Abstract
Practical applications are often affected by uncertainties—more precisely bounded and stochastic disturbances. These have to be considered in robust control procedures to prevent a system from being unstable. Common sliding mode control strategies are often not able to cope with the mentioned impacts simultaneously, because they assume that the considered system is only affected by matched uncertainty. Another problem is the offline computation of the switching amplitude. Under these assumptions, important nonlinear system properties cannot be taken into account within the mathematical model of the system. Therefore, this paper presents sliding mode techniques, that on the one hand are able to consider bounded as well as stochastic uncertainties simultaneously, and on the other hand are not limited to the matched case. Firstly, a sliding mode control procedure taking into account both classes of uncertainty is shown. Additionally, a sliding mode observer for the simultaneous estimation of non-measurable system states and uncertain but bounded parameters is described despite stochastic disturbances. This is possible by using intervals for states and parameters in the resulting stochastic differential equations. Therefore, the Itô differential operator is involved and the system’s stability can be verified despite uncertainties and disturbances for both control and observer procedures. This operator is used for the online computation of the variable structure part gain (matrix of switching amplitudes) which is advantageous in contrast to common sliding mode procedures.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Åström, K.J.: Introduction to stochastic control theory. In: Mathematics in Science and Engineering. Academic Press, London (1970)
Balakrishnan V., Vandenberghe L.: Semidefinite programming duality and linear time-invariant systems. IEEE Trans. Autom. Control 48(1), 30–41 (2003)
Barmish B.R.: New Tools for Robustness of Linear Systems. Macmillan, New York (1994)
Bartoszewicz, A., Nowacka-Leverton, A.: Time-Varying Sliding Modes for Second and Third Order Systems. In: Lecture Notes in Control and Information Sciences, vol. 382. Springer-Verlag, New York (2009)
Dötschel, Th., Rauh, A., Aschemann, H.: Reliable Control and Disturbance Rejection for the Thermal Behavior of Solid Oxide Fuel Cell Systems. In: Paper presented at MATHMOD 2012, Vienna (2012)
Engell, S.: Entwurf nichtlinearer Regelungen. R. Oldenbourg Verlag, München (in German) (1995)
Fridman, L., Moreno, J., Iriarte, R.: Sliding Modes After the First Decade of the 21st Century: State of the Art. In: Lecture Notes in Control and Information Sciences, vol. 412 (2011)
Jaulin L., Kieffer M., Didrit O., Walter É.: Applied Interval Analysis. Springer-Verlag, London (2001)
Kushner H.: Stochastic Stability and Control. Academic Press, New York (1967)
Löfberg, J.: YALMIP: A Toolbox for Modeling and Optimization in MATLAB. In: Proceedings of IEEE International Symposium on Computer Aided Control Systems Design, pp. 284–289, Taipei (2004)
Rauh, A., Aschemann, H.: Interval-based sliding mode control and state estimation for uncertain systems. In Proceedings of IEEE International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje (2012)
Rauh, A., Dittrich, Ch., Aschemann, H., Nedialkov, N.S., Pryce, J.D.: A differential-algebraic approach for robust control design and disturbance compensation of finite-dimensional models of heat transfer processes. In: Proceedings of the IEEE International Conference on Mechatronics ICM 2013, Vicenza (2013)
Rauh, A., Gebhardt, J., Aschemann, H.: Guaranteed stabilizing control strategies for boom cranes in marine applications. In: Proceedings of the 2nd International Conference on Control and Fault-Tolerant Systems, SysTol’13, Nice (2013)
Rump, S.M.: INTLAB—INTerval LABoratory. In: Csendes, T. (Ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht. http://www.ti3.tu-harburg.de/rump/intlab/ (1999). Accessed July 2014
Senkel, L., Rauh, A., Aschemann, H.: Optimal Input Design for Online State and Parameter Estimation using Interval Sliding Mode Observers. In Proceedings of 52nd IEEE Conference on Decision and Control, CDC, Florence (2013)
Senkel, L., Rauh, A., Aschemann, H.:Interval-based sliding mode observer design for nonlinear systems with bounded measurement and parameter uncertainty. In: Proceedings of the IEEE International Conference on Methods and Models in Automation and Robotics, MMAR 2013, Miedzyzdroje (2013)
Senkel, L., Rauh, A., Aschemann, H.: Robust sliding mode techniques for control and state estimation of dynamic systems with bounded and stochastic uncertainty. In: Second International Conference on Vulnerability and Risk Analysis and Managemet 2014, ICVRAM 2014 (to appear). (2014)
Shtessel, Y., Edwards, Ch., Fridman, L., Levant, A. (eds). Sliding Mode Control and Observation, 1st edn. Springer Series, Birkhäuser, Basel (2014)
Sturm J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cone. Optim. Methods Softw. 11–12(1–4), 625–653 (1999)
Universität Karlsruhe (Institut für Angewandte Mathematik, Forschungsschwerpunkt CAVN), Universität Wuppertal (Wissenschaftliches Rechnen/Softwaretechnologie): C-XSC—a C++ class library. http://www2.math.uni-wuppertal.de/~xsc/xsc/cxsc.html (2007). Accessed July 2014
Utkin V.I.: Sliding Modes in Control and Optimization. Springer-Verlag, Berlin (1992)
Utkin V.I.: Sliding Mode Control Design Principles and Applications to Electric Drives. IEEE Trans. Ind. Electron. 40(1), 23–36 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Senkel, L., Rauh, A. & Aschemann, H. Sliding Mode Techniques for Robust Trajectory Tracking as well as State and Parameter Estimation. Math.Comput.Sci. 8, 543–561 (2014). https://doi.org/10.1007/s11786-014-0208-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-014-0208-7
Keywords
- Sliding mode control
- Sliding mode observer
- Parameter estimation
- Bounded and stochastic disturbances
- Uncertainties
- Interval arithmetics