Gauss pseudospectral method
Encyclopedia
The Gauss pseudospectral method (GPM), one of many topics named after Carl Friedrich Gauss
, is a direct transcription method for discretizing a continuous optimal control
problem into a nonlinear program
(NLP). The Gauss pseudospectral method differs from several other pseudospectral methods
in that the dynamics are not collocated at either endpoint of the time interval. This collocation, in conjunction with the proper approximation to the costate, leads to a set of KKT conditions that are identical to the discretized form of the first-order optimality conditions. This equivalence between the KKT conditions and the discretized first-order optimality conditions leads to an accurate costate estimate using the KKT multipliers of the NLP.
where the collocation points (i.e., the points at which the optimal control problem is discretized) are the Legendre
–Gauss (LG) points. The approach used in the GPM is to use a Lagrange polynomial
approximation for the state that includes coefficients for the initial state plus the values of the state at the N LG points. In a somewhat opposite manner, the approximation for the costate (adjoint) is performed using a basis of Lagrange polynomials that includes the final value of the costate plus the costate at the N LG points. These two approximations together lead to the ability to map the KKT multipliers of the nonlinear program (NLP) to the costates of the optimal control problem at the N LG points PLUS the boundary points. The costate mapping theorem that arises from the GPM has been described in several references including two MIT PhD theses and journal articles that include the theory along with applications
pseudospectral method (CPM) the Legendre pseudospectral method
(LPM) and the Gauss pseudospectral method (GPM). The CPM uses Chebyshev polynomials to approximate the state and control, and performs orthogonal collocation at the Chebyshev–Gauss–Lobatto
(CGL) points. An enhancement to the Chebyshev pseudospectral method
that uses a Clenshaw–Curtis quadrature was developed. The LPM uses Lagrange polynomials for the approximations, and Legendre–Gauss–Lobatto (LGL) points for the orthogonal collocation. A costate estimation procedure for the Legendre pseudospectral method
was also developed. Recent work shows several variants of the standard LPM, The Jacobi pseudospectral method is a more general pseudospectral approach that uses Jacobi polynomials to find the collocation points, of which Legendre polynomials are a subset. Another variant, called the Hermite-LGL method uses piecewise cubic polynomials rather than Lagrange polynomials, and collocates at a subset of the LGL points.
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, is a direct transcription method for discretizing a continuous 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...
problem into a nonlinear program
Nonlinear programming
In mathematics, nonlinear programming is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are...
(NLP). The Gauss pseudospectral method differs from several other pseudospectral methods
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,...
in that the dynamics are not collocated at either endpoint of the time interval. This collocation, in conjunction with the proper approximation to the costate, leads to a set of KKT conditions that are identical to the discretized form of the first-order optimality conditions. This equivalence between the KKT conditions and the discretized first-order optimality conditions leads to an accurate costate estimate using the KKT multipliers of the NLP.
Description
The method is based on the theory of orthogonal collocationCollocation method
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations...
where the collocation points (i.e., the points at which the optimal control problem is discretized) are the Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
–Gauss (LG) points. The approach used in the GPM is to use a Lagrange polynomial
Lagrange polynomial
In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x_j and numbers y_j, the Lagrange polynomial is the polynomial of the least degree that at each point x_j assumes the corresponding value y_j...
approximation for the state that includes coefficients for the initial state plus the values of the state at the N LG points. In a somewhat opposite manner, the approximation for the costate (adjoint) is performed using a basis of Lagrange polynomials that includes the final value of the costate plus the costate at the N LG points. These two approximations together lead to the ability to map the KKT multipliers of the nonlinear program (NLP) to the costates of the optimal control problem at the N LG points PLUS the boundary points. The costate mapping theorem that arises from the GPM has been described in several references including two MIT PhD theses and journal articles that include the theory along with applications
Background
Pseudospectral methods, also known as orthogonal collocation methods, in optimal control arose from spectral methods which were traditionally used to solve fluid dynamics problems. Seminal work in orthogonal collocation methods for optimal control problems date back to 1979 with the work of Reddien and some of the first work using orthogonal collocation methods in engineering can be found in the chemical engineering literature. More recent work in chemical and aerospace engineering have used collocation at the Legendre–Gauss–Radau (LGR) points. Within the aerospace engineering community, several well-known pseudospectral methods have been developed for solving optimal control problems such as the ChebyshevPafnuty Chebyshev
Pafnuty Lvovich Chebyshev was a Russian mathematician. His name can be alternatively transliterated as Chebychev, Chebysheff, Chebyshov, Tschebyshev, Tchebycheff, or Tschebyscheff .-Early years:One of nine children, Chebyshev was born in the village of Okatovo in the district of Borovsk,...
pseudospectral method (CPM) the Legendre pseudospectral method
Legendre pseudospectral method
The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger pseudospectral optimal control theory, that was originally proposed by Elnagar and coworkers in 1995. Since then, Ross, Fahroo and co-workers have analysed, extended and...
(LPM) and the Gauss pseudospectral method (GPM). The CPM uses Chebyshev polynomials to approximate the state and control, and performs orthogonal collocation at the Chebyshev–Gauss–Lobatto
Rehuel Lobatto
Rehuel Lobatto was a Dutch mathematician. Lobatto was born in Amsterdam to a Portuguese Marrano family. As a schoolboy Lobatto already displayed remarkable talent for mathematics...
(CGL) points. An enhancement to the Chebyshev pseudospectral method
Chebyshev pseudospectral method
The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. Unlike the Legendre pseudospectral method, the Chebyshev pseudospectral method does not immediately offer high-accuracy quadrature solutions. Consequently, two different versions...
that uses a Clenshaw–Curtis quadrature was developed. The LPM uses Lagrange polynomials for the approximations, and Legendre–Gauss–Lobatto (LGL) points for the orthogonal collocation. A costate estimation procedure for the Legendre pseudospectral method
Legendre pseudospectral method
The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger pseudospectral optimal control theory, that was originally proposed by Elnagar and coworkers in 1995. Since then, Ross, Fahroo and co-workers have analysed, extended and...
was also developed. Recent work shows several variants of the standard LPM, The Jacobi pseudospectral method is a more general pseudospectral approach that uses Jacobi polynomials to find the collocation points, of which Legendre polynomials are a subset. Another variant, called the Hermite-LGL method uses piecewise cubic polynomials rather than Lagrange polynomials, and collocates at a subset of the LGL points.
See also
- PROPT - MATLAB (Gauss and Chebyshev) Optimal Control software with more than 110 examples.
- Gauss Pseudospectral Optimal Control Software (GPOPS) the first software appearing in a peer-reviewed journal article that implements pseudospectral methods.
- JModelica.orgJModelica.orgJModelica.org is a free and open source platform based on the Modelica modeling language for modeling, simulation, optimization and analysis of complex dynamic systems. The platform is maintained and developed by Modelon AB in collaboration with academic and industrial institutions, notably Lund...
(Modelica-based open source platform for dynamic optimization)