1. Harvard - Boston University - University of Maryland Magnetostrictive Models R. Venkataraman, P. S. Krishnaprasad Low dimensional models Presentation to Dr. Randy Zachery, ARO May 25, 2004, Harvard University

2. Harvard - Boston University - University of Maryland Magnetostrictive models Experimental data from actuators coupled PDE’s representing magnetic and mechanical dynamic equilibrium. material properties appear through shape of potential functions. eddy-current losses modeled via Maxwell’s equations. coupled ODE’s or integro- differential equations representing magnetic and mechanical equilibrium. material properties appear through constitutive equations. eddy-current losses modeled via a resistance. validation model validation model simulation Low-dimensional models Micromagnetic model

3. Harvard - Boston University - University of Maryland Derivation of the bulk magnetostriction model. Parameter estimation algorithm. Validation of the model. Discussion of results. Current and future directions. Organization of the talk

4. Harvard - Boston University - University of Maryland Low Dimensional Magnetostrictive Models W.F. Brown derived expressions for work done by a battery in changing the magnetization of a magneto-elastic body. The body was considered to be a continuum. Jiles and Atherton postulated expressions for magnetic hysteresis losses in a ferromagnet. This lead to an ODE with 5 parameters for the evolution of the average magnetization in a thin ferromagnetic rod. Sablik and Jiles extended this result to a quasi-static magnetostriction model. Hom, Shankar et al. have a model for electrostriction that includes inertial effects. But hysteresis was not modeled. Our Work Our model takes account of ferromagnetic hysteresis, magnetostriction, inertial effects, mechanical damping and eddy-current effects. It is low-dimensional with 4 continuous states and 12 parameters. We proposed parameter estimation algorithms that are easy to implement. We have experimentally verified the structure of our model. Current work involves inverting the hysteresis nonlinearity and design of a robust controller. Background

5. Harvard - Boston University - University of Maryland Derivation of the bulk magnetostriction model Langevin’s theory of Paramagnetism Consider a collection of N atomic magnetic moments under the influence of an external magnetic field. Then the average magnetic moment of the ensemble is given by: Weiss’ theory of Ferromagnetism Weiss postulated that an additional “molecular field” experienced by an individual moment in an ideal ferromagnet, where is the average magnetic moment of the ferromagnet. Suppose an external field is applied in the direction of. Then the magnitude of is given by: Weiss considered an ideal ferromagnet without losses. In particular, the curve in the plane is anhysteretic.

6. Harvard - Boston University - University of Maryland Derivation of the bulk magnetostriction model The anhysteretic magnetization curve

7. Harvard - Boston University - University of Maryland Derivation of the bulk magnetostriction model Jiles and Atherton’s assumptions for a lossy ferromagnet The average magnetization is composed of reversible and irreversible components: The losses during a magnetization process occur due to the change in the irreversible component: where and are constants with and. The reversible and irreversible magnetizations are related to the anhysteretic magnetization as:. Further and if Principle of conservation of energy Change in external input Change in internal energylosses Change in kinetic energy

8. Harvard - Boston University - University of Maryland Derivation of the bulk magnetostriction model W. F. Brown’s expression for work done by the battery where F is the external force, x is the displacement of the tip of the actuator, H is the average external magnetic field, and M is the average magnetization in the actuator. Adding the integral of any perfect differential over a cycle does not change the value on the left hand side. Magnetoelastic energy density (following Landau) = Elastic energy = ; Kinetic energy = Expressions for some of the energy terms :

9. Harvard - Boston University - University of Maryland The bulk magnetostriction model The model equations Magnetic dynamic equilibrium equations Mechanical dynamic equilibrium equation otherwise and

10. Harvard - Boston University - University of Maryland The bulk magnetostriction model Schematic diagram of the bulk magnetostriction model with eddy current effects included Eddy currents losses are modeled by a resistor in parallel Voltage source displacement output Mechanical system transfer function Rate-independent hysteresis operator

11. Harvard - Boston University - University of Maryland The bulk magnetostriction model Sufficient condition on parameters Analytical result Theorem : Consider the system of equations (1 - 6). Suppose the matrix A = has eigenvalues with negative real parts and the parameters satisfy conditions (7 - 9). Suppose the input is given by and the initial state is at the origin. Then there exists a such that if then the limit set of the solution trajectory is a periodic orbit. where

12. Harvard - Boston University - University of Maryland Parameter Estimation The parameters to be found are : - electrical circuit parameter (includes lead resistance of the magnetizing coil). - eddy current parameter. - magnetic parameters not pertaining to hysteresis. - magnetic hysteresis pertaining to hysteresis. - mechanical dynamic losses parameter. - inertia parameter. - elasticity parameter. - prestress parameter. Three step algorithm for parameter identification Step 1 : Apply a sinusoidal current input of a very low frequency (0.5 Hz) and measure the voltage and displacement of the actuator as a function of time. This leads to the identification of. Repeat the same experiment, for higher frequencies (200Hz, 350 Hz, 500Hz). This leads to identification of. Step 2 : Obtain the anhysteretic displacement curve of the actuator. This leads to the identification of. Step 3 : Apply a swept sine wave current signal to the actuator and record the displacement versus the frequency. This leads to the identification of.

13. Harvard - Boston University - University of Maryland Parameter Estimation Result of step 2 : Input current waveform Output displacement versus current Result of parameter estimation Parameter Value (in CGS units)

14. Harvard - Boston University - University of Maryland Experiment versus Simulation 500 Hz Amps -1.5 1.5 200 Hz Amps -1.51.5 45 microns 50 microns 100 Hz Amps -1.5 1.5 60 microns 50 Hz Amps -1.5 1.5 80 microns 480 Hz Amps 45 microns -1.51.5 100 Hz Amps 120 microns -1.5 1.5 240 Hz Amps 100 microns -1.5 1.5 50 Hz Amps 80 microns -1.51.5 Frequency(Hz) Peak-Peak current (A) Peak-Peak displacement ( m) 0.25 2.5 53 1 2.5 53 10 2.5 54 50 2.5 54 100 2.5 51 200 2.5 58 350 2.5 66 500 2.3 44 Frequency(Hz) Peak-Peak current (A) Peak-Peak displacement ( m) 1 2.13 71 10 2.26 71 50 2.17 63 100 2.22 54 150 2.19 50 200 2.28 42 350 2.11 54 500 2.39 45

15. Harvard - Boston University - University of Maryland Validation of the structure of the model Original goal : Trajectory tracking by means of an non-identifier based adaptive controller. Why adaptive control? Reasons : (1) Transient effects are unmodeled in our model. (2) System parameters may change with time due to heating heating etc. Basic idea of universal adaptive stabilization : are unknown and Suppose and Then Thereforeis monotonically increasing as decays exponentially andHence for Hence long as

16. Harvard - Boston University - University of Maryland Validation of the structure of the model Universal Adaptive Stabilization result for relative degree one systems. Consider a class of nonlinearly-perturbed, single input, single output, linear systems with nonlinear actuator characteristics : Assumptions: (2) The linear system is minimum phase.is a Caratheodory function and has the (3) (1) for almost alland all property that for some scalar (4) and has the property that, for some scalarand knowncontinuous function is a Caratheodory for almost all and all (5) There exists a map condition below and such that every actuator characteristic is contained in the graph of in the following sense: and every with where satisfying the denotes the restriction of to is a continuous map from to compact intervals ofwith the property that, for some scalars and condition below and such that

17. Harvard - Boston University - University of Maryland Validation of the structure of the model Universal Adaptive Stabilization (contd.) The classof reference signals is the Sobolev space with norm Adaptive Strategy (assuming ) : Theorem (Ryan) : Let be a maximal solution of the initial value problem. 1. 2. 3. 4. is bounded. exists and is finite. Then

18. Harvard - Boston University - University of Maryland Validation of the structure of the model Simulation example for Morse-Ryan controller Reference and output trajectories Input non-linearity Gain evolution Morse-Ryan controller design for relative degree 2 systems

19. Harvard - Boston University - University of Maryland Validation of the structure of the model Experimental Setup :

20. Harvard - Boston University - University of Maryland Validation of the structure of the model Result of trajectory-tracking experiment Reference (sinusoids) vs. actual displacements Control current seconds 0 0.05 amps seconds 0 10 amps seconds00.05 microns seconds00.5 microns seconds00.4 microns seconds010 microns 1 Hz seconds 0 0.4 amps 50 Hz seconds 0 0.1 amps 200 Hz 500 Hz

21. Harvard - Boston University - University of Maryland Discussion of Results 1. We derived a low dimensional model for a thin magnetostrictive actuator that is phenomenology based and models the magneto-elastic effect; ferromagnetic hysteresis; inertial effects; eddy current effects; and losses due to mechanical motion. The model has 12 parameters, 4 continuous states and can be thought to be composed of magnetic and mechanical sub-systems that are coupled. 2. Analytically we showed that for initial conditions at the origin and periodic inputs, the system equations have a unique solution trajectory that is asympotically periodic. This models experimentally observed phenomena. 3. We have proposed a simple parameter estimation algorithm and estimated the parameters for a commercially available actuator. Simulation results show trajectories that are comparable to the actual. 4. We have also validated the structure of the model by designing a trajectory tracking control-law for relative degree 2 systems with input non-linearity. The closed loop system remained stable for all frequencies from 0 to 750 Hz, thus showing that our model structure is correct. There are some differences in the size of the peak-peak displacement predicted by the simulation and actual results. In particular, the predicted peak-peak displacement is larger than the actual for low frequencies while it is smaller than the actual for high frequencies. This can be explained by an eddy-current resistance value that is slightly smaller than the estimated value.

22. Harvard - Boston University - University of Maryland Current Work and Future Directions Drawback of the present model : The rate-independent nonlinearity is defined only for periodic signals. Therefore it is unsuitable for the development of a controller. Solution : Replace the rate-independent nonlinearity by a moving Preisach operator, that is defined as follows : where for continuous inputs is defined as where is a partition such that is monotonic in each sub-interval. for Facts about the Preisach operator: 1. The Preisach operator is Lipschitz continuous and its definition can be extended to the space of functions over the real line that are bounded with integrable derivatives over compact intervals. 2. The Preisach operator is rate-independent and models properties that are observed in bulk ferromagnetic hysteresis like minor-loop closure and saturation. 3. It is invertible in the space of functions defined in 1, under some mild conditions on the measure

23. Harvard - Boston University - University of Maryland Current Work and Future Directions 1. We are currently working on an algorithm for the inversion of the Preisach operator, so that we can approximately linearize the rate-independent nonlinearity. 2. Once this is achieved, we can utilize methods from robust control of linear systems for controller design. Current Work Future Directions While designing complex magnetostrictive systems, one can obtain low dimensional models from the numerical results obtained from PDE model. This will enable us to short circuit the implementation step and design controllers without actual experimental data.