Dynamic Systems and Control Lavi Shpigelman Block Diagram Models, Signal Flow Graphs and Simplification 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 G 1 X 1 G 2 X 2 X 3 G 2 G 1 X 1 X 3 = X 1 + X 2 ± G X 3 + X 2 ± G = X 3 G X 1 Block Diagram Manipulation Rules G X 1 X 2 X 2 X 1 G G X 2 X 2 = X 1 G X 2 X 1 X 1 G X 2 X 1 = G -1
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Block Diagram Models Block Diagram Manipulation · PDF fileBlock Diagram Models, Signal Flo w Gra phs and ... Block Diagram Reduction ... control_3b.pdf Author: Lavi Shpigelman
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• 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”: