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0066_Frame_C24 Page 36 Thursday, January 10, 2002 3:45 PM

The condition R 1 C 1 = R 3 C 3 produces the loss of complete observability, leading to a pole-zero cancel-
lation in the model transfer function, i.e., the pole from the left half of the circuit in Fig. 24.14 is cancelled
by the zero from the right half. There is subtle difference between the transfer functions in (24.210) and
(24.169). The final result is the same, but the order the cancellation is different in each case. The zero-
pole cancellation is connected to the loss of complete observability and the pole-zero cancellation is
connected to the loss of complete controllability. These issues will be discussed in more detail in subsection
“Canonical Decomposition.”
Observability Gramian
The observability test in Theorem 24.4 answers yes or no to the question about completely observability
of a model. However, sometimes we are interested in the degree of observability for a particular model.
So we can quantify the energy of the output signal y(t), when there is no input (u(t) = 0) and the state
is x(0) = x 0 at t = 0

E x 0 = ∫ ∞ y t() t = ∫ ∞ y t() y t() t (24.211)
()
T
2
d
d
0 0
It can be proved that the ouput energy is
()
T
2
E x 0 = ∫ ∞ y t() t = x 0 Qx 0 (24.212)
d
0
where
∞ T
Q = ∫ e A t C C e d t (24.213)
T
At
0
The matrix Q is called observability gramian, and it measures the observability of the state vector
x(0). If this matrix is small, it means that we have a weak contribution of the initial state x 0 in the energy
of the output y(t). Indeed, we can appreciate the effect of each one of the state variables taking, for
T
example, x 0 = [0,…,0,1,0,…,0] .
Note that the existence of the integral defined in (24.213) is guaranteed if and only if the system is
stable, i.e., if and only if the eigenvalues of A have negative real part.
Also, the observability gramian Q defined in (24.213) satisfies the Lyapunov equation

A Q + QA + C C = 0 (24.214)
T
T
For stable discrete-time systems, the controllability gramian is defined by

∞
Q d ∑ ( A d ) C d C d A d k (24.215)
T k
=
T
k=0
which satisfies

T T
A d Q d A d – Q d + C d C d = 0 (24.216)
Example 24.19
We will use the model of Example 24.18, described by the state space models (24.206) and (24.207), to
appreciate the utility of the observability gramian (24.213), especially when the model is close to losing
complete observability, i.e., when R 1 C 1 ≈ R 3 C 3 .

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