(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

This program generates these equations for a given network and pair of nodes.

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

Determine the Transfer Function C(s)/R(s) of the system shown in the Fig. below.