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
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