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Problem 7.51 PP
Consider the control of
Y(s) 10
C(*) = i'£/(«) 4 (« + l) ‘
(a) Let y = x1 and x1 = x2, and write state equations for the system.
(b) Find /C1 and K2 so that u = -K ^x ‘\ -K2x2 yields closed-loop poles with a natural frequency 
ojn = 3 and a damping ratio ^ = 0.5.
(c) Design a state estimator that yields estimator emor poles with o;n1 = 15 and ^1 = 0.5.
(d) What is the transfer function of the controller obtained by combining parts (a) through (c)?
(e) Sketch the root locus of the resulting closed-loop system as plant gain (nominally 10) is 
varied.
Step-by-step solution
The state equation are
:a=[:
;.=[1 o]x
step 1 of 4
(B) jr = [0.9 0.2]
step 2 of 4
(Q £ = [14 21 i f
Step 3 of 4
Step 4 of 4
(D) The transfer function for the controller is
D c {S )= - { S j - F + a K + L H y ' 'L 
-(54.85+202.5)