×

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min-max optimal control problems with uncertainty. (English) Zbl 1397.49035

Summary: The difficulty of solving the Min-Max Optimal Control Problems (M-MOCPs) with uncertainty using generalised Euler-Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the Shifted Jacobi Pseudospectral Method (SJPM) as a numerical technique for solving Two-Point Boundary Value Problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalized Euler-Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite – as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.

MSC:

49K35 Optimality conditions for minimax problems
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
90C47 Minimax problems in mathematical programming

Software:

SOCS; bvp4c
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Avazzadeh, Z., & Heydari, M. (2012). Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind. Computational and Applied Mathematics, 31, 127-142. · Zbl 1247.65159
[2] Basar, T., & Bernhard, P. (1991). H\^{}{∞}-optimal control and related minimax design problems. New Jersey: Birkhauser. · Zbl 0751.93020
[3] Bemporad, A., Borrelli, F., & Morari, M. (2003). Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9), 1600-1606. · Zbl 1364.93181
[4] Bertsimas, D., & Brown, D.B. (2007). Constrained stochastic LQC: A tractable approach. IEEE Transactions on Automatic Control, 52, 1826-1841. · Zbl 1366.93699
[5] Betts, J.T. (2010). Practical methods for optimal control and estimation using nonlinear programming. Philadelphia, Pa: SIAM. · Zbl 1189.49001
[6] Bhrawy, A.H., & Alghamdi, M.A. (2012). A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals. Boundary Value Problem, 62, 1-13. · Zbl 1280.65079
[7] Boyd, J. (2001). Chebyshev and Fourier spectral methods (2nd ed.). Dover: Dover Publications, Inc. · Zbl 0994.65128
[8] Canuto, C., Hussaini, M.Y., Quarteroni, A., & Zang, T.A. (1988). Spectral methods in fluid dynamics. New York, NY: Springer. · Zbl 0658.76001
[9] Diehl, M., & Bjornberg, J. (2004). Robust dynamic programming for min-max model predictive control of constrained uncertain systems. IEEE Transactions on Automatic Control, 49(12), 2253-2257. · Zbl 1365.93131
[10] Doha, E.H., Bhrawy, A.H., & Hafez, R.M. (2012). On shifted Jacobi spectral method for high-order multi-point boundary value problems. Communications in Nonlinear Science and Numerical Simulation, 17, 3802-3810. · Zbl 1251.65112
[11] Dolezal, J. (1978). A gradient-type algorithm for the numerical solution of two-player zero-sum differential game problems. Kybernetika, 14, 429-446. · Zbl 0398.90119
[12] Dosthosseini, R., & Sheikholeslam, F. (2014). Generalisation of Euler-Lagrange equations to find min-max optimal solution of uncertain systems. International Journal of Control, 87, 2535-2548. · Zbl 1305.49028
[13] Engwerda, J.C. (2005). LQ dynamic optimization and differential games. Sussex, England: John Wiley and Sons.
[14] Gao, Y., & Chong, K.T. (2012). The explicit constrained min-max model predictive control of a discrete-time linear system with uncertain disturbances. IEEE Transactions on Automatic Control, 57(9), 277-282. · Zbl 1369.93338
[15] Goulart, P.J., Kerrigan, E.C., & Alamo, T. (2009). Control of constrained discrete-time systems with bounded l_{2} gain. IEEE Transactions on Automatic Control, 54(5), 1105-1111. · Zbl 1367.93244
[16] Johnson, P.A. (2009). Numerical solution methods for differential game problems (MS thesis). Massachusetts Institute of Technology.
[17] Kirk, D.E. (1970). Optimal control theory. Englewood Cliffs, NJ: Prentice Hall.
[18] Long, N.V. (2013). Differential games and resources. In Encyclopedia of Energy, Natural Resource, and Environmental Economics, (pp. 268-276), Montreal, QC: McGill University.
[19] Lyshevski, S.E. (2001). Control systems theory with engineering applications. Springer. · Zbl 1042.93001
[20] Mason, J.C., & Handscomb, D.C. (2003). Chebyshev polynomials. CRC Press LLC. · Zbl 1015.33001
[21] Mayne, D.Q., & Eschroeder, W.R. (1997). Robust time-optimal control of constrained linear systems. Automatica, 33, 2103-2118. · Zbl 0910.93052
[22] Osborne, M. (1969). On shooting methods for boundary value problems. Journal of Mathematical Analysis and Applications, 27, 417-433. · Zbl 0177.20402
[23] Quintana, V.H., & Davison, E.J. (1972). Two numerical techniques to solve differential game problems. International Journal of Control, 16, 465-474. · Zbl 0249.90085
[24] Schmitendorf, W.E. (1997). Minimax control of systems with uncertainty in the initial state and in the state equations. IEEE Transactions on Automatic Control, 22(3), 439-443. · Zbl 0354.49012
[25] Scokaert, P.O.M., & Mayne, D.Q. (1998). Minmax feedback model predictive control for constrained linear systems. IEEE Transactions on Automatic Control, 43, 1136-1142. · Zbl 0957.93034
[26] Shampine, L.F., & Kierzenka, J. (2000). Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c. MATLAB central file exchange. Retrieved from: ftp://ftp.mathworks.com/pub/doc/papers/byp/
[27] Shen, J., Tang, T., & Wang, L.L. (2011). Spectral methods: Algorithms, analysis and applications. Berlin: Springer. · Zbl 1227.65117
[28] Vinter, R.B. (2005). Minimax optimal control. SIAM Journal on Control and Optimization, 44, 939-968. · Zbl 1087.49015
[29] Wu, C., Teo, K.L. & Wu, S. (2013). Min-max optimal control of linear systems with uncertainty and terminal state constraints. Automatica, 49, 1809-1815. · Zbl 1360.49004
[30] Zhang, H., Wei, Q., & Liu, D. (2011). An iterative dynamic programming method for solving a class of nonlinear zero-sum differential games. Automatica, 47, 207-214. · Zbl 1231.91028
[31] Zhao, T., Li, C., Zang, Z., & Wu, Y. (2012). Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation. Applied Mathematical Modelling, 36, 1046-1056. · Zbl 1243.65126
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.