Output variability

Ratio and Multiplicative Feedforward Control. In many physical and chemical processes and portions thereof, it is important to maintain a desired ratio between certain input (independent) variables in order to control certain output (dependent) variables (1,3,6). For example, it is important to maintain the ratio of reactants in certain chemical reactors to control conversion and selectivity the ratio of energy input to material input in a distillation column to control separation the ratio of energy input to material flow in a process heater to control the outlet temperature the fuel—air ratio to ensure proper combustion in a furnace and the ratio of blending components in a blending process. Indeed, the value of maintaining the ratio of independent variables in order more easily to control an output variable occurs in virtually every class of unit operation. 

The relationship between output variables, called the response, and the input variables is called the response function and is associated with a response surface. When the precise mathematical model of the response surface is not known, it is still possible to use sequential procedures to optimize the system. One of the most popular algorithms for this purpose is the simplex method and its many variations (63,64). 

Introduction The model-based contfol strategy that has been most widely applied in the process industries is model predictive control (MFC). It is a general method that is especially well-suited for difficult multiinput, multioutput (MIMO) control problems where there are significant interactions between the manipulated inputs and the controlled outputs. Unlike other model-based control strategies, MFC can easily accommodate inequahty constraints on input and output variables such as upper and lower limits or rate-of-change limits. 

It is a general control strategy for MIMO processes with inequality constraints on input and output variables. 

Inequality constraints on the input and output variables can be included as an option. 

In principle, the step-response coefficients can be determined from the output response to a step change in the input. A typical response to a unit step change in input u is shown in Fig. 8-43. The step response coefficients are simply the values of the output variable at the samphng instants, after the initial value y(0) has been subtracted. Theoretically, they can be determined from a single-step response, but, in practice, a number of bump tests are required to compensate for unanticipated disturbances, process nonhnearities, and noisy measurements. 

Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

Monte Carlo simulation is a numerical experimentation technique to obtain the statistics of the output variables of a function, given the statistics of the input variables. In each experiment or trial, the values of the input random variables are sampled based on their distributions, and the output variables are calculated using the computational model. The generation of a set of random numbers is central to the technique, which can then be used to generate a random variable from a given distribution. The simulation can only be performed using computers due to the large number of trials required. 

Flowever, using these approximations, it is still assumed that the output variable will be a Normal distribution. 

Measurements of the eontrolled variables will be eontaminated with eleetrieal noise and disturbanee effeets. Some sensors will provide aeeurate and reliable data, others, beeause of diffieulties in measuring the output variable may produee highly random and almost irrelevant information. 

Comparing the system shown in Figure 8.12 with the original PD eontroller given in Example 5.10, the state feedbaek system may be eonsidered to be a PD eontroller where the proportional term uses measured output variables and the derivative term uses observed state variables. 

The number of neurons to be used in the input/output layer are based on the number of input/output variables to be considered in the model. However, no algorithms are available for selecting a network structure or the number of hidden nodes. Zurada [16] has discussed several heuristic based techniques for this purpose. One hidden layer is more than sufficient for most problems. The number of neurons in the hidden layer neuron was selected by a trial-and-error procedure by monitoring the sum-of-squared error progression of the validation data set used during training. Details about this proce-... 

Table II summarizes the yields obtained from the CONGAS computer output variable study of the gas phase polymerization of propylene. The reactor is assumed to be a perfect backmix type. The base case for this comparison corresponds to the most active BASF TiC 3 operated at almost the same conditions used by Wisseroth, 80 C and 400 psig. Agitation speed is assumed to have no effect on yield provided there is sufficient mixing. The variable study is divided into two parts for discussion catalyst parameters and reactor conditions. The catalyst is characterized by kg , X, and d7. Percent solubles is not considered because there is presently so little kinetic data to describe this. The reactor conditions chosen for study are those that have some significant effect on the kinetics temperature, pressure, and gas composition.

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

For each of the infimal decision units, °oe has to consider several groups of input and output variables (Fig. 10). Among the inputs are... 

Rate of change of /Input variable - Output Variable output variable / Process time constant /... 

As another example let us consider a complex batch reactor. We may have to consider the concentration of many intermediates and final products in order to describe the system over time. However, it is quite plausible that only very few species are measured. In several cases, the measurements could even be pools of several species present in the reactor. For example, this would reflect an output variable which is the summation of two state variables, i.e., yi=xi+x2... 

In this case, the unknown parameter vector k is the 2-dimensional vector [k, k2]T. There is only one independent variable (xi=t) and only one output variable. Therefore, the model in our standard notation is... 

Input Analysis addresses input mapping approaches that transform input data without knowledge of or interest in output variables. 

Sometimes more meaningful features may be obtained by considering the behavior of both input and output variables together in the analysis. 

Typically, input-output analysis methods extract the most relevant signal features by relating the analyzed variables to process output variables, y,-,... 

As discussed and illustrated in the introduction, data analysis can be conveniently viewed in terms of two categories of numeric-numeric manipulation, input and input-output, both of which transform numeric data into more valuable forms of numeric data. Input manipulations map from input data without knowledge of the output variables, generally to transform the input data to a more convenient representation that has unnecessary information removed while retaining the essential information. As presented in Section IV, input-output manipulations relate input variables to numeric output variables for the purpose of predictive modeling and may include an implicit or explicit input transformation step for reducing input dimensionality. When applied to data interpretation, the primary emphasis of input and input-output manipulation is on feature extraction, driving extracted features from the process data toward useful numeric information on plant behaviors. 

KBSs can be viewed with increasing levels of commitment to problem solving. At the level described in the previous section, a KBS accomplishes symbolic-symbolic mappings between input and output variables analogous to the numeric-symbolic mappings of approaches such as neural networks and multivariate statistical interpreters. For each problem-solving task, the particular numeric-symbolic or symbolic-symbolic approach is based on the task and the knowledge and data available. 

The variables selected as design variables (fixed by the designer) cannot therefore be assigned as output variables from an f node. They are inputs to the system and their edges must be oriented into the system of equations. 

If, for instance, variables r3 and u4 are selected as design variables, then Figure 1.11 shows one possible order of solution of the set of equations. Different types of arrows are used to distinguish between input and output variables, and the variables selected as design variables are enclosed in a double circle. 

We ll learn how to identify input and output variables, how to distinguish between manipulated variables, disturbances, measured variables and so forth. Do not worry about remembering all the terms here. We ll introduce them properly later. 

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