Conference PaperPDF Available

Abstract and Figures

In comparison to classical cascade control architecture of DC motors, the state feedback control offers advantages in terms of design complexity, hardware realization and adaptivity. This paper presents a methodic approach to state space control of a DC motor. The state space model identified from experimental data provides the basis for a linear quadratic regulator (LQR) design. The state feedback linear control is augmented with a feedforward control for compensation of Coulomb friction. The controller is successfully applied and the closed loop behavior is evaluated on the experimental testbed under various reference signals.
Content may be subject to copyright.
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)=J
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
LKe
L
Km
JKd
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=25×106Hisobtained from the
manufacturer datasheet and the remaining veparameters
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+(1w)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) 6.7×1057.2×105
total system inertia J(kg m2)2.9×1063.2×105
Coulombfriction Fc(N m) 0.0200 0.0593
inductance L(H) 25 ×106
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=R1BTP,(8)
is obtained from the solution Pof the algebraic Riccati
equation:
ATP+PA PBR1BTP+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 axed weightmatrix Q,the control
penaltyRis chosen suchthat for the maximum state error,
the feedbackcontrol signal
u=Kx +
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(ABK)1B)1.(11)
Tocompensate disturbances, and model uncertainties of
the DCmodel the integral output error
ε=
t
0
(ωrωa)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
CT0x
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 Kε0.01
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|
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 lter 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.
REFERENCES
B. D. O. Anderson and J. B. Moore. Optimal Control:
Linear Quadratic Methods.Dover Publications, 2007.
17th IFAC World Congress (IFAC'08)
Seoul, Korea, July 6-11, 2008
5800
P.Chevrel, L. Sicot, and S.Siala. Switched LQ controllers
for DCmotor speed and currentcontrol: acomparison
with cascade control. In Proc.Power Electronics Spe-
cialists ConferencePESC’96 Record.,pages 906–912,
Baveno, Italy,June 1996.
R. Isermann. Mechatronische Systeme.Springer, Berlin,
2002.
M. Knudsen and J.G.Jensen. Estimation of nonlinear
DC-motor models using asensitivityapproach. In Proc.
3.EuropeanControlConference ECC95,Rome, Italy,
1995.
J. Paduart, J.Schoukens, K.Smolders, and J.Swevers.
Comparison of twodifferentnonlinear state-space iden-
tification algorithms. InProc.International Conference
on Noise and Vibration Engineering ISMA’06,Leuven,
Belgium, September2006.
A. B. Proca, A. Keyhani, A.El-Antably,W.Lu, and
M. Dai. Analytical model for permanentmagnet motors
with surface mounted magnets. IEEE transactions on
energy conversion,18:386–391, 2003.
A. Rubaai and R.Kotaru. Online identification and con-
trol of aDCmotor using learning adaptation of neural
networks.IEEE transactions on industrial applications,
36:935–942, 2000.
T. Tjahjowidodo, F. Al-Bender, and H. VanBrussel.
Friction identification and compensation in aDCmotor.
In Proc.16th IFACWorld Congress,Prague, Czech
Republic, July 2005.
T. Umeno and Y.Hori. Robust speed control ofDCservo-
motors using modern twodegrees-of-freedom controller
design. IEEE Transactions on industrial electronics,38:
363–368, 1991.
G.-R. Yuand R.-C. Hwang. Optimal PID speed control
of brush less DCmotors using LQR approach. In Proc.
IEEE International Conferenceon Man and Cybernet-
ics,pages 473–478, Hague, Netherlands, October 2004.
17th IFAC World Congress (IFAC'08)
Seoul, Korea, July 6-11, 2008
5801
... This method has resulted in a significant reduction of over 80% in positional error compared to the control without compensation. The feedforward term in the study referenced by [22] includes a component that takes into consideration the velocity error. Unlike feedforward compensation, the feedback compensation technique utilises the measured outputs, such as position and velocity, to estimate the friction force. ...
... The four-state equations can be written as a linear combination of the terms in the candidate term library (Eqs. [19][20][21][22]. Note that the coefficient a 1 is theoretically equal to 1. ...
Article
Full-text available
Friction compensation is critical for robust, dependable, and accurate position and velocity control of motor drives. Large position inaccuracies and vibrations caused by non-characterised friction may be amplified by stick–slip motion and limit cycles. This research study uses two data-driven methodologies to find the governing equations of motor dynamics, which also describe friction. Specifically, data obtained from a brushless DC (BLDC) motor is subjected to Sparse Identification of Nonlinear Dynamics with Control (SINDYc), and low-energy data extraction from time-delayed motor velocity coordinates is done to determine the underlying dynamics. Next, the identified nonlinear model is compared to a linear model without friction and a nonlinear model that contains the LuGre friction model. The optimal friction parameters for the LuGre model are determined using a nonlinear grey box model estimation approach with the collected data. The three validation datasets taken from the BLDC motor are then used to validate the proposed nonlinear motor model with friction characteristics. Over 90%90\% accuracy in predicting the motor states in all input excitation signals under consideration is demonstrated by the proposed model. Additionally, when applied to a model-based feedback friction compensation technique, the proposed model demonstrates a relative improvement in performance in comparison to a system that uses the LuGre model.
... Direct current (DC) motors have been widely used in several applications including robotics, industrial machinery and household applications that require a diverse speed control range [1,2]. These motors are predominantly preferred because of their reduced cost, control ease and resilience versatility [3,4]. ...
Conference Paper
Full-text available
Proportional Integral and Derivative (PID) controllers are widely employed to normalize the response of numerous DC motor-powered systems, such as electric locomotives, traction systems, etc. Specifically, this category of controller is ubiquitous due to their simplicity, ease of control, and low-cost. However, their design parameters need to be carefully tuned to enhance control performance. A novel LHHO-PID controller with a compound objective function that combine Integral of Time multiplied Squared Error (ITSE) with Zwee-Lee Gaing’s (ZLG) time-domain performance criterion is presented in this current study to enhance the control performance of a DC motor system. The system's performance was assessed by analyzing its response across various dimensions, including frequency responses, convergence profile and time. The results demonstrated that the developed PID-based controller exhibited a commendable performance in regulating the DC motor speed as compared to other existing PID-based controllers in the domain. The time response analysis showed an improvement of 7.2% compared to the best performing controller in rise time, a settling time of 0.1204 seconds, zero overshoot and negligible steady state error. These findings indicate that the proposed LHHO-PID controller has the capacity to regulate the speed of a DC motor efficiently in a variety of industrial applications.
... This method has achieved more than 80% reduction in positional error relative to the control without compensation. In [28], the feedforward term also incorporates a term that accounts for the velocity error. In contrast to feedforward compensation, the feedback compensation technique uses the measured outputs i.e. position and velocity to estimate the friction force. ...
Preprint
Full-text available
Friction compensation has become crucial for robust, dependable, and accurate position and velocity control of motor drives. Large position inaccuracies and vibrations caused by non-characterized friction may be amplified by stick-slip motion and limit cycles. In order to find the governing equations of motor dynamics, which also describe friction, this research applies two data-driven methodologies. Specifically, data obtained from a Brushless DC (BLDC) motor is subjected to the Sparse Identification of Nonlinear Dynamics with control (SINDYc) technique and low-energy data extraction from time-delayed coordinates of motor velocity to determine the underlying dynamics. Next, the identified nonlinear model was compared to a linear model without friction and a nonlinear model that contained the LuGre friction model. The optimal friction parameters for the LuGre model were determined using a nonlinear grey box model estimation approach using the collected data. The three validation datasets taken from the BLDC motor were then used to validate the resultant innovative nonlinear motor model with friction characteristics. Over 90% accuracy in predicting the motor states in all input excitation signals under consideration was demonstrated by the innovative model. Additionally, when compared to a system that was identical but used the LuGre model, a model-based feedback friction compensation technique demonstrated a relative improvement in performance.
... To ensure robustness in LQR control, we employ the integral of the output variable within the control system. This type of control function, known in the literature as LQR with integral action, exhibits robustness against system uncertainties and steady-state errors (Ruderman et al., 2008). ...
Article
Full-text available
This study applies these control methods to the DC motor system to examine the robustness and performance of four optimal control methods. Optimal controllers aim to control the system to minimize a selected performance index. These control methods offer advantages such as improving energy efficiency, reducing costs, and enhancing system security. The Linear Quadratic Regulator (LQR) based controller is the primary optimal control method. Two well-known traditional control techniques include the Proportional-Integral-Derivative (PID) and Integral Sliding Mode Controller (ISMC). However, they do not usually contain optimal properties. In this study, the optimal control algorithms, defined by obtaining controller parameters through the Riccati equation, are applied to achieve accurate position-tracking control in a DC motor system using Matlab/Simulink. The integral term-based algorithms seem to be robust and eliminate steady-state errors. The optimal PID controller could not provide the minimum performance index rather than the others. LQR and optimal ISMC algorithms could allow the performance index to be a minimum. An illustrative comparison of the performances of all optimal control algorithms has been presented through graphical representation, along with corresponding interpretations.
... o Superposition principle in LTI input-output systems o Tunable parameters from the response in both, time-and frequency-domain • Advantage of linear time invariant (LTI) systems identification • Example of real applications: joint (q and ) with elasticities source: [12] • Example of real applications: hydraulic cylinder of a crane source: [13] • Example of real applications: DC motor velocity and current source: [14]  source: [15] • Dynamic system of 2 nd order, cont. ...
Presentation
Full-text available
Full lecture slides (10 topics, 4h each) of a guest graduate course for Master and PhD students.
... LQR controllers are designed for multi-variable and dynamic systems that are both linear and sometimes non-linear [5]. It has applications in a variety of fields, including aerospace systems [6], high-performance motion control applications for direct current (DC) motors [7], unmanned aerodynamic vehicles (UAV) [8], control of radar antenna systems [2], and autopilots for racing yachts [9]. The state weighting matrix Q and the control weighting matrix 'R' are two factors that determine its performance. ...
Article
Full-text available
A key factor in the design of a car is the comfort and safety of its passengers. The quarter-car suspension system is a feature of the car that ensures load-carrying capacity as well as comfort and safety. It comprises links, springs, and shock absorbers (dampers). Due to its significance, several research has been conducted, to increase its road handling and holding capability while trying to keep its cost moderate. To enhance customer comfort and load carrying, the road holding capacity of an active quarter car suspension was improved/controlled in this study, using the Global Best Inertia Weight Modified Particle Swarm Optimization Algorithm. The observation of the closed loop and open loop systems after designing and simulating on MATLAB reveals a significant improvement in the closed loop system's road holding ability compared to the open loop, in that, when the system was subjected to pothole, the deflection of sprung mass reached steady state in 37.37 seconds as opposed to 7000 seconds for the open loop.
... And in the contrary case, the usual tendency is to limit oneself only to the nonlinearities introduced by the friction torque of the load. To this end, several models have been proposed and used to accurately approximate the nonlinear behavior of the rotating mechanical load of the DC Motor in order to improve the performance of its control system [1], [10], [12], [15], [23]- [25]. However, Ibbini and Zakaria in [26] demonstrated that DC Motors are generally considered as linear system by neglecting the effect of the induced magnetic reaction or by assuming that the compensating windings completely eliminate such an effect. ...
Article
Full-text available
DC motor’s models that have often been used to test and validate control algorithms with, even the most robust ones, have usually been simplistic: limiting the generalization and applicability of the results obtained. This article presents a modeling approach for a DC motor system consisting of a Buck converter and driving a load at the end of the shaft, in which the nonlinearities of each of its sub-parts are taken into account in the model of the overall system. The model was simulated and its step responses were compared to those of the linear model and conventional nonlinear models with a significant difference (Pvalue<0.001). This model was then used to test the Fast Nonlinear Quadratic Dynamic Matrix Control (FNLQDMC). The letter has demonstrated good performance in simulation in terms of setpoint tracking, constraint handling, and computational load savings during online optimization problem solving
... As a result, all of the eigenvalues of (A-BK) can be used to assess the stability and transient response characteristics of the closed-loop system. The decision to use a feedback gain design is an attempt to K such that the eigenvalues of (A-BK) have negative real values [16]. ...
Article
Full-text available
Tanks have an important role in protecting a region in the event of a battle. The tank's cannon is a very reliable weapon in warfare. However, the cannon's precision while aiming and firing targets is a concern. A control system must be created to increase the cannon's stability and precision. The state feedback control technique with a full-state observer is the control system that can manage cannon disturbances. The control system is built around three DC motors, each of which operates the cannon’s x, y, and z axes. Then performed tests for each axis at an angle of 90 degrees, The state feedback control with a full state observer can produce outstanding performance, with the time required for the cannon to reach the target angle was 0.51 seconds, and the cannon system had 0% overshoot.
Article
Full-text available
This study introduces a comprehensive framework designed to enhance production efficiency by integrating maintenance strategies, energy costs, and production specifications. This integration is achieved through a novel empirical method for estimating state–action costs, suitable for both machines with measurable and non-measurable states-of-health. We address the challenge of under-determination in state–action cost optimization by employing a k-means clustering approach, ensuring robustness and applicability. Utilizing an adapted SARSA algorithm, our framework optimally controls shop-floor machinery to minimize the global cost function. The efficacy of the state–action cost estimation method is validated using NASA’s C-MAPSS dataset. Additionally, the optimization strategy is further corroborated through its successful implementation in an autonomous mining cart model on the shop floor. Our results highlight the framework’s ability to optimize machine lifetime and production processes effectively, providing tailored solutions that adapt to varying operational conditions without depending on predefined machine degradation models and costs.
Article
Full-text available
Direct Current (DC) motor is considered a very critical component of various industrial drive equipment. This is due to their unique advantages including reasonable cost, speed-torque characteristics, ease of control, etc. While most DC motor drive applications often employ PID controllers to regulate the speed of the machine, selecting the optimal design parameters for the controllers used in these applications often posed a serious challenge. Moreover, as the complexity of the industrial process increases, the need for precise speed and position tracking becomes necessary. This current study proposes a novel Leader-based Harris Hawks Optimization (LHHO) algorithm for the design of Proportional-Integral-Derivative (PID) and Fractional Order Proportional-Integral-Derivative (FOPID) controllers to achieve optimal speed regulation of DC motors. The LHHO algorithm is an innovative meta-heuristic algorithm that draw inspiration from the cooperative hunting behavior and leadership prowess of the Harris Hawks called the “seven pounds”. While several error functions were tested in this study, the integral of time multiplied absolute error (ITAE) has been adopted as the error function for obtaining the parameters of PID and FOPID controllers using the LHHO algorithm. Through quantitative evaluations and comparisons with existing techniques, the proposed controllers (LHHO-PID and LHHO-FOPID) have revealed significant improvements in key performance metrics, including rise time, settling time, and maximum overshoot during transient periods when ITAE is used as the error function. Furthermore, the stability response and robustness analyses were carried out by varying the parameters of the DC motor under eight scenarios, which confirmed competitive performance in system response and transient behavior.
Article
Full-text available
Friction modeling and identification is a prerequisite for the accurate control of electromechanical systems. This paper considers the identification and control of friction in a high load torque DC motor to the end of achieving accurate tracking. Model-based friction compensation in the feedforward part of the controller is considered. For this purpose, friction model structures ranging from the simple Coulomb model through the recently developed Generalized Maxwell Slip (GMS) model are employed. The performance of those models is compared and contrasted in regard both to identification and to compensation. It turns out that the performance depends on the prevailing range of speeds and displacements, but that in all cases, the GMS model scores the best. Copyright © 2005 IFAC
Article
Full-text available
In this paper, a comparison between two models for nonlinear systems is made. Both models have a state space nature, but there are some differences in the identification approach and the model structure. The first model that we will discuss is a discrete time model that uses input-output data for the identification. The second model uses explicit measurements of the states of the system and some physical insight to model the relationship between the states. The similarities and differences between the two models are discussed, and their performance is compared utilizing data from an experimental setup.
Article
Full-text available
This paper presents an analytical method of modeling permanent magnet (PM) motors. The model is dependent only on geometrical and materials data which makes it suitable for insertion into design programs, avoiding long finite element analysis (FEA) calculations. The modeling procedure is based on the calculation of the air gap field density waveform at every time instant. The waveform is the solution of the Laplacian/quasi-Poissonian field equations in polar coordinates in the air gap and takes into account slotting. The model allows the rated performance calculation but also such effects as cogging torque, ripple torque, back-EMF form prediction, some of which are neglected in commonly used analytical models.
Article
Integrierte mechanisch-elektronische Systeme entstehen durch eine geeignete Kombination von Mechanik, Elektronik und Informationsverarbeitung. Dabei beeinflussen sich diese Bereiche wechselseitig. Man beobachtet zunächst eine Verlagerung von Funktionen der Mechanik zur Elektronik, dann die Hinzunahme von erweiterten und neuen Funktionen. Schließlich entwickeln sich Systeme mit gewissen intelligenten bzw. autonomen Eigenschaften. Für dieses Gebiet der integrierten mechanisch-elektronischen Systeme wird seit einigen Jahren der Begriff „Mechatronik“ verwendet. Im folgenden wird zunächst die Entwicklung vom mechanischen zum mechatronischen System beschrieben. Es ergeben sich mehrere Aufgabenstellungen, verschiedene Integrationsformen von Prozeß und Elektronik und verschiedene Arten der Informationsverarbeitung. Hierzu sind auch die Entwicklungen bei der Sensorik, Aktorik und der Mikroelektronik von grundsätzlicher Bedeutung. Die einzelnen Entwurfsschritte mechatronischer Systeme werden zusammengefaßt und einige-Entwurfswerk-zeuge zitiert. Schließlich wird noch ein Ausblick auf mi-kromechatronische Systeme gegeben. at - Automatisierungstechnik 43 (1995) 12.
Conference Paper
A novel optimal PID control design is proposed in this paper. The methodology of linear quadratic regulator is utilized to search the optimal parameters of the PID controller. The augmented state vector of performance measure involves output signals only. The weighting functions are determined through poles assignment. The existence criteria of the optimal PID controller are derived. The new PID tuning algorithm is applied to the speed control of BLDC motors. Computer simulations and experimental results show that the performance of the optimal PID controller is better than that of the traditional PID controller.
Conference Paper
That paper presents a methodic approach based on switched quadratic regulators for designing an efficient DC-motor speed controller. This alternative control strategy is compared with two cascade control design methods in terms of performances, robustness and complexity. Experimental results are given for each controller
Article
The authors propose a robust speed control system for DC servomotors based on the parametrization of two-degree-of-freedom controllers. The servosystems can dramatically improve the characteristics of the closed loop systems, i.e. the disturbance torque suppression performance and the robustness to system parameter variations, without changing the command input response. The excellent control performances obtained during laboratory experiments by using a microprocessor-based controller are shown
Article
This paper tackles the problem of the speed control of a DC motor in a very general sense. Use is made of the power of feedforward artificial neural networks to capture and emulate detailed nonlinear mappings, in order to implement a full nonlinear control law. The random training for the neural networks is accomplished online, which enables better absorption of system uncertainties into the neural controller. An adaptive learning algorithm, which attempts to keep the learning rate as large as possible while maintaining the stability of the learning process is proposed. This simplifies the learning algorithm in terms of computation time, which is of special importance in real-time implementation. The effectiveness of the control topologies with the proposed adaptive learning algorithm is demonstrated. It is found that the proposed adaptive leaning mechanism accelerates training speed. Promising results have also been observed when the neural controller is trained in an environment contaminated with noise
Article
A nonlinear model structure for a permanent magnet DC-motor, appropriate for simulation and controller design, is developed. The essential nonlinearities are due to coulomb friction and to voltage and velocity dependent brush resistance. The physical parameters of the nonlinear model are estimated directly, using a sensitivity approach for input design and evaluation of relative accuracy of the estimates. Experimental results demonstrate that linear models, as well as nonlinear models with the parameters determined from traditional static measurements, fit dynamic measurements poorly. A nonlinear model with the parameters estimated from dynamic measurements, however, fits the measurements very well, and the sensitivity measures combined with cross-validation results indicate that this model is robust and accurate.