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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 Eduardo Daniel Sontag (born April 16, 1951, in Buenos Aires, Argentina) is an Argentine-American mathematician, and distinguished university professor at Northeastern University, who works in the fields control theory, dynamical systems, syste ... provided a constructive version of this theorem explicitly exhibiting the feedback. See also * Analysis and control of nonlinear systems * Control-Lyapunov function References Control theory Theorems in dynamical systems {{mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear System
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Locally Lipschitz
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (or '' modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict incl ... [...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 He ... [...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 gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Control
Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dynamical systems with inputs, and how to modify the output by changes in the input using feedback, feedforward, or signal filtering. The system to be controlled is called the " plant". One way to make the output of a system follow a desired reference signal is to compare the output of the plant to the desired output, and provide feedback to the plant to modify the output to bring it closer to the desired output. Control theory is divided into two branches. Linear control theory applies to systems made of devices which obey the superposition principle. They are governed by linear differential equations. A major subclass is systems which in addition have parameters which do not change with time, called '' linear time invariant'' (L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
