Abstract
In advancing our prior work on a unified theory for pseudospectral (PS) optimal control, we present the mathematical foundations for spectral collocation over arbitrary grids. The computational framework is not based on any particular choice of quadrature nodes associated with orthogonal polynomials. Because our framework applies to non-Gaussian grids, a number of hidden properties are uncovered. A key result of this paper is the discovery of the dual connections between PS and Galerkin approximations. Inspired by Polak’s pioneering work on consistent approximation theory, we analyze the dual consistency of PS discretization. This analysis reveals the hidden relationship between Galerkin and pseudospectral optimal control methods while uncovering some finer points on covector mapping theorems. The new theory is used to demonstrate via a numerical example that a PS method can be surprisingly robust to grid selection. For example, even when 60 % of the grid points are chosen to be uniform—the worst possible selection from a pseudospectral perspective—a PS method can still produce satisfactory result. Consequently, it may be possible to choose non-Gaussian grid points to support different resolutions over the same grid.
Similar content being viewed by others
References
Bedrossian, N., Bhatt, S., Lammers, M., Nguyen, L., Zhang, Y.: First ever flight demonstration of zero propellant maneuver attitude control concept. In: Guidance, Navigation, and Control Conference, pp. AIAA-2007-6734. Hilton Head, South Carolina (2007)
Bedrossian, N., Bhatt, S., Lammers, M., Nguyen, L.: Zero propellant maneuver: flight results for 180\(^\circ \) iss rotation. In: 20th International Symposium on Space Flight Dynamics, NASA/CP-2007-214158. Annapolis, MD (2007)
Kang, W., Bedrossian, N.: Pseudospectral optimal control theory makes debut flight–saves NASA $1M in under 3 hrs. SIAM News 40(7), 1–3 (2007)
Bedrossian, N.S., Bhatt, S., Kang, W., Ross, I.M.: Zero-propellant maneuver guidance. IEEE Control Syst. Mag. 29(5), 53–73 (2009)
Bedrossian, N., Karpenko, M., Bhatt, S.: Overclock my satellite. IEEE Spectr. 49(11), 54–62 (2012)
Karpenko, M., Bhatt, S., Bedrossian, N., Fleming, A., Ross, I.M.: First flight results on time-optimal spacecraft slews. AIAA J. Guid. Control Dyn. 35(2), 367–376 (2012)
Karpenko, M., Bhatt, S., Bedrossian, N., Ross, I.M.: Flight implementation of shortest-time maneuvers for imaging satellites. AIAA J. Guid. Control Dyn. 37(4), 1069–1079 (2014)
Ross, I.M., Karpenko, M.: A review of pseudospectral optimal control: from theory to flight. Annu. Rev. Control 36(2), 182–197 (2012)
Ross, I.M.: A Primer on Pontryagin’s Principle in Optimal Control, 2nd edn. Collegiate Publishers, San Francisco, CA (2015)
Paris, S.W., Riehl, J., Sjauw, W.: Enhanced procedures for direct trajectory optimization using nonlinear programming and implicit integration. In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, pp. 21–24, AIAA 2006–6309. Keystone, CO. (2006)
Lu, P., Sun, H., Tsai, B.: Closed-loop endoatmospheric ascent guidance. AIAA J. Guid. Control Dyn. 26(2), 283–294 (2003)
Williams, P., Lansdorp, B., Ockesl, W.: Optimal crosswind towing and power generation with tethered kites. AIAA J. Guid. Control Dyn. 31(1), 81–93 (2008)
Stevens, R., Wiesel, W.: Large time scale optimal control of an electrodynamic tether satellite. AIAA J. Guid. Control Dyn. 31(6), 1716–1727 (2008)
Gong, Q., Kang, W., Bedrossian, N.S., Fahroo, F., Sekhavat, P., Bollino, K.: Pseudospectral optimal control for military and industrial applications. In: The 46th IEEE Conference on Decision and Control, pp. 4128–4142 (2007)
Hawkins, A.M., Fill, T.R., Proulx, R.J., Feron, E.M.: Constrained trajectory optimization for lunar landing. In: AAS Spaceflight Mechanics Meeting, AAS 06–153. Tampa, FL (2006)
Williams, P.: Three-dimensional aircraft terrain-following via real-time optimal control. J. Guid. Control Dyn. 30(4), 1201–1206 (2007)
Bollino, K.P., Lewis, L.R.: Optimal path planning and control of tactical unmanned aerial vehicles in urban environments. In: Proceedings of the AUVSIs Unmanned Systems North America 2007 Conference. Washington, DC (2007)
Ross, I.M., Gong, Q., Sekhavat, P.: The bellman pseudospectral method. In: AIAA/AAS Astrodynamics Specialist Conference and Exhibit, AIAA-2008-6448. Honolulu, Hawaii (2008)
Ross, I.M., Gong, Q., Sekhavat, P.: Low-thrust, high-accuracy trajectory optimization. J. Guid. Control Dyn. 30(4), 921–933 (2007)
Ross, I.M., Fahroo, F.: Pseudospectral methods for optimal motion planning of differentially flat systems. IEEE Trans. Autom. Control 49(8), 1410–1413 (2004)
Ross, I.M., Fahroo, F.: Pseudospectral methods for optimal motion planning of differentially flat systems. In: Proceedings of the IEEE Conference on Decision and Control, Las Vegas, NV. IEEE (2002)
Ross, I.M., Sekhavat, P., Fleming, A., Gong, Q.: Optimal feedback control: foundations, examples, and experimental results for a new approach. J. Guid. Control Dyn. 31(2), 307–321 (2008)
Ross, I.M., Fahroo, F.: Issues in the real-time computation of optimal control. Math. Comput. Model. 43(9), 1172–1188 (2006)
Ross, I.M., Fahroo, F.: A unified computational framework for real-time optimal control, ieee. In: Proceedings of the 42nd IEEE Conference on Decision and Control. IEEE, Maui, Hawaii (2003)
Li, J.S., Ruths, J., Yu, T.Y., Arthanari, H., Wagner, G.: Optimal pulse design in quantum control: a unified computational method. Proc. Nat. Acad. Sci. 108(5), 1879–1884 (2011)
Li, J.S., Khaneja, N.: Control of inhomogeneous quantum ensembles. Phys. Rev. A 73, 030,302-1–030,303–4 (2006)
Ross, I.M., Proulx, R.J., Karpenko, M.: Unscented optimal control for space flight. In: Proceedings of the 24th International Symposium on Space Flight Dynamics. Laurel, MD (2014)
Ross, I.M., Proulx, R.J., Karpenko, M.: Unscented optimal control for orbital and proximity operations in an uncertain environment: A new Zermelo problem. In: Proceedings of AIAA Space and Astronautics Forum and Exposition: AIAA/AAS Astrodynamics Specialist Conference. San Diego, CA (2014)
Bienstock, D., Chertkov, M., Harnett, S.: Chance-constrained optimal power flow: risk-aware network control under uncertainty. SIAM Rev. 56(3), 461–495 (2014)
Ono, M., Kuwata, Y., Balaram, J.: Mixed-strategy chance constrained optimal control. In: Proceedings of the American Control Conference, pp. 4666–4673 (2013)
Phelps, C., Gong, Q., Royset, J.O., Walton, C., Kaminer, I.: Consistent approximation of a nonlinear optimal control problem with uncertain parameters. Automatica 50(12), 2987–2997 (2014)
Walton, C., Gong, Q., Kaminer, I., Royset, J.O.: Optimal motion planning for searching for uncertain targets. In: Proceedings of the 19th IFAC World Congress. South Africa (2014)
Ross, I.M., Proulx, R.J., Karpenko, M., Gong, Q.: Riemann-Stieltjes optimal control problems for uncertain dynamical systems. AIAA J. Guid. Control Dyn. 38(7), 1251–1263 (2015)
Ross, I.M., Karpenko, M., Proulx, R.J.: A lebesgue–stieltjes framework for optimal control and allocation. In: Proceedings of the American Control Conference. IEEE, Chicago, IL (2015)
Elnagar, G., Kazemi, M.A., Razzaghi, M.: The pseudospectral Legendre method for discretizing optimal control problems. IEEE Trans. Autom. Control 40(10), 1793–1796 (1995)
Elnagar, G.N., Kazemi, M.A.: Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems. Comput. Optim. Appl. 11(2), 195–217 (1998)
Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)
Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1998)
Trefethen, L.N.: Spectral methods in MATLAB. Siam, Philadelphia (2000)
Gong, Q., Ross, I.M., Fahroo, F.: Pseudospectral optimal control on arbitrary grids. In: AAS Astrodynamics Specialist Conference, pp. AAS 09–405 (2009)
Fahroo, F., Ross, I.M.: Advances in pseudospectral methods for optimal control. In: AIAA Guidance, Navigation and Control Conference and Exhibit, pp. 18–21 (2008)
Gong, Q., Ross, I.M., Fahroo, F.: Costate computation by a Chebyshev pseudospectral method. AIAA J. Guid. Control Dyn. 33(2), 623–628 (2010)
Fahroo, F., Ross, I.M.: Convergence of the costates does not imply convergence of the control. J. Guid. Control Dyn. 31(5), 1492–1497 (2008)
Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, New York, NY (1997)
Fahroo, F., Ross, I.M.: Pseudospectral methods for infinite-horizon nonlinear optimal control problems. In: Proceedings of the AIAA Guidance, Navigation and Control Conference. San Francisco, CA (2005)
Fahroo, F., Ross, I.M.: Pseudospectral methods for infinite-horizon nonlinear optimal control problems. J. Guid. Control Dyn. 31(4), 927–936 (2008)
Ross, I., Fahroo, F.: A pseudospectral transformation of the covectors of optimal control systems. In: Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, pp. 29–31 (2001)
Fahroo, F., Ross, I.M.: Costate estimation by a Legendre pseudospectral method. AIAA J. Guid. Control Dyn. 24(2), 270–277 (2001)
Ross, I.M., Fahroo, F.: Legendre pseudospectral approximations of optimal control problems. In: Kang, W., et al. (eds.) Lecture Notes in Control and Information Sciences, vol. 295, pp. 327–342. Springer, New York (2003)
Fahroo, F., Ross, I.M.: Direct trajectory optimization by a Chebyshev pseudospectral method. AIAA J. Guid. Control Dyn. 25(1), 160–166 (2002)
Gong, Q., Ross, I.M., Fahroo, F.: A Chebyshev pseudospectral method for nonlinear constrained optimal control problems. In: Proceedings of the 48th IEEE Conference on Decision and Control., pp. 5057–5062. Shanghai, China (2009)
Trefethen, L.N.: Is Gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 50(1), 67–87 (2008)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Corporation, New York (2001)
Gong, Q., Kang, W., Ross, I.M.: A pseudospectral method for the optimal control of constrained feedback linearizable systems. IEEE Trans. Autom. Control 51(7), 1115–1129 (2006)
Hager, W.W.: Numerical analysis in optimal control. In: International Series of Numerical Mathematics, vol. 139, pp. 83–93. Birkhäuser (2001)
Kang, W., Gong, Q., Ross, I.M.: Convergence of pseudospectral methods for a class of discontinuous optimal control. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 2799–2804. Seville, Spain (2005)
Kang, W., Gong, Q., Ross, I., Fahroo, F.: On the convergence of nonlinear optimal control using pseudospectral methods for feedback linearizable systems. Int. J. Robust Nonlinear Control 17(14), 1251–1277 (2007)
Kang, W.: Rate of convergence for the Legendre pseudospectral optimal control of feedback linearizable systems. J. Control Theory Appl. 8(4), 391–405 (2010)
Astolfi, A., Marconi, L. (eds.): Analysis and Design of Nonlinear Control Systems: In Honor of Alberto Isidori, pp. 109–124. Springer, Berlin, Heidelberg (2008)
Boucher, R., Kang, W., Gong, Q.: Galerkin optimal control for constrained nonlinear problems. In: Proceedings of the 2014 American Control Conference, pp. 2432–2437. Portland, OR (2014)
Boucher, R., Kang, W., Gong, Q.: Feasibility of the Galerkin optimal control method. In: Proceedings of the 53rd IEEE Conference on Decision and Control. Los Angeles, CA (2014)
Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37(2), 181–218 (1995)
Vinter, R.: Optimal Control. Springer Science & Business Media, Berlin (2010)
Ross, I.M., Fahroo, F.: Pseudospectral knotting methods for solving nonsmooth optimal control problems. J. Guid. Control Dyn. 27(3), 397–405 (2004)
Gong, Q., Fahroo, F., Ross, I.M.: Spectral algorithm for pseudospectral methods in optimal control. J. Guid. Control Dyn. 31(3), 460–471 (2008)
Williams, P.: Jacobi pseudospectral method for solving optimal control problems. AIAA J. Guid. Control Dyn. 27(2), 293–297 (2004)
Cullum, J.: Finite-dimensional approximations of state-constrained continuous optimal control problems. SIAM J. Control 10(4), 649–670 (1972)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, vol. 330. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. II: Applications, vol. 331. Springer, Berlin (2006)
Pontraygin, L.S.: The Mathematical Theory of Optimal Processes. Interscience Publishers, New York (1962)
Ross, I.M.: A historical introduction to the covector mapping principle. Adv. Astronaut. Sci. 123, 1257–1278 (2006)
Gong, Q., Ross, I.M., Kang, W., Fahroo, F.: Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control. Comput. Optim. Appl. 41(3), 307–335 (2008)
Carpenter, M.H., Gottlieb, D.: Spectral methods on arbitrary grids. J. Comput. Phys. 129(1), 74–86 (1996)
Bosarge, W.E., Johnson, O.G., Mcknight, R.S., Timlake, W.P.: The Ritz–Galerkin procedure for nonlinear control problems. SIAM J. Numer. Anal. 10(1), 94–111 (1973)
Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods in fluid dynamics. Springer, Berlin (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gong, Q., Ross, I.M. & Fahroo, F. Spectral and Pseudospectral Optimal Control Over Arbitrary Grids. J Optim Theory Appl 169, 759–783 (2016). https://doi.org/10.1007/s10957-016-0909-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-016-0909-y