Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
(MASON) تحليل المنظومات المتداخلة باستخدام طريقة
∆+∆+∆
=........2211 TT
lInputSignaalOutputSign
∆=1-(L1+L2+L3+…………) +(Sum of Product of any two nontouching loops) -(Sum of Product of any three nontouching loops)+……..
where:
L = Loops.
T1,T2 = Forward Transmission from Input Signal to Output Signal.
∆1= ∆+L1 =value of ∆ after removing the loops that touch (T1).
∆2= ∆+L2 =value of ∆ after removing the loops that touch (T2).
In block diagram represenattion, we have to apply reduction rules, one after the other to obtain simple of the system and hence overall transfer function. We have to draw the reduction block diagram after every step. This is time consuming. In signal flow graph (SFG) approach, once SFG is obtained, direct use of one formula leads to the overall system transfer function C(s)/R(s). This formula is stated by S.J.Mason (1953) and hence referred as Mason‘s Gain formula.
Lecturer: Dr. Laith Abdullah Mohammed
Example: Find the Response of the following control system at: D2=1, A=1, K1=1, K2=2, KH=0.5, B=-5, D1=1/6.
when:1- v is step function with constant value. (i.e. u=0)2- u is step function with constant value. (i.e. v=0)
AsD
K
1
1
1+ sDK
2
2
1+
B
KH
v c
u
+
-
e
G1
++ G2
H
Input Signal
Output Signal
Solution:
1- v is step function with constant value. (i.e. u=0).
Using MASON method:
L1= -G1G2HT1=G1G2A∆=1-L1=1-(-G1G2H)=1+G1G2H∆1=∆+L1=1+G1G2H-G1G2H=1
tt eetcssss
CsC
sCsc
sRootssss
sc
sonStepFunctivwhen
ssssHGGAGGT
vc
lInputSignaalOutputSign
43
321
221
2111
341)()4(
3)3(
41)4()3(
)(
4,3,0:)4)(3(
12)(
1:
)4)(3(12
12712
1))(1(
−− +−=∴
++
+−=
++
++=
−−=∴++
=∴
==
++=
++=
+=
∆∆
==
v A G1 G2 cL1
T1B
u=0
-H
e
2- when (u) is step function with constant value. (i.e. v=0).
Using MASON method:
L1= -G1G2HT1=BG2∆=1-L1=1-(-G1G2H)=1+G1G2H∆1=∆+L1=1+G1G2H-G1G2H=1
tt eetcssss
CsC
sCsc
sRootssss
ssc
sonStepFunctiuwhen
sss
HGGBGT
uc
lInputSignaalOutputSign
43
321
21
211
5105)()4(
5)3(
105)4()3(
)(
4,3,0:)4)(3(
)6(10)(
1:
)4)(3()6(10
1)1)((
−− −+−=∴
+−
++
−=
++
++=
−−=∴++
+−=∴
==
+++−
=+
=∆∆
==
v=0 A G1 G2 cL1
T1B
u
-H
e
Example: Convert the block diagram representation of a system as shown below into a signal flow graph. Hence find the transfer function C(s)/R(s) of the closed loop system from the signal flow graph.
Example: Find the overall Transfer Function by using Mason‘s gain formula for the single flow graph in the figure below.
Home Work: Find the Response for the speed control system at: D=0.25, K1=1, K2=0.75, u=Load Torque, no=Output Speed, ni =Input Speed, and all initial conditions are zero.
when:1- ni is step function with constant value. (i.e. u=0)2- u is step function with constant value. (i.e. ni =0)
sK1
DsK+1
2nino
u
+
-
e
G1
+- G2
Input Speed
Output Speed
1. no(t)=1+0.5e-3t-1.5e-tAnswer:
2. no(t)=-1.5e-t+1.5e-3t
Mason's Rule MATLAB function
DescriptionMason.m uses mason's rule to simplify signal flow graphs. It takes a file describing the network and produces a symbolic equation relating a dependent output node to an independent input node. The routine requires the user to create a ".txt" file describing a network's signal paths.
s11
s21
s22
s12
R2
Using the ProgramIt is important that the lines in the net file be ordered so that the coefficient numbers count from 1 up. Don't use 0 to number the coefficients or nodes! Once you have made the net file, run 'mason.m' from Matlab, as described below:
USAGE:[Numerator,Denominator] = mason(Netfile,StartNode,StopNode)
Netfile - is a string with the name of the netfile to loadStartNode - is the integer number describing the independent input nodeStopNode - is the integer number describing the dependent output nodeNumerator - is a string containing the equation for the NumeratorDenominator - is a string containing the equation for the Denominator
Try out the example network! To recreate the above examples use:
[Numerator,Denominator] = mason('example.net',1,3)[Numerator,Denominator] = mason('example.net',1,2)
Presentation by studentsEach student prepares a presentation on one of the following topics.No. of Slides: Less than 10 (using Microsoft PowerPoint), Duration: 2 weeks. [Draw the block diagram (The Control System) of the following systems showing the input variables,
the output variables, and inside the block (gain)]1 Potentiometer 11 Automatic
Elevator21 Suspension
system in the car31 Self-Guided
Vehicle41 Active Vibration
Absorber
2 Winders 12 Toaster 22 Actuator 32 Thermostat 42 Tachometer
3 Nuclear reactor 13 DVD Player 23 Hydraulic pump 33 Floppy Disk Drive 43 Antenna Azimuth
4 Control the fluid level in a home tank
14 Remote controlled robot arm
24 Automobile Guidance System
34 Electric Ventricular Assist Device (EVAD)
44 Position control system in NC machine
5 Dynamometer 15 Steam Boiler 25 Furnace 35 Wind turbines 45 Voltage stabilizer
6 Grinder system 16 Automatic ship steering system
26 Walking Robots (Hannibal)
36 Arc Welding Robot
46 High speed rail pantograph
7 High speed proprtional solenoid valve
17 Navigation system of missiles
27 Magnetic Levitation Transportation System
37 Heat Exchanger Process
47 Sunseeker solar system
8 The pupil of human eye
18 Guidance system of Space shuttle
28 Cutting forces during machining operation
38 Coordinate Measuring Machine (CMM)
48 Automatic controlled Load tester
9 A Segway Human Transporter
19 Steel Plate Finishing mill
29 Dynamic Voltage Restorer (DVR)
39 Charge-Coupled Device (CCD)
49 CameraMan (Automatic Presenter Camera system)
10 Automatic Door operating system
20 Continuous Casting machine
30 3D Full body Scanning
40 Anti-lock braking (ABS)
50 conveyor system