Pseudospectral knotting method
Encyclopedia
In the mathematical theory of knots
, a pseudospectral knotting method is a generalization and enhancement of a standard pseudospectral method for optimal control
. 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
known as hybrid optimal control problems.
In a hybrid optimal control problem, an optimal control problem is intertwined with a graph problem
. A standard pseudospectral optimal control
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
.
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. 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
problems.
The PS knotting method was first implemented in the MATLAB
optimal control
software package, DIDO
.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, a pseudospectral knotting method is a generalization and enhancement of a standard pseudospectral method for optimal control
Pseudospectral optimal control
Pseudospectral optimal control is a computational method for solving optimal control problems. Pseudospectral optimal control 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,...
. 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
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
known as hybrid optimal control problems.
In a hybrid optimal control problem, an optimal control problem is intertwined with a graph problem
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
. A standard pseudospectral optimal control
Pseudospectral optimal control
Pseudospectral optimal control is a computational method for solving optimal control problems. Pseudospectral optimal control 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,...
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 (optimal control)
DIDO is a MATLAB program for solving hybrid optimal control problems. Powered by the pseudospectral knotting method, the general-purpose program is named after Dido, the legendary founder and first queen of Carthage who is famous in mathematics for her remarkable solution to a constrained optimal...
.
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. 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
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
problems.
The PS knotting method was first implemented in the MATLAB
MATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
optimal control
Optimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
software package, DIDO
DIDO (optimal control)
DIDO is a MATLAB program for solving hybrid optimal control problems. Powered by the pseudospectral knotting method, the general-purpose program is named after Dido, the legendary founder and first queen of Carthage who is famous in mathematics for her remarkable solution to a constrained optimal...
.