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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 submat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of mathematics that deals with the 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; often with the aim to achieve a degree of 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 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. Control theory is used in control system eng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LTI System Theory
LTI can refer to: * ''LTI – Lingua Tertii Imperii'', a book by Victor Klemperer * Language Technologies Institute, a division of Carnegie Mellon University * Linear time-invariant system, an engineering theory that investigates the response of a linear, time-invariant system to an arbitrary input signal * ''Licensed to Ill'', the 1986 debut album by the Beastie Boys * Lost Time Incident or industrial injury or Occupational injury * Learning Tools Interoperability * Louisiana Training Institute-East Baton Rouge, later known as the Jetson Center for Youth (JCY), a juvenile prison in Louisiana Companies * London Taxis International * Larsen & Toubro Infotech Biology and medicine * Lymphoid tissue-inducer cell, see innate lymphoid cell Innate lymphoid cells (ILCs) are the most recently discovered family of innate immune cells, derived from common lymphoid progenitors (CLPs). In response to pathogenic tissue damage, ILCs contribute to immunity via the secretion of signalling ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 There are two common classes of control action: open loop and closed loop. In an ... [...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] |
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] |
Reachability Problem
Reachability is a fundamental problem that appears in several different contexts: finite- and infinite-state concurrent systems, computational models like 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. In general the reachability problem can be formulated as follows: ''Given a computational (potentially infinite state) 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.'' 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 descrip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Quantum System
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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