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Setpoint Tracking in SS Systems 1. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) Setpoint Tracking in SS Systems 1. In addition to the standard state


  1. Setpoint Tracking in SS Systems 1.

  2. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k )

  3. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action:

  4. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k )

  5. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error,

  6. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error, given by r − y ( k ) ,

  7. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error, given by r − y ( k ) , resulting in, x I ( k + 1) = x I ( k ) + r − y ( k ) =

  8. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error, given by r − y ( k ) , resulting in, x I ( k + 1) = x I ( k ) + r − y ( k ) = x I ( k ) + r − Cx ( k )

  9. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error, given by r − y ( k ) , resulting in, x I ( k + 1) = x I ( k ) + r − y ( k ) = x I ( k ) + r − Cx ( k ) Combine the two and form an augmented state equation:

  10. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error, given by r − y ( k ) , resulting in, x I ( k + 1) = x I ( k ) + r − y ( k ) = x I ( k ) + r − Cx ( k ) Combine the two and form an augmented state equation: � x ( k + 1) � = x I ( k + 1)

  11. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error, given by r − y ( k ) , resulting in, x I ( k + 1) = x I ( k ) + r − y ( k ) = x I ( k ) + r − Cx ( k ) Combine the two and form an augmented state equation: � x ( k + 1) � A � � x ( k ) � � 0 = + x I ( k + 1) − C 1 x I ( k )

  12. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error, given by r − y ( k ) , resulting in, x I ( k + 1) = x I ( k ) + r − y ( k ) = x I ( k ) + r − Cx ( k ) Combine the two and form an augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � 0 = + u ( k ) + x I ( k + 1) − C 1 x I ( k ) 0

  13. Setpoint Tracking in SS Systems 1. In addition to the standard state space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action: x I ( k + 1) = x I ( k ) + e ( k ) e is the error, given by r − y ( k ) , resulting in, x I ( k + 1) = x I ( k ) + r − y ( k ) = x I ( k ) + r − Cx ( k ) Combine the two and form an augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 1 Digital Control Kannan M. Moudgalya, Autumn 2007

  14. Setpoint Tracking with Integral Mode 2.

  15. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � = x I ( k + 1)

  16. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � 0 = + x I ( k + 1) − C 1 x I ( k )

  17. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � 0 = + u ( k ) + x I ( k + 1) − C 1 x I ( k ) 0

  18. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k )

  19. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k ) Closing the loop, the state equation becomes

  20. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k ) Closing the loop, the state equation becomes � A � � x ( k ) � 0 = − − C 1 x I ( k )

  21. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k ) Closing the loop, the state equation becomes � A � � x ( k ) � � x ( k ) � � B � � � 0 = + K p K I − − C 1 x I ( k ) 0 x I ( k )

  22. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k ) Closing the loop, the state equation becomes � A � � x ( k ) � � x ( k ) � � B � � � � 0 � 0 = + r ( k ) K p K I − − C 1 x I ( k ) 0 x I ( k ) 1

  23. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k ) Closing the loop, the state equation becomes � A � � x ( k ) � � x ( k ) � � B � � � � 0 � 0 = + r ( k ) K p K I − − C 1 x I ( k ) 0 x I ( k ) 1 �� A �� � x ( k ) � � B � � � 0 = + K p K I − − C 1 0 x I ( k )

  24. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k ) Closing the loop, the state equation becomes � A � � x ( k ) � � x ( k ) � � B � � � � 0 � 0 = + r ( k ) K p K I − − C 1 x I ( k ) 0 x I ( k ) 1 �� A �� � x ( k ) � � B � � � � 0 � 0 = + r ( k ) K p K I − − C 1 0 x I ( k ) 1

  25. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k ) Closing the loop, the state equation becomes � A � � x ( k ) � � x ( k ) � � B � � � � 0 � 0 = + r ( k ) K p K I − − C 1 x I ( k ) 0 x I ( k ) 1 �� A �� � x ( k ) � � B � � � � 0 � 0 = + r ( k ) K p K I − − C 1 0 x I ( k ) 1 A − BK form.

  26. Setpoint Tracking with Integral Mode 2. Consider the augmented state equation: � x ( k + 1) � A � � x ( k ) � � � B � � 0 � 0 = + u ( k ) + r ( k ) x I ( k + 1) − C 1 x I ( k ) 0 1 Introduce control law: � � x ( k ) � � u ( k ) = − K p K I x I ( k ) Closing the loop, the state equation becomes � A � � x ( k ) � � x ( k ) � � B � � � � 0 � 0 = + r ( k ) K p K I − − C 1 x I ( k ) 0 x I ( k ) 1 �� A �� � x ( k ) � � B � � � � 0 � 0 = + r ( k ) K p K I − − C 1 0 x I ( k ) 1 A − BK form. Recall condition for pole placement. 2 Digital Control Kannan M. Moudgalya, Autumn 2007

  27. Linear Quadratic Regulator - Formulation 3.

  28. Linear Quadratic Regulator - Formulation 3. �� A � � B � � �� 0 When can we place the poles of ? K p K I − − C 1 0

  29. Linear Quadratic Regulator - Formulation 3. �� A � � B � � �� 0 When can we place the poles of ? K p K I − − C 1 0 �� A � � B �� 0 When is controllable! , − C 1 0

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