Distributed Parameter System
In control theory, a distributed-parameter system (as opposed to a lumped-parameter system) is a system whose state space is infinite- dimensional. Such systems are therefore also known as infinite-dimensional systems. Typical examples are systems described by partial differential equations or by delay differential equations. Linear time-invariant distributed-parameter systems Abstract evolution equations Discrete-time With ''U'', ''X'' and ''Y'' Hilbert spaces and ''A\,'' ∈ ''L''(''X''), ''B\,'' ∈ ''L''(''U'', ''X''), ''C\,'' ∈ ''L''(''X'', ''Y'') and ''D\,'' ∈ ''L''(''U'', ''Y'') the following difference equations determine a discrete-time linear time-invariant system: :x(k+1)=Ax(k)+Bu(k)\, :y(k)=Cx(k)+Du(k)\, with ''x\,'' (the state) a sequence with values in ''X'', ''u\,'' (the input or control) a sequence with values in ''U'' and ''y\,'' (the output) a sequence with values in ''Y''. Continuous-time The ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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C0 Semigroup
In mathematical analysis, a ''C''0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations. Formally, a strongly continuous semigroup is a representation of the semigroup (R+, +) on some Banach space ''X'' that is continuous in the strong operator topology. Formal definition A strongly continuous semigroup on a Banach space X is a map T : \mathbb_+ \to L(X) (where L(X) is the space of bounded operators on X) such that # T(0) = I , (the identity operator on X) # \forall t,s \ge 0 : \ T(t + s) = T(t) T(s) # \forall x_0 \in X: \ ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, implies ). In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. A func ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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'' for that system, such as Kalman filters. 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, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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Holomorphic Function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is '' analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term '' analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation. It can be considered a discrete-time equivalent of the Laplace transform (the ''s-domain'' or ''s-plane''). This similarity is explored in the theory of time-scale calculus. While the continuous-time Fourier transform is evaluated on the s-domain's vertical axis (the imaginary axis), the discrete-time Fourier transform is evaluated along the z-domain's unit circle. The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle. In signal processing, one of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by in ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Laplace Transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a function of a Complex number, complex variable s (in the complex-valued frequency domain, also known as ''s''-domain, or ''s''-plane). The transform is useful for converting derivative, differentiation and integral, integration in the time domain into much easier multiplication and Division (mathematics), division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic equation, algebraic polynomial equations, and by simplifyin ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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State Space (controls)
In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values and inputs, while outputs depend on the states and sometimes the inputs too. The state space (also called time-domain approach and equivalent to phase space in certain dynamical systems) is a geometric space where the axes are these state variables, and the system’s state is represented by a state vector. For linear, time-invariant, and finite-dimensional systems, the equations can be written in matrix form, offering a compact alternative to the frequency domain’s Laplace transforms for multiple-input and multiple-output (MIMO) systems. Unlike the frequency domain approach, it works for systems beyond just linear ones with zero initial condi ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): , or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative, so is not necessarily equal to ; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a ''commutative semigroup' ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Unbounded Operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since * "unbounded" should sometimes be understood as "not necessarily bounded"; * "operator" should be understood as "linear operator" (as in the case of "bounded operator"); * the domain of the operator is a linear subspace, not necessarily the whole space; * this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; * in the special case of a bounded operator, still, the domain is usually assumed to be the whole space. In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Lumped-element Model
The lumped-element model (also called lumped-parameter model, or lumped-component model) is a simplified representation of a physical system or circuit that assumes all components are concentrated at a single point and their behavior can be described by idealized mathematical models. The lumped-element model simplifies the system or circuit behavior description into a topology. It is useful in electrical systems (including electronics), mechanical multibody systems, heat transfer, acoustics, etc. This is in contrast to distributed parameter systems or models in which the behaviour is distributed spatially and cannot be considered as localized into discrete entities. The simplification reduces the state space of the system to a finite dimension, and the partial differential equations (PDEs) of the continuous (infinite-dimensional) time and space model of the physical system into ordinary differential equations (ODEs) with a finite number of parameters. Electrical systems ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |