glue lecture 4 slides
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Smriti Chopra ECE Ph.D. Candidate Georgia Tech
Control of Mobile Robots: Glue Lectures
Instructor:
2
Glue Lecture 4: Controllability and Observability Pay attention, this lecture will help you all with Quiz 4!
3
System Stability x =
2
664
✓1
✓1
✓2
✓2
3
775 A =
2
664
0 1 0 00 0 0 00 0 0 10 0 0 0
3
775
x = Ax
In Matlab:
All eigen values of are zero! A Unstable !!!
Introduce Control… x =
2
664
✓1
✓1
✓2
✓2
3
775 A =
2
664
0 1 0 00 0 0 00 0 0 10 0 0 0
3
775
x = Ax+Bu
B =
2
664
0100
3
775
� =⇥B AB A2B A3B
⇤=
2
664
0 1 0 01 0 0 00 0 0 00 0 0 0
3
775 rank(�) = 26= 4
Controllable ??
Say we can control ✓1
4
Introduce Control… x =
2
664
✓1
✓1
✓2
✓2
3
775 A =
2
664
0 1 0 00 0 0 00 0 0 10 0 0 0
3
775
x = Ax+Bu
B =
2
664
0100
3
775
� =⇥B AB A2B A3B
⇤=
2
664
0 1 0 01 0 0 00 0 0 00 0 0 0
3
775 rank(�) = 26= 4
Uncontrollable !!!
Say we can control ✓1
5
In Matlab… x =
2
664
✓1
✓1
✓2
✓2
3
775 A =
2
664
0 1 0 00 0 0 00 0 0 10 0 0 0
3
775
B =
2
664
0100
3
775
Uncontrollable !!! 6
7
Need “more” control… x =
2
664
✓1
✓1
✓2
✓2
3
775 A =
2
664
0 1 0 00 0 0 00 0 0 10 0 0 0
3
775
B =
2
664
0100
3
775
x = Ax+Bu
8
Need “more” control… x =
2
664
✓1
✓1
✓2
✓2
3
775 A =
2
664
0 1 0 00 0 0 00 0 0 10 0 0 0
3
775
x = Ax+Bu
� =⇥B AB A2B A3B
⇤
B =
2
664
0 01 00 00 1
3
775 Say we can control and ✓2✓1
Controllable ??
9
In Matlab again J… x =
2
664
✓1
✓1
✓2
✓2
3
775 A =
2
664
0 1 0 00 0 0 00 0 0 10 0 0 0
3
775
B =
2
664
0 01 00 00 1
3
775
Controllable !!!
10
State feedback… x =
2
664
✓1
✓1
✓2
✓2
3
775 A =
2
664
0 1 0 00 0 0 00 0 0 10 0 0 0
3
775
x = Ax+Bu
B =
2
664
0 01 00 00 1
3
775
u = �Kx
Controllable
11
State feedback… on a simpler system
Unstable !!!
x =
✓1
✓1
�A =
0 10 0
�
x = Ax
12
State feedback… on a simpler system
x = Ax+Bu Controllable ???
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�Say we can control ✓1
13
State feedback… on a simpler system
x = Ax+Bu
u = �Kx
Controllable !!!
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�
14
State feedback… on a simpler system
x = Ax+Bu
u = �Kx
Controllable x =
✓1
✓1
�A =
0 10 0
�
B =
01
�
x =� 0 1
0 0
��
01
� ⇥k1 k2
⇤ �x
x = (A�BK)x
x =
0 1
�k1 �k2
�x
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State feedback… on a simpler system
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�x =
0 1
�k1 �k2
�x
Characteristic polynomial: det(A0 � �I)
A0
�2 + k2�+ k1
Pick your favorite 2 eigen values (LHP) : �1 = �1
�2 = �2(�+ 1)(�+ 2)
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State feedback… on a simpler system
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�x =
0 1
�k1 �k2
�x
Characteristic polynomial: det(A0 � �I)
A0
�2 + k2�+ k1
Pick your favorite 2 eigen values (LHP) : �1 = �1
�2 = �2�2 + 3�+ 2 k2 = 3
k1 = 2
17
State feedback… on a simpler system
x = Ax+Bu
u = �Kx
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�
K =⇥k1 k2
⇤=
⇥2 3
⇤
Stable !!!
18
But we don’t know our state…
x = Ax+Bu
u = �Kx
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�
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Need to estimate our state…
x = Ax+Bu
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�
u = �Kx
C =⇥1 0
⇤
Say we can “see” ✓1
y = Cx Observable ???
20
Need to estimate our state…
x = Ax+Bu
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�C =
⇥1 0
⇤
y = Cx Observable ???
⌦ =
CCA
�=
1 00 1
�rank(⌦) = 2
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Need to estimate our state…
x = Ax+Bu
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�C =
⇥1 0
⇤
y = Cx Observable !!!
e = x� ˙x = (A� LC)e
˙x = Ax+Bu+ L(y � Cx)
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All together… execution x = Ax+Bu
x =
✓1
✓1
�A =
0 10 0
�
B =
01
�C =
⇥1 0
⇤y = Cx
Controllable and Observable K L
a) Wake up b) Start Loop ( increments) c) Read output d) Compute control e) Send control f) Update using dynamics g) Repeat
y = Cx
u = �Kx
dt
x
u
t = t0, x = x0, x = x0
˙x = Ax+Bu+ L(y � Cx)
xk+1 = xk + dt
˙x
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Check the forums, and good luck with Quiz 4!
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