Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control.N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap", ''

IEEE Spectrum ''IEEE Spectrum'' is a magazine edited by the Institute of Electrical and Electronics Engineers. The first issue of ''IEEE Spectrum'' was published in January 1964 as a successor to ''Electrical Engineering''. The magazine contains peer-revie ...

'', November 2012. I. M. Ross and F. Fahroo, A Pseudospectral Transformation of the Covectors of Optimal Control Systems, Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29–31 August 2001. I. M. Ross and F. Fahroo, Legendre Pseudospectral Approximations of Optimal Control Problems, ''Lecture Notes in Control and Information Sciences'', Vol. 295, Springer-Verlag, 2003. I. M. Ross and F. Fahroo, Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems, Proceedings of the American Control Conference, Invited Paper, June 2004, Boston, MA. Examples of the Ross–Fahroo pseudospectral methods are the pseudospectral knotting method, the

flat pseudospectral method The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo. Ross, I. M. and Fahroo, F., �Pseudospectral Methods for the Optimal Motion Planning of Differentially Flat Systems” ...

, the Legendre-Gauss-Radau pseudospectral methodF. Fahroo and I. M. Ross, "Advances in Pseudospectral Methods for Optimal Control," ''Proceedings of the AIAA Guidance, Navigation and Control Conference,'' AIAA 2008-7309.

/ref> and pseudospectral methods for infinite-horizon optimal control. F. Fahroo and I. M. Ross, Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems, AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA.

Overview

The Ross–Fahroo methods are based on shifted Gaussian pseudospectral node points. The shifts are obtained by means of a linear or nonlinear transformation while the Gaussian pseudospectral points are chosen from a collection of Gaussian quadrature, Gauss-Lobatto or Gauss-Radau distribution arising from Legendre or

Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...

. The Gauss-Lobatto pseudospectral points are used for finite-horizon

optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...

problems while the Gauss-Radau pseudospectral points are used for infinite-horizon optimal control problems.

Mathematical applications

The Ross–Fahroo methods are founded on the

Ross–Fahroo lemma Named after I. Michael Ross and F. Fahroo, the Ross–Fahroo lemma is a fundamental result in optimal control theory. I. M. Ross and F. Fahroo, A Pseudospectral Transformation of the Covectors of Optimal Control Systems, Proceedings of the First I ...

; they can be applied to optimal control problems governed by

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s, differential-algebraic equations,

differential inclusion In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form :\frac(t)\in F(t,x(t)), where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point ...

s, and differentially-flat systems. They can also be applied to infinite-horizon optimal control problems by a simple domain transformation technique. The Ross–Fahroo pseudospectral methods also form the foundations for the Bellman pseudospectral method.

Flight applications and awards

The Ross–Fahroo methods have been implemented in many practical applications and laboratories around the world. In 2006, NASA used the Ross–Fahroo method to implement the "zero propellant maneuver" on board the

International Space Station The International Space Station (ISS) is the largest modular space station currently in low Earth orbit. It is a multinational collaborative project involving five participating space agencies: NASA (United States), Roscosmos (Russia), JAXA ...

.N. S. Bedrossian, S. Bhatt, W. Kang, and I. M. Ross, Zero-Propellant Maneuver Guidance, IEEE Control Systems Magazine, October 2009 (Feature Article), pp 53–73. In recognition of all these advances, the AIAA presented Ross and Fahroo, the 2010 Mechanics and Control of Flight Award, for "... changing the landscape of flight mechanics." Ross was also elected AAS Fellow for "his pioneering contributions to pseudospectral optimal control."

Distinctive features

A remarkable feature of the Ross–Fahroo methods is that it does away with the prior notions of "direct" and "indirect" methods. That is, through a collection of theorems put forth by Ross and Fahroo, F. Fahroo and I. M. Ross, Trajectory Optimization by Indirect Spectral Collocation Methods, Proceedings of the AIAA/AAS Astrodynamics Conference, August 2000, Denver, CO. AIAA Paper 2000–4028 they showed that it was possible to design pseudospectral methods for optimal control that were equivalent in both the direct and indirect forms. This implied that one could use their methods as simply as a "direct" method while automatically generating accurate duals as in "indirect" methods. This revolutionized solving optimal control problems leading to widespread use of the Ross–Fahroo techniques.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.

Software implementation

The Ross–Fahroo methods are implemented in the MATLAB optimal control solver,

DIDO Dido ( ; , ), also known as Elissa ( , ), was the legendary founder and first queen of the Phoenician city-state of Carthage (located in modern Tunisia), in 814 BC. In most accounts, she was the queen of the Phoenician city-state of Tyre (t ...

References