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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 (b lack model ) - PowerPoint PPT Presentation
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Automatic Control SystemII.
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) modelThe 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) y(t)
x(j) y(j)1 1
2 2
1 2 1 2
x (t) y (t)
x (t) y (t)
x (t) x (t) y (t) y (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
min
max min
WP1
X(t){dim} X {dim}100 X(t)%
X {dim} X {dim}
X(t) X x(t)
100
0
x(t)
y(t)
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.
WP2
WP1
100
0
min
max min
WP1
Y(t){dim} Y {dim}100 Y(t)%
Y {dim} Y {dim}
Y(t) Y y(t)
Transfer function in frequency domain
Amplitude gain:
Phase shift:
( )( )( ) ( )
( ) jout
in
y jG j A e
x j
( ) 20lg ( )a dB A
Im ( )( )
Re ( )
G jarctg
G j
( ) ( )A 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 Nichols diagram: The horizontal axis is φ(ω) phase shift and the vertical axis is The a(ω) dB.
( ) 20lg ( )a dB A
The basic transfer functionIn the time domain is the
differential equitation
y(t) Ax(t)
y(s) 1
x(s) 1 sT
dy(t)
T y(t) x(t)dt
y(s)A
x(s)
22
2
d y(t) dy(t)T 2 T y(t) x(t)
dt dt 2 2
y(s) 1
x(s) 1 s2 T s T
i
dy(t)T x(t)
dt
i
y(s) 1
x(s) sT
d
dx(t)y(t) T
dt d
y(s)sT
x(s)
y(t) 1(t )x(t ) sy(s)e
x(s)
In the frequency domain is the transfer function
Block representationActuating 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
G2
G1+G2
G1
G2
21
1
1 GG
G
P proportional
Step response Bode diagram
t
I Integral
Step response Bode diagram
D differential
Step response Bode diagram
The step response is an Dirac delta, which isn’t shown
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
plantcontroller
Terms of feedback control
reference input element
reference signal
comparing elementor error detector
error signal
compensator or control task
action signal
feedback signal
actuator
transmitter
manipulated variable
disturbance variable
controlled variable
block model of the plant
Block diagram manipulation
G1
G1 G2 G1G2
G2
G1+G2
G1
G2
21
1
1 GG
G
G1
G1
G1
G1 G1
1
1
G
G1
G1
G1
G1 G1
1
1
G
Block diagram reduction example
G5
G2
G7
G6
G1 G3
G4
)(sx )(sy
765317641532
7417531
1)(
)()(
GGGGGGGGGGGG
GGGGGGG
sx
sysG