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Lyapunov Equation
In control theory , the DISCRETE LYAPUNOV EQUATION is of the form A X A H X + Q = 0 {displaystyle AXA^{H}-X+Q=0} where Q {displaystyle Q} is a Hermitian matrix and A H {displaystyle A^{H}} is the conjugate transpose of A {displaystyle A} . The CONTINUOUS LYAPUNOV EQUATION is of form A X + X A H + Q = 0 {displaystyle 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
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Control Theory
CONTROL THEORY is an interdisciplinary branch of engineering and computational mathematics that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback . The usual objective of control theory is to control a system, often called the plant , so its output follows a desired control signal, called the reference , which may be a fixed or changing value. To do this, a controller is designed, which monitors the output and compares it with the reference. The difference between actual and desired output, called the error signal, is applied as feedback to the input of the system, to bring the actual output closer to the reference. Some topics studied in control theory are stability (whether the output will converge to the reference value or oscillate about it), controllability and observability . Extensive use is usually made of a diagrammatic style known as the block diagram
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Lyapunov Function
In the theory of ordinary differential equations (ODEs), LYAPUNOV FUNCTIONS are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov , Lyapunov functions (also called the 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
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Conformable
In mathematics , a matrix is CONFORMABLE if its dimensions are suitable for defining some operation (e.g. addition, multiplication, etc.). EXAMPLES * If two matrices have the same dimensions (number of rows and number of columns), they are conformable for addition. * Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. That is, if A is an m × n matrix and B is an s × p matrix, then n needs to be equal to s for the matrix product AB to be defined. In this case, we say that A and B are conformable for multiplication (in that sequence). * Since squaring a matrix involves multiplying it by itself (A2 = AA) a matrix must be m × m (that is, it must be a square matrix ) to be conformable for squaring
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International Standard Book Number
The INTERNATIONAL STANDARD BOOK NUMBER (ISBN) is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation (except reprintings) of a book. For example, an e-book , a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, and 10 digits long if assigned before 2007. The method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit STANDARD BOOK NUMBERING (SBN) created in 1966. The 10-digit ISBN format was developed by the International Organization for Standardization (ISO) and was published in 1970 as international standard ISO 2108 (the SBN code can be converted to a ten digit ISBN by prefixing it with a zero)
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Digital Object Identifier
In computing, a DIGITAL OBJECT IDENTIFIER or DOI is a persistent identifier or handle used to uniquely identify objects, standardized by the International Organization for Standardization
International Organization for Standardization
( ISO
ISO
). An implementation of the Handle System , DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos. A DOI aims to be "resolvable", usually to some form of access to the information object to which the DOI refers. This is achieved by binding the DOI to metadata about the object, such as a URL , indicating where the object can be found
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Positive-definite Matrix
In linear algebra , a symmetric n {displaystyle n} × n {displaystyle n} real matrix M {displaystyle M} is said to be POSITIVE DEFINITE if the scalar z T M z {displaystyle z^{mathrm {T} }Mz} is positive for every non-zero column vector z {displaystyle z} of n {displaystyle n} real numbers. Here z T {displaystyle z^{mathrm {T} }} denotes the transpose of z {displaystyle z} . More generally, an n {displaystyle n} × n {displaystyle n} Hermitian matrix M {displaystyle M} is said to be POSITIVE DEFINITE if the scalar z M z {displaystyle z^{*}Mz} is real and positive for all non-zero column vectors z {displaystyle z} of n {displaystyle n} complex numbers. Here z {displaystyle z^{*}} denotes the conjugate transpose of z {displaystyle z}
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Special
SPECIAL or SPECIALS may refer to: CONTENTS * 1 Music * 2 Film and television * 3 Other uses * 4 See also MUSIC * Special (album) , a 1992 album by Vesta Williams * "Special" (Garbage song) , 1998 * "Special" (Mew song) , 2005 * "Special" (Stephen Lynch song) , 2000 * The Specials
The Specials
, a British band * "Special", a song by Violent Femmes on The Blind Leading the Naked * "Special", a song on
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Hermitian Matrix
In mathematics, a HERMITIAN MATRIX (or SELF-ADJOINT MATRIX) is a complex square matrix that is equal to its own conjugate transpose —that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j: a i j = a j i {displaystyle a_{ij}={overline {a_{ji}}}} or A = A T {displaystyle A={overline {A^{text{T}}}}} , in matrix form. Hermitian matrices can be understood as the complex extension of real symmetric matrices . If the conjugate transpose of a matrix A {displaystyle A} is denoted by A H {displaystyle A^{text{H}}} , then the Hermitian property can be written concisely as A = A H
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Conjugate Transpose
In mathematics , the CONJUGATE TRANSPOSE or HERMITIAN TRANSPOSE of an m-by-n matrix A with complex entries is the n-by-m matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each entry (i.e. negating their imaginary parts but not their real parts)
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Lyapunov Stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems . The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov. In simple terms, if the solutions that start out near an equilibrium point x e {displaystyle x_{e}} stay near x e {displaystyle x_{e}} forever, then x e {displaystyle x_{e}} is LYAPUNOV STABLE. More strongly, if x e {displaystyle x_{e}} is Lyapunov stable and all solutions that start out near x e {displaystyle x_{e}} converge to x e {displaystyle x_{e}} , then x e {displaystyle x_{e}} is asymptotically stable. The notion of EXPONENTIAL STABILITY guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge
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Optimal Control
OPTIMAL CONTROL THEORY deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. It is an extension of the calculus of variations , and is a mathematical optimization method for deriving control policies . The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane . Optimal control can be seen as a control strategy in control theory . CONTENTS * 1 General method * 2 Linear quadratic control * 3 Numerical methods for optimal control * 4 Discrete-time optimal control * 5 Examples * 5.1 Finite time * 6 See also * 7 References * 8 Further reading * 9 External links GENERAL METHOD Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved
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Kalman Filter
KALMAN FILTERING, also known as LINEAR QUADRATIC ESTIMATION (LQE), is an algorithm that uses a series of measurements observed over time, containing 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
Rudolf E. Kálmán
, one of the primary developers of its theory. The Kalman filter
Kalman filter
has numerous applications in technology. A common application is for guidance, navigation, and control of vehicles, particularly aircraft and spacecraft. Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics
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Kronecker Product
In mathematics , the KRONECKER PRODUCT, denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix . It is a generalization of the outer product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix of the tensor product with respect to a standard choice of basis . The Kronecker product should not be confused with the usual matrix multiplication , which is an entirely different operation. The Kronecker product is named after Leopold Kronecker
Leopold Kronecker
, even though there is little evidence that he was the first to define and use it. Indeed, in the past the Kronecker product was sometimes called the Zehfuss matrix, after Johann Georg Zehfuss who in 1858 described the matrix operation we now know as the Kronecker product
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Algebraic Riccati Equation
An ALGEBRAIC RICCATI EQUATION is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time . A typical algebraic Riccati equation is similar to one of the following: the continuous time algebraic Riccati equation (CARE): A T X + X A X B R 1 B T X + Q = 0 {displaystyle A^{T}X+XA-XBR^{-1}B^{T}X+Q=0,} or the discrete time algebraic Riccati equation (DARE): X = A T X A ( A T X B ) ( R + B T X B ) 1 ( B T X A ) + Q . {displaystyle X=A^{T}XA-(A^{T}XB)(R+B^{T}XB)^{-1}(B^{T}XA)+Q.,} X is the unknown n by n symmetric matrix and A, B, Q, R are known real coefficient matrices. Though generally this equation can have many solutions, it is usually specified that we want to obtain the unique stabilizing solution, if such a solution exists
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Sylvester Equation
In mathematics , in the field of control theory , a SYLVESTER EQUATION is a matrix equation of the form: A X + X B = C . {displaystyle AX+XB=C.} Then given matrices A,B, and C, the problem is to find the possible matrices X that obey this equation. All matrices are assumed to have coefficients in the complex numbers . For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, A and B must be square matrices of sizes n and m respectively, and then X and C both have n rows and m columns. A Sylvester equation has a unique solution for X exactly when there are no common eigenvalues of A and -B. More generally, the equation AX+XB=C has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space
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