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Pseudospectral Knotting Method
In applied mathematics, the pseudospectral knotting method is a generalization and enhancement of a standard pseudospectral method for optimal control. The concept was introduced by I. Michael Ross and F. Fahroo in 2004, and forms part of the collection of the Ross–Fahroo pseudospectral methods.Ross, I. M. and Fahroo, F., Pseudospectral Knotting Methods for Solving Optimal Control Problems, ''Journal of Guidance, Control and Dynamics,'' Vol. 27, No. 3, pp. 397–405, 2004. Definition According to Ross and Fahroo a pseudospectral (PS) knot is a double Lobatto point; i.e. two boundary points on top of one another. At this point, information (such as discontinuities, jumps, dimension changes etc.) is exchanged between two standard PS methods. This information exchange is used to solve some of the most difficult problems in optimal control known as hybrid optimal control problems.Ross, I. M. and D’Souza, C. N., A Hybrid Optimal Control Framework for Mission Planning, ''Journal o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudospectral Optimal Control
Pseudospectral optimal control is a joint theoretical-computational method for solving optimal control problems. It combines pseudospectral (PS) theory with optimal control theory to produce PS optimal control theory. PS optimal control theory has been used in ground and flight systems in military and industrial applications. The 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, ascent guidance and quantum control. Overview There are a very large number of ideas that fall under the general banner of pseudospectral optimal control. Examples of these are the Legendre pseudospectral method, the Chebyshev pseudospectral method, the Gauss pseudospectral method, the Ross-Fahroo pseudospectral method, the Bellman pseudos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fariba Fahroo
Fariba Fahroo is an American Persian mathematician, a program manager at the Air Force Office of Scientific Research, and a former program manager at the Defense Sciences Office. Along with I. M. Ross, she has published papers in pseudospectral optimal control theory.F. Fahroo and I. M. Ross, “A Second Look at Approximating Differential Inclusions,” Journal of Guidance, Control, and Dynamics, Vol.24, No.1, January–February 2001. F. Fahroo and I. M. Ross, “Costate Estimation by a Legendre Pseudospectral Method,” Journal of Guidance, Control, and Dynamics, Vol.24, No.2, March–April 2001, pp. 270–277F. Fahroo and I. M. Ross, “Direct Trajectory Optimization by a Chebyshev Pseudospectral Method,” Journal of Guidance, Control, and Dynamics, Vol. 25, No. 1, 2002, pp. 160–166F. Fahroo and I. M. Ross, “Pseudospectral Methods for Infinite-Horizon Optimal Control Problems,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, pp. 927–936, 2008. F. Fahroo and I. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Algorithms Boost Satellite Performance on the Cheap", ''IEEE Spectrum'', November 2012. 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, Legendre Pseudospectral Approximations of Optimal Control Problems, ''Lecture Notes in Control and Information Sciences'', Vol. 295, Springer-Verlag, 2003. 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. Examples of the Ross–Fahroo pseudospectral methods are th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DIDO (optimal Control)
DIDO ( ) is a MATLAB optimal control toolbox for solving general-purpose optimal control problems.Ross, I. M. ''A Primer on Pontryagin's Principle in Optimal Control'', Second Edition, Collegiate Publishers, San Francisco, 2015.Eren, H., "Optimal Control and the Software," ''Measurements, Instrumentation, and Sensors Handbook'', Second Edition, CRC Press, 2014, pp.92-1-16. It is widely used in academia, industry, and NASA. Hailed as a breakthrough software, DIDO is based on the pseudospectral optimal control theory of Ross and Fahroo. The latest enhancements to DIDO are described in Ross. Usage DIDO utilizes trademarked expressions and objects that facilitate a user to quickly formulate and solve optimal control problems.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. http://dspace.mit.edu/handle/1721.1/32431 Rapidity in formulation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Pseudospectral Method
The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. A basic version of the Legendre pseudospectral was originally proposed by Elnagar and his coworkers in 1995.G. Elnagar, M. A. Kazemi, and M. Razzaghi, "The Pseudospectral Legendre Method for Discretizing Optimal Control Problems," ''IEEE Transactions on Automatic Control,'' 40:1793–1796, 1995. Since then, Ross, Fahroo and their coworkers have extended, generalized and applied the method for a large range of problems.Q. Gong, W. Kang, N. Bedrossian, F. Fahroo, P. Sekhavat and K. Bollino, "Pseudospectral Optimal Control for Military and Industrial Applications," ''46th IEEE Conference on Decision and Control,'' New Orleans, LA, pp. 4128–4142, Dec. 2007. An application that has received wide publicity is the use of their method for generating real time trajectories for the International Space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Pseudospectral Method
The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first kind. It is part of the larger theory of pseudospectral optimal control, a term coined by Ross. Unlike the Legendre pseudospectral method, the Chebyshev pseudospectral (PS) method does not immediately offer high-accuracy quadrature solutions. Consequently, two different versions of the method have been proposed: one by Elnagar et al., and another by Fahroo and Ross. The two versions differ in their quadrature techniques. The Fahroo–Ross method is more commonly used today due to the ease in implementation of the Clenshaw–Curtis quadrature technique (in contrast to Elnagar–Kazemi's cell-averaging method). In 2008, Trefethen showed that the Clenshaw–Curtis method was nearly as accurate as Gauss quadrature. This breakthrough result opened the door for a covector mapping theorem for Chebyshev PS methods. A complete mathematical theory for Chebyshev PS methods was fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ross–Fahroo Lemma
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'', November 2012. It states that dualization and discretization are, in general, non-commutative operations. The operations can be made commutative by an application of the covector mapping principle. Description of the theory A continuous-time optimal control problem is information rich. A nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 solution must be computed for controllability and stability. This time constant, known as Ross' time constant, is proportional to the inverse of the Lipschitz constant of the vector field that governs the dynamics of a nonlinear control system. Theoretical implications The proportionality factor in the definition of Ross' time constant is dependent upon the magnitude of the disturbance on the plant and the specifications for feedback control. When there are no disturbances, Ross' -lemma shows that the open-loop optimal solution is the same as the closed-loop one. In the presence of disturbances, the proportionality factor can be written in terms of the Lambert W-function. Practical applications In practical applications, Ross' time constant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
