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Unscented Optimal Control
In mathematics, unscented optimal control combines the notion of the unscented transform with deterministic optimal control to address a class of uncertain optimal control problems.Unscented Optimal Control for Orbital and Proximity Operations in an Uncertain Environment: A New Zermelo Problem I. Michael Ross, Ronald Proulx, Mark Karpenko August 2014, American Institute of Aeronautics and Astronautics (AIAA) It is a specific application of Riemmann-Stieltjes optimal control theory, a concept introduced by Ross and his coworkers. Mathematical description Suppose that the initial state x^0 of a dynamical system, \dot = f(x, u, t) is an uncertain quantity. Let \Chi^i be the sigma points. Then sigma-copies of the dynamical system are given by, \dot\Chi^i = f(\Chi^i, u, t) Applying standard deterministic optimal control principles to this ensemble generates an unscented optimal control.Naoya Ozaki and Ryu Funase. "Tube Stochastic Diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unscented Transform
The unscented transform (UT) is a mathematical function used to estimate the result of applying a given nonlinear transformation to a probability distribution that is characterized only in terms of a finite set of statistics. The most common use of the unscented transform is in the nonlinear projection of mean and covariance estimates in the context of nonlinear extensions of the Kalman filter. Its creator Jeffrey Uhlmann explained that "unscented" was an arbitrary name that he adopted to avoid it being referred to as the “Uhlmann filter” though others have indicated that "unscented" is a contrast to "scented" intended as a euphemism for "stinky" Background Many filtering and control methods represent estimates of the state of a system in the form of a mean vector and an associated error covariance matrix. As an example, the estimated 2-dimensional position of an object of interest might be represented by a mean position vector, , y/math>, with an uncertainty given in the form ... [...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] |
Riemann–Stieltjes Integral
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability. Formal definition The Riemann–Stieltjes integral of a real-valued function f of a real variable on the interval ,b/math> with respect to another real-to-real function g is denoted by :\int_^b f(x) \, \mathrmg(x). Its definition uses a sequence of partitions P of the interval ,b/math> :P=\. The integral, then, is defined to be the limit, as the mesh (the length of the longest subinterval) of the partitions approaches 0 , of the approximating sum :S(P,f,g) = \sum_^ f(c_i)\left g(x_) - g(x_i) \right/math> where c_i is in the i-th subinterval _i;x_/math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unscented Transform
The unscented transform (UT) is a mathematical function used to estimate the result of applying a given nonlinear transformation to a probability distribution that is characterized only in terms of a finite set of statistics. The most common use of the unscented transform is in the nonlinear projection of mean and covariance estimates in the context of nonlinear extensions of the Kalman filter. Its creator Jeffrey Uhlmann explained that "unscented" was an arbitrary name that he adopted to avoid it being referred to as the “Uhlmann filter” though others have indicated that "unscented" is a contrast to "scented" intended as a euphemism for "stinky" Background Many filtering and control methods represent estimates of the state of a system in the form of a mean vector and an associated error covariance matrix. As an example, the estimated 2-dimensional position of an object of interest might be represented by a mean position vector, , y/math>, with an uncertainty given in the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tyche
Tyche (; Ancient Greek: Τύχη ''Túkhē'', 'Luck', , ; Roman equivalent: Fortuna) was the presiding tutelary deity who governed the fortune and prosperity of a city, its destiny. In Classical Greek mythology, she is the daughter of Aphrodite and Zeus or Hermes, and at this time served to bring positive messages to people, relating to external events outside their control. During the Hellenistic period, with dramatic socio-political changes starting with Alexander the Great, Tyche increasingly embodied the whims of fate (both negative and positive), eclipsing the role of the Olympic gods. The Greek historian Polybius believed that when no cause can be discovered to events such as floods, droughts, frosts, or even in politics, then the cause of these events may be fairly attributed to Tyche. Other ancient Greek sources corroborate Polybius, such as Pindar who claims Tyche could hand victory to a lesser athlete. This "Hellenistic Tyche" is often featured on coins such as those ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion process, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
