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The process of finding the equivalent discrete-time system from a sampled continuous system, including the use of cayley-hamilton methods, z-transforms, and the fundamental matrix for a discrete-time system. Examples and practice problems are provided.
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from a sampled, continuous system.
find the solution to a first-order matrix
difference equation using Cayley- Hamilton methods.
first-order matrix difference equations.
find the fundamental matrix for a
discrete-time system in terms of Z-transforms.
.
Ref: Sections 5.1-5.6, 5.8, 5.9, 5.11. Fall 2004
i(t) =Ax(t) + Bu(t) y(t) =Cx(t)
Controller
yet) is sampled every T seconds ~rhich generates
u(t)
Example
0 1 0
-1 -2 1
U(s) s2 + 2rco ~ (^) n S + (1)2 n
I+T T
Ans. -T^ 1-^ T]
[
-T-l +eT ]
G = e -T T
\
G = A-I (eAT -I)B (ifAisnonsingular)
0 1 I -2 -
-1 -2 I 1 0
\
Find x(k) if
Example
I.. 0
x(k+l)= 1
(^4) .J
[ ]
T l-k -2k and u(k) = 1 and x(O) = 1 -1/2.. Ans. Ak=(~t_! 2 l+k
Solution
0 1
A k =a I 0 + a 1 A and A-k = a 0 + a 1 A-
J
l
l
021 2 22 2
2 2 2 2
= - k ( - 1. )k + ( - 1. )k = ( - 1. i (1 - k) 2 2 2
al
k-l
i=O
or by transforms
00
j=O
How is the Z-transformof x(k+ 1) related to
x(k)?
00
j=O
Ans.Z[x(k+ 1)] =zX(z).
\
Inverse Z-Transfc)rms
Example
Solution
Example
