1. Événement - date MEA’10, December 09 th, 2010 1/30 Performances analysis of controlled interval systems using H ∞ approach sofiane.khadraoui@femto-st.fr mrakoton@femto-st.fr plutz@femto-st.fr

2. Événement - date MEA’10, December 09 th, 2010 2/30 Introduction Robust control approaches for (parametric) uncertain systems A posteriori performances analysis Example with the control of piezoelectric cantilevers Conclusion Outline

3. Événement - date MEA’10, December 09 th, 2010 3/30 Introduction Robust control approaches for (parametric) uncertain systems A posteriori performances analysis Example with the control of piezoelectric cantilevers Conclusion

4. Événement - date MEA’10, December 09 th, 2010 4/30 1- Introduction Control design requirements Good knowledge on the system Good knowledge on its environment accurate model Système physique ?

5. Événement - date MEA’10, December 09 th, 2010 5/30 1- Introduction Characteristics of physical systems Complexity  nonlinearities (hysteresis, creep: ”piezo systems”)  time varying parameters  etc  vibration, variation of the ambient temperature, etc  disturbance effects and imprecise measurements  etc Sensitivity to the environmental conditions Modeling & control difficulties Non-perfect model Simplifications Model subject to parametric uncertainty

6. Événement - date MEA’10, December 09 th, 2010 6/30 Reference Output Controller Model U  + - uncertainties Control of parametric uncertain systems 1- Introduction Achieve robust performances Closed-loop control robust controller +

7. Événement - date MEA’10, December 09 th, 2010 7/30 Robust controller: a controller that takes into account the uncertainties during the synthesis and ensures the stability and the performances for the closed-loop. Definitions Robust stability: the stability of the closed-loop obtained by a robust controller despite of uncertainties. Robust performances: the performances of the closed-loop ensured by a robust controller in presence of uncertainties. 1- Introduction

8. Événement - date MEA’10, December 09 th, 2010 8/30 Introduction Robust control approaches for (parametric) uncertain systems A posteriori performances analysis Example with the control of piezoelectric cantilevers Conclusion

9. Événement - date MEA’10, December 09 th, 2010 9/30 2- Robust control approaches for parametric uncertain systems H ∞ approach robust controller K(s) Output system Input  (s) uncertainties matrix - Implementation difficulties - Complex controllers µ-synthesis H ∞ -loopshaping H ∞ -LPV H ∞ standard

10. Événement - date MEA’10, December 09 th, 2010 10/30 Interval analysis replacing real numbers by intervals [Moore, 1966] Robust control using interval analysis 2- Robust control approaches for parametric uncertain systems Guaranteed stability & robust performances [Bondia. CDC03][Tchen. CCE00] [Khadraoui. CDC10] Principle: Theory of controlInterval analysis Robust control law Parametric uncertainty Modeling Required specification Accounts - Reduced order controllers - Natural modeling of parametric uncertainties

11. Événement - date MEA’10, December 09 th, 2010 11/30 Introduction Robust control approaches for (parametric) uncertain systems A posteriori performances analysis Example with the control of piezoelectric cantilevers Conclusion

12. Événement - date MEA’10, December 09 th, 2010 12/30 3- A posteriori performances analysis Objective Based on H ∞ standard, verify if the controller C(s) ensures required specifications. y e C(s)C(s) [G](s,[a],[b]) + -  u - : an interval system, - C(s) : a given controller computed with interval control method. Specifications: - Settling time tr ≤  - Static error    ≤  - No (or small) overshoot - etc… ? Does the controller ensures the required specifications.

13. Événement - date MEA’10, December 09 th, 2010 13/30 Reminding H ∞ principle + - x2x2 T1(s)T1(s) T2(s)T2(s) + + x1x1 e1e1 e2e2 y1y1 y2y2 1- Stability: [Zames. 1966] Stability ensured Small gain theorem 3- A posteriori performances analysis The H ∞ norm of a SISO system G(s) is defined by:

14. Événement - date MEA’10, December 09 th, 2010 14/30 Reminding H ∞ principle 2- H ∞ Standard : [Doyle and Glover. 1989] z S W1 z1 z2 C.S W2 W3 d y C(s) G(s) r  u + - + + b Specifications: - Settling time/bandwidth - Static error/static gain - No (or small) overshoot - Control moderation - Disturbances rejection - Stability 3- A posteriori performances analysis

15. Événement - date MEA’10, December 09 th, 2010 15/30 Choice of the Weighting functions bandwidth (min) Overshoot (max) static error (max) 3- A posteriori performances analysis Weighting W 1 ( s )

16. Événement - date MEA’10, December 09 th, 2010 16/30 Choice of the Weighting functions 3- A posteriori performances analysis Weighting W 2 ( s ) Weighting W 3 ( s )

17. Événement - date MEA’10, December 09 th, 2010 17/30 F l (P(s),C(s)) H ∞ standard problem: Compute the controller C ( s ) such as: ( ) - Stability - 3- A posteriori performances analysis W3 d y C(s) G(s) r  u + - + + b S W1 C.S W2 z z1z1 z2z2 P(s): augmented system C(s)C(s) P(s)P(s)  u H ∞ standard formalism

18. Événement - date MEA’10, December 09 th, 2010 18/30 Interval closed-loop performances analysis via H ∞ -standard approach Specifications: - Settling time tr ≤  - Static error    ≤  - No (or small) overshoot Given a controller C ( s ), If the controller satisfies: then, the required specifications are ensured defined in terms of weighting functions 3- A posteriori performances analysis y e C(s)C(s) [G](s,[a],[b]) + -  u z W1(s)W1(s)

19. Événement - date MEA’10, December 09 th, 2010 19/30 H ∞ norm of the sensitivity of an interval system Given an interval system, the maximal H ∞ norm of its sensitivity function is achieved at twelve (out of sixteen) Kharitonov vertices. [Long Wang, 2002] Let consider a sensitivity function of an open-loop interval system : 3- A posteriori performances analysis The four Kharitonov polynomials corresponding to and :

20. Événement - date MEA’10, December 09 th, 2010 20/30 With: The maximal H ∞ norm of the sensitivity function is defined by: 3- A posteriori performances analysis

21. Événement - date MEA’10, December 09 th, 2010 21/30 Introduction Robust control approaches for (parametric) uncertain systems A posteriori performances analysis Example with the control of piezoelectric cantilevers Conclusion

22. Événement - date MEA’10, December 09 th, 2010 22/30 4- Example with the control of piezoelectric cantilevers Fig. 1: a) Manipulation of micro-object; b) Piezocantilever subjected to an electrical excitation (a) (b)

23. Événement - date MEA’10, December 09 th, 2010 23/30 Interval model 4- Example with the control of piezoelectric cantilevers Specifications: - Settling time tr ≤ 30ms - Static error    ≤ 1% - No overshoot Interval Control [Khadraoui, SWIM’10] Based on H ∞ standard, prove the robustness of the PI controller to ensure the required specification

24. Événement - date MEA’10, December 09 th, 2010 24/30 From the required specification: Robustness study of the controller C ( s ) 4- Example with the control of piezoelectric cantilevers After numerical application: H ∞ Norm of

25. Événement - date MEA’10, December 09 th, 2010 25/30 Using the method proposed in [Long Wang, 2002], we obtain: Specifications hold 4- Example with the control of piezoelectric cantilevers Robustness study of the controller C ( s ) H ∞ norm of the sensitivity function [S](s)

26. Événement - date MEA’10, December 09 th, 2010 26/30 The controller has played its role and the required specifications are satisfied Fig. 1: magnitudes of the sensitivity function compared to the magnitude of the weighting function [ S ]( s ) = (1+C(s).[G](s,[a],[b])) -1 W1(s)W1(s) 1 4- Example with the control of piezoelectric cantilevers

27. Événement - date MEA’10, December 09 th, 2010 27/30 Fig. 1: Experimental results using PI controller for two piezocantilevers compared with the desired behaviors 4- Example with the control of piezoelectric cantilevers

28. Événement - date MEA’10, December 09 th, 2010 28/30 Introduction Robust control approaches for (parametric) uncertain systems A posteriori performances analysis Example with the control of piezoelectric cantilevers Conclusion

29. Événement - date MEA’10, December 09 th, 2010 29/30  Experimental results with piezocantilevers prove the efficiency of the proposed method,  H ∞ standard approach confirms the robustness of the computed controller,  Performances analysis for a controlled system with uncertainties,  The interval analysis represents a guaranteed tools for modeling and control system with uncertainties. 5- Conclusion

30. Événement - date MEA’10, December 09 th, 2010 30/30 Thank You For Your Attention