International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 8, No. 4, December 2017, pp. 1814-1821
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v8i4.ppl814-1821
□ 1814
The Control Structure for DC Motor based on the Flatness
Control
Thang Nguyen Trong
Departement of Industrial Electrical Engineering and Automation, Haiphong Private University, Vietnam
Article Info
ABSTRACT
Article history:
Received Sep 8, 2017
Revised Nov 23, 2017
Accepted Nov 30, 2017
Keyword:
Control structure
DC motor
Electric
Flatness cotroller
PID controller
This article presents the new control structure for a Direct Current Motor
(DC Motor) using the flatness-control principle. Basic on the mathematical
model of DC Motors, the author demonstrates the application ability of the
fatness-control theory to control the DC Motor, and then calculates the
parameters and proposes the structure of the flatness-controller. The
proposed structure is built and ran on Matlab-Simulink software to verify the
system efficiency. The simulation results show that the quality of the control
system is very good, especially in case of the flatness controller combined
with PID controller to eliminate static error when the parameters of the DC
Motor have been not known accurately.
Copyright © 2017 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Thang Nguyen Trong,
Departement of Industrial Electrical Engineering and Automation,
Haiphong Private University,
36 Danlap street, Haiphong, Vietnam.
Email: thangnt@hpu.edu.vn
1. INTRODUCTION
DC Motor is one of the traditional electric machines, is appeared in the late of 19th century. In
compared to the other electric machine such as induction machine [1], [2], brushless DC motors [3], the DC
Motor has the internal advantages such as simple control, large electromagnetic torque, the ability to adjust
the speed with the wide range [4]. So, DC Motors are still commonly used in industrial fields such as steel
rolling, transportation, mining, defense, construction [5-7]. Thus, improving the quality of DC Motor control
system is essential. There are many studies to control DC Motors [8-11], the most popular is still the method
using PID controllers. However, in many working modes of DC Motors, the nonlinear of DC Motor is high,
which reduces the quality of control system.
There are several solutions for controlling the nonlinear object such as the input-output linearization
method [12], the sliding mode control technique [13], the backstepping control technique [14], etc. The
drawback of the above methods is the existence of the chattering phenomenon or the difficult problem in the
choice of appropriate Lyapunov function. Therefore, the author proposes a suitable control system to improve
the control quality of a DC Motor, which is a control system based on flatness principle. With this method, it
is easy to decouple the input and output, directly identify each system variables by choosing the appropriate
system output variables.
The Flatness Control Theory is a new method control for nonlinear object [15], [16], promising a
high quality of control [17], being attracted by scientists around the world. Many researchers have come up
with different definitions of flatness control systems, but in general, the flatness control is regarded as a
useful tool for the nonlinear control system [18]. The most specific of the flatness system is the existence of
assemblage of z, through the z variables and the differential of the z variables, all state variables, and input
Journal homepage: http://iaesjournal.com/online/index.php/IJPEDS
IJPEDS
ISSN: 2088-8694
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variables can be determined. So we can determine in advance the trajectory of the input from the desired
trajectory of the output.
2. THE STRUCTURE CONTROL BASED ON FLATNESS CONTROL THEORY
2.1. The Basis of Flatness Control Theory
Flatness control theory is applied to control a lot of nonlinear objects which have the status equation
is written as follows [ 15 ], [ 19 ]:
x = f(x,u) ( 1 )
With U = (u l iU 2 ',-M m ) T is the input variables, X — (x l ,X 2 ,..X n ) T is the status variables.
The system (1) is called a flatness system if there are a set of variables Z — (z l9 Z 2 called
the flat outputs, satisfying three conditions as follows:
a. Existing a function h, which is satisfied:
Z = h(x,u,..u (a) ) (2)
b. All input variables and state variables can be determined from the variable z, which means that
there are function A and function B, which are satisfied:
x=A(z,z,...z (P) ) ( 3 )
w = £(z,z,...z (/?+1) ) (4)
c. All variables of z are independently differential together, which means that there is not the
function G, which is satisfied: G(z, Z, • • . Z^ ) = 0
There are many control systems that satisfy the properties of flatness systems such as electric
motors, chemical reactors, cranes, transmission systems, eg.
2.2. Demonstrating the Flatness of DC Motors
The structure of a separately excited DC Motor is shown in Figure 1 , which includes:
a. The armature coil in the rotor
b. The excitation coil in the stator
Figure 1. The structure of the separately excited DC Motor
To demonstrate a DC Motor is flatness system, the first we have to definite the mathematical model
of the DC Motor, in order to prove DC Motor satisfies the conditions of the flatness system. The
mathematical equations of DC Motors are as follows:
a. The equation of voltage:
di
u a =R a i a+ L a ^ t+ EV)
(5)
With U a is the armature voltage; R a , L a is the armature resistance and inductance.
The Control Structure for DC Motor based on the Flatness Control (Thang Nguyen Trong)
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b. The equation of the Electromotive:
E = K e o = (L af .i f ).o (V) (6)
With K e is the voltage coefficient, co is the rotor speed, is the field armature mutual inductance,
lj is the field current.
c. The motion equation of the DC Motor:
= T-Tr
( 7 )
Where J is the inertia, is the viscous friction coefficient, 7^ is the coulomb friction torque, T e
is the electromechanical torque, T L is the torque applied to the shaft,
d. The equation of electromechanical torque:
T e =K T L a =(L af .i f ).i a (N.m) (8)
With K T is the torque coefficient.
From the equations of DC Motors, changing into the state equations with the state variables are the
armature current and the speed (X — (i a , CD) ,U = U a ):
di a
~dt
dco
~dt
( 9 )
Selecting the flat output variable is Z — CO , it is easy to prove that the DC Motor is flatness under
three conditions:
e. The first condition is existing the function h satisfied : Z = h(x 9 U 9 ..JU^'). From the
Equation (9), it is easy to see that the first condition is satisfied.
f. The second condition, there are the function A and the function B satisfied: X = A(z 9 ,
Transforming the Equation (9), we have:
4 = X (ja>+ B m a>+T L +T f )=X{j i+BmZ+ T L +T f )=G\{z,z)
A.J- A.^
So x = (i a , co) = A(oj, co)
Transforming the Equation (9), we have:
u a=-rT {j(b+ B m a>+ T l + 7 })+ - 7 - (/ .(b+B m cQ+T L ) + K E co
k t k e
= - 7 - (jz + B m z + T l + T f )+ - 7 - (/z + B m z + T L ) + K e z - G2(z, z, z)
Aj’ Aj-
So u = u a =B(z,z,z )
So we can conclude that the second condition is satisfied
g. The third condition, all variables of z are independently differential together, this is obvious
because the flat outputs are selected with only one variable.
So we have concluded that DC Motors are flatness systems with flat output is Z — CO
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ISSN: 2088-8694
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2.3. Designing the Flatness Controller
Based on the equations of DC Motors, we construct the stages of the flatness controller. From the
Equation (9), we design the speed controller (calculating the current i a ). From the Equation (9), we design
the current controller (calculating the voltage U a ).
The equation of speed controller:
C =-
K r
r j^L + B m co*+T L+ T f ^
y at j
( 12 )
The equation of current controller:
A i -f*
T Li t ,, . . *
U a = K l a + L a + K e CO
(13)
The block diagram of flatness control system for DC Motor is shown in Figure 2:
Flatness controller
Figure 2. The block diagram of flatness control system for DC Motor
2.4. Building the Model Simulation
The control system diagram is constructed as shown in Figure 3.
Figure 3. The simulation n diagram of the control system with the flatness controller
In order to achieve objective results, in the diagram we use the DC Motor model available in the
Matlab-Simulink library, the parameters of DC Motor are shown in Table 1.
The Control Structure for DC Motor based on the Flatness Control (Thang Nguyen Trong)
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ISSN: 2088-8694
Table 1. The parameters of DC Motor
Ra(Q)
La(H)
Rf(Q)
Lf(H)
Laf(H)
J(kg.m A 2)
Bm(N.m.s)
Tf(N.m)
0.5
0.015
220
140
1.7
1.2
0.5
20
Based on the model and parameters of DC Motor, the control block is built as follows:
a. The block of speed control is built based on the Equation (12), shown in Figure 4
b. The speed control block is built based on the Equation (13), shown in Figure 5.
Bm Add
Figure 4. The Speed Control Block
Figure 5. The Current Control Block
2.5. The Simulated Results
Running the system with the initial set speed CO = 120(ra^// s ), then changing the value of set
speed at time t = 1.5s and t = 3s ( <39 = 1 5C( 772*7/ s) and <39 = 1 8C(r<2