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Kalman Decomposition
In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's reachable and observable subspaces. Definition Consider the continuous-time LTI control system : \dot(t) = Ax(t) + Bu(t), : \, y(t) = Cx(t) + Du(t), or the discrete-time LTI control system : \, x(k+1) = Ax(k) + Bu(k), : \, y(k) = Cx(k) + Du(k). The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows: : \, = TA^, : \, = TB, : \, = C^, : \, = D, where \, T^ is the coordinate transformation matrix defined as : \, T^ = \begin T_ & T_ & T_ & T_\end, and whose subma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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LTI System Theory
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined in the overview below. These properties apply (exactly or approximately) to many important physical systems, in which case the response of the system to an arbitrary input can be found directly using convolution: where is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication). What's more, there are systematic methods for solving any such system (determining ), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers. Linear time-invariant system theory is also used in image processing, where the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Control System
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial control systems which are used for controlling processes or machines. The control systems are designed via control engineering process. For continuously modulated control, a feedback controller is used to automatically control a process or operation. The control system compares the value or status of the process variable (PV) being controlled with the desired value or setpoint (SP), and applies the difference as a control signal to bring the process variable output of the plant to the same value as the setpoint. For sequential and combinational logic, software logic, such as in a programmable logic controller, is used. Open-loop and closed-loop control Feedback control systems Logic control Logic control systems for indus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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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] [Amazon] |
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Reachability Problem
Reachability is a fundamental problem which can be formulated as follows: ''Given a computational system with a set of allowed rules or transformations, decide whether a certain state of a system is reachable from a given initial state of the system.'' It appears in several different contexts: finite- and infinite-state concurrent systems, cellular automata and Petri nets, program analysis, discrete and continuous systems, time critical systems, hybrid systems, rewriting systems, probabilistic and parametric systems, and open systems modelled as games. Variants of the reachability problem may result from additional constraints on the initial or final states, specific requirement for reachability paths as well as for iterative reachability or changing the questions into analysis of winning strategies in infinite games or unavoidability of some dynamics. Typically, for a fixed system description given in some form (reduction rules, systems of equations, logical formulas, et ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Realization (systems)
In systems theory, a realization of a state space model is an implementation of a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of ( time-varying) matrices (t),B(t),C(t),D(t)/math> such that : \dot(t) = A(t) \mathbf(t) + B(t) \mathbf(t) : \mathbf(t) = C(t) \mathbf(t) + D(t) \mathbf(t) with (u(t),y(t)) describing the input and output of the system at time t. LTI System For a linear time-invariant system specified by a transfer matrix, H(s) , a realization is any quadruple of matrices (A,B,C,D) such that H(s) = C(sI-A)^B+D. Canonical realizations Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system)): Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form: : H(s) = \frac. The coefficients ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Quantum System
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |