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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] |
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] |
Dual Number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0. Dual numbers can be added component-wise, and multiplied by the formula : (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon, which follows from the property and the fact that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. History Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as , where is the angle be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GPOPS-II
GPOPS-II (pronounced "GPOPS 2") is a general-purpose MATLAB software for solving continuous optimal control problems using hp-adaptive Gaussian quadrature collocation and sparse nonlinear programming. The acronym GPOPS stands for "General Purpose OPtimal Control Software", and the Roman numeral "II" refers to the fact that GPOPS-II is the second software of its type (that employs Gaussian quadrature integration). Problem Formulation GPOPS-II is designed to solve multiple-phase optimal control problems of the following mathematical form (where P is the number of phases): ::: \min J = \phi(\mathbf^,\ldots,\mathbf^) :subject to the dynamic constraints ::: \dot^(t)=\mathbf^(\mathbf^(t),\mathbf^(t),t,\mathbf),\quad (p=1,\ldots,P), :the event constraints ::: \mathbf_\leq\mathbf(\mathbf^,\ldots,\mathbf^,\mathbf)\leq\mathbf_, :the inequality path constraints ::: \mathbf_^\leq\mathbf(\mathbf^(t),\mathbf^(t),t,\mathbf)\leq\mathbf_^,\quad (p=1,\ldots,P), :the static parameter constraints : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dido (Queen Of Carthage)
Dido ( ; , ), also known as Elissa ( , ), was the legendary founder and first queen of the Phoenician city-state of Carthage (located in modern Tunisia), in 814 BC. In most accounts, she was the queen of the Phoenician city-state of Tyre (today in Lebanon) who fled tyranny to found her own city in northwest Africa. Known only through ancient Greek and Roman sources, all of which were written well after Carthage's founding, her historicity remains uncertain. The oldest references to Dido are attributed to Timaeus, who was active around 300 BC, or about five centuries after the date given for the foundation of Carthage. Details about Dido's character, life, and role in the founding of Carthage are best known from the account given in Virgil's epic poem, the ''Aeneid,'' written around 20 BC, which tells the legendary story of the Trojan hero Aeneas. Dido is described as a clever and enterprising woman who flees her ruthless and autocratic brother, Pygmalion, after discovering ... [...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] |
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] |
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] |
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] |
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] |
Flat Pseudospectral Method
The flat pseudospectral method is part of the family of the Ross–Fahroo pseudospectral methods introduced by Ross and Fahroo. 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. The method combines the concept of differential flatness with pseudospectral optimal control to generate outputs in the so-called flat space. 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 Optimiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Mapping Principle,” Proceedings of the 2005 AAS/AIAA Astrodynamics Specialist Conference, August 7–11, 2005 Lake Tahoe, CA. AAS 05-332.Ross, I. M. and Fahroo, F., “Legendre Pseudospectral Approximations of Optimal Control Problems,” Lecture Notes in Control and Information Sciences, Vol. 295, Springer-Verlag, New York, 2003, pp 327–342.Ross, I. M. and Fahroo, F., “Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems,” Proceedings of the American Control Conference, June 2004, Boston, MARoss, I. M. and Fahroo, F., “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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to ''quadrature'') is more or less a synonym for ''numerical integration'', especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take ''quadrature'' to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
