Theoretical and Experimental Implementation of DC Motor NonlinearControllers

D.R. Espinoza-Trejo ∗ and D.U. Campos-Delgado †

∗ Facultad de Ingenierıa, CIEP, UASLP, espinoza trejo dr@uaslp.mx† Facultad de Ciencias, UASLP,

Av. Salvador Nava s/n, Zona Univ., C.P. 78290, S.L.P., Mexicoducd@fciencias.uaslp.mx

Abstract— The development of two nonlinear control strate-gies for the velocity regulation of a DC motor are detailed inthe paper. The parallel (shunt) connection of the DC motoris studied. First, a parameter identification was carried outusing experimental input-output data of the motor. One controlalgorithm involves a nonlinear cancelation law (input-outputlinearization) with a PID velocity reference error compensation,and a Luenberger observer to estimate the load torque. Inthis algorithm, the integral compensation and the torqueestimation improve the robustness of the overall control scheme.In addition, a variable-structure control (sliding mode) wasdeveloped also for velocity regulation, that uses the informationof the Luenberger observer to include the estimated load torque.Experimental results in a 2 HP test-bed corroborate the analysisand designs presented.

Index Terms— DC motor, variable-structure control, input-output linearization.

I. INTRODUCTION

Electrical motors are a key piece in almost any automaticprocess. They convert the energy from electrical into me-chanical in order to produce movement. There are two basictypes of electrical machines: DC and AC motors [6]. TheDC motors are pretty common in industrial processes due totheir operational properties and control characteristics. Theyare used for traction, cranes, mills, etc. According with theconnection between armature and field in the DC motor, threeconnection area devised: (i) parallel (shunt) connection, (ii)series connection, and (iii) independent excitation [6],[8].The first two have the advantage that they only need onevariable DC power supply to control the motor. However,they present the disadvantage that the corresponding controlalgorithms are more involved, since the mathematical modelof both systems are nonlinear [2], [3], [7]. The control ofthe parallel configuration will be addressed in the paper.

In some applications, it is necessary to adjust the motorangular velocity constantly despite the motor load. For thispurpose, it is needed the appropriate hardware to be ableto make the adjustments in the input voltage to the motor,according with a control algorithm (variable speed drive).Hence it is necessary to keep the angular velocity of themotor regulated and adjust the motor electrical torque tocompensate the load. This paper addresses the problemof velocity regulation with load torque compensation forthe parallel DC motor. In order to achieve this goal, aninput-output linearizing controller is designed [5], [10], that

includes a PID compensation according with the referenceerror and a load torque estimation. This technique is based ondifferential geometric control [1], [9]. It is observed that theintegral action in conjunction with the torque estimation addrobustness to the control algorithm. Another control method-ology that has been used extensively for nonlinear systemsis variable-structure control [4], [11]. The main advantagesof this algorithm is its robustness against noise and modeluncertainty. For comparison in the paper, a variable-structurecontroller is also developed for the velocity regulation ofthe DC motor. The paper is organized as follows. Section2 presents the mathematical model of the motor, and theparameters identification. The control algorithms are detailedin Section 3, and the experimental implementation is shownin Section 4. The paper ends in Section 5 with final remarksand conclusions.

II. DC MOTOR MODELING

In the following derivations, consider the states of thesystem as x1 = ia the armature current, x2 = if the fieldcurrent, and x3 = ω the angular velocity. It is assumedthat they are all measured on real-time. The control variableu = V is the variable DC voltage delivered to the motor.The parameters of the DC motor are defined as:

Ra Armature resistanceLa Armature inductanceM Mutual inductanceRf Field resistanceLf Field inductanceB Mechanical frictionJ InertiaTl Load torque

In this study compared to previous ones [1], it is notneglected La, since for medium to large size motors, thisparameter could be comparable in magnitude to Ra.

A. Parallel (Shunt) Connection

The mathematical model of this configuration is presented,assuming that it is available a variable resistor Radj to adjust

the maximum velocity in the motor (field weakening) [2]:

x1 = −Ra

Lax1 −

M

Lax2x3 +

1

Lau

x2 = −Rf +Radj

Lfx2 +

1

Lfu (1)

x3 = −B

Jx3 +

M

Jx1x2 −

1

JTl

Note that Radj has to be selected according to the desiredmaximum angular velocity, and the rated maximum fieldcurrent in the motor, in order to calculate the maximumpower dissipated by this resistor. Figure 1 shows the electri-cal diagram of the parallel (shunt) connection.

Rf

Lf

Ra

La

Radj

eTl

J, B

+

-

i f

ia

V

+

-ω

Fig. 1. Parallel Configuration of DC Motor.

B. Experimental Identification

First of all, it is needed to obtain an approximation ofthe parameters of the model in (1), and a least squaresapproximation is carried out [9]. Since the armature electricaland mechanical parameters are the most difficult to estimate,the DC motor is considered in a separated armature-fieldconfiguration. Hence a constant DC voltage source is con-nected to the field, without load torque applied to the motor.Meanwhile, the armature is supplied with a square voltagesignal in order to provide excitation to the system and achievethe identification. Note that in this configuration the DC mo-tor presents linear dynamics with respect to the parameters.The parameters to identify are (La, Ra,Kb, J, B), where theelectromagnetic constant Kb is related to the constant fieldcurrent If and the mutual inductance M , i.e. Kb = MIf .The field parameters Rf and Lf could be identified applyinga step voltage, and measuring the resulting current to identifyits characteristic time and peak response, or directly by aLCR Multimeter. The latter approach was adopted in thispaper. Now, a regressor formulation was used to identify themotor parameters using integral relations to avoid derivativesand improve noise robustness, i.e.

[

ia∫

ia∫

ω 0 00 0 −

∫

ia ω∫

ω

]

La

Ra

Kb

JB

=

[ ∫

u0

]

(2)

The parameters are then identified from collected data of ia,ω and u in order to construct the linear algebraic equationsin (2). Note that the equations must be satisfied at each time

TABLE IDC MOTOR PARAMETERS.

Parameter ValueRa 0.699 ΩLa 0.297 HM 2.134Rf 445 ΩLf 56 HJ 2.79× 10−3 kg ·m2

B 4.45× 10−3 N ·m/rad/s

instant. Thus, assuming that the data is sampled at period Ts,denote the matrices W (nTs) and y(nTs), and the regressorΘ in (5), then to compute a solution a summation for Ntime instants is used [9]. Therefore, define the summationmatrices:

RW ,

N∑

n=1

WT (nTs)W (nTs) ∈ R5×5 (3)

Ry ,

N∑

n=1

WT (nTs)y(nTs) ∈ R5 (4)

RW is always positive semi-definite, but if N is large enoughand under an appropriate excitation of u then RW > 0, anda solution can be deduced Θ = R−1

W Ry . The parametersidentified are shown in Table I.

III. NONLINEAR CONTROL STRATEGIES

The proposed nonlinear control strategies followed in thispaper are detailed next. It is important to mention that thiscontrol problem is very demanding since the mathematicalmodel is nonlinear; There is intrinsically some uncertainty inthe identified parameters; There is an unknown perturbationacting on the system (load torque); and in the test-bed,there are noisy measurements. As a result, a simple PIDor other types of linear controllers can not achieve thecontrol objective, and a more complex algorithm must bepursued. Other model-based control strategies as Lyapunov-based design or adaptive nonlinear control could also beviable tools, however they are not explored in this work.

A. Nonlinear Cancelation Law

This control scheme consists of three parts:1) Input-output linearizing law,2) PID reference compensation, and3) Luenberger observer for load torque estimation.

The control block diagram is presented in Figure 2, wherethe estimated variables are denoted by (·). In the nextsubsections, these three parts are detailed.

1) Input-Output Linearization: The control schemeadopted is based on differential geometric methods [5], [10].For this purpose, the output of interest is the angular velocityω, and it can be proved that the system presents a relativedegree of two. The relative degree is well-defined if thecondition β(x) 6= 0 is satisfied [7], where β(x) is givenby

β(x) =M

J

[

x1

Lf+

x2

La

]

(6)

W (nTs) ,

[

ia(nTs)∫

ia(nTs)∫

ω(nTs) 0 00 0 −

∫

ia(nTs) ω(nTs)∫

ω(nTs)

]

∈ R2×5

y(nTs) ,

[ ∫

u(nTs)0

]

∈ R2 (5)

Θ ,[

La Ra Kb J B]T

∈ R5

PID ReferenceCompensation

Velocity and Load Torque Observer

Input-Output Linearizing Law

Control Actuator DC MOTOR

iaif

u V

v

ω

ωref

ω

Tl^

Fig. 2. Overall Nonlinear Cancelation Law.

Hence, the relative degree is well defined if the sum ofarmature and field currents is different from zero. In apractical setting, this condition is satisfied, assuming thatthe DC motor is operated in just one rotating direction.Therefore, a linearizing control law is given by

u =−α(x) + v

β(x)(7)

for β(x) 6= 0, where v is a desired dynamic added tothe system, and the function α(x) is defined as in (8).The zero dynamics for this configuration were proved tobe asymptotically stable (minimum-phase system), and theproof is not included for brevity.

2) Velocity and Load Torque Luenberger Observer: Inorder to compensate the load torque, it is necessary toestimate this quantity to avoid steady-state error between thevelocity and its reference. A Luenberger observer [9], [10] isproposed assuming that the load torque is roughly constantTl ≈ 0. The observer reproduces the mechanical equationin the DC motor with an error correction term due to thevelocity estimation:

dω

dt= −

B

Jω −

1

JTl +

M

Jiaif + l1(ω − ω)

dTl

dt= −l2(ω − ω) (9)

Note that the input variables to the observer (9) are theangular velocity ω, and the armature and field currents ia, if .The observer gains l1 and l2 must be selected such that thecharacteristic equation:

λ2 + (B/J + l1)λ+ l2 = 0 (10)

has its roots in the left-half plane, in order to guaranteeconvergence to the real values.

3) PID Reference Compensation: A constant reference forthe angular velocity ωref is assumed. In order to providegood reference tracking, the desired dynamic induced in thelaw (7) has a PID action:

v = ωref +Kd(ωref − ω) +Kp(ωref − ω) +

Ki

∫

(ωref − ω)dt−B

J2Tl (11)

where the estimated torque Tl is used to cancel this term andavoid a steady-state error. However, the previous equation(11) can be simplified, since the reference is assumed tobe constant or to change slowly ωref ≈ 0 and ωref ≈ 0.Moreover, it is desirable to avoid the derivative of the angularvelocity, so the mechanical equation in the motor is usedinstead. Therefore, the resulting PID law is proposed:

v = Kd

(

−B

Jω +

M

Jiaif −

1

JTl

)

+ (12)

Kp(ωref − ω) +Ki

∫

(ωref − ω)dt−B

J2Tl

where the estimated value of the velocity ω is used in theproportional and derivative actions to avoid the noise effects.The selection of the PID gains Kp, Kd and Ki must pursuethat the characteristic equation:

λ3 +Kdλ2 +Kpλ+Ki = 0 (13)

had its roots in the left-half plane to provide closed-loopstability.

IV. SLIDING-MODE CONTROLLER

On the other hand, a control scheme using a variable-structure philosophy (sliding mode) [4], [11] consists ondefining an sliding surface that reflects the performanceobjectives, and obtaining the equivalent and approximationcontrol laws. The control block diagram is presented in Fig-ure 3, where the estimated variables are again denoted by (·).In the next subsections, these three parts are detailed. First,consider the following general structure of the mathematicalmodel without external perturbations:

x = f (x) + g (x) · u (x, t) (14)

where for the DC motor (see original model in (1)) f(x) andg(x) are given by:

f (x) =

−Ra

Lax1 −

MLax2x3

−Rf

Lfx2

−BJ x3 +

MJ x1x2

g(x) =

1

La1

Lf

0

(15)

α(x) =B2

J2x3 −

M

J

[

B

J+

Rf +Radj

Lf

+Ra

La

]

x1x2 −M2

JLa

x2

2x3 (8)

ωrefApproximation

Law

EquivalentLaw

ControlActuator

u+

+

DC Motor

Velocity and LoadTorque Observer

V

iaifω

3

3

2

TL^

ω ^

Fig. 3. Overall Variable-Structure Control Strategy.

Therefore, a general control structure following the slidingmode methodology is given by:

u (x, t) = ueq (x, t) + uR (x, t) (16)

where ueq denotes the equivalent control law, and uR theapproximation law [4].

A. Sliding Surface

The sliding surface specifies the desired characteristics ofthe control system as: stability, tracking, regulation, etc. It isthen proposed an sliding surface for velocity regulation:

σ = γe+ e = 0 (17)

where e = x3 − ωref denotes the regulation error and ωref

the velocity reference. Note that when the surface has beenreached (σ = 0), the parameter γ define convergence speedof the regulation error to the origin. Then, substituting thestate x3 in (17), and considering that the reference velocityis a constant (ωref = 0), it is obtained the sliding surfaceas:

σ (x) =

(

γ −B

J

)

x3 +M

Jx1x2 −

1

JTl − γωref (18)

Therefore, in order to avoid the velocity derivative in thesliding surface, it is needed the load torque Tl informationin σ(x).

B. Equivalent Law

The equivalent control constitutes a control input which,when exciting the system, produces that the resulting trajec-tories remain on the sliding surface whenever the initial stateis on the surface [4]. Based on the existence of a slidingmode, and using the chain rule, it is define the equivalentcontrol ueq for systems of the form (14) as:

ueq = −

∂σ

∂x· g(x)

−1

·

∂σ

∂x· f (x)

(19)

As a result, in order to guarantee the existence of anequivalent control law, it is required:

∂σ

∂x· g(x, t) 6= 0 (20)

Therefore, for the DC motor in a parallel configuration, theexistence condition is given by:

1

La

∂σ

∂x1

+1

Lf

∂σ

∂x2

6= 0 (21)

=⇒M

J

[

x2

La+

x1

Lf

]

6= 0 (22)

C. Approximation Law

Now, consider the following approximation control law[4], [11]:

uR (x, t) =

∂σ

∂x· g(x)

−1

· uR (23)

where,uR = −K |σ|

1/2 sign(σ) K > 0 (24)

In the last equation, the parameter K will affect the speedconvergence of the trajectories to the sliding surface. Notethat the approximation law (24) will be large if the systemis far away from the sliding surface. In a practical imple-mentation, there exists noise in the measurements, thereforeto reduce the chattering phenomenon a boundary layer [12]approach can be followed for the approximation control law.Hence, the control (24) is modified as

uR = −K |σ|1/2 sat

(σ

ε

)

(25)

where the parameter ε defines the size of the boundary layer.

V. EXPERIMENTAL IMPLEMENTATION

The control diagrams shown in Figures 2 (nonlinear can-celation law) and 3 (variable-structure control) were imple-mented experimentally in a dSPACE DS1103 system runningat a sampling frequency of 10 kHz. The control parametersfor both strategies are presented in Table II. The test-bedshown in Figure 4 was used, and it consists of a 2 HP ShuntDC Motor that is connected to a 2 HP Permanent MagnetDC Motor utilized as a load. A tacogenerator measures theangular velocity of the shaft, at a proportion of 50 V/RPMwith an error of ±10 %. There are measurements of thearmature and field currents through hall-effect sensors. Itis important to mention that the three measurements arenoisy during the experiments, as it will be observed in theimplementation plots, and this issue presents a challenge forthe control algorithm to show good robustness.

The motor voltage V is controlled by DC-DC chop-per working under a PWM scheme (switching frequency10kHz), where the control parameter is the duty cycle u[6]. The chopper was selected as control actuator due to its

V

DC-DC Chopper

output

+ -

3 HP Shunt DC Motor

DC Motor(permanent

magnet)

R L

+ -Three-phaseVoltage AC

Source

rectifier bridge

tacogenerator

AA

FF

1

1

2

2

+-

T1T2

u

control signal

three-phasetransformer

LoadTorque

hall effectcurrent sensors

Vcd

field-armatureconnection

ω

Fig. 4. Experimental Configuration of the DC Motor.

TABLE IICONTROL PARAMETERS.

Parameter ValueNonlinear Cancelation Law

Kp 227.24Ki 932.46Kd 24.62

Load Torque Observerl1 30.37l2 512.15

Variable-Structure Controlγ 150K 250ε 1.0

fast response and linear dynamics, and it was just modeled asa scaling factor in the control system. The construction of theactuator was carried out in our lab, and it is designed suchthat it is controlled by a voltage signal u in the interval [0, 10]V. This saturation in the control signal u did not limited theperformance of the system, due to the fast dynamics of theactuator related to the time constants of the DC motor. Theadjustment resistor during tests was not used (Radj = 0 Ω),since the open-loop maximum velocity was adequate andthere was no need to apply field-weakening. Two tests werecarried out for the control algorithm:

Case A: tracking of an angular velocity reference changefrom 1400 to 1200 RPM’s (see Figure 5).

Case B: load torque variation from 0 Nm (no-load) to1.5 Nm, and back to 0 N m (Figure 6).

In Figure 5 the results for Case A are presented withthe nonlinear cancelation control. It can be observed thatthe tacogenerator measurement for the angular velocity is

extremely noisy. Nevertheless, the controller is capable ofproviding good tracking for a reference change. Due to thischange, the control signal decreases its value to compensatethe lower reference. Also, the armature and field currentsupdate their value to accommodate the reference modifica-tion. Similarly, for the variable-structure control law goodtracking is observed, and the plots are omitted due to spacelimitations. Finally, in Figure 6 the experimental results forCase B are illustrated, but now for the variable-structurecontrol law. The top plot shows that the angular velocity iscorrectly compensated, since only a transient effect is noticedafter the load change. On the other hand, the armature andfield currents increase due to the load effect, and recovertheir previous value when the load is again removed. Similarresults are derived for the nonlinear cancelation controllaw. Consequently, both techniques show good performanceagainst a reference change and load perturbations.

VI. CONCLUSIONS AND FINAL REMARKS

In the present paper, a theoretical derivation and practicalimplementation of two nonlinear control schemes for ashunt DC motor are detailed. The experimental identificationproposed, with integral relations, was capable to providegood estimates of the real parameters. A nonlinear controlalgorithm is designed departing from input-output lineariza-tion theory (differential geometric tools). This techniqueis recognized to have robustness issues during practicalimplementations. For this purpose, integral action to thereference error and load torque estimation were added toimprove the robustness of the overall structure. Despitenoisy measurements during the practical implementation, thecontrol scheme is able to adjust the motor voltage properly

0 5 10 15 20 251150

1200

1250

1300

1350

1400

1450

time (seconds)

ω (

RP

M)

0 5 10 15 20 256.5

7

7.5

8

8.5

9

9.5

time (seconds)

u (

duty

cycle

)

ωref

=1200 RPMω

ref=1400 RPM ω

ref=1400 RPM

0 5 10 15 20 253.5

4

4.5

5

5.5

6

6.5

7

7.5

8

time (seconds)

i a (

A)

0 5 10 15 20 250.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

time (seconds)

i f (A

)

REFERENCE UPDATE

Fig. 5. Experimental Response for Case A with Nonlinear CancelationControl (TOP) Angular Velocity Measurement, (MIDDLE 1) Control signal(duty cycle), (MIDDLE 2) Armature Current, (BOTTOM) Field Current.

to follow a variable velocity reference and to compensateload torque changes. On the other hand, a variable-structurecontrol was also proposed. The resulting control algorithmproduced good performance against perturbations and noisemeasurement. As a result, both techniques are visualized aspractical tools in this nonlinear setup.

ACKNOWLEDGMENTS

This research was supported in part by a grant fromPROMEP. Diego Espinoza-Trejo acknowledges the financialaid provided by CONACYT through a doctoral scholarship.

REFERENCES

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[3] J. Chiasson. “Nonlinear Differential-Geometric Techniques for Controlof a Series DC Motor”. IEEE Transactions on Control SystemsTechnology, 2(1994), 35–42.

0 5 10 15 20 251050

1100

1150

1200

1250

1300

1350

time (seconds)

ω (

RP

M)

0 5 10 15 20 25

6

6.2

6.4

6.6

6.8

7

7.2

7.4

7.6

7.8

time (seconds)

u (

duty

cycle

)

NO LOAD TORQUETl=1.5 N mNO LOAD TORQUE

0 5 10 15 20 25−1

0

1

2

3

4

5

6

7

8

time (seconds)

i a (

A)

0 5 10 15 20 250.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

0.19

0.195

time (seconds)

i f (A

)

Tl=1.5 N m

Tl=0 N mT

l=0 N m

Fig. 6. Experimental Response for Case B with Variable Structure Control(TOP) Angular Velocity Measurement, (MIDDLE 1) Control signal (dutycycle), (MIDDLE 2) Armature Current, (BOTTOM) Field Current.

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