1                  Reduction of Multiple                     Subsystems•   How to reduce a block diagram of multiple subsy...
2Block diagrams (1)                     Control Systems Engineering, Fourth Edition by Norman S. Nise                     ...
3              Block diagrams (2)Cascaded subsystems                                   Control Systems Engineering, Fourth...
4             Block diagrams (3)Parallel subsystems                                  Control Systems Engineering, Fourth E...
5                            Block diagrams (4) Feedback systems             C ( s)     G ( s)Ge ( s ) =          =       ...
6Moving Blocks to Create Familiar Forms (1)                                Control Systems Engineering, Fourth Edition by ...
7Moving Blocks to Create Familiar Forms (2)                                Control Systems Engineering, Fourth Edition by ...
8Problem          Control Systems Engineering, Fourth Edition by Norman S. Nise            Copyright © 2004 by John Wiley ...
9Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
10Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
11Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
12Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
13Analysis and Design of Feedback Systems                             K … amplifier gain           a2 ⎞     K ∈ 0, ⎟ :    ...
14                  Signal-Flow GraphsBranches – represent systemsNodes – represent signals                               ...
15Problem: Convert to a signal-flow graph                                          Control Systems Engineering, Fourth Edi...
16Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
17Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
18Signal-Flow Graphs of State Equations                            Control Systems Engineering, Fourth Edition by Norman S...
19Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
20Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
21Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
22   Alternative Representations in State SpacePhase variable Form                                    Control Systems Engi...
23 Alternative Representations in State SpaceCascade Form                                  Control Systems Engineering, Fo...
24Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
25       Alternative Representations in State Space     Parallel FormNote that the equations are decoupled(each state equa...
26Parallel Form   -in case of repeated roots the matrix A will not   be diagonal                                          ...
27Controller Canonical Form - used in controller design - obtained from phase variable form by reversing the ordering of t...
28Observer Canonical Form- Used in the design of observers 1         7         2                 9        26         24   ...
29Control Systems Engineering, Fourth Edition by Norman S. Nise  Copyright © 2004 by John Wiley & Sons. All rights reserved.
30                                     Duality   Controller canonical form                           Observer canonical fo...
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05 elec3114

  1. 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. 2Block diagrams (1) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  3. 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. 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. 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. 6Moving 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. 7Moving 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. 8Problem Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  9. 9. 9Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  10. 10. 10Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  11. 11. 11Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  12. 12. 12Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  13. 13. 13Analysis 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. 14 Signal-Flow GraphsBranches – represent systemsNodes – represent signals Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  15. 15. 15Problem: Convert to a signal-flow graph Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  16. 16. 16Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  17. 17. 17Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  18. 18. 18Signal-Flow Graphs of State Equations Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  19. 19. 19Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  20. 20. 20Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  21. 21. 21Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  22. 22. 22 Alternative Representations in State SpacePhase variable Form Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  23. 23. 23 Alternative Representations in State SpaceCascade Form Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  24. 24. 24Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  25. 25. 25 Alternative Representations in State Space Parallel FormNote 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. 26Parallel 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. 27Controller Canonical Form - used in controller design - obtained from phase variable form by reversing the ordering of the phase variablesPhase 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. 28Observer 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.
  29. 29. 29Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
  30. 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.

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