Automatic Control System
II.
Block diagram model
Modelling dynamical systems
Engineers use models which are based upon mathematical
relationships between two variables.
We can define the mathematical equations:
•Measuring the responses of the built process (black model)
•Using the basic physical principles (grey model). In order to
simplification of mathematical model the small effects are
neglected and idealised relationships are assumed.
Developing a new technology or a new construction nowadays
it’s very helpful applying computer aided simulation technique.
This technique is very cost effective, because one can create a
model from the physical principles without building of process.
LTI (Linear Time Invariant) model
The all physical system are non-linear and their parameters
change during a long time.
The engineers in practice use the superposition’s method.
x(t)
x(j)
y(t)
y(j)
x1 (t)  y1 (t)
x 2 (t)  y 2 (t)
x1 (t)  x 2 (t)  y1 (t)  y 2 (t)
First One defines the input and output signal range.
In this range if an arbitrary input signal energize the block
and the superposition is satisfied and the error smaller
than a specified error, than the block is linear.
The steady-state characteristics and the
dynamic behavior
100
Y(t){dim}  Ymin {dim}
100  Y(t)%
Ymax {dim}  Ymin {dim}
y(t)
WP2
Y(t)  YWP1  y(t)
WP1
t
X(t){dim}  X min {dim}
100  X(t)%
X max {dim}  X min {dim}
x(t)
0
0
100
X(t)  X WP1  x(t)
t
The steady-state characteristic. When the transient’s signals have died a new working
point WP2 is defined in the steady-state characteristic.
The dynamic behavior is describe by differential equation or transfer function in
frequency domain.
Transfer function in frequency domain
yout ( j )
G ( j ) 
 A( )e j ( )
xin ( j )
Amplitude gain:
A()  G( j)
a() dB  20lg A()
Phase shift:
Im G ( j )
 ( )  arctg
Re G ( j )
The graphical representation of transfer
function
• The M-α curves: The amplitude gain M(ω) in the frequency
domain. In the previous page M(ω) was signed, like A(ω) The
phase shift α(ω) in the frequency domain. In the previous page
α(ω) was signed, like φ(ω)!
• A Nyquist diagram: The transfer function G(jω) is shown on
the complex plane.
• A Bode diagram: Based on the M-α curves. The frequency is
in logaritmic scale and instead of A(ω) amplitude gain is:
a() dB  20lg A()
• A Nichols diagram: The horizontal axis is φ(ω) phase shift and
the vertical axis is The a(ω) dB.
The basic transfer function
In the time domain is the
differential equitation
y(t)  Ax(t)
T
dy(t)
 y(t)  x(t)
dt
2
d
y(t)
dy(t)
T2

2

T
 y(t)  x(t)
2
dt
dt
In the frequency domain is the
transfer function
y(s)
A
x(s)
y(s)
1

x(s) 1  sT
y(s)
1

x(s) 1  s2T  s 2T 2
dy(t)
 x(t)
dt
y(s)
1

x(s) sTi
dx(t)
y(t)  Td
dt
y(s)
 sTd
x(s)
y(t)  1(t  )x(t  )
y(s)
 e  s
x(s)
Ti
Block representation
Actuating path of signals and variables
One input and one output block represents the context between the
the output and input signals or variables in time or frequency domain
Summing junction
Take-off point (The same signal actuate both path)
G1
G2
G1G2

G1
G1+G2
G2
G1
G2
G1
1  G1G2
P proportional
t
Step response
Bode diagram
I Integral
Step response
Bode diagram
D differential
The step response is an
Dirac delta, which isn’t
shown
Step response
Bode diagram
PT1 first order system
Step response
Bode diagram
PT2 second order system
Step response
Bode diagram
PH delay
Step response
Bode diagram
IT1 integral and first order in cascade
Step response
Bode diagram
DT1 differential and first order in cascade
Step response
Bode diagram
PI proportional and integral in parallel
Step response
Bode diagram
PDT1
Step response
Bode diagram
Terms of feedback control
controller
plant
compensator
or control task
controlled variable
transmitter
manipulated variable
error signal
comparing element
or error detector
reference
signal
reference
input element
actuator
disturbance variable
feedback signal
action signal
block model of the plant
Block diagram manipulation
G1
G2
G1G2
G1
G1
G1
G1
G1+G2
G2

G1
G1
G1
1
G1
G1
1  G1G2
G2
G1
G1
G1
G1
G1
1
G1
Block diagram reduction example
G4
x(s)

G1

G3
G5
G7
y (s)
G2
G6
y(s)
G1G3G5G7  G1G4G7
G ( s) 

x( s) 1  G2G3G5  G1G4G6G7  G1G3G5G6G7