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Model Predictive Control
Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the 1980s. In recent years it has also been used in power system balancing models and in power electronics. Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification. The main advantage of MPC is the fact that it allows the current timeslot to be optimized, while keeping future timeslots in account. This is achieved by optimizing a finite time-horizon, but only implementing the current timeslot and then optimizing again, repeatedly, thus differing from a linear–quadratic regulator ( LQR). Also MPC has the ability to anticipate future events and can take control actions accordingly. PID controllers do not have this predictive ability. MPC is nearly universally implemente ... [...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] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantization (signal Processing)
Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all lossy compression algorithms. The difference between an input value and its quantized value (such as round-off error) is referred to as quantization error, noise or distortion. A device or algorithm function, algorithmic function that performs quantization is called a quantizer. An analog-to-digital converter is an example of a quantizer. Example For example, Rounding#Round half up, rounding a real number x to the nearest integer value forms a very basic type of q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Function
In mathematics, a piecewise linear or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments. Definition A piecewise linear function is a function defined on a (possibly unbounded) interval of real numbers, such that there is a collection of intervals on each of which the function is an affine function. (Thus "piecewise linear" is actually defined to mean "piecewise affine".) If the domain of the function is compact, there needs to be a finite collection of such intervals; if the domain is not compact, it may either be required to be finite or to be locally finite in the reals. Examples The function defined by : f(x) = \begin -x - 3 & \textx \leq -3 \\ x + 3 & \text-3 < x < 0 \\ -2x + 3 & \text0 \leq x < 3 \\ 0.5x - 4.5 & \textx \geq 3 \end is piecewise linear with four pieces. The graph of this function is shown to the right. Since the graph of an affine(*) function is a [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Programming
Parametric programming is a type of mathematical optimization, where the optimization problem is solved as a function of one or multiple parameters. Developed in parallel to sensitivity analysis, its earliest mention can be found in a thesis from 1952. Since then, there have been considerable developments for the cases of multiple parameters, presence of integer variables as well as nonlinearities. Notation In general, the following optimization problem is considered : \begin J^*(\theta) = & \min_ f(x,\theta) \\ & \text g(x,\theta)\leq 0.\\ & \theta \in \Theta \subset \mathbb R^m \end where x is the optimization variable, \theta are the parameters, f(x,\theta) is the objective function and g(x,\theta) denote the constraints. J^* denotes a function whose output is the optimal value of the objective function f. The set \Theta is generally referred to as parameter space. The optimal value (i.e. result of solving ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Preconditioning
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem. The preconditioned problem is then usually solved by an iterative method. Preconditioning for linear systems In linear algebra and numerical analysis, a preconditioner P of a matrix A is a matrix such that P^A has a smaller condition number than A. It is also common to call T=P^ the preconditioner, rather than P, since P itself is rarely explicitly available. In modern preconditioning, the application of T = P^, i.e., multiplication of a column vector, or a block of column vectors, by T = P^, is commonly performed in a matrix-free fashion, i.e., where neither P, nor T = P^ (and often not even A) are explicitly available in a matrix form. Preconditioners are useful in iterative methods to solve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collocation Method
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called ''collocation points''), and to select that solution which satisfies the given equation at the collocation points. Ordinary differential equations Suppose that the ordinary differential equation : y'(t) = f(t,y(t)), \quad y(t_0)=y_0, is to be solved over the interval _0,t_0+h/math>. Choose c_k from 0 ≤ ''c''1< ''c''2< ... < ''c''''n'' ≤ 1. The corresponding (polynomial) collocation method approximates the solution ''y'' by the polynomial ''p'' of degree ''n'' which satisfies the initial condition , and the differential equation at all ''collocation points'' |
Direct Multiple Shooting Method
In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides the interval over which a solution is sought into several smaller intervals, solves an initial value problem in each of the smaller intervals, and imposes additional matching conditions to form a solution on the whole interval. The method constitutes a significant improvement in distribution of nonlinearity and numerical stability over single shooting methods. Single shooting methods Shooting methods can be used to solve boundary value problems (BVP) like y''(t) = f(t, y(t), y'(t)), \quad y(t_a) = y_a, \quad y(t_b) = y_b, in which the time points ''t''a and ''t''b are known and we seek y(t),\quad t \in (t_a,t_b). Single shooting methods proceed as follows. Let ''y''(''t''; ''t''0, ''y''0) denote the solution of the initial value problem (IVP) y''(t) = f(t, y(t), y'(t)), \quad y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shooting Method
In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to an initial value problem. It involves finding solutions to the initial value problem for different initial conditions until one finds the solution that also satisfies the boundary conditions of the boundary value problem. In layman's terms, one "shoots" out trajectories in different directions from one boundary until one finds the trajectory that "hits" the other boundary condition. Mathematical description Suppose one wants to solve the boundary-value problem y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y(t_1) = y_1. Let y(t; a) solve the initial-value problem y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a. If y(t_1; a) = y_1 , then y(t; a) is also a solution of the boundary-value problem. The shooting method is the process of solving the initial value problem for many different values of a until one finds the solution y(t; a) that sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Lagrange Equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
