Input multiplicity

Multivariable control strategies utilize multiple input—multiple output (MIMO) controUers that group the interacting manipulated and controlled variables as an entity. Using a matrix representation, the relationship between the deviations in the n controlled variable setpoints and thek current values,, and the n controUer outputs, is... 

Watt, Universal AC Input, Multiple-output Flyback Converter... 

In the next two examples, we illustrate how state space models can handle a multiple-input multiple output (MIMO) problem. We ll show, with a simple example, how to translate information in a block diagram into a state space model. Some texts rely on signal-flow graphs, but we do not need them with simple systems. Moreover, we can handle complex problems easily with MATLAB. Go over MATLAB Session 4 before reading Example 4.7A. 

There are many advanced strategies in classical control systems. Only a limited selection of examples is presented in this chapter. We start with cascade control, which is a simple introduction to a multiloop, but essentially SISO, system. We continue with feedforward and ratio control. The idea behind ratio control is simple, and it applies quite well to the furnace problem that we use as an illustration. Finally, we address a multiple-input multiple-output system using a simple blending problem as illustration, and use the problem to look into issues of interaction and decoupling. These techniques build on what we have learned in classical control theories. 

In this section, we analyze a multiple input-multiple output (MIMO) system. 

With multiple steadystates, the process outputs can be different with the same process inputs. The reverse of this can also occur. This interesting possibility, called input multiplicity, can occur in some nonlinear systems. In this situation we have the same process outputs, but with different process inputs. For example, we could have the same reactor temperature and concentration but with different values of feed flow rate and cooling water flow rate. 

Here p is the set of characteristic values of the parameters i.e. p(x) = p co(jt) where w(x) has values centered on 1. Often we can set p = / p(x) g(x) dx. The proof is really a statement of what linearity means, for if g(x)djt is the input concentration, g(x)dx.A(p(x)) is the output when the parameter values are p(x). Here x serves merely as an identifying mark, being truly an index variable and the integration in equation (14) follows from the superposition principle for linear systems. The same principle allows an obvious extension to multiple input, multiple output linear systems. A becomes a matrix whose elements are the response at one of the out-ports to a unit input at one of the in-ports, the input at all the others being zero. The detail of this case will not be elaborated here, but rather shall we pass to an application. 

A distillation column provides a good example of multiple input/multiple output (MIMO) control and illustrates well the qualitative methodology involved in determining a suitable control strategy for a process. The first requirement is to decide the primary objective of the process, i.e. what is its principal purpose Let us suppose that, for the column shown in Fig. 7.9, it is required to produce an overhead product D of a particular specification xD without attempting to control... 

Fic. 7.9. Multiple input/multiple output (M1MO) control of a distillation column... 

Interaction can be between two or more processes or between actions produced by different control loops applied to a single process. The former has already been discussed in Section 1.53. Some degree of interaction between control loops will nearly always occur in a multiple-input/multiple-output (MIMO) system. For example, consider the distillation process described in Section 7.3 (Fig. 7.9). Suppose it is desired to control simultaneously the compositions of both the overheads product stream (by manipulating the reflux flowrate) and the bottoms product stream (by regulating the steam flowrate to the reboiler). A typical arrangement is shown in Fig. 7.73. 

This is the most general heat-balance equation for a multiple input, multiple output, multiple reactions (and of course multi-components) system. 

A two-phase heterogeneous system with multiple inputs, multiple outputs, and multiple reactions in each phase and with mass transfer between the two phases... 

Figure 4.8 Input multiplicity of product purity loop.

A plot of xb vs. R at steady state (Figure 4.8) reveals an input multiplicity at low values of the recycle flow rate, an increase in R will yield a decrease in the purity of the product. If R is, however, increased further, it will eventually... 

Empirical Model Identification. In this section we consider linear difference equation models for characterizing both the process dynamics and the stochastic disturbances inherent in the process. We shall discuss how to specify the model structure, how to estimate its parameters, and how to check its adequacy. Although discussion will be limited to single-input, single-output processes, the ideas are directly extendable to multiple-input, multiple-output processes. 

So far we have been discussing output multiplicity as the primary source of open-loop instability. It is also possible to have input multiplicity in distillation systems. Input multiplicity means that we can get the same output for different levels of the input variables. 

The results obtained in the closed-loop control are summarised in Fig. 1 where five time varying curves are presented. The first one is the controlled variable (< B), the second is the input or control variable (A / F) he third is the feed concentration change ( ao), the fourth is the performance index (or objective function) and the last one is the computational time required to get the solution. It should be noted that it happens to the system under consideration to exhibit a steady-state input multiplicity and thus several solutions. This point is well discussed in [12],... 

Sensitivity analysis is a type of uncertainty analysis that is used to consider the impacts of uncertainty. In such analyses, one input is changed at a time to determine how the results of a model will change over the range of possible values of that single input. Multiple inputs can be varied simultaneously, using a sampling technique called Monte Carlo analysis, to obtain an overall distribution of the result. 

In practice, a blend of local and global SA approaches may be employed. Quantitative uncertainty in the form of standard errors may be available only for some model inputs. Multiple executions of the simulation to characterize the global SA of the quantified parameters may be undertaken conditional on a series of fixed-... 

A multiple-input/multiple-output (MIMO) process has two or more inputs and two or more outputs. A two-input/two-output system is shown schematically in Figure 15.69. Note that both Cj and C2 affect both y, and y2- When both inputs affect both outputs, the process is referred to as a coupled process. MIMO processes are frequently encountered in the chemical processing industries. 

Depending on how many controlled outputs and manipulated inputs we have in a chemical process, we can distinguish the control configurations as either single-input, single-output (SISO) or multiple-input, multiple-output (MIMO) control systems. 

In the chemical industry most of the processing systems are multiple-input, multiple-output systems. Since the design of SISO systems is simpler, we will start first with them and progressively cover the design of MIMO systems. 

What manipulated variables should be used A multiple-input, multiple-output system possesses several manipulated variables which can be used for the design of a control system. The selection of the most appropriate manipulations is a very critical problem and should be approached with care. Some manipulations have a direct, fast, and strong effect on the controlled outputs others do not. Furthermore, some variables are easy to manipulate in real life (e.g., liquid flows) others are not (e.g., flow of solids, slurries, etc.). 

VI.11 The following sets of linear differential equations describe the dynamic behavior of various multiple-input, multiple-output processes. 

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