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Separation Principle In Stochastic Control
The separation principle is one of the fundamental principles of stochastic control theory, which states that the problems of optimal control and state estimation can be decoupled under certain conditions. In its most basic formulation it deals with a linear stochastic system :\begin dx & =A(t)x(t)\,dt+B_1(t)u(t)\,dt+B_2(t)\,dw \\ dy & =C(t)x(t)\,dt +D(t)\,dw \end with a state process x, an output process y and a control u, where w is a vector-valued Wiener process, x(0) is a zero-mean Gaussian random vector independent of w, y(0)=0, and A, B_1, B_2, C, D are matrix-valued functions which generally are taken to be continuous of bounded variation. Moreover, DD' is nonsingular on some interval ,T/math>. The problem is to design an output feedback law \pi:\, y \mapsto u which maps the observed process y to the control input u in a nonanticipatory manner so as to minimize the functional : J(u) = \mathbb\left\, where \mathbb denotes expected value, prime (') denotes transpose. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Control Theory
Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system. The system designer assumes, in a Bayesian probability-driven fashion, that random noise with known probability distribution affects the evolution and observation of the state variables. Stochastic control aims to design the time path of the controlled variables that performs the desired control task with minimum cost, somehow defined, despite the presence of this noise. The context may be either discrete time or continuous time. Certainty equivalence An extremely well-studied formulation in stochastic control is that of linear quadratic Gaussian control. Here the model is linear, the objective function is the expected value of a quadratic form, and the disturbances are purely additive. A basic result for discrete-time centralized systems with only additive uncertainty is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener Process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is often also called Brownian motion due to its historical connection with the physical process of the same name originally observed by Scottish botanist Robert Brown (Scottish botanist from Montrose), Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary increments, stationary independent increments) and occurs frequently in pure and applied mathematics, economy, economics, quantitative finance, evolutionary biology, and physics. The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingale (probability theory), martingales. It is a key process in terms of which more complicated sto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous adjective ''Gaussian'' is pronounced . Mathematics Algebra and linear algebra Geometry and differential geometry Number theory Cyclotomic fields *Gaussian period *Gaussian rational *Gauss sum, an exponential sum over Dirichlet characters ** Elliptic Gauss sum, an analog of a Gauss sum **Quadratic Gauss sum Analysis, numerical analysis, vector calculus and calculus of variations Complex analysis and convex analysis *Gauss–Lucas theorem *Gauss's continued fraction, an analytic continued fraction derived from the hypergeometric functions * Gauss's criterion – described oEncyclopedia of Mathematics* Gauss's hypergeometric theorem, an identity on hypergeometric series *Gauss plane Statistics *Gauss–Kuzmi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kalman Filter
For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory. This digital filter is sometimes termed the ''Stratonovich–Kalman–Bucy filter'' because it is a special case of a more general, nonlinear filter developed somewhat earlier by the Soviet mathematician Ruslan Stratonovich. In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before summer 1960, when Kalman met with Stratonovich during a conference in Moscow. Kalman filtering has numerous tech ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Murray Wonham
Walter Murray Wonham (1934–2023) is a Canadian control theorist and professor at the University of Toronto. He dealt with multi-variable geometric control theory, stochastic control and stochastic filters, and more recently the control of discrete event systems from the standpoint of mathematical logic and formal languages. Education Wonham attended a boys’ school and preferred individual to team sports, taking up sailing and tennis. Wonham obtained his bachelor's degree in engineering physics from McGill University in 1956, and then a doctorate in stochastic control from the University of Cambridge in 1961. Career In the 1960s, he worked at Purdue University's Control and Information Systems Laboratory, Martin Marietta's Research Institute for Advanced Studies (RIAS) in Baltimore, Brown University's Department of Applied Mathematics, and in the Office for Control Theory and Applications of NASA's Electronics for Research Center, where he developed a geometric theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girsanov
In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying instrument (such as a share price or interest rate) will take a particular value or values to the risk-neutral measure which is a very useful tool for evaluating the value of derivatives on the underlying. History Results of this type were first proved by Cameron-Martin in the 1940s and by Igor Girsanov in 1960. They have been subsequently extended to more general classes of process culminating in the general form of Lenglart (1977). Significance Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if ''Q'' is a measure that is absolutely continuous with respect to ''P'' then every ''P''-semimartingale is a ''Q''-semimartingale. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Càdlàg
In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space. Two related terms are càglàd, standing for "continue à gauche, limite à droite", the left-right reversal of càdlàg, and càllàl for "continue à l'un, limite à l’autre" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd. Definition Let be a metric space, and let . A function is called a càdlàg function if, for every , * the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. Control theory is used in control system eng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
