Control of systems with hysteresis. Martin Brokate Zentrum Mathematik, TU München

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1 Control of systems with hysteresis Martin Brokate Zentrum Mathematik, TU München Elgersburger Arbeitstagung, Februar 2008

2 Ferromagnetic hysteresis magnetization magnetic field

3 Minor hysteresis loops M H Madelung (1905): Rules for minor loops

4 Elastoplasticity load deformation Berliner (1906): Minor loops Masing (1926): Masing s law

5 Rate independence Input-output system Inputs Outputs Rate-independent processes viewed as operators between function spaces: M.A.Krasnoselskii et al. (1970)

6 Hysteresis operator Input-output system Inputs Outputs The operator is called a hysteresis operator, if it is rate independent, and it has the Volterra property, that is, depends only upon

7 Switch with hysteresis w a b v thermostat

8 Play w r v also called backlash

9 Stop w r v

10 Prandtl s elastoplastic model w v Prandtl (1928)

11 Piezoelectric actuators w v Prandtl s model is used to describe the coupling between electrical and mechanical variables (voltage and current, displacement and force) K. Kuhnen: Kompensation komplexer gedächtnisbehafteter Nichtlinearitäten in Systemen mit aktiven Materialien, Habilitation Thesis, U Saarbrücken 2007

12 The Preisach model w Preisach (1935) v

13 Preisach memory System-theoretic interpretation Nested hysteresis loops

14 Systems with hysteretic elements Bulk models, typical for control settings: Actuators and sensors with hysteretic characteristics piezoelectric, magnetostrictive shape memory materials ( smart materials ) thermostats Backlash (gears), some types of friction Input-state description as ODE coupled to hysteresis operator Typically scalar hysteresis models

15 Systems with hysteretic elements Continuum mechanics, electrodynamics, materials science Hysteresis typically appears in the constitutive law PDE coupled with a hysteresis operator A. Visintin: Differential models of hysteresis, Springer P. Krejci: Hysteresis, convexity and dissipation in hyperbolic equations, Gakkotosho Typically vector (or tensor) hysteresis models

16 Multiscales and Hysteresis Homogenization: simple hysteresis models on microscale lead to complex hysteresis models on macroscale Dimensional reduction via Averaging or Gamma convergence: simple hysteresis models in full dimension lead to complex hysteresis models in reduced dimension

17 Scalar Hysteresis: Piecewise monotone inputs Equivalence of time-continuous and time-discrete description Example: Play initial state can be identified with some

18 Scalar Hysteresis: Piecewise monotone inputs Equivalence of time-continuous and time-discrete description Due to rate independence: monotonicity partition pw linear interpolate

19 Scalar Hysteresis: Piecewise monotone inputs continuous, pw monotone functions Discontinuous inputs: no modification necessary right and left limits have to be considered H. Logemann, A.D. Mawby: Extending hysteresis operators to spaces of piecewise continuous functions, J.Math.Anal.Appl. 282 (2003),

20 Scalar Hysteresis: Piecewise monotone inputs Discontinuous inputs: H. Logemann, A.D. Mawby: Extending hysteresis operators to spaces of piecewise continuous functions, J.Math.Anal.Appl. 282 (2003), Extensions:

21 Scalar Hysteresis: Piecewise monotone inputs Loop nesting, composition, inverse, implicit models I.D. Mayergoyz: Mathematical models of hysteresis, Springer P. Krejci: Hysteresis, convexity and dissipation in hyperbolic equations, Gakkotosho M. Brokate, J. Sprekels: Hysteresis and phase transitions, Springer 1996.

22 w r v Formulation as discontinuous differential equation

23 w r v Formulation as evolution variational inequality:

24 w r v Formulation as differential inclusion:

25 Vector stop and play

26 0

27 Analysis of hysteretic dynamics Properties of hysteresis operators: Continuous in various function spaces (except relay) Best possible: Lipschitz continuous Not smooth. Directional derivatives are possible. Usually leads to wellposedness of ODEs with hysteresis

28 Analysis of hysteretic dynamics Stability, asymptotic stability, bifurcation, resonance, hyperbolicity Focus of investigation of the group of M.A.Krasnoselskii Extensive bibliography in: The Science of Hysteresis, eds. G. Bertotti, I. Mayergoyz (2006) Vol. 1, Chapter 2 (Brokate, Pokrovskii, Rachinskii, Rasskazov)

29 Analysis of hysteretic dynamics PDEs with hysteresis: A. Visintin: Differential models of hysteresis, Springer A. Visintin: vol. 1, chapter 1 of The science of hysteresis P. Krejci: Hysteresis, convexity and dissipation in hyperbolic equations, Gakkotosho Wellposedness based on arguments from convex analysis and hysteresis-specific inequalities Often, open problem.

30 Optimal control: Pontryagin principle Hysteresis operator has the form

31 Optimal control: Pontryagin principle Optimality system:

32 Optimal control: Pontryagin principle Optimality system: M. Brokate, Optimale Steuerung von gewöhnlichen Differentialgleichungen mit Nichtlinearitäten vom Hysteresis-Typ, Verlag Peter Lang, 1987.

33 Optimal control: Pontryagin principle M. Brokate: ODE control problems including the Preisach operator: Necessary optimality conditions. In: Dynamic Economic Models and Optimal Control, Elsevier 1992, S.A. Belbas: Control of systems with hysteresis. In: The Science of Hysteresis, vol. I, Academic Press 2006,

34 Optimal control: HJB and viscosity solutions F. Bagagiolo: Dynamic programming for some optimal control problems with hysteresis, Nonlinear Diff. Eq. Appl. 9 (2002), Hysteresis operator: Play Optimal value function HJB equation

35 Optimal control: HJB and viscosity solutions HJB equation Discontinuous Hamiltonian:

36 Optimal control: HJB and viscosity solutions Alternative form of HJB: Boundary value problem

37 Piezoelectric actuators w v Prandtl s model is used to describe the coupling between electrical and mechanical variables (voltage and current, displacement and force) K. Kuhnen: Kompensation komplexer gedächtnisbehafteter Nichtlinearitäten in Systemen mit aktiven Materialien, Habilitation Thesis, U Saarbrücken 2007 P. Krejci, K. Kuhnen: Inverse control of systems with hysteresis and creep, IEEE Proc. Control Theory and Appl. 148 (2001),

38 Identification of Hysteresis Nonlinearity Discrete approximation

39 Identification of Hysteresis Nonlinearity Discrete approximation Output least squares: Minimize plus linear inequality constraints Needs some excitation property.

40 Compensation of Hysteresis Nonlinearities

41 Compensation of Hysteresis Nonlinearities Prandtl hysteresis operator Inverse Corresponding formula for discretized Prandtl operator

42 Compensation of Hysteresis Nonlinearities Identification of W yields identification of the inverse! Adaptive identification: (projection on boundary) Adaptation of play parameters

43 Compensation of Hysteresis Nonlinearity Preisach operator: Similar discretization and identification procedure. No formulaforinverseisknown. Numerical inversion is necessary. K. Kuhnen X. Tan, J.S. Baras: Modeling and control of hysteresis in magnetostrictive actuators, Automatica 40 (2004), X. Tan, J.S. Baras: Adaptive identification and control of hysteresis in smart materials, IEEE Trans. Automatic Control 50 (2005),

44 Hysteresis in Feedback Loop H. Logemann, E.P. Ryan: Systems with hysteresis in the feedback loop: Existence, regularity and asymptotic behaviour of solutions, ESAIM: COCV 9 (2003),

45 Hysteresis in Feedback Loop

46 Sensitivity estimates Basic result (due to monotonicity), valid for all closed convex sets Z Refined estimates: (Whether they hold, depends on the geometry of the convex set)