Transcript Block Diagrams and Steady State Errors

Block Diagrams and Steady State Errors
Topics
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Block diagrams to represent control systems
Block diagram manipulation
Example
Steady State Error
Block Diagrams
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Block Diagrams provide a pictorial representation of a system
Unidirectional operational block representing individual transfer functions
Three basic elements:
–
–
–
Rectangles = Operators
Lines = Signals
Circles = additional or subtraction
Block Diagrams: Examples
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Y = Ax
x
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y
A
e=r-c
r
+
e
c
Block Diagrams: Examples
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Y = Ax - Bz
x
A
+
Y
-
B
X
Closed loop system
• Simple Closed Loop Control System
Input
Error
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Process
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Sensor
Feedback
Output
Closed Loop System
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Simple Closed Loop Control System
Transfer function from R(s) to C(s)
E(s) = R(s) – B(s)
B(s) = H(s)C(s)
C(s) = G(s)E(s)
So,
C(s)/G(s) = R(s) =H(s)C(s)
E(s) = C(s)/G(s)
C(s)/R(s) =
G(s)
1+G(s)H(s)
Closed Loop System
Transfer function from R(s) to C(s)
C(s)/R(s) =
R(s)
G(s)
1+G(s)H(s)
C(s)
Closed Loop System
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Simple Closed Loop Control System
With unity feedback, H(s) = 1
G(s)
1+G(s)H(s)
G(s)
1+G(s)
Open Loop Transfer Function
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Remove the feedback link from summing junction
E(s)
R(s)
B(s)
G(s)
H(s)
Open Loop Transfer Function given by:
B(s) = G(s)H(s)
E(s)
C(s)
Error is input
Block Diagram Manipulation
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Diagrams can be manipulated using the following transformations
Combining Blocks in Series:
X1
G1G2
X2
X1
X3
G1(s)
G2(s)
Or
X1
G2G1
X3
X3
Block Diagram Manipulation
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Moving a summing junction
Block Diagram Manipulation
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Moving a pickoff point ahead of a block
Block Diagram Manipulation
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Moving a pickoff point behind a block
Example
C =
J
G3
1+G3H3
=
G4
An electrical motor is used in a closed loop system to control the
angular position of an inertial load.
The output signal from the transducer is compared with the input
demand and the resulting error signal is passed to a voltage/current
amplifier.
The input demand is converted from angular displacement to
voltage before being connected to the summing junction.
Steady State Erros
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Feedback control used to reduce steady-state errors
Steady-state error is error after the transient response has decayed
If error is unacceptable, the control system will need modification
Errors are evaluated using standardized inputs
- Step inputs
- Ramp inputs
- Sinusoidal inputs
Ts = setting time
Causes of steady state error
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5.
Errors can be caused by factors including
Instrumentation of measurement errors
System non linearity - saturation etc.
Form of input signal
Form of system transfer function
External disturbances acting on the system, for example: forces or torques
Error Function
E(s) =
R(s)
1
1+ G(s)H(s)
Calculating value
System dependent
Input dependent
• Use the final value theorem:
ess  lim e(t )  lim sF (s)
t 
s 0
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Inputs can be
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Step
Ramp
R(s) = A/s
R(s) = A/s^2
A is step amplitude
A is step velocity