Question 3.8.2: Consider again the inverted pendulum of Example 3.8.1. Desig......
Consider again the inverted pendulum of Example 3.8.1. Design an applied force F(t) such that the response is bounded for kl < mg.
The problem is to find F(t) such that θ satisfying
m l^2 \ddot{\theta}(t)+\left(k l^2-m g l\right) \theta(t)=F(t)is bounded. As a starting point, assume that F(t) has the form
F(t)=-a \theta(t)-b \ddot{\theta}(t)
where a and b are to be determined by the design for stability. This form is attractive because it changes the inhomogeneous problem into a homogeneous problem. The equation of motion then becomes
m l^2 \ddot{\theta}(t)+\left(k l^2-m g l\right) \theta(t)=-a \theta(t)-b \ddot{\theta}(t)
This can be written as a homogenous equation:
m l^2 \ddot{\theta}(t)+b \dot{\theta}+\left(k l^2-m g l+a\right) \theta=0
From Section 1.8, it is known that if each of the coefficients is positive, the response is asymptotically stable, which is certainly bounded. Hence, choose b > 0 and a such that
k l^2-m g l+a>0
and the forced response will be bounded.