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State-transition Equation
The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by \frac = \mathbf(t) + \mathbf(t) + \mathbf(t), with state vector , control vector , vector of additive disturbances, and fixed matrices can be solved by using either the classical method of solving linear differential equations or the Laplace transform method. The Laplace transform solution is presented in the following equations. The Laplace transform of the above equation yields s\mathbf(s) - \mathbf(0) = \mathbf(s) + \mathbf(s) + \mathbf(s) where denotes initial-state vector evaluated at . Solving for gives \mathbf(s) = (s\mathbf - \mathbf)^ \mathbf(0) + (s\mathbf - \mathbf)^[\mathbf(s) + \mathbf(s)]. So, the state-transition equation can be obtained by taking inverse Laplace transform as \begin x(t) &= \mathcal^ \Bigl\ \mathbf(0) + \mathcal^ \Bigl\ \\ &= \mathbf(t) \mathbf(0) + \int_^ \mathbf(t - \tau)[\mathbf(\tau) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Time-invariant System
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 Linear system#Definition, linearity and Time-invariant system, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Inverse Laplace Transform
In mathematics, the inverse Laplace transform of a function F(s) is a real function f(t) that is piecewise- continuous, exponentially-restricted (that is, , f(t), \leq Me^ \forall t \geq 0 for some constants M > 0 and \alpha \in \mathbb) and has the property: :\mathcal\(s) = \mathcal\(s) = F(s), where \mathcal denotes the Laplace transform. It can be proven that, if a function F(s) has the inverse Laplace transform f(t), then f(t) is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same). This result was first proven by Mathias Lerch in 1903 and is known as Lerch's theorem. The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamical systems. Mellin's inverse formula An integral formula for the inverse Laplace transform, called the ''Mellin's inverse formula'', the '' Bromwich integral'', or the '' Fourier– Me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Transition Matrix
In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems. Linear systems solutions The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form : \dot(t) = \mathbf(t) \mathbf(t) + \mathbf(t) \mathbf(t) , \;\mathbf(t_0) = \mathbf_0 , where \mathbf(t) are the states of the system, \mathbf(t) is the input signal, \mathbf(t) and \mathbf(t) are matrix functions, and \mathbf_0 is the initial condition at t_0. Using the state-transition matrix \mathbf(t, \tau), the solution is given by: : \mathbf(t)= \mathbf (t, t_0)\mathbf(t_0)+\int_^t \mathbf(t, \tau)\mathbf(\tau)\mathbf(\tau)d\tau The first term is known as the zero-input response and represents how the system's state would evolve in the absence of any input. The s ... [...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 Feedback control systems Logic control Logic control systems for indus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Control Engineering
Control engineering, also known as control systems engineering and, in some European countries, automation engineering, is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls overlaps and is usually taught along with electrical engineering, chemical engineering and mechanical engineering at many institutions around the world. The practice uses sensors and detectors to measure the output performance of the process being controlled; these measurements are used to provide corrective feedback helping to achieve the desired performance. Systems designed to perform without requiring human input are called automatic control systems (such as cruise control for regulating the speed of a car). Multi-disciplinary in nature, control systems engineering activities focus on implementation of control systems mainly derived by mathematical modeling of a diver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automatic Control
Automation describes a wide range of technologies that reduce human intervention in processes, mainly by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines. Automation has been achieved by various means including mechanical, hydraulic, pneumatic, electrical, electronic devices, and computers, usually in combination. Complicated systems, such as modern factories, airplanes, and ships typically use combinations of all of these techniques. The benefit of automation includes labor savings, reducing waste, savings in electricity costs, savings in material costs, and improvements to quality, accuracy, and precision. Automation includes the use of various equipment and control systems such as machinery, processes in factories, boilers, and heat-treating ovens, switching on telephone networks, steering, stabilization of ships, aircraft and other applications and vehicles with reduced human inter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause and effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled carefully when applied to feedback systems: History Self-regulating mechanisms have existed since antiquity, and the idea of feedback started to enter economic theory in Britain by the 18th century, but it was not at that time recognized as a universal abstraction and so did not have a name. The first ever known artificial feedback device was a float valve, for maintaining water at a constant level, invented in 270 BC in Alexandria, Egypt. This device illustrated the principle of feedback: a low water level opens the valve, the rising water then provides feedback into the system, closing the valve when the required level is reached. This then reoccurs in a circular fashion as the water level fluctuates. Centrifugal governors were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Process Control
Industrial process control (IPC) or simply process control is a system used in modern manufacturing which uses the principles of control theory and physical industrial control systems to monitor, control and optimize continuous Industrial processes, industrial production processes using control algorithms. This ensures that the industrial machines run smoothly and safely in factories and efficiently use energy to transform raw materials into high-quality finished products with reliable consistency while reducing Efficient energy use#Industry, energy waste and economic costs, something which could not be achieved purely by human manual control. In IPC, control theory provides the theoretical framework to understand system dynamics, predict outcomes and design control strategies to ensure predetermined objectives, utilizing concepts like feedback loops, stability analysis and controller design. On the other hand, the physical apparatus of IPC, based on automation technologies, cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PID Loop
PID or Pid may refer to: Medicine * Pelvic inflammatory disease or pelvic inflammatory disorder, an infection of the upper part of the female reproductive system * Primary immune deficiency, disorders in which part of the body's immune system is missing or does not function properly * Prolapsed intervertebral disc, commonly called a herniated disc Science, technology and engineering * BBC Programme Identifier, a unique identifier for a BBC television or radio programme brand, a season or series, or an individual episode * OBD-II PIDs (on-board diagnostics parameter IDs), requests for data through an OBD connector in automotive repair * Packet Identifier, a field in a MPEG transport stream packet * Partial information decomposition, an extension of information theory * Passive infrared detector, a passive infrared sensor * Payload Interface Document (used on space engineering program for example) * Persistent identifier, a long-lasting reference to a document, file, web page ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
