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Optimal State Space Control ofDCMotor
M. Ruderman, J.Krettek, F. Hoffmann, T. Bertram
Chair for ControlSystems Engineering, Technische Universit¨at
Dortmund, D-44221 Germany (Tel. +49/231/755-2496, e-mail:
mykhaylo.ruderman@tu-dortmund.de)
Abstract:
In comparison to classical cascade control architecture ofDCmotors, the state feedbackcontrol
offers advantages interms of design complexity,hardware realization and adaptivity.This
paperpresentsamethodic approachtostate space control of aDCmotor. The state space
model identified from experimental data provides the basis for alinear quadratic regulator
(LQR) design. The state feedbacklinear control is augmented with afeedforward control for
compensation of Coulombfriction. The controller is successfully applied and the closed loop
behavior is evaluated onthe experimental testbedunder various reference signals.
Keywords: model-based control; LQR control method,motor control; state-space models;
feedforward compensation
1. INTRODUCTION
DC motors provide an attractivealternativetoACservo
motors in high-performance motion control applications.
DC motors are in particular popular in low-powerand high
precise servo applications due to their reasonable cost and
ease of control. Traditionally motor controls in industrial
applications employacascade control structure. The outer
speed and inner currentcontrol loops are designed as PD
or PI controllers. However, the cascaded control structure
assumes that the inner loop dynamics are substantially
faster than the outer one (Chevrel etal. (1996)).
In recentyears several publications propose alternative
approaches to identification and control of DC motors.
Umeno and Hori (1991) describeageneralized speed
control design technique ofDCservomotors based on
the parametrization of two-degrees-of-freedom controllers
and apply the design methodof aButterworth filter to
determine the controller parameter. Chevrel etal. (1996)
presentaswitched LQR speed controller, designed from
the linear model of the DCmotor, and compare its
performance with acascade control design in terms of
accuracy,robustness and complexity.Rubaai and Kotaru
(2000) propose analternativewayto identify and control
DC motors bymeans of anonlinear control lawrepresented
byan artificial neural network. Yuand Hwang (2004)
presentanLQR approachtodetermine the optimal PID
speed control ofthe DC motor.
This contribution proposes asystematic approachtove-
locitycontroller design of aDCmotor based onmodel
identification and LQR design augmented with anonlinear
feedforward compensator. The electrical and mechanical
parameters of the DCmotor, i.e., resistance, inertia, back-
EMF, damping are identified from observations ofthe
open loop response. Coulombfriction is considered as
the main cause ofthe nonlinear motor behavior and is
adequately compensated byafeedforward control signal.
The residual steady state error caused byminor nonlin-
earities and uncertainties inthe model is compensated by
an integral error feedbacksignal. The proposed controller
is evaluated for high and lowvelocityreference profiles
including velocityreversal to demonstrate its efficiency
for high-performance servo applications. The proposed
scheme attempts to bridge the currentgap between the
advance of control theory and the practice ofDCactuator
systems.
This paperis structured as follows: Section 2describes the
state-space model of the DCmotor derived from electro-
mechanical relationships including friction. The model
identification is described in Section 3. Section 4details
the LQR design of the optimal state feedbackcontrol with
an integrator and the feedforward friction compensation.
Section 5 analyzes the closed performance on the experi-
mental testbedunder differentvelocityreference profiles.
Finally,Section 6summarizes the major conclusions ofthe
paper.
2. DC MOTOR MODEL
2.1 Linearstate-space model
Under the assumption of ahomogenous magnetic field,
the direct current(DC) motor is modeled as alinear
transducer from motor currentto electrical torque. The
classical model of the DCmotor, described byIsermann
(2002) is composed of acoupled electrical and amechanical
subsystem.
The angular velocityωis controlled bythe input voltage u
with aconstantvoltage drop attributed to the brush and
rotor resistance, and aback-electromotiveforce (EMF)
caused bythe rotary armature. The motor inductance
contributes proportional to the change inmotor current
i.The motor currentcouples the electrical component
with the mechanical one, as it generates the driving
torque. This torque isantagonized bythe motor inertia,
Proceedings of the 17th World Congress
The International Federation of Automatic Control
Seoul, Korea, July 6-11, 2008
978-1-1234-7890-2/08/$20.00 © 2008 IFAC 5796 10.3182/20080706-5-KR-1001.3625
structure damping, friction, and the external load. The
motor dynamics are described by:
u(t)=Ldi
dt +Rmi(t)+Keω(t),(1)
Kmi(t)=Jdω
dt +Kdω(t)+τl+τf,(2)
where Km,Keand Kddenote the motor torque, the back-
EMF and the damping constants. Jdenotes the mechan-
ical inertia including the motor armature and shaft. L
and Rmrepresent the inductance and the total connection
resistance of the motor. The system load and friction are
denoted byτland τf.
However, for manyapplications this structure isnot suffi-
cient. The main drawbackof the linear state-space model
is anegligence of nonlinear effects, whose properties can
significantly affect the dynamic behavior of amodeled sys-
tem. Tocomplete the representation of essential physical
phenomena effecting inthe motor structure the frictional
nonlinearitymust beincluded.
2.2 System nonlinearity
According toPaduart et al. (2006)) the linear state-
space model with amultivariable nonlinear input function
f(x(t),u(t)) assumes the general form:
˙x=Ax+Bu+Hf(x,u),(3)
in whichAis the system matrix, Bis the input vector, and
the coupling vector Hlinks the nonlinearitywith the linear
part. Incontext of permanentmagnet DC motor Coulomb
friction constitutes the major source ofnonlinear behavior
(Knudsen and Jensen (1995)). Additional nonlinearities
emerge from the inhomogeneityof the stator magnetic
field and transfer characteristics ofthe amplifier and IO
elements as well as motor cogging and ripple effects (see
Procaetal. (2003)).
Tjahjowidodoetal. (2005) describeadvanced friction mod-
els whichintroduce auxiliary internal states tocapture
friction dynamics. As these auxiliary states are not observ-
able customized identification techniques are required for
the identification of their associated parameters. For many
applications astatic friction model that includes Coulomb
and viscous parts suffices to capture the main frictional
phenomena.
The linear viscous friction is already comprehended in the
damping term inequation (2). Considering the nonlinear
Coulombfriction whichdepends on the rotation direction
and introducing the state vector x=[i, ω]Tresults in:
˙x=
−Rm
L−Ke
L
Km
J−Kd
J
x+
1
L
0
u+
0
−Fc
J
sgn(x2)(4)
in whichFcdenotes the Coulombfriction coefficient. The
overall model has six independentparameters, ofwhich
the inductance L=25×10−6Hisobtained from the
manufacturer datasheet and the remaining fiveparameters
are identified from experimental data.
12.542,6 12.542,7 12.542,8 12.542,9
8.8
9
9.2
9.4
9.6
9.8
10
ISE velocity
ISE current
pareto−optimal solutions
Fig. 1. Pareto-optimal compromises between ISE for cur-
rentand angular velocity
3. MODEL IDENTIFICATION
The signals for identification are generated from the open
loop step response of the DCmotor at differentampli-
tudes. The identification yields the set of optimal parame-
ters that minimize the squared error between model output
and data.
Eω=(ω(t)−ˆω(t))2dt
Ei=(i(t)−ˆ
i(t))2dt .
(5)
As the model is linear inthe unknown parameters, these
are identified bymeans ofleast squares. The remaining
choice isthe trade-off between the twoerrors. This trade-
off is specified bytheir relativeweightwin
E=wEω+(1−w)Ei.(6)
In the context of state feedbackcontrol the model should
not only reflect the input-output behavior but also accu-
rately describethe dynamics ofinternal states, in our case
the motor current.
Fig. 1visualizes the set of pareto-optimal solutions ob-
tained from variations ofthe weightw∈[0,1]. The squared
error inthe angular velocityis rather insensitivetopa-
rameter variations asthe stickslip effect at lowvelocities
causes an oscillation in the angular velocity(see upper left
graph inFig. 2) not captured bythe model. This deviation
causes alarge offset insquared error compared whichthe
residual error contributions inthe rising edge and steady
state are negligible. The compromise solution is marked
byan arrowinFig. 1.
The actual step responses are compared with the model
output in Fig. 2for asubset of six out of sixteen signals
taken into accountfor identification. The graphs show
that the identified parameters correctly capture the steady
state behavior as well as the characteristic time constants
in the rising edge. The oscillatory behavior at lowfre-
quencies does not correspond to aneigen frequency of the
system but merely reflects the variation of friction during
acomplete rotation of the motor shaft.
The identified parameters are listed inTable 1 and com-
pared with the nominal values provided bythe manufac-
turer. The differences between the nominal and the iden-
17th IFAC World Congress (IFAC'08)
Seoul, Korea, July 6-11, 2008
5797
Fig. 2. Comparison of real and model step-responses for
velocityand currentat differentvoltages
tified inertia and friction are explained bythe additional
inertia and bearing of the rotary encoder. The increase of
identified resistance is explained byadditional contacts of
the motor connection and structure changes of the motor
brushes and commutator. The other identified parameters
are inaccordance with the nominal values.
Table 1.System parameter identification
nominal identified
total motor resistance Rm(Ω) 0.35 0.98
torque constantKm(N m/A) 0.0296 0.0274
EMF constantKe(V s/rad) 0.0296 0.0297
damping constantKd(N s/rad) 6.7×10−57.2×10−5
total system inertia J(kg m2)2.9×10−63.2×10−5
Coulombfriction Fc(N m) 0.0200 0.0593
inductance L(H) 25 ×10−6
4. CONTROLDESIGN
State feedbackcontroller design is accomplished either
bypole placementorinthe context ofoptimal control
bymeans oflinear quadratic regulator (LQR) design
(see Anderson and Moore (2007)). In pole placement the
designer specifies the desired eigenvalues of the closed
loop system in the left half plane. LQR design minimizes
aweighted squared state error and control effort. The
optimal feedbackstate regulation, minimizes the quadratic
cost function
J=
∞
0xT(t)Qx(t)+uT(t)Ru(t),(7)
in whichQand Rare symmetric, positivesemi-definite
respectively positivedefinite weightmatrices. The optimal
feedbackgain
K=R−1BTP,(8)
is obtained from the solution Pof the algebraic Riccati
equation:
ATP+PA −PBR−1BTP+Q=0.(9)
The weightmatrices are specified suchthat the closed loop
system is able to trackthe reference signal with acontrol
signal that does not significantviolates the saturated
actuator limits. For afixed weightmatrix Q,the control
penaltyRis chosen suchthat for the maximum state error,
the feedbackcontrol signal
u=−Kx +Vω
r(10)
is in accordance with the actuator bounds. Tocompensate
the steady state error ofthe closed control loopafeedfor-
ward term is included in the control:
V=−(CT(A−BK)−1B)−1.(11)
Tocompensate disturbances, and model uncertainties of
the DCmodel the integral output error
ε=
t
0
(ωr−ωa)dt (12)
is introduced as an additional state variable, inwhichωr
and ωadenote the reference and actual velocities. The
linear part ofthe state space model is augmented bythe
auxiliary integral state variable:
˙
x
˙ε=A0
−CT0x
ε+B
0u+0
1ωr.(13)
Correspondingly the weightmatrix is augmented with a
small weightfor the integral error.
Q=100
010
000.001 ,R=10.(14)
4.1 Feedforwardfriction compensation
The closed loop behavior is further improved byafeed-
forward control for immediate compensation of Coulomb
friction. The friction is constantand it sign is opposite
to the direction of rotation. The discontinuityat velocity
reversal is smoothed byreplacing the step at ωr=0bya
linear segmentfor small velocities inthe range ofσ=±1
rad/s. The add-on feedforward control gain
Kf=RmFc
Km
,(15)
determined from equations (1) and (2) bythe elimination
of all dynamic terms compensates the static friction phe-
nomenon.
Table 2.Controller parameters
friction gain Kf1.06
feedforward gain V0.3166
feedbackgain Ki0.0984
feedbackgain Kω0.3003
feedbackgain Kε−0.01
17th IFAC World Congress (IFAC'08)
Seoul, Korea, July 6-11, 2008
5798
4.2 Control law
The overall control lawincluding feedforward compensa-
tion
Γ=
Kfsgn(ωr),if|ωr|>σ
Kf
ωr
σ,else(16)
becomes
u=−[Ki,ω Kε]x
ε+[ΓV]1
ωr,(17)
with the corresponding gains listed inTable 2.
The entire control structure instate space representation
is depicted in Fig. 3.
Fig. 3. Blockdiagram: proposed control structure with a
state-space system representation
5. CONTROLSYSTEM BEHAVIOR
5.1 Disturbancerejection
The designed velocitycontroller isvalidated onthe nom-
inal model in simulation for differentreference signals.
Toanalyze the robustness ofthe controller anexternal
periodic disturbance torque with amplitude of 0.06 Nm
and pulse width 0.01 sisapplied tomotor. The disturbance
has the same magnitude as the constantDCmotor friction.
00.5 1
−3
−2
−1
0
1
2
3
simulation time (s)
(V) / (Nm)
contol voltage
disturbance
00.5
1
−30
−20
−10
0
10
20
30
simulation time (s)
velocity (rad/s)
reference
actual
Fig. 4. Simulation results: torque disturbance and control
voltage (left), reference and actual velocity(right)
The controller responds immediately to the disturbance
resulting inarapid compensation of the error inoutput
velocityas showninFig. 4. Notice, that the designed
controller is specifically adopted tothe unloaded operation
of the DCmotor. The identification and control system
design has toberepeated ifthe DC motor is utilized
to drivean permanentexternal load. Ifthe applied load
is not known in advance, an adaptive control scheme is
advocated.
5.2 Experimental setup
The experimental testbedofthe DC motor for identifica-
tion and control is showninFig. 5. The sample rate of
the real time controller onboard the host computer is 5
kHz. The control signal uout with arange of ±5Vapplied
to the DCmotor is amplified. The motor is an AXEM
DC servomotor with ashrunk-on-disk rotor, F9M2 with
rated poweroutput of 63 Wand rated speed 3000 r.p.m.
(=314.1593 rad/s).
Fig. 5. Laboratory testbed: DC motor with arotary
encoder (left), and system overview (right)
The motor currentis measured bythe voltage drop uI
across ashunt Rm,and the motor shaft position ϕand
direction of rotation are provided byadigital single turn
rotary encoder Swith 13 bit resolution. The angular
velocityis obtained from time derivation of the shaft
position. Alowpass filter with windowsize 3is applied
to smooth the velocitysignal and reduce the quantization
errors.
5.3 Velocity control
The tracking behavior of the velocitycontroller isevalu-
ated for asinusoidal signal with amplitude of 200 rad/s
and periodof0.4sand asequence of up and downstep
signals with reference velocities inthe absolute range of5
up to 220 rad/s atintervals of 0.4s.
The reference and observedvelocities aswell as simulated
response of the closed loop system are depicted in Fig.
6 a)and b). The controller isable totrackthe reference
signal and with no residual steady state error athigh as
well as lowvelocities. The closed loop system exhibits a
lag characteristic with atime constantdetermined bythe
slowest eigenvalue attributed to the mechanical subsystem.
Torecognize isthe overlap in acceleration phase of the
closed loop behavior and the open loop response (bysatu-
rated control signal) depicted in Fig. 6c), whichindicates
amaximal achievable controller performance bounded by
actuator properties. Plotting the signals from Fig. 6 at
larger scale in Fig. 7reveals that the actual velocityoscil-
lates with small amplitude around the reference velocity.
This jitter iscaused bythe limited resolution of the rotary
encoder. The amplitude of the jitter corresponds to the
magnitude equivalentto asingle bit and is independentof
progress ofthe angular velocity.
The control signal saturates during the acceleration and
deceleration phases, suchthat the abilitytotrackthe ref-
erence is effectively dominated bythe actuator limitations
rather than the control. Fig. 8compares the observedcur-
rentwith the currentpredicted bythe model in case of step
17th IFAC World Congress (IFAC'08)
Seoul, Korea, July 6-11, 2008
5799
0.5 11.5 22.5 3
−200
−100
0
100
200
velocity (rad/s)
time (s)
actual
simulation
reference
a)
00.5 11.5
2
−200
−100
0
100
200
velocity (rad/s)
time (s)
actual
simulation
reference
b)
0.2 0.25 0.3 0.35 0
.4
0
50
100
150
200
250
time (s)
velocity (rad/s)
closed loop
open loop
c)
Fig. 6. Experimental results: a)stair shaped signal, b)
sinusoidal signal, c) step response of the openand
closed loops
1.59 1.6 1.61 1.62 1.63 1.6
4
0
10
20
30
40
50
velocity (rad/s)
time (s)
actual
simulation
reference
1.45 1.5 1.55 1
.6
−200
−190
−180
−170
−160
−150
velocity (rad/s)
time (s)
actual
simulation
reference
Fig. 7. Zoom in of angular velocity
responses. The currentpeaks incase ofabrupt changes in
reference velocities causing large state errors. The actual
peak currentsare about twoto three times larger than the
predicted ones whereas the steady state currentsmatch.
Weassume that the excess in peak currentsisexplained
bythe initial breakaway force (static friction) that the
motor has to overcome. Nevertheless, these peaks are not
critical as the maximal allowedmotor currentof 60 Aisnot
exceeded. The high-frequency jitter ofthe currentsignal at
steady state is attributed to the cogging and ripple effects
and the stick-slip motion.
0.5 11.5 22.5 3
−40
−30
−20
−10
0
10
20
30
time (s)
current (A)
actual
simulation
a)
0.4 0.5 0.6 0.7 0.8
−5
0
5
10
15
20
25
30
time (s)
current (A)
actual
simulation
b)
Fig. 8. Model predicted and measured motor currentby
the stair shaped reference signal a) and zoominof
motor currentb)
6. CONCLUSIONS
This paperproposes anovelapproachtocontrol design
of aDCservomotor based onsystem identification and
LQR control design. The feedbackcontroller isaugmented
with afeed-forward friction compensation. The mechanical
and electrical parameters of the DCmotor are identified
from the openloopresponses with respect to motor current
and angular velocity.The LQR design provides an optimal
state feedbackcontrol minimizes the quadratic state error
and control effort. The auxiliary integral error state and
feedforward compensation of the nonlinear friction reduce
the residual error across the entire range of reference veloc-
ities. The experimental results demonstrate the feasibility
of the controller design for high precision servo applica-
tions. The proposed methodiswell suited for the controller
design of highly dynamic DC motors. Future researchis
concerned with the design of adaptive controllers to han-
dle variations ofload and the identification of periodical
disturbances and advanced friction models.
ACKNOWLEDGEMENTS
The authors are grateful tothe K¨ubler GmbH company
for providing the sensor. Wealso thank HeikoPreckwinkel
for controller implementation and experiments.
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17th IFAC World Congress (IFAC'08)
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