Theoretical and Experimental Implementation of DC Motor Nonlinear Controllers

Transcription

1 Theoretical and Experimental Implementation of DC Motor Nonlinear Controllers D.R. Espinoza-Trejo and D.U. Campos-Delgado Facltad de Ingeniería, CIEP, UASLP, espinoza trejo Facltad de Ciencias, UASLP, Av. Salvador Nava s/n, Zona Univ., C.P. 89, S.L.P., México Abstract The development of two nonlinear control strategies for the velocity reglation of a DC motor are detailed in the paper. The parallel (shnt) connection of the DC motor is stdied. First, a parameter identification was carried ot sing experimental inpt-otpt data of the motor. One control algorithm involves a nonlinear cancelation law (inpt-otpt linearization) with a PID velocity reference error compensation, and a Lenberger observer to estimate the load torqe. In this algorithm, the integral compensation and the torqe estimation improve the robstness of the overall control scheme. In addition, a variable-strctre control (sliding mode) was developed also for velocity reglation, that ses the information of the Lenberger observer to inclde the estimated load torqe. Experimental reslts in a HP test-bed corroborate the analysis and designs presented. Index Terms DC motor, variable-strctre control, inptotpt linearization. I. INTRODUCTION Electrical motors are a key piece in almost any atomatic process. They convert the energy from electrical into mechanical in order to prodce movement. There are two basic types of electrical machines: DC and AC motors [6]. The DC motors are pretty common in indstrial processes de to their operational properties and control characteristics. They are sed for traction, cranes, mills, etc. According with the connection between armatre and field in the DC motor, three connection area devised: (i) parallel (shnt) connection, (ii) series connection, and (iii) independent excitation [6],[8]. The first two have the advantage that they only need one variable DC power spply to control the motor. However, they present the disadvantage that the corresponding control algorithms are more involved, since the mathematical model of both systems are nonlinear [], [], []. The control of the parallel configration will be addressed in the paper. In some applications, it is necessary to adjst the motor anglar velocity constantly despite the motor load. For this prpose, it is needed the appropriate hardware to be able to make the adjstments in the inpt voltage to the motor, according with a control algorithm (variable speed drive). Hence it is necessary to keep the anglar velocity of the motor reglated and adjst the motor electrical torqe to compensate the load. This paper addresses the problem of velocity reglation with load torqe compensation for the parallel DC motor. In order to achieve this goal, an inpt-otpt linearizing controller is designed [5], [1], that incldes a PID compensation according with the reference error and a load torqe estimation. This techniqe is based on differential geometric control [1], [9]. It is observed that the integral action in conjnction with the torqe estimation add robstness to the control algorithm. Another control methodology that has been sed extensively for nonlinear systems is variable-strctre control [4], [11]. The main advantages of this algorithm is its robstness against noise and model ncertainty. For comparison in the paper, a variable-strctre controller is also developed for the velocity reglation of the DC motor. The paper is organized as follows. Section presents the mathematical model of the motor, and the parameters identification. The control algorithms are detailed in Section, and the experimental implementation is shown in Section 4. The paper ends in Section 5 with final remarks and conclsions. II. DC MOTOR MODELING In the following derivations, consider the states of the system as x 1 = the armatre crrent, x = the field crrent, and x = the anglar velocity. It is assmed that they are all measred on real-time. The control variable = is the variable DC voltage delivered to the motor. The parameters of the DC motor are defined as: R a Armatre resistance Armatre indctance M Mtal indctance R f Field resistance Field indctance B Mechanical friction J Inertia Load torqe In this stdy compared to previos ones [1], it is not neglected, since for medim to large size motors, this parameter cold be comparable in magnitde to R a. A. Parallel (Shnt) Connection The mathematical model of this configration is presented, assming that it is available a variable resistor R adj to adjst

2 the maximm velocity in the motor (field weakening) []: ẋ 1 = R a x 1 M x x 1 ẋ = R f R adj x 1 (1) ẋ = B J x M J x 1x 1 J Note that R adj has to be selected according to the desired maximm anglar velocity, and the rated maximm field crrent in the motor, in order to calclate the maximm power dissipated by this resistor. Figre 1 shows the electrical diagram of the parallel (shnt) connection. Fig. 1. J, B - R a B. Experimental Identification e R adj R f Parallel Configration of DC Motor. First of all, it is needed to obtain an approximation of the parameters of the model in (1), and a least sqares approximation is carried ot [9]. Since the armatre electrical and mechanical parameters are the most difficlt to estimate, the DC motor is considered in a separated armatre-field configration. Hence a constant DC voltage sorce is connected to the field, withot load torqe applied to the motor. Meanwhile, the armatre is spplied with a sqare voltage signal in order to provide excitation to the system and achieve the identification. Note that in this configration the DC motor presents linear dynamics with respect to the parameters. The parameters to identify are (, R a, K b, J, B), where the electromagnetic constant K b is related to the constant field crrent I f and the mtal indctance M, i.e. K b = MI f. The field parameters R f and cold be identified applying a step voltage, and measring the reslting crrent to identify its characteristic time and peak response, or directly by a LCR Mltimeter. The latter approach was adopted in this paper. Now, a regressor formlation was sed to identify the motor parameters sing integral relations to avoid derivatives and improve noise robstness, i.e. [ ia ia ] R a K b J B - = [ The parameters are then identified from collected data of, and in order to constrct the linear algebraic eqations in (). Note that the eqations mst be satisfied at each time ] () TABLE I DC MOTOR PARAMETERS. Parameter ale R a.699 Ω.9 H M.14 R f 445 Ω 56 H J.9 1 kg m B N m/rad/s instant. Ths, assming that the data is sampled at period T s, denote the matrices W (nt s ) and y(nt s ), and the regressor Θ in (5), then to compte a soltion a smmation for N time instants is sed [9]. Therefore, define the smmation matrices: N W T (nt s )W (nt s ) R 5 5 () R W R y n=1 N W T (nt s )y(nt s ) R 5 (4) n=1 R W is always positive semi-definite, bt if N is large enogh and nder an appropriate excitation of then R W >, and a soltion can be dedced Θ = R 1 W R y. The parameters identified are shown in Table I. III. NONLINEAR CONTROL STRATEGIES The proposed nonlinear control strategies followed in this paper are detailed next. It is important to mention that this control problem is very demanding since the mathematical model is nonlinear; There is intrinsically some ncertainty in the identified parameters; There is an nknown pertrbation acting on the system (load torqe); and in the test-bed, there are noisy measrements. As a reslt, a simple PID or other types of linear controllers can not achieve the control objective, and a more complex algorithm mst be prsed. Other model-based control strategies as Lyapnovbased design or adaptive nonlinear control cold also be viable tools, however they are not explored in this work. A. Nonlinear Cancelation Law This control scheme consists of three parts: 1) Inpt-otpt linearizing law, ) PID reference compensation, and ) Lenberger observer for load torqe estimation. The control block diagram is presented in Figre, where the estimated variables are denoted by (ˆ ). In the next sbsections, these three parts are detailed. 1) Inpt-Otpt Linearization: The control scheme adopted is based on differential geometric methods [5], [1]. For this prpose, the otpt of interest is the anglar velocity, and it can be proved that the system presents a relative degree of two. The relative degree is well-defined if the condition β(x) is satisfied [], where β(x) is given by β(x) = M [ x1 x ] (6) J

3 W (nt s) y(nt s) [ ] ia(nt s) ia(nt s) (nts) R (nt s) (nt 5 s) (nts) [ ] (nts) R (5) Θ [ R a K b J B ] T R 5 v PID Reference Compensation ^ Inpt-Otpt Linearizing Law ^ Fig.. elocity and Load Torqe Observer Control Actator DC MOTOR Overall Nonlinear Cancelation Law. Hence, the relative degree is well defined if the sm of armatre and field crrents is different from zero. In a practical setting, this condition is satisfied, assming that the DC motor is operated in jst one rotating direction. Therefore, a linearizing control law is given by = α(x) v β(x) for β(x), where v is a desired dynamic added to the system, and the fnction α(x) is defined as in (8). The zero dynamics for this configration were proved to be asymptotically stable (minimm-phase system), and the proof is not inclded for brevity. ) elocity and Load Torqe Lenberger Observer: In order to compensate the load torqe, it is necessary to estimate this qantity to avoid steady-state error between the velocity and its reference. A Lenberger observer [9], [1] is proposed assming that the load torqe is roghly constant T l. The observer reprodces the mechanical eqation in the DC motor with an error correction term de to the velocity estimation: dˆ dt d ˆ dt = B J ˆ 1 J ˆ M J l 1 ( ˆ) () = l ( ˆ) (9) Note that the inpt variables to the observer (9) are the anglar velocity, and the armatre and field crrents,. The observer gains l 1 and l mst be selected sch that the characteristic eqation: λ (B/J l 1 )λ l = (1) has its roots in the left-half plane, in order to garantee convergence to the real vales. ) PID Reference Compensation: A constant reference for the anglar velocity is assmed. In order to provide good reference tracking, the desired dynamic indced in the law () has a PID action: v = K d ( ) K p ( ) K i ( )dt B J ˆ (11) where the estimated torqe ˆ is sed to cancel this term and avoid a steady-state error. However, the previos eqation (11) can be simplified, since the reference is assmed to be constant or to change slowly and. Moreover, it is desirable to avoid the derivative of the anglar velocity, so the mechanical eqation in the motor is sed instead. Therefore, the reslting PID law is proposed: v = K d ( B J ˆ M J 1 J ˆ ) (1) K p ( ˆ) K i ( )dt B J ˆ where the estimated vale of the velocity ˆ is sed in the proportional and derivative actions to avoid the noise effects. The selection of the PID gains K p, K d and K i mst prse that the characteristic eqation: λ K d λ K p λ K i = (1) had its roots in the left-half plane to provide closed-loop stability. I. SLIDING-MODE CONTROLLER On the other hand, a control scheme sing a variablestrctre philosophy (sliding mode) [4], [11] consists on defining an sliding srface that reflects the performance objectives, and obtaining the eqivalent and approximation control laws. The control block diagram is presented in Figre, where the estimated variables are again denoted by (ˆ ). In the next sbsections, these three parts are detailed. First, consider the following general strctre of the mathematical model withot external pertrbations: ẋ = f (x) g (x) (x, t) (14) where for the DC motor (see original model in (1)) f(x) and g(x) are given by: Ra L a x 1 M x x 1 f (x) = R L f a x g(x) = 1 (15) B J x M J x 1x

4 α(x) = B J x M J [ B J R ] f R adj Ra x 1 x M x JL x (8) a ^ T L ^ Fig.. Approximation Law Eqivalent Law elocity and Load Torqe Observer Control Actator DC Motor if Overall ariable-strctre Control Strategy. Therefore, a general control strctre following the sliding mode methodology is given by: (x, t) = eq (x, t) R (x, t) (16) where eq denotes the eqivalent control law, and R the approximation law [4]. A. Sliding Srface The sliding srface specifies the desired characteristics of the control system as: stability, tracking, reglation, etc. It is then proposed an sliding srface for velocity reglation: σ = γe ė = (1) where e = x denotes the reglation error and the velocity reference. Note that when the srface has been reached (σ = ), the parameter γ define convergence speed of the reglation error to the origin. Then, sbstitting the state ẋ in (1), and considering that the reference velocity is a constant ( = ), it is obtained the sliding srface as: σ (x) = ( γ B ) x M J J x 1x 1 J γ (18) Therefore, in order to avoid the velocity derivative in the sliding srface, it is needed the load torqe information in σ(x). B. Eqivalent Law The eqivalent control constittes a control inpt which, when exciting the system, prodces that the reslting trajectories remain on the sliding srface whenever the initial state is on the srface [4]. Based on the existence of a sliding mode, and sing the chain rle, it is define the eqivalent control eq for systems of the form (14) as: { } 1 { } σ σ eq = x g(x) x f (x) (19) As a reslt, in order to garantee the existence of an eqivalent control law, it is reqired: σ g(x, t) () x Therefore, for the DC motor in a parallel configration, the existence condition is given by: 1 σ 1 x 1 = M J C. Approximation Law [ x x 1 σ (1) x ] () Now, consider the following approximation control law [4], [11]: { } 1 σ R (x, t) = x g(x) û R () where, û R = K σ 1/ sign(σ) K > (4) In the last eqation, the parameter K will affect the speed convergence of the trajectories to the sliding srface. Note that the approximation law (4) will be large if the system is far away from the sliding srface. In a practical implementation, there exists noise in the measrements, therefore to redce the chattering phenomenon a bondary layer [1] approach can be followed for the approximation control law. Hence, the control (4) is modified as ( û R = K σ 1/ σ ) sat (5) ɛ where the parameter ɛ defines the size of the bondary layer.. EXPERIMENTAL IMPLEMENTATION The control diagrams shown in Figres (nonlinear cancelation law) and (variable-strctre control) were implemented experimentally in a dspace DS11 system rnning at a sampling freqency of 1 khz. The control parameters for both strategies are presented in Table II. The test-bed shown in Figre 4 was sed, and it consists of a HP Shnt DC Motor that is connected to a HP Permanent Magnet DC Motor tilized as a load. A tacogenerator measres the anglar velocity of the shaft, at a proportion of 5 /RPM with an error of ±1 %. There are measrements of the armatre and field crrents throgh hall-effect sensors. It is important to mention that the three measrements are noisy dring the experiments, as it will be observed in the implementation plots, and this isse presents a challenge for the control algorithm to show good robstness. The motor voltage is controlled by DC-DC chopper working nder a PWM scheme (switching freqency 1kHz), where the control parameter is the dty cycle [6]. The chopper was selected as control actator de to its

5 rectifier bridge three-phase transformer T 1 T - control signal tacogenerator - cd DC-DC Chopper otpt - field-armatre connection F F1 A HP Shnt DC Motor hall effect crrent sensors A 1 R L DC Motor (permanent magnet) Three-phase oltage AC Sorce Load Torqe Fig. 4. Experimental Configration of the DC Motor. TABLE II CONTROL PARAMETERS. Parameter ale Nonlinear Cancelation Law K p.4 K i 9.46 K d 4.6 Load Torqe Observer l 1. l ariable-strctre Control γ 15 K 5 ɛ 1. fast response and linear dynamics, and it was jst modeled as a scaling factor in the control system. The constrction of the actator was carried ot in or lab, and it is designed sch that it is controlled by a voltage signal in the interval [, 1]. This satration in the control signal did not limited the performance of the system, de to the fast dynamics of the actator related to the time constants of the DC motor. The adjstment resistor dring tests was not sed (R adj = Ω), since the open-loop maximm velocity was adeqate and there was no need to apply field-weakening. Two tests were carried ot for the control algorithm: Case A: tracking of an anglar velocity reference change from 14 to 1 RPM s (see Figre 5). Case B: load torqe variation from Nm (no-load) to 1.5 Nm, and back to N m (Figre 6). In Figre 5 the reslts for Case A are presented with the nonlinear cancelation control. It can be observed that the tacogenerator measrement for the anglar velocity is extremely noisy. Nevertheless, the controller is capable of providing good tracking for a reference change. De to this change, the control signal decreases its vale to compensate the lower reference. Also, the armatre and field crrents pdate their vale to accommodate the reference modification. Similarly, for the variable-strctre control law good tracking is observed, and the plots are omitted de to space limitations. Finally, in Figre 6 the experimental reslts for Case B are illstrated, bt now for the variable-strctre control law. The top plot shows that the anglar velocity is correctly compensated, since only a transient effect is noticed after the load change. On the other hand, the armatre and field crrents increase de to the load effect, and recover their previos vale when the load is again removed. Similar reslts are derived for the nonlinear cancelation control law. Conseqently, both techniqes show good performance against a reference change and load pertrbations. I. CONCLUSIONS AND FINAL REMARKS In the present paper, a theoretical derivation and practical implementation of two nonlinear control schemes for a shnt DC motor are detailed. The experimental identification proposed, with integral relations, was capable to provide good estimates of the real parameters. A nonlinear control algorithm is designed departing from inpt-otpt linearization theory (differential geometric tools). This techniqe is recognized to have robstness isses dring practical implementations. For this prpose, integral action to the reference error and load torqe estimation were added to improve the robstness of the overall strctre. Despite noisy measrements dring the practical implementation, the control scheme is able to adjst the motor voltage properly

6 NO LOAD TORQUE =1.5 N m NO LOAD TORQUE (RPM) 15 1 =14 RPM =1 RPM =14 RPM (RPM) (dty cycle) (dty cycle) REFERENCE UPDATE = N m = N m 5 =1.5 N m Fig. 5. Experimental Response for Case A with Nonlinear Cancelation Control (TOP) Anglar elocity Measrement, (MIDDLE 1) Control signal (dty cycle), (MIDDLE ) Armatre Crrent, (BOTTOM) Field Crrent. Fig. 6. Experimental Response for Case B with ariable Strctre Control (TOP) Anglar elocity Measrement, (MIDDLE 1) Control signal (dty cycle), (MIDDLE ) Armatre Crrent, (BOTTOM) Field Crrent. to follow a variable velocity reference and to compensate load torqe changes. On the other hand, a variable-strctre control was also proposed. The reslting control algorithm prodced good performance against pertrbations and noise measrement. As a reslt, both techniqes are visalized as practical tools in this nonlinear setp. ACKNOWLEDGMENTS This research was spported in part by a grant from PROMEP. Diego Espinoza-Trejo acknowledges the financial aid provided by CONACYT throgh a doctoral scholarship. REFERENCES [1] M. Bodson and J. Chiasson, Differential Geometric Methods for Control of Elecrtical Motors, International Jornal of Robst and Nonlinear Control, ol. 8, pp , [] J. Chiasson and M. Bodson. Nonlinear Control of a Shnt DC Motor. IEEE Transactions on Atomatic Control, 8(199), [] J. Chiasson. Nonlinear Differential-Geometric Techniqes for Control of a Series DC Motor. IEEE Transactions on Control Systems Technology, (1994), 5 4. [4] J.Y. Hng, W. Gao and J.C. Hng, ariable Strctre Control: A Srvey, IEEE Transactions on Indstrial Engineering, ol. 4, No. 1, pp. -, 199. [5] A. Isidori. Nonlinear Control Systems. Springer-erlag London Limited, (1995). [6] R. Krishnan. Electric Motor Drives: Modleing, Analysis and Control. Prentice-Hall, Upper Saddle River, NJ, (1). [] P.D. Oliver. Feedback Linearization of DC Motors. IEEE Transactions on Indstrial Electronics, 8(1991), [8] W. Leonhard. Control of Electrical Drives. Springer-erlag Berlin (1). [9] S. Mehta and J. Chiasson, Nonlinear Control of a Series DC Motor: Theory and Experiment, IEEE Transactions on Indstrial Electronics, 45(1998), [1] S. Sastry. Nonlinear Systems: Analysis, Stability, and Control. Springer-erlag New York, Inc. (1999). [11]. I. Utkin, Sliding Mode Control Design Principles and Applications to Electric Drives, IEEE Transactions on Indstrial Electronics, vol. 4, No. 1, 199. [1] K.D. Yong,.I. Utkin and U. Ozgner, A Control Engineer s Gide to Sliding Mode Control, IEEE Transaction on Control Systems Technology, ol., No., pp. 8-4, 1999.