduction (MOR) metho d is proposed for reducing higher order system into lower order s ystem. Speed controller design is carried out to th e lower order system by Genetic Algorithm (GA) approa ch and this controller is used to higher order system. Approach: This study is used to find a solution to a given o bjective function employing different procedures and computational techniques. The problem area chosen is that of lower order system modelling used in design of speed cont roller for Permananent Magnet Synchronous Motor (PMSM) drive. Results: Genetic Algorithm obtains a better controller valu es that reflects the characteristics of the original higher order sy stem and the performance evaluated using this method are compared with the existing approximation method. Conclusion: Performance of this Speed controller has been verified through Simulati on using MATLAB package.


INTRODUCTION
During the early 1950s researchers studied evolutionary systems as an optimisation tool, with an introduction to the basics of evolutionary computing (Sivanandam and Deepa, 2009). Until 1960s evolutionary systems was working in parallel with Genetic Algorithm (GA) research. At this stage, evolutionary programming was developed with the concepts of evolution, selection and mutation. Holland (1992) introduced the concept of Genetic Algorithm as a principle of Charles Darwinian theory of evolution to natural biology. The working of genetic algorithm starts with a population of random chromosomes. The algorithm then evaluates these structures and allocates reproductive opportunities such that chromosomes, which have a better solution to the problem, are give more chance to reproduce. While selecting the best candidates, new fitter offspring are produced and reinserted and the less fit is removed. The exchange of characteristics of chromosomes takes place-using operators like crossover and mutation. The solution is defined with respect to the current population. GA operation basically depends on the Schema theorem. GAs are recognized as best function optimisers and is used broadly in pattern discovery, image processing, signal processing and in training Neural Networks.
Many control system applications, such as satellite altitude control, fighter aircraft control, model-based predictive control, control of fuel injectors, automobile spark timer, possess a mathematical model of the process with higher order, due to which the system defined becomes complex. These higher order models are cumbersome to handle (Sivanandam and Deepa, 2009). As a result, lower order system modelling can be performed, which helps in alleviating computational complexity and implementation difficulties involved in the design of controllers and compensators for higher order systems. Further, the development and usage of micro controllers and microprocessors in the design and implementation of control system components has increased the importance of lower order system modeling (Prasad, 2000;2003a;2003b). Thus, in this study, Genetic Algorithm is used independently to higher order systems and a suitable lower order system is modelled (Sivanandam and Deepa, 2009).
The availability of modern Permanent Magnets (PM) with considerable energy density led to the development of dc machines with PM field excitation in the 1950's. Introduction of PM to replace electromagnets, which have windings and require an external electric energy source, resulted in compact dc machines (Islam et al., 2011). The synchronous machine, with its conventional field excitation in the rotor, is replaced by the PM excitation; the slip rings and brush assembly are dispensed with. With the advent of switching power transistor and silicon-controlled rectifier devices in later part of 1950s, the replacement of the mechanical commutator with an electronic commutator in the form of an inverter was achieved. These two developments contributed to the development of PM synchronous and brushless dc machines (Islam et al., 2011). The armature of the dc machine need not be on the rotor if the mechanical commutator is replaced by its electronic version. Therefore, the armature of the machine can be on the stator, enabling better cooling and allowing higher voltages to be achieved: significant clearance space is avilable for insulation in the stator. The excitation field that used to be on the stator is transferred to the rotr with the PM poles. These machines are nothing but 'an inside out dc machine' with the field and armature interchanged from the stator to rotor and rotor to stator respectively (Pillay and Krishnan, 1989). In this study contains design of speed controller for permanant magnet synchronous machines using Genetic algorithm based lower order modelling.

Speed controller design:
The design of the speedcontroller is important from the point of view of imparting desired transient and steady state characteristics to the speed-controlled PMSM drive systems (Islam et al., 2011). A proportional-plus-integral controller is sufficient for many industrial applications; hence, it is considered in this work. Selection of the gain and time constants of such a controller (Talebi et al., 2007) by using the symmetricoptimum principle is straightforward if the d axis stator current is assumed to be zero (Wallace, 1994). In the presence of a d axis stator current, the d and q current channels are cross-coupled and the model is non-linear, as a result of the torque term . Under the assumption that the d axis current being zero (i.e., r ds i =0), then the system becomes linear and resembles that of a separately-excited dc motor with constant excitation (Sharma et al., 2008). From then on, the block-diagram derivation, current loop approximation, speed-loop approximation and derivation of the speed controller by using symmetric optimum are identical to those for a dc motor drive speed controller design.

Block diagram derivation:
The motor q axis voltage equation with the d axis current being zero becomes (Sharma et al., 2008): And the electromechanical Eq. 2 is: where, the electromagnetic torque is given by Eq. 3: And if the load is assumed to be frictional, then Eq. 4: Which, upon substitution, gives the electromechanical Eq. 5 as: The frictional torque coefficient is Eq. 6: And torque consatnt is Eq. 7: The Eq. 1 and 5, when combined into a block diagram with the current-and speed-feedback loops added (Sharma et al., 2008) are shown in Fig. 1.
The inverter is modeled as a gain with a time lag (Talebi et al., 2007) by Eq. 8-10: where, V dc is the dc-link voltage input to the inverter (Islam et al., 2011), V cm is the maximum control voltage and f c is the switching (carrier) frequency of the inverter. The induced emf due to rotor flux linkages, e a , is Eq. 11: Where: This current-loop transfer function (Krishnan and Ramaswami, 1974) is substituted in the design of the speed controller as follows.

Proposed method of model reduction:
The proposed method of model reduction is Cross Multiplication of Polynomials Model order reduction method. It consists of the following steps in the system approximation process.
Step-1: The denominator and numerator polynomial constant terms in the reduced order model are obtained through Pade approximation: The transfer function of higher order (n th ) is considered as Eq. 14: G (S) can be expanded into a power series about S = 0 of the form Eq. 15-17 (Shamash, 1975): Where: With: The d i are directly proportional to the time moments of the system, assuming the system is stable Eq. 18 (Shamash, 1975 Then for R (s) to be Pade approximant of G (S), the following Eq. 19 and 20 are obtained: From the Eq. 16 and 20: From the Eq. 21, let Eq. 22: Step-2: The unknown coefficients of different powers of 's' remaining in reduced order model are determined: The given higher order system transfer function is equated and cross multiplied with k th order general transfer function. This process yields (n+2) equations with (2r-1) unknown reduced order transfer function coefficients. This step is similar to the model order reduction method proposed in Manigandan et al. (2005), The coefficients of same power of 's' on both side of the Eq. 24 equated with each other (Ramesh et al., 2008) and is given by Eq. 25: The (n+2) Step-3: The cumulative error index (J) for initial reduced order model is calculated ( where, ζ is the damping ratio and ω n is the undamped natural frequency of oscillation in rad/sec. The values of T 1 and T 2 corresponding to Eq. 28 can be computed as T 1 = T g and The unit step input time response of the initial second order approximant G ri (S) is analyzed with a computer program and its characteristics are noted. The cumulative error index J using the integral square error of the unit step time responses of the given higher order system G(s) represented by Eq. 15 and the initial second order approximant G ri (S) represented by Eq. 29 is calculated. The cumulative error index J is calculated using the formula Eq. 30: where, y(t) is the output response of the higher order system at the N th instant of time, y r (t) is the output response of the second order model at the N th instant of time and N is the time interval in seconds over which the error index is computed.   After giving the above parameters to GA the PID controllers can be easily tuned and thus system performance can be improved (Thomas and Poongodi, 2009).
Step-4: Find the PID Controller Constants using GA: GA can be applied to the tuning of PID controller gains to ensure optimal control performance at nominal operating conditions. The block diagram for the entire system is given below in Fig. 4 and also the genetic algorithm parameters chosen (Thomas and Poongodi, 2009) for the tuning purpose are shown below in Table 1.The constants K p , K i and K d are determined using Genetic Algorithm (GA) approach (Mahony et al., 2000). The Controller design for resultant reduced order model will closely match with the corresponding higher order model. After giving the above parameters to GA the PID controllers can be easily tuned and thus system performance can be improved.

Speed controller design by proposed method:
Transfer function Approach: Original Higher order system without speed controller and filter: Let G(s) be the transfer function of the original higher order system. The transfer function of PMSM drive system without speed controller and filter is as follows Eq. 31: Reduced order system: Let G r (s) be the transfer function of the reduced order system (Portone, 1997 From Eq. 29, the reduced order model is obtained as Eq. 33 (Ravichandran, 2007 Speed controller: To obtain an optimum transient response of the system, a PID controller is chosen with transfer function Eq. 34 (Kuo and Golnaraghi, 2003;Ogata, 2010): Where: K p = Proportional gain K i = Integral gain K d = Derivative gain The Values of K p , K i and K d are obtained by Genetic Algorithm (GA) approach. The resultant K p , K i and K d values are tabulated as shown in Table 2.
Speed controller design for reduced system: The block diagram of reduced system with speed controller is shown in Fig. 5. where, K p , K i and K d values are 17.70713, 31.7933 and 0.00407 respectively.
Speed controller design for original system: The K p , K i and K d values of speed controller of Original system is same as that of reduced system. Using this value the speed controller of original system is done. Fig. 6 shows the block diagram of original system with speed controller.

Design specifications:
The system is tested with unit step input and the design procedure is followed based on the following design specifications: Maximum peak overshoot = less than 3% Settling time = less than 3 sec Steady state error = 2% (assumed for optimum response) Before proceeding on to the simulation, the starting values of the parameters of controller are deduced using newly proposed procedures Step response of original system with speed controller Simulation results: The effectiveness of the newly proposed scheme for the design of PID Speed controller for PMSM drives are demonstrated using computer simulations. The system is simulated for step input using MATLAB-SIMULINK software with and without controllers (Chapman, 2002). The output responses of the above simulation studies are given in the following Figures. Step response of original system: The step response of the PMSM drive system is shown in Fig. 7.
Step response of reduced system: The step response of the reduced order PMSM drive system is shown in Fig. 8.
Step response of original system with controller: The step response of the PMSM drive system with Speed controller by proposed method is shown in Fig. 9. Compared with PMSM drive with conventional speed controller it gives better performance as listed in Table 3.
Step response of reduced system with controller: The step response of the reduced order PMSM drive system with Speed controller by proposed method is shown in Fig. 10.
Step response of original system with conventional controller: Step response of original system with conventional speed controller of PMSM drive is shown in Fig. 11. Step response of original system with conventional speed controller

MATERIALS AND METHODS
For an ideal control performance by the PID controller, an appropriate PID parameter tunning is necessary (Oi et al., 2008). Mostly used PID controller tunning methods for drive controls are Zigler-Nichols method and symmetric optimum tunning method.These tunning methods are very simple, but cannot guarantee to be always effective. To surmount this inconvenience, optimization procedure may be used for the better design of controllers.
Genetic algorithm (GA) methods have been widely used in control applications.The GA method have been employed successfully to solve complex problems.The use of GA methods in the determination of the different controller parameters is effective due to their fast convergence and reasonable accuracy.This work the parameters of the PID speed controller is tuned using Genetic algorithm.

RESULTS
In this study, the performance of a PMSM drive with MOR based speed controller is evaluated on the basis of rise time, settling time and maximum overshoot. The performance of the drive system with MOR based controller has been improved as compared with the conventional PI speed controller (Singh, 2006). Table 3 gives the response of the drive system. The simulation result (Fig. 9) shows that the response of the PMSM drive system with MOR based speed controller is better as compared to conventional method ( Table 3). The speed control loop of the drive is simulated with a Conventional controller; in order to compare the performances to those obtained from the respective MOR based drive system (Rahman, 2003).

DISCUSSION
The dynamic and steady state performance of the MOR based speed controller for permanent magnet synchronous motor drive is much better than the Conventional PI speed controller. All the comparisons for the different cases are tabulated in Table 3.

CONCLUSION
The model order reduction method proposed in this study gives better approximated reduced order model for the given PMSM drive system. Because of this we get the reduced order system performance as close as possible to the higher order system response. This will result in reduction in design cost and system complexity. The method proposed in this study are applied for the Speed controller design of PMSM drive. This study focuses on the reduction of models it minimizes the complexity involved in direct design of PID Speed Controller. The approximate values for PID Controller parameters are calculated from the Genetic algorithm approach and suitably tuned to meet the required performance specifications. The tuned values of these controller parameters are attached with the original system and its closed loop response for a unit step input is found to be in good accord with the response of reduced order model.