Abstract
Hysteresis in smart materials hinders their wider applicability in actuators. The low dimensional hysteresis models for these materials are hybrid systems with both controlled switching and autonomous switching. In particular, they belong to the class of Duhem hysteresis models and can be formulated as systems with both continuous and switching controls. In this paper, we study the control methodology for smart actuators through the example of controlling a commercially available magnetostrictive actuator. For illustrative purposes, an infinite horizon optimal control problem is considered. We show that the value function satisfies a Hamilton-Jacobi-Bellman equation (HJB) of a hybrid form in the viscosity sense. We further prove uniqueness of the viscosity solution to the (HJB), and provide a numerical scheme to approximate the solution together with a sub-optimal controller synthesis method. Numerical and experimental results based on this approach are presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Gorbet, R.B., Wang, D.W.L., Morris, K.A.: Preisach model identification of a two-wire SMA actuator. In: Proceedings of IEEE International Conference on Robotics and Automation. (1998) 2161–2167
Hughes, D., Wen, J.T.: Preisach modeling and compensation for smart material hysteresis. In: Active Materials and Smart Structures. Volume 2427 of SPIE. (1994) 50–64
Tan, X., Venkataraman, R., Krishnaprasad, P.S.: Control of hysteresis: theory and experimental results. In: Smart Structures and Materials 2001, Modeling,Signal Processing, and Control in Smart Structures. Volume 4326 of SPIE. (2001) 101–112
Jiles, D.C., Atherton, D.L.: Theory of ferromagnetic hysteresis. Journal of Magnetism and Magnetic Materials 61 (1986) 48–60
Venkataraman, R.: Modeling and Adaptive Control of Magnetostrictive Actuators. PhD thesis, University of Maryland, College Park (1999)
Venkataraman, R., Krishnaprasad, P.S.: Qualitative analysis of a bulk ferromagnetic hysteresis model. In: Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, Florida. (1998) 2443–2448
Smith, R.C., Hom, C.L.: A domain wall theory for ferroelectric hysteresis. Technical Report CRSC-TR99-01, CRSC, North Carolina State University (1999)
Witsenhausen, H.S.: A class of hybrid-state continuous-time dynamic systems. IEEE Transactions on Automatic Control 11 (1966) 161–167
Piccoli, B.: Hybrid systems and optimal control. In: Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, Florida (1998) 13–18
Riedinger, P., Kratz, F., Iung, C., Zanne, C.: Linear quadratic optimization for hybrid systems. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, Arizona (1999) 3059–3064
Hedlund, S., Rantzer, A.: Optimal control of hybrid systems. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, Arizona (1999) 3972–3977
Rantzer, A., Johansson, M.: Piecewise linear quadratic optimal control. IEEE Transactions on Automatic Control 45 (2000) 629–637
Yong, J.: Systems governed by ordinary differential equations with continuous, switching and impulse controls. AppliedMathematics and Optimization 20 (1989) 223–235
Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Transactions on Automatic Control 43 (1998) 31–45
Crandall, M.G., Lions, P.L.: Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society 277 (1983) 1–42
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997)
Tan, X., Baras, J.S.: Control of smart actuators: A viscosity solutions approach. Technical Report TR 2001-39, Institute for Systems Research,University of Maryland, College Park (2001) Available at http://www.isr.umd.edu/TechReports/ISR/2001/TR 2001-39/TR2001-39.phtml.
Venkataraman, R., Krishnaprasad, P.S.: A model for a thin magnetostrictive actuator. In: Proceedings of the 32nd Conference on Information Sciences and Systems, Princeton, NJ, Princeton (1998)
Branicky, M.S.: Studies in hybrid systems: modeling, analysis, and control. PhD thesis, MIT, Cambridge (1995)
Visintin, A.: Differential Models of Hysteresis. Springer (1994)
Crandall, M.G., Evans, L.C., Lions, P.L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Transactions of the American Mathematical Society vn282 (1984) 487–502
Ishii, H.: Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations. Indiana University Mathematics Journal 33 (1984) 721–748
Soner, H.M.: Optimal control with state-space constraint. SIAM Journal on Control and Optimization 24 (1986) 552–561
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tan, X., Baras, J.S. (2002). Optimal Control of Hysteresis in Smart Actuators: A Viscosity Solutions Approach. In: Tomlin, C.J., Greenstreet, M.R. (eds) Hybrid Systems: Computation and Control. HSCC 2002. Lecture Notes in Computer Science, vol 2289. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45873-5_35
Download citation
DOI: https://doi.org/10.1007/3-540-45873-5_35
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43321-7
Online ISBN: 978-3-540-45873-9
eBook Packages: Springer Book Archive