Dynamic Systems and ControlLavi Shpigelman

Block Diagram Models,Signal Flow Graphs andSimplification Methods

Block Diagram Models

• Visualize input output relations

• Useful in design and realization of (linear) components

• Helps understand flow of information between internal variables.

• Are equivalent to a set of linear algebraic equations

(of rational functions ).

• Mainly relevant where there is a cascade of information flow

Block Diagram Manipulation Rules

G1X1

G2X2 X3

G2 G1X1 X3

=

X1 +

X2

±G

X3

+

X2

±

G

=

X3G

X1

Block Diagram Manipulation RulesG

X1 X2

X2

X1G

G

X2

X2

=

X1G

X2

X1

X1G

X2

X1

= G-1

Block Diagram Manipulation Rules

GX2

X1

+

-

H

X2(I+GH)-1G=

GX1 X3

X2

X1G

X3

X2

+

±

+

±=

G-1

G1U(s)

+ -G2

+

+G3 G4

Y(s)

H3

H1

H2

+

-

Block Diagram Reduction(with SISO components)

+G2

+

+G3 G4

H3

H1

H2!G4

G1

+ -U(s) Y(s)

-

Block Diagram Reduction(with SISO components)

+G2

+

+G3 G4

H3

H1

H2!G4

G1

+ -U(s) Y(s)

-

Block Diagram Reduction(with SISO components)

+G2

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

H3

H2!G4

G1

+ -U(s) Y(s)

-

Block Diagram Reduction(with SISO components)

+

H3

H2!G4

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

G1

+ -U(s) Y(s)

-

Block Diagram Reduction(with SISO components)

H3

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

G1

+U(s) Y(s)

-

Block Diagram Reduction(with SISO components)

H3

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

+U(s) Y(s)

-

Block Diagram Reduction(with SISO components)

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

U(s) Y(s)

Block Diagram Reduction(with SISO components)

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 (?)

Block Diagram Vs. SFG

• Blocks ⇒ Edges (aka branches)

(representing transfer functions)

• Edges + junctions ⇒ Vertices (aka nodes)

(representing variables)

Gx2+

±

H

x1 1x1

x1 G(s) x2 1x2

±H(s)

Algebraic Eq representation

• x = Ax + r

x1 = a11x1+a12x2+r1

x2 = a21x1+a22x2+r2

• y(s) = G(s)u(s) u1(s)

u2(s)

y1(s)

y2(s)

G11(s)

G22(s)

G12(s)

G21(s)

Another SFG Examplea12

y1 y2 y3 y4 y5

a32

y2 = a12y1 + a32y3

a12

y1 y2

a23

y3 y4 y5

a32 a43

y3 = a23y2 + a43y4

a12

y1 y2

a23

y3

a34

y4 y5

a24

a44

a32 a43

y4 = a24y2 + a34y3+a44y4

a12

y1 y2

a23

y3

a34

y4

a45

y5

a25

a24

a44

a32 a43

y5 = a25y2 + a45y4

Input / Output

• Input (source) has only outgoing edges

• Output (sink) has only incoming edges

• any variable can be made into an output by adding a sink with “1” edge

a12

y1 y2

a23

y3

a32

a12

y1 y2

a23

y3 y3

a32

1

1

y2

Definitions

• 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

• Non-touching Loops: Loops that have no vertex in common (and, therefore, no edge.)

Mason’s Gain Rule (1956)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!kTij = !!!!

!

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

P1 = G(s)

L1 = -G(s)H(s)

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

!1 = 1

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

1x1

x1 G(s) x2 1x2

-H(s)

Forward paths:P1 = G1G2G3G4G5G6 P2 = G1G2G7G6 P3 = G1G2G3G4G8

Feedback loops:L1 = -G2G3G4G5H2 L2 = -G5G6H1 L3 = -G8H1

L4 = -G7H2G2 L5 = -G4H4 L6 = -G1G2G3G4G5G6H3

L7 = -G1G2G7G6H3 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+P3T(s) = !! = !!!!!`

x(s) !

G3 G4 G5 G6

-H3

-H1

G8

G1 G2

-H2

-H4

G7

1x(s) y(s)

A Feedback Loop Reduces Sensitivity To Plant Variations

1 G(s) 1y(s)

-H(s)

u(s)

G=10000

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

G=20000

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

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

= 0.01