PWM Speed Control of DC Motor Based On Singular Perturbation Technique
PWM Speed Control of DC Motor Based On Singular Perturbation Technique
PWM Speed Control of DC Motor Based On Singular Perturbation Technique
Abstract – The problem of regulation for a DC motor with In this paper the cascaded control system structure is
autonomous voltage inverter is discussed. The considered control discussed [7], where in the first control loop the armature current
system consists of two feedback loops. In the first one, the armature control for DC motor is provided. In the second one the DC
current control for a DC motor is provided by means of pulse-width motor speed control is maintained. Another example of the
modulated control of such autonomous voltage inverter as the H-
cascaded control system structure can be found in [11] where
bridge. In the second one, DC motor speed control is maintained.
Proportional-integral (PI) controllers are designed for armature sliding mode control design methodology to control the speed of
current and motor speed control based on singular perturbation the DC motor was presented.
technique such that multi-time-scale motions are artificially induced The switching control strategy for the armature current
in the closed-loop system. Multi-time-scale motions analysis allows control loop is provided in this paper based on pulse-width
getting analytical expressions for selection of controller parameters. modulation and the existence of equivalence between sliding
Simulation results are presented as well. modes of variable structure control and PWM control responses
under the high frequency sampling is taken into account [15,16],
Keywords – DC motor speed control, DC-DC converters, pulse- where it is assumed that PWM controller is not saturated and the
width modulation, PI controller, multi-time-scale motions.
sampling frequency is high enough, such that the response of
discontinuously controlled system coincides with average model
[17] where control variable is represented by continuous-time
I. INTRODUCTION duty ratio function.
In this paper, the parameters of proportional-integral (PI)
DC motor is the key element of mobile robots and
controllers of inner and outer control loops are selected based on
autonomous systems [1-3]. There is a set of various robotics
singular perturbation technique [18,19] where two-time-scale
applications where it is necessary to control the speed of the DC
motions are artificially induced in the closed-loop system. The
motor subject to the DC motor load variations over a speed range
analysis of the closed-loop system properties is provided via the
and demand of high-speed control accuracy and good dynamic
method of singular perturbations [20,21]. The advantage of time-
responses. It is well known, the speed of the DC motor can be
scale separation technique for closed-loop system analysis is that
controlled by the using a variable resistor in series with the
analytical expressions for parameters of controller can be easily
motor. However, this approach to a problem of speed control
derived.
leads to a lot of energy waste and one generates a great deal of
The paper is organized as follows. First, the cascaded
heat in the resistor. From point of view of energy efficiency and
control system structure with PWM in control loop is discussed.
flexibility of practical implementation using a microcontroller,
Second, the Filippov's average model of the armature circuit
the most widely used approach at present to control the speed of
equipped by the H-bridge voltage converter is introduced and PI
the DC motor is based on Pulse-Width Modulation (PWM) [4-6].
armature current controller with an additional low-pass filtering
By using PWM the average power delivered to a load can be
is designed via singular perturbation technique. Third, the outer
easily controlled and by that the desired speed of the DC motor
DC motor speed controller is designed based on the discussed
can be effectively provided. In order to implement the PWM
two-time-scale design methodology. Finally, simulation results
technique, the DC motor should be equipped by voltage
of the designed control system are included from which it follow
converter, for example, such as the H-bridge. Particular features
that the desired transients performance specifications are
caused by pulse-width modulated control for DC-DC voltage
provided for the motor speed.
converter (H-bridge) should be taken into account during the
control system design.
Despite that PWM controllers for DC motors are widely II. CONTROL PROBLEM STATEMENT
used in industry, the most part of the recent publications in this
area is related with issues of practical hardwire implementation, The discussed control system consists of two feedback loops
computer simulations and experimental results [7-10]. as shown on Fig.1. In the first one the armature current control
There is a set of research works devoted to control design for DC motor is provided by means of pulse-width modulated
methodologies such as DC motor sliding mode control [11], DC control of the H-bridge voltage converter. In the second one the
motor control based on optimization technique [12], fuzzy logic DC motor speed control is maintained. Both of controllers for
controllers design methodology [13], Ziegler-Nichols method armature current and motor speed are designed based on singular
[14]. However, theoretical issues of control law structure perturbation technique such that multi-time-scale motions are
justification, controller design methodology, tuning of controller artificially induced in the closed-loop system.
parameters contain the great number of still open problems. The DC motor is connected with H-bridge convertor and
PWM as shown on Fig.2, where control signal of the each
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𝑈 = 1, 𝑆 = 𝑆 = 1, 𝑆 = 𝑆 = 0,
𝑖𝑓 𝑡 < 𝑡 ≤ 𝑡 + |𝜒(𝑡 )|𝑇 , 𝜒(𝑡 ) > 0;
𝑈 = 0, 𝑆 = 𝑆 = 1, 𝑆 = 𝑆 = 0,
𝑖𝑓 𝑡 < 𝑡 ≤ 𝑡 + 𝑇 , 𝜒(𝑡 ) = 0;
𝑈 = −1, 𝑆 = 𝑆 = 0, 𝑆 = 𝑆 = 1,
𝑖𝑓 𝑡 < 𝑡 ≤ 𝑡 + |𝜒(𝑡 )|𝑇 , 𝜒(𝑡 ) < 0;
𝑈 = 0, 𝑆 = 𝑆 = 0, 𝑆 = 𝑆 = 1,
𝑖𝑓 𝑡 + |𝜒(𝑡 )|𝑇 < 𝑡 ≤ 𝑡 + 𝑇 , 𝜒(𝑡 ) < 0.
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results, where 𝐸, 𝐼 , 𝐸 are treated as the frozen variables during. where 𝜇 is a small positive parameter of the controller, 𝜇 > 0,
Remark 1: The stability of FMS transients of (7) is provided by 𝜏 > 0.
selection of the gain 𝑘 such that the condition 𝑘 𝐸 > 0 holds The control law (12) can be expressed in terms of the Laplace
transform that is the structure of the conventional PI controller
given that 𝜇 > 0 and 𝑑 > 0.
given by
Assume that the control law parameters 𝑘 , 𝜇 , and 𝑑 have
been selected such that the FMS (7) is stable as well as time-
scale decomposition is maintained in the closed-loop system (6). 𝑖 (𝑠) = [ [𝜔 (𝑠) − 𝜔(𝑠)] − 𝜔(𝑠)].
Take, for example,
The closed-loop system analysis is provided below based on
𝑘 = 𝐿 /𝐸, the consideration of the reduced model (11) with controller (12).
The replacement of 𝜔 ( ) in (12) by the right member of (11)
then the FMS (7) characteristic polynomial is given by yields the closed-loop system
μa2 s 2 + da μa s + 1. (8)
𝜔( ) =
𝑘 𝑇
𝑖 − −
𝑇
,
𝐽 𝐽 𝐽
From (6), the averaged slow-motion subsystem (SMS) given by
( )
𝜇 𝑖 =𝑘 (𝜔 − 𝜔) − 𝑖 + + . (13)
( )
𝐼 = (𝑖 − 𝐼 ). (9)
The systems of equations (13) are the singularly perturbed
results, where the parameter 𝜏 is selected in accordance with the differential equations where fast and slow modes are artificially
desired settling time 𝑡 for the average armature current 𝐼 forced as 𝜇 → 0. The degree of time-scale separation between
such that 𝜏 ≈ 𝑡 /3. these modes depends on the parameter 𝜇 .
The time-scale decomposition is maintained in the system (6) From (13), the fast-motion subsystem (FMS)
by selection of the parameter 𝜇 such that the condition 𝜇 ≈
( )
𝜏 /𝜂 holds, where 𝜂 is the degree of time-scale separation 𝜇 𝑖 =− 𝑘 𝑖 +𝑘 (𝜔 − 𝜔) + + (14)
between fast and slow modes, for instance, 𝜂 ≥ 10. The
desired damping of the FMS transients is provided by selection
results, where ω is treated as the frozen variable.
of the parameter 𝑑 , for example, 𝑑 = 2.
Assume that the control law parameter 𝑘 has been selected
such that 𝑘 = 𝐽/𝑘 ,then the FMS (14) characteristic polynomial
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is given by
𝜇 𝑠 + 1. (15)
From (13) it follows that the averaged slow-motion subsystem
(SMS) is given by
𝜔 = (𝜔 − 𝜔). (16)
So, after damping of fast transients of (14), we get from (13),
the slow-motion subsystem (16). As the result, the behavior of
the motor speed ω is prescribed by the stable reference equation
(16) and by that, the requirement (2) is maintained.
Note, time-scale decomposition between the both control loops
is maintained by selection of controller parameters such that the
conditions
𝜇 ≪𝜏 ≪𝜇 ≪𝜏
hold. Fig. 5 Simulation results: Plot of 𝑖 (A)
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VII. CONCLUSION
The advantage of the presented singular perturbation technique
of controller design for the discussed DC motor equipped by the
H-bridge voltage converter is that the desired transients of the
motor speed are provided. The increase of the time-scale
separation degree allows providing the desired output transients
specifications with more high accuracy.
The other advantage is that analytical expressions for selection
of controller parameters are derived, where controller parameters
depend explicitly on the specifications of the desired behavior of
motor speed.
REFERENCES
[1] Power Electronics Handbook, Ed. by Muhammad H. Rashid, Academic
Press, 2001.
[2] R.W. Erikson, M. Dragan, Fundamentals of Power Electronics, Springer, 2-
nd ed, 2001.
[3] J. Lovine, Robots, androids, and animations, McGraw-Hill, 2001.
[4] J. Holtz, “Pulsewidth modulation: a survey,” IEEE Trans. Industrial
Electronics, 1992, vol. IE-39, no.5 pp.410-420.
[5] S.D. Barman, A. Hussain, T. Ahmed, Speed Control of DC Motor Using
PWM Technique: Pulse Width Modulated DC Motor Control, LAP Lambert
Academic Publishing, 2012.
Fig. 18 Simulation results: Plot of 𝜒
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