Named after
I. Michael Ross and
F. Fahroo, the Ross–Fahroo lemma is a fundamental result in
optimal control theory.
[
I. M. Ross and F. Fahroo, A Pseudospectral Transformation of the Covectors of Optimal Control Systems, Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29–31 August 2001.][
I. M. Ross and F. Fahroo, Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems, ''Proceedings of the American Control Conference, Invited Paper'', June 2004, Boston, MA.][N. Bedrossian, M. Karpenko, and S. Bhatt, "Overclock My Satellite: Sophisticated Algorithms Boost Satellite Performance on the Cheap", '']IEEE Spectrum
''IEEE Spectrum'' is a magazine edited by the Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical e ...
'', November 2012.
It states that dualization and
discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...
are, in general, non-commutative operations. The operations can be made commutative by an application of the
covector mapping principle
The covector mapping principle is a special case of Riesz' representation theorem, which is a fundamental theorem in functional analysis. The name was coined by Ross and co-workers,Ross, I. M., “A Historical Introduction to the Covector Mappin ...
.
Description of the theory
A continuous-time optimal control problem is information rich. A number of interesting properties of a given problem can be derived by applying the
Pontryagin's minimum principle
Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that i ...
or the
Hamilton–Jacobi–Bellman equation In optimal control theory, the Hamilton-Jacobi-Bellman (HJB) equation gives a necessary and sufficient condition for optimality of a control with respect to a loss function. It is, in general, a nonlinear partial differential equation in the valu ...
s. These theories implicitly use the continuity of time in their derivation.
[B. S. Mordukhovich, Variational Analysis and Generalized Differentiation: Basic Theory, Vol.330 of Grundlehren der Mathematischen Wissenschaften undamental Principles of Mathematical SciencesSeries, Springer, Berlin, 2005.]
When an optimal control problem is discretized, the Ross–Fahroo lemma asserts that there is a fundamental loss of information. This loss of information can be in the primal variables as in the value of the control at one or both of the boundary points or in the dual variables as in the value of the Hamiltonian over the time horizon.
[F. Fahroo and I. M. Ross, Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems, AIAA Guidance, Navigation and Control Conference, August 15–18, 2005, San Francisco, CA.] To address the information loss, Ross and Fahroo introduced the concept of closure conditions which allow the known information loss to be put back in. This is done by an application of the
covector mapping principle
The covector mapping principle is a special case of Riesz' representation theorem, which is a fundamental theorem in functional analysis. The name was coined by Ross and co-workers,Ross, I. M., “A Historical Introduction to the Covector Mappin ...
.
Applications to pseudospectral optimal control
When pseudospectral methods are applied to discretize optimal control problems, the implications of the Ross–Fahroo lemma appear in the form of the discrete covectors seemingly being discretized by the transpose of the differentiation matrix.
When the
covector mapping principle
The covector mapping principle is a special case of Riesz' representation theorem, which is a fundamental theorem in functional analysis. The name was coined by Ross and co-workers,Ross, I. M., “A Historical Introduction to the Covector Mappin ...
is applied, it reveals the proper transformation for the adjoints. Application of the transformation generates the
Ross–Fahroo pseudospectral method
Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control.N. Bedrossian, M. Karpenko, and S. Bhatt,
"Overclock My Satellite: Sophisticated Algorit ...
s.
[A. M. Hawkins, ''Constrained Trajectory Optimization of a Soft Lunar Landing From a Parking Orbit,'' S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2005.](_blank)
/ref>[J. R. Rea, ''A Legendre Pseudospectral Method for Rapid Optimization of Launch Vehicle Trajectories,'' S.M. Thesis, Dept. of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2001.](_blank)
/ref>
See also
*Ross' π lemma Ross' lemma, named after I. Michael Ross, is a result in computational optimal control. Based on generating Carathéodory- solutions for feedback control, Ross' -lemma states that there is fundamental time constant within which a control soluti ...
*Ross–Fahroo pseudospectral method
Introduced by I. Michael Ross and F. Fahroo, the Ross–Fahroo pseudospectral methods are a broad collection of pseudospectral methods for optimal control.N. Bedrossian, M. Karpenko, and S. Bhatt,
"Overclock My Satellite: Sophisticated Algorit ...
s
References
{{DEFAULTSORT:Pseudospectral Optimal Control
Optimal control
Numerical analysis
Control theory