Control Systems
EE 4314
Lecture 11
February 18, 2014
Spring 2014
Woo Ho Lee
[email protected]
Announcement
• Lab #2 report due is extended by Feb. 25th at
3:30PM. Submit a hard copy at class.
Woo Ho Lee Control Systems EE 4314, Spring 2014
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
.
−𝑎
𝑛
𝑥𝑛
1
0.
.
0
−𝑎𝑛−1
0
1.
.
0
−𝑎𝑛−2
⋯ 0
⋯ 0.
⋯
. ,B =
⋯
1
−𝑎
⋯
1
0
0.
.
.
1
𝒚 = 𝑪𝑿 + 𝐷𝑢
Where 𝑪 = 1 0 ⋯ 0 , output matrix
•
Transfer function
𝑌(𝑠)
1
= 𝑛
𝑈(𝑠) 𝑠 + 𝑎1 𝑠 𝑛−1 + ⋯ + 𝑎𝑛−1 𝑠 + 𝑎𝑛
Woo Ho Lee Control Systems EE 4314, Spring 2014
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
Woo Ho Lee Control Systems EE 4314, Spring 2014
Canonical Form
• Canonical form: each state variable is connected by the feedback to
the control input
• Consider a system
𝑏(𝑠)
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
Canonical Form
𝑦 + 6𝑦 + 11𝑦 + 6𝑦 = 6𝑢
Define 𝑥1 = 𝑦, 𝑥2 = 𝑦, 𝑥3 = 𝑦
Woo Ho Lee Control Systems EE 4314, Spring 2014
Canonical Form
𝑦 + 6𝑦 + 11𝑦 + 6𝑦 = 6𝑢
Define 𝑥1 = 𝑦, 𝑥2 = 𝑦, 𝑥3 = 𝑦
Woo Ho Lee Control Systems EE 4314, Spring 2014
Transfer Function from State Space
Equations
• State space equation
𝑿 = 𝑨𝑿 + 𝑩𝑢
• Find transfer function from state space equation
Woo Ho Lee Control Systems EE 4314, Spring 2014
Transfer Function from State Space
Equations
• State space equation
𝑿 = 𝑨𝑿 + 𝑩𝑢
• Find transfer function from state space equation
– Taking the Laplace transform with I.C.=0
𝑠𝑋 𝑠 = 𝐴𝑋(𝑠) + 𝐵𝑈(𝑠)
– Combining with 𝑋 𝑠
𝑠𝐼 − 𝐴 𝑋 𝑠 = 𝐵𝑈 𝑆
– Premultiplying 𝑠𝐼 − 𝐴 −1
𝑋 𝑆 = 𝑠𝐼 − 𝐴 −1 𝐵𝑈(𝑆)
where 𝐼: identity matrix
– Since 𝑦 = 𝐶𝑋
𝑌 𝑆 = 𝐶𝑋 𝑆 = 𝐶 𝑠𝐼 − 𝐴 −1 𝐵𝑈(𝑆)
𝑌(𝑆)
– Transfer function 𝐺 𝑆 = 𝑈(𝑆)
𝐺 𝑆 = 𝐶 𝑠𝐼 − 𝐴
Woo Ho Lee Control Systems EE 4314, Spring 2014
−1
𝐵
Transfer Function from State Space
Equations
• Find transfer function from state space equation
−7 −12
1
𝐴=
,𝐵 =
,𝐶 = 1 2 ,D = 0
1
0
0
From 𝐺 𝑆 = 𝐶 𝑠𝐼 − 𝐴
−1 𝐵
Woo Ho Lee Control Systems EE 4314, Spring 2014
Transfer Function from State Space
Equations
• Find transfer function from state space equation
−7 −12
1
𝐴=
,𝐵 =
,𝐶 = 1 2 ,D = 0
1
0
0
– Transfer function 𝐺 𝑆 = 𝐶 𝑠𝐼 − 𝐴 −1 𝐵
𝑠𝐼 − 𝐴 =
𝑠𝐼 − 𝐴
𝑠
−1
𝑠+7
−1
12
𝑠
𝑠 −12
= 1 𝑠+7
𝑠 𝑠 + 7 + 12
−12
1
1 2 1 𝑠+7 0
𝐺 𝑆 =
=
𝑠 𝑠 + 7 + 12
– MATLAB functions
•
•
[num,den]=ss2tf(A,B,C,D)
[A,B,C,D]=tf2ss(num,den)
Woo Ho Lee Control Systems EE 4314, Spring 2014
(𝑠 + 2)
𝑠 + 3 (𝑠 + 4)
Transfer Function Poles from State
Space Equations
• The eigenvalues of an 𝑛 × 𝑛 matrix 𝑨 are the roots of the
characteristic equation
det 𝑠𝑰 − 𝑨 = 0
Where 𝑰: 𝑛 × 𝑛 identity matrix
−6 −11 −6
• Find poles when 𝐴 = 1
0
0
0
1
0
Woo Ho Lee Control Systems EE 4314, Spring 2014
Transfer Function Poles from State
Space Equations
−6 −11 −6
• Find poles when 𝐴 = 1
0
0
0
1
0
𝑠 0 0
−6 −11 −6
𝑠 + 6 11 6
𝑠𝐼 − 𝐴 = 0 𝑠 0 − 1
0
0 = −1
𝑠 0
0 0 𝑠
0
1
0
0
−1 𝑠
det 𝑠𝑰 − 𝑨 = 𝑠 2 𝑠 + 6 + 6 + 11𝑠 = 0
𝑠+1 𝑠+2 𝑠+3 =0
Poles=-1, -2, -3
• MATLAB command: eig(A)
Woo Ho Lee Control Systems EE 4314, Spring 2014