Exponential Mapping Function for Nonlinear Control*
Hildebrando DE CASTRO*, Pedro PAGLIONE, Carlos RIBEIRO
Aeronautics Technological Institute,
Praça Marechal Eduardo Gomes, 50, São José dos Campos, 12.228-900, Brazil
email@example.com; firstname.lastname@example.org; email@example.com
* Corresponding author
Abstract: Industrial control engineers, who are often faced with the problem of dealing with projects involving unknown or poorly modelled systems, usually have at their disposal a limited number of options to develop, implement, and maintain controllers, namely PID and, lately, fuzzy-based controllers. The large use of the PID controller comes from the widespread knowledge of its theory and easy tuning methods and its prompt availability in control equipment and supervisory systems. Nevertheless, depending on the manufacturer’s discretion, at least five gains have to be set. Fuzzy controllers for a SISO system usually demand between fifteen and twenty parameters. That’s also a problem for the maintenance personnel. The proposed SISO controller needs two heuristically defined gains to be set, based on knowledge about the free response of the plant and expected disturbances. Its simplicity allows its implementation on devices with memory and processor constraints.
Keywords: Nonlinear control, Sliding-Mode Control, Fuzzy Logic Control.
CITE THIS PAPER AS:
Hildebrando DE CASTRO, Pedro PAGLIONE, Carlos RIBEIRO, Exponential Mapping Function for Nonlinear Control, Studies in Informatics and Control, ISSN 1220-1766, vol. 24 (4), pp. 449-460, 2015. https://doi.org/10.24846/v24i4y201509
The fundamental control problem, according to  is “… to find a technically feasible way to act on a given system or process so that it adheres, as closely as possible, to some desired behaviour. This approximate behaviour should be achieved in the face of uncertainty of the process and in the presence of uncontrollable external disturbances acting on the process” (italics belong to the authors). Worth noting here is that control should exhibit or approximate some behaviour without a perfect model of the process, defined by uncertainty in the amount and value of parameters and degree of modeling abstraction. A perfect model, furthermore, is practically impossible to obtain and, when a model is used, one should have in mind the constraints and trade-offs that are taken into consideration in its design.
This paper is then organized as follows:
- A brief discussion about issues with model-based control;
- A way to streamline Sliding Mode Control (SMC);
- Prior works on how to dynamically tune an SMC;
- A brief explanation of how a Fuzzy Logic Controller (FLC) can be simplified;
- The EMC (Exponential Mapping Function) derivation and its use in control;
- A procedure showing how to implement it in a real system;
- An example of its implementation on a rigorously simulated system;
- CASTRO FILHO, H. F., P. Paglione, C. H. C. Ribeiro, Exponential Mapping Function for Control¸ XIX Congresso Brasileiro de Automática, Campina Grande, Paraíba, Brazil, 2012.
- GOODWIN, G. C.; S. F. GRAEBE, M. E. SALGADO, Control System Design¸ Upper Saddle River, New Jersey, Prentice Hall, 2001.
- SLOTINE, J. J., W. LI, Applied Nonlinear Control. New Jersey, Prentice Hall, 1991.
- UTKIN, V. I., Sliding Regimes and Their Applications in Variable Structure Systems. Moscow, 1978, MIR Publ.
- SONG, F., S. M. SMITH, Design of Sliding Mode Fuzzy Controllers for an Autonomous Underwater Vehicle without System Model. OCEAN 2000 MTS/IEEE Conf. and Exhibition, 2000.
- KOSKO, B., Fuzzy Systems as Universal Approximators. IEEE Transactions on Computers, vol. 43(11), Nov. 1994.
- IYANAGA, S., Y. KAWADA, Pontrjagin’s [sic] Maximum Principle. Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, §88C, 1980, pp. 295-296.
- CHOI, B.-J., S.-W. KWAK, B. K. KIM, Design and Stability Analysis of Single-Input Fuzzy Logic Controller. IEEE Transactions on Systems, Man, and Cybernetics 2000 – Part B, Cybernetics. vol. 30, no 2., pp 303-309.
- YAMAMOTO, S., I. HASHIMOTO, Present Status And Future Needs: The View From Japanese Industry. Chemical Process Controls CPC IV: Proceedings of the Fourth International Conference on Chemical Process Control, AIChE: New York: 1991. Arkun, Y., Ray, W. H., Eds.
- CHOI, S. B., D. W. PARK, Moving Sliding Surfaces for Fast Tracking Control of Second-order Dynamical systems. ASME Journal of Dynamic Systems, Measurement, and Control, vol. 116, 1994, pp. 154-158.
- TABATABAEI, E., A. GUEZ, H. CHOI, Adaptive Sigmoidal Molten Metal Pouring Control. IEEE Transactions on Control Systems Technology, 1998, Vol. 6, No. 2.
- HA, Q. P., D. C. RYE, H. F. DURRANT-WHYTE, Fuzzy Moving Sliding Mode Control with Application to Robotic Manipulators. Automatica, vol. 35, 1999, pp. 607-616.
- TOKAT, S., I. EKSIN, M. GUZELKAYA, Sliding Mode Control Using A Nonlinear Time-Varying Sliding Surface. Lisbon, Portugal: 2002. Proceedings of the 10th Mediterranean Conference on Control and Automation – MED2002.
- KIM, Y.-K., G. J. JEON, Error Reduction of Sliding Mode Control Using Sigmoid-Type Nonlinear Interpolation in the Boundary Layer. International Journal of Control, Automation, and Systems, 2004. Vol. 2, no. 4, pp 523-520.
- YAGIZ, N., Y. HACIOGLU, Fuzzy Sliding Modes with Moving Surface for the Robust Control of a Planar Robot. Journal of Vibration and Control, 2005. Vol. 11, No. 7, pp. 903-922.
- YORGANCIOĞLU, F. H. KÖMÜRCÜGIL, Single-input Fuzzy-like Moving Sliding Surface Approach to the Sliding Mode Control. Electrical Engineering, 2008, vol. 90(1-2), pp. 199-207.
- BARTOSZEWICS, A. A. NOWACKA-LEVERTON, Time-Varying Sliding Modes for Second and Third Order Systems. Lecture Notes in Control and Information Sciences, Springer-Verlag, Heidelberg, 2009.
- ZADEH, L., Fuzzy Sets. Information and Control, vol. 8, 1965, pp. 338-353.
- ZADEH, L., Fuzzy Logic and Approximate Reasoning. Synthese, vol. 30, 1975, pp. 407-428.
- JAGER, R. Fuzzy Logic in Control. PhD Thesis Technische Universiteit Delft. Netherlands, 1995.
- FRANKLIN, G. F., J. D. POWELL, A. EMAMI-NAEINI, Feedback Control of Dynamic Systems. Seventh Edition. Chapter 8. Prentice-Hall, 2014.
- MATHWORKS, Inc. Control of a Two-Tank System. Robust Control Toolbox. Available on http://www.mathworks.com/ help/robust/examples/control-of-a-two-tank-system.html. Accessed on Dec. 14, 2014.
- QUANSER, Inc. Quanser_Coupled_Tanks_System_Specificatons_Generic.pdf. Available on http://www.quanser.com/ Products/coupled_tanks. Accessed on November, 25th, 2014.
- ÅSTRÖM, K. J., B. WITTENMARK, Computer-Controlled Systems: Theory and Design. 3rd Ed., Dover Books, 2013.
* This paper is based on a previous presentation of the authors at the CBA 2012 (Brazilian Automation Congress) . The simulation on this paper, however, incorporates all the necessary electrical signals conversions and incorporates enhancements in the controller developed thenceforth.