Real Poles Research Articles

A time-domain finite element formulation of the equivalent fluid model for the acoustic wave equation

Sound-absorptive materials such as foam can be described by the equivalent fluid (EF) model. The homogenized fluid’s acoustic behavior is thereby described by complex-valued, frequency-dependent acoustic material parameters. When transforming the acoustic wave equation for the EF model from the frequency domain to the time domain, convolution integrals arise. The auxiliary differential equation (ADE) method is used to circumvent the direct calculation of these convolution integrals. The wave equation and the coupled set of ordinary ADEs are solved in the time domain using the finite element (FE) method. The approach relies on approximating the complex-valued frequency response functions of the inverse equivalent bulk modulus and density by a sum of rational functions consisting of real and complex poles. The order of the rational function approximation defines the number of additionally introduced auxiliary variables per nodal degree of freedom. The presented FE formulation includes a narrow-band non-reflecting boundary condition (NRBC) for normal incidence. The implementation in openCFS shows optimal temporal and spatial convergence for a semi-infinite duct based on the analytic plane wave solution for harmonic excitation. The simulation of a pressure pulse propagating in an infinite EF domain with a scatterer demonstrates the capability for multidimensional, actual transient problems.

A robust numerical scheme for solving Riesz-tempered fractional reaction–diffusion equations

The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this equation numerically is challenging due to the need to discretize complicated integral operators which increase the computational costs. These complexities are exacerbated by nonlinear source terms, nonsmooth data and irregular domains. In this study, we propose a second order Exponential Time Differencing Finite Element Method (ETD-RDP-FEM) to efficiently solve nonlinear FDE, posed in irregular domains. This approach discretizes matrix exponentials using a rational function with real and distinct poles, resulting in an L-stable scheme that damps spurious oscillations caused by non-smooth initial data. The method is shown to outperform existing second-order methods for FDEs with a higher accuracy and faster computational time.

Oscillation rheology, calorimetry and computed tomography: A practical toolkit combination of qualification for real poly(ethylene terephthalate) waste

The conventional qualification methodologies, such as internal viscosity measurement and Melt Flow Index (MFI) tests, which are routinely applied to pristine materials and homogenous waste, are not directly transferrable to the qualification process of plastic mixtures. The current investigation is oriented towards the development of a pragmatic suite of measurements techniques aimed at facilitating the qualification of diverse real polyethylene terephthalate (PET) waste streams. This is primarily achieved through the integration of oscillatory rheology, calorimetry and computed tomography (CT). Samples derived from both selective income (SI) and sorting residue (SR) of a manual selective waste sorting plant are included, along with an examination of PET material originating from refuse-derived fuel (RDF). To establish references for comparison, manually collected bottles with (BCL) and without (B) caps and labels are employed, leveraging their known composition and purity. Calorimetry reveals that the glass transition temperature (Tg) is the most sensitive parameter for tracking the structural and compositional changes in waste PET-based streams. The utilization of temperature sweep test in rheology for determining the melting temperature (Tm) proves advantageous, offering a significantly shorter measurement time (∼1:24) compared to calorimetry. Importantly, this approach yields the Tm of the genuine composition rather than individual components. Detection of metal content contaminations in PET waste streams is made possible through CT, while frequency tests in rheological measurements demonstrate effectiveness in discerning their effects.

Analytical solutions for dispersions and waveforms of acoustic logging in a cased hole

Acoustic logging is one of the most promising methods for the quantitative evaluation of cement bond conditions in cased holes. However, inefficient use of full-wave information yields unsatisfactory interpretation accuracy. Fundamentally, this is because the wavefield characteristics have not been thoroughly investigated under various cement bonding conditions. Thus, this study derives analytical solutions of wavefields for a single-cased-hole model and emphasizes the dispersion calculation algorithm. To solve the dispersion equation when solving for the poles of the propagating modes with real wavenumbers, we renormalize the Bessel function related to the borehole fluid by multiplying it with an attenuation factor. For leaky modes with complex wavenumbers, we develop a novel method to find local peaks of the matrix condition number (LPMCN) in the frequency domain to determine dispersion poles, avoiding the local optimization issues resulting from the traditional Gauss-Newton iteration method. Combining these two methods, we establish a fast and accurate workflow for evaluating the dispersion of all modes in cased holes using a relatively fast bisection method to manage the dispersion of the propagating modes and using the LPMCN method to derive the dispersion curves of leaky modes. Furthermore, all propagating modes are individually investigated in the monopole measurement by evaluating the residues of the real poles in a casing-free model. The analysis demonstrates that the first-order pseudo-Rayleigh wave (PR1) and inner Stoneley wave (ST1) are the two strongest modes. Finally, we focus on the waveforms and dispersion characteristics of the outer Stoneley wave (ST2) related to the fluid channel in the cement annulus. The results reveal that as the fluid thickness increases, the phase velocity of the ST2 mode decreases, and its amplitude increases. Therefore, the ST2 mode can potentially evaluate the thickness of the fluid channel in a cement annulus if an effective weak-signal-extraction method is used.

Design of simple nonovershooting controllers for linear high order systems with or without time delay

In this paper, we mainly considered the problem of nonovershooting control of high order systems with or without time delay by simple controllers. As basic principles for nonovershooting control systems, three propositions are offered and proved. Under direction of these principles, a nonovershooting dominant pole control structure having three dominant poles, i.e., one real pole and a pair of complex conjugate poles on its left, is proposed. While its zeroes and nondominant poles are on the left side of these three dominant poles with sufficient distance. The controllers adopted are composed by first order filter and PD-PID controller. Dominance of the three dominant poles can be checked and ensured through the computational method we offered. Two illustrating examples are given to show the effectiveness of our method.

Constrained Series PI, PID and PIDA Controller Design Inspired by Ziegler–Nichols

Abstract The present paper complements the results of several recent papers on higher-order (HO) controllers with automatic-reset. A modification of the two-step tuning of the constrained second-order derivative controllers based on integrator-plus-dead-time (IPDT) models is proposed. In the first step, the linear controller is designed using the multiple real dominant poles (MRDPs) method to avoid the slowdown of the closed-loop dynamics due to the presence of slow poles. In the second step, the smallest time constant of the numerator of the MRDP-optimal controller transfer function is selected as the automatic-reset time constant. The derived control method was tested on a thermal system for the filament disc dryer to demonstrate the deployment, tuning, use and impact of controllers with increasing derivative degree in practical applications. It is shown that the use of HO controllers is similar to the traditional hyper-reset controllers (i.e. series proportional-integral-derivative [PID] controllers) from the user’s point of view. However, the advantages are faster transient responses while maintaining sufficiently smooth input and output shapes of the process with a minimum number of monotonic intervals. The overall design can be seen as a generalisation and discretisation of the Ziegler and Nichols graphical tuning method. One of the main new features is the consideration of a constrained control signal, as is typical for a pulse width modulated (PWM) actuator. Such actuators are often used in speed-controlled electric drives and in power electronics, among other applications.

Runge–Kutta type Time Stepping Methods for Space Fractional Reaction Diffusion Model with Restricted Padé Approximation

A new numerical method that solves a time-space-fractional reaction-diffusion equation effectively is presented. The space-fractional Riesz operator is first discretized using fourth order compact finite differences, resulting in a system with a linear stiff term. The system of differential equations is then solved through time integration using an exponential time difference method, which explicitly handles the nonlinear non-stiff term. Restricted Padé rational approximation of a single real pole is used to approximate the matrix exponential e-A. The problems concerning the stability and computational efficiency of this new approach are tackled by means of a splitting technique. Based on compact finite differences with third-order accuracy in time and restricted Padé approximation, this novel efficient technique reduces computing cost and effectively tackles the problems with non-smooth initial data. The superiority of new method in terms of accuracy, reliability, and computational efficiency is finally shown by numerical examples.

A generalization of Floater–Hormann interpolants

In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on γ, an additional positive integer parameter. For γ=1, the original Floater–Hormann interpolants are obtained. When γ>1 we prove that the new rational functions share a lot of the nice properties of the original Floater–Hormann functions. Indeed, for any configuration of nodes in a compact interval, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum (h∗) and maximum (h) distance between two consecutive nodes. It turns out that, in contrast to the original Floater–Hormann interpolants, for all γ>1 we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when h∼h∗). For such configurations, as the number of nodes tends to infinity, we prove that the new interpolants (γ>1) uniformly converge to the interpolated function f, for any continuous function f and all γ>1. The same is not ensured by the original FH interpolants (γ=1). Moreover, we provide uniform and pointwise estimates of the approximation error for functions having different degrees of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater–Hormann interpolants.

Observer-predictor based on a PD controller for high-order delayed systems with one unstable pole and n pairs of complex conjugate poles

This paper considers the stability of continuous linear systems with time delay. In particular, systems with one unstable pole and n pairs of complex conjugate and/or real poles are analyzed. The proposed solution consists of using an estimated state observer with a closed-loop PD type controller to access to the internal variables of the system. A partition of the time delay is proposed in the observer to addmit the double of the maximum size that when a simple PD controller is considered. The necessary and sufficient conditions for the stability of the proposed control scheme are presented and an example is presented with simulations to demostrate the usefulness of the proposed control strategy.

Optimal tuning of PID controllers with derivative filter for stable processes using three points from the step response

This paper proposes a novel approach for tuning Proportional Integral Derivative (PID) controllers, utilizing experimental data obtained from an open-loop step input. We propose to use the measurements of the times taken to reach 5%, 35.3% and 85.3% of the final output, as well as the process static gain, to tune the controller. The tuning equations are applicable for a wide range of stable and over damped systems. Starting from an approximate model with three real poles and time delay (obtained from the measurements), the tuning equations approximate the controller that minimizes the Integral of Absolute Error (IAE) of the disturbance response. The designer can freely decide the Sensitivity Margin (Ms) to fix a desired robustness, as a difference with respect to other known methods where the user chooses among few predefined robustness values. The user can also select freely the derivative filter parameter, N, to find the required compromise between speed response and measurement noise amplification. For N=0 a PI controller is chosen. As N increases, a faster PID controller is selected, with higher noise amplification. We have also developed a web-based application and an Android application, downloadable for free, that implement the tuning equations as well as the whole required procedure.

On the computation of stable coupled state-space models for dynamic substructuring applications

This paper aims at introducing a methodology to compute stable coupled state-space models for dynamic substructuring applications by introducing two novel approaches targeted to accomplish this task: (a) a procedure to impose Newtons’s second law without relying on the use of undamped RCMs (residual compensation modes) and (b) a novel approach to impose stability on unstable coupled state-space models. The enforcement of stability is performed by dividing the unstable model into two different models, one composed by the stable poles (stable model) and the other composed by the unstable ones (unstable model). Then, the poles of the unstable state-space model are forced to be stable, leading to the computation of a stabilized state-space model. If this model is composed by real poles, it should be divided into two different ones, one composed by the pairs of complex conjugate poles and the other composed by the real poles. Afterwards, to make sure that the Frequency Response Functions (FRFs) of the stabilized model well match the FRFs of the unstable model, the Least-Squares Frequency Domain (LSFD) method is exploited to update the modal parameters of the stabilized model composed by the pairs of complex conjugate poles. The validity of the proposed methodologies is presented and discussed by exploiting experimental data. Indeed, by exploiting the FRFs of a real system, accurate state-space models respecting Newton’s second law are computed. Then, decoupling and coupling operations are performed with the identified state-space models, no matter the models resultant from the decoupling/coupling operations are unstable. Stability is then imposed on the computed unstable coupled model by following the approach proposed in this paper. The methodology proved to work well on these data. Moreover, the paper also shows that the coupled state-space models obtained using this methodology are suitable to be exploited in time-domain analyses and simulations.

Parametrization and Optimal Tuning of Constrained Series PIDA Controller for IPDT Models

The new modular approach to constrained control of higher-order processes with dominant first-order dynamics using generalized controllers with automatic resets (ARCs) is addressed. The controller design is based on the multiple real dominant pole (MRDP) method for the integrator plus dead time (IPDT) process models. The controller output constraints are taken into account by inserting the smallest numerator time constant of the controller transfer function into the positive feedback loop representing the automatic reset (integral) term. In the series realization of the proportional–integral–derivative–acceleration (PIDA) controller (and other controllers with even derivative degree), the time constant mentioned is complex, so only the real part of the time constant has been used so far. Other possible conversions of a complex number to a real number, such as the absolute value (modulus), can be covered by introducing a tuning parameter that modifies the calculated real time constant and generalizes the mentioned conversion when designing controllers with constraints. In this article, the impact of the tuning parameter on the overall dynamics of the control loop is studied by simulation. In addition, an evaluation of the stability of the closed-loop control system is performed using the circle criterion in the frequency domain. The analysis has shown that the approximation of the complex zero by its real part and modulus leads to a near optimal response to the set point tracking. The disturbance rejection can be significantly improved by increasing the tuning parameter by nearly 50%. In general, the tuning parameter can be used to find a compromise between servo and regulatory control. The robustness and applicability of the proposed controller is evaluated using a time-delayed process with first-order dominant dynamics when the actual transfer function is much more complicated than the IPDT model. A comparison of the proposed MRDP-PIDA controller with series PI, PID and PIDA controllers based on a modified SIMC method has shown that the MRDP-PIDA controller performs better than the SIMC method, although the SIMC uses a more complex process model.

The black hole behind the cut

We study the analytic structure of the heavy-heavy-light-light holographic correlators in the supergravity approximation of the AdS3 × S3/CFT2 duality. As an explicit example, we derive the correlator where the heavy operator is a classical microstate of the 5D supersymmetric black hole and its dual geometry interpolates as a function of a continuous parameter between global AdS3 and the extremal BTZ black hole. The simplest perturbation of this interpolating geometry by a light field is described by the Heun equation and we exploit the relation of its connection coefficients to the Liouville CFT to analytically compute the correlator in the two limits, focusing in particular on the black hole regime. In this limit we find that the real poles of the correlator become dense and can be approximated by a cut. We show that, when the charges of the heavy state are in the black hole regime, the discontinuity across the cut has complex poles corresponding to the quasi-normal modes of BTZ. This behaviour is qualitatively similar to what is expected for the large central charge limit of a typical black hole microstate.

Parameterization of minimal order strongly stabilizing controllers for two-link underactuated planar robot with single sensor

This paper investigates the minimal order of strongly stabilizing controllers for a two-link planar robot moving in a vertical plane with a single actuator and sensor, focusing on stabilizing the robot around its upright equilibrium point where both links are upright. By designing an output as the horizontal displacement of a point along the second link’s straight line, existing literature demonstrates that the robot’s strong stabilization problem is equivalent to that of a fourth-order linear plant, featuring two pairs of symmetric real poles and adjustable zeros. In contrast to an existing second-order strongly stabilizing controller tailored to a pair of symmetric real zeros around the origin within a given range, this paper achieves four main contributions. First, this paper obtains a parameterization of second-order strongly stabilizing controllers allowing the positive real zero in an expanded range. Second, this paper considers the plant with a pair of symmetric real zeros around the origin, a double origin zero, and a pair of conjugate imaginary zeros in a unified manner, and presents a more general parameterization of second-order strongly stabilizing controllers accommodating these three types of zeros in a more expanded range. Third, this paper provides a necessary and sufficient condition on these zeros (equivalently, all ranges of the observation point) for the existence of a second-order strongly stabilizing controller. Fourth, this paper demonstrates that a first-order stabilizing controller does not exist for the plant, regardless of its types of zeros, confirming that the minimal order of the strongly stabilizing controllers is two. This paper obtains all feasible regions for the parameters in each parameterization by explicitly expressing the range of each parameter using inequalities in a cascade structure, which makes the inequalities easy to check and verify. The results are applicable to both the Acrobot and Pendubot. The paper validates the theoretical findings through numerical investigations and simulations.

Real-microcellular poly(ether-block-amide)/carbon fiber foams with improved mechanical property and surface quality for high-performance electromagnetic interference shielding applications

Challenges remain in large-scale and successive preparation of porous polymer composite component, which limits its practical applications for high-performance electromagnetic interference (EMI) shielding. Herein, a scalable and efficient method based on mold-opening foam injection molding was developed to fabricate carbon fiber (CF) reinforced polymer composite foams for EMI shielding applications. Thanks to the novel foaming process and heterogeneous nucleation effect of CF, real microcellular poly(ether-block-amide) (PEBA)/CF composite foams with an average cell size of 6.6 μm and a cell density of up to 109 cells/cm3 were achieved. Due to the significantly refined cellular morphology, the PEBA/CF composite foams exhibit superior EMI shielding performance over the unfoamed composites which was characterized by absorption-dominated shielding characteristics and significantly enhanced EMI shielding effectiveness. Furthermore, compared with the unfoamed case, the foamed PEBA/CF composites with refined cellular morphology shows greatly improved cold toughness and duality, and the tensile strain at break and specific tensile toughness are increased by 88.6% and 109.1%, respectively. Moreover, real-microcellular PEBA/CF composite foams have remarkably enhanced surface quality than those fabricated by regular foam injection molding process, and the surface roughness can be maximumly increased by 91.6%. Thus, this study provides a promising strategy for producing lightweight and strong components for high-performance EMI shielding applications.

Nonlinear swing-down control of the Acrobot: Analysis and optimal gain design

In this paper, we address the swing-down control of the Acrobot, a two-link planar robot operating in a vertical plane with only the second joint being actuated. The control objective is to rapidly stabilize the Acrobot around the downward equilibrium point, with both links in the downward position, from almost all initial states. Under the conditions of no friction and measurability of only the angle and angular velocity of the actuated joint, we present a sinusoidal-derivative (SD) controller. This controller consists of a linear feedback of the sinusoidal function of the angle of the actuated joint and a linear feedback of its angular velocity. We prove that the control objective is achieved if the sinusoidal gain is greater than a negative constant and the derivative gain is positive. We establish crucial relationships between the relative stability of the Acrobot under the SD controller and its physical parameters, presenting analytically all optimal control gains. These gains minimize the real parts of the dominant poles of the linearized model of the resulting closed-loop system around the downward equilibrium point. We demonstrate that the resulting dominant closed-loop poles can be double complex conjugate poles, or a quadruple real pole, or a triple real pole, depending on the Acrobot’s physical parameters. Simulation studies indicate that the proposed SD controller outperforms the derivative (D) controller in rapidly stabilizing the Acrobot at the downward equilibrium point.

Analytic solutions of linear neutral and non-neutral delay differential equations using the Laplace transform method: featuring higher order poles and resonance

In this article, we extend the Laplace transform method to obtain analytic solutions for linear RDDEs and NDDEs which have real and complex poles of higher order. Furthermore, we present first-order linear DDEs that feature resonance phenomena. The procedure is similar to the one where all of the poles are order one, but requires one to use the appropriate modifications when using Cauchy’s residue theorem for the poles of higher order. The process for obtaining the solution relies on computing the relevant infinite sequence of poles and then determining the Laplace inverse, via the Cauchy residue theorem. For RDDEs, the poles can be obtained in terms of the Lambert W function, but for NDDEs,the complex poles, in most cases, must be computed numerically. We found that an important feature of first-order linear RDDES and NDDES with poles of higher order is that it is possible to incite the resonance phenomena, which in the counterpart ordinary differential equation cannot occur. We show that despite the presence of higher order poles or resonance phenomena, the solutions generated by the Laplace transform method for linear RDDEs and NDDEs that have higher order poles are still accurate.

Uncertainty quantification in the assessment of human exposure to pulsed or multi-frequency fields

Objective: pulsed fields or waveforms with multi-frequency content have to be assessed with suitable methods. This paper deals with the uncertainty quantification associated to these methods. Approach: among all possible approaches, the weighted peak method (WPM) is widely employed in standards and guidelines, therefore, in this paper, we consider its implementation both in time domain and frequency domain. For the uncertainty quantification the polynomial chaos expansion theory is used. By means of a sensitivity analysis, for several standard waveforms, the parameters with more influence on the exposure index are identified and their sensitivity indices are quantified. The output of the sensitivity analysis is used to set up a parametric analysis with the aim of evaluating the uncertainty propagation of the analyzed methods and, finally, also several measured waveforms generated by a welding gun are tested. Main results: it is shown that the time domain implementation of the weighted peak method provides results in agreement with the basilar mechanisms of electromagnetic induction and electrostimulation. On the opposite, the WPM in frequency domain is found to be too sensitive to parameters that should not influence the exposure index because its weight function includes sharp variations of the phase centered on real zeros and poles. To overcome this issue, a new definition for the phase of the weight function in frequency domain is proposed. Significance: it is shown that the time domain implementation of the WPM is the more accurate and precise. The standard WPM in frequency domain has some issues that can be avoided with the proposed modification of the phase definition of the weight function. Finally, all the codes used in this paper are hosted on a GitHub and can be freely accessed at https://github.com/giaccone/wpm_uncertainty.

IPDT Model-Based Ziegler-Nichols Tuning Generalized to Controllers with Higher-Order Derivatives.

The paper extends the earlier work entitled "Making the PI and PID Controller Tuning Inspired by Ziegler and Nichols Precise and Reliable", to higher-order controllers and a broader range of experiments. The original series PI and PID controllers, based on automatic reset calculated by filtered controller outputs, are now augmented by higher-order output derivatives. This increases the number of degrees of freedom that can be used to modify the resulting dynamics, accelerates transient responses, and increases robustness to unmodeled dynamics and uncertainties. The fourth order noise attenuation filter used in the original work allows for the addition of an acceleration feedback signal, thus resulting in a series PIDA controller or even a jerk feedback that leads to a PIDAJ series controller. Such a design can further use the original process and filter approximation of the step responses through the integral-plus-dead-time (IPDT) model, while allowing experimentation with disturbance and setpoint step responses of the series PI, PID, PIDA and PIDAJ controllers, and thus, evaluating the role of output derivatives and noise attenuation from a broader perspective. All controllers considered are tuned using the Multiple Real Dominant Pole (MRDP) method, which is complemented by a factorization of the controller transfer functions to achieve the smallest possible time constant for automatic reset. The smallest time constant is chosen to improve the constrained transient response of the considered controller types. The obtained excellent performance and robustness allow the proposed controllers to be applied to a wider range of systems with dominant first-order dynamics. The proposed design is illustrated on a real-time speed control of a stable direct-current (DC) motor, which is approximated (together with a noise attenuation filter) by an IPDT model. The transient responses obtained are nearly time-optimal, with control signal limitations active for most setpoint step responses. Four controllers with different degrees of derivative with generalized automatic reset were used for comparison. It was found that controllers with higher-order derivatives may significantly improve the disturbance performance and virtually eliminate overshoots in the setpoint step responses in constrained velocity control.

Linear Time-Invariant Model Identification Algorithm for Mechatronic Systems Based on MIMO Frequency Response Data

This article describes a new frequency- domain multi-input multioutput linear time-invariant (LTI) system identification algorithm for accurate model construction suitable for motion control systems. The proposed method can capture the effects of time-delay, lightly damped poles (structural resonances), as well as highly damped complex or real poles, and direct or derivative-like terms. The effectiveness of the algorithm has been validated using experimental frequency response measurements obtained from different types of motion control mechanisms. The proposed algorithm has also been compared with “ <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tfest</monospace> ” and “ <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">modalfit</monospace> ” functions in MATLAB. Over these methods, nearly two orders of magnitude improvement is observed in the closeness of the model prediction, in terms of root mean square of the frequencywise modeling error.