IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: 2278-684,p-ISSN: 2320-334X, Volue 3, Issue 6 Ver. II (Nov. - Dec. 206), PP 46-56 www.iosrjournals.org Modeling and Paraeter Identification of a DC Motor Using Constraint Optiization Technique Surajudeen Adewusi (Mechanical Engineering Departent, Jubail University College, Royal Coission for Jubail and Yanbu, P. O. Box 0074, Jubail Industrial City 396, Saudi Arabia). Abstract: This paper presents atheatical odeling and siulation, and experiental study of an electroechanical syste (a direct current otor) to identify the echanical and electrical paraeters (ass oent of inertia, daping, arature resistance and otor constant) of the otor and its accessories. Laboratory experients were perfored to obtain the step responses of the otor. The paraeters were identified in two stages viz: ) deterination of the tie constant and gain of the first order transfer function of the otor s odel using MATLAB Paraeter Identification Toolbox. 2) MATLAB codes were written to invoke MATLAB Constraint Optiization Toolbox and iniize the root ean squared error between the experiental and siulated step responses using the relationships between the otor paraeters and the tie constant and gain as constraint equations. The coparisons between the experiental and siulated step responses showed excellent agreeent. The echanical and electrical paraeters of the otor were found to be the following: ass oent of inertia = 9.85 x 0-3 kg 2, arature resistance = 6.29 Ohs, daping = 2.52 x 0-3 Ns/rad, and otor constant =.57 x 0-2 N/Ap. Keywords: dc otor odel, constraint optiization, paraeter identification, tie constant, step response. Noenclature DC Direct current ν(t) The otor's input voltage (Volts). R The otor's arature resistance (Ohs). L The arature's inductance (Henry) i The arature's current (Ap) k DC otor constant (N/Ap.) v b The otor's back electrootive force (ef) (Volts). τ Motor s supplied torque (N) T Tie constant (seconds) b Daping of the DC otor and accessories (Ns/rad). I Mass oent of inertia of the flywheel (kg 2 ). J Cobined ass oent of inertia of the flywheel and shaft and arature (kg 2 ). G Motor gain or proportionality constant ω Angular speed (rad/s) PI Proportional and integral controller TQ TecQuipent (Tradeark for servo otor anufacturer) I. Introduction Electroechanical devices (DC and alternating current otors) are widely used as prie overs for echanical systes and achines. In soe cases where autoation and/or control of the echanical systes is required, a control unit is attached to the electric otors and these type of otors are generally referred to as servo otor. Due to the iportance of the servo otors in the autoation of systes and processes, research studies on the characterization, atheatical odeling and paraeter identification of electroechanical devices have been published [-6]. Modeling and siulation of physical systes, a realistic coproise, are widely used in engineering for better understanding of the characteristics of systes in order to control and/or iprove systes perforance and reduce costs by building and testing a prototype at the preliinary stage instead of the exact achine. The concepts of odeling and paraeter identification are ore useful during odification of existing systes when little or no inforation about the existing syste is available. This paper is the outcoe of the author s efforts during the developent of laboratory anual for Systes Dynaic and Control course at the Departent of Mechanical Engineering, Jubail University College, Saudi Arabia. The anual for the TQ servo trainer contains no inforation about the echanical and electrical paraeters or DOI: 0.9790/684-306024656 www.iosrjournals.org 46 Page
properties of the syste, which are needed to teach students about designing and testing of PI controller for servo otor to siulate speed control and responses. The echanical and electrical properties of electroechanical devices are needed to derive atheatical odels. Even if these paraeters are docuented in the user anuals, it ay be necessary to deterine the if the syste has been used for a few years. The reason for this is that these paraeters will change due to aging, wear and tear of the syste. Different paraeter identification techniques and algoriths have been developed and used by different researchers to deterine soe iportant echanical and electrical properties of various odels of electroechanical systes. For exaples, a ulti-objective elitist genetic algorith has been used to iniize the error between the odel and easured responses of a DC otor [7]. Artificial Neutral Network has also been used to deterine odel paraeters and control dynaic systes [8]. One of the advantages of neutral network is that paraeter identification could be done online while the syste is working by using adaptive learning ethod. Furtherore, a paraeter estiation ethod using a block pulse function has been used to deterine the paraeters of DC servo otor systes [9]. The block pulse approxiates a function as a linear cobination of sets of orthogonal basis functions. The ost coon technique for paraeter identification is the least square error (LSE) ethod [0]. LSE involves the iniization of the su of the squared error between the odel and experiental responses. It should be noted that the different ethods of odel paraeter identification reported [, 2, 3, 5, 0] are siilar to a curve fitting, which involves changing the odel paraeters until a good atch between the odel and experiental responses is achieved. However, it is well-known that curve-fitting technique has ultiple solutions. This eans that there are infinite sets of values that would yield a good atch between the odel and experiental responses. Therefore, care needs to be taken to ensure that the obtained set of paraeters are realistic, that is the paraeters can give a unique solution, which can be related to the actual physical syste. A unique solution iplies that regardless of the assued initial paraeters while solving the error iniization of the odel and experiental responses, the final values of the paraeters will always be the sae. One of the ways to ensure the uniqueness of the identified paraeters is by using constraint optiization ethod, which checks the relationship between the paraeters in addition to the iniization of error between experiental and siulated responses. The constraint optiization technique has been used in the paraeter identification of odels for the huan hand-ar syste []. The reported studies [, 2, 3, 5, 0] did not check the uniqueness of the identified paraeters. The present study uses constraint optiization technique in addition to the conventional root-ean-square error ethod in order to ensure that the identified DC otor paraeters yield unique solution. In this study, the echanical and electrical paraeters (ass oent of inertia, daping, arature resistance and otor constant) of an electroechanical device (TQ DC servo otor and accessories) are deterined fro a linear odel of the syste and experiental responses of the device. Both the MATLAB Syste Identification and Constraint Optiization Toolboxes are used for the paraeter identification of the odel for the electroechanical device in order to ensure the uniqueness of the identified paraeters. The paraeters of the electroechanical device are needed in order to design and siulate a PI controller for speed control of the DC otor for deonstration to students in Systes Dynaic and Control course. II. Methods 2. Experiental Set-up Figure shows the electroechanical device that was used in the present study, the TecQuipent (TQ) CE0 Servo Trainer Apparatus [2, 3]. The CE0 Servo Trainer Apparatus is designed for DC servo otor speed and position control using a typical PID controller (TQ CE 20 Controller). The relevant parts of the apparatus, shown in Fig., for the experient coprises a DC otor, a DC generator and a flywheel ounted on a coon shaft. An analogue input signal to the otor circuit in the range 0 to ±0V enables variable shaft speed of rotation in either direction to be achieved. Shaft speeds, up to 999 revolutions per inute (RPM), are continually sensed optically and indicated on a panel ounted digital eter. An analogue voltage signal proportional to speed, in the range 0 to ±0V, is available at an adjacent socket. The DC otor ay be loaded, statically or dynaically, using the DC generator. The full range of loads ay be applied by inputting an analogue voltage in the range 0 to 0V. This electronically varies the load on the generator. The DC generator was not used in the present study. The CE 0 servo trainer has two additional reovable inertia discs which ay be added to the flywheel peranently fixed on the coon shaft. An electrically operated clutch, enabled by a toggle switch, is used to study angular position responses by engaging an output shaft via a 30: reduction gearbox. The apparatus ay also be used for the practical introduction into the design, operation and application of controllers with linear and non-linear paraeters and inputs. Figure 2 shows the TQ CE20 external analogue controller for the CE0 Servo Trainer. The CE 20 Controller board consists of signal generator, proportional, integral, derivative controllers and their DOI: 0.9790/684-306024656 www.iosrjournals.org 47 Page
(a) (b) Figure : (a) TQ CE0 Servo Trainer; (b) scheatic diagra of the servo trainer showing all coponents. Figure 2: TQ CE 20 analogue controller. DOI: 0.9790/684-306024656 www.iosrjournals.org 48 Page
cobinations, and other accessories necessary to build either an open loop or closed loop control actions for the CE 0 servo trainer. Since all of the power supplies for the built-in devices and systes are included, only low power (0 to ±0V) connections between the CE0 and the external controller are required. Finally, Fig. 3 shows the PC in which the TQ CE2000 software is installed for input and output data acquisitions, analysis, display and storage. The CE 2000 software has a dongle without which the software will not work. Figure 3: PC with CE 2000 software to easure and display responses of the DC servo otor. 2.2 Experiental Setup to Measure Responses of the DC Motor to Step Input Figure 4 shows the scheatic diagra for the experiental set-up to easure the open-loop responses of the DC otor to step inputs. The equipent used include CE0 Servo Trainer, CE20 Controller (only signal generation unit), CE 2000 Software installed on PC and accessories (connecting wires/cables). Prior to To TQ CE2000 software Figure 4: Experiental set-up for step response [3]. DOI: 0.9790/684-306024656 www.iosrjournals.org 49 Page
the experient, the CE0 s clutch for the gear syste was disengaged and the additional two reovable inertia discs were also reoved. Step input to the DC otor was achieved by using square wave signal of 0.5 Hz fro the signal generator on the CE20 Controller. The transitions in the square wave signal provide step changes in the input, and the output of the speed sensor will be a series of step responses, as shown in Fig. 5 fro the CE 2000 Software onitor. Figure 5: Experiental responses to step input. III. Matheatical Modeling The atheatical odel of the DC servo otor is derived fro the scheatic diagra shown in Fig. 6. The Load Generator in Fig. 6(a) was not used in the study hence the load control voltage v is zero, however the ass of the rotating parts of the generator shaft is considered in the total ass oent of inertia J of the syste. Fig. 6(b) shows a siplified scheatic diagra of the DC otor and its accessories. For the echanical coponents of the DC otor, Newton s second law of otion can be applied to obtain the atheatical odel equation of otion. Newton s second law can be applied as: d b J () dt Figure 6: (a) Scheatic diagra of DC otor and accessories [3]; (b) Siplified scheatic diagra of DC otor and accessories. DOI: 0.9790/684-306024656 www.iosrjournals.org 50 Page
where τ represents the DC otor s supplied torque (N), b is the effective viscous daping of the bearings in Ns/rad, J is the effective ass oent of inertia of the flywheel, and the shaft and arature of the DC otor and generator in kg 2, dω/dt is the angular acceleration of the shaft of the DC otor in /s 2, and ω is the angular speed of the DC otor in rad/s. The atheatical odel of the electrical circuit of the DC otor can be obtained by applying Kirchoff s Voltage Law (KVL). The electrical circuit equation is: di v( t) Ri L dt where v(t) is the input voltage to the DC otor in volts, R and i are the resistance and the current of the arature, respectively, L is the inductance of the arature windings, and v b is the back ef. According to Faraday s laws of electroagnetic induction, the torques generated by the DC otor is proportional to the arature current, and the otor's back ef is proportional to the angular speed, where the proportionality constant is the otor constant k. Faraday s laws can be expressed as: k i (3) vb k (4) Taking Laplace transfors and cobining Equations () through (4), the transfer function relating the output speed ω(s) to the input voltage V(s) can be expressed, in frequency doain, as: k V ( s) ( s) ( sj b)( sl R) k 2 The transfer function can be siplified by using the fact that the inductance L of the arature circuit is usually sall copared with the inertia of the flywheel. This will give the following first order transfer function: ( s) G V ( s) Ts where T, the tie constant, is given as: JR T = 2 br k and G, the gain of the DC otor, is given as: k G 2 br k 3. Paraeter Identification Techniques The echanical paraeters (the effective ass oent of inertia J and daping b) and the electrical paraeters (arature resistance R and servo otor constant k ) are identified in two stages. The first stage involves the use of the Syste Identification Toolbox in MATLAB to deterine the tie constant T (Eq. (7)) and the otor gain G (Eq. (8)) using the transfer function of the atheatical odel expressed in Eq. (6). It should be noted that the tie constant T of the electroechanical device and the otor gain G are related to the echanical and electrical paraeters of the servo otor that are to be deterined, as shown in Eqs. (7) and (8), respectively. The second stage of the analysis of the experiental response of the servo otor was the writing of codes for Constraint Optiization Toolbox in MATLAB to deterine the echanical and electrical paraeters (J, b, R and k related by Eqs. (7) and (8)) by iniizing the error between the experiental and odel v b responses using the root-ean-square ethod. These two stages are to ensure the uniqueness of the paraeter identification procedures. 3.. Deterination of tie constant and gain of the DC otor The tie constant T and the proportionality constant or gain G of the DC otor and accessories were first estiated fro the experiental step response graph shown in Fig. 5 using the established concept presented in Fig. 7. The tie constant is the tie, in seconds, that the syste response takes to reach 63.2 % of the steady-state response and the proportionality constant or gain can be deterine fro the steady-state value of the response, such that: (5) (6) (7) (8) (2) DOI: 0.9790/684-306024656 www.iosrjournals.org 5 Page
U G = steady-state value of response (9) Figure 8 shows the experiental output response to step input after the DC offset was reoved fro Fig. 5 to ensure that the response starts fro zero at tie equals to zero to confor to Fig. 7. The aplitude of the step input U is obtained fro Fig. 8 as 3.25 Volts, the steady state output response was also obtained fro Fig. 8 as 3.045 Volts. Fro Eq. (9), G was calculated to be 0.974. Siilarly, the tie constant T was calculated to be 0.5 seconds. These values of T and G were used as the starting values (initial guess) in MATLAB Syste Identification Toolbox, which is discussed in the next subsection. u(t) Figure 7: Standard step response graph [3]. Figure 8: Experiental step response of the DC servo otor. 3..2 Deterination of tie constant and gain of DC otor using MATLAB syste identification toolbox In other to ensure accurate deterination of the tie constant T and the gain G of the DC otor and accessories, MATLAB Syste Identification Toolbox was also used. The Syste Identification Toolbox in MATLAB is used for building accurate, siplified odels of coplex systes fro tie-series data. It provides tools for creating atheatical odels of dynaic systes based on easured output and input data, and an expression for transfer function. The toolbox features a flexible graphical user interface (GUI) that aids in the organization of data and odels. Figure 9 shows the ain graphical user interface of the syste identification toolbox in MATLAB. The toolbox has the option of specifying the starting values and the range within which the variable paraeters could be changed during the error iniization coputations. The values of T and G obtained in section 3.. above were used as the starting values (initial guess values) for the MATLAB Syste Identification Toolbox as shown in Fig. 9. DOI: 0.9790/684-306024656 www.iosrjournals.org 52 Page
Figure 9: MATLAB Syste Identification Toolbox GUI The experiental input and output data, shown in Fig. 8, corresponding to the first step response were iported into the MATLAB Syste Identification Toolbox using the GUI shown in Fig. 9. First order transfer function, siilar to Eq. (6), was selected in the MATLAB Syste Identification Toolbox. In Fig. 9, K corresponds to G, and T p corresponds to T in Eq. (6). The values of G and T obtained fro the MATLAB Syste Identification Toolbox are 0.9723 and 0.3846, respectively. The coparison of the experiental and siulated response graphs is presented and discussed in section 4 titled Results and Discussion. 3..3 DC otor paraeters identifaction using MATLAB constraint optiization toolbox The four paraeters of the DC otor and its accessories are related to the two paraeters obtained in subsection 3..2 by Eqs. (7) and (8). In order to ensure the uniqueness of the paraeters, Constraint Optiization Toolbox in MATLAB was used. As stated earlier, the ajor challenge with syste paraeters identification is finding a unique solution since there are several cobinations of the syste s paraeters that could ake the error between easured and siulated responses a iniu. In order to ensure the uniqueness of the solution of the optiization proble, the paraeter J was estiated by calculating the ass oent of inertia of various coponents that could be easily calculated (flywheel and shaft). Furtherore, reported values for R, b and k [2, 6] were used as starting values (initial guess values) for the constraint optiization algorith to deterine the paraeters of the DC servo otor for speed control. The paraeters are deterined such that the root-ean-square error between the odel and easured responses are iniized and the values of J, R, b and k satisfy the relationships between paraeters of the DC otor and the tie constant T and the gain G as expressed in Eqs. (7) and (8). To accoplish this task, three MATLAB -files were written. The first -file is the ain file that was used to call the other two -files. The second file contains the objective function to be iniized, which is basically the MATLAB codes for the root-ean-square error between the odel and easured responses. The results of the second -file is indicated as f(x) in Fig. 0. The third -file contains the constraint equations, Eqs. (7) and (8), the results of which is shown as Max Constraint in Fig. 0. Figure 0 shows a screen shot of the MATLAB files and output window for the paraeter identification using constraint optiization technique. The iniized value of f(x) was found to be 0.0678 and the value of the constraint equation was found to be 8.274 x 0-3. These values indicate that the solution of the constraint optiization algorith converged. The coparison of the experiental and siulated response graphs is presented and discussed in section 4 titled Results and Discussion. DOI: 0.9790/684-306024656 www.iosrjournals.org 53 Page
Figure 0: MATLAB files and output windows for constraint optiization. IV. Results and Discussion Figure shows the coparison of the siulated and easured responses based on the values of G = 0.9723 and T = 0.3846 seconds that were obtained fro the MATLAB Syste Identification Toolbox. The two responses are siilar except when tie is less than.5 seconds where there is a slight difference in the easured and siulated responses. The corresponding values of G and T that were directly calculated fro the experiental response graph (Fig. 8) are 0.974 and 0.5 seconds, respectively. These values were only used as initial guess values since they are expected to be less accurate than those values obtained fro MATLAB Syste Identification Toolbox, which iniizes the error between the easured and siulated responses. Therefore, the transfer function for the first order odel of the DC servo otor is found to be as follows: ( s) G 0.9723 (0) V ( s) Ts 0.3846 s It should be noted that the output of the DC servo otor is speed in revolution per inute (RPM) or radians per second but the response graphs in Figure are expressed in voltage (volts) for convenience. The output voltage of the DC otor could be converted to speed in RPM by using a sensitivity factor of 93.75 RPM/volt, which is reported in the user anual for the equipent [3]. The steady-state output response of the DC servo otor is 3.25 volts (Fig. ), this corresponds to 605.5 RPM. The coparison of the easured and siulated step responses of the DC otor using the identified paraeters fro MATLAB Constraint Optiization Toolbox are presented in Figure 2. The figure shows good agreeent between the experiental and siulated responses of the syste above.5 seconds, siilar to Fig.. A slight deviation between the experiental and siulated responses is observable below.5 seconds in both Figs. and 2. This ay be attributed to error in siulation when there is a sudden change in the value of step input signal fro zero to 3.25 volts (Fig. 8). The echanical and electrical paraeters of the DC otor that were used to obtained the siulated response in Fig. 2 are: ass oent of inertia J = 9.85 x 0-3 kg 2, arature resistance R = 6.29 Ohs, daping b = 2.52 x 0-3 Ns/rad, and otor constant k =.57 x 0-2 N/Ap. These values were obtained fro the constraint optiization. DOI: 0.9790/684-306024656 www.iosrjournals.org 54 Page
Responses (Volts) Resposes (Volts) Modeling and Paraeter Identification of a DC Motor Using Constraint Optiization Technique 3 2.5 Measured Siulated 2.5 0.5 0 0 0.5.5 2 2.5 3 3.5 Tie (seconds) Figure : Coparison of experiental and siulated responses using MATLAB Syste Identification Toolbox for deterination of G and T. 3 2.5 Siulation Experient 2.5 0.5 0 0 0.5.5 2 2.5 3 3.5 Tie (seconds) Figure 2: Coparison of experiental and siulated responses using MATLAB constraint optiization toolbox V. Conclusion The echanical and electrical paraeters (ass oent of inertia J, daping b, arature winding resistance R and direct current otor constant k ) of a DC servo otor (TQ CE 0 servo trainer) and its accessories were obtained. These paraeters are not reported in the user anual for the device by the anufacturer but the paraeters are needed for a proper design and testing of speed controller for the DC otor to enhance students learning. Laboratory experients were perfored to obtain the step responses of the DC otor and a atheatical odel was also derived for the DC otor in ters of the four unknown paraeters. The paraeters were obtained by iniizing the root-ean-square error between the odel response and experiental step response using Syste Identification and Constraint Optiization Toolboxes in MATLAB. The results of coparison between the experiental and siulated step responses showed excellent agreeent and the corresponding echanical and electrical paraeters of the DC otor were found to be ass oent of inertia J = 9.85 x 0-3 kg 2, arature resistance R = 6.29 Ohs, daping b = 2.52 x0-3 Ns/rad, and otor DOI: 0.9790/684-306024656 www.iosrjournals.org 55 Page
constant k =.57 x 0-2 N/Ap. These paraeters were used to prepare laboratory anual on design and siulation of Proportional-Integral controller for speed control, which is a topic in the syllabus for Systes Dynaic and Control course at the Departent of Mechanical Engineering, Jubail University College, Saudi Arabia. This work could be extended to study the position control and responses of the TQ CE 0 servo trainer when the clutch is engaged to include the gearbox. References []. N Sinha, C Dicenzo and B Szabados, Modeling of DC otors for control applications, IEEE Trans. Industrial Electronics and Control Instruentation, 2, 974, 84-88. [2]. W. Lord and J. H. Hwang, DC servootors odeling and paraeter deterination. IEEE Trans. Industrial Applications, 3, 973, 234-243. [3]. S Zahidi, R Dhaouadi and S Meber, Siultaneous identification of the linear and nonlinear characteristics of otor drives using Dynaic Wavelet Networks, Proc. of 0 th IEEE Conf. on Industrial Electronics and Applications (ICIEA), Auckland, New Zealand, 205, 888-892. [4]. J V Prisco and P A Voglewede, Dynaic odeling of a belt driven electroechanical XY plotter cutter, Journal of Coputational Nonlinear Dynaics, 0(2), 205, 2450-24507. [5]. C S Chen, S C Lin, S M Wang, Y M Hong, Adaptive control based on reduced-order paraeter identification and disturbance observer for linear otor, Proc. of ASME 2007 Conf. on International Manufacturing Science & Engineering, Atlanta, Georgia, USA, 2007, 75-758 [6]. S Saab and R Abi Kaed-Bey, Paraeter identification of a DC otor: an experiental approach, IEEE Trans. on Electronics, Circuits and Systes, 2, 200, 98-984. [7]. A Dupuis, M Ghribi and A Kaddouri, Multiobjective genetic estiation of DC otor paraeters and load torque, Proc. of IEEE International Conference on Industrial Technology, Busan, Korea (South), 2004, 5-54. [8]. S. Weerasooriya and M. El-Sharkawi, Identification and control of a DC otor using back-propagation neural networks, IEEE Trans. on Energy Conversion, 6, 99, 663-669. [9]. T Hanaoto, Y Tanaka, I Karube, T Mochizuk and Z Xu, A paraeter estiation ethod using block pulse functions for DC servo otor systes, Proc. IEEE International Conference on Industrial Electronics, Control, Instruentation, and Autoation, Power Electronics and Motion Control 3, 992,288-293. [0]. R Krneta, S Antic and D Stojanovic, Recursive least squares ethod in paraeters identification of DC otors odels. ELEC. ENERG, 8(3), 2005, 467-478. []. S Adewusi, S Rakheja, and P Marcotte, Bioechanical Models of the Huan Hand-ar to Siulate Distributed Biodynaic Responses for Different Postures International Journal of Industrial Ergonoics 42, 202, 249-260. [2]. http://www.tecquipent.co/control/control-engineering/ce0.aspx [3]. User Manual for TQ CE 0 Servo Trainer, (TecQuipent Ltd., Bonsall Street, Long Eaton, Nottingha NG0 2AN, England). DOI: 0.9790/684-306024656 www.iosrjournals.org 56 Page