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Event-based Control: Theory and Application J AN L UNZE Ruhr-Universitt Bochum email: Lunze@atp.rub.de Overview Introduction to event-based control 1 Event-based state feedback 2 Analysis of the event-based control loop 3 Experimental


  1. Event-based Control: Theory and Application J AN L UNZE Ruhr-Universität Bochum email: Lunze@atp.rub.de

  2. Overview Introduction to event-based control 1 Event-based state feedback 2 Analysis of the event-based control loop 3 Experimental evaluation 4 Event-based PI control 5 Conclusion and outlook 6

  3. Introduction to event-based control

  4. � Event-based control � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Aim: Reduction of the network traffic.

  5. Event-based control Why is the theory of sampled-data control not applicable? No zero-order hold input. No periodic sampling → no discrete-time model available. Sampling theorem possibly violated. It is necessary to develop a new theory of event-based control.

  6. � � Literature survey • Deadband control (O TANEZ , M OYNE , T ILBURY , American Control Conf. , 2002) Do not send new data as long as || x ( t ) − x ( t k ) || < ¯ e � � � � � � � � � � � � � � � � � � • Quantised state feedback • Self-triggered control (T ABUADA , IEEE Trans. Autom. Control , 2007) Estimate the next event time: t k + 1 = h ( x ( t k ) , t k )

  7. � � Literature survey • Deadband control • Quantised state feedback (G RÜNE , J UNGE , Syst. Control Lett. , 2005) (D E P ERSIS , IFAC World Congress , 2008) � � � � � � � � � � • Self-triggered control (T ABUADA , IEEE Trans. Autom. Control , 2007)

  8. � � Literature survey • Deadband control • Quantised state feedback • Self-triggered control (T ABUADA , IEEE Trans. Autom. Control , 2007) Estimate the next event time: t k + 1 = h ( x ( t k ) , t k ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

  9. Literature survey Evaluation of existing approaches : Many approaches result from an extension of sampled-data control and do not answer the three basic questions Many approaches do not consider any kind of disturbance and model uncertainties. Almost all approaches use a zero-order hold as control input generator.

  10. � Experiment What is a typical behaviour of an event-based control loop? � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

  11. Experiment Discrete-time control: 2

  12. Experiment Event-based control: 2 2 -

  13. Control aim „Ultimate boundedness“ („practical stability“): Hold the state x ( t ) inside a set Ω d : x ( t ) ∈ Ω d , ∀ t ≥ T ( x 0 ) , x 0 ∈ Ω 1 , d ( t ) ∈ [ d min , d max ] 1 d Then the set Ω d is said to be robust positively invariant.

  14. Event-based state feedback

  15. Event-based state feedback Why do we need information feedback? ... to stabilise an unstable plant, ... to compensate model uncertainties, ... to attenuate disturbances. � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

  16. Event-based state feedback Plant : x ( t ) ˙ = Ax ( t ) + Bu ( t ) + Ed ( t ) , x ( 0 ) = x 0 y ( t ) = Cx ( t ) Assumptions: Linear, asymptotically stable plant, no model uncertainties, synchronised clocks, no transmission delays, no computational restrictions.

  17. � � Event-based state feedback � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Main idea: The event-based controller should mimic the behaviour of a continuous state-feedback controller with adjustable precision. ! || x ( t ) − x SF ( t ) || ≤ e max

  18. � � Control input generator � � � � � � � � x SF ( t ) ˙ = ( A − BK ) x SF ( t ) + Ed ( t ) � �� � � � � � � � � � � � ¯ A � � � � � � � � � � � � � � u ( t ) = − Kx SF ( t ) � � � � � � � � � � � � � � � � � � � � Behaviour of the continuous feedback loop after the state x SF ( t k ) at time t k is known: � t ¯ ¯ A ( t − t k ) x SF ( t k ) − A ( t − τ ) Ed ( τ ) d τ, u ( t ) = − K e K e t ≥ t k t k

  19. Control input generator � t ¯ ¯ A ( t − t k ) x SF ( t k ) − A ( t − τ ) Ed ( τ ) d τ, u ( t ) = − K e K e t ≥ t k t k ↓ Control input generator mimics the continuous state feedback: � t ¯ ¯ A ( t − t k ) x ( t k ) − A ( t − τ ) E ˆ u ( t ) = − K e K e d k d τ, t ≥ t k t k

  20. � � � Control input generator This input is generated by the following system: Ax s ( t ) + E ˆ ¯ x s ( t + x s ( t ) ˙ = d k , k ) = x ( t k ) , t ≥ t k u ( t ) = − Kx s ( t ) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

  21. � � � � Event generator � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Event generator initiates an information exchange whenever − ) || x ( t k ) − x s ( t k || = ¯ e � �� � x ∆ ( t k )

  22. � � � � � � � Disturbance estimator � � � � � � � � � � � � � � � � � � � � For constant disturbances d ( t ) = ¯ d , t ∈ [ t k − 1 , t k ) the state difference is � t k e A ( t − τ ) E (¯ d − ˆ x ( t k ) − x s ( t k ) = d k − 1 ) d τ t k − 1 A − 1 � � e A ( t k + 1 − t k ) − I E (¯ d − ˆ = d k − 1 ) .

  23. Disturbance estimator The „mean“ magnitude ¯ d of the disturbance in the time interval t ∈ [ t k − 1 , t k ] is used as disturbance estimate ˆ d k in the time interval t ≥ t k : Disturbance estimator ˆ d 0 = 0 � A − 1 � � � + � e A ( t k − t k − 1 ) − I � ˆ ˆ x ( t k ) − x s ( t − d k = d k − 1 + E k )

  24. Event-based control algorithm � � � � � � � � � � � � � � � � � � � � � � Summary of the event-based control algorithm Check the difference || x ( t ) − x s ( t ) || until an event is detected. Then: Send the information x ( t k ) from the event generator to the 1 control input generator. Determine the disturbance estimate ˆ d k . 2 Reinitialise the control input generator: x s ( t + k ) = x ( t k ) 3

  25. Event-based control algorithm Behaviour of the event-based control loop: x t s ( ), x t ( ) { e t t 0 t 1 t 2

  26. Summary Three novelties of this methods with respect to literature: The control input generator is not a zero-order hold, but determines exponential inputs u ( t ) . The event generator compares the behaviour of the event-based control loop with some reference system (model of the continuous feedback loop). A disturbance estimate is used to adapt the event-based loop to the unknown disturbance d ( t ) .

  27. Analysis of the event-based control loop

  28. Analysis of the event-based control loop Closed-loop system between two consecutive events t ∈ [ t k , t k + 1 ) : � ˙ � � A � � x ( t ) � � E � � O � x ( t ) − BK ˆ = + d ( t ) + d k ¯ x s ( t ) ˙ O A x s ( t ) O E � x ( t k ) � � x ( t k ) � = x s ( t + k ) x ( t k ) � x ( t ) � y ( t ) = ( C O ) x s ( t ) State transformation � x ∆ ( t ) � � I � � x ( t ) � − I = x s ( t ) O I x s ( t )

  29. Analysis of the event-based control loop Transformed state-space model � ˙ � � A � � x ∆ ( t ) � � E � � O � x ∆ ( t ) O ( d ( t ) − ˆ ˆ = + d k ) + d k ¯ x s ( t ) ˙ O A x s ( t ) O E � �� � d ∆ ( t ) � � � � x ∆ ( t k ) 0 = x s ( t + k ) x ( t k )

  30. � � � � � � � Analysis of the event-based control loop � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � d ∆ ( t ) = d ( t ) − ˆ d k affects the (uncontrolled) plant. For a good approximation d ( t ) − ˆ d k ≈ 0 , for t ≥ t k no communication is necessary.

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