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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
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Control Theory
Control theory
Control theory
in control systems engineering deals with the control of continuously operating dynamical systems in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability. To do this, a controller with the requisite corrective behaviour 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
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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, 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.[1] 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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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
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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
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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
Lev Pontryagin
and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane.[1] Optimal control can be seen as a control strategy in control theory.Contents1 General method 2 Linear quadratic control 3 Numerical methods for optimal control 4 Discrete-time optimal control 5 Examples5.1 Finite time6 See also 7 References 8 Further reading 9 External linksGeneral method[edit] 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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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992 album by Vesta Williams "Special" (Garbage song), 1998 "Special
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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 strictly positive for every non-zero column vector z displaystyle z of n displaystyle n real numbers
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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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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 di
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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.).[1] Examples[edit]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. Thus for example only a square matrix can be idempotent. Only a square matrix is conformable for matrix inversion
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International Standard Book Number
"ISBN" redirects here. For other uses, see ISBN (other).International Standard Book
Book
NumberA 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar codeAcronym ISBNIntroduced 1970; 48 years ago (1970)Managing organisation International ISBN AgencyNo. of digits 13 (formerly 10)Check digit Weighted sumExample 978-3-16-148410-0Website www.isbn-international.orgThe International Standard Book
Book
Number (ISBN) is a unique[a][b] numeric commercial book identifier. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.[1] 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
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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).[1] An implementation of the Handle System,[2][3] 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. Thus, by being actionable and interoperable, a DOI differs from identifiers such as ISBNs and ISRCs which aim only to uniquely identify their referents
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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
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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 )
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Sylvester Equation
In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form:[1] 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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