Parameter Estimation of DC Motor using Adaptive Transfer Function based on Nelder-Mead Optimisation

ABSTRACT


INTRODUCTION
The mathematical model such as state-space form or transfer function of any system is required to know the dynamic behaviour and design a proper controller to get the specific requirement responses without affecting the stability of the system. This demands the exact value of model parameters. Some parameters of the transfer function model of the system can be determined through conducting different experiments, but some other parameters are difficult to be determined. In addition to that, the accuracy of determined parameters is mainly depends on the accuracy of measuring instruments which is connected in the experimental system. In order to overcome the above difficulties, the replica of the actual system can be designed in the form of transfer function and the values are optimised, so that the responses of replica model named as adaptive model will exactly match with the corresponding response of actual (reference) system model.
In this paper, effort has been made to find the five parameters of separately excited dc motor named as: armature resistance, armature inductance, Back emf constant, Moment of Inertia and Viscous friction coefficient. The experimental data of armature current and speed response with respect to time are collected by running the separately excited motor at full rated voltage with load torque of 5 N-m using current sensor and voltage sensor respectively. The transfer functions of dc motors are built in MATLAB environment with randomly chosen initial parameters. The error between response of experimental data and that of randomly chosen initial parameters transfer function is used to formulate the objective function. The objective function is based on different integral criterion well defined in control engineering [1]. The Simpson's one-third rule is used for integration of objective function. The minimisation of the objective function in order to adapt the response of adaptive model to the experimental response can be realised by Nelder-Mead simplex direct  [2]. A Neuro-Genetic controller is used to control the dc motor speed by obtaining the controller parameter for each load torque [3]. An adaptive parameter adjustment mechanism is combined with fuzzy controller solve various problems associated with brushless dc motor drive [4]. A fractional order PID controller for dc motor is designed using Genetic Algorithm and Simulated Annealing and the result is compared with the PID [5].

Motivation, Objectives and Procedures
The main motivation factor has to find out the unknown parameters of system through adaption method. This can be achieved by simulating the unknown mathematical model repeatedly applying search method for fulfilment of the objective function. The objective is to design the adaptive model of the actual unknown model, which can be used to develop the control algorithms for controlling the actual system. a. The procedures set up by the authors can be summed up as follows b. Built the transfer function of DC motor model in simpower system in MATLAB environment with unknown variables and it will act as adaptive model. The experimental data of speed response and armature current response with respect to time are collected and placed in one dimensional lookup table in same MATLAB simulink file and it will be regarded as reference model response .The model should be stored with name. The model is shown in Figure 1. parameters are optimised till the matching occurs between the adaptive response and experimental data. The iterative optimised searching method is discussed briefly in subsequent sections.

MODEL OF SEPARETELY EXCITED DC MACHINE
The two equations are used to develop the transfer function of separately excited DC motor. The equations are as follows: (1) The above equation can be written in state space form as: where i a , ω m , and V are the armature current ( ampere) and mechanical speed (rad/sec) and armature voltage (Volt) respectively. The five unknown parameters R a L a , K b , J and B are armature resistance (Ω) and armature inductance (H), back emf constant (Volt-sec/rad.), moment of inertia (kg-m 2 ) and viscous friction (Volt-sec/rad.) respectively. In this paper efforts are made to determine the above five parameters.

OBJECTIVE FUNCTION
The objective function, also named as performance index is to solve a mathematical problem such that a system satisfies a performance specification expressed depends on the requirement of control engineer.

NELDER-MEAD "SIMPLEX" DIRECT SEARCH METHOD
In the mid-1960s, two English statisticians invented the Nelder-Mead "simplex" direct search method used for solving the unconstrained optimization problem [9]. The Nelder-Mead method iteratively generates a sequence of interested vertex points which converge to an optimal vertex point of objective function f(x) [10]. At each iteration, the vertices xi are ordered according to the objective function values (6) where x1 is the best vertex and xn+1 is the worst vertex. The algorithm uses four possible operations: reflection, expansion, contraction and shrink, each being associated with a scalar parameter: α (reflection), β (expansion), γ (contraction), and δ (shrink).The values of α, β, γ and δ are lying in the range of >0, >1 and 0 to 1 in both γ and δ respectively.
The one iteration of Nelder-Mead algorithm is as follows [ f. shrink: for define (11) and proceed to the next iteration.

RESULTS AND DISCUSSION
The parameters of separately excited dc motor can be determined by performing the various experiments in laboratory. The dc motor specification is given in machine specification data sheet. But, actually in some complicated system, it is difficult to ascertain the exact value of parameters because of inaccuracy of measuring instruments. Therefore, the adaptive method is the best solution for determination of unknown parameters of the system.
The data of speed encoder and current sensor corresponding to time are collected by running the dc motor at rated voltage and with load torque of 5N-m by using the speed encoder and current sensor. Two look up tables are formed based on the speed and current response data in MATLAB simulink environment. The transfer function of DC motor is also placed in same MATLAB simulink file. The transfer function of dc motor model will act as adaptive model because of unknown parameters are updated on each iteration. The responses of lookup table model will act as reference model. The current responses of reference and adaptive are stored in the workspace y_out1, whereas the speed responses are stored in y_out. The model should be stored in certain name. By calling this name the responses are passed to the objective function. The objective function is built in m-file of the MATLAB [11][12]. The simulink model is shown in Figure 1.
The optimised adaptive values are found out by considering the two objective functions.
where, is the difference of reference speed and adaptive speed and t is the time at that instant. Another two cases are also performed by adapting both current response and voltage response for considering both objective functions.
f 1  where, is the difference of reference speed and adaptive speed and reference armature current and adaptive current respectively. The results of 4 cases are compared and given in tabular form in Table 1. One of the optimisation results are given in appendix for reference. The program of objective function of ITAE used for to adapt the reference response and main calling function are also given in appendix.

CONCLUSIONS
In this paper a novel approach for estimation the parameters of separately excited DC motor is presented for enhance the students' knowledge. The approach is the adaption of response of transfer function of DC motor with unknown parameters with the experimental responses. Two well known objective functions are considered for error estimations. Nelder-Mead simplex direct search method is used to minimisation of objective function. Since the results will not match with the machine data sheet and also the armature current response is also deviated from the experimental data in both chosen objective function, the efforts are made to adapt both current and speed responses. The adaptive responses are shown in Figure 3 (a and b) confirms the accuracy of adapting if both current and speed responses are adapted, particularly in ITAE optimisation. The accuracy can be enhanced by using the other well known optimisation techniques. The content of this paper is not for accurate estimation, but to explain the procedures for adaption. The results of the adaptive algorithms are quite encouraging and therefore, suggest that it is a helpful not only for electrical engineer students but for other students, who works on finding the unknown parameters of their system.