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Minimum Energy Control
In control theory, the minimum energy control is the control u(t) that will bring a linear time invariant system to a desired state with a minimum expenditure of energy. Let the linear time invariant (LTI) system be : \dot(t) = A \mathbf(t) + B \mathbf(t) : \mathbf(t) = C \mathbf(t) + D \mathbf(t) with initial state x(t_0)=x_0 . One seeks an input u(t) so that the system will be in the state x_1 at time t_1, and for any other input \bar(t), which also drives the system from x_0 to x_1 at time t_1, the energy expenditure would be larger, i.e., : \int_^ \bar^*(t) \bar(t) dt \ \geq \ \int_^ u^*(t) u(t) dt. To choose this input, first compute the controllability Gramian : W_c(t)=\int_^t e^BB^*e^ d\tau. Assuming W_c is nonsingular (if and only if the system is controllable), the minimum energy control is then : u(t) = -B^*e^W_c^(t_1) ^x_0-x_1 Substitution into the solution :x(t)=e^x_0+\int_^e^{A(t-\tau)}Bu(\tau)d\tau verifies the achievement of state x_1 at t_1. See al ... [...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] |
Linear Time Invariant
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 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, as is frequently employed by the symbol in computer languages). 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Controllability Gramian
In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are, respectively, n\times n, n\times p, q\times n and q\times p matrices. One of the many ways one can achieve such goal is by the use of the Controllability Gramian. Controllability in LTI Systems Linear Time Invariant (LTI) Systems are those systems in which the parameters \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are invariant with respect to time. One can observe if the LTI system is or is not controllable simply by looking at the pair (\boldsymbol,\boldsymbol). Then, we can say that the following statements are equivalent: 1. The pair (\boldsymbol,\boldsymbol) is controllable. 2. The n\times n matrix \boldsymbol(t)=\int_^e^\boldsymbole^d\tau=\int_^e^\boldsymbole^d\tau is nonsingular for any t>0. 3. The n\times n ... [...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 Engineering
Control engineering or control systems 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 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 diverse range of systems. Overview Modern day control engineering is a relatively new field of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space (controls)
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve over time in a way that depends on the values they have at any given time and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables. The "state space" is the Euclidean space in which the variables on the axes are the state variables. The state of the system can be represented as a ''state vector'' within that space. To abstract from the number of inputs, outputs and states, these variables are expressed as vectors. If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. The state-space method is characterized by significant algebraization of general syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Calculus
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
