Control Systems
EE 4314
Lecture 10
February 13, 2014
Spring 2014
Woo Ho Lee
whlee@uta.edu
Block Diagram Simplification
• Example: Simplify the block diagram
Woo Ho Lee Control Systems EE 4314, Spring 2014
Block Diagram Simplification
• Example: Simplify the block diagram
Woo Ho Lee Control Systems EE 4314, Spring 2014
Example
Woo Ho Lee Control Systems EE 4314, Spring 2014
Closed-Loop System subject to
Disturbance
• Obtain transfer functions of C(s)/R(s) and
C(s)/D(s)
Woo Ho Lee Control Systems EE 4314, Spring 2014
Transfer Functions
Woo Ho Lee Control Systems EE 4314, Spring 2014
Block Diagram Simplification
• Simplify the block diagram and obtain transfer
functions C(s)/R(s) and C(s)/D(s)
Woo Ho Lee Control Systems EE 4314, Spring 2014
Block Diagram Simplification
• Simplify the block diagram and obtain transfer
functions C(s)/R(s) and C(s)/D(s)
Woo Ho Lee Control Systems EE 4314, Spring 2014
Example
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State Space Representation
• Idea: An nth-order differential equation can be expressed as a first order
vector-matrix differential equation.
Consider nth-order system
𝑦 (𝑛) + 𝑎1 𝑦 (𝑛−1) + ⋯ + 𝑎𝑛−1 𝑦 + 𝑎𝑛 𝑦 = 𝑢
with I.C.=0 and define
𝑥1 = 𝑦
𝑥2 = 𝑦
⋮
𝑥𝑛 = 𝑦 (𝑛−1)
Then
𝑥1 = 𝑥2
𝑥2 = 𝑥3
⋮
𝑥𝑛−1 = 𝑥𝑛
𝑥𝑛 = −𝑎𝑛 𝑥1 − ⋯ −𝑎1 𝑥𝑛 +𝑢
Woo Ho Lee Control Systems EE 4314, Spring 2014
State Space Representation
•
Matrix form
𝑿 = 𝑨𝑿 + 𝑩𝒖
𝑥1
0
1
0
⋯ 0
𝑥2
0.
0.
1.
⋯ 0.
.
⋯
Where 𝑿 = . , 𝑨 = .
.
.
. ,𝐁 =
⋯
0
0
0
1
.
−𝑎𝑛 −𝑎𝑛−1 −𝑎𝑛−2 ⋯ −𝑎1
𝑥𝑛
𝑿: state vector, 𝑨: system matrix, 𝑩: input matrix
• Output can be
𝑥1
𝑥2
.
𝑦 = 1 0 ⋯0 .
.
𝑥𝑛
𝒚 = 𝑪𝑿 + 𝐷𝑢
Where 𝑪 = 1 0 ⋯ 0 , output matrix,
Woo Ho Lee Control Systems EE 4314, Spring 2014
0
0.
.
.
1
State Space Representation
•
Nth-order system
𝑦 (𝑛) + 𝑎1 𝑦 (𝑛−1) + ⋯ + 𝑎𝑛−1 𝑦 + 𝑎𝑛 𝑦 = 𝑢
•
State space representation
𝑿 = 𝑨𝑿 + 𝑩𝒖
𝑥1
0
𝑥2
0.
.
where 𝑿 = . , 𝐴 = .
0
.
−𝑎𝑛
𝑥𝑛
•
1
0.
.
0
−𝑎𝑛−1
0
1.
.
0
−𝑎𝑛−2
⋯ 0
⋯ 0.
⋯
. ,B =
⋯
1
⋯ −𝑎1
Transfer function
𝑌(𝑠)
1
=
𝑈(𝑠) 𝑠 𝑛 + 𝑎1 𝑠 𝑛−1 + ⋯ + 𝑎𝑛−1 𝑠 + 𝑎𝑛
Woo Ho Lee Control Systems EE 4314, Spring 2014
0
0.
.
.
1
Modern Control vs. Classic Control
• Modern control theory (State space model)
–
–
–
–
Multi Input Multi Output (MIMO) system
Linear and Nonlinear system
Time invariant and Time varying system
Internal variables (states)can be monitored
• Classic control theory (Transfer function)
– Only applied to Linear Time Invariant, Single Input Single
Output (LTI SISO) system
Woo Ho Lee Control Systems EE 4314, Spring 2014
State Space Representation
• State space representation is not unique.
• The number of state variables is the same for any
representation. You need n state variables for nth-order
system.
– First order system: 𝑥1
– Second order system: 𝑥1 , 𝑥2
– Third order system: 𝑥1 , 𝑥2 , 𝑥3
• State variables determines the behavior of the system.
– Mechanical systems: position, velocity, angle, angular velocity
– Electrical systems: voltage, current, charge
• Three variables
– Input variables: u
– Output variables: y
– State variables: X
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mass Spring Dashpot System
• Dynamic equations of motion
𝑚𝑦 + 𝑏𝑦 + 𝑘𝑦 = 𝑢
• Find state space equation and draw a block diagram
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mass Spring Dashpot System
• Dynamic equations of motion
𝑚𝑦 + 𝑏𝑦 + 𝑘𝑦 = 𝑢
• Find state space equation and draw a block diagram
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mass Spring Dashpot System
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mass Spring Dashpot System
• Ex] Obtain state space form
𝑚 = 1.5, 𝑏 = 2 𝑘 = 3 𝑢 = 1
0
𝑘
𝐴=
−
𝑚
𝐴=
0
−2
1
0
𝑏 ,𝐵 = 1 ,𝐶 = 1 0 ,𝐷 = 0
−
𝑚
𝑚
1
0
,𝐵 =
,𝐶 = 1 0 ,𝐷 = 0
−1.33
0.67
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mass Spring Dashpot System
•
MATLAB
–
–
–
–
m=1.5, b=2,k=3
num=1
den=[m b k]
[A B C D]=tf2ss(num,den)
𝐴=
−1.33
1
−2
1
,𝐵 =
, 𝐶 = 0 0.67 , 𝐷 = 0
0
0
State space form from MATLAB is different from previous
form.
𝐴=
0
−2
1
0
,𝐵 =
,𝐶 = 1 0 ,𝐷 = 0
−1.33
0.67
State space form is not unique. There are several different
representations.
Woo Ho Lee Control Systems EE 4314, Spring 2014
Mass Spring Dashpot System
• Previously, we defined
𝑥1 = 𝑦, 𝑥2 = 𝑦
• At this time, 𝑥1 = 𝑦, 𝑥2 = 𝑦
• Then, we obtain
𝑥1 = −
𝑏
𝑥
𝑚 1
−
𝑘
𝑥
𝑚 2
+
𝑥2 = 𝑥1
𝑏
−
𝐴=
𝑚
1
𝐴=
−1.33
1
Woo Ho Lee Control Systems EE 4314, Spring 2014
𝑘
𝑚
0
−
−2
0
𝑢
𝑚
DC Motor in State Space Form
• Dynamic equations of motion
𝐽𝑚 𝜃𝑚 + 𝑏𝜃𝑚 = 𝐾𝑡 𝑖𝑎
𝑑𝑖𝑎
𝐿𝑎
+ 𝑅𝑎 𝑖𝑎 = 𝑣𝑎 − 𝐾𝑒 𝜃𝑚
𝑑𝑡
• Find state space equation
– Define 𝑥1 = 𝜃𝑚 , 𝑥2 = 𝜃𝑚 , 𝑥3 = 𝑖𝑎
Woo Ho Lee Control Systems EE 4314, Spring 2014
DC Motor in State Space Form
Woo Ho Lee Control Systems EE 4314, Spring 2014
DC Motor in State Space Form
•
Draw a block diagram
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Canonical Form
• Consider a
𝑏(𝑠)
system
𝑎(𝑠)
𝑏(𝑠) = 𝑏1 𝑠 𝑛−1 + 𝑏2 𝑠 𝑛−2 + ⋯ + 𝑏𝑛
𝑎(𝑠) = 𝑠 𝑛 + 𝑎1 𝑠 𝑛−1 + 𝑎2 𝑠 𝑛−2 + ⋯ + 𝑎𝑛
−𝑎1 −𝑎3 −𝑎3 ⋯ −𝑎𝑛
1.
0.
0. ⋯ 0.
𝐴= .
.
,B =
. ⋯ .
0
0
1 ⋯ 0
0 1 0
0
0
𝐶 = 𝑏1 𝑏2 ⋯ 𝑏𝑛 , D = 0
• MATLAB symbolic canonical form
Woo Ho Lee Control Systems EE 4314, Spring 2014
0
0.
.
.
1
Canonical Form
• Ex] Find a state space representation and draw a
block diagram
𝑦 + 6𝑦 + 11𝑦 + 6𝑦 = 6𝑢
Define 𝑥1 = 𝑦, 𝑥2 = 𝑦, 𝑥3 = 𝑦
Woo Ho Lee Control Systems EE 4314, Spring 2014