Simulation and Modeling of DC Motor by Current Response Model Abstract––Pasek described a testing procedure for determining the dc parameters from the current response to a step in the armature voltage motor. Time moments have been introduced in automatic control because of the analogy between the impulse response of a linear system and a probability function. This basic idea has generated applications in identification, model order reduction and controller design. In this paper, two developed identification algorithms based on the moments and Pasek’s methods, are introduced and applied to the parameter identification of a dc motor. The simulation and experimental results are presented and compared. Index Terms––Identification, moments method , Pasek’s identification method, dc motor. INTRODUCTION The DC motor is the obvious proving ground for advanced control algorithm in electric drives due to the stable and straight forward characteristics associated with it[1], [2]. It is also ideally suited for trajectory control applications. The requirement for high performance speed control of dc motors has produced great research efforts in the application of modern control theory[1], [2], [4]. System identification is the subject which deals with the problem of building mathematical models of dynamical systems based on the observed system data [2]. The actual identification of models from data involves decision making on the part of the person in search of models, as well as fairly demanding computations to furnish bases for these decisions. There are situations when identification is necessary even though a relatively accurate mathematical model is available. For example, the dc motor parameters might be subject to sometime variations, in the cases, a mathematical model that is accurate at the time of the design may not accurate at a later time. In the literature there are many classical methods to identify theses parameters [1]-[6]. Time moments have been introduced in automatic control because of the analogy between the impulse response of a linear system and a probability function [6],[7]. Thus, an impulse response is characterized by infinity of moments, practically, only the first ones are necessary as for a probability density function. This basic idea has generated applications in identification, model order reduction and controller design, known as the method of moments. Pasek described a testing procedure for determining the parameters of a dc motor linear model from the current response to a step in the armature voltage of the motor. The Pasek’s equation for the parameter determination has been derived in his work [8]. This paper is structured as follows. Section II, describes the dynamic of the separately excited dc motor. Section III, describes a new identification algorithm for dc motor parameters by Pasek’s technique. Section IV, describes the method of moments and analyzes the developed identification algorithm, applied to the parameter identification of a dc motor. In section V, experimental results are illustrated and compared to the simulation results. Finally, conclusions of the paper are summarized in section VI. DC MOTOR MODEL The dynamic of the separately excited dc motor may be expressed by the following equations (1) (2) Where K, Ra , La , J and f are respectively, the torque and back-EMF constant, the armature resistance, the armature inductance, the rotor mass moment of inertia and the viscous friction coefficient. ω(t), ia(t), Ua(t) and TL(t) respectively denote the rotor angular speed, the armature current, the terminal voltage and the load torque. DEVELOPED MODEL BASED ON THE PASEK METHOD Pasek proposed a simple method, which requires one test for the identification of the dynamic model of the separately excited dc motor, with the assumption: f=0 (viscous friction coefficient null)[8]. Accordingly we thought of developing a model of identification based on the Pasek model, by introducing the viscous friction coefficient. It is possible from a terminal voltage step ΔUa to determine the majority of the dc motor parameters that is also possible from the abrupt terminal voltage variation. We measure the initial and final currents and speeds values shown on table III. At the steady state we can write (3) (4) Where Ua0, ia0 and ω0 respectively denote the terminal voltage armature current and the rotor speed at initial regime subscripted "0". Ua1, ia1 and ω1 respectively denote the terminal voltage, armature current and the rotor speed at final regime subscripted "1". According to (3) and (4) the torque and back-EMF constant K can be written as (5) the steady state check (6) where  ΔUa, Δia and Δω respectively denote the terminal voltage variation, armature current variation and rotor speed variation. The armature resistance is given from (6) as (7) according to equations (8) and (9) we can obtain the two transfer functions of the armature current and rotor speed (8) (9) the armature current transfer function is given as (10) the rotor speed transfer function is given as (11) Where Electrical time constant Mechanical time constant Usually small coefficient Let us pose (12) for a terminal voltage step ΔUa the current form is given as Where are the transfer function poles With We determine the moment t1 when the current pass by its maximum, noted Δia(t1) we have (14) By dividing the two terms of equation (14) by τe we find (15) Let us pose (16) Then (17) Where L1, L2, L3 and L4 are shown on the appendix. From two equations (15) and (17) we deduce the abacuses(Fig. 7) which gives δ and according to λ. Measurements of t1, Δia(t1) and Δia(2t1) define the dc motor parameters thanks to the following step We calculate δ and an abacus gives λ The other abacus gives We deduce τe and λ τe = τm Finally from τe and λ τe we deduce respectively La and J the calculation of output current gain gives (18) from (18) we calculate (19) The static torque can be calculated from steady state as (20) METHOD OF MOMENTS The moments constitute the basis for a non classical representation of linear systems. The characterization of an impulse response by its moments is equivalent to the moment characterization of a probability density function[7]. Impulse response moments are system invariants. Like for a probability density function, it is not necessary to compute an infinity of moments to characterize with a good approximation the shape of the impulse response only the first ones are necessary to perform this characterization. A. Temporal Moment of a Function Let us consider a stable linear system, characterized by its impulse h(t) then, (21) H(s) can be expanded in Taylor series in the vicinity of (22) where is the n th order frequency moment of h(t) for ω=ω0, notice that is complex. In the particular case ω0=0, frequency moments correspond to classical time moments (23) they permit the characterization of H(jω) around ω0=0, as well as that of the impulse response h(t). A0(h) is the area of h(t), A1(h) defines mean time of h(t) and A2(h) deals with the dispersion of h(t) around its mean time[7]. Equation (22) is rewritten as (24) Let Then, time moments can be expressed as (25) B. Moments and Parameters of a Transfer Function Let y(t) the step response of the studied system. We proposes to identify the system by the model (26) from the final value theorem, as time approaches infinity for a stable linear system, the system response approaches a steady state value K1 given by (27) if a step input is applied to the system described in equation (26), by taking the Laplace transform of the normalized response gives (28) let us consider ε(t) an error function with (29) by introducing the Laplace transform in equation (29), (26) can be written as (30) the development of (30) gives (31) Then, using (24) we can write (32) according to (24) and (32) we can deduce the coefficients of the transfer function H(s) by solving the following matrix system (33) where An(ε) is the nth order temporal moment. C. Dc Motor Transfer Function and its Moments For our cases, when n=2 and m=1, the transfer function (8) becomes (34) system (33) is reduced to the following matrix system (35) the resolution of this matrix system (35), gives the following coefficients: , , (36) D. Parametric Identification After having deduced the mathematical forms which are used for calculation of the transfer function coefficients and which enable us at the same time to calculate the electric and mechanical motor parameters, we present here, the stages to be followed at the time of the determination of these parameters. the calculation of Ki and Kv gains of the tow outputs ia(t) and ω(t) respectively, by taking account equations (8) and (9) gives (37) (38) according to (37) and (38) we deduce f and μ (39) (40) by identification of H1(s) and H2(s) denominators with H(s) denominator we obtain (41) (42) according to (41) and (42) we can obtain a second order equation (43) The resolution of the equation (43) gives two roots one is positive, the other is negative (rejected). According to (41) and (42) we deduce τm The deduction of τm and τe gives La and J. The static torque can be calculated from steady state as (44) EXPERIMENTAL RESULTS The separately excited dc motor used for experimental tests, has the nominal characteristics shown on table IV. the first experiment is the determination of electric dc motor parameters according to direct tests, like armature resistance Ra and armature inductance La, as well as the back-EMF constant K. Mechanical parameters are also determined by direct tests ( static torque Tst and viscous friction coefficient f ). The deceleration test, enables us to determine the moment of inertia J. The second experiment to be carried out is to identify the dc motor parameters from the dynamic test. According to a step amplitude ΔUa of terminal voltage applied to the armature circuit of the dc motor. The initial and final values of the armature current and the angular speed obtained from this test, are shown on table III. The recording of ia(t) gives t1=0.026s where the current pass by its maximum. For the two identification methods ( developed Pasek’s and moments methods), the back-EMF constant K, armature resistance Ra, and static torque Tst can be determined by using respectively equations (5), (7) and (20). the viscous friction coefficient f, can be determined by using equation (19) for the first method (developed Pasek’s method) and by using (39) for the second (moments method). By using the first method and let us know Δia(t1) and Δia(2t1) , δ is then calculated from equation (16),( δ=0.79). from the abacus of Fig. 7 we deduce respectively λ = 4.5 and t1/ τe =1.82. let us know λ and τe we deduce τm and consequently J and La. Table I shows the first, second and third order moments values ,as well as the transfer function coefficients values successively calculated from the trapezoids method. Let us know a1, a2 and μ then, we have the second order equation , the resolution of this equation gives and ( τe2 is a rather large time constant, is thus rejected). The deduction of τe and τm enables to calculate J and La. Table II summarizes the values of the parameters calculated from the two identification methods (direct tests, moments and Pasek’s developed methods). Finally to check the precision of each method, we have simulated the dynamic test applying a step amplitude of terminal voltage ΔUa=188V, to the armature circuit of dc motor, as well as, deceleration test and mechanical characteristic. According to Figs. 3, 4 and 5 the curves simulated from the dynamic test parameters with the moment method are close to the real curve (experimental curve) that those simulated from the direct tests parameters and Pasek’s developed method. With regard to the steady state (Fig.6) the curves simulated from the dynamic test and direct tests parameters are almost identical and close to real measurement (experimental curve). CONCLUSION In this work, we tried to contribute our share in the discipline of the dc motors modeling. This contribution, which can be classified in a very wide field of identification methods can be summarized in the following results: we have developed a dynamic model based on the moments and Pasek’s methods. The moments method especially makes it possible to have a model closer to reality in transitory mode. We have proposed a comparison between various models based on the identification methods(direct tests, moments and Pasek’s methods), this comparison is made on the basis of real measurements taken in laboratory on a separately excited dc motor, with 180 W of rated power. It shows the advantage of the only dynamic test for identification, coupled to the moments method REFERENCES [1] A. Rubaai, R. Kotaru , “Online identification and control of a dc motor using learning adaptation of neural networks,’’ IEEE Trans. Ind. Aplicat., vol. 36, pp. 935-942, May/June 2008. [2] J.C. Basilio and M.V. 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