Linear process model

In order to make the problem solvable, a linearized process model has been derived. This enables the use of standard Mixed Integer Linear Programming (MILP) techniques, for which robust solvers are commercially available. In order to ensure the validity of the linearization approach, the process model was verified with a significant amount of real data, collected from production databases and production (shift) reports. 

We now develop a mathematical statement for model predictive control with a quadratic objective function for each sampling instant k and linear process model in Equation 16.1 ... 

The targets for the MPC calculations are generated by solving a steady-state optimization problem (LP or QP) based on a linear process model, which also finds the best path to achieve the new targets (Backx et al., 2000). These calculations may be performed as often as the MPC calculations. The targets and constraints for the LP or QP optimization can be generated from a nonlinear process model using a nonlinear optimization technique. If the optimum occurs at a vertex of constraints and the objective function is convex, successive updates of a linearized model will find the same optimum as the nonlinear model. These calculations tend to be performed less frequently (e.g., every 1-24 h) due to the complexity of the calculations and the process models. 

For the case where perturbations from a cyclic steady are moderate in magnitude, it is convenient to employ a linearized process model when designing a regulatory system. In particular, if the effluent profiles are linearized locally about the upstream and downstream cut points, and if the yield is maximized by enforcing the restriction Puic = P Equations 1-5 lead to the following result ... 

The linear process model used at each scale remains unchanged. 

This problem is extensively studied for refinery operations planning leading to non-linear process models, see Zhang et al. (2001), Li et al. (2005), or Alhajri et al. (2008). 

Key concepts relating to linear process models are reviewed in the first section of the chapter. For deeper coverage, the reader is referred to the following undergraduate-level texts ... 

Explains how to generate linear process models in their standard forms. 

Most work on the development of dynamic process models has been empirical this work is usually referred to as process identification. As mentioned earlier, two classes of empirical identification techniques are available one uses deterministic (step, pulse, etc.) functions, the other stochastic (random) identification functions. With either technique, the process is perturbed and the resulting variations of the response are measured. The relationship between the perturbing variable and the response is expressed as a transfer function. This function is the process model. Empirical identification of process models by the deterministic method has been reported by various workers [55-58]. A drawback of this method is the difficulty in obtaining a measurable response while restricting the process to a linear response (small perturbation). If the perturbation is large, the process response will be nonlinear and the representations of the process with a linear process model will be inaccurate. 

If the prediction is performed using a linear process model then the superposition principle can be used and the vector y can be decomposed as ... 

The non-linear and linear process models are simulated in Matlab Simulink and stored in file F0606.mdl. The design of the simulation will be explained in chapter 8. 

One could choose as file name, for example out.mat and as variable name out. The level would then be stored as follows row one in outmat would contain values of the time, row two would contain the level from the non-hnear process simulation and row three the values of the level from the linearized process model. If one would like to plot the results, one could type ... 

As already discussed in the previous chapters, process behavior is usually non-Unear. Whether or not the empirical model to be developed should also be non-linear depends on the operating range in which the model will be used. If the process is controlled and the operating range is small, a linear process model may be an adequate approximation of reality. The application of the model will determine whether the model needs to be dynamic or static. For control and prediction type applications, models are usually dynamic. 

In this chapter, discrete linear-state space models will be discussed and their similarity to ARX models will be shown. In addition Wiener models are introduced. They are suitable for non-linear process modeling and consist of a linear time variant model and a non-linear static model. Several examples show how to develop both types of models. 

Khandalekar, P.D. and Riggs, J.B. (1995) Non-linear process model-based control and optimization of a model IV FCC unit. Computers and Chemical Engineering, 19 (11), 1153-68. 

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