Process system input-output relationships

The term model-based can be a source of confusion because descriptions of any aspects of reality can be considered to be models. Any KBS is model based in this sense. For some time, researchers in KBS approaches (Venkatasubramanian and Rich, 1988 Finch and Kramer, 1988 Kramer and Mah, 1994 McDowell and Davis, 1991,1992) have been using model-based to refer to systems that rely on models of the processes that are the objects of the intent of the system. This section will avoid confusion by using the term model to refer to the type of model in which the device under consideration is described largely in terms of components, relations between components, and some sort of behavioral descriptions of components (Chandrasekaran, 1991). In other words, model-based is synonymous with device-centered. Figure 27 shows a diagram displaying relationships among components. The bubble shows a local model associated with one of the components that relates input-output relationships for flow, temperature, and composition. 

The Laplace transforms allowed us to develop simple input-output relationships for a process and provided the framework for easy analysis and design of loops with continuous analog controllers. For discrete-time systems we need to introduce new analytical tools. These will be provided by the z-transforms. 

We shall call discrete those dynamic systems that process their input signals only at the sampling instants, thus producing output signals that are only defined at particular time instants. In other words, discrete is a system whose input and output are discrete-time signals. The input-output relationship for such systems is given by a discrete-time model (i.e., by a difference equation). 

Neural networks are processing systems that work by feeding in some variables and get an output as response to these inputs. The accuracy of the desired output depends on how well the network learned the input-output relationship during training. 

There are a number of modeling approaches that can be used with process control systems. Whereas mathematical models based on the chemistry and physics of the system represent one alternative, the typical process control model utilizes an empirical input/output relationship, the so-called black-box model. These models are found by experimental tests of the process. Mathematical models of the control system may include not only the process but also the controller, the final control element, and other electronic components such as measurement devices and transducers. Once these component models have been determined, one can proceed to analyze the overall system dynamics, the effect of different controllers in the operating process configuration, and the stability of the system, as well as obtain other usefid information. 

In Chapter 5 is was stated that a linear system based on deviation variables, is asymptotically stable if the roots of the characteristic eqnation of an inpnt-output relationship have negative real parts. The roots of the characteristic eqnation correspond to the poles of the transfer function or the eigen valnes of the homogenons part of the differential equation. Therefore, it is necessary to derive the characteristic linear input-output relationship for the process in an operating point. Because only the homogenous part of the differential equation is of interest, the inputs of the differential equations can be ignored. Next, fiom the roots of the characteristic equation, conditions for stability can be formulated. The following three steps will be performed ... 

We know how to find the z transformations of functions. Let us now turn to the problem of expressing input-output transfer-function relationships in the z domain. Figure 18.9a shows a system with samplers on the input and on the output of the process. Time, Laplace, and z-domain representations are shown. G(2, is called a pulse transfer function. It will be defined below. 

A neural-network-based simulator can overcome the above complications because the network does not rely on exact deterministic models (i.e., based on the physics and chemistry of the system) to describe a process. Rather, artificia] neural networks assimilate operating data from an industrial process and learn about the complex relationships existing within the process, even when the input-output information is noisy and imprecise. This ability makes the neural-network concept well suited for modeling complex refinery operations. For a detailed review and introductory material on artificial neural networks, we refer readers to Himmelblau (2008), Kay and Titterington (2000), Baughman and Liu (1995), and Bulsari (1995). We will consider in this section the modeling of the FCC process to illustrate the modeling of refinery operations via artificial neural networks. 

The nonlinear nature of these mixed-integer optimization problems may arise from (i) nonlinear relations in the integer domain exclusively (e.g., products of binary variables in the quadratic assignment model), (ii) nonlinear relations in the continuous domain only (e.g., complex nonlinear input-output model in a distillation column or reactor unit), (iii) nonlinear relations in the joint integer-continuous domain (e.g., products of continuous and binary variables in the schedul-ing/planning of batch processes, and retrofit of heat recovery systems). In this chapter, we will focus on nonlinearities due to relations (ii) and (iii). An excellent book that studies mixed-integer linear optimization, and nonlinear integer relationships in combinatorial optimization is the one by Nemhauser and Wolsey (1988). 

In Table 5.1, where the statistical model is presented in a polynomial state, a rapid increase in the number of identifiable coefficients can be observed as the number of factors and the degree of the polynomial also increase. Each process output results in a new identification problem of the parameters because the complete model process must contain a relationship of the type shown in Eq. (5.3) for each output (dependent variable). Therefore, selecting the Ne volume and particularizing relation (5.5), allows one to rapidly identify the regression coefficients. When Eq. (5.5) is particularized to a single algebraic system we take only one input and one output into consideration. With such a condition, relations (5.3) and (5.5) can be written as ... 

Such a model, describing directly the relationship between the input and output variables of a process, is called an input-output model. It is a very convenient form since it represents directly the cause-and-effect relationship in processing systems. For this reason it is also appealing to process engineers and control designers. 

A transfer function is the relationship between the input and output of a system. In classical control systems literature that makes use of Laplace transforms, extensive use is made of Laplace transfer functions. Table 3.2 presents the transfer functions of common process systems dynamics. 

Constant steps are not necessary, but they simplify the matrix g of eq.(6). Eq.(5) and eq.(6) respectively show the relationship between input and output signal for discrete signal processing. It is given by a linear equation system, which can easily be solved. 

A system is an ordered set of ideas, principles, and theories or a chain of operations that produces specific results to be a chain of operations, the operations need to work together in a regular relationship. A quality system is not a random collection of procedures (which many quality systems are) and therefore quality systems, like air conditioning systems, need to be designed. All the components need to fit together, the inputs and outputs need to be connected, sensors need to feed information to processes which cause changes in performance and all parts need to work together to achieve a common purpose i.e. to ensure that products conform to specified requirements. You may in fact already have a kind of quality system in place. You may have rules and methods which your staff follow in order to ensure product conforms to customer requirements, but they may not be documented. Even if some are documented, unless they reflect a chain of operations that produces consistent results, they cannot be considered to be a system. 

Equation 7.52 is the standard form of a second-order transfer function arising from the second-order differential equation representing the model of the process. Note that two parameters are now necessary to define the system, viz. r (the time constant) and (the damping coefficient). The steady-state gain KMT represents the steady-state relationship between the input to the system AP and the output of the system z (cf. equation 7.50). 

---