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Observability
Observability is concerned with the issue of what can be said on the state if we measure the plant output.
Example 24.15
If we look at the system defined by state space model

x ˙ 1 t() x 1 t() x 1 t()
= – 1 0 , yt() = [ 10] (24.192)
x ˙ 2 t() 1 – 1 x 2 t() x 2 t()

we can see that the output y(t) only is determined by x 1 (t), and the other state variable x 2 (t) has no
influence on the output. So the system is not completely observable.
A formal definition is as follows:
Definition 24.3 The state x o ≠ 0 is said to be unobservable if given x(0) = x o , and u(t) = 0 for t ≥ 0,
then y(t) = 0 for t ≥ 0, i.e., we cannot see any effect of x o on the system output.
The system is said to be completely observable if there exists no nonzero initial state that it is
unobservable.
Reconstructibility
There is another concept, closely related to observability, called reconstructibility. Reconstructibility is
concerned with what can be said about x(T), having observed the past values of the output, y, for
0 ≤≤ T . For linear time invariant, continuous-time systems, the distinction between observability and
t
reconstructibility is unnecessary. However, the following example illustrates that in discrete time, the two
concepts are different. Consider

[
x t + 1] = 0, x 0[] = x o (24.193)
y t[] = 0 (24.194)

This system is clearly reconstructible for all T ≥ 1, since we know for certain that x[T] = 0 for T ≥ 1.
However, it is completely unobservable since y[t] = 0, k∀ irrespective of x o .
In view of the subtle difference between observability and reconstructibility, we will use the term
observability in the sequel to cover the stronger of the two concepts.
Observability Test
A test for observability of a system is established in the following theorem.
Theorem 24.4 Consider the linear, continuous, time-invariant, state space model where A ∈ n×n

x ˙ t() = Ax t() + Bu t() (24.195)

y t() = Cx t() + Du t() (24.196)
i) The set of all unobservable states is equal to the null space of the observability matrix ΓΓ ΓΓ [A, C] where
o

C
[
ΓΓ Γ Γ A, C] ∆ CA (24.197)
=
o
M
CA n−1
ii) The system is completely observable if and only if ΓΓ ΓΓ [A, C] has full column rank n.
o

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