Dynamic system nonlinear modeling

Hernandez, E., and Arkun, Y., A study of the control relevant properties of backpropagation neural net models of nonlinear dynamical systems. Comput. Chem. Eng. 16, 227 (1992). 

T. Gross and U. Feudel, Generalized models as an universal approach to the analysis of nonlinear dynamical systems. Phys. Rev. E 73, 016205 (2006). 

In the fifth chapter, a general overview of temperature control for batch reactors is presented the focus is on model-based control approaches, with a special emphasis on adaptive control techniques. Finally, the sixth chapter provides the reader with an overview of the fundamental problems of fault diagnosis for dynamical systems, with a special emphasis on model-based techniques (i.e., based on the so-called analytical redundancy approach) for nonlinear systems then, a model-based approach to fault diagnosis for chemical batch reactors is derived in detail, where both sensors and actuators failures are taken into account. 

To give an illustration of our concept, we want to present some rather simple model systems. These models are examples of a typical nonlinear behaviour and they might perhaps serve as an explanation of some specific effects found in experiments (vid. Refs. J, 2, 3). It should be emphasized that the models are rather speculative. However, they are based on both, the physical properties and the relevant physical laws, which determine the dynamics of the system under consideration. 

The book targets graduate students and researchers interested in dynamics and control, as well as practitioners involved in advanced control in industry. It can serve as a reference text in an advanced process systems engineering or process control course and as a valuable resource for the researcher or practitioner. Written at a basic mathematical level (and largely self-contained from a mathematical point of view), the material assumes some familiarity with process modeling and an elementary background in nonlinear dynamical systems and control. 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

Consider the dynamic systems whose mathematical modeling often yields nonlinear differenhal-algebraic equations as shown below ... 

Significance of these models is that the complicated solutions are shown to exist even for simple nonlinear dynamic systems. Recently some of these models have been applied to explain the oscillations in experimental systems, e.g. Olsen and Degn (1977). 

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. 

Nonlinear dynamics of complex processes is an active research field with large numbers of publications in basic research and broad applications from diverse fields of science. Nonlinear dynamics as manifested by deterministic and stochastic evolution models of complex behaviour has entered statistical physics, physical chemistry, biophysics, geophysics, astrophysics, theoretical ecology, semiconductor physics and -optics etc. This research has induced a new terminology in science connected with new questions, problems, solutions and methods. New scenarios have emerged for spatio-temporal structures in dynamical systems far from equilibrium. Their analysis and possible control are intriguing and challenging aspects of the current research. 

The analysis is limited to linear dynamic systems. This may seem incompatible with the fact that most of the chemical engineering processes are modeled by nonlinear equations. However, linear techniques are very valuable and of great practical importance for the following reasons (1) There is no general theory for the analytic solution of nonlinear differential equations, and consequently no comprehensive analysis of nonlinear dynamic systems. (2) A nonlinear system can be adequately approximated by a linear system near some operating conditions. (3) Significant advances in the linear control... 

---