Systems with multiple inputs

Systems with multiple inputs 4.3.1 Principle of superposition... 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

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

For a discrete system with multiple inputs and outputs, we define the discrete transfer function matrix D(z) as follows ... 

For a system with multiple-inputs, multiple-outputs and multiple reactions (the most general case), as shown in Figure 2.22, the mass balance equation is... 

We note that the Volterra-Wiener approach has been extended to the case of nonlinear systems with multiple inputs and multiple outputs [Marmarelis and McCann, 1973 Marmarelis and Naka, 1974 Westwick and Kearney, 1992 Marmarelis, 2004] where functional terms are introduced, involving crosskernels which measure the nonlinear interactions of the inputs as reflected on the output This extension has led to a generalization for nonlinear systems with spatio-temporal inputs that has found applications to the visual system [Yasui et al., 1979 Citron et al., 1981]. Extension of the Volterra-Wiener approach to systems with spike (action potential) inputs encountered in neurophysiology also has been made, where the GWN test input is replaced by a Poisson process of impulses [Krausz, 1975] this approach has found many applications including the study of the hippocampal formation in the brain [Sclabassi et al., 1988]. Likewise, the case of neural systems with spike outputs has been explored in the context of the Volterra-Wiener approach, leading to efficient modeling and identification methods [Marmarelis et al., 1986 Marmarelis and Orme, 1993 Marmarelis, 2004]. 

Simple linear or sectorally linear models do not enable accurate description of a series of physical phenomena. Some very important limitations are noticeable in the case of highly nonlinear phenomena or systems with multiple inputs (variables), where the scope of input data varies to a large extent. The rapid development of computer techniques drew attention toward the possibility of using computationally simple models that had good non-linear properties and were relatively adept at identifying solutions. This cleared the way for research on strictly biologically inspired models. 

Notice that for a distributed system, the multiple input problem changes into an artificial idle stage mass balance with no reaction or mass transfer at the input for each phase as shown in Figure 6.13. 

If a consensus is reached, the user is more confident. A critical situation may appear in case of conflicting results. A simple approach is to adopt a conservative strategy and get the worst case. A different strategy may be to associate a higher uncertainty to the predictions in case of disagreement. A third strategy is to develop a suitable system capable to deal with multiple inputs. This is the case of the above-mentioned hybrid models, which needs a specific study for the optimization of the final results. 

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. 

From the discussion in Chapter 23, two characteristics should be clear concerning the design of control systems for processes with multiple inputs and multiple outputs ... 

Chemical and biochemical units with multiple-input multiple-output (MIMO) and with multiple reactions (MRs) for all of the above-mentioned systems are also covered. Nonreacting systems and single-input single-out-put (SISO) systems are treated as special cases of the more general MIMO, MR cases. The systems approach helps to establish a solid platform on which to formulate and use these generalized models and their special cases. 

Let us first review the most general material and heat balance equations and all of the special cases which can be easily obtained from these equations. This will be followed by the basic idea of how to transform these material and energy balance equations into design equations, first for lumped systems and then followed by the same for distributed systems. We will use homogeneous chemical reactors with multiple inputs, multiple outputs, and multiple reactions. It will be shown in Chapter 6 how to apply the same principles to heterogeneous system and how other rates (e.g., rates of mass transfer) can systematically replace (or is added to) the rates of reactions. 

The general mass balance equation for the reacting system with single input-single output (SISO) and N multiple reactions is given by... 

LAMMPS [225] is a classical MD program implementing potentials for soft materials (biomolecules, polymers), solid-state materials (metals, semiconductors), and coarse-grained or mesoscopic systems. The code is designed to be easy to modify or extend with new functionalities. The comprehensive manual compensates for the somewhat clumsy input script syntax. Most of its model potentials have been parallelized and run on systems with multiple CPUs and GPUs, granting very good speedups, especially for the most compUcated pair potential styles, like the Gay-Beme and other CG potentials. 

Use an opto-mechanical switch to direct all of laser to each channel sequentially. There are several types of fiber switches commercially available. The only major drawback to these devices are their maximum power capacity. Careful selection of components is required to generate a reliable device which is usable over a wide range of excitation wavelengths and input laser powers. Switches with multiple inputs and outputs can be put to use in systems configured with backup lasers. 

Extensions of residence time distributions to systems with multiple inlets and outlets have been described (27-29). If the system contains M inlets and N outlets one can define a conditional density function E. (t) as the normalized tracer impulse response in outlet j to input in inlet i as shown schematically in Figure 2. 

Karow et al. (Kl), Rushton et al. (R12), and Oldshue (02) studied rates of absorption with multiple impellers, and their results indicate that incorrect spacing and operating conditions result in decreased capacities. Rushton et al. (R12) found that for ratios of liquid depth to tank diameter less than 2.5 it was not possible to alter the absorption coefficients more than +10% by the use of multiple impellers. They achieved a 25 % increase in the absorption coefficient by the use of multiple impellers for a system with liquid depth equal to 4 tank diameters, at a power input of about 3.5 hp/1000 gal. In the same system, the coefficient could be decreased by 50% if the impellers were not properly positioned. Advantages of multiple turbines accrue only at high air flows or at high power levels (1.8-3.6 hp/1000 gal). Disadvantages of multiple impellers may come about at low air flows, low power inputs, and with improper spacing of impellers. 

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. 

When dealing with an entire fire detection system that utilizes more than one type of detector, a Detonator Module greatly expands the flexibility and capability of the system. An individual Detonator Module can accept multiple inputs from UV and IR controllers, other Detonator Modules, manual alarm stations, heat sensors, smoke detectors or any contact closure device. In the event of a fire, any of these devices will cause the internal fire circuitry of the module to activate the detonator circuit, sound alarms, and identify the zone that detected the fire. When properly used, a Detonator Module will add only one millisecond to the total system response time. See Figure 8 for an illustration of a fire detection system with a Detonator Module. 

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. 

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