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By investing in a premium Word - Press theme, you're investing in the future of your website. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Should you go with simple HTML or use a platform like Wordpress. Keep reading for some great Word - Press ideas you can start using today. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. This plugin allows a blogger get more Facebook fans on the related fan page. Apart from these, you are also required to give some backlinks on other sites as well. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help. <br><br>It is very easy to install Word - Press blog or website. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. To turn the Word - Press Plugin on, click Activate on the far right side of the list. There are plenty of tables that are attached to this particular database. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. If you adored this short article and you would certainly such as to obtain even more info pertaining to [http://rlpr.co/wordpress_backup_486822 backup plugin] kindly see our web-page. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Word - Press is the most popular open source content management system (CMS) in the world today. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>Instead, you can easily just include it with our bodies integration field in e - Panel. When you sign up with Wordpress, you gain access to several different templates and plug-in that allow you to customize your blog so that it fits in with your business website design seamlessly. You can select color of your choice, graphics of your favorite, skins, photos, pages, etc. Change the entire appearance of you blog using themes with one click. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.
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According to [[I. Michael Ross|Ross]] et al.,<ref name="ReviewPOC">[[I. Michael Ross|I. M. Ross]] and M. Karpenko, "A Review of Pseudospectral Optimal Control: From Theory to Flight," ''Annual Reviews in Control,'' Vol. 36, pp. 182-197, 2012. [http://www.sciencedirect.com/science/article/pii/S1367578812000375]</ref><ref name="RossAcademy">I. M. Ross, "A Roadmap for Optimal Control: The Right Way to Commute,"  ''Annals of the New York Academy of Sciences,'' Vol. 1065,  pp. 210–231, January 2006.</ref><ref name="advances">F. Fahroo and I. M. Ross, "Advances in Pseudospectral Methods for Optimal Control," ''Proceedings of the AIAA Guidance, Navigation and Control Conference,'' AIAA 2008-7309.
[http://www.elissarglobal.com/wp-content/uploads/2012/04/Advances-in-Pseudospectral-Methods-for-Optimal-Control.pdf]</ref><ref name="unified-ross">
I. M. Ross and F. Fahroo, "A Unified Computational Framework for Real-Time Optimal Control," ''Proceedings of the 42nd IEEE Conference on Decision and Control,'' Vol.3, 2003, pp.2210-2215.[http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1272946&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D1272946]
</ref> '''pseudospectral optimal control''' is a joint theoretical-computational method for solving [[optimal control]] problems. It combines [[pseudo-spectral method|pseudospectral (PS) theory]] with [[optimal control]] theory to produce PS optimal control theory. PS optimal control theory has been used in ground and flight systems<ref name="ReviewPOC"/> in military and industrial applications.<ref name="GKBFSB">Q. Gong, W. Kang, N. Bedrossian, [[Fariba Fahroo|F. Fahroo]], P. Sekhavat and K. Bollino, Pseudospectral Optimal Control for Military and Industrial Applications, 46th IEEE Conference on Decision and Control, New Orleans, LA, pp. 4128-4142, Dec. 2007.</ref> The techniques have been extensively used to solve a wide range of problems such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, ascent guidance and quantum control.<ref name="GKBFSB" /><ref name ="unified-li">Jr-S Li, J. Ruths, T-Y Yu, H. Arthanari and G. Wagner, "Optimal Pulse Design in Quantum Control: A Unified Computational Method,"  ''Proceedings of the National Academy of Sciences,'' Vol.108, No.5, Feb 2011, pp.1879-1884. http://www.pnas.org/content/108/5/1879.full</ref>
 
==Overview==
There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. Examples of these are the [[Legendre pseudospectral method]], the [[Chebyshev pseudospectral method]], the [[Gauss pseudospectral method]], the [[Ross-Fahroo pseudospectral method]], the [[flat pseudospectral method]] and many others.<ref name="ReviewPOC"/><ref name="advances"/> Solving an optimal control problem requires the approximation of three types of mathematical objects: the integration in the cost function, the differential equation of the control system, and the state-control constraints.<ref name="advances"/> An ideal approximation method should be efficient for all three approximation tasks. A method that is efficient for one of them, for instance an efficient ODE solver, may not be an efficient method for the other two objects. These requirements make PS methods ideal because they are efficient for the approximation of all three mathematical objects.<ref name="GKS"/><ref name="HGG">J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral methods for time-dependent problems, ''Cambridge University Press'', 2007. ISBN 978-0-521-79211-0</ref><ref name="GRKF">Q. Gong, I. M. Ross, W. Kang and [[Fariba Fahroo|Fahroo, F.]], Connections between the [[covector mapping principle|covector mapping theorem]] and convergence of pseudospectral methods for optimal control, ''Computational Optimization and Applications'', Springer Netherlands, published online: 31 October 2007, to appear in Journal, 2008.</ref> In a pseudospectral method, the continuous functions are approximated at a set of carefully selected [[Gaussian quadrature|quadrature nodes]]. The quadrature nodes are determined by the corresponding orthogonal polynomial basis used for the approximation. In PS optimal control, [[Legendre polynomials|Legendre]] and [[Chebyshev polynomials]] are commonly used. Mathematically, quadrature nodes are able to achieve high accuracy with a small number of points. For instance, the [[Lagrange polynomial|interpolating polynomial]] of any smooth function (C<sup><math>\infty</math></sup>) at Legendre–Gauss–Lobatto nodes converges in L<sup>2</sup> sense at the so-called spectral rate, faster than any polynomial rate.<ref name="HGG"/>
 
==Details==
 
A basic pseudospectral method for optimal control is based on the [[covector mapping principle]].<ref name="RossAcademy" /> Other pseudospectral optimal control techniques, such as the [[Bellman pseudospectral method]], rely on node-clustering at the initial time to produce optimal controls.  The node clusterings occur at all Gaussian points.<ref name="GKS">Q. Gong, W. Kang and I. M. Ross, A Pseudospectral Method for The Optimal Control of Constrained Feedback Linearizable Systems, ''IEEE Trans. Auto. Cont.'', Vol.~51, No.~7, July 2006, pp.~1115–1129.</ref><ref name="Elnagar1">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</ref><ref>F. Fahroo and I. M. Ross, Costate Estimation by a [[Legendre pseudospectral method|Legendre Pseudospectral Method]], ''Journal of Guidance, Control and Dynamics'', Vol.24, No.2, March–April 2001, pp.270–277.</ref><ref>Gong, Q., [[Fariba Fahroo|Fahroo, F.]], and [[I. Michael Ross|Ross, I.M.]], "Spectral Algorithm for Pseudospectral Methods in Optimal Control," "Journal of Guidance, Control, and Dynamics," Vol. 31, No. 3, 2008.</ref><ref name="Elnagar2">Elnagar, G. and Kazemi, "[[Chebyshev pseudospectral method|Pseudospectral Chebyshev Optimal Control]] of Constrained Nonlinear Dynamical Systems," ''Computational Optimization and Applications'', Vol. 17, No. 2, pp. 195–217</ref><ref name="Fahroo2">[[Fariba Fahroo|Fahroo, F.]], and [[I. Michael Ross|Ross, I. M.]], "Direct Trajectory Optimization via a [[Chebyshev pseudospectral method|Chebyshev Pseudospectral Method]]," ''Journal of Guidance, Control, and Dynamics'', Vol. 25, pp. 160–166</ref><ref name="Benson1">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</ref><ref name="Rao1">Rao, A. V., Benson, D. A., Darby, C. L., Patterson, M. A., Francolin, C., Sanders, I., and Huntington, G. T., "Algorithm 902:  GPOPS, A Matlab Software for Solving Multiple-Phase Optimal Control Problems Using the Gauss Pseudospectral Method", ''ACM Transactions on Mathematical Software,'' Vol. 37, No. 2, April–June, 2010, Article 22, pages 22:1-22:39.</ref><ref name="Garg1">Garg, D. A., Patterson, M. A., Darby, C. L., Francolin, C., Huntington, G. T., Hager, W. W., and Rao, A. V., "Direct Trajectory Optimization and Costate Estimation of Finite-Horizon and Infinite-Horizon Optimal Control Problems Using a Radau Pseudospectral Method," ''Computational Optimization and Applications,'' Vol., 49, No. 2, June 2011, pp. 335-358.</ref>
 
In pseudospectral methods, integration is approximated by quadrature rules, which provide the best [[numerical integration]] result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to <math>2N-1</math>. In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems.
 
==See also==
*[[Bellman pseudospectral method]]
*[[Chebyshev pseudospectral method]]
*[[Covector mapping principle]]
*[[Flat pseudospectral method]]s
*[[Gauss pseudospectral method]]
*[[Legendre pseudospectral method]]
*[[Pseudospectral knotting method]]
*[[Ross–Fahroo lemma]]
*[[Ross–Fahroo pseudospectral method]]s
*[[Ross' π lemma]]
 
==References==
{{Reflist}}
 
==External links==
* [http://computer.howstuffworks.com/dido.htm How Stuff Works]
 
==Software==
* [http://www.mathworks.com/products/connections/product_detail/product_61633.html DIDO - MATLAB tool for optimal control] named after [[DIDO (optimal control)|Dido]], the first [[Dido (Queen of Carthage)|queen of Carthage]].
* [http://www.astos.de/products/gesop GESOP – Graphical Environment for Simulation and OPtimization]
{{Use dmy dates|date=September 2010}}
* [http://www.gpops2.com GPOPS-II, General Pseudospectral Optimal Software.]
* [http://tomdyn.com/ PROPT – MATLAB Optimal Control Software]
* [https://sites.google.com/a/psopt.org/psopt/ PSOPT – Open Source Pseudospectral Optimal Control Solver in C++]
 
{{DEFAULTSORT:Pseudospectral Optimal Control}}
[[Category:Optimal control]]

Latest revision as of 17:02, 29 November 2013

Template:Multiple issues

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According to Ross et al.,[1][2][3][4] pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems. It combines pseudospectral (PS) theory with optimal control theory to produce PS optimal control theory. PS optimal control theory has been used in ground and flight systems[1] in military and industrial applications.[5] The techniques have been extensively used to solve a wide range of problems such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, ascent guidance and quantum control.[5][6]

Overview

There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. Examples of these are the Legendre pseudospectral method, the Chebyshev pseudospectral method, the Gauss pseudospectral method, the Ross-Fahroo pseudospectral method, the flat pseudospectral method and many others.[1][3] Solving an optimal control problem requires the approximation of three types of mathematical objects: the integration in the cost function, the differential equation of the control system, and the state-control constraints.[3] An ideal approximation method should be efficient for all three approximation tasks. A method that is efficient for one of them, for instance an efficient ODE solver, may not be an efficient method for the other two objects. These requirements make PS methods ideal because they are efficient for the approximation of all three mathematical objects.[7][8][9] In a pseudospectral method, the continuous functions are approximated at a set of carefully selected quadrature nodes. The quadrature nodes are determined by the corresponding orthogonal polynomial basis used for the approximation. In PS optimal control, Legendre and Chebyshev polynomials are commonly used. Mathematically, quadrature nodes are able to achieve high accuracy with a small number of points. For instance, the interpolating polynomial of any smooth function (C) at Legendre–Gauss–Lobatto nodes converges in L2 sense at the so-called spectral rate, faster than any polynomial rate.[8]

Details

A basic pseudospectral method for optimal control is based on the covector mapping principle.[2] Other pseudospectral optimal control techniques, such as the Bellman pseudospectral method, rely on node-clustering at the initial time to produce optimal controls. The node clusterings occur at all Gaussian points.[7][10][11][12][13][14][15][16][17]

In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to . In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems.

See also

References

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External links

Software

30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí.

  1. ↑ 1.0 1.1 1.2 I. M. Ross and M. Karpenko, "A Review of Pseudospectral Optimal Control: From Theory to Flight," Annual Reviews in Control, Vol. 36, pp. 182-197, 2012. [1]
  2. ↑ 2.0 2.1 I. M. Ross, "A Roadmap for Optimal Control: The Right Way to Commute," Annals of the New York Academy of Sciences, Vol. 1065, pp. 210–231, January 2006.
  3. ↑ 3.0 3.1 3.2 F. Fahroo and I. M. Ross, "Advances in Pseudospectral Methods for Optimal Control," Proceedings of the AIAA Guidance, Navigation and Control Conference, AIAA 2008-7309. [2]
  4. ↑ I. M. Ross and F. Fahroo, "A Unified Computational Framework for Real-Time Optimal Control," Proceedings of the 42nd IEEE Conference on Decision and Control, Vol.3, 2003, pp.2210-2215.[3]
  5. ↑ 5.0 5.1 Q. Gong, W. Kang, N. Bedrossian, F. Fahroo, P. Sekhavat and K. Bollino, Pseudospectral Optimal Control for Military and Industrial Applications, 46th IEEE Conference on Decision and Control, New Orleans, LA, pp. 4128-4142, Dec. 2007.
  6. ↑ Jr-S Li, J. Ruths, T-Y Yu, H. Arthanari and G. Wagner, "Optimal Pulse Design in Quantum Control: A Unified Computational Method," Proceedings of the National Academy of Sciences, Vol.108, No.5, Feb 2011, pp.1879-1884. http://www.pnas.org/content/108/5/1879.full
  7. ↑ 7.0 7.1 Q. Gong, W. Kang and I. M. Ross, A Pseudospectral Method for The Optimal Control of Constrained Feedback Linearizable Systems, IEEE Trans. Auto. Cont., Vol.~51, No.~7, July 2006, pp.~1115–1129.
  8. ↑ 8.0 8.1 J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral methods for time-dependent problems, Cambridge University Press, 2007. ISBN 978-0-521-79211-0
  9. ↑ Q. Gong, I. M. Ross, W. Kang and Fahroo, F., Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control, Computational Optimization and Applications, Springer Netherlands, published online: 31 October 2007, to appear in Journal, 2008.
  10. ↑ 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
  11. ↑ F. Fahroo and I. M. Ross, Costate Estimation by a Legendre Pseudospectral Method, Journal of Guidance, Control and Dynamics, Vol.24, No.2, March–April 2001, pp.270–277.
  12. ↑ Gong, Q., Fahroo, F., and Ross, I.M., "Spectral Algorithm for Pseudospectral Methods in Optimal Control," "Journal of Guidance, Control, and Dynamics," Vol. 31, No. 3, 2008.
  13. ↑ Elnagar, G. and Kazemi, "Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems," Computational Optimization and Applications, Vol. 17, No. 2, pp. 195–217
  14. ↑ Fahroo, F., and Ross, I. M., "Direct Trajectory Optimization via a Chebyshev Pseudospectral Method," Journal of Guidance, Control, and Dynamics, Vol. 25, pp. 160–166
  15. ↑ 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
  16. ↑ Rao, A. V., Benson, D. A., Darby, C. L., Patterson, M. A., Francolin, C., Sanders, I., and Huntington, G. T., "Algorithm 902: GPOPS, A Matlab Software for Solving Multiple-Phase Optimal Control Problems Using the Gauss Pseudospectral Method", ACM Transactions on Mathematical Software, Vol. 37, No. 2, April–June, 2010, Article 22, pages 22:1-22:39.
  17. ↑ Garg, D. A., Patterson, M. A., Darby, C. L., Francolin, C., Huntington, G. T., Hager, W. W., and Rao, A. V., "Direct Trajectory Optimization and Costate Estimation of Finite-Horizon and Infinite-Horizon Optimal Control Problems Using a Radau Pseudospectral Method," Computational Optimization and Applications, Vol., 49, No. 2, June 2011, pp. 335-358.
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