O PTIMAL control problems are often solved numerically via direct methods [1]. In recent years, considerable attention has been focused on a class of direct transcription methods called pseudospectral [2–4] or orthogonal collocation [5,6] methods. In a pseudospectral method, a finite basis of global interpolating polynomials is used to approximate the state at a set of discretization points. The time derivative of the state is approximated by differentiating the interpolating polynomial and constraining the derivative to be equal to the vector field at a finite set of collocation points. Although any set of unique collocation points can be chosen, generally speaking, an orthogonal collocation is chosen, i.e., the collocation points are the roots of an orthogonal polynomial (or linear combinations of such polynomials and their derivatives). Because pseudospectral methods are commonly implemented via orthogonal collocation, the terms pseudospectral and orthogonal collocation are interchangeable (thus researchers in onefieldmay use the term pseudospectral [3], whereas others may use the term orthogonal collocation [5]). One advantage to pseudospectral methods is that for smooth problems, pseudospectral methods typically have faster convergence rates than other methods, exhibiting “spectral accuracy” [4]. Pseudospectral methods were traditionally used to solve fluid dynamics problems [3], whereas orthogonal collocation methods were first established in the chemical engineering community [6]. Seminal work in the mathematics of orthogonal collocation methods for optimal control dates back to 1979 [7] and, in recent years, the following orthogonal collocationmethods have risen to prominence: the Legendre pseudospectral method [8], the Chebyshev pseudospectral method [9], the Radau orthogonal collocation method [10,11], and the Gauss pseudospectral method [12]. Within the class of pseudospectral methods, there are two very different and widely used implementation strategies that can be best described as local and global approaches. In a local approach, the time interval is divided into a large number of subintervals called segments or finite elements [6] and a small number of collocation points are usedwithin each segment. The segments are then linked via continuity conditions on the state, the independent variable, and possibly the control. The rationale for using local collocation is that a local method provides so-called local support [13] (i.e., the discretization points are located so that they support the local behavior of the dynamics) and is both computationally simple and efficient. Although local methods have a long history in solving optimal control problems, much of the recent work has shown great success in the application of global collocation [14,15] (i.e., collocation using a global polynomial across the entire time interval). Researchers are also looking into higher-order local methods [16,17], which lie in between global and local methods. In light of the recent results that promote global orthogonal collocation and the long history of the use local collocation, it is important to gain a better understanding as to how these two different philosophies compare in practice. This Note provides a comparison between global and local orthogonal collocation solutions for two optimal control problems.A recently developed orthogonal collocation method called the Gauss pseudospectral method (GPM) [12] is employed in both a global format and a local format. In the global approach, the GPM is implemented on a single segment and the number of collocation points is varied. As a local approach, the GPM is implemented such that a number of equal width segments are varied, while a small fixed number of collocation points are used in each segment. We note that the local application of the GPM is similar to the approaches of [6,10,18]. The results obtained in this research suggest that, except in special circumstances, global orthogonal collocation is preferable to local orthogonal collocation. For the smooth example in this study, the global GPM is much more accurate than the local GPM for a given number of total collocation points. Furthermore, for a desired accuracy, the global approach is computationally more efficient than the local approach in smooth problems. For nonsmooth problems, as in the second example, the local and global approach are quite similar in terms of accuracy.
[1]
B. Silva,et al.
Dynamic Trajectory Optimization Between Unstable Steady-States of a Class of CSTRs
,
2002
.
[2]
A. Rao,et al.
A State Approximation-Based Mesh Refinement Algorithm for Solving Optimal Control Problems Using Pseudospectral Methods
,
2009
.
[3]
Bengt Fornberg,et al.
A practical guide to pseudospectral methods: Introduction
,
1996
.
[4]
I. Michael Ross,et al.
Pseudospectral Knotting Methods for Solving Optimal Control Problems
,
2004
.
[5]
L. S. Pontryagin,et al.
Mathematical Theory of Optimal Processes
,
1962
.
[6]
Waldy K. Sjauw,et al.
Enhanced Procedures for Direct Trajectory Optimization Using Nonlinear Programming and Implicit Integration
,
2006
.
[7]
Anil V. Rao,et al.
Constrained Trajectory Optimization Using Pseudospectral Methods
,
2008
.
[8]
Richard G. Cobb,et al.
Three-Dimensional Trajectory Optimization Satisfying Waypoint and No-Fly Zone Constraints
,
2009
.
[9]
Anil V. Rao,et al.
EXTENSION OF A PSEUDOSPECTRAL LEGENDRE METHOD TO NON-SEQUENTIAL MULTIPLE-PHASE OPTIMAL CONTROL PROBLEMS
,
2003
.
[10]
G. Reddien.
Collocation at Gauss Points as a Discretization in Optimal Control
,
1979
.
[11]
A. L. Herman,et al.
Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules
,
1996
.
[12]
Geoffrey T. Huntington,et al.
A Comparison between Global and Local Orthogonal Collocation Methods for Solving Optimal Control Problems
,
2007,
2007 American Control Conference.
[13]
Philip Rabinowitz,et al.
Methods of Numerical Integration
,
1985
.
[14]
Anil V. Rao,et al.
Practical Methods for Optimal Control Using Nonlinear Programming
,
1987
.
[15]
L. Bittner.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishechenko, The Mathematical Theory of Optimal Processes. VIII + 360 S. New York/London 1962. John Wiley & Sons. Preis 90/–
,
1963
.
[16]
J. Betts.
Survey of Numerical Methods for Trajectory Optimization
,
1998
.
[17]
Michael A. Saunders,et al.
SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization
,
2002,
SIAM J. Optim..
[18]
I. Michael Ross,et al.
Pseudospectral Methods for Infinite-Horizon Nonlinear Optimal Control Problems
,
2005
.
[19]
R. V. Dooren,et al.
A Chebyshev technique for solving nonlinear optimal control problems
,
1988
.
[20]
Gamal N. Elnagar,et al.
The pseudospectral Legendre method for discretizing optimal control problems
,
1995,
IEEE Trans. Autom. Control..
[21]
Donald E. Kirk,et al.
Optimal control theory : an introduction
,
1970
.
[22]
J. E. Cuthrell,et al.
Simultaneous optimization and solution methods for batch reactor control profiles
,
1989
.
[23]
I. Michael Ross,et al.
A Spectral Patching Method for Direct Trajectory Optimization
,
2000
.
[24]
Anil V. Rao,et al.
Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method
,
2006
.
[25]
L. Trefethen.
Spectral Methods in MATLAB
,
2000
.
[26]
T. A. Zang,et al.
Spectral methods for fluid dynamics
,
1987
.
[27]
J. Villadsen,et al.
Solution of differential equation models by polynomial approximation
,
1978
.
[28]
I. Michael Ross,et al.
A Direct Method for Solving Nonsmooth Optimal Control Problems
,
2002
.