Discrete-Time LTI-Systeriis The second-order discrete system in Figure 13.1 I is described by the followiilg matrices of the stat,e-space description:.. This can he confimed by insertion
Trang 1350 14 Discrete-Time LTI-Systeriis
The second-order discrete system in Figure 13.1 I is described by the followiilg matrices of the stat,e-space description:
A = [ 1 0 : B = [ i ] , C = [ O f ] , D = 1
W e calculate the response of the system for ~ [ k ] = 0, and the initial values y[0] and y(Z], using the formula (cornpare (7.64) - (7.66)):
1
I
G ( z ) = C(zE - A)-' = -.A 11 z]
22 - 1
Y ( z ) = G ( z ) z ~ ,
4 The conriection between the given initial values y[O] arid yjl] mid the statc vector
zo = z[/], which caiiseb the same output signal, can be wad froin Figure 14.11
21/11 = y[O], z2[1] = 4y[l] I
IE;lroin that we obt>ain
1
- yt4 = + (Y[OI z 4 ~ ~ 1 ) (14.32)
4
and splitbirig into partial fractions
Finally, the output signal is foiind by inverse z-transformation
As we bid taken the initial state at XiJ = 1 , thc calculated uiitput, 4gnal also starts
a t k = 1, and delivers the expected value g[11 This can he confimed by insertion
iril o the givw diffeience equation
In Clinpter 8 we introduced the iiripiilse rcsponse c2s a second irriportant charac- terist ic for contimious systems, in addition to thc system function Ii,s irnportancr
can be expressed in three significant properties: