Digital Signal Processing Reference
In-Depth Information
(h)
Give the transfer function for the closed-loop system.
(i)
Write the differential equation for the closed-loop system.
(j)
Verify the results of part (h), using MATLAB.
(k)
It can be shown that the closed-loop transfer function is given by
H(s)
H
c
(s)H
p
(s)
1 + H
c
(s)H
p
.
Y(s)
U(s)
= H(s) =
Verify your results in part (h) by showing that this equation is satisfied by the
derived transfer functions in parts (b) and (e).
8.9.
In Figure P8.8, let the compensator transfer function be
H
c
(s) = 2,
a pure gain. Solve
all parts of Problem 8.8 for this system.
8.10.
Given a system described by the state equations
-45
01
0
1
x
#
(t) =
B
R
B
R
x
(t) +
u(t);
y(t) = [1
1]
x
(t) + 2u(t).
(a)
Draw a simulation diagram of this system.
(b)
Find the transfer function directly from the state equations.
(c)
Use MATLAB to verify the results in part (b).
(d)
Draw a different simulation diagram of the system.
(e)
Write the state equations of the simulation diagram of part (d).
(f)
Verify the simulation diagram of part (d) by showing that its transfer function is
that of part (b)
(g)
Verify the results of part (f), using MATLAB.
(h)
Repeat parts (a) through (g) for the state equations
x
#
(t) =-2x(t) + 4u(t);
y(t) = x(t).
(i)
Repeat parts (a) through (g) for the state equations
01 0
00 1
11-1
2
0
0
x
#
(t) =
C
S
x
(t) +
C
S
u(t);
y(t) =
B
100
R
x
(t).
8.11.
Consider the
RL
circuit of Figure P8.1. Parts of this problem are repeated from Prob-
lem 8.1. Use those results if available.
(a)
Write the state equations of the circuit, with both the state variable and the output
equal to the resistance voltage
(b)
Use the results of part (a) to find the circuit transfer function.
(c)
Use the
s
-plane impedance approach to verify the transfer function of part (b).
Recall that for the impedance approach, the impedance of the resistance is
R
and
of the inductance is
sL
.
v
R
(t).
