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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] |
Systems Theory
Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structure, function and role, and expressed through its relations with other systems. A system is "more than the sum of its parts" by expressing synergy or emergent behavior. Changing one component of a system may affect other components or the whole system. It may be possible to predict these changes in patterns of behavior. For systems that learn and adapt, the growth and the degree of adaptation depend upon how well the system is engaged with its environment and other contexts influencing its organization. Some systems support other systems, maintaining the other system to prevent failure. The goals of systems theory are to model a system's dynamics, constraints, conditions, and relations; and to elucidate principles (such as purpose, measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighting Pattern
A weighting pattern for a linear dynamical system describes the relationship between an input u and output y. Given the time-variant system described by : \dot(t) = A(t)x(t) + B(t)u(t) : y(t) = C(t)x(t), then the output can be written as : y(t) = y(t_0) + \int_^t T(t,\sigma)u(\sigma) d\sigma, where T(\cdot,\cdot) is the weighting pattern for the system. For such a system, the weighting pattern is T(t,\sigma) = C(t)\phi(t,\sigma)B(\sigma) such that \phi is the state transition matrix. The weighting pattern will determine a system, but if there exists a realization for this weighting pattern then there exist many that do so. Linear time invariant system In a LTI system then the weighting pattern is: ; Continuous : T(t,\sigma) = C e^ B where e^ is the matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie grou ... [...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 of experiments for efficiently generating informative data for 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 system identification. Overview A dynamical 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 processes such as the movement of a falling body under the influence of gravity; * economic processes such as stock markets that react to external influences. One ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model represents, often in considerably idealized form, the data-generating process. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman J. Adèr, Herman Adèr quoting Kenneth A. Bollen, Kenneth Bollen). All Statistical hypothesis testing, statistical hypothesis tests and all Estimator, statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. Introduction Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grey Box Model
In mathematics, statistics, and computational modelling, a grey box modelKroll, Andreas (2000). Grey-box models: Concepts and application. In: New Frontiers in Computational Intelligence and its Applications, vol.57 of Frontiers in artificial intelligence and applications, pp. 42-51. IOS Press, Amsterdam.Sohlberg, B., and Jacobsen, E.W., 2008Grey box modelling - branches and experiences Proc. 17th World Congress, Int. Federation of Automatic Control, Seoul. pp 11415-11420 combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of results, to models that need only parameter values from data or existing literature.Whiten, B., 2013Model completion and validation using inversion of grey box models ANZIAM J.,54 (CTAC 2012) pp C187–C199. Thus, almost all models are grey box models as opposed to black box where no model form is assumed or white box models that are purely theoretical. Some models ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Domain Decomposition
The frequency domain decomposition (FDD) is an output-only system identification technique popular in civil engineering, in particular in structural health monitoring. As an output-only algorithm, it is useful when the input data is unknown. FDD is a modal analysis technique which generates a system realization using the frequency response given (multi-)output data. Algorithm # Estimate the power spectral density matrix \hat_(j\omega) at discrete frequencies \omega = \omega_i. # Do a singular value decomposition of the power spectral density, i.e. \hat_(j \omega_i) = U_i S_i U_i^H where U_i = _,u_,...,u_/math> is a unitary matrix holding the singular values u_, S_i is the diagonal matrix holding the singular values s_. # For an n degree of freedom system, then pick the n dominating peaks in the power spectral density using whichever technique you wish (or manually). These peaks correspond to the mode shape A normal mode of a dynamical system is a pattern of motion in which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigensystem Realization Algorithm
The Eigensystem realization algorithm (ERA) is a system identification technique popular in civil engineering, in particular in structural health monitoring. ERA can be used as a modal analysis technique and generates a system realization using the time domain response (multi-)input and (multi-)output data. The ERA was proposed by Juang and Pappa and has been used for system identification of aerospace structures such as the Galileo spacecraft, turbines,Sanchez-Gasca, J. J. "Computation of turbine-generator subsynchronous torsional modes from measured data using the eigensystem realization algorithm." Power Engineering Society Winter Meeting, 2001. IEEE. Vol. 3. IEEE, 2001. civil structures and many other type of systems. Uses in structural engineering In structural engineering the ERA is used to identify natural frequencies, mode shapes and damping ratios. The ERA is commonly used in conjunction with the Natural Excitation Technique (NExT) to identify modal parameters fr ... [...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 sec ... [...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] |
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] |
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] |
Strictly Proper
In mathematics, mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of Inequality (mathematics), inequality and Monotonic function, monotonic functions. It is often attached to a technical term to indicate that the exclusive meaning of the term is to be understood. The opposite is non-strict, which is often understood to be the case but can be put explicitly for clarity. In some contexts, the word "proper" can also be used as a mathematical synonym for "strict". Use This term is commonly used in the context of inequality (mathematics), inequalities — the phrase "strictly less than" means "less than and not equal to" (likewise "strictly greater than" means "greater than and not equal to"). More generally, a Partially_ordered_set#Correspondence of strict and non-strict partial order relations, strict partial order, strict total order, and Strict weak ordering, strict weak order exclude equality and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
