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Kalman–Yakubovich–Popov Lemma
The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number \gamma > 0, two n-vectors B, C and an n x n Hurwitz matrix A, if the pair (A,B) is completely controllable, then a symmetric matrix P and a vector Q satisfying :A^T P + P A = -Q Q^T : P B-C = \sqrtQ exist if and only if : \gamma+2 Re ^T (j\omega I-A)^Bge 0 Moreover, the set \ is the unobservable subspace for the pair (C,A). The lemma can be seen as a generalization of the Lyapunov equation in stability theory. It establishes a relation between a linear matrix inequality involving the state space constructs A, B, C and a condition in the frequency domain. The Kalman–Popov–Yakubovich lemma which was first formulated and proved in 1962 by Vladimir Andreevich Yakubovich where it was stated that for the strict frequency inequality. The case of nonstrict frequency inequality was published in 1963 by Rudolf E. Kálmán Rudolf Emil Kálmán (May 19, 1930 – ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Analysis
System analysis in the field of electrical engineering characterizes electrical systems and their properties. System analysis can be used to represent almost anything from population growth to audio speakers; electrical engineers often use it because of its direct relevance to many areas of their discipline, most notably signal processing, communication systems and control systems. Characterization of systems A system is characterized by how it responds to input signals. In general, a system has one or more input signals and one or more output signals. Therefore, one natural characterization of systems is by how many inputs and outputs they have: * '' SISO'' (Single Input, Single Output) * ''SIMO'' (Single Input, Multiple Outputs) * ''MISO'' (Multiple Inputs, Single Output) * ''MIMO'' (Multiple Inputs, Multiple Outputs) It is often useful (or necessary) to break up a system into smaller pieces for analysis. Therefore, we can regard a SIMO system as multiple SISO systems (on ... [...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] |
Hurwitz Matrix
In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial. Hurwitz matrix and the Hurwitz stability criterion Namely, given a real polynomial :p(z)=a_z^n+a_z^+\cdots+a_z+a_n the n\times n square matrix : H= \begin a_1 & a_3 & a_5 & \dots & \dots & \dots & 0 & 0 & 0 \\ a_0 & a_2 & a_4 & & & & \vdots & \vdots & \vdots \\ 0 & a_1 & a_3 & & & & \vdots & \vdots & \vdots \\ \vdots & a_0 & a_2 & \ddots & & & 0 & \vdots & \vdots \\ \vdots & 0 & a_1 & & \ddots & & a_n & \vdots & \vdots \\ \vdots & \vdots & a_0 & & & \ddots & a_ & 0 & \vdots \\ \vdots & \vdots & 0 & & & & a_ & a_n & \vdots \\ \vdots & \vdots & \vdots & & & & a_ & a_ & 0 \\ 0 & 0 & 0 & \dots & \dots & \dots & a_ & a_ & a_n \end. is called Hurwitz matrix corresponding to the polynomial p. It was established by Adolf Hurwitz in 1895 that a real polynomial with a_0 > 0 is stable (that is, all its roots have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Controllability
Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. Controllability and observability are dual aspects of the same problem. Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied. The following are examples of variations of controllability notions which have been introduced in the systems and control literature: * State controllability * Output controllability * Controllability in the behavioural framework State controllability The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Equation
In control theory, the discrete Lyapunov equation is of the form :A X A^ - X + Q = 0 where Q is a Hermitian matrix and A^H is the conjugate transpose of A. The continuous Lyapunov equation is of the form :AX + XA^H + Q = 0. The Lyapunov equation occurs in many branches of control theory, such as stability analysis and optimal control. This and related equations are named after the Russian mathematician Aleksandr Lyapunov. Application to stability In the following theorems A, P, Q \in \mathbb^, and P and Q are symmetric. The notation P>0 means that the matrix P is positive definite. Theorem (continuous time version). Given any Q>0, there exists a unique P>0 satisfying A^T P + P A + Q = 0 if and only if the linear system \dot=A x is globally asymptotically stable. The quadratic function V(x)=x^T P x is a Lyapunov function that can be used to verify stability. Theorem (discrete time version). Given any Q>0, there exists a unique P>0 satisfying A^T P A -P + Q = 0 if and only if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Matrix Inequality
In convex optimization, a linear matrix inequality (LMI) is an expression of the form : \operatorname(y):=A_0+y_1A_1+y_2A_2+\cdots+y_m A_m\succeq 0\, where * y= _i\,,~i\!=\!1,\dots, m/math> is a real vector, * A_0, A_1, A_2,\dots,A_m are n\times n symmetric matrices \mathbb^n, * B\succeq0 is a generalized inequality meaning B is a positive semidefinite matrix belonging to the positive semidefinite cone \mathbb_+ in the subspace of symmetric matrices \mathbb{S}. This linear matrix inequality specifies a convex constraint on ''y''. Applications There are efficient numerical methods to determine whether an LMI is feasible (''e.g.'', whether there exists a vector ''y'' such that LMI(''y'') ≥ 0), or to solve a convex optimization problem with LMI constraints. Many optimization problems in control theory, system identification and signal processing can be formulated using LMIs. Also LMIs find application in Polynomial Sum-Of-Squares. The prototypical primal and dual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the toy problem Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are the natural numbers starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped pendulum is a continuous (and therefore infinite) state space. Definition In the theory of dynamical systems, the state space of a discrete system defined by a function ''ƒ'' can be modeled as a directed graph where each possible state of the dynamical system is represented by a vertex with a directed edge from ''a'' to ''b'' if and only if ''ƒ''(''a'') = ''b''. This is known as a state diagram. For a cont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal. A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourier transform, which converts a time function into a complex valued sum or integral of sine waves of different frequencies, with amplitudes and phases, each of which represents a frequency component. The "spectrum" of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Andreevich Yakubovich
Vladimir Andreevich Yakubovich (October 21, 1926 in Novosibirsk – August 17, 2012 in the Gdov region) was a notable Russian control theorist and head of the Department of Theoretical Cybernetics at Saint Petersburg State University (formerly Leningrad University). In 1996 he received the IEEE Control Systems Award for his contributions to control theory, including the Kalman–Yakubovich–Popov lemma The Kalman–Yakubovich–Popov lemma is a result in system analysis and control theory which states: Given a number \gamma > 0, two n-vectors B, C and an n x n Hurwitz matrix A, if the pair (A,B) is completely controllable, then a symmetric matrix .... References External links *Personal web page (English*Personal web page (Russian*On-line C(includes photo) S. Abramovich, N. Kuznetsov, G. Leonov, V. A. Yakubovich — mathematician, “father of the field”, and herald of intellectual democracy in science and society, IFAC-PapersOnLine, 48(11), 2015, 1–3 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vasile M
The male name Vasile is of Greek origin and means "King". Vasile is a male Romanian given name or a surname. It is equivalent to the English name Basil. As a given name As a surname *Cristian Vasile (1908–1985), Romanian tango-romance singer *Nicolae Vasile (born 1995), Romanian professional footballer *Niculina Vasile (born 1958), former Romanian high jumper * Radu Vasile (1942–2013), Romanian politician and Prime Minister *Ștefan Vasile (born 1982), Romanian Olympic canoer Places *Pârâul lui Vasile, a river in Romania * Valea lui Vasile, a river in Romania * Vasile Aron (Sibiu district) See also * Vasiliu (surname) * Vasilescu (surname) * Vasilievca (other) * Vasile Alecsandri (other) * Vasileuți Vasileuți is a commune in Rîșcani District, Moldova Moldova ( , ; ), officially the Republic of Moldova ( ro, Republica Moldova), is a Landlocked country, landlocked country in Eastern Europe. It is bordered by Romania to the west ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemmas , part of a neuron
{{Disambiguation ...
Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a part of a grass plant * Lemma (mathematics), a type of proposition Other uses * ''Lemma'' (album), by John Zorn (2013) * Lemma (logic), an informal contention See also *Analemma, a diagram showing the variation of the position of the Sun in the sky *Dilemma *Lema (other) * Lemmatisation *Neurolemma Neurilemma (also known as neurolemma, sheath of Schwann, or Schwann's sheath) is the outermost nucleated cytoplasmic layer of Schwann cells (also called neurilemmocytes) that surrounds the axon of the neuron. It forms the outermost layer of the ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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