05 elec3114
- 1. 1
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.
- 2. 2
Block diagrams (1)
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 3. 3
Block diagrams (2)
Cascaded subsystems
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 4. 4
Block diagrams (3)
Parallel subsystems
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 5. 5
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)
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 6. 6
Moving Blocks to Create Familiar Forms (1)
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 7. 7
Moving Blocks to Create Familiar Forms (2)
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 8. 8
Problem
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 13. 13
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
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 14. 14
Signal-Flow Graphs
Branches – represent systems
Nodes – represent signals
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 15. 15
Problem: Convert to a signal-flow graph
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 18. 18
Signal-Flow Graphs of State Equations
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 22. 22
Alternative Representations in State Space
Phase variable Form
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 23. 23
Alternative Representations in State Space
Cascade Form
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 25. 25
Alternative Representations in State Space
Parallel Form
Note that the equations are decoupled
(each state equation is function of only one state variable)
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 26. 26
Parallel Form
-in case of repeated roots the matrix A will not
be diagonal
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 27. 27
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
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 28. 28
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
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.
- 30. 30
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
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.