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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 In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser .... The Kalman–Popov–Yakubovich lemma which was first formu ... [...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 (one for eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, 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 theory, stability; often with the aim to achieve a degree of Optimal control, 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 Setpoint (control system), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurwitz-stable Matrix
In mathematics, a Hurwitz-stable matrix, or more commonly simply Hurwitz matrix, is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix. Such matrices play an important role in control theory. Definition A square matrix A is called a Hurwitz matrix if every eigenvalue of A has strictly negative real part, that is, :\operatorname lambda_i< 0\, for each eigenvalue . is also called a stable matrix, because then the differential equation : is , that is, as If is a (matrix-valued) |
Controllability
Controllability is an important property of a control system and plays a crucial role in many regulation problems, such as the stabilization of unstable systems using feedback, tracking problems, obtaining optimal control strategies, or, simply prescribing an input that has a desired effect on the state. Controllability and observability are dual notions. Controllability pertains to regulating the state by a choice of a suitable input, while observability pertains to being able to know the state by observing the output (assuming that the input is also being observed). Broadly speaking, the concept of controllability relates to the ability to steer a system around in its configuration space using only certain admissible manipulations. The exact definition varies depending on the framework or the type of models dealt with. The following are examples of variants of notions of controllability that have been introduced in the systems and control literature: * State controllability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Equation
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. In particular, the discrete-time Lyapunov equation (also known as Stein equation) for X is :A X A^ - X + Q = 0 where Q is a Hermitian matrix and A^H is the conjugate transpose of A, while the continuous-time Lyapunov equation is :AX + XA^H + Q = 0. 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 the linear ... [...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 sem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space
In computer science, a state space is a discrete space representing 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 State spaces are useful in computer science as a simple model of machines. Formally, a state space can be defined as a tuple [''N'', ''A'', ''S'', ''G''] where: * ''N'' is a Set (mathematics), set of states * ''A'' is a set of arcs connecting the states * ''S'' is a nonempty subset of ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Domain
In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time series. While a time-domain graph shows how a signal changes over time, a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A complex valued frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids (or other basis waveforms) at the frequency components of the signal. Although it is common to refer to the magnitude portion (the real valued frequency-domain) as the frequency response of a signal, the phase portion is required to uniquely define the 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 transfo ... [...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 matri .... 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
Vasile is a male Romanian given name or a surname. It is equivalent to the English name Basil which is of Greek origin and means "King". It is also used by the Megleno-Romanians. 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 The Pârâul lui Vasile is a left tributary of the river Miletin in Romania Romania is a country located at the crossroads of Central Europe, Central, Eastern Europe, Eastern and Southeast Europe. It borders Ukraine to the north and east, H ..., a river in Romania * Valea lui Vasile, a river in Romania * Vasile Aron (Sibiu district) See also * Vasiliu (surname) * Vasilescu (surname) * Vasilievca (other) * Vasile Alecs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemmas , part of a neuron
{{Disambiguation ...
Lemma (from Ancient Greek ''premise'', ''assumption'', from Greek ''I take'', ''I get'') 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 proven proposition used as a step in a larger proof Other uses * ''Lemma'' (album), by John Zorn (2013) 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 cell nucleus, nucleated cytoplasmic layer of Schwann cells (also called neurilemmocytes) that surrounds the axon of the neuron. It forms the outermost la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
