Non-linear feedback

Zames, G. (1966) On the Input-Output Stability of Time-Varying Non-Linear Feedback Systems. Parts I and II. IEEE Trans, on Automat. Contr., AC-11(2 3), pp. 228-238, 465-476. 

This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

TES suffer from some limitations such as the small useful temperature range and the non-linearity of the transition curve. The latter drawback is especially evident in roughly patterned TES, as in the case shown in Fig. 15.5 [25], Feedback techniques, similar to those used in electronic amplifiers, minimize these drawback, reducing also the TES time response [26], The superconducting transition temperature (sometimes quite different from those of the bulk metal) of a TES made with one metal layer (single layer) depends on the metal used and on the film thickness. 

A unity feedback control loop consists of a non-linear element N and a number of linear elements in series which together approximate to the transfer function ... 

This step provides a mechanism for feedback within the model, since a product of the reaction has an accelerating influence on the reaction rate. The feedback is also non-linear with a b2 factor leading to an overall dependence of the reaction rate on concentration that is cubic in form. 

This chapter and chapter 5 study the prototypical thermokinetic oscillator. Thermal feedback replaces autocatalysis, and the Arrhenius temperature dependence of rate coefficients supplies non-linearity in the scheme P - A - B + heat. After careful study of this chapter the reader should be able to ... 

The feedback in this model arises from the thermal coupling represented in the form k1(T). An increase in the temperature Tincreases the value of /c1 and hence of the rate of production of heat. The feedback is highly non-linear, following the Arrhenius temperature dependence given by eqn(4.1). 

Fig. 7.86. Equivalent unity feedback block diagram of control system illustrated in Fig. 7.42 including non-linear element...

The concept of adaptive control, in that it can be considered as a kind of non-linear feedback action, is not new. For example, Kalman<45> described a self-optimising controller in 1958. However, it was impossible to implement Kalman s procedure at that time due to digital computer limitations. The more recent... 

As shown notably by the thermodynamic school of Brussels,9,10 systems maintained far from equilibrium and endowed with appropriate non-linearities and feedback interactions may display such nontrivial behaviors as sustained oscillations and multiple steady states (see papers by I. Prigogine and G. Nicolis in this volume). Even though the structures studied are much simpler than biological systems, this type of work provides a firm fundamental basis, not sufficient but absolutely necessary, for the future understanding of such processes as cell differentiation. Clearly, the sustained oscillations and multiple steady states displayed by simpler systems are related, respectively, with the two types of biological regulation described above. 

Non-linearity with Memory. The AR-MNL model is clearly somewhat restrictive in that most distortion mechanisms will involve memory. For example an amplifier with a non-linear output stage will probably have feedback so that the memoryless non-linearity will be included within a feedback loop and the overall system could not be modelled as a memory less non-linearity. The general NARMA model incorporates memory but its use imposes a number of analytical problems. A special case of the NARMA model is the NAR (Non-linear AutoRegressive) model in which the current output x[w] is a non-linear function of only past values of output and the present input s [n ]. Under these conditions equation 4.27 becomes ... 

Thus if the multiplicity of steady states for the catalyst surface manifesting itself in the multiplicity of steady-state catalytic reaction rates has been found experimentally and for its interpretation a three-step adsorption mechanism of type (4) and a hypothesis about the ideal adsorbed layer are used, the number of concrete admissible models is limited (there are four). It can be claimed that some types of adsorption mechanism have "feedbacks , but for the appearance of the multiplicity of steady states these "feedbacks must possess sufficient "strength . The analysis of these cases (mechanisms 4-7 in Table 2) shows that, to achieve multiplicity, the reaction conditions must "help the non-linear step. 

Phenomena such as El Ninos or thermocline circulation in the North Atlantic lead to global instability in the processes of energy and matter exchange, which should be reflected by parameterizing non-linear feedbacks. 

We can view this mode as combination of the Counter mode and the Cipher Feedback mode (CTR+CFB). After processing all message blocks the cipher internal state (.S, ) depends linearly on the last block and non-linearly on all other message blocks (Mi), encryption key (K) and initialization vector (IV). In order to get Message Authentication Code (MAC) it is necessary to perform yet one blank encryption. The obtained base cipher output block or its part can be used as MAC. 

NON-LINEARITY AND FEEDBACK IN CHEMICAL KINETICS STOICHIOMETRY AND ELEMENTARY STEPS... 

Fig. 5.1. Variation of reaction rate R with extent of reaction (a) linear relationship for first-order reaction (b) non-linear deceleratory reactions of overall order n (c) reactions showing chemical feedback in the form of autocatalysis (d) comparison of chemical and...

---