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Control-Lyapunov Function
In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is ''(Lyapunov) stable'' or (more restrictively) ''asymptotically stable''. Lyapunov stability means that if the system starts in a state x \ne 0 in some domain ''D'', then the state will remain in ''D'' for all time. For ''asymptotic stability'', the state is also required to converge to x = 0. A control-Lyapunov function is used to test whether a system is ''asymptotically stabilizable'', that is whether for any state ''x'' there exists a control u(x,t) such that the system can be brought to the zero state asymptotically by applying the control ''u''. The theory and application of control-Lyapunov functions were developed by Zvi Artstein and Eduardo D. Sontag in the 1980s and 1990s. Definition Consider an autonomous dynamical system with inputs where x\in\mathbb ... [...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] |
Differentiable Function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If is an interior point in the domain of a function , then is said to be ''differentiable at'' if the derivative f'(x_0) exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function f. Generally speaking, is said to be of class if its first k derivatives f^(x), f^(x), \ldots, f^(x) exist and are continuous over the domain of the func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Optimization
This article describes Lyapunov optimization for dynamical systems. It gives an example application to optimal control in queueing networks. Introduction Lyapunov optimization refers to the use of a Lyapunov function to optimally control a dynamical system. Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi-dimensional vector. A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state. Typically, the function is defined to grow large when the system moves towards undesirable states. System stability is achieved by taking control actions that make the Lyapunov function drift in the negative direction towards zero. Lyapunov drift is central to the study of optimal control in queueing networks. A typical goal is to stabilize all network queues while optimizing some performance objective, such as minimizing average energy or maximizing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If ''T'' is a tensor field and ''X'' is a vector field, then the Lie derivative of ''T'' with respect to ''X'' is denoted \mathcal_X(T). The differential operator T \mapsto \mathcal_X(T) is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francis Clarke (mathematician)
Frank "Francis" H. Clarke (born 30 July 1948, in Montreal) is a Canadian and French mathematician. Biography Francis Clarke graduated in 1969 from McGill University with a B.Sc. degree in 1969 and in 1973 from the University of Washington with a Ph.D. with thesis advisor R. Tyrrell Rockafellar. In 1978 Clarke became a full professor at the University of British Columbia and gave an invited lecture at the International Congress of Mathematicians (ICM) in Helsinki. In 1984 he was appointed director of the ''Centre de Recherches Mathématiques'' (CRM) of the University of Montreal. During the nine years of his directorship, CRM became Canada's leading national research center for mathematics and its applications. The successes of Clarke's directorship included the creation of workshops and postdoctoral fellowships, thematic years, two series of publications, research awards, and an endowment fund. Francis Clarke is also the founding director of the ''Institut des Sciences Mathématiq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artstein's Theorem
Artstein's theorem states that a nonlinear dynamical system in the control-affine form \dot = \mathbf + \sum_^m \mathbf_i(\mathbf)u_i has a differentiable control-Lyapunov function if and only if it admits a regular stabilizing feedback ''u''(''x''), that is a locally Lipschitz function on Rn\. The original 1983 proof by Zvi Artstein proceeds by a nonconstructive argument. In 1989 Eduardo D. Sontag provided a constructive version of this theorem explicitly exhibiting the feedback. See also * Analysis and control of nonlinear systems *Control-Lyapunov function In control theory, a control-Lyapunov function (CLF) is an extension of the idea of Lyapunov function V(x) to systems with control inputs. The ordinary Lyapunov function is used to test whether a dynamical system is ''(Lyapunov) stable'' or (more ... References Control theory Theorems in dynamical systems {{mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two vectors in the space is a Scalar (mathematics), scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite Dimension (vector space), dimension are widely used in functional analysis. Inner product spaces over the Field (mathematics), field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autonomous System (mathematics)
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems. Many laws in physics, where the independent variable is usually assumed to be time, are expressed as autonomous systems because it is assumed the laws of nature which hold now are identical to those for any point in the past or future. Definition An autonomous system is a system of ordinary differential equations of the form \fracx(t)=f(x(t)) where takes values in -dimensional Euclidean space; is often interpreted as time. It is distinguished from systems of differential equations of the form \fracx(t)=g(x(t),t) in which the law governing the evolution of the system does not depend solely on the system's current state but also the parameter , again often interpreted as time; such systems are by definition not autonomous. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduardo D
Eduardo is the Spanish and Portuguese form of the male given name Edward. Another version is Duarte. It may refer to: Association football * Eduardo Bonvallet, Chilean football player and sports commentator * Eduardo Carvalho, Portuguese footballer * Eduardo "Edu" Coimbra, Brazilian footballer * Eduardo Costa, Brazilian footballer * Eduardo da Conceição Maciel, Brazilian footballer * Eduardo da Silva, Brazilian-born Croatian footballer * Eduardo Adelino da Silva, Brazilian footballer * Eduardo Ribeiro dos Santos, Brazilian footballer * Eduardo Gómez (footballer), Chilean footballer * Eduardo Gonçalves de Oliveira, Brazilian footballer * Eduardo Jesus, Brazilian footballer * Eduardo Martini, Brazilian footballer * Eduardo Ferreira Abdo Pacheco, Brazilian footballer Music * Eduardo (rapper), Carlos Eduardo Taddeo, Brazilian rapper * Eduardo De Crescenzo, Italian singer, songwriter and multi-instrumentalist Politicians * Eduardo Año, Filipino politician and retired army g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zvi Artstein
Zvi ( he, צְבִי and , ''Tzvi'', Ṣvi, "gazelle") is a Jewish masculine given name. Notable people with this name include: * Zvi Aharoni (1921–2012), Israeli Mossad agent * Zvi Arad (1942–2018), Israeli mathematician, acting president of Bar-Ilan University, president of Netanya Academic College * Zvi Ben-Avraham (born 1941), Israeli geophysicist * Zvi Bodie, American academic * Zvi Hirsch Chajes (1805–1855), Orthodox Polish rabbi * Zvi Chalamish, Israeli financier * Zvi Elpeleg (1926–2015), Israeli academic * Zvi Galil (born 1947), Israeli computer scientist, mathematician, and President of Tel Aviv University * Zvika Greengold (born 1952), Israeli officer during the Yom Kippur War, awarded the Medal of Valor * Zvi Griliches (1930–1999), Jewish-American economist * Zvi Hirsch Grodzinsky (born 1857), American rabbi * Zvi Elimelech Halberstam (born 1952), Israeli rebbe * Zvi Hecker (born 1931), Israeli architect * Zvi Heifetz (born 1956), Israeli diplomat * Zvi Hende ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
