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The Inverted Pendulum on a Linear Cart
Lab 8 Pre Lab
Goals
• Model the dynamics of the cart and the pendulum
• Find a suitable PID controller for stabilizing the pendulum while ignoring the cart stability
• Use an ad hoc approach to stabilizing the pendulum and the cart
Model the cart and the pendulum
x
θ
Newton or Energy Method nonlinear equation of motion:
2( ) 6 cos sin2 2
cos sin 02 2
c r r r c
c r r
l lm m x x m m F
l lI m x m g
2
2
is cart mass: 0.94 kg
m is rod mass: 0.23 kg
b is the cart damping coefficient
l is the rod length: 0.6413
1I is the rod moment about the cart
3
is the gravity 9.18 m/s
c
r
c r
m
m l
g
Observations
sin 02
c c
c
m x bx F
lI mg
Uncoupled: Linearize:
ˆˆ( ) ( )
ˆ( ) ( )
x c
c
x s G s F
s G s F
We need damping coefficient b
Matlab to design the closed loop controllers:
ˆ ( )( )ˆ
( )ˆ( )
ˆ ˆ( ) ( )
x
c
G sG sF
G ss
X s H s F
Find Damping Coefficient b1. Remove the pendulum
2. Use in feedback loop to get b c cm x bx F
𝐶𝑠 𝐺𝑥00
0
0
2
2
1
/
/ /
p x p
x
p x c p
p c
c p c
K G KH
K G m s bs K
K m
s b m s K m
2 2 2p
n c c p c
c
Kb m m K m
m
Critical damped condition, then ζ = 1
b = 2 𝐾𝑝𝑚𝑐
Confirm the Full Nonlinear Model
Experimental (LabVIEW):
Initialcondition
Numerical Integration ofNonlinear equations (Matlab)
Cart Position
Pendulum Position
Match?
Stabilize the Pendulum in the upright position
Disturbance
Apply force toKeep θ ≈ π
𝐶𝑠 𝐺𝑥0
r(s)
ˆ ( )df s
( )ˆˆ( ) ( )
1 ( ) ( )d
G ss f s
C s G s
Closed loop root locus
X X XO
PI controller PID controller
( )( )
( )
I
p
p I
p
p
I
I
p
s zC s K
s
KK
s
K Ks
s K
K K
K Kz
Kz
K
1 2
2
1 2
1 2
( )( )( )
( )
/
( )
I
p D
p p I
D D
p
I
D
s z s zC s K
s
KK K s
s
K K Ks s
s K K
K z z K
K z z K
K K
Stabilize Both the Pendulum and Cart
+
+
+
+
u(s)ˆ( )s
ˆ( )x s
-
-
Since φ is small, ϕ is small, and ϕ 0
( ) 2
02 2
c r r c
c r r
lm m x bx m F
l lI m x m g
(1)
(2)
From (1), take Laplace transform:2 2
2
2
ˆ ˆˆ[( ) ]2
ˆˆ2ˆ
( )
c r r c
c r
c r
lm m s bs x m s F
lF m s
xm m s bs
Small perturbationfrom unstable
solution
φ
Stabilize Both the Pendulum and CartPlug into Laplace of (2):
2 2
2 4 4
2
2 2
3 2
ˆ ˆ[ ] 02 2
ˆ ˆ( )2 2ˆ[ ]
2 ( ) ( )
2ˆ ˆ( ) ( )( )
2 2
c r r
r r c
c r
c r c r
r
c
c r c r r
l lI s m g m s x
l lm s m s F
lI s m g
m m s bs m m s bs
m ls
Rs F sI b m gl m m m gl
s s s bR R R
This is the open loop transfer function
𝑃𝐼𝐷𝑥
𝑃𝐼𝐷θ
+
+
+
+
𝐺ϕ 𝑠
𝐺𝑥(𝑠)
u(s)ˆ( )s
ˆ( )x s
-
-
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