Stability nonlinear model

Ya.B. s more recent papers have been devoted to the study of nonlinear problems. In 1966 Ya.B. turned his attention to the stabilizing effect of accelerated motion through a hot mixture of a boundary of intersection of two flame fronts, convex in the direction of propagation, and proposed an approximate model of a steady cellular flame. G. I. Sivashinsky, on the basis of this work, proposed a nonlinear model equation of thermodiffusional instability which describes the development of perturbations of a bent flame in time and, together with J. M. Michelson, studied its solution near the stability boundary Le = Lecrit. It was shown numerically that the flat flame is transformed into a three-dimensional cellular one with a non-steady, chaotically pulsating structure. The formation of a two-dimensional cellular structure was also the subject of a numerical investigation by A. P. Aldushin, S. G. Kasparyan and K. G. Shkadinskii, who obtained steady flames in a wider parameter interval. 

It can be easily argued that the choice of the process model is crucial to determine the nature and the complexity of the optimization problem. Several models have been proposed in the literature, ranging from simple state-space linear models to complex nonlinear mappings. In the case where a linear model is adopted, the objective function to be minimized is quadratic in the input and output variables thus, the optimization problem (5.2), (5.4) admits analytical solutions. On the other hand, when nonlinear models are used, the optimization problem is not trivial, and thus, in general, only suboptimal solutions can be found moreover, the analysis of the closed-loop main properties (e.g., stability and robustness) becomes more challenging. 

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. 

Intensity changes in the natural fluorescence of fulvic acid (FA) caused by the binding of metal ions have been well documented. Various quantitative models have been developed relating the measured fluorescence signal to the amount of metal ion bound to fulvic acid. Stem-Volmer, linear, and nonlinear models developed for 1 1 binding between metal ions and fulvic acid ligand sites have been used to calculate concentrations of FA binding sites (CJ, and conditional stability constants (K). However, the ability of these models to describe metal complexation by the polydispersed fulvic acid system is somewhat limited. 

After determining the simplified equation 9, Ventry (23) postulates that Ires Iq where d is related to the residual and initial fluorescence intensity. Manipulation of mass balance and stepwise formation constant relationships, and application of a similar derivation procedure used in the nonlinear model, yields equation 10. Equation 10, like the nonlinear model equation, relates observed changes in FA fluorescence intensity I, to total metal, with a conditional stability constant (for the metal ion and FA) and the degree of complexation of the FA. The modified Stem-Volmer equation is ... 

With best fidelity, the undertaken nonlinear model fitting for the stabilized samples of PE and PE-n-MMT has provided a triple-stage model scheme of successive reactions, wherein an nth-order autocatalysis reaction (Cn) was used at the first step, while a general nth-order (Fn) reaction was used for both the second and the third steps of the overall process of thermal oxidative degradation (Table 1) ... 

A simple two-variable theoretical model is proposed. It exhibits most of the temporal dynamical behaviours reported for nonlinear chemical systems [1] oscillations, excitability, multi stability. This autocatalytic model, part of our nonlinear model of calcium metabolism [2], has been associated with bone calcification processes (nucleation and crystal growth). It is described by the differential system ... 

The subject of liquid jet and sheet atomization has attracted considerable attention in theoretical studies and numerical modeling due to its practical importance.[527] The models and methods developed range from linear stability models to detailed nonlinear numerical models based on boundary-element methods 528 5291 and Volume-Of-Fluid (VOF) method. 530 ... 

There are several control problems in chemical reactors. One of the most commonly studied is the temperature stabilization in exothermic monomolec-ular irreversible reaction A B in a cooled continuous-stirred tank reactor, CSTR. Main theoretical questions in control of chemical reactors address the design of control functions such that, for instance (i) feedback compensates the nonlinear nature of the chemical process to induce linear stable behavior (ii) stabilization is attained in spite of constrains in input control (e.g., bounded control or anti-reset windup) (iii) temperature is regulated in spite of uncertain kinetic model (parametric or kinetics type) or (iv) stabilization is achieved in presence of recycle streams. In addition, reactor stabilization should be achieved for set of physically realizable initial conditions, (i.e., global... 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

Asymptotic server Ob- Process model, (process kinetics as well as yield coefficients can be estimated on-line), process inputs. It takes specifically into account the nonlinear structure of the system Simplicity of the method Stability and convergence are guaranteed if the inputs are persistent and bounded. Partial model knowledge Inputs knowledge Non-adjustable convergence rate. [6]... 

In summary, the origin of the chiral amplification is basically the difference in stability of the homochiral and heterochiral dinuclear Zn complexes. These complexes act as catalyst precursors, but differences in their kinetic behavior also affect the degree of the nonlinear effect. This investigation is probably the first example of elucidation of a molecular mechanism of catalytic chiral amplification (41) and may provide a chemical model of one means of propagation of chirality in nature. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

Integration of a time-dependent thermal-capillary model for CZ growth (150, 152) also has illuminated the idea of dynamic stability. Derby and Brown (150) first constructed a time-dependent TCM that included the transients associated with conduction in each phase, the evolution of the crystal shape in time, and the decrease in the melt level caused by the conservation of volume. However, the model idealized radiation to be to a uniform ambient. The technique for implicit numerical integration of the transient model was built around the finite-element-Newton method used for the QSSM. Linear and nonlinear stability calculations for the solutions of the QSSM (if the batchwise transient is neglected) showed that the CZ method is dynamically stable small perturbations in the system at fixed operating parameters decayed with time, and changes in the parameters caused the process to evolve to the expected new solutions of the QSSM. The stability of the CZ process has been verified experimentally, at least... 

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