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Problem 7.50PP
Suppose a DC drive motor with motor current u is connected to the wheels of a cart in order to 
control the movement of an inverted pendulum mounted on the cart. The linearized and 
normalized equations of motion corresponding to this system can be put in the form 
e = e + v + u.
v = 6 - V - u, 
where
6 = angle of the pendulum,
V = velocity of the cart.
(a) We wish to control 6 by feedback to u of the fonn 
u = -K -\e -K 2 d -K 3 v .
Find the feedback gains so that the resulting closed-loop poles are located at —1» —1 
u = -K -\e -K 2 d -K 3 v .
Find the feedback gains so that the resulting closed-loop poles are located at —1, —1
(b) Assume that 6 and v are measured. Construct an estimator for d and & of the form 
t = A l - ( - L ( y - C t ) .
where x = [0 d\T and y = d. Treat both v and u as known. Select L so that the estimator poles are
it -2. and -2.
(c) Give the transfer function of the controller, and draw the Bode plot of the closed-loop system, 
indicating the corresponding gain and phase margins.
(d) Using Matlab, plot the response of the system to an initial condition on d. and give a physical 
explanation for the initial motion of the cart.
Step-by-step solution
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