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

ABSTRACT 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 generalised 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.

[1]  Sergey Edward Lyshevski.  Control Systems Theory with Engineering Applications , 2001 .

[2]  Huaguang Zhang,et al.  An iterative adaptive dynamic programming method for solving a class of nonlinear zero-sum differential games , 2011, Autom..

[3]  Richard B. Vinter,et al.  Minimax Optimal Control , 2005, SIAM J. Control. Optim..

[4]  Tamer Başar,et al.  H1-Optimal Control and Related Minimax Design Problems , 1995 .

[5]  Edward J. Davison,et al.  Two numerical techniques to solve differential game problems , 1972 .

[6]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[7]  Moritz Diehl,et al.  Robust dynamic programming for min-max model predictive control of constrained uncertain systems , 2004, IEEE Transactions on Automatic Control.

[8]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[9]  Farid Sheikholeslam,et al.  Generalisation of Euler–Lagrange equations to find min–max optimal solution of uncertain systems , 2013, Int. J. Control.

[10]  Tamer Basar,et al.  H∞-Optimal Control and Related , 1991 .

[11]  Zakieh Avazzadeh,et al.  Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind , 2012 .

[12]  Jaroslav Dolezal.  A gradient-type algorithm for the numerical solution of two-player zero-sum differential game problems , 1978, Kybernetika.

[13]  Kil To Chong,et al.  The Explicit Constrained Min-Max Model Predictive Control of a Discrete-Time Linear System With Uncertain Disturbances , 2012, IEEE Transactions on Automatic Control.

[14]  Emanuel Todorov,et al.  Optimal Control Theory , 2006 .

[15]  Lawrence F. Shampine,et al.  Solving Boundary Value Problems for Ordinary Differential Equations in M atlab with bvp 4 c , 2022 .

[16]  John T. Betts,et al.  Practical Methods for Optimal Control and Estimation Using Nonlinear Programming , 2009 .

[17]  W. Schmitendorf,et al.  Minmax control of systems with uncertainty in the initial state and in the state equations , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[18]  Yujiang Wu,et al.  Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation , 2012 .

[19]  N. Long.  Differential Games and Resources , 2013 .

[20]  Ali H. Bhrawy,et al.  A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals , 2012 .

[21]  Philip A. Johnson.  Numerical solution methods for differential game problems , 2009 .

[22]  Kok Lay Teo,et al.  Min-max optimal control of linear systems with uncertainty and terminal state constraints , 2013, Autom..

[23]  D. Mayne,et al.  Min-max feedback model predictive control for constrained linear systems , 1998, IEEE Trans. Autom. Control..

[24]  Jacob Engwerda,et al.  LQ Dynamic Optimization and Differential Games , 2005 .

[25]  J. Dicapua.  Chebyshev Polynomials , 2019, Fibonacci and Lucas Numbers With Applications.

[26]  David Q. Mayne,et al.  Robust time-optimal control of constrained linear Systems , 1997, Autom..

[27]  Eric C. Kerrigan,et al.  Control of Constrained Discrete-Time Systems With Bounded $\ell_{2}$ Gain , 2009, IEEE Transactions on Automatic Control.

[28]  Ali H. Bhrawy,et al.  On shifted Jacobi spectral method for high-order multi-point boundary value problems , 2012 .

[29]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[30]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[31]  Dimitris Bertsimas,et al.  Constrained Stochastic LQC: A Tractable Approach , 2007, IEEE Transactions on Automatic Control.

[32]  M. R. Osborne.  On shooting methods for boundary value problems , 1969 .

[33]  Alberto Bemporad,et al.  Min-max control of constrained uncertain discrete-time linear systems , 2003, IEEE Trans. Autom. Control..