Pair Of Complex Conjugate Poles Research Articles

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.

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.

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.

Control based on a hybrid predictor for linear unstable delayed systems with a pair of complex conjugate poles

This paper presents a methodology to design a hybrid predictor of continuous signals to solve the stabilization problema of unstable linear systems with time delay, specifically systems with an unstable pole and a pair of stable conjugate complex poles. Among the main contributions of this work is to guarantee the existence of the predictor device, regardless of the size of the delay. To solve the stabilization problem, the necessary and sufficient conditions are obtained for a Proportional-Integral (PI) controller that can guarantee the stability of the system using the designed predictor. The performance of this control strategy is evaluated by means of a numerical simulation.

Closed-Loop Microbial Fuel Cell Control System Designed for Online Monitoring of TOC Dynamic Characteristics in Public Swimming Pool.

Total organic carbon (TOC) in the water of public swimming pools (PSPs) must be monitored online for public health. In order to address the shortcomings of conventional microbial fuel cell biosensor (MFC-biosensor), an innovative biosensor with peculiar closed-loop structure was developed for online monitoring of TOC in PSPs. Its design was based on experimental data, model identification, cybernetics, and digital and real-time simulation. The outcomes of the digital simulation demonstrated that the closed-loop MFC control system possesses the desired structure with a pair of dominant complex-conjugate closed-loop poles (−15.47 ± 7.73j), and the real-time simulation showed that its controller output signals can automatically and precisely track the variation in TOC concentration in PSP water with the desired dynamic response performances; for example, mean delay time was 0.06 h, rise time was 0.12 h, peak time was 0.18 h, maximum overshoot was 7.39%, settling time was 0.22 h, and best fit 0.98. The proposed principle and method of the closed-loop MFC-biosensor control system in the article can also be applied for online monitoring of other substances in water, such as heavy metal ions, chemical toxicants, and so forth, and lay a theoretical foundation for MFC-based online monitoring substances in an aquatic environment.

LTP Modeling and Stability Assessment of Multiple Second-Order Generalized Integrator-Based Signal Processing/Synchronization Algorithms and Their Close Variants

A second-order generalized integrator (SOGI) is a resonant regulator with a pair of complex-conjugate poles, and therefore, with infinite magnitude at its center frequency. Thanks to this property, one can put together a set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> SOGIs in an elegant way and decompose a single-phase signal into its constituent frequency components and detect their amplitude and phase angle. Such a configuration, which is often referred to as the multi-SOGI (MSOGI) structure, requires a frequency estimator to adapt the center frequency of SOGIs to frequency changes. This frequency estimation is often provided by interconnecting the MSOGI structure with a basic frequency-locked loop (FLL) or phase-locked loop (PLL). The resulting structures are known as the MSOGI-FLL and MSOGI-PLL. These structures and their close variants are mathematically difficult to analyze probably because of the lack of a linear model for these systems. This article aims to bridge this research gap. First, it is shown how linear time-periodic (LTP) models of the MSOGI-FLL and MSOGI-PLL can be obtained. The model verification, obtaining open-loop harmonic transfer function from the LTP model, and LTP stability assessment of MSOGI-based synchronization systems are the next parts of this article. Finally, some close variants of the MSOGI-FLL/PLL are considered and their modeling and stability assessment are briefly discussed.

Complex dynamics of a bi-directional N-type locally-active memristor

This paper presents a bi-directional N-type locally-active memristor (LAM) model, which has two symmetrical locally-active regions with respect to the origin. For this memristor, the locally-active regions coincide with the edge of chaos regimes, where complex dynamic behaviors may occur. By deriving the small-signal admittance of the memristor, it is found that the LAM may be pure resistive, inductive or capacitive in terms of biasing voltages. When the LAM operates at the edge of chaos and connects with an external inductor in series, a second-order periodic oscillator is established. In this case, it is possible to move the poles of the system from the left-half plane to the right-half plane, thereby generating periodic oscillations. Complex dynamics of the second-order memristor-based oscillator is analyzed via Hopf bifurcation analysis and based on the theories of local activity and edge of chaos. Using the parasitic capacitor of the LAM in the second-order oscillator, a third-order chaotic oscillator is designed, which has a negative real pole and a pair of complex conjugate poles with positive real parts, satisfying the Shilnikov criterion. It is found that both periodic oscillation and chaos appear only at the edge of chaos of the LAM, thus verifying the complexity theorem, no-complexity theorem and edge of chaos theorem. Finally, circuit experiments are performed, which verify the dynamics and the effectiveness of the LAM model, the periodic oscillator and the chaotic oscillator.

Effects of a quark chemical potential on the analytic structure of the gluon propagator

We perform complex analyses of the gluon propagator at nonzero quark chemical potential in the long-wavelength limit, using an effective model with a gluon mass term of the Landau-gauge Yang-Mills theory, which is a Landau-gauge limit of the Curci-Ferrari model with quantum corrections being included within the one-loop level. We mainly investigate complex poles of the gluon propagator, which could be relevant to confinement. Around typical values of the model parameters, we show that the gluon propagator has one or two pairs of complex conjugate poles depending on the value of the chemical potential. In addition to a pair similar to that in the case of zero chemical potential, a new pair appears near the real axis when the chemical potential is roughly between the effective quark mass and the effective gluon mass of the model. We discuss possible interpretations of these poles. Additionally, we prove the uniqueness of analytic continuation of the Matsubara propagator to a class of functions that vanish at infinity and are holomorphic except for a finite number of complex poles and singularities on the real axis.

Analytic structure of the lattice Landau gauge gluon and ghost propagators

Starting from the lattice Landau gauge gluon and ghost propagator data we use a sequence of Pad\'e approximants, identify the poles and zeros for each approximant and map them into the analytic structure of the propagators. For the Landau gauge gluon propagator the Pad\'e analysis identifies a pair of complex conjugate poles and a branch cut along the negative real axis of the Euclidean ${p}^{2}$ momenta. For the Landau gauge ghost propagator the Pad\'e analysis shows a single pole at ${p}^{2}=0$ and a branch cut also along the negative real axis of the Euclidean ${p}^{2}$ momenta. The method gives precise estimates for the gluon complex poles that agree well with other estimates found in the literature. For the branch cut the Pad\'e analysis gives, at least, a rough estimate of the corresponding branch point.

Piezoelectric mirror shifter transfer function measurement, modelling, and analysis using feedback based synthetic-heterodyne Michelson interferometry

Laser vibrometry has many applications in non-contact dynamic displacement and vibration measurement. A test beam reflected from a target and a reference beam are combined and detected by a photodiode; the photodetected signal is then processed to determine the target displacement and vibration. This paper describes the use of a 9 kHz measurement bandwidth system, consisting of a Michelson interferometer and self-correcting feedback synthetic-heterodyne signal processing technique, to measure the displacement impulse response of a commercial piezoelectric mirror shifter (PMS), consisting of a mirror mounted on a Piezoelectric transducer and a connecting 50 Ω electrical coaxial cable. The actual non-ideal applied impulse and measured impulse response data were used in conjunction with the instrument variable method to determine a Laplace domain linear transfer function approximation to the actual PMS transfer function. The best transfer function fitting, having a 84% normalized root mean square goodness of fit, was obtained using a 5-th order transfer function having two complex conjugate pole pairs, with associated natural frequencies of 6.29 and 6.79 kHz, and a single real pole. The transfer function zeros consisted of a single complex conjugate zero pair, having an antiresonance frequency of 6.38 kHz and a single real zero. Knowing the analytic transfer function of PMS based nanopositioners is useful for example in the design of closed-loop phase-locked interferometers for wideband sensing.

Pole identification method to extract the equivalent fluid characteristics of general sound-absorbing materials

A method is presented to characterize general sound-absorbing materials through a pole-based identification of the equivalent fluid. This is accomplished by 1) determining the extended equivalent fluid of the material sample through the transfer function method (TFM), 2) identification of the acoustic response of the material through the poles of the extended effective density and compressibility, and 3) build the effective density and compressibility from the poles associated to the local acoustic response. Real pole pairs describe a dissipative medium (or equivalently an over-damped resonating medium), which is the natural behavior of rigid-frame porous materials, while complex-conjugate pole pairs describe a locally-resonant medium typical of metamaterials. Complex-conjugate poles associated to elastic resonances of the sample are discarded. We test the method for several non-conventional porous materials. In general, a better fit to the measured surface impedance is obtained than with an acoustics-based identification to the Johnson-Champoux-Allard-Pride-Lafarge model (JCAPL), and the method appears also to be robust to errors of the TFM.

Complex poles and spectral functions of Landau gauge QCD and QCD-like theories

In view of the expectation that the existence of complex poles is a signal of confinement, we investigate the analytic structure of the gluon, quark, and ghost propagators in the Landau gauge QCD and QCD-like theories by employing an effective model with a gluon mass term of the Yang-Mills theory, which we call the massive Yang-Mills model. In this model, we particularly investigate the number of complex poles in the parameter space of the model consisting of gauge coupling constant, gluon mass, and quark mass for the gauge group $SU(3)$ and various numbers of quark flavors $N_F$ within the asymptotic free region. Both the gluon and quark propagators at the best-fit parameters for $N_F=2$ QCD have one pair of complex conjugate poles, while the number of complex poles in the gluon propagator varies between zero and four depending on the number of quark flavors and quark mass. Moreover, as a general feature, we argue that the gluon spectral function of this model with nonzero quark mass is negative in the infrared limit. In sharp contrast to gluons, the quark and ghost propagators are insensitive to the number of quark flavors within the current approximations adopted in this paper. These results suggest that details of the confinement mechanism may depend on the number of quark flavors and quark mass.

Reflection positivity and complex analysis of the Yang\u2013Mills theory from a viewpoint of gluon confinement

In order to understand the confining decoupling solution of the Yang–Mills theory in the Landau gauge, we consider the massive Yang–Mills model which is defined by just adding a gluon mass term to the Yang–Mills theory with the Lorentz-covariant gauge fixing term and the associated Faddeev–Popov ghost term. First of all, we show that massive Yang–Mills model is obtained as a gauge-fixed version of the gauge-invariantly extended theory which is identified with the gauge-scalar model with a single fixed-modulus scalar field in the fundamental representation of the gauge group. This equivalence is obtained through the gauge-independent description of the Brout–Englert–Higgs mechanism proposed recently by one of the authors. Then, we reconfirm that the Euclidean gluon and ghost propagators in the Landau gauge obtained by numerical simulations on the lattice are reproduced with good accuracy from the massive Yang–Mills model by taking into account one-loop quantum corrections. Moreover, we demonstrate in a numerical way that the Schwinger function calculated from the gluon propagator in the Euclidean region exhibits violation of the reflection positivity at the physical point of the parameters. In addition, we perform the analytic continuation of the gluon propagator from the Euclidean region to the complex momentum plane towards the Minkowski region. We give an analytical proof that the reflection positivity is violated for any choice of the parameters in the massive Yang–Mills model, due to the existence of a pair of complex conjugate poles and the negativity of the spectral function for the gluon propagator to one-loop order. The complex structure of the propagator enables us to explain why the gluon propagator in the Euclidean region is well described by the Gribov–Stingl form. We try to understand these results in light of the Fradkin–Shenker continuity between confinement-like and Higgs-like regions in a single confinement phase in the complementary gauge-scalar model.

Fault detection optimization for controllable dynamic systems

Introduction: When diagnosing the deviations of controllable dynamic system parameters, it is convenient in terms of control simplicity to apply the Schreiber method which uses a set of rectangular pulses of equal duration as a test signal. Since for a single object you can construct many test signals which differ in the number of pulses, the problem arises how to minimize the number of test pulses when using the Schreiber method. Purpose: Simplification of test control and diagnostics of linear controllable dynamic systems. Results: It has been shown that a set of test pulse amplitude vectors is a kernel of the controllability matrix of a discrete analogue of the object under test. The problem is formulated of finding the optimal length of a test pulse in order to minimize the number of pulses in the test signal. For a given pulse length, the pulse amplitudes of an optimal test signal are equal to the coefficients of the control vector minimal polynomial for the discrete analog of the object relative to its system matrix. The number of test pulses can be reduced by choosing the pulse duration calculated from the imaginary component of the object poles. In particular, if an object has at least one pair of complex-conjugate poles, the number of test pulses does not at least exceed the order of the object. An algorithm has been developed for calculating a test signal for linear controllable object FDI by the Schreiber method. The input to the algorithm is the system matrix of the object, and the output is the length of the test pulse and the pulse amplitude vector. The efficiency of the algorithm is illustrated by FDI for two technical objects. Practical relevance: The results of the study can be applied to static parameter FDI of controllable dynamical objects which allow a linear description in their state space.

Parametric Synthesis of a Robust Controller Based on the Method of Dominant Poles

In the paper a linear control system described by its characteristic polynomial with interval coefficients including parameters of controller linearly is considered. Problem of the research is finding parameters of a controller guaranteeing dynamic characteristics of a system despite interval parametric uncertainty of its object. It is proposed to base a controller synthesis on root quality indices: minimal stability degree and maximal oscillability degree. Desired values of these indices will be provided with the help of dominant poles method. Applying this method consists in placing a pair of complex-conjugate dominant poles; all other poles — unrestricted poles — will be placed by defining a right border of their allocation area on a complex plane. To apply dominant poles method, a feature of stability degree and oscillability degree to be determined by images of certain vertices of a parametric polytope was used. To synthesize a controller, it is proposed to divide its parameters in two groups: dependent ones and unrestricted ones. The first group of controller parameters is to provide desired allocation of dominant poles in one of vertices of parametric polytope (a dominant vertex). Unrestricted parameters of a controller are to provide desired distance between dominant poles and allocation area of unrestricted poles. To find coordinates of a dominant vertex and verifying vertices providing unrestricted poles allocation, an interval extension of basic phase equation of a root locus theory was developed. This resulted in interval phase inequalities, whose solution allows finding coordinates of desired vertices of characteristic polynomials coeffi cients polytope. Knowing a dominant vertex polynomial and dominant poles allows expressing dependent parameters of a controller from unrestricted ones. Obtained expressions allow placing unrestricted poles in a desired area of a complex plane. To do this, a D-partition by unrestricted parameters of a controller is performed in all verifying vertices of parametric polytope of a system. After choosing values of unrestricted parameters from intersection of all stability domains obtain during D-partition, dependent parameters of a controller can be calculated. An example of synthesizing a PID-controller guaranteeing desired values of dynamics characteristics for an interval control system of the fourth order is provided.

Spectral functions of confined particles

We determine the gluon and ghost spectral functions along with the analytic structure of the associated propagators from numerical data describing gauge correlators at space-like momenta obtained by either solving the Dyson-Schwinger equations or through lattice simulations. Our novel reconstruction technique shows the expected branch cut for the gluon and the ghost propagator, which, in the gluon case, is supplemented with a pair of complex conjugate poles. Possible implications of the existence of these poles are briefly addressed.

Complex poles and spectral function of Yang-Mills theory

We derive general relationships between the number of complex poles of a propagator and the sign of the spectral function originating from the branch cut in the Minkowski region under some assumptions on the asymptotic behaviors of the propagator. We apply this relation to the mass-deformed Yang-Mills model with one-loop quantum corrections, which is identified with a low-energy effective theory of the Yang-Mills theory, to show that the gluon propagator in this model has a pair of complex conjugate poles or "tachyonic" poles of multiplicity two, in accordance with the fact that the gluon field has a negative spectral function, while the ghost propagator has at most one "unphysical" pole. Finally, we discuss implications of these results for gluon confinement and other non-perturbative aspects of the Yang-Mills theory.

Envelope-Domain Analysis and Modeling of Super-Regenerative Oscillators

An envelope-domain methodology for the numerical modeling of super-regenerative oscillators (SROs) is presented. The main advantage is its generality of application to transistor-based oscillators with arbitrary topology. Initially, a stability analysis of the nonoscillatory steady-state solution, forced by the quench signal, is performed. It is based on the calculation of a linear-time-variant (LTV) transfer function, obtained by linearizing the circuit envelope-domain equations about the nonoscillatory regime. Under moderate quench frequencies, it will be possible to estimate the SRO normalized envelope and sensitivity function from the detected dominant pair of complex-conjugate poles. In the general case, the SRO oscillatory response is modeled with a numerical method, valid under linear operation with respect to the input signal. This is based on the calculation of the LTV impulse response from a time–frequency transfer function obtained under a small-signal sinusoidal excitation. The LTV impulse response enables a straightforward determination of the sensitivity time interval and time distance to the envelope maximum. An integral expression, in terms of the LTV transfer function, will provide the SRO response to any small-signal input with any arbitrarily carrier frequency and modulation. The methodology has been successfully validated through its application to an SRO at 2.7 GHz, which has been manufactured and measured.

Damped Nyquist Plot for a pole placement design of power system stabilizers

This paper proposes a new frequency domain control design method through the use of a modified Nyquist diagram with an embedded partial pole-placement capability. The proposed Damped Nyquist Plot (DNP) method evaluates the open loop transfer function (OLTF) along a line of constant damping ratio to provide a graphical tool to design power system stabilizers (PSS). The graphical tool shows how the closed loop poles move around this damping ratio line for different choices of PSS parameters, all of them placing a selected pair of complex-conjugate poles at the same desired point in the complex plane. New formulas are developed to determine the exact PSS parameters that promote the several possibilities of partial pole placement considering the parameter ranges used in tuning practice. These formulas can also be used for phase compensation of conventional methods of PSS design, such as GEP or Residue Angle Compensation. Multiple PSSs can be tuned using the DNP method sequentially. The proposed method is applied to two stabilization examples, the first of tutorial nature and the second is the large actual Brazilian Interconnected Power System.

Stability Analysis of Digital Microwave Power Amplifiers

This paper presents a stability analysis of microwave power amplifiers (PAs) driven by binary pulse trains, as in the case of class-S PAs. First, using a simplified digital PA test bench in class-D configuration, different qualitative behaviors are obtained when varying the pulsewidth, including subharmonic and incommensurable oscillations. The mechanisms affecting the stability properties are studied with a harmonic balance-based formulation, by means of pole-zero identification and bifurcation detection. A sufficiently high number of harmonic components must be considered, together with the Krylov decomposition for an efficient computation of the inverse of the Jacobian matrix. It is demonstrated that, when varying the pulsewidth, the distinct pairs of complex-conjugate poles may shift to the right-hand side of the complex plane and, therefore, lead to different kinds of unstable behavior. This phenomenon is related to the dependence of the critical poles on the average value of the input signal. Boundaries of the various types of unstable behavior are traced in the plane defined by the pulse repetition rate and pulsewidth, using bifurcation detection techniques. All the predicted phenomena have been confirmed experimentally. In a second step, the algorithms derived from the simple class-D circuit are transferred to study the stability of a more complex tri-band class-S amplifier. It has been analyzed versus the input bitrate, obtaining a fully stable behavior that has been validated experimentally.