Setpoint Tracking in SS Systems 1. Setpoint Tracking in SS Systems - PowerPoint PPT Presentation

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 space model, x ( k + 1) = Ax ( k ) + Bu ( k ) y ( k ) = Cx ( k ) consider including an integral action:

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 )

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,

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 ) ,

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 ) =

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 )

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:

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)

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 )

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

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

Setpoint Tracking with Integral Mode 2.

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

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 )

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

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 )

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

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 )

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 )

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

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 )

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

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.

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

Linear Quadratic Regulator - Formulation 3.

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

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