Gauss pseudospectral method

<h2 id="description">Description</h2>
The method is based on the theory of orthogonal <a href="/facts/Collocation_method/YMVkNS7m">collocation</a> where the collocation points (i.e., the points at which the optimal control problem is discretized) are the <a href="/facts/Adrien-Marie_Legendre/2a7CAolM">Legendre</a>–Gauss (LG) points. The approach used in the GPM is to use a <a href="/facts/Lagrange_polynomial/MnpMmhtC">Lagrange polynomial</a> approximation for the state that includes coefficients for the initial state plus the values of the state at the N LG points. In a somewhat opposite manner, the approximation for the <a href="/facts/Costate/KgNTRV4h">costate</a> (adjoint) is performed using a basis of Lagrange polynomials that includes the final value of the costate plus the costate at the N LG points. These two approximations together lead to the ability to map the KKT multipliers of the nonlinear program (NLP) to the costates of the optimal control problem at the N LG points PLUS the boundary points. The costate mapping theorem that arises from the GPM has been described in several references including two PhD theses<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> and journal articles that include the theory along with applications<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="background">Background</h2>
Pseudospectral methods, also known as orthogonal collocation methods, in optimal control arose from spectral methods which were traditionally used to solve fluid dynamics problems.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> Seminal work in orthogonal collocation methods for optimal control problems date back to 1979 with the work of Reddien<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> and some of the first work using orthogonal collocation methods in engineering can be found in the chemical engineering literature.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> More recent work in chemical and aerospace engineering have used collocation at the Legendre–Gauss–Radau (LGR) points.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> Within the aerospace engineering community, several well-known pseudospectral methods have been developed for solving optimal control problems such as the <a href="/facts/Pafnuty_Chebyshev/LzRqLqRm">Chebyshev</a> pseudospectral method (CPM)<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> the <a href="/facts/Legendre_pseudospectral_method/XXJapl1u">Legendre pseudospectral method</a> (LPM)<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> and the Gauss pseudospectral method (GPM).<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> The CPM uses Chebyshev polynomials to approximate the state and control, and performs orthogonal collocation at the Chebyshev–Gauss–<a href="/facts/Rehuel_Lobatto/7aKBKvju">Lobatto</a> (CGL) points. An enhancement to the <a href="/facts/Chebyshev_pseudospectral_method/qfHJv5aJ">Chebyshev pseudospectral method</a> that uses a Clenshaw–Curtis quadrature was developed.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> The LPM uses Lagrange polynomials for the approximations, and Legendre–Gauss–Lobatto (LGL) points for the orthogonal collocation. A costate estimation procedure for the <a href="/facts/Legendre_pseudospectral_method/XXJapl1u">Legendre pseudospectral method</a> was also developed.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> Recent work shows several variants of the standard LPM, The Jacobi pseudospectral method<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> is a more general pseudospectral approach that uses Jacobi polynomials to find the collocation points, of which Legendre polynomials are a subset. Another variant, called the Hermite-LGL method<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> uses piecewise cubic polynomials rather than Lagrange polynomials, and collocates at a subset of the LGL points.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/APMonitor/8I0EfqH1">APMonitor</a> software for dynamic optimization</li>
<li><a href="http://tomdyn.com/">PROPT</a> - MATLAB (Gauss and Chebyshev) Optimal Control software with more than 110 examples.</li>
<li><a href="/facts/GPOPS-II/mDCHTYvx">GPOPS-II</a>: <a href="http://www.gpops2.com/">General Pseudospectral Optimal Control Software</a> (peer-reviewed journal article that implements variable-order Gaussian quadrature collocation methods).</li>
<li><a href="/facts/JModelica.org/Ym67Ym2N">JModelica.org</a> (Modelica-based open source platform for dynamic optimization)</li></ul>
<h2 id="references-and-notes">References and notes</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Benson, D.A., A Gauss Pseudospectral Transcription for Optimal Control, Ph.D. Thesis, Dept. of Aeronautics and Astronautics, MIT, November 2004, <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Huntington, G.T., Advancement and Analysis of a Gauss Pseudospectral Transcription for Optimal Control, Ph.D. Thesis, Dept. of Aeronautics and Astronautics, MIT, May 2007 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Benson, D.A., Huntington, G.T., Thorvaldsen, T.P., and Rao, A.V., "Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method", Journal of Guidance, Control, and Dynamics. Vol. 29, No. 6, November–December 2006, pp. 1435–1440., <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Huntington, G.T., Benson, D.A., and Rao, A.V., "Optimal Configuration of Tetrahedral Spacecraft Formations", The Journal of The Astronautical Sciences. Vol. 55, No. 2, March–April 2007, pp. 141–169. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Huntington, G.T. and Rao, A.V., "Optimal Reconfiguration of Spacecraft Formations Using the Gauss Pseudospectral Method", Journal of Guidance, Control, and Dynamics. Vol. 31, No. 3, March–April 2008, pp. 689–698. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A., Spectral Methods in Fluid Dynamics, Springer–Verlag, New York, 1988. <a href="/wiki/M._Yousuff_Hussaini" target="_blank">/wiki/M._Yousuff_Hussaini</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Fornberg, B., A Practical Guide to Pseudospectral Methods, Cambridge University Press,
1998. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Reddien, G.W., "Collocation at Gauss Points as a Discretization in Optimal Control,"SIAM Journal on Control and Optimization, Vol. 17, No. 2, March 1979. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Cuthrell, J.E. and Biegler, L.T., “Simultaneous Optimization and Solution Methods for Batch Reactor Control Profiles,” Computers and Chemical Engineering, Vol. 13, Nos. 1/2, 1989, pp.49–62. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Hedengren, J.D.; Asgharzadeh Shishavan, R.; Powell, K.M.; Edgar, T.F. (2014). "Nonlinear modeling, estimation and predictive control in APMonitor". Computers & Chemical Engineering. 70 (5): 133–148. doi:10.1016/j.compchemeng.2014.04.013. S2CID 5793446. <a href="https://scholarsarchive.byu.edu/facpub/1667" target="_blank">https://scholarsarchive.byu.edu/facpub/1667</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Fahroo, F. and Ross, I., “Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems,” 2005 AIAA Guidance, Navigation, and Control Conference, AIAA Paper 2005–6076, San Francisco, CA, August 15–18, 2005. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Kameswaran, S. and Biegler, L.T., “Convergence Rates for Dynamic Optimization Using Radau Collocation,” SIAM Conference on Optimization, Stockholm, Sweden, 2005. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Kameswaran, S. and Biegler, L.T., “Convergence Rates for Direct Transcription of Optimal Control Problems at Radau Points,” Proceedings of the 2006 American Control Conference, Minneapolis, Minnesota, June 2006. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Vlassenbroeck, J. and Van Doreen, R., “A Chebyshev Technique for Solving Nonlinear Optimal Control Problems,” IEEE Transactions on Automatic Control, Vol. 33, No. 4, 1988, pp. 333–340. <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Vlassenbroeck, J., “A Chebyshev Polynomial Method for Optimal Control with State Constraints,” Automatica, Vol. 24, 1988, pp. 499–506. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Elnagar, J., Kazemi, M. A. and Razzaghi, M., The Pseudospectral Legendre Method for Discretizing Optimal Control Problems, IEEE Transactions on Automatic Control, Vol. 40, No. 10, 1995, pp. 1793–1796 <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Benson, D.A., Huntington, G.T., Thorvaldsen, T.P., and Rao, A.V., “Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 6, November–December 2006, pp. 1435–1440. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Fahroo, F. and Ross, I.M., “Direct Trajectory Optimization by a Chebyshev Pseudospectral Method,” Journal of Guidance, Control, and Dynamics, Vol. 25, No. 1, January–February 2002, pp. 160–166. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Ross, I. M., and Fahroo, F., Legendre Pseudospectral Approximations of Optimal Control Problems, Lecture Notes in Control and Information Sciences, Vol.295, Springer-Verlag, New York, 2003 <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Williams, P., “Jacobi Pseudospectral Method for Solving Optimal Control Problems”, Journal of Guidance, Vol. 27, No. 2, 2003 <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Williams, P., “Hermite–Legendre–Gauss–Lobatto Direct Transcription Methods In Trajectory Optimization,” Advances in the Astronautical Sciences. Vol. 120, Part I, pp. 465–484. 2005 <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
</ol>

Gauss pseudospectral method open-in-new

Gauss pseudospectral method