closed and open loop control systems representation and applied Laplace transforms- 3.pdf

1. Mathematical model: Identify governing parameters of a plant/process (input/output,
capacity, physical properties, etc)
2. Analysis: understand interaction of the system with the environment and other subsystems
by deriving the steady state and transient time response, stability, frequency response
(Bode plot) and stability (Nyquist)
3. Design of a controller: design specification of type of controller, stability operation using
Routh-Hurwitz method, Roots locus, frequency domain, etc…
4. Implementation
Control systems
System/Plant
R
OUTPUT
variable
-
+
error
Controller
u
sensor
Implementation of a control strategy
1. Mathematical modelling
2. Analysis
3. Controller design
4. implementation

Example 1: Temperature control in buildings
T Heat
loss
Qc
Set point
(Tref =20oC)
Ta
Qo
Control System Continuous-Time Model
• Heater properties : heat rate output, Qo
• Building characteristics
▪ Dimensions, mass, specific heat capacity of materials
▪ Heat loss coefficient (U-value) – thermal resistance
R=1/UA
• Initial conditions:
Ambient temperature, Ta and space temperature, Ti
Block diagram
Input
(manipulated
variable, Q)
Output
(controlled
variable, T)
Control system
(building/heating system)
Deriving the mathematical model of the system (The building thermal dynamics)
• A dynamic system is usually defined by a mathematical model equivalent –
Ordinary Differential Equations ( 1st , 2nd , order or higher).
inside
outside

T
Heat loss
Qc
Set point
(Tref =20oC)
Ta
Qo
Dynamic model - The mathematical relationship of
temperature and heat input.
Deriving the mathematical model of the system (Building) thermal dynamics
1) Show that the building space temperature variation has the
following form:
 =mcp/UA Time constant
*
)
(
)
(
Q
t
dt
t
d
=
+



Q*(t) =Qo / mcp Heat input
(t) = T(t) - Ta Temperature
Temperature / heater output relationship
)
1
(
)
( 
t
o
a e
UA
Q
T
t
T
−
−
+
=
)
1
(
)
( * 


t
e
Q
t
−
−
=

(t)Q*
t
0.5
0.25
1
0.75
 
 
0.632
*
)
(
)
(
Q
t
dt
t
d
=
+



settling time
T
Heat loss
Qc
Set point
(Tref =20oC)
Ta
Qo
)
1
(
)
( * 


t
e
Q
t
−
−
=
2% error
5% error
settling time
Temperature
profile
2) Draw the temperature time response

Voltage change through the capacitor when connected to a DC voltage source
through a resistor.
C
R
E(t) v(t)
RC circuit The voltage at the terminals of the capacitor
is given by:
)
1
(
)
( 
t
e
E
t
v
−
−
=
Examples:
a) Control of water level in a storage tank
b) Control of temperature in an oven
c) Control of a lift (elevator) position in a building
d) Control of a fan motor speed
e) Charging a capacitor (electronics appl.)

Closed loop (negative feedback) control Systems
Building
temperature

Temperature
Set point
r
radiator
Hot water flow
q
Controller
Example: Single variable Control system

Example: Multivariable modern Control systems for boiler-generator
Closed loop (negative feedback) control Systems

Introduction to Laplace transforms

• A dynamic system is usually defined by a mathematical model equivalent
– Ordinary Differential Equations ( 1st , 2nd , order or higher).
LAPLACE TRANSFORM
Mathematical modelling of a dynamic system
• The mathematical model of a dynamic system is often complex to
solve directly
• Use a TRANSFORM method – make the problem easier to solve

• Operator, s, is usually used to refer to Laplace transformation
• The Laplace transform is a mathematical tool for solving systems of linear
differential equations with constant coefficients.
• The Laplace transform allows original functions of time (i.e., a function
describing the relationship of a physical system’s input and output
variables) to be represented (mirrored) in a new domain called the s-
domain (or frequency domain).
• The transformation provides a different but simpler method for
analysing the behaviour of a dynamic system.
Laplace transform
(Pierre Simon De Laplace 1749-1827)
Definition

Laplace transform ( L)
Time
domain
f(t)
s-domain
F(s)
L
f(t)= 0 for t<0
f(t) : is continuous for t>0
Laplace transform
  

−
=
=
0
)
(
)
(
)
( dt
t
f
e
t
f
s
F st
L
Apply Laplace
transform

Laplace transform
)
(
)
( t
f
t
y =
)}
(
{
)
( t
f
s
Y =
)
(
)
( s
F
s
Y =
)
(
)
( t
f
t
y =
1) Derive a mathematical model of
dynamic system/process
(differential equation) in the time
domain
2) Write the differential equation
in s-domain
3) Solve the resulting equation
in s-domain
4) Find the solution in t-domain by
apply inverse Laplace transform
time
domain
L
)}
(
{
)
( s
Y
t
y = L-1
time
domain
s-domain
s-domain

Linear differential
equation
(Time domain: f(t))
Time domain
solution
f(t)
L
L -1
f(t) = L -1
{ F(s) }
Laplace transform
Laplace Transform application methodology
ODE solution
s-domain solution
F(s)
Laplace transform
equation
(s-domain: F(s))
Laplace
transform
Algebraic
solution
Inverse Laplace
transform
F(s)= L { f(t)}

t ≥ to
t < 0
0
)
(
=
= a
t
f
s
a
s
ae
dt
e
a
dt
e
t
f
t
f
L
st
st
st
=






−
=
=
=

−

−

−


0
0
0
)
(
)
(
)]
(
[
1) Step signal (Heaviside)
a
f(t)
t
f(t)
t
a
0
t
0
0
)]
(
[ 0
st
t
st
e
s
a
s
e
a
t
t
f
L −

−
=






−
=
−


−
=
=
0
st
dt
e
)
t
(
f
)
s
(
F
)]
t
(
f
[
L
L
2) Delayed step signal (Heaviside)
Laplace transform
Example of common signals
a
t
f =
)
(
L
L

2
0
st
0
st
0
st
s
a
dt
e
s
a
s
ate
dt
ate
)]
t
(
r
[
L =
−
−





−
=
= 


−

−

−
0
t
,
at
)
t
(
r 
=
f(t)
t
Laplace transform
Examples
3) Ramp signal
L


−
=
=
0
st
dt
e
)
t
(
f
)
s
(
F
)]
t
(
f
[
L
L

Laplace transform
Common Laplace transforms

Laplace transform
Common signals

Common Laplace transforms
Laplace transform

Linearity properties of Laplace transform
)]
t
(
f
[
L
)
s
(
F 1
1 =
)]
t
(
f
[
L
)
s
(
F 2
2 =
ts
tan
Cons
c
,
c 2
1 =
)
(
.
)
(
.
)]
(
[
.
)]
(
[
.
)]
(
.
)
(
.
[
2
2
1
1
2
2
1
1
2
2
1
1
s
F
c
s
F
c
t
f
L
c
t
f
L
c
t
f
c
t
f
c
L
+
=
+
=
+
Laplace transform
L L L

)
0
(
)
0
(
)
(
.
)]
(
[
]
[
)]
(
"
[ '
2
..
2
2
+
+
−
−
=
=
= f
sf
s
F
s
t
f
dt
f
d
t
f
Derivatives
)
0
(
f
)
s
(
F
.
s
)]
t
(
f
[
L
]
dt
df
[
L
)]
t
(
'
f
[
L +
•
−
=
=
=
)
1
(
)
1
(
2
1
)
0
(
.....
)
0
(
)
0
(
)
(
]
)
(
[
−
−
−
−
−
−
=
n
n
n
n
n
n
f
f
s
f
s
s
F
s
dt
t
df
)
1
i
(
n
1
i
i
n
n
)
n
(
)
0
(
f
.
s
)
s
(
F
s
]
)
t
(
f
[
L
−
=
−

−
=
Laplace transform
L L L
L L
L
L
L

 =
t
0
s
)
s
(
F
]
du
)
u
(
f
[
L
Integral
Laplace transform
L

Example 1
a) Use Laplace transform to obtain the time response y(t)
for the following cases
i) f(t)=1
ii) f(t)=t
b) Use Laplace transform to find the time response of:
c) Find the mathematical model (relationship between input and output as a differential
equation) in time domain of the following system
)
(
2 t
f
y
y +
=

2
12
8 =
+
+ y
y
y 


Y(s)
3
1
10
4
+
s
R(s)
R(s)
Laplace transform

closed and open loop control systems representation and applied Laplace transforms- 3.pdf

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closed and open loop control systems representation and applied Laplace transforms- 3.pdf