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0066_Frame_C24 Page 27 Thursday, January 10, 2002 3:45 PM

discrete-time case. Consider the system and the output

t
0.5 1 0.5 1
x t + 1] = x t[] ⇒ x t[] = x 0[] (24.153)
[
– 0.25 – 0.5 – 0.25 – 0.5
      
A d
We can see that this system is completely controllable since x[t] = 0, t ≥ 2 and x 0[] ∈ 2 . This
∀
∀
implies that every initial state is controllable. However, no nonzero state is reachable.
In view of the distinction between controllability and reachability in discrete time, we will use the
term controllability in the sequel to cover the stronger of the two concepts.
Usually, in the context of linear time invariant systems, controllability and reachability are used
interchangeably.
Controllability Test
We now present a systematic way to determine the complete controllability of a system.
Theorem 24.3 Consider the linear, time-invariant, state space model where A ∈ n×n :

x ˙ t() = Ax t() + Bu t() (24.154)
y t() = Cx t() + Du t() (24.155)

i) The set of all controllable states is the range space of the controllability matrix ΓΓ ΓΓ [A, B] where
c
∆
[
2
[
Γ c A, B] = B AB A B … A n−1 B] (24.156)
ii) The model is completely controllable if and only if ΓΓ ΓΓ [A, B] has full row rank.
c
Example 24.12
Consider the state space model given in (24.152), with state space matrices

A = 01 , B = 1 (24.157)
00 0
The controllability matrix for this system, is given by

Γ c A, B] = [ B AB] = 10 (24.158)
[
00
Clearly, rank ΓΓ ΓΓ [A, B] = 1, thus the system is not completely controllable.
c
The result above applies to continuous-time models, and it holds equally well for reachability of discrete-
time models.
Also we can see that the controllability of a system is a property that does not depend on the choice
of state variables. To see that, consider the similarity transformation defined in subsection “State Similarity
i – 1 i
Transformation.” Then, observing that A = T A T , we have

Γ c A, B] = T Γ c A, B] (24.159)
[
[
1
–
which implies that Γ c A, B] and Γ c A, B] have the same rank.
[
[
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