Chapter 4 System Analysis
Xmath Control Design Module 4-2 ni.com
The time-response of discrete systems is found directly as a summation of
the information from preceding time points in the state and input histories.
Using * to indicate discrete convolution, you can express the time domain
output as a function of the discrete impulse response:
(4-2)
System Stability: Poles and Zeros
After you have general expressions for the response of a system over time,
according to Equations 4-1 and 4-2, you can assess the stability of the
system. For the purposes of system analysis within Xmath, you define a
stable system as one where output does not grow without bound for any
bounded input or initial condition. A necessary and sufficient condition for
this type of bounded-input bounded-output (BIBO) stability is:
Continuous systems are BIBO stable if and only if all poles of the system
are in the left half of the complex plane; discrete systems are BIBO stable
if and only if all poles are within the unit circle in the complex plane.
For a coprime transfer function H(q) (one having no root cancellations
between the numerator and the denominator), the poles are the roots of the
denominator of H(q). H(q) is infinite at these values. Values of q for which
the numerator of H(q) is zero are termed the zeros of the system.
The poles of a system in transfer-function form are identical to the
eigenvalues of the A matrix in that system’s equivalent state-space
representation.
For systems in transfer-function form, zeros are easily defined as the
polynomial roots of the numerator. You define the system matrix for a
state-space system as
yk=CAkx0hk*uk
()+
hk=CAk1–Bk0>()
Dk0=()
ht() M∞<<
0
∞
∫
Sλ() λIA–B
C–D
=