Robust extremum seeking for a second order uncertain plant using a sliding mode controller

Solis, Cesar ; Clempner, Julio ; Poznyak, Alexander

International Journal of Applied Mathematics and Computer Science, Tome 29 (2019) no. 4, pp. 703-712.

Résumé

This paper suggests a novel continuous-time robust extremum seeking algorithm for an unknown convex function constrained by a dynamical plant with uncertainties. The main idea of the proposed method is to develop a robust closed-loop controller based on sliding modes where the sliding surface takes the trajectory around a zone of the optimal point. We assume that the output of the plant is given by the states and a measure of the function. We show the stability and zone-convergence of the proposed algorithm. In order to validate the proposed method, we present a numerical example.

Détail

Keywords: convex optimization, extremum seeking, continuous time gradient algorithm, dynamical constrained optimization, unknown function
Mots-clés : optymalizacja wypukła, poszukiwanie ekstremum, algorytm gradientowy, optymalizacja ograniczona

Bibliographie

[1] Alnejaili, T., Drid, S., Mehdi, D., Chrifi-Alaoui, L. and Sahraoui, H. (2015). Sliding mode control of a multi-source renewable power system, 3rd International Conference on Control Engineering Information Technology, Tlemcen, Algeria, pp. 1–6.

[2] Apkarian, P. and Tuan, H.D. (2000). Robust control via concave minimization local and global algorithms, Transactions on Automatic Control 45(2): 299–305.

[3] Armstrong, E.H. (1914). Operating features of the audion, Electrical World (December 12): 1149–1152.

[4] Bartoszewicz, A. and Leśniewski, P. (2014). An optimal sliding mode congestion controller for connection-oriented communication networks with lossy links, International Journal of Applied Mathematics and Computer Science 24(1): 87–97, DOI: 10.2478/amcs-2014-0007.

[5] Bazzi, A.M. and Krein, P.T. (2011). Concerning “Maximum power point tracking for photovoltaic optimization using ripple-based extremum seeking control”, IEEE Transactions on Power Electronics 26(6): 1611–1612.

[6] Belkaid, A., Colak, I. and Kayisli, K. (2016). Optimum control strategy based on an equivalent sliding mode for solar systems with battery storage, IEEE International Conference on Power Electronics and Motion Control (PEMC), Varna, Bulgaria, pp. 1262–1268.

[7] Cassandras, C.G. and Lin, X. (2013). Optimal control of multi-agent persistent monitoring systems with performance constraints, in D.C. Tarraf (Ed.), Control of Cyber-Physical Systems, Lecture Notes in Control and Information Sciences, Vol. 449, Springer, Cham, pp. 281–299.

[8] Davila, J. and Poznyak, A. (2010). Attracting ellipsoid method application to designing of sliding mode controllers, 11th International Workshop on Variable Structure Systems (VSS), Mexico City, Mexico, pp. 83–88.

[9] Dimitrova, N. and Krastanov, M. (2009). Nonlinear stabilizing control of an uncertain bioprocess model, International Journal of Applied Mathematics and Computer Science 19(3): 441–454, DOI: 10.2478/v10006-009-0036-0.

[10] Eichfelder, G., Krüger, C. and Schöbel, A. (2017). Decision uncertainty in multiobjective optimization, Journal of Global Optimization 69(2): 485–510.

[11] Ghadimi, S. and Lan, G. (2012). Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization. I: A generic algorithmic framework, SIAM Journal on Optimization 22(4): 1469–1492.

[12] Jignesh, D.J., Sripati, U. and Kulkarni, M. (2013). Performance of QPSK modulation for FSO geo-synchronous satellite communication link under atmospheric turbulence, International Conference Emerging Research Areas, Kanjirapally, India, pp. 1–5.

[13] Liu, X., Chen, X. and Kong, F. (2015). Utilization Control and Optimization of Real-Time Embedded Systems, https://ieeexplore.ieee.org/document/8187024.

[14] Liu, X., Hu, F. and Su, X. (2018). Sliding mode control of a class of nonlinear systems, 7th IEEE Conference on Data Driven Control and Learning Systems (DDCLS), Hubei, China, pp. 1069–1072.

[15] Mills, G. and Krstic, M. (2015). Maximizing higher derivatives of unknown maps with extremum seeking, 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan, pp. 5648–5653.

[16] Montesinos-García, J.J. and Martínez-Guerra, R. (2017). A fractional exponential polynomial state observer in secure communications, 14th International Conference on Electrical Engineering, Mexico, Mexico, pp. 1–6.

[17] Nana, S., Yugang, N. and Bei, C. (2012). Optimal integral sliding mode for uncertain discrete time systems, 31st Chinese Control Conference, Hefei, China, pp. 3155–3159.

[18] Perruquetti,W. and Barbot, J.P. (2002). Sliding Mode Control in Engineering, M. Dekker, New York, NY.

[19] Poznyak, A. (2018). Stochastic super-twist sliding mode controller, IEEE Transactions on Automatic Control 63(5): 1538–1544.

[20] qun Mei, W. (2013). Optimal control algorithm of multivariate second-order distributed parameter systems based on Fourier transform, 25th Chinese Control and Decision Conference (CCDC), Guiyang, China, pp. 4623–4627.

[21] Raju, B.V.S.S.N. and Rao, K.D. (2009). Blind robust multiuser detection in synchronous chaotic modulation systems, Annual IEEE India Conference, Gujarat, India, pp. 1–4.

[22] Sahneh, F.D., Hu, G. and Xie, L. (2012). Extremum seeking control for systems with time-varying extremum, 31st Chinese Control Conference, Hefei, China, pp. 225–231.

[23] Sarkar, M.K., Arkdev and Singh, S.S.K. (2017). Sliding mode control: A higher order and event triggered based approach for nonlinear uncertain systems, 8th Annual Industrial Automation and Electromechanical Engineering Conference (IEMECON), Bangkok, Thailand, pp. 208–211.

[24] Shi, P., Xia, Y., Liu, G. and Rees, D. (2006). On designing of sliding-mode control for stochastic jump systems, IEEE Transactions on Automatic Control 51(1): 97–103.

[25] Shtessel, Y., Edwards, C., Fridman, L. and Levant, A. (2014). Birkhäuser Basel, Springer Science+Business Media, New York, NY.

[26] Solis, C., Clempner, J.B. and Poznyak, A.S. (2019). Extremum seeking by a dynamic plant using mixed integral sliding mode controller with stochastic synchronous detection gradient estimation, International Journal of Robust and Nonlinear Control 29(3): 702–714, DOI: 10.1002/rnc.4408.

[27] Solis, C.U., Clempner, J.B. and Poznyak, A.S. (2018a). Constrained extremum seeking with function measurements disturbed by stochastic noise, 15th International Conference on Electrical Engineering, Mexico City, Mexico, pp. 1–4.

[28] Solis, C.U., Clempner, J.B. and Poznyak, A.S. (2018b). Continuous-time extremum seeking with function measurements disturbed by stochastic noise: A synchronous detection approach, 15th International Conference Electrical Engineering, Mexico City, Mexico, pp. 1–5.

[29] Stade, E. (2005). Fourier Analysis, Wiley-Interscience, Hoboken, NJ.

[30] Ulusoy, A., Liu, G., Trasser, A. and Schumacher, H. (2011). An analog synchronous QPSK demodulator for ultra-high rate wireless communications, German Microwave Conference (GeMiC), Darmstadt, Germany, pp. 1–4.

[31] Wang, L., Chen, S. and Zhao, H. (2014). A novel fast extremum seeking scheme without steady-state oscillation, 33rd Chinese Control Conference, Nanjing, China, pp. 8687–8692.

[32] Zhang, C. and Ordóñez, R. (2012). Extremum-seeking Control and Applications, Springer, London.