Models stochastic disturbance

Empirical Model Identification. In this section we consider linear difference equation models for characterizing both the process dynamics and the stochastic disturbances inherent in the process. We shall discuss how to specify the model structure, how to estimate its parameters, and how to check its adequacy. Although discussion will be limited to single-input, single-output processes, the ideas are directly extendable to multiple-input, multiple-output processes. 

Comparing this with equation (3) shows that this can be considered as the output of a first order transfer function in response to a random input sequence. More generally, most stochastic disturbances can be modelled by a general autoregressive-integrated moving-average (ARIMA) time series model of order (p,d,q), that is,... 

It is well-known that, from a practical view point, it is always interesting to be aware of the behaviour of a process near the boundaries of validity. The same statement can be applied to the stochastic model of a process for small stochastic disturbances which occur at large intervals of time. In this situation, we can expect the real process and its model not to be appreciably modified for a fixed time called system answer time or constant time of the system . This statement can also be taken into account in the case of random disturbances with measurements realized at small intervals of time. 

Stochastic disturbance model There are various possibilities for using stochastic disturbance models other than the zero-order model shown in Eq. (13) (Ljung, 1987). 

Feedback inventory controllers Eq. 7-8 Stochastic disturbance model... 

Linear discrete-time models can also be developed that include the effects of unmeasured stochastic disturbances. For example, separate process models and disturbance models, also called noise models, can be obtained from data (Ljung, 1999). In this case the error term e in Eq. 7-1 is not white noise but colored noise that is autocorrelated. In other words, there are underlying disturbance dynamics that causes e to depend on previous values. 

There are several components in an intervention model a deterministic component describing the intervention(s), the associated response of the system to the intervention, and a stochastic disturbance term. The overall modeling strategy is to obtain reasonable initial representations for these components and to iterate to a final model based on intermediate estimates, diagnostic checks, and model interpretations (Liu and Hudak 2004). 

Stochastic Models for the Disturbances The type of stochastic process disturbances N-t occurring in practice can usually be modelled quite conveniently by statistical time series models (Box and Jenkins (k)). These models are once again simple linear difference equation models in which the input is a sequence of uncorrelated random Normal deviates (a. ) (a white noise sequence)... 

Watanabe K, Yoshimura C, Omura T. 2005. Stochastic model for recovery prediction of macroinvertebrates following a pulse-disturbance in river. Ecol Model 189 396 412. 

Sources of disturbances considered in this example are categorized in three classes. First, the production plants are stochastic transformers, i.e. the transformation processes are modelled by stationary time series models with normally distributed errors. The plants states are modelled by Markov models as introduced before. The corresponding transition matrices are provided in the appendix in Table A.15 and Table A.16. Additionally, normally distributed errors are added to simulate the inflovj rates with e N (O, ) where oj is the current state of the plant. 

A real-world plant can be usually characterized by time-varying dynamical properties, which affect the plant behavior. Stochastic models are used to represent the disturbances acting at the plant output because of the large number and different nature of the factors disturbing the normal plant operation. 

Adaptive control is usually used to cope with an unknown or/and changing plant to be controlled (Astrom and Wittenmark, 1995). Analysis and synthesis of such a control system is possible only under some assumptions concerning the nature of the plant and its dynamics. In this chapter only linear, discrete-time plants disturbed in a stochastic manner will be considered. The following plant model will be used (Moscinski and Ogonowski, 1995) ... 

The first comment is that the stmcture of the locally hnear model is a locally linear trend with stochastic level, fixed slope and fixed seasonal, and this is the case for all five models. This resirlts from the fact that the variances of the disturbances associated with the slopes and the seasonal component are close to zero. It should be noted that the values of the level, measiued at the end of the period (December 2012), differ according to the model as the loeally linear stmctme applies to a different variable in each case. The correlation to the explanatory variable is in all cases significant, but the estimated coefficients are to be interpreted differently according to the model. ... 

Random disturbance interfacial model, in which the disturbed points are stochastic. 

Stochastic models play an important role in understanding chaotic phenomena such as Brownian motion and turbulence. They are also used to describe highly heterogeneous systems, e.g. transport in fractured media. Stochastic models are used in control theory to account for the irregular nature of disturbances. 

The theory of stochastic processes began in the nineteenth century when physicists were trying to show that heat in a medium is essentially a random motion of the constituent molecules. At the end of that century, some researches began to adopt more direct mathematical models of random disturbances instead of considering random motion as due to collisions between objects having a random distribution of initial positions and velocities. In this context several physicists, among which Fok-ker (1914) and Planck (1915), developed partial differential equations, which were versions of what was subsequently called the Fokker-Planck equation, to study the theory of Brownian motion. 

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