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tracking control; nonlinear polynomial systems; augmented error system approach; block pulse functions; practical stability
In this paper, tracking control design for a class of nonlinear polynomial systems is investigated by augmented error system approach and block pulse functions technique. The proposed method is based on the projection of the close loop augmented system and the associated linear reference model that it should follow over a basis of block pulse functions. The main advantage of using this tool is that it allows to transform the analytical differential calculus into an algebraic one relatively easy to solve. The developments presented have led to the formulation of a linear system of algebraic equations depending only on parameters of the feedback control. Once the control gains are determined by solving the latter optimization problem in least square sense, the practical stability of the closed loop augmented system is checked through given conditions. A double inverted pendulums benchmark is used to validate the proposed tracking control method.
[1] Balatif, O., Rachik, M., Labriji, E., Rachik, Z.: Optimal control problem for a class of bilinear systems via block pulse functions. J. Math. Control Inform. 34 (2017), 973-986. DOI 10.1093/imamci/dnw005 | MR 3746086
[2] Belhaouane, M. M., Braiek, N. Benhadj: Design of stabilizing control for synchronous machines via polynomial modelling and linear matrix inequalities approach. Int. J. Control, Automat. Systems 9 (2011), 425-436. DOI 10.1007/s12555-011-0301-5
[3] Braiek, N. Benhadj: Feedback stabilization and stability domain estimation of nonlinear systems. J. Franklin Inst. 332 (1995), 183-193. DOI 10.1016/0016-0032(95)00038-x | MR 1367190
[4] Brewer, J. W.: Kronecker products and matrix calculus in systems theory. IEEE Trans. Circuits Systems 25 (1978), 772-781. DOI 10.1109/tcs.1978.1084534 | MR 0510703
[5] Chen, W., Ge, Sh. S., Wu, J., Gong, M.: Globally stable adaptive backstepping neural network control for uncertain strict-feedback systems with tracking accuracy known a priori. IEEE Tran. Neural Networks Learning Systems 26 (2015), 1842-1854. DOI 10.1109/tnnls.2014.2357451 | MR 3453124
[6] Chen, M., Wu, Q. X., Cui, R. X.: Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISA Trans. 52 (2013), 198-206. DOI 10.1016/j.isatra.2012.09.009
[7] Cui, Y., Zhang, H., Wang, Y., Gao, W.: Adaptive control for a class of uncertain strict-feedback nonlinear systems based on a generalized fuzzy hyperbolic model. Fuzzy Sets Systems 302 (2016), 52-64. DOI 10.1016/j.fss.2015.11.015 | MR 3543818
[8] Cui, Y., Zhang, H., Qu, Q., Luo, C.: Synthetic adaptive fuzzy tracking control for MIMO uncertain nonlinear systems with disturbance observer. Neurocomputing 249 (2017), 191-201. DOI 10.1016/j.neucom.2017.03.064
[9] Deb, A., Roychoudhury, S., Sarkar, G.: Analysis and Identification of Time Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions. Theory and Algorithms with MATLAB. Studies in Systems, Decision and Control, 2016. DOI 10.1007/978-3-319-26684-8 | MR 3727125
[10] Deb, A., Sarkar, G., Sengupta, A.: Triangular Orthogonal Functions for the Analysis of Continuous Time Systems. Anthem Press, 2012. DOI 10.7135/upo9781843318118 | MR 3586384
[11] Ghorbel, H., Hajjaji, A. El, Souissi., M., Chaabane, M.: Robust tracking control for TS fuzzy systems with unmeasurable premise variables: Application to tank system. J. Dynamic Systems, Measurement Control 136 (2014), 4. DOI 10.1115/1.4026467
[12] Ginoya, D., Shendge, P. D., Phadke, S. B.: Disturbance observer based sliding mode control of nonlinear mismatched uncertain systems. Comm. Nonlin. Sci. Numer. Simul. 26 (2015), 98-107. DOI 10.1016/j.cnsns.2015.02.008 | MR 3332038
[13] Gruyitch, L. T.: Nonlinear Systems Tracking. CRC Press. Taylor and Francis Group, 2016. DOI 10.1201/b19258
[14] Hwang, C., Shih, Y. P.: Optimal control of delay systems via block pulse functions. J. Optim. Theory Appl. 45 (1985), 101-112. DOI 10.1007/bf00940816 | MR 0778160
[15] Jarzebowska, E.: Model-Based Tracking Control of Nonlinear Systems. CRC Press. Taylor and Francis Group, 2012. DOI 10.1201/b12262 | MR 2964364
[16] Jemai, W. J., Jerbi, H., Abdelkrim, M. N.: Nonlinear state feedback design for continuous polynomial systems. Int. J. Control Automat. Systems 9 (2011), 566-573. DOI 10.1007/s12555-011-0317-x
[17] Lakshmikantham, V., Leela, S., Martynyuk, A. A.: Practical Stability of Nonlinear Systems. World Scientific Publishing, 1990. DOI 10.1142/1192 | MR 1089428
[18] Marzban, H. R.: Parameter identification of linear multi-delay systems via a hybrid of block-pulse functions and Taylor's polynomials. Int. J. Control 90 (2016), 504-518. DOI 10.1080/00207179.2016.1186288 | MR 3604170
[19] Han, Y. Qun: Adaptive tracking control of nonlinear systems with dynamic uncertainties using neural network. Int. J. Systems Sci. 49 (2018), 1391-1402. DOI 10.1080/00207721.2018.1453955 | MR 3825710
[20] Rehan, M., Hong, K. Shik, Ge, Sh. Sam: Stabilization and tracking control for a class of nonlinear systems. Nonlinear Analysis: Real World Appl. 12 (2011), 1786-1796. DOI 10.1016/j.nonrwa.2010.11.011 | MR 2781896
[21] Rotella, F., Dauphin-Tanguy, G.: Nonlinear systems: Identification and optimal control. Int. J. Control 48 (1988), 525-544. DOI 10.1080/00207178808906195 | MR 0957420
[22] Shao, X., Wang, H.: A Novel Method of Robust Trajectory Linearization Control Based on Disturbance Rejection. Math. Problems Engrg. 2014 (2014), 1-10. DOI 10.1155/2014/129247 | MR 3200860
[23] Tang, Y., Li, N., Liu, M., Lu, Y., Wang, W.: Identification of fractional-order systems with time delays using block pulse functions. Mech. Systems Signal Process. 91 (2017), 382-394. DOI 10.1016/j.ymssp.2017.01.008
[24] Tong, S., Wang, T., Li, H. X.: Fuzzy robust tracking control for uncertain nonlinear systems. Int. J. Approx. Reason. 30 (2002), 73-90. DOI 10.1016/s0888-613x(02)00061-0 | MR 1906629
[25] Wang, H., Shi, P., Li, H., Zhou, Q.: Adaptive neural tracking control for a class of nonlinear systems with dynamic uncertainties. IEEE Trans. Cybernet. 47 (2017), 3075-3087. DOI 10.1109/tcyb.2016.2607166 | MR 3410700
[26] Warrad, B. Iben, Bouafoura, M. K., Braiek, N. Benhadj: Tracking control synthesis of nonlinear polynomial systems. In: 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Colmar 2015.
[27] Warrad, B. Iben, Bouafoura, M. K., Braiek, N. Benhadj: Static output tracking control for non-linear polynomial time-delay systems via block-pulse functions. J. Chinese Inst. Engineers 41 (2018), 194-205. DOI 10.1080/02533839.2018.1459849
[28] Xingling, S., Honglun, W.: Trajectory Linearization Control Based Output Tracking Method for Nonlinear Uncertain System Using Linear Extended State Observer. Asian J. Control 18 (2016), 316-327. DOI 10.1002/asjc.1053 | MR 3458911
[29] Xu, Y., Zhang, J., Liao, F.: Augmented error system approach to control design for a class of neutral systems. Advances Diff. Equations (2015), 1-17. DOI 10.1186/s13662-015-0596-2 | MR 3393013
[30] Yu, Y., Zhong, Y. Sh.: Semiglobal Robust Backstepping Output Tracking for Strict-feedback Form Systems with Nonlinear Uncertainty. Int. J. Control Automat. Systems 9 (2011), 366-375. DOI 10.1007/s12555-011-0219-y
[31] Yu, Y., Zhong, Y.: Semi-global robust output tracking for non-linear uncertain systems in strict-feedback form. IET Control Theory Appl. 6 (2012), 751-759. DOI 10.1186/s13662-015-0596-2 | MR 2952999
[32] Zhang, H., Li, C., Liao, X.: Stability analysis and h infinity controller design of fuzzy large-scale systems based on piecewise Lyapunov functions. IEEE Trans. Syst. Man Cybernet. 36 (2006), 685-698. DOI 10.1109/tsmcb.2005.860133
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