Time series modeling output error model

Time series models of the output error such as Eq. 9.5 can be used to identify the dynamic response characteristics of e k) [148]. Dynamic response characteristics such as overshoot, settling time and cycling can be extracted from the pulse response of the fitted time series model. The pulse response of the estimated e k) can be compared to the pulse response of the desired response specification to determine if the output error characteristics are acceptable [148]. 

One source of mis-specification is the type of the error process. Homogeneity of variances is typically assumed by standard time series models. However, in the context of chemical production processes e.g. an adjustment of a plant s production rate may lead to an imbalance of the underlying chemical reaction(s). This instability may materialize in fluctuations of the output flow rate(s) which may also affect flow rates in subsequent... 

This chapter will focus on practicable methods to perform both the model specification and model estimation tasks for systems/models that are static or dynamic and linear or nonlinear. Only the stationary case win be detailed here, although the potential use of nonstationary methods will be also discussed briefly when appropriate. In aU cases, the models will take deterministic form, except for the presence of additive error terms (model residuals). Note that stochastic experimental inputs (and, consequently, outputs) may stiU be used in connection with deterministic models. The cases of multiple inputs and/or outputs (including multidimensional inputs/outputs, e.g., spatio-temporal) as well as lumped or distributed systems, will not be addressed in the interest of brevity. It will also be assumed that the data (single input and single output) are in the form of evenly sampled time-series, and the employed models are in discretetime form (e.g., difference equations instead of differential equations, discrete summations instead of integrals). 

Even if no losses or gross errors have occurred, there will be variations among successive material balance or MUF measurements about the process mean. A succession of such material balance measurements forms a time series. As with the control of a product specification in a production process, the output may be monitored for outliers which are defined as significant deviations from the process model or time series pattern in a direction away from the target value. Thus an outlier indicates a process change to be investigated. 

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