A Tutorial on Dynamic Simulation of DC Motor and Implementation of Kalman Filter on a Floating Point DSP

Transcription

1 A Tutorial on Dynamic Simulation of DC Motor an Implementation of Kalman Filter on a Floating Point DSP Pamakumar S., Vivek Agarwal, an Kallol Roy Abstract With the avent of inexpensive 32 bit floating point igital signal processor s availability in market, many computationally intensive algorithms such as Kalman filter becomes feasible to implement in real time. Dynamic simulation of a self excite DC motor using secon orer state variable moel an implementation of Kalman Filter in a floating point DSP TMS32C673 is presente in this paper with an objective to introuce an implement such an algorithm, for beginners. A fractional hp DC motor is simulate in both Matlab an DSP an the results are inclue. A step by step approach for simulation of DC motor in Matlab an C routines in CC Stuio is also given. CC stuio project file etails an environmental setting requirements are aresse. This tutorial can be use with 673 DSK, which is base on floating point DSP an CC Stuio either in harware moe or in simulation moe. Keywors DC motor, DSP, Dynamic simulation, Kalman Filter. NOMENCLATURE R m Armature resistance in Ohms; L m Inuctance in mh, K b Back EMF constant (Volt-sec/Ra) K t Torque constant (Nm/A) J m Rotor inertia (Kg m 2 ) B m Mechanical amping factor K k Kalman gain P k Posterior co-variance Prior co-variance P ( k ) Q R T State noise Measurement noise. I. INTRODUCTION HIS paper presents a tutorial on ynamic simulation of DC motor on a Digital Signal Processor (DSP). The 32 bit floating point DSP processors are reaily available in the market an have high computing power an performance. This has helpe in implementing some of the computationally complex observer algorithms such as Kalman filter which offers excellent estimates of states even in the presence of system an measurement noise. Sensor-less estimation of inuction motor spee an GPS tracking are just a few examples, where these observer algorithms have been successfully use. S. Pamakumar an Kallol Roy are with the BARC, Mumbai, Inia 485 ( spama@iitb.ac.in; kallol.roykallol@gmail.com). Vivek Agarwal is with the Department of Electrical Engineering, IIT Bombay, Mumbai-476, Inia ( agarwal@ee.iitb.ac.in). The objective of this paper is to acquaint a beginner with what all is involve in the esign an implementation of such systems. As an example, a self excite fractional hp DC motor is moele using Matlab an the simulation results are presente for open loop time response of the state variables. The measurements are then corrupte by super imposing a noise, which follow a Gaussian istribution. A Kalman filter is implemente in Matlab to observe the states from the corrupte measurement signal. A comparison of the noisy measurement signal an the estimate states through Kalman Filter is presente. The same moel is implemente in TI floating point DSP C673 an the results incluing the Kalman Filter responses are also presente. The Matlab coe for all the simulations an the Coe Composer Stuio project file incluing the require heaer files are available for ownloa, from a link provie in Appenix-I. II. DC MOTOR MODEL IN MATLAB A secon orer, linear state space moel of DC motor is consiere for this stuy. The two states consiere are the armature current (I a ) an the spee (ω) of the DC motor. The input supply to the self excite DC motor is 2 V an the outputs are same as states. A. Continuous Time State Variable Moel of DC Motor / L B = m Rm / Lm K b / Lm A= Kt / Jm Bm / Jm (2); C = () (3) an D = [ ] (4) The continuous time state variable moel, which is use for this work has the system matrices as given by (-4). A= ; B = (5) C = an D = [ ] (6) Numerically, the system matrices are as given by (5-6). Continuous time state (I a an ω) response in Matlab for an input of 2 V DC supply is as shown in Fig.. It can be observe that the starting current is of the orer of 5 amps, which is taken as p.u an no loa current is.25 78

2 amps an the steay state spee is 5 rpm, which is again scale to p.u. PU spee an currrent Time in millisec Fig. Current an spee (scale) response of DC motor in Matlab B. Discrete Time DC Motor Moel The continuous time state space moel is iscretise with a sampling time of microsec, using the Matlab function c2. Discrete time state variable DC motor moel is as shown by equation (6) an is obtaine by Matlab function c A = ; B ; C [ ] =.352 = (9) The iscrete time simulation is as given in Fig. 2 which shows per unit combine current an spee response. against the actual p.u. value. III. APPLICATION OF KALMAN FILTER FOR DC MOTOR Kalman Filter is an optimal stochastic state observer for estimation of states which can take care of moel uncertainties calle system or process noise an measurement uncertainties or measurement noise [2]. X = ( ) A X ( k) + k+ B U ( k) T P = A P A Q ( k ) + + ( k) T ( T K ( k) = P C C P ( ) C R) ( k) k + X ˆ ( ) X k = + ( ) [ ( ) - ( ( ) )] ( k) K k Z k C X k P ( k) = ( I K C ) P k ( k) () (2) (3) (4) (5) It consists of a set of preiction equations (-2) an a set of correction equations (3-5). The Kalman filter algorithm flowchart is as per Fig. 3. It starts with initialization of states an initialization of process an measurement co-variance matrices. The process co-variance matrix has the imension of the A matrix of the system an the measurement co-variance matrix has a imension of the measurement or observation matrix. In this particular case, Q is taken as e-3 an R is taken as. This is base on an intuitive approach, with apriory knowlege of the system [3]. Initialise state variables an co-cariance matrices PU spee an current PU spee PU current K=K+ Time upate Compute X projection An P projection Time in millisec Fig. 2 Discrete time estimate of current an spee If the sampling time is increase to 2 millisecons, the system matrices in iscrete time omain are as shown below: A ; B ; C = = 6.34 = [ ] () By reucing the sampling time, the system may become nonobservable, as can be seen in Fig. 2, where the current estimate is not accurate. The peak value attaine is only.82 p.u. as Make measurement Measurement upate Compute Kalman gain Correct state variables Correct P projection Fig. 3 Kalman filter flowchart The DC motor is moele in an open loop moe an the time response an output ata set is generate. The measurement so obtaine is corrupte by a white noise of mean zero, which can be generate using the Matlab function 782

3 rann. With this corrupte measurement, the Kalman filter can estimate the correct amplitue of the state variable. Fig. 4 shows a corrupte measurement an Fig. 5 shows the estimate current an spee, in presence of noisy observations. The Matlab coe use in this tutorial is available for ownloa through the link provie. 3 PU measurent an states Measurement noise(a) samples Fig. 4 Measure current (corrupte by noise) PU spee PU measure current PU estimate current Time in millisec Fig. 5 Estimate current an spee from noisy measurements IV. DC MOTOR SIMULATIONS ON 673 DSP DC motor moeling an simulations on a floating point DSP is iscusse in this section. Spectrum igital DSK base on TI TMS32C673 processor is use for simulations. This DSK is base on 32 bit floating point processor, with both built in flash of 52 kb an 6 MB SDRAM. Besies, it has other aitional features which inclue a coec an a built in one channel 8 bit ADC an DAC. This DSK works on a single 5 volts power supply, with 4 number of on boar LEDs an a set of DIP switches. These DIP switches can be use for simulating limite external events, an the LEDs can be programme to inicate any specific conitions base on the outcome of the computations. The programming in one through USB interface an the evelopment platform for this prouct is coe composer stuio from TI. Fig. 6 gives the photo of DSK [4]. Fig. 6 DSK 673 (courtesy: Spectrum Digital ) The essential pre requisites are the coe composer stuio an the DSK. Initial training can be ha on the CC stuio itself, as there is a evice simulator moe available, which can simulate the results exactly as the DSK woul have given. During the setup, the specific DSK (673) is to be selecte for the simulator to function properly. The CC stuio [5] provies an integrate evelopment environment, primarily base on project concept. A project file will have many types of files incluing system, GEL, heaer, library an the user programs, which is written in either C or assemble language. User has to correctly select a list of system files pertaining to the specific processor, its library files an the heaer files. Simple tasks are provie in CC stuio to start with. After this, the project menu allows the user to compile iniviual files or buil the while project with all the inclue files to generate a COFF or an OUT file, which can be loae to the processor s flash an execute. The following are the steps to work on a DSK 673 environment. The pre-requisites are also given (first two steps). The project file can also be ownloae from the link provie.. A DSK 673 kit (Spectrum Digital). 2. Coe composer stuio (An evaluation version is available at TI site for ownloa) is to be installe in the PC, preferably in efault C rive. 3. The DSK can be power up an connecte to a winows base PC through a USB port. 4. Driver files are installe properly for the DSK. 5. It is avisable to loa both DSK 673 an the evice simulator at the setup stage, so that even if the DSK is not available reaily, the simulator can be use. 6. On starting CC stuio, the DSK will be etecte initially an no error messages will be given if everything is properly installe. An interactive pop up menu will turn on in case of any issue, which can be ignore, so that the simulator can be starte. 7. The project file, which is provie in the link can be ownloae an copie to my project foler within CC stuio. 783

4 8. Check all the project epenencies an compile through Buil all option. Select 67X option in compiler if it oesn t work properly. 9. The project will be compile an an out file will be generate by the compiler, with no errors an warnings.. From the File menu, loa this program, which is usually resiing in the efault project foler, uner the ebug sub-foler, with an.out extension.. The assembly coe will appear in the active right pane. From the ebug menu, give a run comman to execute the assembly coe of DSP. After successful execution, test passe message will appear an various parameters can be viewe through view menu. 2. Check the main program to an familiarize with variables such as observ for noisy observations, cur an sp for current an spee etc. From the view menu an the graph option (time/frequency) the tren of these variables can be seen by proviing the variable name, the number of ata (25) an the format of ata (32 bit floating point format). 3. CC stuio can hanle both simulation of 673 processor to a great exten an the DSK itself. The selection of the type of environment is to be given uring starting of the program. Fig. 7 gives the listing of the main C file, which is separately given in Appenix-II also. Typical project environment for the c motor simulation is shown in Fig. 8, which gives a list of files require to be inclue in the project environment. All the associate files are provie in the project file. The native compiler of CC stuio environment support C an C++. All the functions an main programs are to be written in C++ or assembly language for it to compile properly. Matlab can hanle the matrix operations easily, but for a DSP system evelopment, all the functions an programs are to be evelope in C or C++ only. Fig. 7 List of associate files require for compilation The coe for c motor simulations an the Kalman filter coe are provie in the annexure-. Fig. 8 shows the corrupte measurements provie to Kalman filter through a ata set an Fig. 9 shows the current estimation by Kalman filter in DSP an Fig. shows both the estimates as compute by the recursive Kalman filter iterations. Fig. 8 Measure current, which is corrupte by noise 784

5 Fig. 9 Estimate an actual current of c motor in DSP Fig. Estimate spee an current by Kalman filter (DSP) 785

6 V. CONCLUDING REMARKS A tutorial on DC motor moeling an simulations in Matlab an implementation of Kalman filter for estimation of states from corrupte or noisy measurement is presente. A step by step approach for the implementation of the same in a 32 bit floating point TI TMS32C673 base DSP an the results are also provie. This tutorial gives etails of how to implement ynamic moel simulations in iscrete omain an a stochastic filter implementation both in harware an simulation moe for a floating point DSP. The associate coes are available for ownloa, which are expecte to be helpful for the beginners, to get an experience with a floating point DSP. APPENDIX-I Web links for coe ownloa APPENDIX-II C routine for c motor simulation an Kalman filter in Coe Composer stuio P[][]= Pproj[][] * KC[][] + Pproj[][] * KC[][]; CUR[cnt]= xhat[][]; SPD[cnt]= xhat[][]; INNV[cnt]=innov; X[][]=xhat[][]; X[][]=xhat[][]; /* Exit from the program when certain count is reache */ if (cnt > TIMER_CNT) { TIMER_pause(hTimer); TIMER_close(hTimer); printf("\ntest PASSED\n"); exit(); } printf("\n Count : %3 ",cnt); APPENDIX-III DC motor ata use for the coe R m 2.6 Ω, Armature resistance; L m.238 mh, Inuctance K b.2352, Back EMF constant (Volt-sec/ra) K t.235, Torque constant (Nm/A) J m.7e-6, Rotor inertia (Kg m 2 ) B m 2e-7, Mechanical amping factor voi TimerEventHanler(voi) { /* process timer event here */ cnt++; AT[][]=A[][];AT[][]=A[][];AT[][]=A[][];AT[][]=A[][] ; Xhatproj[][] = A[][] * X[][] + A[][] * X[][] + B[][] * 2.; Xhatproj[][] = A[][] * X[][] + A[][] * X[][] + B[][] * 2.; AP[][]=A[][] * P[][] + A[][] * P[][]; AP[][]=A[][] * P[][] + A[][] * P[][]; AP[][]=A[][] * P[][] + A[][] * P[][]; AP[][]=A[][] * P[][] + A[][] * P[][]; APAT[][]=AP[][] * AT[][] + AP[][] * AT[][]; APAT[][]=AP[][] * AT[][] + AP[][] * AT[][]; APAT[][]=AP[][] * AT[][] + AP[][] * AT[][]; APAT[][]=AP[][] * AT[][] + AP[][] * AT[][]; Pproj[][]= APAT[][] + Q[][]; Pproj[][]= APAT[][] + Q[][]; Pproj[][]= APAT[][] + Q[][]; Pproj[][]= APAT[][] + Q[][]; INVCPCT =./(Pproj[][] + R); PprojCT[][]= Pproj[][]; PprojCT[][]= Pproj[][]; K[][]= PprojCT[][] * INVCPCT; K[][]= PprojCT[][] * INVCPCT; innov =(observ[cnt]- C[][] * Xhatproj[][]); xhat[][]= Xhatproj[][] + K[][] * innov; xhat[][]= Xhatproj[][] + K[][] * innov; KC[][]=.-K[][]; KC[][]=.; KC[][]= -K[][]; KC[][]=.; P[][]= Pproj[][] * KC[][] + Pproj[][] * KC[][]; P[][]= Pproj[][] * KC[][] + Pproj[][] * KC[][]; P[][]= Pproj[][] * KC[][] + Pproj[][] * KC[][]; REFERENCES [] Simulink Simpowersystem ocumentation, pp [2] Grewal M. S an Anrews A. P, Kalman Filtering theory an practice, book by Englewoo Cliffs, Prentice Hall, 993. [3] Peter. S Maybeck, Stochastic moels-estimation an control Vol., Acaemy press, New York, 982. [4] Spectrum Digital prouct catalog for DSK 673: [5] CC stuio programmer s manual by TI Pamakumar.S has grauate from Kerala University in 985, complete his post grauation from IIT Bombay in 987 an is unergoing Ph.D program at IIT Bombay. He is presently working as Engineer in charge of electrical systems refurbishment cell, research reactor maintenance ivision, BARC. He is associate with electrical systems maintenance of nuclear research reactors. Vivek Agarwal receive the bachelor s egree in Physics from St. Stephen s college, Delhi University, Inia. He then obtaine an integrate master s egree in Electrical Engineering from Inian Institute of Science, Bangalore. Subsequently he pursue a Ph.D. egree in the ept. of Electrical an Computer Engineering, University of Victoria, Canaa. After obtaining the Ph.D. egree, he briefly worke for Statpower Technologies, Burnaby, Canaa as a research engineer. In 995 he joine the epartment of Electrical Engineering, Inian Institute of Technology-Bombay, where he is currently a Professor. Kallol Roy complete his B.Tech (Electrical) from NIT, Calicut in 984. He obtaine his M.Tech. from IISc., Bangalore in 99 an PhD (Systems an Control) from IIT, Powai, Mumbai in 2. He has been Post-Doctoral Fellow at University of Alberta, Canaa, from April 25 to March 26. His expertise is in the Maintenance of Research Reactors, especially Planning, Retrofitting an Development of Preictive Maintenance tasks. 786