In
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
, the pseudospectral knotting method is a generalization and enhancement of the standard
pseudospectral method for optimal control. Introduced by
I. Michael Ross and
F. Fahroo in 2004, it forms part of the collection of the
Ross–Fahroo pseudospectral method
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 Algorit ...
s.
[Ross, I. M. and Fahroo, F., Pseudospectral Knotting Methods for Solving Optimal Control Problems, ''Journal of Guidance, Control and Dynamics,'' Vol. 27, No. 3, pp. 397–405, 2004.]
Definition
According to Ross and Fahroo, a pseudospectral (PS) knot is a double Lobatto point; i.e. two boundary points coinciding.
At this point, information (such as discontinuities, jumps, dimension changes etc.) is exchanged between two standard PS methods. This information exchange is used to solve some of the most difficult problems in
optimal control
Optimal control theory is a branch of control theory 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 operations ...
, known as hybrid optimal control problems.
[Ross, I. M. and D’Souza, C. N., A Hybrid Optimal Control Framework for Mission Planning, ''Journal of Guidance, Control and Dynamics,'' Vol. 28, No. 4, July–August 2005, pp. 686–697.]
In a hybrid optimal control problem, an optimal control problem is intertwined with a
graph problem. A standard
pseudospectral optimal control
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 a PS optimal control theory. ...
method is incapable of solving such problems; however, through the use of pseudospectral knots, the graph information can be encoded at the double Lobatto points, thereby allowing a hybrid optimal control problem to be discretized and solved using powerful software such as
DIDO
Dido ( ; , ), also known as Elissa ( , ), was the legendary founder and first queen of the Phoenician city-state of Carthage (located in Tunisia), in 814 BC.
In most accounts, she was the queen of the Phoenician city-state of Tyre (located ...
.
Applications
PS knots have found applications in various aerospace problems such as the ascent guidance of launch vehicles and advancing the Aldrin Cycler using solar sails.
[Stevens, R. and Ross, I. M., Preliminary Design of Earth–Mars Cyclers Using Solar Sails, ''Journal of Spacecraft and Rockets,'' Vol. 41, No. 4, 2004.][ Stevens, R., Ross, I. M. and Matousek, S. E., "Earth-Mars Return Trajectories Using Solar Sails," 55th International Astronautical Congress, Vancouver, Canada, IAC-04-A.2.08, October 4–8, 2004.]
PS knots have also been used for anti-aliasing of PS optimal control solutions and for capturing critical information in switches when solving bang-bang-type
optimal control
Optimal control theory is a branch of control theory 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 operations ...
problems.
[Gong, Q., Fahroo, F. and Ross, I. M., A Spectral Algorithm for Pseudospectral Methods in Optimal Control, ''Journal of Guidance, Control and Dynamics,'' Vol. 31, No. 3, pp. 460–471, 2008.]
Software
The PS knotting method was first implemented in th
MATLAB optimal controlsoftware package,
DIDO
Dido ( ; , ), also known as Elissa ( , ), was the legendary founder and first queen of the Phoenician city-state of Carthage (located in Tunisia), in 814 BC.
In most accounts, she was the queen of the Phoenician city-state of Tyre (located ...
.
See also
*
Legendre pseudospectral method Legendre, LeGendre or Le Gendre is a French surname. It may refer to:
* Adrien-Marie Legendre (1752–1833), French mathematician
** Associated Legendre polynomials
** Legendre's equation
** Legendre polynomials
** Legendre symbol
In number t ...
*
Chebyshev pseudospectral method The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. Unlike the Legendre pseudospectral metho ...
*
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 ...
*
Ross' π lemma Ross' lemma, named after I. Michael Ross, is a result in computational optimal control. Based on generating Carathéodory- solutions for feedback control, Ross' -lemma states that there is fundamental time constant within which a control solutio ...
*
Ross–Fahroo pseudospectral method
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 Algorit ...
s
References
{{DEFAULTSORT:Pseudospectral Optimal Control
Optimal control
Numerical analysis
Control theory