Flat pseudospectral method
The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo.[1][2] The method combines the concept of differential flatness with pseudospectral optimal control to generate outputs in the so-called flat space.[3][4]
Concept
Because the differentiation matrix, [math]\displaystyle{ D }[/math], in a pseudospectral method is square, higher-order derivatives of any polynomial, [math]\displaystyle{ y }[/math], can be obtained by powers of [math]\displaystyle{ D }[/math],
- [math]\displaystyle{ \begin{align} \dot y &= D Y \\ \ddot y & = D^2 Y \\ &{} \ \vdots \\ y^{(\beta)} &= D^\beta Y \end{align} }[/math]
where [math]\displaystyle{ Y }[/math] is the pseudospectral variable and [math]\displaystyle{ \beta }[/math] is a finite positive integer. By differential flatness, there exists functions [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] such that the state and control variables can be written as,
- [math]\displaystyle{ \begin{align} x & = a(y, \dot y, \ldots, y^{(\beta)}) \\ u & = b(y, \dot y, \ldots, y^{(\beta + 1)}) \end{align} }[/math]
The combination of these concepts generates the flat pseudospectral method; that is, x and u are written as,
- [math]\displaystyle{ x = a(Y, D Y, \ldots, D^\beta Y) }[/math]
- [math]\displaystyle{ u = b(Y, D Y, \ldots, D^{\beta + 1}Y) }[/math]
Thus, an optimal control problem can be quickly and easily transformed to a problem with just the Y pseudospectral variable.[1]
See also
References
- ↑ 1.0 1.1 Ross, I. M. and Fahroo, F., “Pseudospectral Methods for the Optimal Motion Planning of Differentially Flat Systems,” IEEE Transactions on Automatic Control, Vol.49, No.8, pp. 1410–1413, August 2004.
- ↑ Ross, I. M. and Fahroo, F., “A Unified Framework for Real-Time Optimal Control,” Proceedings of the IEEE Conference on Decision and Control, Maui, HI, December, 2003.
- ↑ Fliess, M., Lévine, J., Martin, Ph., and Rouchon, P., “Flatness and defect of nonlinear systems: Introductory theory and examples,” International Journal of Control, vol. 61, no. 6, pp. 1327–1361, 1995.
- ↑ Rathinam, M. and Murray, R. M., “Configuration flatness of Lagrangian systems underactuated by one control” SIAM Journal on Control and Optimization, 36, 164,1998.
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