Hardware Reference
In-Depth Information
where x[i]=x(iT f ), z = e sT f , and according to sampled-data control theory,
the multirate input and output vectors u and y are defined as,
=
δ [u 1 [i], ···,u N [i]] T ,
= u(kT y ),u((k +1)T y ), ···,u((k + N −1)T y )] T ,
u[i]
(3.115)
= y 1 [i], ···,y N [i]] T ,
= y(kT y ),y((k +1)T y ), ···,y((k + N −1)T y )] T ,
y[i]
(3.116)
and matrices A, B, C, D can be calculated by
A s
A N− s B s
A N− s B s
···
B s
C s
D s
0
···
0
¸
A B
C D
C s A s
C s B s
D s
···
0
=
,
(3.117)
.
.
.
.
···
C s A N−1
C s A N− s B s C s A N− s B s
···
D s
s
where P [z s ]=A s ,B s ,C s ,D s is the plant discretized by the zero order hold on
T y (= T u )andz s =e sT y . A s =e A c T N ,B s =
R
T f /N
0
e A c τ B c dτ.
Toverifytheabove,wehave
x[k +1] = A s x[k]+B s u[k],
(3.118)
x[k +2] = A s x[k +1]+B s u[k +1],
= A s x[k]+A s B s u[k]+B s u[k +1],
(3.119)
x[k +3] = A s x[k +2]+B s u[k +2],
= A s x[k]+A s B s u[k]+A s B s u[k +1]+B s u[k +2], (3.120)
.
x[k + N]=A s x[k]+A N− s B s u[k]+···
+A s B s u[k + N −2] + B s u[k + N −1],
(3.121)
y[k]=C s x[k]+D s u[k],
(3.122)
y[k +1] = C s x[k +1]+D s u[k +1],
= C s A s x[k]+C s B s u[k]+D s u[k +1],
(3.123)
y[k +2] = C s x[k +2]+D s u[k +2],
= C s A s x[k]+C s A s B s u[k]+C s B s u[k +1]+D s u[k +2],
(3.124)
.
(3.125)
y[k + N −1] = C s A N−1
s x[k]+C s A N− s B s u[k]+
···+ C s B s u[k + N −2] + D s u[k + N −1].
(3.126)
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