Course Outline
Week
1
2
3
4
5
6
7
8
Amme 3500 :
System Dynamics and Control
Block Diagrams
9
10
11
12
13
14
Dr Stefan Williams
Date
1 Mar
8 Mar
15 Mar
22 Mar
29 Mar
5 Apr
12 Apr
19 Apr
26 Apr
3 May
10 May
17 May
24 May
31 May
Dr. Stefan B. Williams
Block Diagrams
Assign 1 Due
Assign 2 Due
No Tutorials
Assign 3 Due
Assign 4 Due
Amme 3500 : Introduction
Slide 2
•  Many systems are composed of multiple
subsystems
•  In this lecture we will examine methods for
combining subsystems and simplifying
block diagrams
Y(s)
H(s)
Input: control surfaces
(flap, aileron), wind gust
Assignment Notes
Block Diagrams
•  As we saw in the introductory lecture, a
subsystem can be represented with an
input, an output and a transfer function
U(s)
Content
Introduction
Frequency Domain Modelling
Transient Performance and the s-plane
Block Diagrams
Feedback System Characteristics
Root Locus
Root Locus 2
Bode Plots
BREAK
Bode Plots 2
State Space Modeling
State Space Design Techniques
Advanced Control Topics
Review
Spare
Aircraft output: pitch, yaw, roll
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 3
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 4
1
Examples of subsystems
Examples of Subsystems
•  Automobile control:
Desired
course
of travel
Actual
course
of travel
Steering
Mechanism
Control
+
•  Antenna control
Automobile
Sensing
Measurements
Dr. Dunant Halim
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Amme 3500 : Block Diagrams
Slide 5
Mathematical Modelling
u(t)
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 6
Mathematical Modelling
U(s)
y(t)
h(t)
Y(s)
H(s)
•  In the time domain, the input-output relationship is
usually expressed in terms of a differential equation
•  In the Laplace domain, the input-output relationship is
usually expressed in terms of an algebraic equation in
terms of s
d n y(t )
d n−1 y(t )
d mu (t )
+ an−1
+ L + a0 y(t ) = bm
+ L + b0u (t )
n
n −1
dt
dt
dt m
s nY ( s ) + an −1s n −1Y ( s ) + L + a0Y ( s ) = bm s mU ( s ) + L + b0U ( s )
•  (an-1, …, a0, bm, … b0) are the system s parameters, n ≥
m
•  The system is LTI (Linear Time Invariant) iff the
parameters are time-invariant
•  n is the order of the system
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 7
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 8
2
Cascaded systems
•  In time, a cascaded
system requires a
convolution
A General Control System
u(t )∗ h(t ) ≡ ∫−∞ u(τ ) h(t − τ ) dτ
∞
u(t)
h (t)
y1(t)
h1(t)
y(t)
y(t ) = (u(t )* h(t ))* h1 (t )
•  In the Laplace
domain, this is simply
a product
•  Many control systems can be
characterised by these
components/subsystems Plant
Reference
D(s)
Error
E(s)
+
U(s)
H(s)
Y1(s)
H1(s)
Y(s)
Control
Control
Signal
U(s)
Disturbance
Output
Y(s)
Actuator
Process
Feedback
H(s)
Sensor
Y (s) = U (s) H (s) H1 (s)
Sensor Noise
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 9
Simplifying Block Diagrams
Dr. Dunant Halim
–  To provide an understanding of the signal
flows in the system
–  To enable modelling of complex systems from
simple building blocks
–  To allow us to generate the overall system
transfer function
•  A block
diagram is
made up of
signals,
systems,
summing
junctions and
pickoff points
•  A few rules allow us to simplify complex
block diagrams into familiar forms
Amme 3500 : Block Diagrams
Slide 10
Components of a Block Diagram
•  The objectives of a block diagram are
Dr. Dunant Halim
Amme 3500 : Block Diagrams
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Slide 11
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 12
3
Typical Block Diagram Elements
Typical Block Diagram Elements
•  Cascaded Systems
•  Parallel Systems
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 13
Typical Block Diagram Elements
•  Feedback Form
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 14
Moving Past A Summation
a)  To the left past
a summing
junction
b)  To the right
past a summing
junction
E ( s ) = R ( s ) − H ( s )C ( s )
C (s) = G (s) E (s)
C (s)
= R ( s ) − H ( s )C ( s )
G (s)
G ( s ) R ( s ) = (1 + H ( s )G ( s ) ) C ( s )
C (s)
G (s)
=
R ( s ) 1 + H ( s )G ( s )
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Dr. Dunant Halim
Amme 3500 : Block Diagrams
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Slide 15
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 16
4
Moving Past Pick-Off Points
a)  To the left past
a summing
junction
b)  To the right
past a summing
junction
Example I : Simplifying Block
Diagrams
•  Consider this
block diagram
•  We wish to find
the transfer
function
between the
input R(s) and
C(s)
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 17
Example I: Simplifying Block
Diagrams
a)  Collapse the summing
junctions
b)  Form equivalent
cascaded system in the
forward path and
equivalent parallel
system in the feedback
path
c)  Form equivalent
feedback system and
multiply by cascaded G1
(s)
* N.S. Nise (2004)
Dr. Dunant Halim
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 18
Example II : Simplifying Block
Diagrams
•  Form the equivalent
system in the
forward path
•  Move 1/s to the left
of the pickoff point
•  Combine the parallel
feedback paths
•  Compute C(s)/R(s)
using feedback
formula
Control Systems Engineering Wiley & Sons
Amme 3500 : Block Diagrams
Slide 19
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 20
5
Example III: Shuttle Pitch Control
•  The control
mechanisms
available include
the body flap,
elevons and
engines
•  Measurements
are made by the
vehicle s inertial
unit, gyros and
accelerometers
•  Manipulating block
diagrams is important
but how do they relate
to the real world?
•  Here is an example of
a real system that
incorporates feedback
to control the pitch of
the vehicle (amongst
other things)
Dr. Dunant Halim
Example III: Shuttle Pitch Control
Amme 3500 : Block Diagrams
Slide 21
Example III : Shuttle Pitch Control
•  A simplified model of
a pitch controller is
shown for the space
shuttle
•  Assuming other
inputs are zero, can
we find the transfer
function from
commanded to
actual pitch?
Dr. Dunant Halim
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Slide 22
Amme 3500 : Block Diagrams
Example III : Shuttle Pitch Control
Pitch Reference
+
-
+
+
K1K2
-
Output
G1(s)G2(s)
s2
_s_
K1
1
•  Combine G1 and G2.
•  Push K1 to the right past the summing junction
* N.S. Nise (2004) Control Systems Engineering Wiley & Sons
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 23
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 24
6
Example III: Shuttle Pitch Control
Pitch Reference
+
-
+
+
-
Output
K1K2 G1(s)G2(s)
_s2_
K1K2
_s_
K1
Example III: Shuttle Pitch Control
•  Matlab provides us with a tool called
Simulink that allows us to represent
systems by their block diagram
•  We ll take a bit of time to see how we
construct a Simulink model of this system
1
•  Push K1K2 to the right past the summing junction
K1 K 2G1 ( s )G2 ( s )
•  Hence
T (s) =
⎛
s
s2 ⎞
1 + K1 K 2G1 ( s )G2 ( s ) ⎜1 +
+
⎟
⎝ K1 K1 K 2 ⎠
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 25
Dr. Dunant Halim
Single-Loop Feedback System
Closed Loop Feedback
•  We have suggested that many control
systems take on a familiar feedback form
•  Intuition tells us that feedback is useful –
imagine driving your car with your eyes
closed
•  Let us examine why this is the case from a
mathematical perspective
•  Error Signal e(t ) = d (t ) − f (t ) = d (t ) − Ky(t )
•  The goal of the Controller K(s) is:
Ø To produce a control signal u(t) that
drives the error e(t) to zero
d (t )
Desired +
Value
e(t )
-
K ( s)
error
Controller
f (t )
Amme 3500 : Block Diagrams
Slide 27
Dr. Dunant Halim
u (t )
Control
Signal
G (s)
y (t )
Output
Plant
H
Feedback
Signal
Dr. Dunant Halim
Slide 26
Amme 3500 : Block Diagrams
Transducer
Amme 3500 : Block Diagrams
Slide 28
7
Feedforward & Feedback
Controller Objectives
•  Controller cannot drive error to zero
instantaneously as the plant G(s) has dynamics
•  Clearly a large control signal will move the
plant more quickly
•  The gain of the controller should be large so that
even small values of e(t) will produce large
values of u(t)
•  However, large values of gain will cause
instability
Dr. Dunant Halim
Slide 29
Amme 3500 : Block Diagrams
•  Driving a Formula-1 car
•  Feedforward control: pre-emptive/predictable
action: prediction of car s direction of travel
(assuming the model is accurate enough)
•  Feeback control: corrective action since the
model is not perfect + noise
Dr. Dunant Halim
Why Use Feedback ?
Why Use Feedback ?
•  Perfect feed-forward controller: Gn
d (t )
1/ Gn
•  Output and error:
•  Only zero when:
Dr. Dunant Halim
+ l (t )
+
G
demand
y (t )
l =0
Amme 3500 : Block Diagrams
d
+−
No dynamics
Here !
G
y=
d + Gl
Gn
⎛
G⎞
e = d − y = ⎜1 − ⎟ d − Gl
G
n ⎠
⎝
Gn = G
Slide 30
Amme 3500 : Block Diagrams
e
K
controller
e = d − y,
(Perfect Knowledge)
(No load or disturbance)
Slide 31
load
l
u+
+
G
y
output
plant
A sensor
unity feedback
y = G (u + l ), u = Ke
So:
e = d − y = d − G (u + l ) = d − G ( Ke + l )
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 32
8
Closed Loop Equations
Closed Loop Equations
e = d − GKe − Gl
Collect terms:
Similarly
(1 + KG ) e = d − Gl
y=
GK
G
d−
l
1 + KG
1 + KG
Demand to Output
Transfer Function
Load to Output
Transfer Function
Equivalent Open Loop Block Diagram
Or:
e=
1
G
d−
l
1 + KG
1 + KG
Demand to Error
Transfer Function
Dr. Dunant Halim
d
Load to Error
Transfer Function
Slide 33
Amme 3500 : Block Diagrams
Rejecting Loads and Disturbances
y=
G
l
1 + KG
If K is big:
K
Dr. Dunant Halim
l +
G
1 + KG
+
Dr. Dunant Halim
Tracking Performance
y=
Load to Output
Transfer Function
KG >> 1
GK
d
1 + KG
If K is big:
Demand to Output
Transfer Function
KG >> 1
If K is big:
Perfect Disturbance Rejection
Independent of knowing G
Regardless of l
Amme 3500 : Block Diagrams
Slide 34
Amme 3500 : Block Diagrams
If K is big:
G
1
y≈
l = l ≈0
KG
K
y
Slide 35
GK
y≈
d =d
KG
Dr. Dunant Halim
Perfect Tracking of Demand
Independent of knowing G
Regardless of l
Amme 3500 : Block Diagrams
Slide 36
9
Two Key Problems
d
+−
e
K
u+
+
l
G
Two Key Problems
d
+−
y
e
K
u+
+
l
G
y
+n
Power:
u = K (d − y )
Large K requires large
actuator power u
Noise:
u = K (d − y + n)
u ∝ Kd
Large K amplifies
sensor noise
In practise a Compromise K is required
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 37
Dr. Dunant Halim
Conclusions
Amme 3500 : Block Diagrams
Slide 38
Further Reading
•  We have examined methods for
computing the transfer function by
reducing block diagrams to simple form
•  We have also presented arguments for
using feedback in control systems
•  Next week, we will look at more closely on
feedback control
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 39
•  Nise
–  Sections 5.1-5.3
•  Franklin & Powell
–  Section 3.2
Dr. Dunant Halim
Amme 3500 : Block Diagrams
Slide 40
10