A = T-1 block diag. [C, C, ., C] T (2-31)
p blocks
then for some no € Rpr, z(t) can be generated as follows
z(t) = K2n (t)
A*(t) An*(t) , n (0) = no (2-32)
*0
Furthermore, the initial state no is unique for any given z(t).
Proof: By the conditions given in Proposition 2.2 each of the p components of z(t) must satisfy (2-30) and hence the entire vector z(t) can be expressed uniquely in terms of a pr dimensional initial condition vector. By assumption, this vector can lie anywhere in pr dimensional space. From the definition of the A matrix, each element of the vector K2n (t) must also satisfy (2-30). Hence, if we show the initial
condition vector representing K2n (t) can be made to lie anywhere in pr dimensional space by appropriate choice of the initial state no,
the proof will be complete. Such an initial state can be shown to exist by noting that any state n0 can be observed through the output
K2n (t). Consequently, a linearly independent set of initial states must result in a linearly independent set of outputs K2n*(t). Thus, since n0 spans pr dimensional space, the initial condition vector defining K2n (t) will span pr dimensional space.
1
To prove uniqueness, let n0 be another initial state such that 1(t 1 1 1 1
z(t) = K2n , 1 (t) = An (t) , n (0) = no (2-33)