A fluid level control system includes a tank, a level sensor, a fluid source and an actuator to control
fluid inflow. Consult any classical control text1

... [Show More] to obtain a block diagram of an analog fluid control
system. Modify the block diagram to show how the fluid level could be digitally controlled.
Water
Level
Reference
Level
DAC
ADC
Computer Tank
Level
Sensor
Actuator &
InflowValve
Block diagram of water level digital control system.
1.2 If the temperature of the fluid of Problem 1.1 is to be regulated together with its level, modify the
analog control system to achieve the additional control (Hint: an additional actuator and sensor are
needed). Obtain a block diagram for the two-input-two-output control system with digital control.
Reference Water Level
Level
DAC
ADC
Computer Tank
Level
Sensor
Actuator &
InflowValve
Reference
Temperature
Temperature
Sensor
Heater Temperature
Block diagram of water level and temperature digital control system.
Note that the DAC and ADC can have more than one input and output channel.
1.3 Position control servos are discussed extensively in classical control texts. Draw a block diagram for a
DC motor position control system after consulting your classical control text. Modify the block
diagram to obtain a digital position control servo.
For the angular position sensor we could use a potentiometer, which is often packaged with an ADC to
give a digital output.
1See for example: J. Van deVegte, Feedback Control Systems, Prentice Hall, Englewood Cliffs, NJ, 1994.
2
Angular
Position
Reference
Position
DAC
ADC
Motor
Computer & Load
Angular
Position
Sensor
Block diagram of DC motor digital position control system.
1.4 Repeat Problem 1.3 for a velocity control servo.
For the angular velocity sensor we could use a tachometer, which is often combined with an ADC to
give a digital output. Alternatively, we could use an optical encoder , which has a digital output.
Angular
Velocity
Reference
Velocity
DAC
ADC
Motor
Computer & Load
Angular
Velocity
Sensor
Block diagram of DC motor digital velocity control system.
1.5 A ballistic missile is required to follow a predetermined flight path by adjusting its angle of attack
(the angle between its axis and its velocity vector v). The angle of attack is controlled by adjusting the
thrust angle (angle between the thrust direction and the axis of the missile). Draw a block diagram
for a digital control system for the angle of attack including a gyroscope to measure the angle and a
motor to adjust the thrust angle .
Fig. P1.1 Missile angle of attack control.
3
Reference
Angle
DAC
ADC
Computer Missile
Angle
Sensor
Angle
Actuator
Thruster
Block diagram of digital missile control system.
1.6 A system is proposed to remotely control a missile from an earth station. Due to cost and technical
constraints, the missile coordinates would be measured every 20 seconds for a missile speed of up to
500 m/s. Is such a control scheme feasible? What would the designers need to do to eliminate
potential problems?
If the missile is only observed every 20 seconds with speeds of up to 500 m/s, the missile position
could change drastically between measurements. This makes the control scheme unrealistic. The
missile coordinates need to be measured at a much higher rate.
1.7 The control of the recording head of a dual actuator hard disk drive (HDD) requires two types of
actuators to achieve the required high areal density. The first is a coarse voice coil motor (VCM) with
large stroke but slow dynamics and the second is a fine piezo-electric transducer (PZT) with a small
stroke and fast dynamics. A sensor measures the head position and the position error is fed to a
separate controller for each actuator. Draw a block diagram for a dual actuator digital control system
for the HDD2.
Control
Computer
Reference
Position
Position
Sensor
+
Fine
Controller
DAC
ADC
DAC
Coarse
Controller
VCM
PZT
Recording
Head
2 J. Ding, F. Marcassa, S.-C. Wu, and M. Tomizuka, “Multirate control for Computational Saving”, IEEE
Trans. Control Systems Tech., Vol. 14, No. 1, January 2006, pp. 165-169.
1
Chapter 2 Solutions
2.1 Derive the discrete-time model of Example 2.1 from the solution of the system differential equation
with initial time kT and final time(k+1)T.
The volumetric fluid balance gives the analog mathematical model
C
h q
dt
dh i
where = R C is the fluid time constant for the tank. The solution of this equation is
t
t i
t t e t q d
C
h t e h t
0
( ) 0 ( ) 1 ( ) / ( )
0
( ) /
Let qi be constant over each sampling period T, i.e. qi(t) = qi(k) = constant, for t in the interval
[kT, (k+1)T). Then
(i) Let t0 = kT, t = (k + 1)T
(ii) Simplify the integral as follows with : (k 1)T
1 ( )
1 ( ) ( )
1 ( )
1 ( )
/
0 /
( 1) [( 1) ] /
( 1) [( 1) ] /
e q kT
C
e d q kT
C
e d q kT
C
e q kT d
C
i
T
T i
i
k T
kT
k T
k T
kT i
k T
k T
T kT
d d
0, ( 1)
,
:
We thus reduce the differential equation to the difference equation
h(k 1) e / h(k) R1 e / qi (k)
T T
2.2 For each of the following equation, determine the order of the equation then test it for
(i) Linearity. (ii) Time-invariance. (iii) Homogeneousness.
(a) y(k+2) = y(k+1) y(k) + u(k)
(b) y(k+3) + 2 y(k) = 0
(c) y(k+4) + y(k-1) = u(k)
(d) y(k+5) = y(k+4) + u(k+1) u(k)
(e) y(k+2) = y(k) u(k)
The results are summarized below
Problem Order Linear Time-invariant Homogeneous
(a) 2 No Yes No
(b) 3 Yes Yes Yes
(c) 5 Yes Yes No
(d) 5 Yes Yes No
(e) 2 No Yes No
2
2.3 Find the transforms of the following sequences using Definition 2.1
(a) {0, 1, 2, 4, 0, 0,...} (b) {0, 0, 0, 1, 1, 1, 0, 0, 0,...}
(c) {0, 20.5 , 1, 20.5 , 0, 0, 0, ... }
From Definition 2.1, {u0, u1 , u2 , ... , uk , ... } transforms to U z ukz
k
k
( )
0
. Hence:
(a) Z 0,1,2,4,0,0,... z 1 2z 2 4z 3 (b) Z 0,0,0,1,1,1,0,0,... z 3 z 4 z 5
(c) Z 0,20.5 ,1,20.5 ,0,0,... 20.5 z 1 z 2 20.5 z 3
2.4 Obtain closed forms of the transforms of Problem 2.3 using the table of z-transforms and the time
delay property.
Each sequence can be written in terms of transforms of standard functions
(a) {0, 1, 2, 4,0,0,...} = {0, 1, 2, 4, 8, 16,...} {0, 0, 0, 0, 8, 16,...}={f(k)}{g(k)}
where
0, 0
f ( ) 2 , 0
1
k
k
k
k
0, 4
g( ) 8 2 , 4
4
k
k
k
k
( 2)
8
2
8
2
0,1,2,4,0,0,... 3
3
1 4
z z
z
z
z z
z
Z z z
(b) {0, 0, 0, 1, 1, 1, 0, 0,...} = {0, 0, 0, 1, 1, 1, 1, 1,...} {0, 0, 0, 0, 0, 0, 1, 1, 1, 1,...}
= {f(k)} {g(k)}
where
0, 3
1, 3
f ( )
k
k
k
0, 6
1, 6
g( )
k
k
k
( 1)
1
1 1
0,0,0,1,1,1,0,0,... 5
3
3 6
z z
z
z
z z
z
Z z z
(c) {0,2-0.5,1,2-0.5,0,0,...} = {0,2-0.5,1,2-0.5,0,-2-0.5,-1,-2-0.5,0,...}+ {0,0,0,0,2-0.5,1,2-0.5,0,-2-0.5,-1,-2-0.5,0,...}
= {f(k)} + {g(k)}
where
0, 0
sin( 4) , 0
f ( )
k
k k
k
0, 4
sin( 4) , 4
g( )
k
k k
k
2 1
2 1
2 cos( 4) 1
sin( 4)
2 cos( 4) 1
0,2 ,1,2 ,0,0,0,... sin( 4) 3 2 0.5
0.5 4
2
4
2
0.5 0.5
z z z
z
z z
z z
z z
z
Z
2.5 Prove the linearity and time delay properties of the z-transform from basic principles.
To prove linearity, we must prove homogeneity and additivity using Definition 2.1,
(i) Homogeneity: Z f (k) Z f (k)
3
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