BLOCK DIAGRAMS

SYSTEMS
The basic unit of the block
diagram is the system block.

x(s) H(s) y(s)

input output
 SYSTEMS
The output = the input multiplied
by the transfer function.

x(s) H(s) y(s)

input output
y(s) = H(s) x(s)
 SYSTEMS
…or convolved, if you’re working in
the time domain.

x(t) h(t) y(t)

input output
y(t) = h(t) * x(t)

(which we probably won’t do very

often.)
CASCADE CONNECTION
= one system connected after another, so the
output of the first becomes the input to the next.

x(s) H1(s) H2(s) y(s)

= H2(s)H1(s)x(s)
= H1(s)x(s)
 CASCADE CONNECTION
effective transfer function: H(s) = H2(s)H1(s)

x(s) H1(s) H2(s) y(s)

= H2(s)H1(s)x(s)
Effective transfer function = transfer function of
a single system that has the same effect as the
combination of systems.
 TAKE-OFF POINT
= point where a signal splits into different paths.

x(s)

The signal will be the same anywhere along either

of these arrows as it is at the start.
 PARALLEL CONNECTION
= two systems along separate
paths that join at a summing
junction.
H1(s)
+
x(s) y(s)
+
H2(s)
effective transfer function:
H(s) = H1(s) + H2(s)
PARALLEL CONNECTION
alternative
sum symbols:
H1(s)
-
x(s) y(s)
+
H2(s)
PARALLEL CONNECTION
alternative
sum symbols:
H1(s)
x(s) - y(s)
+
H2(s)
PARALLEL CONNECTION
alternative
sum symbols:
H1(s)
x(s) y(s)
H2(s)
PARALLEL CONNECTION
What I’ll be using:
H1(s)
-
x(s) y(s)
+
H2(s)
PARALLEL CONNECTION
representing sums with >2 signals:

+ + - + + -
- + + + y(s) +- + y(s)
These are both valid ways to
represent sums, but I find
this one easier to read.
 LOOP
= connecting the output of
a system to its input.

x(s)
r(s) + - H(s) y(s)
 feedback
FEEDBACK
Feedback = the ability of a system to
monitor its own output.

x(s)
r(s) + - H(s) y(s)

Putting a system in feedback gives it the ability to

regulate itself by responding to its present output.
 feedback
FEEDBACK
The shower is a simple example of where we use
feedback in daily life:

shower
 feedback
FEEDBACK
If you just turn the tap to a position, you have no way of knowing
whether the water is at the right temperature before you get in.

shower
This is the equivalent of using an open-loop system.
 feedback
FEEDBACK
Even if you know the exact position that gets the water
temperature just right…

shower
 feedback
FEEDBACK

-
shower
+
A disturbance, such as your roommate waking up earlier and using
most of the hot water, could cause the temperature to come out
different from what you expect.
 feedback
FEEDBACK
…so, we close the loop by adding a sensor to
detect the output i.e. feeling the water

shower
-
 feedback
FEEDBACK
output > setpoint
+
shower
- input - -

If the water’s too hot i.e. the

temperature is greater than our
desired value (setpoint), you turn
the tap more towards the cold side.
 feedback
FEEDBACK
output < setpoint
+
shower
- input + +

If the temperature is less than the

setpoint, you turn the tap more
towards the hot side.
 feedback
FEEDBACK
Even though it might oscillate a few times between states of too hot
and too cold, you eventually get the temperature that you want.

+
shower
-
 feedback
FEEDBACK
Notice that the corrections were always
output < setpoint input + + opposite to the direction of the error…
output > setpoint input - -
+
shower
-
This is because the system is
in negative feedback.
 feedback
FEEDBACK
negative feedback: output is subtracted from setpoint

+
shower
-
 feedback
FEEDBACK
positive feedback: output is added to setpoint

+
shower
+
 feedback
FEEDBACK
positive feedback: output is subtracted from setpoint

But it tends to
Positive feedback reinforce, rather than
might be useful for regulate deviations
modelling an from the ideal, so we
outbreak of mass won’t see it often in
hysteria… control.
 feedback
FEEDBACK
positive feedback: output is subtracted from setpoint

+
shower
+
It probably wouldn’t help
you get the shower
temperature right…
Control system
plant = thing we’re trying to control

plant
 Control system
process = part of the plant that produces the output
plant

process y(s)
output
 Control system
actuator = drives the process in response to the input
plant
actuator input x(s)
u(s) actuator process y(s)
 Control system
actuator = drives the process in response to the input
plant
actuator input x(s)
u(s) actuator process y(s)
(you) (shower)

In the example, the shower is the process, which produces water

temperature as an output, and you are the actuator, since you are
the one controlling the input to the process (the tap position).
 Control system
plant
+ u(s) x(s)
actuator process y(s)
-
 Control system
“input” =
actuator input plant
+ u(s) x(s)
actuator process y(s)
-
 Control system
process input
plant
+ u(s) x(s)
actuator process y(s)
-
setpoint Control system
r(s) setpoint = desired value for the output

+ u(s)
plant y(s)
-
 Control system
r(s) error = difference between setpoint and output

error
+ e(s) u(s)
plant y(s)
-
 Control system
r(s) sensor = system that measures the output

+ e(s) u(s)
plant y(s)
-

sensor
 Control system
r(s) sensor = system that measures the output

+ e(s) u(s)
plant y(s)
-

no sensor block = perfect sensing

i.e. output of the sensor is exactly the true output
 Control system
r(s) controller = system that adjusts plant input in response to error

+ e(s) u(s)
controller plant y(s)
-

sensor
If you don’t like the definition from the first lecture, you can think
of control engineering as “the science and art of choosing inputs to
get the outputs that we want.”
 Control system
r(s) Once the system reliably reaches the setpoint value, the next
objective is to improve how it gets there.
+ e(s) u(s)
controller plant y(s)
-

sensor
 Control system
r(s) Once the system reliably reaches the setpoint value, the next
objective is to improve how it gets there.
+ e(s) u(s)
controller plant y(s)
- E.g. making it get
there in less time…

sensor
 Control system
r(s) Once the system reliably reaches the setpoint value, the next
objective is to improve how it gets there.
+ e(s) u(s)
controller plant y(s)
- …or with less
oscillation.

sensor
 Control system
r(s) closed loop transfer function = effective transfer
function that maps the setpoint to the output
+ e(s) u(s)
controller plant y(s)
-

sensor
 Control system
r(s) closed loop transfer function = effective transfer
function that maps the setpoint to the output
+ e(s) u(s)
K(s) P(s) y(s)
-

H(s)
 Control system
r(s)
G(s) = K(s)P(s)
+ e(s) u(s)
K(s) P(s) y(s)
-

H(s)
 Control system
r(s) forward path = path from setpoint to output, without loop
G(s) = K(s)P(s)
+ e(s) u(s)
K(s) P(s) y(s)
-

H(s)
 Control system
r(s) open loop transfer function = transfer function of forward path

+ e(s)
G(s) y(s)
-

H(s)
 Control system
r(s)

+ e(s)
G(s) y(s)
-
= Hy
H(s)
 Control system
r(s)

+ e(s) = r - Hy
G(s) y(s)
-
= Hy
H(s)
 Control system
r(s)

+ e(s) = r - Hy
G(s) y(s)
- y = G(r – Hy)

= Hy
H(s)
 Control system
r(s)

+ e(s) = r - Hy
G(s) y(s)
- y = Gr – GHy

= Hy
H(s)
 Control system
r(s)

+ e(s) = r - Hy
G(s) y(s)
- y + GHy = Gr

= Hy
H(s)
 Control system
r(s)

+ e(s) = r - Hy
G(s) y(s)
- y(1 + GH) = Gr

= Hy
H(s)
 Control system
r(s)

+ e(s) = r - Hy
G(s) y(s)
- y(1 + GH) = Gr
G
y= r
= Hy 1 + GH
H(s)
 Control system
r(s)

+ e(s)
G(s) y(s)
-
G
GCL =
H(s)
1 + GH
Control system

+ G(s) effective transfer

- function for basic loop
G
H(s) GCL =
1 + GH
Control system
Some more complicated loop formations:

+ + A(s)
- -
B(s)

nested loops
C(s)
Control system
Some more complicated loop formations:

cascaded loops
+ A(s)
+ C(s)
- -
B(s) D(s)
 Control system
interlocking loops
E(s)

+ A(s)
+ B(s) C(s)
- -
D(s)
Cascaded Loops
CASCADED LOOPS

+ A(s)
+ C(s)
- -
B(s) D(s)
Cascaded Loops
CASCADED LOOPS
A C
= =
1 + AB 1 + CD
+ A(s)
+ C(s)
- -
B(s) D(s)
Cascaded Loops
CASCADED LOOPS

A C
1 + AB 1 + CD
Cascaded Loops
CASCADED LOOPS
A C
= x 1 + CD
1 + AB
A C
1 + AB 1 + CD
Cascaded Loops
CASCADED LOOPS

AC
(1 + AB)(1 + CD)
NESTED LOOPS

+ + A(s)
- -
B(s)

C(s)
NESTED LOOPS
A
=
1 + AB
+ + A(s)
- -
B(s)

C(s)
NESTED LOOPS

+ A
- 1 + AB

C(s)
NESTED LOOPS

+ A
- 1 + AB
A
= 1 + AB
A
1 + 1 + AB C
C(s)
NESTED LOOPS

A
1 + AB + AC
 INTERLOCKING LOOPS
E(s)

+ A(s)
+ B(s) C(s)
- -
D(s)

We need some new techniques to deal with interlocking loops…

 First, Moving a

TAKE-OFF POINT
From this
G(s)

…to this
G(s)
 Moving a
TAKE-OFF POINT
…without changing the outputs of the original system.
x(s)

x(s) G(s) G(s)x(s)

G(s)x(s)

x(s) G(s) G(s)x(s)

 Moving a
TAKE-OFF POINT
…without changing the outputs of the original system.
x(s)

x(s) G(s) G(s)x(s)

Solution: 1
G(s)
x(s)

x(s) G(s) G(s)x(s)

 INTERLOCKING LOOPS
E(s)
We want to move this to here
+ A(s)
+ B(s) C(s)
- -
D(s)
 INTERLOCKING LOOPS
E(s)
Let’s call this signal x(s)
+ A(s)
+ B(s) C(s)
- - x(s)

D(s)
 INTERLOCKING LOOPS
x(s) The input to E must still be x(s)
E(s)

+ A(s)
+ B(s) C(s)
- - x(s)

D(s)
 INTERLOCKING LOOPS
x(s) 1 solution
E(s)
C(s)

+ A(s)
+ B(s) C(s)
- - x(s)

D(s)
 INTERLOCKING LOOPS
1
E(s)
C(s)

+ A(s)
+ B(s) C(s)
- -
Now to move this summing
junction from here… D(s)
 INTERLOCKING LOOPS
1
E(s)
C(s)

+ A(s) B(s) C(s)

-
D(s)
…to here.
 Moving a
SUMMING JUNCTION
From this
G(s) +
+

…to this + G(s)

+
 Moving a
SUMMING JUNCTION
x1(s) G(s) + G(s)x1(s) + x2(s)
+
x2(s)

x1(s)
+ G(s) G(s)x1(s) + G(s)x2(s)
+
x2(s) As before, we need the output to stay the same.
 Moving a
SUMMING JUNCTION
x1(s) G(s) + G(s)x1(s) + x2(s)
+
x2(s)
x1(s)/G(s) + x2(s)

x1(s)
+ G(s) G(s)x1(s) + x2(s)
solution
+
1
x2(s) G(s) x1(s)/G(s)
 INTERLOCKING LOOPS
1
E(s)
C(s)

+ x1(s)
A(s) B(s) C(s)
-
x2(s)
D(s)
This is the new
configuration we want.
 INTERLOCKING LOOPS
1
E(s)
C(s)
A(s)x1(s) + x2(s)
+ x1(s)
A(s)
+ B(s) C(s)
- -
x2(s)
D(s)
The input to B must be the same
as in the original configuration.
 INTERLOCKING LOOPS
1
E(s)
C(s)
A(s)x1(s) + x2(s)
+ x1(s)
A(s) B(s) C(s)
-
1 x2(s)
D(s)
A(s)
solution
 INTERLOCKING LOOPS
1 E
E(s) =
C(s) C
= ABC
+ A(s) B(s) C(s)
-
1 D
D(s) =
A(s) A

Now each of these cascade connections can be combined…

 INTERLOCKING LOOPS
E
C

+ ABC
-
D
A
 INTERLOCKING LOOPS
E
C To see the next step, we
ignore the middle branch.
+ ABC
-
D
A
 INTERLOCKING LOOPS
E
C

+ ABC
-
A signal starting at the
D star would flow through
A
both branches and
combine again at the sum.
 INTERLOCKING LOOPS
E
C

+ ABC
- = parallel connection
D E
A C -
E D
=− + D -
C A A
 INTERLOCKING LOOPS

So now we have a simple loop again:

+ ABC
-
EA + DC
CA
 INTERLOCKING LOOPS

+ ABC
- ABC
=
EA + DC EA
1 + ABC CA + DC
CA
INTERLOCKING LOOPS

ABC
1 + BEA + BDC