A General Framework for Optimal Control of Fractional Nonlinear Delay Systems by Wavelets
Abstract
An iterative procedure to find the optimal solutions of general fractional nonlinear delay systems with quadraticperformance indices is introduced. The derivatives of state equations are understood in the Caputo sense. By presenting and applying a general framework, we use the Chebyshev wavelet method developed for fractional linear optimal control to convert fractional nonlinear optimal control problems as a sequence of quadratic programming ones. The concepts and computational procedure that are used for fractional linear optimal control are applied on fractional nonlinear optimal control. Different types of nonlinear optimal control problems with fractional or integer order can be solved. To see this, some numerical examples are solved. Another operational property of Chebyshev wavelets is presented.References
M. Malek-Zavarei, M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, Elsevier Science Inc., North-Holland, 1978.
D.E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004.
D.S. Naidu, Optimal Control Systems, Idaho State University. Pocatello. Idaho. USA, CRC PRESS, 2003.
A.P. Sage, Optimum Systems Control, Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1968.
T. C¸ imen, S.P. Banks, Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria, Systems & Control Letters, vol. 53, no. 5, pp. 327–346, 2004.
M. Tomas-Rodrigueza, S.P. Banks, An Iterative Approach to Eigenvalue Assignment for Nonlinear Systems, International Journal of Control, vol. 86, no. 5, pp. 883–892, 2013. https://doi.org/10.1080/00207179.2013.765037
H. Jaddu, A. Majdalawi, An Iterative Technique for Solving a Class of Nonlinear Quadratic Optimal Control Problems Using Chebyshev Polynomials, International Journal of Intelligent Systems and Applications, vol. 6, no. 6, pp. 53–57, 2014. DOI: 10.5815/ijisa.2014.06.06
E. Ziaei, M.H. Farahi, The approximate solution of non-linear time-delay fractional optimal control problems by embedding process, IMA Journal of Mathematical Control and Information, vol. 36, no. 3, pp. 713–727, 2019. DOI: 10.1093/imamci/dnx063
F. Kheyrinataj, A. Nazemi, Fractional power series neural network for solving delay fractional optimal control problems,
Connection Science, vol. 32, no. 1, pp. 53–80, 2020. https://doi.org/10.1080/09540091.2019.1605498
H.R. Marzban, F. Malakoutikhah, Solution of delay fractional optimal control problems using a hybrid of block-pulse
functions and orthonormal Taylor polynomials, Journal of the Franklin Institute, vol. 356, no. 15, pp. 8182–8215, 2019. doi: https://doi.org/10.1016/j.jfranklin.2019.07.010
I. Malmir, A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems, Fractal and Fractional, vol. 3, no. 3, 2019: 46. Doi:10.3390/fractalfract3030046
M.A. Johnson, F.C. Moon, Experimental characterization of quasiperiodicity and chaos in a mechanical system with delay, International journal of Bifurcation and Chaos, vol. 9, no. 01, pp. 49–65, 1999. http://dx.doi.org/10.1142/S0218127499000031
M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Annals of Geophysics, vol. 19, no. 4, pp. 383–393, 1966.
I. Malmir, S.H. Sadati, Transforming linear time-varying optimal control problems with quadratic criteria into quadratic programming ones via wavelets, Journal of Applied Analysis, vol. 26, no. 1, pp. 131–152, 2020. https://doi.org/10.1515/jaa-2020- 2011
M. Barrios, G. Reyero, An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational
Problems with Classical Derivatives, Statistics, Optimization & Information Computing, vol. 8, no. 2, pp. 590–601, 2020.
https://doi.org/10.19139/soic-2310-5070-865
I. Malmir, Legendre wavelets with scaling in time-delay systems, Statistics, Optimization & Information Computing, vol. 7, no. 1, pp. 235–253, 2019. https://doi.org/10.19139/soic.v7i1.460
A. Kheirabadi, A. Mahmoudzadeh Vaziri, S. Effati, Liner optimal control of time delay systems via Hermite wavelet, Numerical Algebra, Control and Optimization, vol. 10, no. 2, pp. 143–156., 2020. doi:10.3934/naco.2019044
M.A. Sarhan, S. Shihab, M. Rasheed, A New Boubaker Wavelets Operational Matrix of Integration, Journal of Southwest Jiaotong University, vol. 55, no. 2, 2020.
E. Keshavarz, Y. Ordokhani, A fast numerical algorithm based on the Taylor wavelets for solving the fractional integrodifferential equations with weakly singular kernels, Mathematical Methods in the Applied Sciences, vol. 42, no. 13, pp. 4427–4443, 2019.
M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism, Pure and Applied Geophysics, vol. 91, no. 1, pp. 134–147, 1971. doi: https://doi.org/10.1007/BF00879562
F. Brauer, Perturbations of nonlinear systems of differential equations, II, Journal of Mathematical Analysis and Applications, vol. 17, no. 3, pp. 418–434, 1967.
B.G. Pachpatte A Note on Gronwali-Bellman Inequality, Journal of Mathematical Analysis and Applications, vol. 44, no. 3, pp. 758–762, 1973.
R. Luus, X. Zhang, F. Hartig, F. J. Keil, Use of piecewise linear continuous optimal control for time-delay systems, Industrial & engineering chemistry research, vol. 34, no. 11, pp. 4136–4139, 1995.
A.Y. Lee, Numerical solution of time-delayed optimal control problems with terminal inequality constraints, Optimal Control Applications and Methods, vol. 14, no. 3, pp. 203–210, 1993.
A. Nazemi, E. Fayyazi, M. Mortezaee, Solving optimal control problems of the time-delayed systems by a neural network framework Connection Science, vol. 31, no. 4, pp. 342–372, 2019. DOI: 10.1080/09540091.2019.1604627
Y.H. Roh, J.H. Oh, Sliding Mode Control for Robust Stabilization of Uncertain Input-Delay Systems, The Institute of Control, Automation and Systems Engineers, vol. 2, no. 2, pp. 98–103, 2000.
D. Bresch-Pietri, N. Petit, M. Krstic, Prediction-based control for nonlinear state- and input-delay systems with the aim of delayrobustness analysis, 54th IEEE Conference on Decision and Control (CDC 2015), Osaka, Japan, pp. 6403–6409, 2015. DOI: 10.1109/CDC.2015.7403228
B. Zhou, Truncated Predictor Feedback for Time-Delay Systems, Springer-Verlag, 2014. DOI: 10.1007/978-3-642-54206-0-1
I. Estrada-S´anchez, M. Velasco-Villa, H. Rodr´ıguez-Cort´es, Prediction-Based Control for Nonlinear Systems with Input Delay, Mathematical Problems in Engineering Volume 2017, Article ID 7415418. https://doi.org/10.1155/2017/7415418
I. Malmir, Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models, Applied Mathematical Modelling, vol. 69, pp. 621–647, 2019. doi:10.1016/j.apm.2018.12.009.
I. Malmir, Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets, Statistics, Optimization & Information Computing, vol. 5, no. 4, pp. 302–324 2017. http://dx.doi.org/10.19139/soic.v5i4.341
H.T. Banks, Approximation of Nonlinear Functional Differential Equation Control Systems, Journal of Optimization Theory and Applications, vol. 29, no. 3, pp. 383–408, 1979.
K.H. Wong, D.J. Clements, K.L. Teo Optimal Control Computation for Nonlinear Time-Lag Systems, Journal of Optimization Theory and Applications, vol. 47, no. 1, pp. 91–107, 1985.
G.N. Elnagar, M.A. Kazemi, Numerical solution of time-delayed functional differential equation control systems, Journal of computational and applied mathematics, vol. 130, no. 1–2, pp. 75–90, 2001.
S.M. Hoseini, H.R. Marzban, Costate Computation by an Adaptive Pseudospectral Method for Solving Optimal Control Problems with Piecewise Constant Time Lag, Journal of Optimization Theory and Applications, vol. 170, no. 3, pp. 735–755, 2016. DOI: 10.1007/s10957-016-0957-3
X. Tang, H. Xu, Multiple-Interval Pseudospectral Approximation for Nonlinear Optimal Control Problems with Time-Varying Delays, Applied Mathematical Modelling, vol. 68, pp. 137–151, 2018. doi: https://doi.org/10.1016/j.apm.2018.09.039
M.K. Bouafoura, N.B. Braiek, Hybrid Functions Direct Approach and State Feedback Optimal Solutions for a Class of Nonlinear Polynomial Time Delay Systems, Complexity, 2019, Article ID 9596253. doi: http://dx.doi.org/10.1155/2019/9596253
C. Hua, P.X. Liu, X. Guan, Backstepping control for nonlinear systems with time delays and applications to chemical reactor systems, IEEE Transactions on Industrial Electronics, vol. 56, no. 9, pp. 3723–3732, 2009.
J.T. Betts, S. L. Campbell, K.C. Thompson, Optimal Control Software for Constrained Nonlinear Systems with Delays, CACSD, IEEE Multi Conference on Systems and Control, Denver, USA, 2011. 444–449. DOI: 10.1109/CACSD.2011.6044560
J.T. Betts, S. L. Campbell, K.C. Thompson, Solving optimal control problems with control delays using direct transcription, Applied Numerical Mathematics, vol. 108, pp. 185–203, 2016.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
