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Problem 7.46PP
The linearized equations of motion of the simple pendulum in Fig. are 
6 + w2d = u.
(a) Write the equations of motion in state-space form.
(b) Design an estimator (observer) that reconstructs the state of the pendulum given 
measurements of &. Assume w = 5 rad/sec, and pick the estimator roots to be at s = -10 ± 10j.
(c) Write the transfer function of the estimator between the measured value of & and the 
estimated value of d.
6Stiffiaiea vaiue dr d.
(d) Design a controller (that is, determine the state feedback gain K) so that the roots of the 
closed-loop characteristic equation are at s = -4 ± 4/.
Figure Pendulum diagram
1
Step-by-step solution
A) We have
y =
y =
0 \]x
0 T]X
Step 1 of 4
: ■ 'm
step 2 of 4
B) d e t ( £ f l - i ? + i« ) = 0
£r^+ /jS '+ ii)’ ( ; i - l ) = 0
Step 3 of 4
C) We use the estimator equation
i = F ^ + a u + L l ^ y - m )
= (F -L H )X + O u + Iiy 
M = [ l 0 ] (S ^ i i ' + £ « ) - ‘ £
- 7 (S '-20/7)
~ S ’ + 20S+200
Step 4 of 4
D) d e t(S l-J i '+ O £ r) = 0
s ’ +£:jSf+oci‘ + i : , = o
K , = l , K ^ = S