© National Instruments Corporation 4-1 Xmath Control Design Module
4
System Analysis
This chapter discusses time-domain solutions of the equations underlying
transfer functions and state-space system models, and what these solutions
tell us about the stability of the system. Xmath provides a number of
functions for performing this system analysis and computing the
time-domain system response to both general and specific “standard”
inputs.

Time-Domain Solution of System Equations

Given the state-space equations:
you obtain:
letting x0 denote any initial conditions on the system states. The integral
term in the preceding equation defines a convolution integral. Using * to
represent the convolution operator, the time-domain system output for all
time t≥0 is:
(4-1)
The response Y(s) of the system (with zero initial conditions) to a unit
impulse input δ(t) is H(s), the transfer function representation from the
Transfer Function System Models section of Chapter 2, Linear System
Representation. You accordingly term h(t), the inverse Laplace transform
of H(s), the impulse response.
x
·Ax Bu+=
yCxDu+=
xt() eAtx0eAτBu t τ–()dτ
0
t
∫
+=
yt() ceAtx0ht()
*ut()()+=
ht() CeAtBDδt()+=