Nonlinearities discretized models

The paper addresses the problem of optimising design of a grinding mill-classifier system in order to satisfy the product requirements expressed as a combination of the mean and variance of particle size. A discrete model is used for describing the material transport in the mill, and the optimal conditions are chosen by numerical experiments and nonlinear optimisation. 

Another possibility to quantify the response of a stochastic system to periodic signals is to generalize the notion of synchronization, which is known from deterministic nonlinear oscillators. We will pursue this idea in what follows. To this end we review in section 2.2 the notion of effective synchronization in stochastic systems. The mean number of synchronized system cycles turns out to be an appropriate quantity to characterize the synchronization properties of the system to the periodic signal. However the task remains to calculate this quantity. This calculation will be based on discrete renewal models for bistable and excitable dynamics. These discrete models are introduced in section 2.3. We first recapitulate the well known two state model for the stochastic dynamics of an overdamped particle in a doublewell system [10] and afterwards introduce a phenomenological discrete model for excitable dynamics. In section 2.4 a theory to calculate the mean frequency and effective diffusion coefficient in periodically driven renewal processes is presented. These two quantities allow to calculate the mean number of synchronized cycles. Finally in section 2.5 we apply this theory to investigate synchronization in bistable and excitable systems. 

VII.10 Derive the discrete-time equivalent of the nonlinear continuous model for a stirred tank heater developed in Example 4.4 and given by eqs. (4.4a) and (4.5b). 

The discrete model has the advantage that it can be easily generalized to include various nonlinearities such as the kinetic term F p) and the dependence of the jump kernel w on the density p. In this case we have a nonlinear recurrence equation... 

An appropriate solver needs to be selected to numerically solve discretized conservation equations explained in Section 6.2. There are two main types of solvers, a stationary or steady-state solver for solving steady-state linear or nonlinear computational models and time-dependent or transient solver for solving transient linear or nonlinear computational models. 

This section discusses strategies for including Q—parametrization within an optimization framework of the type described earlier. Since Q is infinite dimensional, a useful first step is to approximate Q using a finite number of parameters. Both continuous and discrete-time approximations will be shown. The integration of the approximation into the optimization framework can be done in different ways. Two different approaches for doing this will be described - direct inclusion of the closed-loop transfer function, and inclusion of the individual components of the feedback system with appropriate interconnections between them. The latter approach admits nonlinear plant models. 

A variety of nonlinear discrete-time models have also been used in process control (Pearson, 1999). They include the neural net models discussed in Section 7.3 as well as nonlinear models obtained by adding nonlinear terms to the linear models of the previous section. 

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). 

A sound decomposition strategy should be applicable to any type of mathematical model of a physical process. Therefore, the set of system equations might include linear or nonlinear equations algebraic, differential, difference, or integral equations continuous or discrete variables with the following restrictions ... 

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