Input-output behaviour

As complexity is not considered here, the internal state of a component is not really interesting. Hence a notion that describes the pure input-output behaviour of a component might be more appropriate, e.g., as in [Gray92, Schu94]. However, the behaviour of connected components is much easier to describe with state-transition functions. 

Since this work deals with the aggregated simulation and planning of chemical production processes, the focus is laid upon methods to determine estimations of the process models. For process control this task is the crucial one as the estimations accuracy determines the accuracy of the whole control process. The task to find an accurate process model is often called process identification. To describe the input-output behaviour of (continuously operated) chemical production plants finite impulse response (FIR) models are widely used. These models can be seen as regression models where the historical records of input/control measures determine the output measure. The term "finite" indicates that a finite number of historical records is used to predict the process outputs. Often, chemical processes show a significant time-dynamic behaviour which is typically reflected in auto-correlated and cross-correlated process measures. However, classic regression models do not incorporate auto-correlation explicitly which in turn leads to a loss in estimation efficiency or, even worse, biased estimates. Therefore, time series methods can be applied to incorporate auto-correlation effects. According to the classification shown in Table 2.1 four basic types of FIR models can be distinguished. 

A DEVS model of a system takes a sequence of discrete events, i.e. of instantaneous system changes, as inputs and produces an output sequence of events according its initial conditions. Figure 2.22 displays this input/output behaviour of a DEVS model. Events can be characterised by a value and the time point of their occurrence. Accordingly, they are indicated by perpendicular strokes in Fig. 2.22. A sequence of events is called an event trajectory. It is assumed that the number of state changes in any finite time interval is finite. 

The input-output behaviour of the operational amplifier is captured by the simple commonly used model depicted in Fig. B.7. That is, the example circuit contains a dependent controlled source. 

For illustration, consider the circuit schematic of a simple functional model of an analog integrator depicted in Fig. 4.26. In reality, an integrated operational amplifier is built by means of a number of transistors. The macro-model in Fig. 4.26 reproduces the input-output behaviour of an operational amplifier. It is sufficiently accurate for low frequencies. Its parameters that can be tuned are the gain. A, the input resistance Ri, and the output resistance Ro- The measurement at internal nodes of a real integrated circuit requires special equipment such as a probe station. The output voltage Vo of a bonded and packaged operational amplifier chip can be measured at one of its pins and may be used for the detection of possible failures in the circuit [36]. 

Performance indices for state-space models Model-based control is considered for this type of model as a modelling goal, where the requirements of a process system to satisfy the modelling goal is naturally given in terms of properties and/or parameters of a desired input-output behaviour. 

Minimality for nonlinear state-space models Lumped process models with differential index 1 can be described by input-affine state-space models, where the notion of minimality is related to the number of state variables necessary and sufficient to produce a given input-output behaviour. This means, that one naturally uses the single quality index = dim x = n for both linear and nonlinear state-space models and orders the functionally equivalent models accordingly. 

Controllability measures, such as the RGA and CLDG, can sometimes be interpreted on a physical basis [6], but the insight they can give into the dynamics of a process are limited because they only consider input-output behaviour. For process design it is useful to have a measure of dynamic structure which retains information about the internal structure of the model states. The process model states are often closely related to the physical structure of a... 

In this respect the UPSR measures the dynamics of a system s states, rather than a system s input-output behaviour. This focuses attention on the dynamics within a system, especially if the states have a physical significance, but neglects important characteristics of the input-output behaviour. 

Some input-output behaviour can be included in the UPSR by applying it to closed-loop systems. The coefficient matrix A of closed-loop systems will include some of the input-output structure from the open-loop B, C, and D matrices. 

The transfer matrices Ta and Tb are computed from the transfer matrices of the plant and the base functions in the series expansion of the Youla parameter. Eq. (17) is a complete and convex description of all possible input-output behaviours of stable linear closed loops with the given plant. x,tj) depends on the choice of the external inputs Mf) and the external outputs z(0 as well as on the signals used for feedback and the available control inputs. 

Model-based testing (MBT) [20] provides techniques to automatically generate test cases from specification models. These test cases are used to check conformance of the DUT to the specification. Most MBT approaches rely on the input/output behaviour of the DUT and do not need any access to its internal structure. They are therefore called black-box methods. Usually MBT techniques generate test cases until the test suite meets a predefined coverage criteria (e.g. transition coverage, where the test cases have to reach every transition of the model at least once). 

Thus the transfer function is basically an input-output mathematical relationship. This is a most appropriate concept to use in conjunction with block diagrams (Section 7.2.1) which are themselves basically input-output schematic diagrams so that each block may be represented by the transfer function describing its behaviour. 

Thus, the behaviour of the non-linear on-off element is approximated by the input/output relationship ... 

A novel gradient-based optimisation framework for large-scale steady-state input/output simulators is presented. The method uses only low-dimensional Jacobian and reduced Hessian matrices calculated through on-line model-reduction techniques. The typically low-dimensional dominant system subspaces are adaptively computed using efficient subspace iterations. The corresponding low-dimensional Jacobians are constructed through a few numerical perturbations. Reduced Hessian matrices are computed numerically from a 2-step projection, firstly onto the dominant system subspace and secondly onto the subspace of the (few) degrees of freedom. The tubular reactor which is known to exhibit a rich parametric behaviour is used as an illustrative example. 

A probabilistic interactive function is a probabilistic state-transition function, usually on a countable domain. Hence it is characterized by three sets /, O, and 5, the sets of possible inputs, outputs, and states, respectively, and a probabilistic function (see above) / 7 x 5 — O x 5. (In contrast to the field of non-cryptologic protocol specification, the functions themselves, and not finite descriptions of them, are used here. Anyway, they are mainly used to model the unknown behaviour of computationally unrestricted attackers.)... 

A drastic change in the output behaviour over small variations of the input parameters is not unusual for real-world models and therefore needs further studying. Hence we analyse the following test function... 

The reactive distillation processes which combine reaction and gas liquid separation are of increasing interest for scientific investigation and industrial application. Nowadays, simulation and design of multi component reactive distillation is carried out using the non equilibrium stage model (NEQ model) due to the limitation of conventional equilibrium stage efficiency calculations for equilibrium model (Lee Dudukovic (1998), Baur al. (2000), Taylor Krishna (1993), and Wesselingh (1997)). So, the NEQ model is developed by numerous authors. But there is a lack of experimental data in order to validate the model. Some input/output measurements are available but they provide little information about the behaviour inside the column. With this in mind, our paper is focus on the NEQ models and experimental validation. 

Whenever equation (2) has a solution one must consider the possibility of the existence of input multiplicity behaviour and cannot be confident that the inverse of the system exists as a one-to-one mapping from the values of the output to the values of the input. In this case there is a possibility of a large move of the input for an inverse-based control framework. Equation (2) can be used to determine the locus of the input multiplicity conditions assuming pseudo steady state conditions using continuation methods. 

For example, the subsystem might be a pump, which can be modelled with the help of bondgraphs and easily be introduced in the system as a whole, also modelled with bondgraphs. Subsystems, modelled by means of other computational techniques, can also be introduced by means of and adaptation of the subsystem s behaviour in terms of the input/output relations at the ports in the total system s bondgraph. 

Especially in an operating environment where quality has become more important than quantity, there is a strong desire to develop input-output models that can be used in advanced control applications, in order to develop control strategies for quality improvement. These models are usually discrete hnear transfer function (difference equation) type models, which provide a representation of the dynamic behaviour of the process at discrete sampling... 

When the system state vector v(t) is difficult to determine or totally unavailable, then the system behaviour must be described on the basis of the system input-output relation, i.e. the relations between controlled input signals and the system response. A frequently used approach is the description of the object dynamics by an equation which combines the previous outputs and inputs directly with the fijture outputs from the object... 

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