Nonlinear Output Relationship

When the output vector is nonlinearly related to the state vector (Equation 6.3) then substitution of x ,+l from Equation 6.74 into the Equation 6.3 followed by substitution of the resulting equation into the objective function (Equation 6.4) yields the following equation after application of the stationary condition (Equation 6.78) 

The above equation represents a set of p nonlinear equations which can be solved to obtain k Jf h. The solution of this set of equations can be accomplished by two methods. First, by employing Newton s method or alternatively by linearizing the output vector around the trajectory xw(t). Kalogerakis and Luus (1983b) showed that when linearization of the output vector is used, the quasilinearization computational algorithm and the Gauss-Newton method yield the same results. 

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THE GAUSS-NEWTON METHOD - NONLINEAR OUTPUT RELATIONSHIP... 

Implementation Guidelines for ODE Models The Gauss-Newton Method — Nonlinear Output Relationship The Gauss-Newton Method - Systems with Unknown Initial Conditions Examples... 

Now, to be sure, McCulloch-Pitts neurons are unrealistically rendered versions of the real thing. For example, the assumption that neuronal firing occurs synchronously throughout the net at well defined discrete points in time is simply wrong. The tacit assumption that the structure of a neural net (i.e. its connectivity, as defined by the set of synaptic weights) remains constant over time is known be false as well. Moreover, while the input-output relationship for real neurons is nonlinear, real neurons are not the simple threshold devices the McCulloch-Pitts model assumes them to be. In fact, the output of a real neuron depends on its weighted input in a nonlinear but continuous manner. Despite their conceptual drawbacks, however, McCulloch-Pitts neurons are nontrivial devices. McCulloch-Pitts were able to show that for a suitably chosen set of synaptic weights wij, a synchronous net of their model neurons is capable of universal computation. This means that, in principle, McCulloch-Pitts nets possess the same raw computational power as a conventional computer (see section 6.4). 

PPR is a linear projection-based method with nonlinear basis functions and can be described with the same three-layer network representation as a BPN (see Fig. 16). Originally proposed by Friedman and Stuetzle (1981), it is a nonlinear multivariate statistical technique suitable for analyzing high-dimensional data, Again, the general input-output relationship is again given by Eq. (22). In PPR, the basis functions 9m can adapt their shape to provide the best fit to the available data. 

This representation is also called normal form and it is graphically depicted in Figure 3. It can be seen that the normal form is composed of three parts respectively given by the subsystems (4a), (4b) and (4c). The first part presents a linear structure and it is given by a chain of r — 1 integrators, whereas the second part has a nonlinear structure, where the input-output relationship explicitly appears. Finally, the last part is conformed by the dynamics of the n — r complementary functions. This part is called internal dynamics because it cannot be seen from the input-output relationship (see Figure 3) and whose structure can be linear or nonlinear. 

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). 

Model-free adaptive (MFA) control does not require process models. It is most widely used on nonlinear applications because they are difficult to control, as there could be many variations in the nonlinear behavior of the process. Therefore, it is difficult to develop a single controller to deal with the various nonlinear processes. Traditionally, a nonlinear process has to be linearized first before an automatic controller can be effectively applied. This is typically achieved by adding a reverse nonlinear function to compensate for the nonlinear behavior so that the overall process input-output relationship becomes somewhat linear. It is usually a tedious job to match the nonlinear curve, and process uncertainties can easily ruin the effort. 

An experiment involving a complex computer model or code may have tens or even hundreds of input variables and, hence, the identification of the more important variables (screening) is often crucial. Methods are described for decomposing a complex input-output relationship into effects. Effects are more easily understood because each is due to only one or a small number of input variables. They can be assessed for importance either visually or via a functional analysis of variance. Effects are estimated from flexible approximations to the input-output relationships of the computer model. This allows complex nonlinear and interaction relationships to be identified. The methodology is demonstrated on a computer model of the relationship between environmental policy and the world economy. 

Many biological happenings are nonlinear. They may oscillate, but not with any set frequency. They may form exponential-like responses, but cannot be characterized by one time constant. Input-output relationships may not follow idealized forms. In these cases, the biological engineer must either resort to nonlinear equations or to numerical solutions to describe these phenomena. 

In this section, the general Bayesian framework is presented. It was originally presented for structural model updating using input-output measurements in Beck and Katafygiotis [19]. Consider a linear or nonlinear dynamical system with input-output relationship ... 

Considering a network with a single hidden layer, where the hidden and output layers are denoted by superscripts (1) and (2) respectively, then for r = 1, 2,. .., H hidden nodes the nonlinear functional relationship is represented by equation (8) ... 

The input-output relationship will be decomposed into two or more interconnected elements, when the output of a system depends nonlinearly on its inputs. So, we can describe the relationship by a linear transfer function and a nonlinear function of inputs. The Hammerstein-Wiener model uses this configuration as a series connection of static nonlinear blocks with a dynamic linear block. 

The functional form of the HDMR expansion and its use for global sensitivity analysis was aheady discussed in Sect. 5.5.5, but a similar approach can also be taken to develop reduced model representations. The purpose is to create a fast equivalent operational model (FEOM) based on the HDMR, giving sufficient accuracy with respect to the full chemical model, but with much lower computational expense. HDMR builds approximations recursively, based on the assumption that high-order-correlated effects of the inputs are expected to have negligible impact on the output. Applications have shown that the order of the correlations between the independent variables dies off rapidly, and therefore, only a few terms are usually required to represent even highly nonlinear input-output relationships. 

Compared to other methods, the use of NNs is often effective because they can simultaneously address nonlinear dependences and complex physical behavior with reduced computational efforts and without requiring any a priori information (Bishop 1995). Moreover, NNs learn the input-output relationships exclusively from the training patterns therefore, no explicit expression addressing the interactions between the parameters of interest is needed with such an approach. 

As indicated earlier, the model specification task is generally pursued via two approaches the parametric or hypothesis-based approach and the nonparametric or data-based approach. Ideally, these two approaches should be used in a synergistic manner. In the parametric approach, specific sets of algebraic and/or differential/difference equations are postulated to represent the input-output relationship. These equations are linear or nonlinear depending on whether the subject system/model is hnear or nonhnear. The parametric models contain a number of unknown parameters (e.g., the coefficients... 

The high-field output of laser devices allows for a wide variety of nonlinear interactions [17] between tire radiation field and tire matter. Many of tire initial relationships can be derived using engineering principles by simply expanding tire media polarizability in a Taylor series in powers of tire electric field ... 

The second classification is the physical model. Examples are the rigorous modiiles found in chemical-process simulators. In sequential modular simulators, distillation and kinetic reactors are two important examples. Compared to relational models, physical models purport to represent the ac tual material, energy, equilibrium, and rate processes present in the unit. They rarely, however, include any equipment constraints as part of the model. Despite their complexity, adjustable parameters oearing some relation to theoiy (e.g., tray efficiency) are required such that the output is properly related to the input and specifications. These modds provide more accurate predictions of output based on input and specifications. However, the interactions between the model parameters and database parameters compromise the relationships between input and output. The nonlinearities of equipment performance are not included and, consequently, significant extrapolations result in large errors. Despite their greater complexity, they should be considered to be approximate as well. 

Neural networks can also be classified by their neuron transfer function, which typically are either linear or nonlinear models. The earliest models used linear transfer functions wherein the output values were continuous. Linear functions are not very useful for many applications because most problems are too complex to be manipulated by simple multiplication. In a nonlinear model, the output of the neuron is a nonlinear function of the sum of the inputs. The output of a nonlinear neuron can have a very complicated relationship with the activation value. 

When the output vector (measured variables) are related to the state variables (and possibly to the parameters) through a nonlinear relationship of the form y(t) = h(x(t),k), we need to make some additional minor modifications. The sensitivity of the output vector to the parameters can be obtained by performing the implicit differentiation to yield ... 

The water-oil ratio is a complex time-dependent function of the state variables since a well can produce oil from several grid cells at the same time. In this case the relationship of the output vector and the state variables is nonlinear of the form y(t,)=h(x(t,)). 

Furthermore, a variable classification strategy based on an output set assignment algorithm and the symbolic manipulation of process constraints is discussed. It manages any set of unmeasured variables and measurements, such as flowrates, compositions, temperatures, pure energy flows, specific enthalpies, and extents of reaction. Although it behaves successfully for any relationship between variables, it is well suited to nonlinear systems, which are the most common in process industries. 

Figure 7.4c shows a AP transmitter used with an orifice plate as a flow transmitter. The pressure drop over the orifice plate (the sensor) is converted into a control signal. Suppose the orifice plate is sized to give a pressure drop of 100 in H2O at a process flow rate of 2000 kg/h. The AP transmitter converts inches of HjO into milliamperes, and its gain is 16 mA/100 in HiO. However, we really want flow rate, not orifice-plate pressure drop. Since AP is proportional to the square of the flow rate, there is a nonlinear relationship between flow rate F and the transmitter output signal ... 

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