Transcendental Transfer Functions Research Articles

Current-Mode Control of a Distributed Buck Converter with a Lossy Transmission Line

This article presents a buck converter in which the inductor has been replaced by a transmission line. This approach would be practically conceivable if the power supply and load had a greater spatial distance. Alternatively, the model derived in this way could also be regarded as an intermediate model in order to replace a power coil via discretization with a larger number of smaller coils and capacitors. In the time domain, this new converter can be described by a system of coupled partial and ordinary differential equations. In the frequency domain, a transcendental transfer function is obtained. For comparison with an equivalently parameterized conventional converter, Padé approximants are derived. A linear controller is designed for the converter topology under consideration.

Feedback enhanced Dyakonov–Shur instability in graphene field-effect transistors

Graphene devices are known to have the potential to operate THz signals. In particular, graphene field-effect transistors (gFETs) have been proposed as devices to host plasmonic instabilities in the THz realm; for instance, Dyakonov–Shur (DS) instability which relies upon dc excitation. In this work, starting from a hydrodynamical description of the charge carriers, we extend the transmission line description of gFETs to a scheme with a positive feedback loop, also considering the effects of delay, which leads to the transcendental Laplace-transform transfer function, with complex frequency s, with terms of the form e−assechk(s)/s , for a given a∈R0+ arising from the delay time and with k∈N . Applying the conditions for the excitation of DS instability, we report an enhanced voltage gain in the linear regime that is corroborated by our simulations of the nonlinear hydrodynamic model for the charge carriers. This translates to both greater saturation amplitude—often up to 50% increase—and faster growth rate of the self-oscillations. Thus, we bring forth a prospective concept for the realization of a THz oscillator suitable for future plasmonic circuitry.

Haar wavelet operational matrix based numerical inversion of Laplace transform for irrational and transcendental transfer functions

Irrational and transcendental functions can often be seen in signal processing or physical phenomena analysis as consequences of fractional-order and distributed order models that result in fractional or partial differential equations. In cases when finding solution in analytical form tends to be difficult or impossible, numerical calculations such as Haar wavelet operational matrix method can be used. Haar wavelet establishes a direct procedure for transfer function inversion using the wavelet operational matrix for orthogonal function set integration. In this paper an inverse Laplace transform of irrational and transcendental transfer functions using Haar wavelet operational matrix is proposed. Results for a number inverse Laplace transforms are numerically solved and compared with the analytical solutions and solutions provided by commonly used Invlap and NILT algorithms. This approach is useful when the original cannot be represented by an analytical formula and validity of the obtained result needs to be crosschecked and error estimated.

Pressure pattern for fault diagnosis in gas pipelines through graph tools

This paper deals with the detection and localization of damage in gas-filled pipelines by means of an oriented-signals graph. The purpose is to search for transient patterns in a faulty main conduit, when an external acoustic source excites the viscous fluid. The study assumes a linearized model of distributed parameters for the healthy sections of the pipe and two-port lumped models for leaks and blockages. Based on these models, the exact transcendental transfer function of an acoustic source at the upstream end P(0,s) to the pressure P(z0,s) at point z0 is derived and transformed through graph tools in an oriented-signals graph for one and two faults. Thus, one establishes the transient response patterns associated with the faults from the graph paths without numerical parameters. This shown that graph theory is another suitable tool to visualize pressure wave propagation in a damage conduit. To validate the responses for one and two leaks, one compares the features with the analytical solution and the digital simulated pressure.

Analytical distribution function of relaxation times for porous electrodes and analysis of the distributions of time constants

Based on the work by Boukamp (Boukamp, 2017), the method of Fuoss and Kirkwood (Fuoss and Kirkwood, 1941) is applied to derive an analytical distribution function of relaxation times for physics based porous electrode impedance cases. These impedance models are typically described by transcendental transfer functions. The porous electrode impedance treated here reflects a balance of the effective ionic and electronic impedances inside a porous electrode consisting of particles. Therefore, first the DFRT of the single particle interface impedance is derived. This includes treatment of charge transfer, double layer charging, solid state diffusion inside the particles, open-circuit voltage variations due to solid-state concentration, and insulating layers surrounding the particles. The resulting single particle DFRT relations are then incorporated into a mathematical description of the porous electrode DFRT. The results show that the DFRT of the porous electrode can be clearly separated into distributions of time constants corresponding to charge transfer, solid state diffusion and in case of intercalating particles, like in lithium-ion batteries, a third distribution of time constants is identified. A novelty of this work is the explicit treatment of the low-frequency capacitance and the resulting distribution of time constants in porous electrode systems. Analytical relations for the individual time constants are derived and reported. Since the ideal distribution of time constants can be represented by a series of R||C circuit elements, validation is performed by reconstruction of the impedance spectra, based on the analytical results.

Modelling of Electric Vehicle with PID Controller Transfer Function using GA and Model-Reduced Order DRA Algorithm

In this paper, a model of an electric vehicle transfer function using GA and a model-reduced order discrete time realization (DRA) algorithm is presented. The electric vehicle (EV) control system regulates vehicle speed according to the driver’s command signal and brings the vehicle to its equilibrium point, i.e., the desired speed under any abnormal conditions. The controller transfer function is designed based on EV's dynamic differential equations. An infinite-order transcendental transfer function for the EV model is approximated to find high-reliability discrete-time state-space reduced-order models (ROMs). Reduced the robustness and increased the predictions' accuracy of ROM output in terms of accurate numerical simulations of the EV transfer function. Also, the design of the PID controller parameter using optimization techniques (OT) for the tuning of conventional controllers for the desired speed is done using a linearized model. In this study, the performance of a conventional PID controller is compared with that of a GA-based PID controller. In terms of maximum overshoot, peak time, rising time, settling time, and steady state error, the simulation results show that the suggested controller is preferable.

A Reduced-Order Electrochemical Model for All-Solid-State Batteries

All-solid-state batteries (ASSBs) have been considered as the next generation of lithium-ion batteries. Physics-based models have the advantage of providing internal electrochemical information. To promote physics-based models in real-time applications, in this study, a series of model reduction methods are applied to obtain a reduced-order model (ROM) for ASSBs. First, analytical solutions of the partial differential equations (PDEs) are derived by the Laplace transform. Then, the Padé approximation method is used to convert the transcendental transfer functions into lower order fractional transfer functions. Next, the concentration distributions in electrodes and electrolytes are approximated by parabolic and cubic functions, respectively. Due to the fast calculation of concentration distributions in real time, the equilibrium potential, overpotentials, and battery voltage can now be directly calculated. Compared with the original PDE-based model, the voltage errors of the proposed ROM are less than 2.6 mV. Compared with the voltage response of experimental data, a good agreement can be observed for the ROM under three large C-rates discharging conditions. The calculation time of ROM per step is within 0.2 ms, which means that it can be integrated into a battery management system. The proposed ROM achieves excellent performance and a better tradeoff between model fidelity and computational complexity.

Model Simplification Techniques for Li-Ion Battery Thermal-Electrochemical Models

The pseudo two-dimensional (P2D) model proposed by Fuller, Doyle and Newman [1] has taken a predominant role in the modelling of lithium-ion batteries. This model gives accurate predictions of the electrochemical behaviour over a wide range of C-rate conditions but requires further development when simulating the cell voltage with temperature dependences, for characterisation or control of batteries. This is mainly due to the complex system of partial differential equations required for the simulation. For this reason, it is important to develop simplified models which capture the relevant features of the physical system but allow faster and simpler simulations. Here we present an asymptotic reduction and compare model order reduction techniques as one way of model simplification. The asymptotic method allows us to systematically reduce a full P2D thermal-electrochemical model down to much simpler model (in some cases a system of three ordinary differential equations). This method has the advantage of keeping only the phenomena (and thus the parameters) that are relevant and, therefore, provides insight to the experimentalists on which experiments to focus. Within the model reduction methods the residue grouping method is compared with the balance truncation method to reduce the order of a single partial model with electrolyte (SPMe). The residue grouping method can be applied either analytically (from a transcendental transfer function) or numerically (from a finite element model). Residues with similar eigenvalues are grouped together to reduce the model order. The simplified and reduced order model can be used for model based state estimation, and optimal fast charging control. Such simplifications of the model, which are complementary techniques, will also allow more efficient data fitting methods that lead to faster and more accurate characterisations as well as the possibility of running real-time simulations for control purposes. Obtaining a valid set of parameters is crucial for the model development and validation. One problem the models can suffer is the lack of accuracy in parameters, as values are often obtained from literature, and the experiments are not necessarily designed for the parameterisation of the specific battery of interest. In this work, we have chosen a commercially available cylindrical battery, LG M50, to obtain a full set of parameters, specifically for the asymptotic reduction model. The combination of electrochemical testing together with material characterization provides a powerful tool to understand the physical-chemical properties of the cells. We believe that the methods introduced in this work are not only valuable for the model simplifications developed herein, but will also provide valuable insights to the parameterization of other P2D models.

Implementation and Model Order Reduction of a Lithium-Ion Battery Single Particle Model with Electrolyte Dynamics

Electrochemical models of batteries are typically developed from first principles and aim to accurately capture the internal battery dynamics such as diffusion, intercalation kinetics, and electric potentials. Underpinned by such models, battery state or parameter behaviour can be better understood and manipulated by sophisticated estimation and control approaches. However, the use of electrochemical models makes it computationally intractable for online implementation as the model is subject to a complicated mathematical structure including partial-differential equations (PDE), ordinary-differential equations (ODE) and algebraic equations. In order to devise a control-oriented model, to support real-time monitoring and control, model order reduction techniques can be used to reduce the model order while maintaining the desired level of accuracy. This paper is based on the single particle model with electrolyte dynamics (SPMe) derived by Scott J. Moura in [1]. The PDEs in the governing equation of the SPMe are solved by finite difference method (FDM) and finite element method (FEM). The process and differences of applying these two methods are analysed. After solving the PDEs, the full order model would be high (e.g. 350th order). Model order reduction techniques are used to develop a low order Li-ion battery model derivative that is suitable for real-time implementation, for example within a battery control system. Residue grouping method is an order reduction technique for high order systems. It can be applied either analytically (from a transcendental transfer function) or numerically (from finite element method). In this paper, the residue grouping method is applied to the SPMe model to reduce the model order from 350 to 13. The solid-state diffusion equations of the two electrodes employ residue grouping analytically; conversely, the liquid-state diffusion applied residue grouping numerically. Both the full order SPMe model (FOM) and the reduced order model (ROM) have been implemented within the commercially available software Matlab and Simulink. From the simulation results presented in the figure below, it is observed that the ROM correlates well with the SPMe model in the time domain, with a peak relative difference in the order of 10-3 and the root mean square (RMS) of the relative error of 3.09x10-4 in the terminal voltage. In addition to time domain analysis, the ROM is analysed in the frequency domain, by comparing the Bode graphs for each model type. Finally, the application of the ROM within a battery state estimation strategy for the purpose of estimating the lithium ion concentrations and the optimal order number of the model will be discussed.

Dynamics of cortical activity eigenmodes including standing, traveling, and rotating waves

The eigenfrequencies of natural eigenmodes of cortical activity are calculated from corticothalamic neural field theory. These correspond to poles of a transcendental transfer function, which is accurately approximated here as a rational function. In this approximation, the least damped frequencies of the lowest order spatial eigenmodes dominate the dynamics and correspond to slow-wave ($\ensuremath{\lesssim}1$ Hz) and alpha ($\ensuremath{\sim}10$ Hz) resonances. The eigenmodes exhibit standing and rotating wave patterns in which eigenmode beating can give the appearance of sudden changes in dynamics. This provides a physical explanation of sudden transitions that have been argued to occur between artificially discretized brain activity patterns, sometimes termed microstates, and allows the dynamics to be analyzed by replacing these phenomenologically constructed activity patterns with well-defined eigenmodes. An eigenmode interpretation of large-scale brain activity can thus potentially supplant phenomenological approaches and place analysis on a systematic physical footing.

Modal testing of hydraulic pipeline systems

Dynamic models of fluid power systems require accurate descriptions of hydraulic pipeline systems. For laminar flow in a rigid pipeline, modal approximations of transcendental transfer functions lead to a multi-degrees-of-freedom description. This suggests the application of experimental modal analysis to investigate fluid dynamics in hydraulic pipeline systems. The concept of modal testing is adapted accordingly and demonstrated for a straight pipeline, the same pipeline with a single side branch, and a pipeline system with three side branches. Frequency response functions are determined by injecting a defined flow rate excitation and measuring pressure responses along the pipelines. The underlying theory is confirmed by comparisons between calculated transcendental, measured, and estimated rational frequency response functions. Natural frequencies, damping ratios, and pressure mode shapes are identified. Although the experiments are made for low flow rates and stiff pipeline walls, they indicate the way to perform modal testing in practical applications of fluid power.

A Structure-preserving Model Reduction Algorithm for Dynamical Systems with Nonlinear Frequency Dependence

Very large-scale dynamical systems, even linear time-invariant systems, can present significant computational difficulties when used in numerical simulation. Model reduction is one response to this challenge but standard methods often are restricted to systems that are presented as standard first-order realizations; in the frequency domain such systems will be linear in the frequency parameter. We consider here dynamical systems with a nonlinear frequency dependence; systems for which either a standard first-order realization is unknown or inconvenient to obtain. Such systems may nonetheless have realizations that reflect important structural features of the model and we may wish to retain this structure in any reduced model used as a surrogate. In this work, we present a structure-preserving model reduction algorithm for systems having quite general nonlinear frequency dependence. We take advantage of recent algorithms that produce high quality rational interpolants to transfer functions that only require transfer function evaluation, thus allowing for nonstandard realizations that are nonlinear in the frequency parameter. However, our final reduced model will have a structure that reflects the structure of the original system, and indeed, may not have a rational transfer function. We illustrate our approach on a benchmark problem that offers a transcendental transfer function.

Model Order Reduction of 1D Diffusion Systems Via Residue Grouping

A model order reduction method is developed and applied to 1D diffusion systems with negative real eigenvalues. Spatially distributed residues are found either analytically (from a transcendental transfer function) or numerically (from a finite element or finite difference state space model), and residues with similar eigenvalues are grouped together to reduce the model order. Two examples are presented from a model of a lithium ion electrochemical cell. Reduced order grouped models are compared to full order models and models of the same order in which optimal eigenvalues and residues are found numerically. The grouped models give near-optimal performance with roughly 1∕20 the computation time of the full order models and require 1000–5000 times less CPU time for numerical identification compared to the optimization procedure.

Neural net versus classical models for the detection and localization of leaks in pipelines

Four models of a pipeline are compared in the paper: a nonlinear distributed-parameter model, a linear distributed-parameter model, a simplified lumped-parameter model and an extended neural-net-based model. The transcendental transfer function of the linearized model is obtained by a Laplace transformation and corresponding initial and boundary conditions. The lumped-parameter model is obtained by a Taylor series extension of the transencdental transfer function. Based on the experience of linear models the structure of the neural net model, as an addendum to the nonlinear distributed-parameter model, is obtained. All four models are tested on a real pipeline data with an artificially generated leak.

On the Passivity of a Galerkin Finite Element Model for Transient Flow in Hydraulic Pipelines

The Galerkin finite method for the simulation of transient, laminar flow in hydraulic pipelines with an interlacing grid has gained considerable popularity because of the relatively good accuracy achieved with a moderate model order. However, the implementation used in a number of software packages has recently been shown not to guarantee model passivity. This paper reviews the concept of using an interlacing grid. Two modifications of the previously published interlacing grid method are proposed and the resulting model is shown to fulfil the passivity requirement in an example where the previously published method fails. The new model also improves the approximation error in terms of the deviation of the frequency response from the theoretical frequency response given by the transcendental transfer functions of the well-known ‘frequency-dependent friction’ model. This improvement is bought at the expense of a slight increase in model order due to additional nodal values for the friction model at the boundary nodes. Furthermore, the new model is given in the form of a descriptor system in order to maintain the sparse structure of the system matrices.

ROBUST CONTROL OF CHAOTIC VIBRATIONS FOR IMPACTING HEAT EXCHANGER TUBES IN CROSSFLOW

A robust feedback controller design to suppress flutter-type chaotic vibrations in baffled heat exchanger tubes is presented. The vibrations are the result of the fluid dynamic forces on the tube which behave as a negative damping element. A consequence of these vibrations is that the heat exchanger tubes impact with the baffle plates thereby reducing the service life of the heat exchanger. To eliminate the resulting tube impacts, a feedback control strategy is proposed. The heat exchanger tube and the fluid dynamic forces acting on the tube are modeled with linear delayed differential equations. Due to the presence of the delay, these equations do not have a rational realization. The feedback controller is realized using a frequency domain loop shaping approach which is well suited for systems with transcendental transfer functions. The control effector is a magnetic force transducer that acts on the heat exchanger tube. The design strategy is based upon the premise that stabilizing the linear instability about the undeformed tube position will preclude the formation of the nonlinear chaotic vibrations that arise from impacting. The feedback controller is shown to provide robust stability and performance over a large flow velocity regime

Modelling the Nonlinear Dynamics of Combustion Instabilities in Gas Turbines Towards Active Control

A main obstacle to the use of premixed flames in gas turbine combustors is the occurrence of combustion instabilities. In this paper, we propose a model for the description of the coupling between the acoustic field and the heat release process. While the gasdynamics is modelled by means of transcendental transfer functions, the flame description, worked out on physical basis, is a nonlinear lumped parameter system. The developed model is intended for active control, and it accounts for the limit-cycle phenomenon.

Modal analysis and control of a rotating Euler-Bernoulli beam part I: Control system analysis and controller design

Control of the vibration modes become critical when one wants to push the state of the art with faster, lighter, and more accurate flexible link. There are three steps which are necessary for the control of the flexible link. First, a good design based model of the plant must exist. Second, a good controller which is also realizable must be designed. Third, input to the controller must be constructed using knowledge of the system dynamic response. In this paper, involving a complete control strategy, pertaining to design based model, control, and dealing with the shaping of system input is presented. In Part I, a single-input single-output transcendental transfer function, pole-zero pattern, controllability, observability, and system type for distributed parameter system is illustrated by application to feedback control of an Euler-Bernoulli beam. The eigenfunctions, orthogonality condition, and mode summation method have been investigated in order to get the system analytical solution. A new control scheme, which depends on the pole-zero plot of the infinite-dimensional system and uses a realizable actuator and sensor without involving truncation of the higher-frequency modes, shows that good stability, robustness, and efficient tracking property can be achieved by moving all the poles of the corresponding closed-loop system further into the left half-plane.

Closed-Loop Behavior of a Feedback-Controlled Flexible Arm: A Comparative Study

The dynamics of mechanical systems with distributed flexi bility are described by infinite-dimensional mathematical models. In order to design afinite-dimensional controller, a finite-dimensional model of the system is needed. The con trol problem of a flexible beam is a typical example. The general practice in obtaining a finite-dimensional model is to use modal approximation for distributed flexibility, retain a finite number of modes, and truncate the rest. In this approx imation, the appropriate selection of the mode shape func tions and the number of modes is not clearly known. Mostly standard pinned-free and clamped-free mode shapes are used for the flexible beam model, retaining only two or three modes and truncating the rest. The actual system, on the other hand, is infinite-dimensional, and the modes describing its flexible behavior under feedback control would be neither pinned-free nor clamped-free boundary condition modes. Rather, the mode shapes themselves are a function of the feedback control. The infinite-dimensional transcendental transfer functions for a flexible beam are formulated without any modal ap proximation. Finite-dimensional transfer functions with different shapes and numbers of modes are formulated. The closed-loop performance predictions of different models under the same colocated and noncolocated controllers, which attempt to achieve high closed-loop bandwidth, are compared. Results are surprisingly consistent in all cases; the predictions of clamped-free mode shape models are much more accurate than the predictions of the pinned-free mode shape models.

Estimation of uncertainty in an open-channel network mathematical model

A mathematical model is presented for open-channel networks that expresses the dynamic relationships, in terms of transcendental transfer functions, between the gate opening sections and the corresponding stored water volume variations in the different canal reaches with respect to an initial reference condition of uniform flow. Series expansion around s = 0 gives a state variable linear and time-invariant model as well as the corresponding rational transfer matrix. With this last model, taken as the nominal model, is associated a measure of its “distance” from the first model, described by the maximum singular value of a suitable matrix, which characterizes the uncertainty and can be used in the synthesis of a robust control system based on the nominal model.