Pseudospectral optimal control
Encyclopedia
Pseudospectral optimal control is a computational method for solving optimal control
problems. Pseudospectral (PS) 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, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, and ascent guidance.
. The quadrature nodes are determined by the corresponding orthogonal polynomial basis used for the approximation. In PS optimal control, Legendre and Chebyshev polynomials
are commonly used. Mathematically, quadrature nodes are able to achieve high accuracy with few number of points. For instance, the interpolating polynomial
of any smooth function (C
) at Legendre–Gauss–Lobatto nodes converges in L2 sense at the so-called spectral rate, i.e., faster than any polynomial rate.
), and collocation at Legendre–Gauss–Radau points (known as the Radau Pseudospectral Method). It is also noted that versions of the Gauss and Radau pseudospectral methods have been developed for solving infinite-horizon optimal control problems. It is important to note that the Lobatto pseudospectral method has the property that the differentiation matrix is square and singular whereas the Gauss and Radau pseudospectral methods have the property that the differentiation matrices are non-square and full rank. This last property of the Gauss and Radau pseudospectral methods leads to the fact that either of these latter two methods can be written equivalently in either differential or implicit integral form.
In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration
result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to
. In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems. One interesting property of pseudospectral optimal control is that, if done correctly, it permits commutativity between discretization and dualization. Specifically, this commutativity exists if the Gauss pseudospectral method
(GPM, which uses Legendre–Gauss points) or the Radau pseudospectral method (RPM, which uses Legendre–Gauss–Radau points) are used. For either the GPM or RPM, the KKT multipliers are related to the costates of the continuous problem in an algebraically simple manner. In the case of Gauss-Lobatto points, this commutativity is lost because the transformed adjoint system is singular in the discretized costate.
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. Pseudospectral (PS) 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, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, and ascent guidance.
Overview
Solving an optimal control problem requires the approximation of three types of mathematical objects: the integration in the cost function, the differential equation of the control system, and the state-control constraints. An ideal approximation method should be efficient for all three approximation tasks. A method that is efficient for one of them, for instance an efficient ODE solver, may not be an efficient method for the other two objects. These requirements make PS methods ideal because they are efficient for the approximation of all three mathematical objects as proved in, and. In a pseudospectral method, the continuous functions are approximated at a set of carefully selected quadrature nodesGaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration....
. The quadrature nodes are determined by the corresponding orthogonal polynomial basis used for the approximation. In PS optimal control, Legendre and Chebyshev polynomials
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
are commonly used. Mathematically, quadrature nodes are able to achieve high accuracy with few number of points. For instance, the interpolating 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...
of any smooth function (C
) at Legendre–Gauss–Lobatto nodes converges in L2 sense at the so-called spectral rate, i.e., faster than any polynomial rate.Details
These methods include forms of the collocation at Legendre–Gauss–Lobatto points, collocation at Chebyshev–Gauss–Lobatto points, Legendre–Gauss points (known as the Gauss Pseudospectral MethodGauss pseudospectral method
The Gauss pseudospectral method , 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 . The Gauss pseudospectral method differs from several other pseudospectral methods in that the...
), and collocation at Legendre–Gauss–Radau points (known as the Radau Pseudospectral Method). It is also noted that versions of the Gauss and Radau pseudospectral methods have been developed for solving infinite-horizon optimal control problems. It is important to note that the Lobatto pseudospectral method has the property that the differentiation matrix is square and singular whereas the Gauss and Radau pseudospectral methods have the property that the differentiation matrices are non-square and full rank. This last property of the Gauss and Radau pseudospectral methods leads to the fact that either of these latter two methods can be written equivalently in either differential or implicit integral form.
In pseudospectral methods, integration is approximated by quadrature rules, which provide the best numerical integration
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
result. For example, with just N nodes, a Legendre-Gauss quadrature integration achieves zero error for any polynomial integrand of degree less than or equal to
. In the PS discretization of the ODE involved in optimal control problems, a simple but highly accurate differentiation matrix is used for the derivatives. Because a PS method enforces the system at the selected nodes, the state-control constraints can be discretized straightforwardly. All these mathematical advantages make pseudospectral methods a straightforward discretization tool for continuous optimal control problems. One interesting property of pseudospectral optimal control is that, if done correctly, it permits commutativity between discretization and dualization. Specifically, this commutativity exists if the Gauss pseudospectral methodGauss pseudospectral method
The Gauss pseudospectral method , 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 . The Gauss pseudospectral method differs from several other pseudospectral methods in that the...
(GPM, which uses Legendre–Gauss points) or the Radau pseudospectral method (RPM, which uses Legendre–Gauss–Radau points) are used. For either the GPM or RPM, the KKT multipliers are related to the costates of the continuous problem in an algebraically simple manner. In the case of Gauss-Lobatto points, this commutativity is lost because the transformed adjoint system is singular in the discretized costate.
Further reading
- Gauss pseudospectral methodGauss pseudospectral methodThe Gauss pseudospectral method , 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 . The Gauss pseudospectral method differs from several other pseudospectral methods in that the...
- GPOPS, General Pseudospectral Optimal Control Software. http://www.gpops.org.
- DIDO (optimal control)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...