SOUTHERN TAIWAN UNIVERSITY OF
SCIENCE AND TECHNOLOGY
Department of Mechanical Engineering
Final Project of Modern Control Theory:
DC Motor
Name: Jeffrey Levin
Student ID: MA11Y207
Adviser: Prof. Yung-Peng Wang
February 15, 2015
1. Model Description
A DC motor with the equivalent electric circuit of the armature and the free body
diagram of the rotor is shown in the figure below:
The differential equations for the systems are:
 J mm (t )  bm (t )  K t ia (t )

 dia (t )

 La dt  Raia (t )  va (t )  K e m (t )
Assume the parameters' values are:
 J m  0.01 kg  m 2

b  0.001 N  m  sec
 K t  0.02 N  m/A
 K  0.02 V  sec
 e
 Ra  10 Ω

 La  0.01 H
Input and output for the following system are:
Input (u ) = v a (t )
.
Output ( y ) = q m (t ) ,q m (t ) , i a (t )
Let,
x1 ( t ) = q m ( t )
.
x 2 (t ) = q m (t )
x3 (t ) = i a (t )
So that,
.
x (t ) = x (t )
K
b
x (t ) = - J x (t ) + J
2
1
.
t
2
2
m
Ke
.
x (t ) = - L
3
a
x3 (t )
m
x 2 (t ) -
Ra
1
x 3 (t ) + v a (t )
La
La
The state space equations for the DC motor system are:
.
x
-
ì
ï
ï
ï
=í
ï
ï
ï
î
q m (t )
..
q (t )
i (t )
m
a
ì
ï
ï
ï
t =í
ï
ï
ï
î
y( )
-
.
ü é
ï ê
1
ï ê 0
ê
ï
ý =ê 0 - b
Jm
ï ê
ï ê
K
ï ê 0 - e
La
þ êë
ùì
úï
úï
0
úï
Kt
úí
J m úï
úï
R
úï
- a
La úûî
ü
ì
ï
t
q m ( ) ï é 1 0 0 ùïï
.
úï
ï ê
t
=
0
1
0
ý
ê
úí
(
)
qm ï ê
0 0 1 úï
ûï
ï ë
t
(
)
ia ï
ï
þ
î
ü
t
q m ( ) ïï éê 0 ùú
.
ï ê
ú
t
+
0
ý
(
)
ê
úv a ( t )
qm ï
ê 1
ú
ï ê
L
i a (t ) ï ë a úû
þ
ü
t
q m ( ) ïï é 0 ù
.
ú
ï ê
t
+
0
ý
ê
(
)
q m ï ê úúva (t )
0
û
ï ë
t
(
)
ia ï
þ
Input the parameters' values to the state space equations, such that:
.
x
-
ì
ï
ï
ï
=í
ï
ï
ï
î
q (t )
q (t )
m
..
m
i (t )
a
ì
ï
ï
ï
t =í
ï
ï
ï
î
y( )
-
.
ü
ì
ï
ï
ù
ï é 0
ï
1
0
úï
ï ê
2
ý = ê 0 -0.1
úí
ï ê 0 -2 -1000 úï
ûï
ï ë
ï
ï
þ
î
q (t )
q (t )
m
.
m
i (t )
a
ü
ì
ï
ï
ï é 1 0 0 ùï
úï
ï ê
ý = ê 0 1 0 úí
ï ê 0 0 1 úï
ûï
ï ë
ï
ï
þ
î
q (t )
q (t )
m
.
m
i (t )
q (t )
q (t )
m
.
m
i (t )
a
a
ü
ï
ï é 0 ù
ú
ï ê
ý + ê 0 úv a t
ï ê 100 ú
û
ï ë
ï
þ
()
ü
ï
ï é 0 ù
ú
ï ê
ý + ê 0 úv a t
ï ê 0 ú
û
ï ë
ï
þ
(1)
()
Compute the A,B,C and D matrices from Equation 1 to MATLAB in order to get the
transfer function of the system. The MATLAB's commands are:
>> [b,a]=ss2tf(A,B,C,D)
From MATLAB computation, the transfer functions are:
é
200
ê 3
ê s +1000.1s 2 +104 s
ê
200 s
G s =ê 3
2
ê s +1000.1s +104 s
ê
100 s 2 +10 s
ê 3
2
ë s +1000.1s +104 s
()
ù
ú
ú
ú
ú
ú
ú
ú
û
2. Open-loop System Analysis
 Eigenvalues
The eigenvalues of matrix A, can easily be found in MATLAB eig
command:
>> lambda= eig(A)
thus, the eigenvalues of the system are:
lambda =
0
-0.1040
-999.9960
 Poles
Poles are the roots the system's transfer function denumerator. In MATLAB, the
command to find the roots of the denumerator is roots (d), where d is the denumerator. In
this case, d = [1 1000.1 104 0].
>> roots (d)
ans =
0
-999.9960
-0.1040
 Stability
From the stability theory (Theorem 5.4), "the state space equation is marginally stable
if and only if all eigenvalues of A matrix have zeros or negative real parts and those with
zero real parts are simple roots of the minimal polynomial of A matrix". One of the
eigenvalues of the A matrix is equal to zero and the others are negative real parts, thus the
system is marginally stable.
 Controllability
For LTI system, a system is controllable if and only if the controllability matrix "W"
has full rank (i.e. rank (W) = n, where n is the number of states). In order to compute the
controllability matrix for the state space system, ctrb command was used. rank computes
the rank of a matrix.
>> W = ctrb (A,B)
W=
0
0
0
200
100
200
-200020
-100000
99999600
>> rank (W)
ans =
3
From the results above, controllability matrix "W" has full rank of 3, where the number
of states in our system also 3. Thus, the system is controllable.
 Observability
For LTI system, a system is observable if and only if the observability matrix "OB" has
full rank equals to n, where n is the number of states. The observability matrix of an LTI
model and its rank can be determined in MATLAB using obsv and rank.
>> OB = obsv (A,C)
OB =
1.0e+05 *
0.0000
0
0
0
0.0000
0
0
0
0
0.0000
0.0000
0 -0.0000
0
0.0000
0 -0.0000 -0.0100
0 -0.0000
0.0000
0 -0.0000 -0.0200
0
0.0200 10.0000
>> rank (OB)
ans =
3
Computation results exhibit the rank of the observability matrix "OB" is equal to the
number of states, which are 3. Based on this, the system is observable.
 Simulations
By using Simulink software, we can observe the behavior of the system, in this case
motor position, motor speed and currrent.
Figures below are the block diagram of the system and the state space block parameters:
The block diagram used a constant input and the value was 1. The figures below are the
output of system in relation with time:
Next, the constant input was change to sine wave, with parameters as follow:
The behavior of the system is like the pictures below:
3. Close-loop System Analysis
For close-loop system, the gain need to be considered into the system, thus effect the
state space equation. The gain values are 0.5, 1, and 2. The feedback gain is given by:
(
v a (t ) = K q r ( t ) - q m ( t )
)
Substitute the equation above to the state space equation, thus the new state space
equation becomes:
x = (éë A ùû - éëB ùûéëK ùû) x + éë B ùûéë K ùûq
.
-
r
 Eigenvalues, Poleas and Stability
The new A matrix for the system is:
é AA ù = é A ù - é B ùé K ù
ë û ë û ë ûë û
The new A matrix and eigenvalues for each gain are:
K
0.5
[A] Matrix
é 0
ù
1
0
ê
ú
é AA 1ù = ê 0 -0.1
2
ú
ë
û
ê -50 -2 -1000 ú
ë
û
Eigenvalues
1.0e+02 *
-10.0000 + 0.0000i
-0.0005 + 0.0031i
-0.0005 - 0.0031i
1
é 0
1
0 ù
ê
ú
é AA 2ù = ê 0
-0.1
2
ú
ë
û
ê -100 -2 -1000 ú
ë
û
1.0e+02 *
-10.0000 + 0.0000i
-0.0005 + 0.0044i
-0.0005 - 0.0044i
2
é 0
ù
1
0
ê
ú
é AA 3ù = ê 0
-0.1
2
ú
ë
û
ê -200 -2 -1000 ú
ë
û
1.0e+02 *
-10.0000 + 0.0000i
-0.0005 + 0.0063i
-0.0005 - 0.0063i
100
é
ù
0
1
0
ê
ú
é AA 4ù = ê
0
0
2
ú
ë
û
ê -10000 -2 -1000 ú
ë
û
1.0e+03 *
-1.0000 + 0.0000i
-0.0000 + 0.0045i
-0.0000 - 0.0045i
The eigenvalues of matrix AA is the pole of the system. From all K values (0.5, 1, 2,
100), the eigenvalues have real part <0. Therefore, it is concluded that the close-loop
system is BIBO stable.
From the variation of the eigenvalues from the table above, it is not clear the difference
between them until the K value increased to 100. When K values are increased the real part
of eigenvalues also increased ( from -0.05 to -0.00). Thus, the system is more unstable if
the K values were increased.
 Simulation
The close loop system of DC motor will be simulated in Simulink with different
feedback gain (0.5, 1, 2, 100) to see the effect of K on the behavior of the system.
The block diagrams, which were built in the Simulink is shown below.
K Value
0.5
1
System Response
2
100
* Yellow curve describes the motor position. Purple curve describes the motor velocity. The blue curve
describes the current in circuit.
Figures above exhibit the effect of the K in the A matrix on the response of the system.
The response of the system becomes a dynamic response. If K value was increased, the
system will be very unstable (K = 100). The peak time of the motor position will be faster
when K values were increased.
Now, we changed the source signal with a sine wave with amplitude is 1 and sample
time is 1.
K value
0.5
1
System Response
2
100
* Yellow curve describes the motor position. Purple curve describes the motor velocity. The blue curve
describes the current in circuit.
4. Conclusions
1. Open loop system
By using MATLAB, we can easily check the eigenvalues, poles, stability,
controllability and observability of a system.
By typing lambda = eig (A) and root (d); where d is the denumerator of the system's
transfer function, we can check the eigenvalues and poles of a system. If the real part of
eigenvalues is lower or equal to zero, the system is stable. If the real part of eigenvalues is
above zer, the system is unstable. In this study, one of the eigenvalues is equal to zero.
Based on Theorem 5.4, it is concluded that the open loop system is marginally stable.
The controllability and observability of a system can be checked easily by using the ctrb
(A*B) and obsv (A*C), respectively. These commands will compute the controllability and
observability matrix. In case the rank of the controllability and observability matrix are the
same with the number of state, the system is controllable and observable.
The simulation in the open loop system showed that when the input signal was constant,
the output response was increased in relation with time. When the input signal was sine
wave, the output response value will go up and down between a certain value like a sine
wave.
2. Close loop system
Feedback gain (K) values were considered to analyze the behavior of a close loop
system. As a result, the A matrix from this system is different compared to the open loop
system. The K values were 0.5, 1, 2, 100.
For the eigenvalues, all of the K values will have negative real part, thus the system is
BIBO stable. The bigger the K value, the more unstable the system will be. It is proved by
the increasing values of the eigenvalues' real part every time the K value was increased.
On simulation part, with constant input and sine wave input, when K values were
increased, the peak time and rise time were decreased. The bigger the K values, the system
will become more unstable.