Gauss pseudospectral method
The Gauss Pseudospectral Method (abbreviated "GPM") is a direct transcription method for solving optimal control problems. The method is based on the theory of orthogonal collocation where the collocation points (i.e., the points at which the optimal control problem is discretized) are the Legendre-Gauss (LG) points. The history of orthogonal collocation at Gauss points dates back to the famous work of Carl deBoor[1]. Several years later, the excellent work using splines as approximating functions and LG points for collocation was developed by Reddien[2]. The topic of LG collocation in optimal control became prominent again in the mid 1980s in the chemical engineering community in the work of Cuthrell and Biegler.[3] Cuthrell and Biegler's work differed from that of Reddien in two ways. First, Cuthrell and Biegler used Lagrange polynomials as basis functions (as opposed to splines). Second, Cuthrell and Biegler placed a key focus on demonstrating the equivalence between the indirect (i.e., calculus of variations) and direct (i.e., nonlinear programming) forms of the optimal control problem. In a related paper, Cuthrell and Biegler introduced the concept of finite elements and arrived at a concept that they called a knot[4]. A knot, as defined by Cuthrell, is a location where two finite elements are connected (i.e., the end of one finite element is connected to the beginning of the next finite element).
In the 1990s and 2000s, the topic of orthogonal collocation in optimal control became a topic of interest. In 1995, Elnagar, et. al.[5]. In Elnagar 1995 work, the term pseudospectral (as opposed to orthogonal collocation) was used.[6]
- ^ deBoor, C. and Swartz, B., "Collocation at Gaussian Points," SIAM Journal on Numerical Analysis, Vol. 10, No. 4, September 1973, pp. 582-606
- ^ Reddien, G. W., "Collocation at Gaussian Points as a Discretization in Optimal Control," SIAM Journal on Control and Optimization, Vol. 17, No. 2, 1979, pp. 298--306.
- ^ Cuthrell, J.E. and Biegler, L.T., " Simultaneous Optimization and Solution Methods for Batch Reactor Control Profiles, Computers and Chemical Engineering, Vol. 13, No. 2., pp. 49-62, 1989
- ^ Cuthrell, J.E. and Biegler, L.T., "On the Optimization of Differential Algebraic Processes, AiChE Journal, Vol. 33, No. 8, 1987, pp. 1257-1270.
- ^ Elnagar. G., Kazemi, M.A., and Razzaghi, M., "The Pseudospectral Legendre Method for Discretizing Optimal Control Problems," IEEE Transactions on Automatic Control, Vol. 40, No. 10, October 1995, pp. 1793-1796.
- ^ Strictly speaking, pseudospectral methods (which had previously been used in computational fluid dynamics) and orthogonal collocation methods are not the same because "pseudospectral" refers to the form of the approximating functions and the fact that the dynamics are only satisfied at a finite number of points whereas "orthogonal collocation" refers to the way that the points are chosen (e.g., LG points which are the roots of a Legendre polynomial). As it turns out, however, pseudospectral methods are always implemented using orthogonal collocation, hence the two terms can effectively be used interchangeably. One other important distinction is that orthogonal collocation methods have typically been employed using local collocation whereas pseudospectral methods are employed using global (or semi-global) collocation. Since each approach has reached different communities, researchers in one field (e.g., chemical engineering) may use one term whereas researchers in another field (e.g., aerospace or electrical engineering) may use another term.