Homework #9
1.
Consider the single-output system 
, 
, where ( A,c ) is assumed to be observable. Express x(t) as a function of y(t),
u(t), and their derivatives.
Hint: write y(t), y(1)(t), y(2)
(t), … , y(n-1) (t) in terms of x(t)
and y(t), u(1)(t), u(2)
(t), … , u(n-1) (t), where n is the size
of x(t).
Note that in general this is not a practical way of determining x(t),
since this method requires the differentiation of signals, which is very
susceptible to measurement noise.
2. Problem 6.11 of Chen’s book.
Hint: Start by showing that
if P2 is chosen so
that P = [ P1 P2
] is square and nonsingular then, partitioning the matrix Q := P-1
as
,
we have P1Q1
= In1. Once this is done,
show that the n1-dimensional system in the problem is
precisely the controllable system in the Controllable Decomposition Theorem
proved in class (Theorem 6.6 of Chen’s book).
Use this theorem to finish solving the problem.
3. a) Given an n´n matrix A and an n´m matrix B, show that the pair (A, B) is controllable if and only if the pair (A+BF, B) is also controllable. Here F can be any m´n matrix.
b) Given an n´n matrix A and an m´n matrix C, show that the pair (A, C) is observable if and only if the pair (A+K C, C) is also observable. Here K can be any n´m matrix.
Hint: For a) show that the rows of [A+lI B] are linearly dependent if and only if the rows of [A+BF +lI B] are linearly dependent. Once you proved a), use duality to prove b).