In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
, the pseudospectral knotting method is a generalization and enhancement of a standard
pseudospectral method for optimal control. The concept was introduced by
I. Michael Ross and
F. Fahroo in 2004, and 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 on top of one another.
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 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 ...
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 pseudospectral (PS) theory with optimal control theory to produce PS optimal control theory. PS optimal control theory ...
method is incapable of solving such problems; however, through the use of pseudospectral knots, the information of the graph 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 modern Tunisia), in 814 BC.
In most accounts, she was the queen of the Phoenician city-state of Tyre (t ...
.
Applications
PS knots have found applications in aerospace problems such as the ascent guidance of a launch vehicles, and advancing the Aldrin Cycler through the use of 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 in solving bang-bang-type
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.
[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 modern Tunisia), in 814 BC.
In most accounts, she was the queen of the Phoenician city-state of Tyre (t ...
.
See also
*
Legendre pseudospectral method The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. A basic version of the Legendre pseudospectral was origi ...
*
Chebyshev pseudospectral method
*
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