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Unformatted text preview: e value of p. Aept✲ G (s ) G(p)Aept
✲ A special case of the theorem is p = 1.
If the input is constant, u(t) = A, one
solution is that the output is the constant
y (t) = G(0)A = G(0)u.
So G(0) is the D.C. gain of the system. Linearity
Let y1(t) and y2 denote the output of a
system resulting from the inputs u1(t) and
u2(t) respectively.
u1(t)✲ u2(t)✲ G (s ) y1(t)
✲ G (s ) y2(t)
✲ The system is said to be Linear if the input
u3(t) = c1u1(t) + c2u2(t)
results in the output
y3(t) = c1y1(t) + c2y2(t)
for all constants c1 and c2, and all inputs
u1(t) and u2.
c1u1(t) + c2u2(t) ✲ G (s ) c1y1(t) + c2y2(t)
✲ Example  Ampliﬁer or Gain
u(t) ✲ K y (t)
✲ This system is linear since
y3(t) = K (c1u1(t) + c2u2(t))
= c1 (c1u1(t)) + c2 (Ku2(t))
= c1y1(t) + c2y2(t)
Example
Consider the system
y (t) = (u(t))2
i.e. the output is the square of the input.
y3(t) = (c1u1(t) + c2u2(t))2
y3(t) = c2u2(t) + c2u2(t)
11
22
= c1y1(t) + c2y2(t) = c1u2(t) + c2u2(t)
1
2
So this system is nonlinear. Example  The Integrator
u(t) ✲
y (t) = y (t)
✲ u(t)dt dy (t)
= u(t)
dt
1
Transfer function is
s
This system is linear because
y3(t) = (c1u1(t) + c2u2(t)) dt
= c1
u1(t)dt + c2
u2(t)dt
= c1y1(t) + c2y2(t) Example  The Diﬀerentiator
u(t) ✲ d dt y (t)
✲ du(t)
y (t) =
dt
Transfer function is s
This system is linear because
d
y3(t) = (c1u1(t) + c2u2(t))
dt
du1(t)
du2(t)
= c1
+ c2
dt
dt
= c1y1(t) + c2y2(t) Theorem 2
Systems described by
n
m
dk y dk u
ak k =
bk k
dt
dt
k =0 (1) k =0 are linear.
Proof Suppose that the inputs u1 and u2 give rise
to the outputs y1(t) and y2.
This means that
n
m
dk y1 dk u1
ak k =
bk k
dt
dt
k =0
n
k =0 dk y2
ak k =
dt k =0
m
k =0 dk u2
bk k
dt (2) (3) Multiply (2) by c1 and (3) by c2, and add. ⇒
= n
k =0
m
k =0 n dk y1 dk y2
ak k +
ak k
dt
dt dk u1
bk k +
dt k =0
m
k =0 dk u2
bk k
dt dk
Using the fact that k is linear gives
dt
n
dk
⇒
ak k ( c 1 y 1 + c...
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This note was uploaded on 01/22/2014 for the course ENG 282IN taught by Professor Haines during the Spring '12 term at Pima CC.
 Spring '12
 Haines
 Volt

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