Showing papers on "Transfer function published in 2017"

Online Identification of Power System Equivalent Inertia Constant

Abstract: This paper proposes a closed-loop identification method to estimate the equivalent inertia constant of a power system at the connection bus. A microperturbation is first performed with a well-designed multisine signal probed through any power electronic devices in the internal system. Then, responses of frequency and active power measured by the phase measurement unit at the connection bus are used for the closed-loop identification. Compared to the conventional transient signal based method, the proposed method has simple implementation and minimum impacts on the system security, and thus could be carried out in real-time to identify the time-varying and nonlinear equivalent inertia constant in modern power systems with complex heterogeneous components. The effectiveness of the proposed method is validated in an 8-generator 36-bus simulation system and an actual power system.

All-optical regenerator of multi-channel signals.

Abstract: One of the main reasons why nonlinear-optical signal processing (regeneration, logic, etc.) has not yet become a practical alternative to electronic processing is that the all-optical elements with nonlinear input–output relationship have remained inherently single-channel devices (just like their electronic counterparts) and, hence, cannot fully utilise the parallel processing potential of optical fibres and amplifiers. The nonlinear input–output transfer function requires strong optical nonlinearity, e.g. self-phase modulation, which, for fundamental reasons, is always accompanied by cross-phase modulation and four-wave mixing. In processing multiple wavelength-division-multiplexing channels, large cross-phase modulation and four-wave mixing crosstalks among the channels destroy signal quality. Here we describe a solution to this problem: an optical signal processor employing a group-delay-managed nonlinear medium where strong self-phase modulation is achieved without such nonlinear crosstalk. We demonstrate, for the first time to our knowledge, simultaneous all-optical regeneration of up to 16 wavelength-division-multiplexing channels by one device. This multi-channel concept can be extended to other nonlinear-optical processing schemes. Nonlinear optical processing devices are not yet fully practical as they are single channel. Here the authors demonstrate all-optical regeneration of up to 16 channels by one device, employing a group-delay-managed nonlinear medium where strong self-phase modulation is achieved without nonlinear inter-channel crosstalk.

Smith Predictor Based Fractional-Order-Filter PID Controllers Design for Long Time Delay Systems

Abstract: In this paper, an original model-based analytical method is developed to design a fractional order controller combined with a Smith predictor and a modified Smith predictor that yield control systems which are robust to changes in the process parameters. This method can be applied for integer order systems and for fractional order ones. Based on the Bode's ideal transfer function, the fractional order controllers are designed via the internal model control principle. The simulation results demonstrate the successful performance of the proposed method for controlling integer as well as fractional order linear stable systems with long time delay.

Investigating Continuous Class-F Power Amplifier Using Nonlinear Embedding Model

Abstract: This letter investigates the design space for feasible load impedances of broadband continuous class-F power amplifier (PA) using the nonlinear embedding transfer function. This can overcome the limitation of anticlockwise trajectory of second-harmonic impedance on Smith chart. The approach starts with the design at intrinsic plane based on generic representation of drain voltage and current waveforms. Nonlinear embedding transfer network is then used to project intrinsic loads to the package plane. The proposed design strategy is verified with the implementation of a 15-W GaN HEMT-based PA operating over the frequency range of 1.5 to 2.5 GHz with drain efficiency between 60%–75%. This corresponds to the fractional bandwidth of 50%.

Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems

Abstract: We construct two error feedback controllers for robust output tracking and disturbance rejection of a regular linear system with nonsmooth reference and disturbance signals. We show that for sufficiently smooth signals the output converges to the reference at a rate that depends on the behavior of the transfer function of the plant on the imaginary axis. In addition, we construct a controller that can be designed to achieve robustness with respect to a given class of uncertainties in the system, and we present a novel controller structure for output tracking and disturbance rejection without the robustness requirement. We also generalize the internal model principle for regular linear systems with boundary disturbance and for controllers with unbounded input and output operators. The construction of controllers is illustrated with an example where we consider output tracking of a nonsmooth periodic reference signal for a two-dimensional heat equation with boundary control and observation, and with periodi...

DQ Current Control of Voltage Source Converters With a Decoupling Method Based on Preprocessed Reference Current Feed-forward

Abstract: Axes cross-coupling in dq-frame will deteriorate the current control performance, especially for applications with low switching/sampling frequency. Decoupling methods based on inductor current state feedback and complex vector proportional-integrator controllers are usually performed to solve this problem. In this paper, an alternative decoupling method based on preprocessed reference current feed-forward is proposed, which can provide excellent decoupling performance even with considerable control delay. The current control of grid-connected voltage source converters is analyzed with complex-coefficient transfer functions. It is revealed that the decoupling performance of current control schemes is closely related to the symmetry of current closed-loop transfer function bode diagrams. The current closed-loop transfer function should show a standard symmetrical bode diagram with respect to 0 Hz, if axes cross-coupling is well decoupled. A quantitative index named cross-coupling function is then proposed to analyze and comprehensively compare the decoupling performance of the proposed method and existing ones in terms of the influence of the control delay, inductance estimation error, grid frequency deviation, and voltage disturbance. Experimental tests are implemented to validate the performance of the proposed method and comparison results.

Efficient quantitative phase microscopy using programmable annular LED illumination.

Abstract: In this work, we present an efficient quantitative phase imaging (QPI) approach using programmable annular LED illumination. As a new type of coded light source, the LED array provides flexible illumination control for noninterferometric QPI based on a traditional microscopic configurations. The proposed method modulates the transfer function of system by changing the LED illumination pattern, which provides noise-robust response of transfer function and achieves twice resolution limit of objective NA. The quantitative phase can be recovered from slightly defocused intensity images through inversion of transfer function. Moreover, the weak object transfer function (WOTF) of axis-symmetric oblique source is derived, and the noise-free and noisy simulation results validate the predicted theory. Finally, we experimentally confirm accurate and repeatable performance of our method by imaging calibrated phase samples and cellular specimens with different NA objectives.

Model-based framework for multi-axial real-time hybrid simulation testing

Abstract: Real-time hybrid simulation is an efficient and cost-effective dynamic testing technique for performance evaluation of structural systems subjected to earthquake loading with rate-dependent behavior. A loading assembly with multiple actuators is required to impose realistic boundary conditions on physical specimens. However, such a testing system is expected to exhibit significant dynamic coupling of the actuators and suffer from time lags that are associated with the dynamics of the servo-hydraulic system, as well as control-structure interaction (CSI). One approach to reducing experimental errors considers a multi-input, multi-output (MIMO) controller design, yielding accurate reference tracking and noise rejection. In this paper, a framework for multi-axial real-time hybrid simulation (maRTHS) testing is presented. The methodology employs a real-time feedback-feedforward controller for multiple actuators commanded in Cartesian coordinates. Kinematic transformations between actuator space and Cartesian space are derived for all six-degrees-offreedom of the moving platform. Then, a frequency domain identification technique is used to develop an accurate MIMO transfer function of the system. Further, a Cartesian-domain model-based feedforward-feedback controller is implemented for time lag compensation and to increase the robustness of the reference tracking for given model uncertainty. The framework is implemented using the 1/5th-scale Load and Boundary Condition Box (LBCB) located at the University of Illinois at Urbana- Champaign. To demonstrate the efficacy of the proposed methodology, a single-story frame subjected to earthquake loading is tested. One of the columns in the frame is represented physically in the laboratory as a cantilevered steel column. For realtime execution, the numerical substructure, kinematic transformations, and controllers are implemented on a digital signal processor. Results show excellent performance of the maRTHS framework when six-degrees-of-freedom are controlled at the interface between substructures.

Some a posteriori error bounds for reduced-order modelling of (non-)parametrized linear systems

Abstract: We propose a posteriori error bounds for reduced-order models of non-parametrized linear time invariant (LTI) systems and parametrized LTI systems. The error bounds estimate the errors of the transfer functions of the reduced-order models, and are independent of the model reduction methods used. It is shown that for some special non-parametrized LTI systems, particularly efficiently computable error bounds can be derived. According to the error bounds, reduced-order models of both non-parametrized and parametrized systems, computed by Krylov subspace based model reduction methods, can be obtained automatically and reliably. Simulations for several examples from engineering applications have demonstrated the robustness of the error bounds.

On Stabilization of 2D Roesser Models

Abstract: This note is devoted to the stabilization of 2D Roesser models which are discrete, continuous, or mixed continuous-discrete. A recent linear matrix inequalities (LMIs) necessary and sufficient condition for stability of such models is used to derive a quasi non conservative technique for state feedback stabilization.

Rotary flexible joint control by fractional order controllers

Abstract: In this paper, a fractional order control law is proposed and implemented for the evaluation of trajectory tracking performance of a rotary flexible-joint system. A state feedback based fractional integral control scheme is used in this proposed method. In this scheme, state feedback is responsible for stabilizing the system. The compensator, in series with the fractional integrator leads to obtain a similar closed-loop transient response like Bode’s ideal transfer function. The effectiveness of the proposed controller in tracking and being robust against parameter uncertainties is demonstrated through simulation. In addition, to show the usefulness of the proposed control scheme, the fractional controller is compared to an integer state feedback control by simulation and through experimentation on the Quanser’s rotary flexible-joint system.

Full-duplex Communication on the Shared Channel of a Capacitively Coupled Power Transfer System

Abstract: This paper proposed a novel wireless power transfer system with full-duplex communication on a shared channel for capacitively coupled power transfer systems. For the analysis of power and signal transmission, a frequency-domain model of the power and signal channels is established. Based on this model, the signal transfer characteristic of the channel and the influence of power flow on the signal channel are analyzed. Moreover, to ensure the power and signal transfer without unacceptable interference or attenuation, a parameters selection method of the communication channel is developed. In addition, an interference suppression strategy by taking the interference from the ipsilateral channel into consideration is proposed. To suppress the interference effectively, an estimation of the ipsilateral channel output signal is made. Then, the signal from the opposite channel is demodulated by removing the estimated values of the interference. Both simulation and experimental results showed have proven the correctness and effectiveness of the proposed wireless power and signal transfer method. Finally, it has demonstrated that the designed channel can transfer 100 W of power, and a full-duplex communication can be well achieved with different data rates in two directions when both two data rates are set within 200 kb/s.

Symbolic Regression for the Estimation of Transfer Functions of Hydrological Models

Abstract: Current concepts for parameter regionalization of spatially distributed rainfall-runoff models rely on the a priori definition of transfer functions that globally map land surface characteristics (such as soil texture, land use, digital elevation, etc.) into the model parameter space. However, these transfer functions are often chosen ad hoc or derived from small-scale experiments. This study proposes and tests an approach for inferring the structure and parametrization of possible transfer functions from runoff data to potentially circumvent these difficulties. The concept uses context free grammars to generate possible proposition for transfer functions. The resulting structure can then be parametrized with classical optimization techniques. Several virtual experiments are performed to examine the potential for an appropriate estimation of transfer function, all of them using a very simple conceptual rainfall-runoff model with data from the Austrian Mur Catchment. The results suggest that a priori defined transfer functions are in general well identifiable by the method. However, the deduction process might be inhibited e.g. by noise in the runoff observation data, often leading to transfer function estimates of lower structural complexity.

Simplified Unified Analysis of Switched-RC Passive Mixers, Samplers, and $N$ -Path Filters Using the Adjoint Network

Abstract: Recent innovations in software defined CMOS radio transceiver architectures heavily rely on high-linearity switched- RC sampler and passive-mixer circuits, driven by digitally programmable multiphase clocks Although seemingly simple, the frequency domain analysis of these linear periodically time variant (LPTV) circuits is often deceptively complex This paper uses the properties of sampled LPTV systems and the adjoint (inter-reciprocal) network to greatly simplify the analysis of the switched- RC circuit We first derive the transfer function of the equivalent linear time-invariant filter relating the input to the voltage sampled on the capacitor in the switched- RC kernel We show how a leakage resistor across the capacitor can be easily addressed using our technique A signal-flow graph is then developed for the complete continuous-time voltage waveform across the capacitor, and simplified for various operating regions We finally derive the noise properties of the kernel The results we derive have largely been reported in prior works, but the use of the adjoint network simplifies the derivation, while also providing circuit insight

Fundamental performance limitations of networked control systems with novel trade-off factors and constraint channels

Abstract: This paper focuses on the optimal tracking performance issues for linear time invariant system with bandwidth limited and additive colored white Gaussian noise (ACGN) simultaneously. The nonminimal phase and unstable plant are considered, and multi-repeated zeros and poles is investigated. The objective function of tracking response is minimized jointly with the control effort. In order to more fully reflect the performance of the network control systems (NCSs), the performance index is measured by the tracking error energy, input channel energy and plant input energy using novel trade-off factors. The novel trade-off factors can be measured each frequency band for each signal, which are stable and minimal phase transfer function. To obtain the optimal performance, the two-parameter controller is adopted. The tracking performance is given by explicit expression, which is critically dependent on the intrinsic characteristics of the given plant (unstable poles and nonminimal phase zeros), communication parameters (bandwidth and statistical characteristics of network noise) and statistical characteristics of reference signal. Finally, the simulation results demonstrate the effectiveness of the proposed control scheme.

Self-organized synchronization of digital phase-locked loops with delayed coupling in theory and experiment.

Abstract: Self-organized synchronization occurs in a variety of natural and technical systems but has so far only attracted limited attention as an engineering principle. In distributed electronic systems, such as antenna arrays and multi-core processors, a common time reference is key to coordinate signal transmission and processing. Here we show how the self-organized synchronization of mutually coupled digital phase-locked loops (DPLLs) can provide robust clocking in large-scale systems. We develop a nonlinear phase description of individual and coupled DPLLs that takes into account filter impulse responses and delayed signal transmission. Our phase model permits analytical expressions for the collective frequencies of synchronized states, the analysis of stability properties and the time scale of synchronization. In particular, we find that signal filtering introduces stability transitions that are not found in systems without filtering. To test our theoretical predictions, we designed and carried out experiments using networks of off-the-shelf DPLL integrated circuitry. We show that the phase model can quantitatively predict the existence, frequency, and stability of synchronized states. Our results demonstrate that mutually delay-coupled DPLLs can provide robust and self-organized synchronous clocking in electronic systems.

Quantum Noises, Physical Realizability and Coherent Quantum Feedback Control

Abstract: Arbitrary linear time invariant systems can be implemented as quantum systems if additional quantum noises are permitted in the implementation. We give several results concerning how many additional quantum noise channels are necessary to implement state space realizations and transfer functions as quantum systems. We also give algorithms to do so. We demonstrate the utility of these results with an algorithm for obtaining a suboptimal solution to a coherent quantum LQG control problem.

Transfer function characterization for HFCTs used in partial discharge detection

Abstract: High frequency current transformers (HFCTs) are widely employed to detect partial discharge (PD) induced currents in high voltage equipment. This paper describes measurements of the wideband transfer functions of HFCTs so that their influence on the detected pulse shape in advanced PD measurement applications can be characterized. The time-domain method based on the pulse response is a useful way to represent HFCT transfer functions as it allows numerical determination of the forward and reverse transfer functions of the sensor. However, while the method is accurate at high frequencies it can have limited resolution at low frequencies. In this paper, a composite time-domain method is presented to allow accurate characterization of the HFCT transfer functions at both low and high frequencies. The composite method was tested on two different HFCTs and the results indicate that the method can characterize their transfer functions ranging from several kHz to tens of MHz. Results are found to be in good agreement with frequency-domain measurements up to 50 MHz. Measurement procedures for using the method are summarized to facilitate further applications.

Assessment of GIC Based On Transfer Function Analysis

Abstract: Transfer functions are calculated for periods between 2 and 1000 minutes between geomagnetically induced currents (GIC) measured at three transformers in the South Island of New Zealand and variations in the horizontal components of the geomagnetic field measured at the Eyrewell Observatory near Christchurch. Using an inverse Fourier Transform, the transfer functions allow the GIC expected in these transformers to be estimated for any variation of the inducing magnetic field. Comparison of the predicted GIC with measured GIC for individual geomagnetic storms shows remarkable agreement, although the lack of high frequency measurements of GIC and the need for interpolation of the measurements leads to a degree of underestimation of the peak GIC magnitude. An approximate correction for this is suggested. Calculation of the GIC for a magnetic storm in November 2001 which led to the failure of a transformer in Dunedin suggests that peak GIC were as large as about 80 A. Use of spectral scaling to estimate the likely GIC associated with a geomagnetic storm of the magnitude of the 1859 Carrington Event indicate that GIC of at least 10 times this magnitude may occur at some locations. Although the impact of changes to the transmission network on calculated transfer functions remains to be explored, it is suggested that the use of this technique may provide a useful check on estimates of GIC produced by other methods such as thin-sheet modelling. Plain Language Summary Rapid changes in Earth’s magnetic field, such as occur during a magnetic storm, induce electric currents in the ground. These currents, known as geomagnetically induced currents (GIC), are able to enter a power transmission network through the ground connection of a substation transformer. Not only can such currents cause damage to transformers, but in extreme situations they may cause failure of the entire power transmission network. We relate measurements of GIC in the New Zealand power transmission network to variations in the magnetic field at a local magnetic observatory. This allows us to construct mathematical relationships between GIC and magnetic field variations which enable us to predict the magnitude of GIC that might occur in the event of a magnetic storm such as the so-called Carrington Event of 1859 – the largest such storm ever recorded. It is found that GIC of almost 1000 A might occur.

Sensitive Load Voltage Compensation Performed by a Suitable Control Method

Abstract: This paper proposes the usage of a repetitive-based control to dynamically restore the voltage applied to sensitive and critical loads of power system. The control intrinsically is able to wipe off harmonic distortion and relies on simple transfer function. As a consequence, there is no need to apply harmonic selective filters. Furthermore, the control system is able to work out on sinusoid references and, thus, avoids the need of employing the dq transform. A recursive least-squares is also included to the control system in order to assure the synchronization of the voltages to be restored. The design of the control parameters along with the system stability is discussed. The experimental results are produced with a setup of a three-phase series compensator. The scenarios for emulating faulty voltages are the same for experimental and simulated results. The results corroborates the usage of the proposed method.

Simplified symbolic transfer function factorization using combined artificial bee colony and simulated annealing

Abstract: Display Omitted A swarm intelligence based simplified transfer function factorization methodology is proposed.We extend the traditional root splitting technique for factorization of the expanded transfer function.A hybrid global and local search algorithm based on ABC and SA (named GLABCSA) is introduced.Our method guarantees to limit pole/zero displacements in both factorization and simplification phases via user-specified thresholds. Symbolic circuit analysis inherits the exponential growth of transfer function complexity with the circuit size. Therefore, symbolic simplification is an NP-hard problem. Although many simplification techniques have been presented, the simplified transfer functions are not written in a factorized form, and consequently, it is difficult to assess the contribution of poles and zeros on the circuit behavior. In this paper, a swarm intelligence based methodology is presented for the simplified factorized symbolic analysis of analog circuits. In this method, an extension of the root splitting technique is utilized to rewrite the expanded transfer function of the circuit into a factorized form comprising DC-gain, poles, and zeros. Then, the derived factorized transfer function is simplified using a hybrid Global and Local search algorithm based on Artificial Bee Colony and Simulated Annealing (named GLABCSA). The objective function is defined to minimize the complexity of the symbolic factorized transfer function while minimizing the DC-gain error and pole/zero displacements. The presented approach has been successfully developed in MATLAB. The program can derive the simplified factorized symbolic transfer function automatically from the input text netlist of the circuit. Symbolic and numerical results over two analog amplifiers are given to illustrate the efficiency of the presented methodology.

Frequency domain based DC fault analysis for bipolar HVDC grids

Abstract: This paper proposes a frequency domain based methodology to analyse the influence of High Voltage Direct Current (HVDC) configurations and system parameters on the travelling wave behaviour during a DC fault. The method allows us to gain deeper understanding of these influencing parameters. In the literature, the majority of DC protection algorithms essentially use the first travelling waves initiated by a DC fault for fault discrimination due to the stringent time constraint in DC grid protection. However, most protection algorithms up to now have been designed based on extensive time domain simulations using one specific test system. Therefore, general applicability or adaptability to different configurations and system changes is not by default ensured, and it is difficult to gain in-depth understanding of the influencing parameters through time domain simulations. In order to analyse the first travelling wave for meshed HVDC grids, voltage and current wave transfer functions with respect to the incident voltage wave are derived adopting Laplace domain based component models. The step responses obtained from the voltage transfer functions are validated by comparison against simulations using a detailed model implemented in PSCAD™. Then, the influences of system parameters such as the number of parallel branches, HVDC grid configurations and groundings on the first travelling wave are investigated by analysing the voltage and current transfer functions.

Reduced-Order Models of Series Resonant Inverters in Induction Heating Applications

Abstract: From the controller design framework, a simple analytical model that captures the dominant behavior in the range of interest is the optimal. When modeling resonant circuits, complex mathematical models are obtained. These high-order models are not the most suitable for controller design. Although some assumptions can be made for simplifying these models, variable frequency operation or load uncertainty can make these premises no longer valid. In this study, a systematic modeling order reduction technique, slowly varying amplitude and phase (SVAP), is considered for obtaining simpler analytical models of resonant inverters. SVAP gives identical results as the classical model-order residualization technique from automatic control theory. A slight modification of SVAP, slowly varying amplitude derivative and phase (SVADP) is applied in this paper to obtain a better validity range. SVADP is validated for a half-bridge series resonant inverter and for a high-order plant, a dual-half bridge series resonant inverter giving analytical second-order transfer functions for both topologies. Simulation and experimental results are provided to show the validity range of the reduced-order models.

Plug-In Repetitive Control Strategy for High-Order Wide-Output Range Impedance-Source Converters

Abstract: High-order wide-output (HOWO) impedance-source converters (ISCs) have been presented for ac inverter applications that require voltage step-up ability. With intrinsic passive impedance networks as energy sources, these converters are able to achieve voltage boosting with either polarity, leading to improved dc-link voltage utilization compared with the conventional two-level converter. However, HOWO-ISCs suffer from transfer functions giving low bandwidth, a penalty of increased passive devices and right-half-plane zeros, which result in lower order distortion of the ac output power. In this paper, a modified plug-in repetitive control scheme is presented for HOWO-ISCs with accurate reference tracking (hence low distortion), fast dynamic response, and enhanced robustness. By using zero-phase-shift finite impulse response filters in both the internal model of the repetitive controller and its compensation network, the proposed method achieves zero steady-state error and an extended closed-loop bandwidth. For HOWO-ISC cases, this method outperforms conventional proportional-integral (PI) control, which has considerable steady-state error. It also eliminates the need of parallel loops for several frequencies when proportional resonant control or orthogonal transformation-based PI schemes are used to remove lower order distortion. The design process and performance analysis of the proposed repetitive control strategy are based on a novel three-phase HOWO-ISC configuration with a reduced number of switches. Simulation and experimental results confirm the feasibility and effectiveness of the proposed control approach.

Low-Gain Integral Control for Multi-Input Multioutput Linear Systems With Input Nonlinearities

Abstract: We consider the inclusion of a static antiwindup component in a continuous-time low-gain integral controller in feedback with a multi-input multi-output stable linear system subject to an input nonlinearity (from a class of functions that includes componentwise diagonal saturation). We demonstrate that the output of the closed-loop system asymptotically tracks every constant reference vector, which is “feasible” in a natural sense, provided that the integrator gain is sufficiently small. Robustness properties of the proposed control scheme are investigated and three examples are discussed in detail.

Dynamic Modeling and Control of Self-Oscillating Parallel Resonant Converters Based on a Variable Structure Systems Approach

Abstract: This paper describes a novel modeling approach for self-oscillating resonant power converters operating under a control method based on the variable structure of the system (VSS). Using a fundamental harmonic approximation, the link between control and switching frequency in steady state is found to be accurately described by an affine function that depends on the load. Besides, this link has been extended in order to characterize the control-to-switching frequency dynamics, resulting in a novel small-signal continuous-time model. The resulting control-to-switching frequency transfer function can be appended to the well-known models for frequency modulation, in order to obtain a complete control-to-output system. The new model exposes that the dynamics of the converter under the VSS-based control approach presents advantages with respect to the conventional methods as frequency modulation: 1) reduced dc gain variation for uncertain loads and 2) improved phase margin at high frequencies. The dynamic modeling is complemented with the design of a controller for regulating the output voltage of a parallel resonant converter, whose performance and robustness are compared with the standard frequency modulation. Numerical simulations and experimental results confirm and verify the analytical derivations.

On generalized fractional vibration equation

Abstract: In this paper, a generalized fractional vibration equation with multi-terms of fractional dissipation is developed to describe the dynamical response of an arbitrary viscoelastically damped system. It is shown that many classical equations of motion, e.g., the Bagley–Torvik equation, can be derived from the developed equation. The Laplace transform is utilized to solve the generalized equation and the analytic solution under some special cases is derived. Example demonstrates the generalized transfer function of an arbitrary viscoelastic system.