On the Real Time Implementation of a Controller for an Electromechanical System

Transcription

1 On the Real Time Implementation of a Controller for an Electromechanical System Ruben Salas-Cabrera, Jonathan C. Mayo-Malonao, Erika Y. Renon-Fraga, Euaro N. Salas-Cabrera an Aaron Gonzalez-Roriguez Abstract This paper eals with the experimental control of the rotor position of a DC Motor. A real time Linux-base software is use for implementing this close loop system. A setup for measuring the rotor position is implemente. This esign consists of several harware/software components. A iscretetime state observer is employe for calculating the transient an steay state values of the state variables. A loa torque ynamic moel is propose in orer to improve the close loop performance. Several tests are carrie out to ientify the parameters of the augmente ynamic moel that inclues the loa torque state equation. A custom-mae Pulse With Moulation-base Mosfet H-Brige converter is implemente as a power interface. Keywors: real time, ata acquisition, electric machine, electronics, microcontroller 1 Introuction The contribution of this work is not about the theoretical analysis an esign of an observer-base control law but its implementation an experimental results. In particular, this work eals with the experimental implementation of a state space controller for a DC motor that has separate wining excitation. In other wors, the fiel an armature winings of the electric machine are fe from inepenent sources, [1]. The output to be controlle is the rotor position. A brief review of the literature follows. An interesting contribution is presente in [2]. It is relate to the spee control of a series DC motor. Authors in [2] use a nonlinear ynamic moel for eriving a backstepping-base control law. The emphasis is on the analysis of the controller. Author in [3] uses RTAI-Lab an Scilab for implementing a transfer function-base controller for a DC motor. The main contribution in [3] is relate to testing several open source real time tools (RTAI, RTAI-Lab, Authors wish to thank Michael Griffin an Carlos A. Cruz- Villar for their valuable suggestions uring the early stage of this work. This work was supporte in part by the Instituto Tecnologico e C. Maero an the Fono Mixto e Fomento a la Investigacion Cientifica y Tecnologica CONACyT-Gobierno el Estao e Tamaulipas. Authors are with the Instituto Tecnologico e C. Maero, Departamento e Ingenieria Electrica y Electronica an Division e Estuios e Posgrao e Investigacion, Av. 1o. e Mayo S/N, C. Maero, MEXICO. salascabrera@aol.com Comei, Scilab, xrtailab). Experimental results regaring high gain observers are presente in [4]. RTAI-Lab an Scicos/Scilab are use for the real time implementation. The system to be teste in [4] was a series DC motor. Valuable information about a problem associate with the analog outputs of the acquisition car was presente in [4]. Linux an RTAI-Lab are the open source real time platform that we employe in this implementation [5], [6]. The real time program that we implemente consists of a iscrete-time state feeback incluing an integrator, a full orer iscrete-time state observer an a rotor position sensing algorithm. In this control law implementation the only state variable that is measure is the rotor position of the motor. Important components of the experimental system are: a personal computer, a National Instruments PCI-6024E ata acquisition car [7], comei river library for Linux [8], a free open source real time platform [6], a custom-mae power electronics converter, an incremental encoer an signal conitioning circuits for measuring the rotor position. 2 Moeling The equations that establish the DC motor behavior are obtaine by using funamental electrical an mechanical laws [1], this is V f = r f i f + L ff t i f (1) V a = r a i a + L aa t i a + L af i f ω r (2) J t ω r + T L = L af i f i a (3) t θ r = ω r (4) Notation for parameters an variables is given in Table 1. Parameters involve in (1)-(4) were calculate once we analyze the experimental results of several transient an steay state tests. 2.1 Loa Torque Moel There are ifferent approaches for moeling the loa torque in electrical machines. Some contributions use an algebraic escription to represent the loa torque an

2 ω r i a i f (0.46 Amps) V f V a r f ( Ohms) r a (6.615 Ohms) L aa ( H) L ff ( H) L af ( H) J( kg/m 2 ) T L K 0 ( ),K 1 ( ) Table 1: Notation Rotor spee Armature current Fiel current Fiel voltage Armature voltage Fiel resistance Armature resistance Armature inuctance Fiel inuctance Mutual inuctance Inertia Loa torque Coefficients of Loa torque the associate terms (friction an amping terms) [9]. Other works utilize a etaile ynamic representation for the loa torque-relate terms [10]. In this work, we propose an experimental-base loa torque ynamic moel an ientify the parameters involve in the new ynamic equation. The iea of proposing this moel is to improve the performance of the close loop system. Let us propose a state equation for the loa torque, this is t T L = K 0 ω r + K 1 T L (5) This moel is a lineal ynamic approximation of a more etaile representation, [10]. For the purpose of obtaining parameters in (5), we rewrite equations (3) an (5) as [ ] ωr = t T L [ ] [ ] [ 0 1 Laf i J ωr f ] + J i K 0 K 1 T a (6) L 0 where the armature current i a an rotor spee w r are interprete as the input an output of this subsystem, respectively. To ientify parameters in (5) an experimental test was carrie out. During this test, the fiel wining was fe with a constant current at i f = 0.46 amps. It is clear that uner these conitions state equation in (6) becomes a linear time-invariant escription. Basically, the test consists of applying a ranom sequence to the armature terminals. FIFO is a tool that is inclue in the real time platform an was use to obtain the experimental transient variables, [6], [11]. The experimental setup use for ientifying parameters in state equation (6) consists of an ATmega8 microcontroller for efining the ranom sequence, a Nana Electronics SHR-100 hall effect sensor for measuring the armature current, a Pepperl+Fuchs incremental encoer for measuring the rotor spee, a ata acquisition car for recoring the transient variables, RTAI-Lab real time platform an a custommae PWM H-brige converter. Diagrams of the experimental setup for parametric ientification can be foun in [12]. Reference [12] also contains the source program of the microcontroller for efining the ranom sequence. Due to the lack of the space, reaer is referre to [12], where the measure armature current an spee transient traces are shown. Once we obtaine the transient experimental ata, we utilize Matlab for the off-line processing of the ata. In particular, preiction-error approach is use to calculate the numerical version of (6). By comparing (6) an its corresponing numerical version, loa torque parameters K 0 an K 1 are efine. Before proceeing to the main contribution presente in Section 3, we have to escribe briefly the space state representation, state observer an control law. 2.2 State Space Representation Mostly of the physical systems presents non-linear ynamic behavior. However, uner some conitions they may be consiere to have a linear time-invariant representation. This is the case of a DC motor with a fiel wining that is fe from a constant source. Substituting the numerical parameters specifie in Table 1 into (2),(3),(4) an (5) an employing 5 khz as a sample rate, it is possible to obtain the nominal linear time-invariant iscrete time ynamic moel. Thus we have θ r (k + 1) ω r (k + 1) i a (k + 1) = T L (k + 1) x θ r (k) 4.4x10 9 ω r (k) i a (k) V a(k) T L (k) 9.201x10 10 (7) θ r (k) = [ ] [ θ r (k) ω r (k) i a (k) T L (k) ] T (8) where (7) is the iscrete time state equation an (8) is the iscrete time output equation. It is clear the following efinition for the state x = [ ] T θ r ω r i a T L, the output to be controlle y = θ r an the input of the system u = V a. State equation (7) assumes that parameters are linear time-invariant. Similarly, the fiel wining is suppose to be fe by a voltage source that keeps constant the fiel current at 0.46 amps. This is the reason for not incluing the state equation of the fiel wining. Experimental results in Section 4 will show that the nominal moel in (7)-(8) contains the funamental ynamic characteristics of the system necessary to esign a successful experimental control law. Parametric uncertainty will be aresse by using an integrator, i.e. x I (k + 1) = x I (k) + e(k) = x I (k) + y(k) r(k) (9) where x I is the integrator state variable, r is the setpoint an y = θ r is the output to be controlle.

3 2.3 State Observer The state space equation of a stanar linear timeinvariant iscrete time observer can be written as [13] thus the corresponing controller gains become [K I, K] = [ , , , , ] (13) x(k + 1) = G x(k) + Hu(k) + K e [y(k) C x(k)] (10) The observer gain K e is a 4x1 constant vector associate with the output error y(k) C x(k) = θ r (k) θ r (k). As it is explaine in the next subsection, esire poles are basically compute by specifying the esire time constant. Once the set of esire poles are obtaine, a stanar proceure for calculating the gain K e is applie. This gain can also be compute by using the Scilab/Scicos ppol comman. In this particular case, gains in (10) were chosen such that the esire observer poles become z 1,2,3,4 = , , , , this is K e = (11) It is important to emphasize that for the purpose of implementing the real time close loop system the rotor position is the only state variable to be measure. The state feeback uses the state variables (incluing the loa torque) provie by the experimental iscrete-time state observer. Armature current an rotor spee were measure just for the off-line parametric ientification of the loa torque moel. 2.4 Controller Pole placement technique is use to calculate the gains associate with the state feeback. Armature voltage u(k) = V a (k) is now efine by the following stanar state feeback u(k) = [ K I K ] [ ] x I (k) (12) x(k) where K I is the gain corresponing to the integral state variable, K is the gain vector associate with the state variable of the original system x. These gains are calculate following a stanar proceure [13]. Gains can also be compute by using the Matlab comman place. On the other han, it is well known that the relationship between a s-plane pole having a particular time constant an the corresponing z-plane pole is given by z = e 1 τ T, where τ is the time constant an T is the sampling perio. If τ = 0.1 secons an T = secons, thus the corresponing pole in the z-plane becomes z 1 = e (0.0002) = Similarly, the rest of esire controller poles can be obtaine z 1,2,3,4,5 = , , , , Next section presents the main contribution of this work, i.e. the esign of several harware/software components employe in the close loop system. 3 Experimental Setup 3.1 Real-time platform As it was establishe earlier, a free open source real-time platform is employe to solve the control algorithm. The resulting executable program is able to provie correctness of the calculate values an to accomplish strict time requirements [14]. In our case, the control algorithm was compute every secons (5 khz). Furthermore, this software is in charge of reaing/writing igital an analog signals at the terminals of the acquisition car. Knoppix 5.0 an RTAI-Lab are use as the real-time platform. Knoppix 5.0 is a Linux istribution which inclues RTAI-Lab tool-chain [6]. The Real Time Application Interface (RTAI) is a patch of the Linux kernel that is able to carry out real-time tasks. The RTAI-Lab inclues Comei, which is a collection of Linux rivers for a variety of common ata acquisition cars [8]. RTAI-Lib an Comei, allows the real-time program to access the signals at the terminals of the ata acquisition car. In this particular work, a National Instruments PCI-6024E ata acquisition car is use. Scilab/Scicos is a graphical environment that inclues a library of blocks that can be employe to simulate the close loop system, [11]. The library of real-time blocks is efine in RTAI-Lib. We use Scilab/Scicos to simulate the close-loop system as a first step of the implementation. Then, the Scilab/Scicos simulation blocks are switche to the corresponing RTAI- Lib real-time blocks. The executable program can be create by compiling the RTAI-Lib/Scilab/Scicos-base source program by employing the RTAIcoegen tool locate in Scicos. The main source program is shown in Fig. 1. In orer to be able to present it in just one figure we use several subroutines that are calle super blocks. They are a tool for organizing the program structure. A super block looks like any other block, however its operation is efine by a custom-mae Scicos coe. The main program can be ivie in three parts. The first part is the super block that inclues the position measurement algorithm. A general escription of that algorithm is presente in the next subsection. The secon part is the super block that computes the state observer. The last part calculates the state feeback an solves the integral state space equation. Close loop gains K I, K 1, K 2, K 3 an K 4 are efine in (13). In aition, the program inclues the set-point block, the Comei block (DAC) an a gain enote by A. This gain scales the numerical value of the calculate armature voltage. The scale voltage is the

4 numerical representation of the actual voltage applie to the power converter. It is enote by control voltage, (see Fig. 5). Aitional information regaring the program coe can be foun in [12]. Figure 1: Main Scicos program incluing several super blocks (subroutines) 3.2 Rotor position measurement esign A esign for measuring the rotor position is implemente. There are several components of this harware/software esign. One of them is the Pepperl+Fuchs encoer that is physical attache to the motor shaft. The secon one is a ATmega8535 microcontroller-base signal conitioning circuit an the thir one is the real time software that calculates the rotor position from the igital signals provie by the microcontroller. The ATmega8535 microcontroller is programme for being use as a 16-bit binary UP/DOWN counter. A signal conitioning circuit is necessary to etermine the shaft irection, which is implemente by using a flip-flop integrate circuit. The flip-flop uses the pulses of channels A an B of the encoer an provies a binary 1 or 0 epening on the shaft irection. Channel A (or B) of the encoer is then connecte to a micro-controller terminal without moifying the voltage as they both the encoer an the microcontroller have 5 volts as a nominal voltage. The microcontroller is in charge of counting the pulses provie by the encoer. In this case, the encoer generates 1024 pulses per revolution. The micro-controller count goes up or own epening on the flip-flop binary signal. The complete electronics iagram of the rotor position measurement setup is shown in Fig. 2. The micro-controller provies a 16-bit binary count that represents the rotor position. On the other han, the National Instruments PCI-6024E ata acquisition car has only 8 igital inputs [7]. In orer to be able to rea the 16-bit binary ata, 8 aitional igital inputs are obtaine by employing 8 analog inputs of the acquisition car an using them as igital inputs. The algorithm to interpret those analog inputs as igital inputs is coe in the Scilab/Scicos source program. The routine that obtains the binary ata by using igital inputs is shown in Fig. 3. The algorithm con- Figure 2: Rotor position measurement setup sists of multiplying each bit (1 or 0) by its corresponing value in the ecimal system, then the results are ae. The same algorithm is employe to convert bits that were obtaine by using the analog inputs. In aition, a signal conitioning routine is employe to use those analog inputs as igital inputs, see Fig. 4. A comparator etermines the binary value (1 or 0) of the signal that was rea at the terminals of the analog input. This routine is implemente for each one of the 8 signals obtaine by using the analog inputs. Combining the results provie by these routines a ecimal measurement of the rotor position is obtaine. A final super block calle Position Figure 3: Subroutine associate with the igital inputs of the position measurement algorithm Measurement Algorithm (see Fig. 1) was create to inclue the escribe coe. It is important to mention that the rotor position setup is able to measure positive an negative values. In orer to accomplish this feature, the microcontroller was programme to have an initial count that is locate at the mile of the 16-bit count range. Due to the lack of space, reaer is referre to [12], where

5 information regaring the ATmega8535 micro-controller assembler coe can be foun. Figure 4: Software-base signal conitioning for one of the eight analog inputs use as igital inputs 3.3 Power converter A Pulse With Moulation-base MOSFET H-Brige converter was esigne an implemente. This type of power electronics evice is commonly use to rive DC motors when biirectional spee/position control is neee. The numerical value of the armature voltage is efine by the state feeback an calculate by the computer. This value is written by the real time software to one of the analog output channels of the ata acquisition car. The actual armature voltage is applie by the H- Brige to the DC motor an is efine by the following equation V a = δv cc, where V cc is the DC power supply voltage of the converter an δ is the uty cycle of the power MOSFETs. The uty cycle is clearly associate to the signal at the analog output of the ata acquisition car. This signal is enote by control voltage in iagram of the power electronics converter Fig. 5. This converter consists of four parts. Part (a) is implemente for conitioning the signal provie by the ata acquisition car. Part (b) generates the PWM signals epening on the voltage in Part (a). This part also implements a switching elay to provie a recovery time of the power components uring every switching perio. Part (c) isolates power from instrumentation Parts (a) an (b). Finally, Part () is the actual MOSFET H-Brige circuit which is irectly connecte to the armature terminals of the DC motor. 4 Experimental Results This section presents the experimental transient characteristics of the close loop system. The real time program calculates the iscrete-time state feeback in (12), which inclues the iscrete-time integrator state variable. Gains of the controller are efine in (13). In our case, all of the parts of the close loop control algorithm was compute every secons (5 khz). Here, we implement the armature voltage as a function of estimate states excepting Figure 5: Power Converter. a) Signal Conitioning. b) PWM an switching elay. c) Isolation. ) H Brige an DC motor. the rotor position, which is the only state variable that is measure. Estimate states, incluing the loa torque, are obtaine by solving the full orer iscrete-time state observer in (10). By comparing (7) an (10) it is easy to efine matrix G an vector H. Gains associate to the observer are specifie in (11). Transient behavior of the integrator state variable was compute by solving the state equation in (9). Initial conitions of the integrator an observer state variables were set numerically equal to zero. The experimental trace shown in Fig. 6 illustrates the ynamic characteristic of the rotor position following a raian (4 revolution) reference comman. The Position (raians) Time (secons) Figure 6: Rotor Position: Experimental measure, ** Simulate moel-base simulate moel-base trace of the rotor position is also shown in Fig. 6. It is clear that these signals (simulate an measure) are similar to each other. Initially, the rotor position was at zero raians. The position begins to increase immeiately until the position error is close to zero, which occurs approximately at 1.2 sec. In or-

6 er to save these transient ata, a real-time block calle FIFO was use, [6] [11]. Experimental observer-base variables were also save by using FIFO. One of those observe states is shown in Fig. 7. For the purpose of Spee (ra/s) Time (secons) Figure 7: Rotor spee: Experimental observer-base, ** Simulate moel-base testing the experimental close loop system in a more emaning operating conition, we installe an inertial isk. The original value of the inertia was kg/m 2, see Table 1. The new inertia is kg/m 2. In other wors, the new parameter (inertia) is about 10.5 times the original value. It means a significant increase of a particular parameter, of the mechanical subsystem, that is clearly connecte to the output to be controlle (rotor position). In aition, the inertial isk was intentionally slightly loose to the shaft, therefore the parameter varie uring the test aroun the non-original value. Even uner that emaning conition the experimental close loop system was able to follow a raian (3.5 revolution) reference comman, see Fig. 8. Position (raians) Time (secons) Figure 8: Experimental measure rotor position when a significant change of a parameter ocurrs (10.5 times the original value aprox.) 5 Conclusions an Future Work An experimental setup was esigne an use in this work. A funamental part of the setup is the free open source real time software that is able to accomplish the control algorithm uner strict time requirements. This setup also provies an effective framework for rapi testing of more complex control algorithms. A future work might be relate to the iea of implementing a new nonlinear control scheme, keeping the harware platform an changing just the source program running uner the same real time software platform. References [1] P. C. Krause, O. Wasynczuk an S. D. Suhoff, Analysis of Electrical Machinery an Drive Systems. John Wiley an Sons, IEEE Press Power Engineering; [2] D. Zhao, N. Zhang. An improve nonlinear spee controller for series DC motors. IFAC 17th Worl Congress, [3] J. Jugo. Real-time control of a DC motor using Scilab an RTAI. Scilab INRIA Rocquencourt, [4] N. Boizot, E. Busvelle, an J. Sachau. High-gain observers an Kalman filtering in har real-time. RTL 9th Workshop, [5] Open Source Linux Distribution. Knoppix 5.0, [6] R. Bucher, S. Mannori an T. Netter. RTAI-Lab tutorial: Scilab, Comei an real-time control [7] National Instruments. PCI-6024E National Instruments Manual [8] D. Schleef, F. Hess, an H. Bruyninckx. The Control an Measurement Device Interface hanbook [9] M. Ruerman, J. Krettek, F. Hoffmann, T. Bertram. Optimal state space control of DC motor. IFAC 17th Worl Congress, [10] T. Tjahjowioo, F. Al-Bener, H. Van Brussel. Friction ientification an compensation in a DC motor. IFAC 16th Worl Congress, [11] S. L. Campbell, J. P. Chancellier, an R. Nikoukhah. Moeling an Simulation in Scilab/Scicos. Springer, [12] E. Y. Renon-Fraga, Rotor Position Control of a DC Motor using a real time platform, MSc. thesis, in spanish. Instituto Tecnologico e C. Maero. C. Maero, Mexico; [13] G. F. Franklin, J. D. Powell an M. Workman, Digital Control of Dynamic Systems, Aison Wesley, Menlo Park, CA; [14] P. A. Laplante. Real-Time systems esign an analysis an engineer s hanbook. IEEE Computer Society Press, 1997.