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Problem 7.27PP
Consider the system
0 4 0 0 ■ ■ 0 ■
- I - 4 0 0 0
s 7 15 x +
0 0 3 - 3 0 _
(a) Find the eigenvalues of this system. {Hint Note the block-triangular structure of A .)
(b) Find the controllable and uncontrollable modes of this system.
(c) For each of the uncontrollable modes, find a vector v such that 
v T B = 0. v T A = / \ v 7 .
(c) For each of the uncontrollable modes, find a vector v such that 
v T B = 0, v T A = / \ v 7 .
(d) Show that there are an infinite number of feedback gains K that will relocate the modes of the 
system to -5, -3, -2, and -2.
(e) Find the unique matrix K that achieves these pole locations and prevents Initial conditions on 
the uncontrollable part of the system from ever affecting the controllable part.
Step-by-step solution
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