Digital Signal Processing Reference
In-Depth Information
(13.47) is also the characteristic equation of a multivariable system, which has more
than one input or more than one output. We now consider an example illustrating
stability.
Stability from state equations
EXAMPLE 13.9
We consider the system of Example 13.8. The state equation is given by
01
-65
B
R
x
[n + 1] =
x
[n].
We have ignored the input and output terms, since stability is independent of these terms.
From (13.47), the characteristic equation is given by
z
-1
det(z
I
-
A
) = det
B
R
6
z - 5
= z
2
- 5z + 6 = (z - 2)(z - 3) = 0.
The roots are at and Both poles are outside the unit circle; hence, the system is
unstable, as is noted following Example 13.4. A MATLAB program that calculates the sys-
tem characteristic equation is given by
z = 2
z = 3.
A=[0,1;-6 5]
charpoly=poly(A)
charroots=roots (charpoly)
■
In this section, a procedure is developed for calculating the transfer function
of a system from its state equations. The procedure can be implemented on a com-
puter and is used extensively in the practice of engineering for calculating transfer
functions of high-order systems. As a final point, it is shown that the characteristic
polynomial of a system is equal to hence, this determinant calculates
the modes of a system. These modes determine the stability of a system and the
characteristics of the transient response of a stable system.
det(z
I
-
A
);
13.6
SIMILARITY TRANSFORMATIONS
In this chapter so far, procedures have been presented for finding a state-variable
model from the system difference equations, the system-transfer function, or a
system-simulation diagram. In this section, a procedure is given for finding different
state models from a given state model. It is seen that a system has an unlimited num-
ber of state models. The state models have the same input-output characteristics
(same transfer function), whereas the internal characteristics are different.
The procedure is identical to that developed in Section 8.6 for a continuous-time
system; thus, only the results are reviewed. The state model for a discrete-time
