1. 1 A Dynamic Model for Magnetostrictive Hysteresis Xiaobo Tan, John S. Baras, and P. S. Krishnaprasad Institute for Systems Research and Department of Electrical and Computer Engineering University of Maryland, College Park, MD 20742 2003 American Control Conference June 4, 2003

2. 2 Outline  Introduction  The Preisach Operator  A Dynamic Model for Magnetostrictive Hysteresis  Analysis of the Dynamic Model  Well-posedness  System-theoretic properties  Existence of periodic solutions  Numerical integration methods  Conclusions

3. 3  Magnetostriction: coupling between the magnetic properties and the mechanical properties. Magnetostrictive Actuators Magnetostrictive Actuators A Terfenol-D actuator manufactured by Etrema Inc.

4. 4 Rate-dependent magnetostrictive hysteresis.

5. 5 The Preisach Operator The Preisach Operator An elementary Preisach hysteron. The Preisach operator.

6. 6 The Preisach Operator Memory curve in the Preisach plane   P-(t)P-(t) P+(t)P+(t) == 00 00 The Preisach operator is  Rate-independent;  Piecewise monotone increasing if the Preisach measure is nonnegative.

7. 7 Model structure of a magnetostrictive actuator (Venkataraman, 1999). The Dynamic Hysteresis Model The Dynamic Hysteresis Model W (·) 2 G(s) I MM2M2 y (Tan & Baras, CDC 2002)

8. 8 Model validation. Solid line: experimental measurement; dashed line: numerical prediction based on the dynamic model (Tan & Baras, CDC 2002).

9. 9 Well-posedness of the Model Well-posedness of the Model Theorem: If the Preisach measure is nonnegative and nonsingular, and I(·) is piecewise continuous, then for any initial condition  0, for any T > 0, there exists a unique pair {H (·), M (·)}  C([0,T])  C([0,T]) satisfying the above equation almost everywhere. The solution depends continuously on the initial conditions and the parameters. Proof. Using the Euler polygon method.

10. 10 An Alternative Perspective An Alternative Perspective

11. 11 Stability of Equilibria Stability of Equilibria Proposition: If the Preisach measure is nonnegative, and nonsingular with a piecewise continuous density. Then every equilibrium is stable but not asymptotically stable.

12. 12 Other System-Theoretic Properties Other System-Theoretic Properties We have studied  Input-output stability;  Reachability and approximate reachability;  Observability.

13. 13 Existence of Periodic Solutions Existence of Periodic Solutions Theorem: If the Preisach measure is nonnegative and nonsingular, then for any T-periodic I(·), there exists an initial condition ψ 0, such that the solution is also T- periodic. Proof. Define a map Ξ T on the space of memory curves. Then show that Ξ T has a fixed point using the Schauder fixed point theorem.

14. 14 Numerical Integration Methods Numerical Integration Methods

15. 15 Comparison of the implicit Euler scheme and the explicit Euler scheme.

16. 16 Conclusions Conclusions In this talk, we have analyzed a special dynamic hysteresis model:  Well-posedness  System-theoretic properties  Existence of periodic solutions  Numerical simulation schemes

17. 17 Reachability and Observability Reachability and Observability Proposition: If the Preisach measure is nonnegative and nonsingular, the state space for ($) is not reachable, but approximately reachable. Proposition: If the Preisach measure is nonnegative and nonsingular, the system ($) is observable if and only if the Preisach measure of any connected set of nonzero Lebesgue measure is positive.

18. 18 Input-Output Stability Input-Output Stability