feedback linearization of dc motors - theory and experiment

134 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 45, NO. 1, FEBRUARY 1998

Nonlinear Control of a Series DC Motor:Theory and Experiment

Samir Mehta,Member, IEEE, and John Chiasson,Member, IEEE

Abstract—The control problem for a series dc motor is con-sidered. Based on a nonlinear mathematical model of a series-connected dc motor, it is shown that the combination of anonlinear transformation and state feedback (feedback lineariza-tion) reduces the nonlinear control design to a linear controldesign. To demonstrate its effectiveness, an experimental study ofthis controller is presented. These experimental results are alsocompared with a simulation of the closed-loop system. Finally, itis shown that a nonlinear observer (with linear error dynamics)for speed and load torque can be constructed based only onmeasurements of the motor current. Experimental results of thisspeed and load-torque estimator are also presented.

Index Terms—Feedback linearization, nonlinear control, seriesdc motor.

I. INTRODUCTION

A DC MOTOR in which the field circuit is connected inseries with the armature circuit is referred to as aseries dc

motor. Due to this electrical connection, the torque producedby this motor is proportional to the square of the current (belowfield saturation), resulting in a motor that produces more torqueper ampere of current than any other dc motor. Such a motoris used in applications that require high torque at low speed,such as subway trains and people movers [1]. In fact, the seriesmotor is the most widely used dc motor for electric tractionapplications [7].

The mathematical model of the series dc motor is nonlinear.The nonlinearities consist of the torque being proportional tothe square of the current and the back EMF being propor-tional to the product of the current and speed. Based on thisnonlinear model, the work in [10] and [12] has shown thatdifferential-geometric methods of nonlinear control [2]–[5],[16], [17], [19] are applicable to series dc motors. In [10],both feedback-linearizing and input–output linearizing controlalgorithms for speed control were explicitly constructed forthe series dc motor. That is, by a combination of a nonlineartransformation and state feedback (feedback linearization) it isshown that the nonlinear control design is reduced to designinga linear control law. An experimental study is done using thiscontroller and compared with a simulation of the closed-loopsystem.

Manuscript received October 18, 1996; revised July 14, 1997.S. Mehta was with the University of Pittsburgh, Pittsburgh, PA 15261 USA,

and also with ABB Daimler-Benz Transportation Inc., Pittsburgh, PA 15236-1491 USA. He is now with Teradyne Inc., Agoura Hills, CA 91301 USA.

J. Chiasson was with the University of Pittsburgh, Pittsburgh, PA 15261USA. He is now with ABB Daimler-Benz Transportation Inc., Pittsburgh, PA15236-1491 USA.

Publisher Item Identifier S 0278-0046(98)00900-9.

Fig. 1. Simplified picture of separately excited dc motor.

Based on the work of [17]–[19], it was shown in [10] thata nonlinear observer for speed and load torque for the seriesdc motor can be constructed based only on measurements ofthe motor current. The constructed observer has linear errordynamics, so that the rate of convergence of the speed andload-torque estimates can be arbitrarily specified. Experimentalresults of this speed and load torque are presented. Based onthese efforts, it was found that the A/D converter used for thecurrent measurements requires a higher resolution than 8 b, inorder to obtain a speed estimate accurate enough for feedbackcontrol.

For applications of the differential-geometric techniquesof nonlinear control to stepper motors, shunt-connected dcmotors, and induction motors, the reader is referred to [6],[9], and [11].

II. M ATHEMATICAL MODEL

Fig. 1 is a simplified picture of a separately excited dcmotor. A series dc motor is configured by simply connectingthe field circuit in series with the armature circuit

That is, by connecting terminal to terminal inFig. 1, so that one obtains a series-connecteddc motor. The input voltage is then applied between terminals

andIn Fig. 2, the armature inductance is denoted by and

the field flux is given by , where is themagnetization curve given in Fig. 3. As seen in Fig. 3, themagnetization curve is strictly increasing. It is also symmetricwith respect to the origin and satisfies forBelow the knee of the magnetization curve, may bemodeled as a linear function of i.e.,where In a typical series-connected dc motor,the condition holds, where is the armatureinductance. In Fig. 2, and denote the resistance ofthe armature and field windings, respectively. The constant

denotes the torque/back EMF constant, so that

0278–0046/98$10.00 1998 IEEE

MEHTA AND CHIASSON: NONLINEAR CONTROL OF A SERIES DC MOTOR 135

Fig. 2. Equivalent circuit.

Fig. 3. Magnetization curve.

Finally, denotes the viscous-friction coefficient.

Therefore, with , the equations of a series-connected dc motor are [1]

(1)

Note that the torque produced by the motoris always positive, which is a direct result of the seriesconnection. That is, when the armature current is reversed,so is the field current (as they are the same current), thusreversing the magnetic field in the airgap and keeping thetorque positive. If a negative torque is required, the terminal

of the field circuit (see Fig. 1) must be disconnected fromterminal and reconnected to terminal

If the field circuit is not in magnetic saturation, so thatthe system (1) reduces to

(2)

where The motor used in theexperimental results reported below has the saturation curveshown in Fig. 3. As seen in the figure, the field flux goes intosaturation at approximately 1.5 A. However, the experimentalmotor is rated for only 1.7 A, so that the effects of saturation

are negligible under these conditions. Therefore, the nonlinearsystem (2) is used to model the motor.

III. N ONLINEAR SPEED CONTROLLERS

FOR THE SERIES DC MOTOR

A speed controller for the series dc motor is now describedusing the concept of feedback linearization [10], [12], [14],that is, by a combination of a change of coordinates and statefeedback, the original nonlinear system (3) can be made linearfrom input to state. To do so, consider the nonlinear changeof coordinates

In these new coordinates, the system is represented by

where the load torque is assumed to be constant.To linearize the system using feedback, the input voltageis set as

(3)

resulting in thelinear system

where is a new control input.The controller (3) is singular when , which is simply a

consequence of the fact that the motor cannot produce torquewithout current.

With a reference trajectory given byand

the motor’s acceleration, tracking of this referencetrajectory is achieved by choosing the inputas

(4)

Setting the gains as

places the closed-loop poles at


IV. SPEED AND LOAD-TORQUE OBSERVER BASED ON

OPTICAL ENCODER MEASUREMENTS

In the experimental setup, an optical encoder is used todetermine the motor’s speed. Typically, a simple backwarddifference algorithm is used to compute the speed from theposition measurements. However, an observer can be usedto provide a smoother estimate of the speed, as well as anestimate of the load torque. To do so, the load torque ismodeled as a constant, so that the system model becomes

(5)

A speed and load-torque observer is then given by

(6)

Subtracting (6) from (5), the resulting observer error systemis then

(7)

where and It iseasily seen that choosing the gains as

places the poles of the observer error system (7) at

Even if a load torque is not present, the observer (5) shouldstill be used to eliminate any steady error on the speed estimatedue to modeling errors. For example, when equation (6) wassubtracted from equation (5), it was assumed that the motortorque terms cancelledexactly. However, if this is notthe case, for example, the difference is (constant),then (7) must be modified to

Denoting the Laplace transform of as it is easilyfound that

Fig. 4. Block diagram of feedback linearization controller.

where However, by the final value theoremit follows that

That is, an unknown constant disturbance in the torque modelis rejected from the speed estimate.

A block diagram of the feedback linearization controlleralong with the speed and load-torque observer is given inFig. 4.

V. SPEED ESTIMATOR WITHOUT AN

ENCODER (SENSORLESSOBSERVER)

It is also theoretically possible to design an observer forspeed and load torque basedonly on the current measurement[16]–[19]. The idea, as described in [10], is to find a coordinatetransformation that will transform the original system (2) intoa system that is linear in theunmeasuredstate variable

For the system (5), consider the change of coordinates

Then, the system equations become

(8)

which is nowlinear in the unmeasured state variableThe observer is then defined by

(9)


The observer error system is found by subtracting (9) from(8), i.e.,

(10)

where andChoosing the gains and as

results in the characteristic equation of (10) being given by

That is, these gains put the poles of the observer atand where and are positive.

VI. L EAST-SQUARES PARAMETER IDENTIFICATION

A standard least-squares approach [8] was used to identifythe parameters of the model (2). Specifically, the mathematicalmodel (2) is rewritten as

(11)

where is given at the bottom of the page, with

being the sampling period and

The least-squaressolution is then

(12)

where

(13)

In the identification experiment, the current and voltage arechosen to ensure that is invertible. In this case, it is wellknown that (12) minimizes the squared error

(14)

VII. EXPERIMENTAL RESULTS

The experimental setup used in this paper consisted ofa single-pole-pair dc motor (Bodine Electric) with the fieldwinding connected in series with the armature circuit. Themotor was fitted with a 2000 pulse/rev optical encoder. AMotorola DSP56001 fixed-point processor was used to executethe control algorithms. A data acquisition board to collectdata consisted of four 8-b A/D’s to measure the currents andvoltages and two 12-b D/A’s to command voltages to theamplifier. The linear amplifiers were restricted to 40 V and5 A continuous. The currents were scaled, so that a full A/Dreading corresponds to 5 A. As one of the 8 b is a sign bit,the resolution of the current measurements was A.The sample rate was set at 5 kHz.

Using the least-squares approach described in Section VI,the parameter values for the experimental motor were deter-mined to be H, , N m/rad/s,

N m/Wb A, kg-mIn order to have a load torque on the motor, a permanent

magnet dc motor was coupled to the series-connected dc motorand run as a generator with an external resistorconnectedto its armature terminals. The back EMF/torque constant ofthe load motor was N m/A and the armatureresistance of the load-motor The voltageproduced by the load motor is so that the steady-state current across the armature and load resistor is

The steady-state load torque is thenThe inertia identified

for the series-connected motor also includes the inertia of theload motor.

The system’s closed-loop poles using the feedback lineariza-tion controller were set at ,while the poles of the observer’s error system were set at

The above controller has a singularity when the current iszero and, thus, at the startup of the motor. To get around thisdifficulty, the current in the controller algorithm was boundedbelow by 0.04 A. That is to say, if A, then wasset equal to 0.04 sign in the controller algorithms.

Using the simulation packageSIMNON [15] along withthe above parameter values, controller gains, and amplifierconstraints, the system model (2), along with the feedbacklinearization controller and observer, was simulated for com-parison with the experimental results.


(a) (b)

(c) (d)

(e) (f)

Fig. 5. Speed control up to 190 rad/s. (a)!ref ; !̂; !sim (rad/s) versus time (s). (b)!ref � !̂ (rad/s) versus time (s). (c)�ref ; �; �sim (rad/sec2) versustime (s). (d)�L; �̂L (N � m) versus time (s). (e)i; isim (A) versus time (s). (f)V; Vsim (V) versus time (s).

A. Speed Control of the Motor Up to 190 rad/s

The first experimental set was chosen such that the speedwas brought up to 190 rad/s with no load torque on themotor. Fig. 5(a) is a plot of the (estimated) speed usingthe feedback linearization controller along with the speedreference and the simulated speed. The tracking capability ismore readily seen by a plot of the speed error, as shownin Fig. 5(b), where it is seen that the error is no largerthan 1.5 rad/s. Fig. 5(c) is a plot of the desired acceleration,estimated acceleration, and simulated acceleration. Note howwell the estimated acceleration tracks the desired acceleration.

It is also interesting to note that the simulated accelerationmatches the estimated acceleration quite well. The staircaseeffect in Fig. 5(c) is due to the quantization of the currentmeasurement (used in the calculation of the acceleration) bythe 8-b A/D. Fig. 5(d) is a plot of the predicted load torqueand the estimated load torque. Note that the estimated loadtorque is not zero as expected, but does oscillate around zero.This is most likely due to inaccuracies in the assumed modelon which the observer is based. Fig. 5(e) is a plot of the actualcurrent and the simulated current where, again, the staircaseeffect is due to the limited (8-b) resolution of the A/D used to


(a) (b)

(c) (d)

(e) (f)

Fig. 6. Speed control with steady-state load torque. (a)!ref ; !̂ (rad/s) versus time (s). (b)!ref � !̂ (rad/s) versus time (s). (c)�ref ; � (rad/s2) versustime (s). (d) �L; �̂L (N � m) versus time (s). (e)i (A) versus time (s). (f)V (V) versus time (s).

measure the current. By examining this figure, it can be seenthat the motor does not hit the current limit. Fig. 5(f) is a plotof the applied voltage and the simulated voltage. Note that theapplied voltage is just under the limit of 40 V.

B. Speed Control of the Motor with a Steady-State Load Torque

In this experimental run, the feedback linearization con-troller was used to force the motor to track a trajectory thattakes the motor up to a speed of 100 rad/s with a steady-state load torque of 0.17 Nm on the motor. Fig. 6(a) is aplot of the (estimated) speed along with the speed reference.Again, the tracking capability is more readily seen by a plotof the speed error as shown in Fig. 6(b), which shows thatthe error is no greater than 1 rad/s. Fig. 6(c) is a plot of

the desired acceleration and estimated acceleration. Again,note how well the estimated acceleration tracks the desiredacceleration. Fig. 6(d) is a plot of the predicted load torqueand the estimated load torque. Note that although the estimatedload torque does not track the predicted load torque perfectly,it is not too far off. Again, errors are most likely due to nothaving an exact model of the system. Fig. 6(e) is a plot of theactual current, where it is seen that the motor does not hit thecurrent limit. Fig. 6(f) is a plot of the actual voltage and thesimulated voltage. Note that the applied voltage never hits thelimit of 40 V.

VIII. SENSORLESSOBSERVER

Here, we present some of the preliminary results at imple-menting the encoderless observer given in Section V.


(a) (b)

(c) (d)

Fig. 7. Sensorless observer. (a)!̂standard; !̂sensorless (rad/s) versus time (s). (b)!ref � !̂sensorless (rad/s) versus time (s). (c)�L; �̂L (N � m) versustime (s). (d) �L; �̂L (N � m) versus time (s).

The gains used in the sensorless observer (9) were asfollows:

gain value

These gains place the closed-loop poles of the observer’serror system (10) at and

The observer’s measurement is singular atFor this reason, the observer was not initiated until the currentreached a value of .1 A. This corresponds to just over 0.05s in Fig. 6(e). Before this time, both the speed estimate andload-torque estimate are taken to be zero. Fig. 7 shows theexperimental results of the sensorless observer, showing thatthe observer was able to track the speed to some extent.

Fig. 7(a) is a plot of the estimated speed using the standardobserver (Section IV) and the estimated speed using thesensorless observer (Section V). Again, note the speeds track,except for the slight oscillations on the sensorless observerspeed. Fig. 7(b) represents the error between the desired speedand the estimated speed. The speed error is quite large at thebeginning, due to the fact that the observer does not get acurrent measurement until 0.05 s have elapsed. Fig. 7(c) is aplot of the predicted load torque, estimated load torque fromthe sensorless observer, and the estimated load torque fromthe standard observer described in Section IV. Fig. 7(d) is a

close-up of Fig. 7(c). Note that the load-torque estimate fromthe sensorless observer is similar to that of the load-torqueestimate from the observer based on the position measurement.Even though the speed estimates and load-torque estimatestracked to some degree when not used for feedback, theseestimates (based on the sensorless observer) seriously deteri-orated the tracking performance of the motor when used forfeedback. This appears to be due to the lack of resolution inthe current measurements, since only an 8-b A/D was used.

IX. CONCLUSIONS

It has been shown that feedback linearization can be suc-cessfully implemented as a high-performance feedback con-troller for speed control of a series dc motor. This implemen-tation was compared with a simulation of the system whichshowed a considerable degree of correspondence. It was alsoshown that a speed and load-torque observer results in an accu-rate speed estimate that is insensitive to constant disturbancesin the torque model. Preliminary results of a sensorless speedobserver were also presented. More work remains to be doneto be able to use this speed estimate for feedback in a high-performance motion control. In particular, an A/D with higherthan 8-b resolution for the current measurement will be needed.

ACKNOWLEDGMENT

The authors are grateful to Prof. M. Bodson for making theseries dc motor available to them for this work, to M. Aiello


of Aerotech Incorporated for his hardware expertise, and tothe Motorola Corporation for providing the DSP56001 digitalsignal processor used in this work. Finally, the authors wouldalso like to thank R. Novotnak for providing his MotorolaDSP expertise.

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Samir Mehta (S’94–M’96) received the B.S. andM.S. degrees in electrical engineering from theUniversity of Pittsburgh, Pittsburgh, PA, in 1993 and1996, respectively.

From 1995 to 1997 he was an Engineer with ABBDaimler-Benz Transportation Inc. in the PropulsionSoftware Group. He is currently with Teradyne Inc.,Agoura Hills, CA, writing embedded C++ softwarefor automatic test equipment.

John Chiasson(S’82–M’84) received the B.S. de-gree in mathematics from the University of Arizona,Tucson, the M.S. degree in electrical engineeringfrom Washington State University, Pullman, and thePh.D. degree in control sciences from the Universityof Minnesota, Minneapolis.

From 1988 to 1996, he was with the Departmentof Electrical Engineering, University of Pittsburgh,Pittsburgh, PA. He is currently with the SystemsEngineering Group, ABB Daimler-Benz Transporta-tion Inc., Pittsburgh, PA.

feedback linearization of dc motors - theory and experiment

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