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Structural Identifiability
In the area of system identification, a dynamical system is structurally identifiable if it is possible to infer its unknown parameters by measuring its output over time. This problem arises in many branch of applied mathematics, since dynamical systems (such as the ones described by ordinary differential equations) are commonly utilized to model physical processes and these models contain unknown parameters that are typically estimated using experimental data. However, in certain cases, the model structure may not permit a unique solution for this estimation problem, even when the data is continuous and free from noise. To avoid potential issues, it is recommended to verify the uniqueness of the solution in advance, prior to conducting any actual experiments. The lack of structural identifiability implies that there are multiple solutions for the problem of system identification, and the impossibility of distinguishing between these solutions suggests that the system has poor forec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Identification
The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design#System identification and stochastic approximation, optimal design of experiments for efficiently generating informative data for regression analysis, fitting such models as well as model reduction. A common approach is to start from measurements of the behavior of the system and the external influences (inputs to the system) and try to determine a mathematical relation between them without going into many details of what is actually happening inside the system; this approach is called Black box (systems), black box system identification. Overview A dynamic mathematical model in this context is a mathematical description of the dynamic behavior of a system or process in either the time or frequency domain. Examples include: * physical system, physical processes such as the movement of a falling bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Control
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory. Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Order Reduction
Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with applications in all areas of mathematical modelling. Overview Many modern mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and large size (dimension). Model order reduction aims to lower the computational complexity of such problems, for example, in simulations of large-scale dynamical systems and control systems. By a reduction of the model's associated state space dimension or degrees of freedom, an approximation to the original model is computed which is commonly referred to as a reduced order model. Reduced order models are useful in settings where it is often unfeasible to perform numerical simulations using the complete full order model. This can be due to limitations in computational resources or the requir ... [...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'' 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] |
System Identification
The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design#System identification and stochastic approximation, optimal design of experiments for efficiently generating informative data for regression analysis, fitting such models as well as model reduction. A common approach is to start from measurements of the behavior of the system and the external influences (inputs to the system) and try to determine a mathematical relation between them without going into many details of what is actually happening inside the system; this approach is called Black box (systems), black box system identification. Overview A dynamic mathematical model in this context is a mathematical description of the dynamic behavior of a system or process in either the time or frequency domain. Examples include: * physical system, physical processes such as the movement of a falling bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia (programming Language)
Julia is a high-level programming language, high-level, general-purpose programming language, general-purpose dynamic programming language, dynamic programming language, designed to be fast and productive, for e.g. data science, artificial intelligence, machine learning, modeling and simulation, most commonly used for numerical analysis and computational science. Distinctive aspects of Julia's design include a type system with parametric polymorphism and the use of multiple dispatch as a core programming paradigm, a default just-in-time compilation, just-in-time (JIT) compiler (with support for ahead-of-time compilation) and an tracing garbage collection, efficient (multi-threaded) garbage collection implementation. Notably Julia does not support classes with encapsulated methods and instead it relies on structs with generic methods/functions not tied to them. By default, Julia is run similarly to scripting languages, using its runtime, and allows for read–eval–print loop, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Likelihood Function
A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the joint probability distribution of the random variable that (presumably) generated the observations. When evaluated on the actual data points, it becomes a function solely of the model parameters. In maximum likelihood estimation, the argument that maximizes the likelihood function serves as a point estimate for the unknown parameter, while the Fisher information (often approximated by the likelihood's Hessian matrix at the maximum) gives an indication of the estimate's precision. In contrast, in Bayesian statistics, the estimate of interest is the ''converse'' of the likelihood, the so-called posterior probability of the parameter given the observed data, which is calculated via Bayes' rule. Definition The likelihood function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. , MATLAB has more than four million users worldwide. They come from various backgrounds of engineering, science, and economics. , more than 5000 global colleges and universities use MATLAB to support instruction and research. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was base ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PottersWheel
PottersWheel is a MATLAB toolbox for mathematical modeling of time-dependent dynamical systems that can be expressed as chemical reaction networks or ordinary differential equations (ODEs). It allows the automatic calibration of model parameters by fitting the model to experimental measurements. CPU-intensive functions are written or – in case of model dependent functions – dynamically generated in C. Modeling can be done interactively using graphical user interfaces or based on MATLAB scripts using the PottersWheel function library. The software is intended to support the work of a mathematical modeler as a real potter's wheel eases the modeling of pottery. Seven modeling phases The basic use of PottersWheel covers seven phases from model creation to the prediction of new experiments. Model creation The dynamical system is formalized into a set of reactions or differential equations using a visual model designer or a text editor. The model is stored as a MATLAB *. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Informativity
Data ( , ) are a collection of discrete or continuous values that convey information, describing the quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted formally. A datum is an individual value in a collection of data. Data are usually organized into structures such as tables that provide additional context and meaning, and may themselves be used as data in larger structures. Data may be used as variables in a computational process. Data may represent abstract ideas or concrete measurements. Data are commonly used in scientific research, economics, and virtually every other form of human organizational activity. Examples of data sets include price indices (such as the consumer price index), unemployment rates, literacy rates, and census data. In this context, data represent the raw facts and figures from which useful information can be extracted. Data are collected using techniques such as m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...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'' 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] |
