An Approximate Transfer Function Model for a Double-Pipe Counter-Flow Heat Exchanger

The transfer functions G(s) for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions G^(s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix Gn(s), n=1,2,…,N. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix G^(s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix G^(s), the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model G(s). The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection.

Structure of the fungal hydroxylase, CYP505A30, and rational transfer of mutation data from CYP102A1 to alter regioselectivity

CYP505A30 is a fungal, self-sufficient cytochrome P450 monooxygenase that can selectively oxyfunctionalise n-alkanes, fatty alcohols, and fatty acids. From alkanes, it produces a mixture of non-vicinal diols by two sequential...

Realization of Cole–Davidson Function-Based Impedance Models: Application on Plant Tissues

The Cole–Davidson function is an efficient tool for describing the tissue behavior, but the conventional methods of approximation are not applicable due the form of this function. In order to overcome this problem, a novel scheme for approximating the Cole–Davidson function, based on the utilization of a curve fitting procedure offered by the MATLAB software, is introduced in this work. The derived rational transfer function is implemented using the conventional Cauer and Foster RC networks. As an application example, the impedance model of the membrane of mesophyll cells is realized, with simulation results verifying the validity of the introduced procedure.

Design of Cascaded and Shifted Fractional-Order Lead Compensators for Plants with Monotonically Increasing Lags

This paper concerns cascaded, shifted, fractional-order, lead compensators made by the serial connection of two stages introducing their respective phase leads in shifted adjacent frequency ranges. Adding up leads in these intervals gives a flat phase in a wide frequency range. Moreover, the simple elements of the cascade can be easily realized by rational transfer functions. On this basis, a method is proposed in order to design a robust controller for a class of benchmark plants that are difficult to compensate due to monotonically increasing lags. The simulation experiments show the efficiency, performance and robustness of the approach.

Rational Transfer Function Model for a Double-Pipe Parallel-Flow Heat Exchanger

Transfer functions of typical heat exchangers, resulting from their partial differential equations, usually contain irrational functions which quite accurately describe the spatio-temporal nature of the processes occurring therein. However, such an accurate but complex mathematical representation is often not convenient from the practical point of view, and some approximation of the original model would be more useful. This paper discusses approximate rational transfer functions for a typical thick-walled double-pipe heat exchanger working in the parallel-flow configuration. Using the method of lines with the backward difference scheme, the original symmetric hyperbolic partial differential equations describing the heat transfer phenomena are transformed into a set of ordinary differential equations and expressed in the form of N subsystems representing spatial sections of the exchanger. Each section is described by a rational transfer function matrix and their cascade interconnection results in the overall approximation model expressed by a matrix of rational transfer functions of high order. Based on the rational transfer function representation, the frequency and steady-state responses of the approximate model are evaluated and compared with those resulting from its original irrational transfer function model. The presented results show better approximation quality for the “crossover” input–output channels where the in-domain heat conduction effects prevail as compared to the “straightforward” channels, where the transport delay associated with the heat convection dominates.

Optimal robustness of passive discrete-time systems

We study different representations of a given rational transfer function that represents a passive (or positive real) discrete-time system. When the system is subject to perturbations, passivity or stability may be lost. To make the system robust, we use the freedom in the representation to characterize and construct optimally robust representations in the sense that the distance to non-passivity is maximized with respect to an appropriate matrix norm. We link this construction to the solution set of certain linear matrix inequalities defining passivity of the transfer function. We present an algorithm to compute a nearly optimal representation using an eigenvalue optimization technique. We also briefly consider the problem of finding the nearest passive system to a given non-passive one.

Computational Algorithms for Reducing Rational Transfer Functions’ Order

A problem of reducing a linear time-invariant dynamic system is considered as a problem of approximating its initial rational transfer function with a similar function of a lower order. The initial transfer function is also assumed to be rational. The approximation error is defined as the standard integral deviation of the transient characteristics of the initial and reduced transfer function in the time domain. The formulations of two main types of approximation problems are considered: a) the traditional problem of minimizing the approximation error at a given order of the reduced model; b) the proposed problem of minimizing the order of the model at a given tolerance on the approximation error. Algorithms for solving approximation problems based on the Gauss-Newton iterative process are developed. At the iteration step, the current deviation of the transient characteristics is linearized with respect to the coefficients of the denominator of the reduced transfer function. Linearized deviations are used to obtain new values of the transfer function coefficients using the least-squares method in a functional space based on Gram-Schmidt orthogonalization. The general form of expressions representing linearized deviations of transient characteristics is obtained. To solve the problem of minimizing the order of the transfer function in the framework of the least squares algorithm, the Gram-Schmidt process is also used. The completion criterion of the process is to achieve a given error tolerance. It is shown that the sequence of process steps corresponding to the alternation of coefficients of polynomials of the numerator and denominator of the transfer function provides the minimum order of transfer function. The paper presents an extension of the developed algorithms to the case of a vector transfer function with a common denominator. An algorithm is presented with the approximation error defined in the form of a geometric sum of scalar errors. The use of the minimax form for error estimation and the possibility of extending the proposed approach to the problem of reducing the irrational initial transfer function are discussed. Experimental code implementing the proposed algorithms is developed, and the results of numerical evaluations of test examples of various types are obtained.