Input-Output Models of Dynamic Processes

The prediction error e k) is the difference between the observed and the predicted values at a specific time e k) = y k) —y k). An autoregressive - integrated - moving average model is represented as ARIMA(p, d, r). For example, AR1MA(0,1,1) = 1MA(1,1) indicates  

Model identification is an iterative process. There are several software packages with modules that automate time series model development. When a model is developed to describe data that have stochastic variations, one has to be cautious about the degree of fit. By increasing model complexity (adding extra terms) a better fit can be obtained. But, the model may describe part of the stochastic variation in that particular data which will not occur identically in other data sets. Consequently, although the fit to the training data may be improved, the prediction errors may get worse. 

Inputs, outputs and disturbances will be denoted as u, y, and d, respectively. For multivariable processes where ui k), U2 k), , Um k) are the m inputs, the input vector u k) at time k is written as a column vector. Similarly, the p outputs are defined by a column vector  

Disturbances d k) and residuals e k) are also represented by column vectors with appropriate dimensions in a similar manner. 

A general linear discrete-time model for a variable y k) can be written 

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Various multivariate regression techniques are outlined in Section 4.1. Section 4.2 introduces PCA-based regression and its extension to capture dynamic variations in data. PLS regression is discussed in Section 4.3. Input-output modeling of dynamic processes with time series models is introduced in Section 4.4 and state-space modeling techniques are presented in Section 4.5. 

In the previous chapters deterministic models were derived. They were designed based on the chemical and physical balances and mechanisms of the process and consequently the model described the internal functional behavior. Black-box models, on the other hand, are designed based on the input-output behavior of the process and consequently the model describes the overall behavior. A black-box model consists of a certain stmcture of which the parameters are determined by means of experimental results. Therefore, they often are called experimental models. The main properties of black-box models are the stmcture characteristics, which are level of detail, degree of non-linearity and the stmctural way in which dynamics are composed. 

The model of a process is a relation between the outputs and inputs (feed conditions, design parameters and adjustable parameters of the process) with a view to scaling-up the process from laboratory to industrial scale, predicting the process dynamics (case of this concrete example) and optimizing the operating conditions. 

Part III (Chapters 6 through 12) is devoted to the analysis of static and dynamic behavior of processing systems. The emphasis here is on identifying those process characteristics which shape the dynamic response for a variety of processing units. The results of such analysis are used later to design effective controllers. Input-output models have been employed through the use of Laplace transforms. 

Since this work deals with the aggregated simulation and planning of chemical production processes, the focus is laid upon methods to determine estimations of the process models. For process control this task is the crucial one as the estimations accuracy determines the accuracy of the whole control process. The task to find an accurate process model is often called process identification. To describe the input-output behaviour of (continuously operated) chemical production plants finite impulse response (FIR) models are widely used. These models can be seen as regression models where the historical records of input/control measures determine the output measure. The term "finite" indicates that a finite number of historical records is used to predict the process outputs. Often, chemical processes show a significant time-dynamic behaviour which is typically reflected in auto-correlated and cross-correlated process measures. However, classic regression models do not incorporate auto-correlation explicitly which in turn leads to a loss in estimation efficiency or, even worse, biased estimates. Therefore, time series methods can be applied to incorporate auto-correlation effects. According to the classification shown in Table 2.1 four basic types of FIR models can be distinguished. 

Kiranoudis et al. [20] developed a dynamic model for the simulation of conveyor-belt dryers and proposed a SISO (single-input, single output) control scheme for the regulation of material moisture content. In a subsequent work, Kiranoudis et al. [21] extended the dynamic model of this process to include MIMO (multiple-input, multiple output) schane to control the material moistme content and temperature. In both works, PI controllers were appropriately tuned and nonlinear simulations were performed. 

Especially in an operating environment where quality has become more important than quantity, there is a strong desire to develop input-output models that can be used in advanced control applications, in order to develop control strategies for quality improvement. These models are usually discrete hnear transfer function (difference equation) type models, which provide a representation of the dynamic behaviour of the process at discrete sampling... 

The environmental model is the product of a problem analysis. It forms the basis for the development of behavioral models of the process. This holds for physical models, in which the internal dynamic mechanism is described by pltysical laws, as well as empirical (black box) models, in which only overall dynamic relationships between process inputs and outputs are formulated. The environmental model can also be used for the development of a process control scheme. By evaluating the static (power of control) and dynamic relationships (speed of control) between process inputs and outputs, the process control scheme can be selected. In general no dynamic model of the process is required yet at this stage. This is the starting point that is used in the chapters on process control. 

In this chapter we consider model predictive control (MPC), an important advanced control technique for difficult multivariable control problems. The basic MPC concept can be summarized as follows. Suppose that we wish to control a multiple-input, multiple-output process while satisfying inequality constraints on the input and output variables. If a reasonably accurate dynamic model of the process is available, model and current measurements can be used to predict future values of the outputs. Then the appropriate changes in the input variables can be calculated based on both predictions and measurements. In essence, the changes in the individual input variables are coordinated after considering the input-output relationships represented by the process model. In MPC applications, the output variables are also referred to as controlled variables or CVs, while the input variables are also called manipulated variables or MVs. Measured disturbance variables are... 

The identification of plant models has traditionally been done in the open-loop mode. The desire to minimize the production of the off-spec product during an open-loop identification test and to avoid the unstable open-loop dynamics of certain systems has increased the need to develop methodologies suitable for the system identification. Open-loop identification techniques are not directly applicable to closed-loop data due to correlation between process input (i.e., controller output) and unmeasured disturbances. Based on Prediction Error Method (PEM), several closed-loop identification methods have been presented Direct, Indirect, Joint Input-Output, and Two-Step Methods. 

The APS system is supported by an independent modem data model that is often based on generic input/output nodes and their dynamic combination in complex networks and thus better suited for algorithmic processing and optimization than the transaction-oriented business data model of the ERP system. 

Our analysis also provides clues regarding the manipulated inputs that are available for use in each of the control layers. Recall that the flow rates u1 of the material streams not connected with the process impurity input and output are present only in the model of the fast dynamics (4.20). u1 thus represent the... 

Clearly, the general model (5.10) contains terms of very different magnitudes, corresponding, respectively, to the process input and output flows and to the chemical reactions, to the presence of the large material recycle stream, and to the presence of the impurity and the purge stream used for its removal. While (as we have argued in the previous chapters of the book) the presence of these terms is purely a steady-state, design feature of the process, it is intuitive that their impact on the process dynamics will also be very different. 

The four experiments done previously with Rnp (= 0.5, 1, 3, 4) were used to train the neural network and the experiment with / exp = 2 was used to validate the system. Dynamic models of process-model mismatches for three state variables (i.e. X) of the system are considered here. They are the instant distillate composition (xD), accumulated distillate composition (xa) and the amount of distillate (Ha). The inputs and outputs of the network are as in Figure 12.2. A multilayered feed forward network, which is trained with the back propagation method using a momentum term as well as an adaptive learning rate to speed up the rate of convergence, is used in this work. The error between the actual mismatch (obtained from simulation and experiments) and that predicted by the network is used as the error signal to train the network as described earlier. 

This is a first order discrete transfer function of the sampled process. In general, the dynamic behaviour between a single input variable X and an output variable y in most polymer reactor systems can be adequately modelled at the sampling instants by a difference equation model of the form y = < iyt l + 2yt-2 + ... 

Given the main characteristics of the pulp process, it was possible to list seven different types of disturbance that can affect the dynamic behaviour of the process the input and output variables, the autoregressive parameter and the residuals of the model (p, N,). 

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