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Problem 9.26PP
Consider the nonlinear upright pendulum with a motor at its base as an actuator. Design c 
feedback controller to stabilize this system.
Step-by-step solution
step 1 of 2
The giTen sjrstem is a non-linear p endiilum with a motor at its base as an actuator. 
We know that 
9 = sin ̂ + a .
When a lead network is used we get 
We know that
6 = sin ̂ + a .
When a lead network is used we get
We have
x ^ = 9 
x̂ = 9
X, =
step 2 of 2
Obtain die matrix form of the sjrstem equations.
X '
= sin Xj - 4xj - 4xj
-3x3 “ 2xi
Hence, die equation for the lineaiized system is
0 1 0 
-3 0 - 4 
- 2 0 -3
The poles of the system are at det(s’I - F ) * (s +1)*
If die system starts near the iq u i^ position dien die system is asynqitotically stable near the 
oiigia