1. Événement - date SWIM’09, Jun 10 th -11 th, 2009 1/30 Design of a Robust Controller for Guaranteed Performances: Application to Piezoelectric Cantilevers sofiane.khadraoui@femto-st.fr mrakoton@femto-st.fr plutz@femto-st.fr SWIM’09 10-11 June 2009, EPFL Lausanne

2. Événement - date SWIM’09, Jun 10 th -11 th, 2009 2/30 Outline Why Control System Using Intervals theory? Basic Concepts on intervals. Direct synthesis of the controller. Computation of the interval Controller. Application to piezoelectric cantilevers. Controller implementation and results. Conclusion.

3. Événement - date SWIM’09, Jun 10 th -11 th, 2009 3/30 1- Why Control Systems Using Intervals theory ? Context Systems are subject to various uncertainties due to varying parameters, nonlinearities,...etc (Piezocantilevers). Obtaining an appropriate model is often difficult because of incomplete knowledge of the system and its environment Approach Robust control methods: H 2, H and  -synthesis. Robust control methods require the check of some conditions and lead to high order controller.

4. Événement - date SWIM’09, Jun 10 th -11 th, 2009 4/30 Our solution Seek for simple methods for control system ensuring the stability and performances. Solution: Using Interval Methods. These methods represent a reliable computation to achieve a high-level purposes in control systems 1- Why Control Systems Using Intervals theory ?

5. Événement - date SWIM’09, Jun 10 th -11 th, 2009 5/30 Definitions 2- Basic Concepts on intervals A closed interval, denoted by, is the set of real numbers given by: and are the lefts and right endpoints of the interval The width of : The midpoint of : The radius of :

6. Événement - date SWIM’09, Jun 10 th -11 th, 2009 6/30 Operations on intervals Given two intervals and 2- Basic Concepts on intervals

7. Événement - date SWIM’09, Jun 10 th -11 th, 2009 7/30 3- Direct synthesis of the controller Computation of the controller C(s) from a known closed-loop H(s) and the model transfer G(s). We define the closed-loop transfer by:

8. Événement - date SWIM’09, Jun 10 th -11 th, 2009 8/30 Controller transfer function The controller transfer is derived from the equation of H: 3- Direct synthesis of the controller K: a static gain. B - : a stable polynomial. B + : an unstable polynomial. A: the denominator of G(s). Let us written G(s) as follow:

9. Événement - date SWIM’09, Jun 10 th -11 th, 2009 9/30 Controller transfer function Finally we get the controller: Let us choose the following structure for the closed-loop: Where: D is the wanted transient behavior. 3- Direct synthesis of the controller

10. Événement - date SWIM’09, Jun 10 th -11 th, 2009 10/30 4- Definition of the interval Controller Interval model Given a second order interval system denoted by [G](s) as follow: Such as: Direct synthesis method Interval theory Interval controller design Guaranteed Stability & performances

11. Événement - date SWIM’09, Jun 10 th -11 th, 2009 11/30 Interval closed-loop model The closed-loop model with interval parameters [H](s) is defined as: Where [D](s) can be deduced from the required specifications. 4- Definition of the interval Controller  Overshoot:  Settling time: Specifications for the closed-loop:

12. Événement - date SWIM’09, Jun 10 th -11 th, 2009 12/30 Interval closed-loop model Such as the interval parameters [  ] and [ w n ] are calculated from: In result, the closed-loop is given as: From the 1 st specification From the 2 nd specification and the interval  4- Definition of the interval Controller

13. Événement - date SWIM’09, Jun 10 th -11 th, 2009 13/30 Deduction of the interval controller Finally, we obtain the interval controller: 4- Definition of the interval Controller

14. Événement - date SWIM’09, Jun 10 th -11 th, 2009 14/30 5- Application to piezoelectric cantilevers  Control the deflection of the piezocantilevers  Introducing the model uncertainties, using intervals  Two cantilevers with the same dimensions are used to define the interval model Fig. 1: a-b: Presentation of the setup; c: Piezocantilever subjected to an electrical excitation L*b*h=15mm*2mm*0.3mm

15. Événement - date SWIM’09, Jun 10 th -11 th, 2009 15/30 5- Application to piezoelectric cantilevers The interval model is derived from the two model G 1 and G 2. We obtain: Model with interval parameters After identification, we obtain the following models: Piezocantilever -1 Piezocantilever -2 Small differences on the dimensions (some tens of microns) Parameters model are slightly different.

16. Événement - date SWIM’09, Jun 10 th -11 th, 2009 16/30 We propose to expand the interval width of each parameter of [G](s) by 10% of its radius, which lead to: Model with an expanded width of the intervals 5- Application to piezoelectric cantilevers Expanding the width of the intervals The inclusion of the two models G 1 and G 2 inside the interval model ensures

17. Événement - date SWIM’09, Jun 10 th -11 th, 2009 17/30 5- Application to piezoelectric cantilevers Fig. 2: Models simulation compared with the experimental results

18. Événement - date SWIM’09, Jun 10 th -11 th, 2009 18/30 Harmonic analysis To confirm the inclusion of G 1 and G 2 inside the interval model [G](s) (between two bounds G inf and G sup ), a harmonic analysis was performed on the two piezocantilevers. Where: 5- Application to piezoelectric cantilevers

19. Événement - date SWIM’09, Jun 10 th -11 th, 2009 19/30 5- Application to piezoelectric cantilevers Fig. 3: Magnitudes of interval model compared with the experimental results

20. Événement - date SWIM’09, Jun 10 th -11 th, 2009 20/30 Interval controller transfer The wanted specifications for the closed-loop are:  overshoot:  settling time: 5- Application to piezoelectric cantilevers The interval of the ratio damping and natural frequency are given by: From the specifications

21. Événement - date SWIM’09, Jun 10 th -11 th, 2009 21/30 With the choice of the following polynomials for [K], [B - ] and [B + ]: Interval controller transfer 5- Application to piezoelectric cantilevers Using the controller transfer with interval parameters defined earlier: We get:

22. Événement - date SWIM’09, Jun 10 th -11 th, 2009 22/30 Transfer controller implemented 6- Controller implementation and results Taken the midpoint of each interval parameters of [C](s) for the implementation of the controller. The experimental results are compared with the wanted step responses H inf and H sup  H inf (s) represents the transfer function with the maximum settling time tr max =30ms and minimum overshoot d min =1%.  H sup (s) represents the transfer function with the minimum settling time tr max =15ms and maximum overshoot d min =5% Where:

23. Événement - date SWIM’09, Jun 10 th -11 th, 2009 23/30 6- Controller implementation and results Fig. 4: Step responses of the wanted closed-loop compared with the experimental results

24. Événement - date SWIM’09, Jun 10 th -11 th, 2009 24/30 Noises become non- negligible compared to the signal. 6- Controller implementation and results Fig. 5: Magnitudes of the wanted closed-loop compared with the experimental results

25. Événement - date SWIM’09, Jun 10 th -11 th, 2009 25/30 Application to a new piezocantilever  approximately the same dimensions previously used.  and having a model G’(s) such as: Application of the same law control C mid (s) to a new piezocantilever with: After identification, we obtain: 6- Controller implementation and results

26. Événement - date SWIM’09, Jun 10 th -11 th, 2009 26/30 Stability is always ensured 6- Controller implementation and results Fig. 6: Step response of the new piezocantilever compared with the simulation responses and the previous results

27. Événement - date SWIM’09, Jun 10 th -11 th, 2009 27/30  The interval analysis represents a reliable computation to achieve a high- level purposes in control systems.  The model uncertainties have been taken into account using intervals.  An application to the control of piezocantilevers was performed to show the efficiency of the interval computation in control systems.  The experimental results have effectively proved the accuracy of the proposed controller design method. 7- Conclusion

28. Événement - date SWIM’09, Jun 10 th -11 th, 2009 28/30 Thank You… & Questions are welcome

29. Événement - date SWIM’09, Jun 10 th -11 th, 2009 29/30 Design of a Robust Controller for Guaranteed Performances: Application to Piezoelectric Cantilevers sofiane.khadraoui@femto-st.fr mrakoton@femto-st.fr plutz@femto-st.fr