4.12 Consider the second-order plant
1
G(s) =
( s + 1)(5s + 1)
(a) Determine the system type and error constant with respect to tracking polynomial reference
inputs of the system for P, PD, and PID controllers (as configured in Fig.4.28). Let k p = 19,
4
1
. Consider the following types of controllers
s s +1
(a ) C ( s ) = k p , (b) C ( s ) = k p + k I / s, (c) C ( s ) = k D s + k p , (d ) C ( s ) = k D s + k p + k I / s
Given G ( s ) =
2
For each type, is it possible to stabilize the closed-loop system b
16.413 Linear Feedback Systems
Midterm Exam
Name: _
ID: _
Total 26 points. 20 is a full score.
1. (2) Match time domain specifications and regions of pole locations.
Im
Im
Re
Im
Re
Re
(b)
(a)
Im
Re
(d)
(c)
(d ) M p M * , t s t*
p
s
( c ) t s t* , tr t*
s
3.2 Find the Laplace transform of the following time functions:
Solution: (b) f (t ) = 3 + 7t + t 2 + (t )
cfw_ f (t ) = cfw_3 + cfw_7t + cfw_t 2 + cfw_ (t )
3 7 2!
= + 2 + 3 +1
ss
s
3
2
s + 3s + 7 s + 2
=
s3
3.3 Find the Laplace transform of the followi
3.19 Find the transfer functions for the block diagrams in Fig. 3.54:
G1
1 + G1
Solution: Simplify the block diagram as above,
Y ( s)
G1
T (s) =
=
+ G2
R( s ) 1 + G1
3.22 Use block-diagram algebra to determine the transfer function between R(s) and Y(s) i
3.32 In aircraft control systems, an ideal pitch response (qo ) versus a pitch command (qc ) is
described by the transfer function
Q0 ( s )
2 ( s + 1/ )
=2n
2
Qc ( s ) s + 2n s + n
The actual aircraft response is more complicated than this ideal transfer
4.4 The DC-motor speed control in Fig. 4.38 is described by the differential equation
y + 60 y = 600va 1500 w,
where y is the motor speed, va is the armature voltage, and w is the load torque. Assume the
armature voltage is computed using the PI control l