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Reduction of Multiple
Subsystems
• How to reduce a block diagram of multiple subsystems to a single
block representing the transfer function from input to output

• How to analyze and design transient response for a system consisting
of multiple subsystems

• How to represent in state space a system consisting of multiple
subsystems

• How to convert between alternate representations of a system in state
space

Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.

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Block diagrams (1)


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Block diagrams (2)

Cascaded subsystems


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Block diagrams (3)

Parallel subsystems


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Block diagrams (4)

Feedback systems

C ( s) G ( s)
Ge ( s ) = =
R( s) 1 ± G ( s) H ( s)

“-” negative feedback
“+” positive feedback

G(s)H(s) … open-loop transfer function
(loop gain)


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Moving Blocks to Create Familiar Forms (1)


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Moving Blocks to Create Familiar Forms (2)


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Problem


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10


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Analysis and Design of Feedback Systems

K … amplifier gain

a2 ⎞
K ∈ 0, ⎟ : … overdamped response
4⎟ ⎠

a2 … critically damped
K= :
4

a2 … underdamped response
K> :
4


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Signal-Flow Graphs

Branches – represent systems

Nodes – represent signals


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Problem: Convert to a signal-flow graph


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Signal-Flow Graphs of State Equations


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Alternative Representations in State Space
Phase variable Form


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Cascade Form


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Parallel Form

Note that the equations are decoupled
(each state equation is function of only one state variable)


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Parallel Form

-in case of repeated roots the matrix A will not
be diagonal


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Controller Canonical Form
- used in controller design
- obtained from phase variable form by reversing the ordering of the phase
variables

Phase variable form:

Note: System matrices that contain the coefficients of the characteristic
polynomial are called companion matrices

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Observer Canonical Form
- Used in the design of observers

1 7 2 9 26 24
R( s) + 2 R( s) + 3 R( s) = C ( s) + C ( s) + 2 C ( s) + 3 C ( s)
s s s s s s


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Duality
Controller canonical form Observer canonical form

⎡ x1 ⎤ ⎡− 9 − 26 − 24⎤ ⎡ x1 ⎤ ⎡1⎤
& ⎡ x1 ⎤ ⎡ − 9 1 0⎤ ⎡ x1 ⎤ ⎡1⎤
&
⎢x ⎥ = ⎢ 1
&2 0 0 ⎥ ⎢ x 2 ⎥ + ⎢0 ⎥ r ⎢ x ⎥ = ⎢− 26 0 1⎥ ⎢ x ⎥ + ⎢7⎥ r
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ &
⎢ 2⎥ ⎢ ⎥⎢ 2 ⎥ ⎢ ⎥
⎢ x3 ⎥ ⎢ 0
⎣& ⎦ ⎣ 1 0 ⎥ ⎢ x3 ⎥ ⎢0⎥
⎦⎣ ⎦ ⎣ ⎦ ⎢ x3 ⎥ ⎢− 24 0 0⎥ ⎢ x3 ⎥ ⎢2⎥
⎣& ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
⎡ x1 ⎤ ⎡ x1 ⎤
y = [1 7 2]⎢ x2 ⎥
⎢ ⎥ y = [1 0 0]⎢ x2 ⎥
⎢ ⎥
⎢ x3 ⎥
⎣ ⎦ ⎢ x3 ⎥
⎣ ⎦

If a system is described by A, B, C then its dual system is described by: AD = AT
BD = CT
x = Ax + Bu
& x = A D x + B Du
& CD = BT
y = Cx y = CD x


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