Causal Stable LW-Systenis 3813 system function W z for a causal, stable, discrete IXLsystem must lie within t,he rwi circle of the z-pli3ne.. Their sixigularities are siinplc or inulti
Trang 116.2 Causal Stable LW-Systenis 3813
system function W ( z ) for a causal, stable, discrete IXLsystem must lie within t,he rwi( circle of the z-pli3ne
f
The genctal statemeiits C R U IN> inndc more precise for svsteizis for which the transfer function i s rational Their sixigularities are siinplc or inultiple poles i lial cfelint tlie charact eristic fr ecluencies of the system We caii describe t l m e systems with pole- zcro diagrams
1x1 functioii H (s) for 8 caiisal and stable roritinuous left half of t lit s-plane _.1
'PO illustrate this property w e m i s t recall the interilitl term ylzlT ( 6 ) of (be output, sigiial from Chapter 7 3 3 It describes the part of the output sigiial tltat is caused
by the initial slate of the system The Laplace transform of the int,ernal tcrrri can
be represented (set (7.92)) by pwti tioris with poles s, In the time-tlornein this coiresponds t i t the sum of the 1's characteris< ic freipencics, whit-li we
have only written uiit here for N simple poles:
(16.1.3)
a-1
= (T, -/- J W ? , tlic real part CT, < 0 is in the lef% h d f of thc plane, ing characteri~~,ic frequency decays
the
The condiliorr for stability, that all poles lie in the left half of the s - p i m c , means that, in the t i i ~ ~ - d ~ ~ I ~ a i ~ ~ , the response to the iiiitial coIiditions decay5 with time, just mhat we expect for a stable system This IS ctcar from Figure 3.3 which shows that only poles in the left half of the s-plane coxrespond to decaying cqmrcxntial
that ttie iiiital state of a system dors riot define the system Feknviour, a s long 22s one is prepaxecl to w a i t long enough
The location of the zcros in the cornplcx plane does not influence t Irt charitc- teristic frequencies mtl thus hi$% no influence on the stability of t h ~ system illalions The decay of ttie inttcrrral tcruri mc