Block Diagram Models, Signal Flow Graphs and Simplification ...

Block Diagram Models, Signal Flow Graphs and Simplification Methods

Block Diagram Manipulation Rules X1

X2

G1

X3

G2

X1 = X1

Dynamic Systems and Control Lavi Shpigelman

X1 + ± X2

G

(of rational functions ).

• Mainly relevant where there is a cascade of information flow

X3 ±

X3

G

=

X2

X1

X2

G

X1

G

= X2

X2

G

X2

variables.

• Are equivalent to a set of linear algebraic equations

+

Block Diagram Manipulation Rules

Block Diagram Models • Visualize input output relations • Useful in design and realization of (linear) components • Helps understand flow of information between internal

G

X3

G2 G1

X1

G

X1

X2

X1 =

X2

G G-1

X1

Block Diagram Reduction

Block Diagram Manipulation Rules X1

G

+

X1 +

X3 ±

=

±

X2

(with SISO components)

G

X3

G-1

U(s) +

-

X2

G1

+

-

H2 ! G4 G2

+ +

G3

G4

Y(s)

H1

+

X2

G

-

=

X1

(I+GH)-1G

X2

H3

H




(with SISO components) H2 ! G4

H2 U(s) +

-

G1

+

-

G2

+ +

G3

G4

Y(s)

U(s) +

-

G1

+

-

G2

H1 H3

+ +

G3 G4 H1

H3

Y(s)




U(s) +

-

G1

+

-


H2 ! G4 G3 G4 !!!!!! 1-G3 G4H1

G2

Y(s)

U(s) +

-

H3



-

G1

+

Y(s)

H3


U(s) +

G2 G3 G4 !!!!!!!!!!!! 1-G3 G4H1+ G2 G3H2

G1


H2 ! G4

-

G2 G3 G4 !!!!!! 1-G3 G4H1

H3

Y(s)

U(s) +

G1G2 G3 G4 !!!!!!!!!!!! 1-G3 G4H1+ G2 G3H2

-

H3

Y(s)


Block Diagram Vs. SFG


x1 + ± U(s)

G1 G2 G3 G4 !!!!!!!!!!!!!!!!!!!!!!! 1-G3 G4H1+ G2 G3H2+ G1 G2 G3 G4H3

G

x2

x1

1

x1

• Blocks

x2

±H(s)

H

Y(s)

G(s)

Edges (aka branches)

(representing transfer functions)

• Edges + junctions

Vertices (aka nodes)

(representing variables)

Signal Flow Graphs • Alternative to block diagrams • Do not require iterative reduction to find

transfer functions (using Mason’s gain rule)

• Can be used to find the transfer function

between any two variables (not just the input and output).

• Look familiar to computer scientists (?)

Algebraic Eq representation • x = Ax + r

x1 = a11x1+a12x2+r1 x2 = a21x1+a22x2+r2

• y(s) = G(s)u(s)

G11(s) u1(s) G21(s) G12(s) u2(s) G22(s)

y1(s)

y2(s)

1

x2

Definitions

Another SFG Example a32

y2 = a12y1 + a32y3

a12 y1

y2

y3 a32

y3 = a23y2 + a43y4

a12

y5

y4

y5

a23

y1

y2

y3

a43

a32 a12

y4 = a24y2 + a34y3+a44y4

y4 a43

a23

y1

y2

a44 a34

y3

y4

y5

a24

a43

a32

y5 = a25y2 + a45y4

a12

a23

y1

y2

a44 a34

y3

a45 y4

y5

• Input: (source) has only outgoing branches • Output: (sink) has only incoming branches • Path: (from node i to node j) has no loops. • Forward-path: path connecting a source to a sink • Loop: A simple graph cycle. • Path Gain: Product of gains on path edges • Loop Gain: Product of gains on loop Loops: Loops that have no vertex • Non-touching in common (and, therefore, no edge.)

a24 a25

Mason’s Gain Rule (1956)

Input / Output • Input (source) has only

a32

outgoing edges

• Output (sink) has only incoming edges

a12 y1

a23 y2

• any variable can be

made into an output by adding a sink with “1” edge

Given an SFG, a source and a sink, N forward paths between them and K loops, the gain (transfer function) between the source-sink pair is !Pk!k Tij = !!!! !

y2 1 a12 y1

y3

a32 a23

y2

1 y3

y3

Pk is the gain of path k, ! is the “graph determinant”: ! = 1- !(all loop gains) + !(products of non-touching-loop gain pairs) - !(products of non-touching-loop gain triplets) + ... !k = ! of the SFG after removal of the kth forward path

Mason’s Rule for Simple Feedback loop

A Feedback Loop Reduces Sensitivity To Plant Variations

P1 = G(s) L1 = -G(s)H(s)

1

x1

x1

! = 1 - (-G(s)H(s))

x2

G(s)

1

x2

-H(s)

G8 G2

G3

G4

G5

G6

y(s)

-H1

-H4 -H2

Forward paths:

-H3

P1 = G1G2G3G4G5G6 P2 = G1G2G7G6 P3 = G1G2G3G4G8

Feedback loops: L1 = -G2G3G4G5H2 L4 = -G7H2G2 L7 = -G1G2G7G6H3

G(s)

G=20000 y(s)/u(s)=20000/(1+20000*0.01)=99.50

G7

G1

1

-H(s)

P1 !1 G(s) G(s) T(s) = !!! = !! = !!!!! ! ! 1+G(s)H(s)

1

u(s)

G=10000 y(s)/u(s)=10000/(1+10000*0.01)=99.01

!1 = 1

x(s)

y(s)/u(s)=G/(1+GH)

L2 = -G5G6H1 L3 = -G8H1 L5 = -G4H4 L6 = -G1G2G3G4G5G6H3 L8 = -G1G2G3G4G8H3

Loops {3,4},{4,5} and {5,7} don’t touch ! = 1-(L1+L2+L3+L4+L5+L6+L7+L8)+(L3L4+L4L5+L5L7) !1 = !3 = 1 , !2 = 1-L5 = 1 - G4H4 y(s) P1+P2!2+P3 T(s) = !! = !!!!!` x(s) !

= 0.01

1

y(s)