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Ackermann's Formula
In control theory, Ackermann's formula is a control system design method for solving the pole allocation problem for invariant-time systems by Jürgen Ackermann. One of the primary problems in control system design is the creation of controllers that will change the dynamics of a system by changing the eigenvalues of the matrix representing the dynamics of the closed-loop system. This is equivalent to changing the poles of the associated transfer function in the case that there is no cancellation of poles and zeros. State feedback control Consider a linear continuous-time invariant system with a state-space representation : \dot(t)=Ax(t)+Bu(t) : y(t)=Cx(t) where ''x'' is the state vector, ''u'' is the input vector, and ''A'', ''B'' and ''C'' are matrices of compatible dimensions that represent the dynamics of the system. An input-output description of this system is given by the transfer function : G(s) = C(sI-A)^B=C\ \frac\ B. Since the denominator of the right equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ackermann Function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function is defined as follows for nonnegative integers ''m'' and ''n'': : \begin \operatorname(0, n) & = & n + 1 \\ \operatorname(m+1, 0) & = & \operatorname(m, 1) \\ \operatorname(m+1, n+1) & = & \operatorname(m, \operatorname(m+1, n)) \end Its value grows rapidly, even for small inputs. For example, is an integer o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Stability
:''See Lyapunov stability, which gives a definition of asymptotic stability for more general dynamical systems. All exponentially stable systems are also asymptotically stable.'' In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative real parts. (i.e., in the left half of the complex plane).David N. Cheban (2004), ''Global Attractors Of Non-autonomous Dissipative Dynamical Systems''. p. 47 A discrete-time input-to-output LTI system is exponentially stable if and only if the poles of its transfer function lie strictly within the unit circle centered on the origin of the complex plane. Exponential stability is a form of asymptotic stability. Systems that are not LTI are exponentially stable if their convergence is bounded by exponential decay. Practical consequences An exponentially stable LTI system is one that will not "blow up" ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Concepts
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Systems/State Feedback
Control may refer to: Basic meanings Economics and business * Control (management), an element of management * Control, an element of management accounting * Comptroller (or controller), a senior financial officer in an organization * Controlling interest, a percentage of voting stock shares sufficient to prevent opposition * Foreign exchange controls, regulations on trade * Internal control, a process to help achieve specific goals typically related to managing risk Mathematics and science * Control (optimal control theory), a variable for steering a controllable system of state variables toward a desired goal * Controlling for a variable in statistics * Scientific control, an experiment in which "confounding variables" are minimised to reduce error * Control variables, variables which are kept constant during an experiment * Biological pest control, a natural method of controlling pests * Control network in geodesy and surveying, a set of reference points of known geospatial coo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Observability
Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. In control theory, the observability and controllability of a linear system are mathematical duals. The concept of observability was introduced by the Hungarian-American engineer Rudolf E. Kálmán for linear dynamic systems. A dynamical system designed to estimate the state of a system from measurements of the outputs is called a state observer or simply an observer for that system. Definition Consider a physical system modeled in state-space representation. A system is said to be observable if, for every possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors). In other words, one can determine the behavior of the entire system from the system's outputs. On the other hand, if the system is not observable, there ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Observer
In control theory, a state observer or state estimator is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. It is typically computer-implemented, and provides the basis of many practical applications. Knowing the system state is necessary to solve many control theory problems; for example, stabilizing a system using state feedback. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer. Typical observer model Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Hamilton Theorem
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation. If is a given matrix and is the identity matrix, then the characteristic polynomial of is defined as p_A(\lambda)=\det(\lambda I_n-A), where is the determinant operation and is a variable for a scalar element of the base ring. Since the entries of the matrix (\lambda I_n-A) are (linear or constant) polynomials in , the determinant is also a degree- monic polynomial in , p_A(\lambda) = \lambda^n + c_\lambda^ + \cdots + c_1\lambda + c_0~. One can create an analogous polynomial p_A(A) in the matrix instead of the scalar variable , defined as p_A(A) = A^n + c_A^ + \cdots + c_1A + c_0I_n~. The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encyclopedia Of Life Support Systems
The Encyclopedia of Life Support Systems (EOLSS) is an integrated compendium of twenty one encyclopedias. The first Earth Summit of 1992, held in Rio de Janeiro, issued a document that is now famous as Agenda 21. This document refers to the Earth's life support systems, considering the whole of our planet as a grand intensive care unit that supports all forms of life (both natural and human-engineered systems). The Encyclopedia of Life Support Systems (EOLSS) is based on this concept and the above definition of 'life support systems'. It is intended to be a source of knowledge conveniently accessible at a single location, knowledge that could be useful in providing an understanding of global systems and issues with the potential to render decisions well informed to ensure the ability of our planet to support life. Unlike most encyclopedias, the contents of which are alphabetically arranged, EOLSS has a thematic organization (Theme Level, Topic Level and Article Level where Theme ... [...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] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ackermann (other)
Ackermann may also refer to the following: *Ackermann (surname), for many people with this name *Several mathematical objects named after Wilhelm Ackermann **Ackermann function **Ackermann ordinal **Ackermann set theory *Ackermann steering geometry, in mechanical engineering * Ackermann's formula, in control engineering *''Der Ackermann aus Böhmen'', or "The Ploughman from Bohemia", a work of poetry in Early New High German by Johannes von Tepl, written around 1401 *''Ackermannviridae'', virus family named in honor of H.-W. Ackermann See also *Ackerman (other) *Ackermans (other) *Akkerman (other) Akkerman may refer to: Places * Bilhorod-Dnistrovskyi, a city in Ukraine * Akkerman Oblast, a region in Ukraine *Akerman Fortress; see Tyras Other uses * Akkerman (surname) * Akkerman Inc., a construction equipment manufacturer in Brownsdale, ... * Åkerman {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
