Infinite Number Of Poles Research Articles

Design of delayed PID‐type controllers for high‐order recycling systems with time delays using model reduction

AbstractIn even the simplest recycling system with a first‐order plus dead time model for the direct and recycling paths, the characteristic equation for the open‐loop transfer function contains an infinite number of poles. This is due to the presence of time delays in the internal dynamics. The task of designing a controller is therefore challenging from the perspective of control. This paper considers the problem of stabilising a high‐order delayed recycling system (with a possible unstable direct path) through the use of a reduced model in conjunction with a new PID (proportional‐integral‐derivative)‐type delayed controller. We state the condition under which the stability of the closed‐loop system is guaranteed and determine the sets of stabilising control parameters. The performance of our proposed strategy is illustrated using several numerical simulations of industrial processes, including an industrial quench column, two continuous stirred tank reactor systems with recycling and a reactor–separator process.

Small-signal stability assessment of power systems with multiple transport delays based on s-domain modelling

Wide area monitoring systems are becoming increasingly important to power systems security. Control and protection schemes that use remote sensing are being developed and becoming a reality. The delay introduced by remote sensing should be accurately modelled in the analysis tools. A precise representation of transport delays turns the system model into an infinite-dimensional system, with an infinite number of poles. As a result, usual representations such as state-space and descriptor systems become unfeasible. This paper presents novel mathematical advances and computational methodologies to allow a comprehensive modal analysis using exact representation of multiple transport delays. The proposed approach is based on a non-linear formulation in the Laplace domain and uses numerical methods founded on the general complex variable theory. The proposed methodologies are analysed through simulations in a didactical example and in the Brazilian Interconnected Power System.

Asymptotic behaviour of the third Painlevé transcendents in the space of initial values

We study the asymptotic behavior of solutions of the fourth Painleve equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any solution that is not rational has an infinite number of poles and infinite number of zeros.

Exact summation of leading infrared logarithms in 2D effective field theories

A method of exact all-order summation of leading infrared logarithms in two dimensional massless Φ4-type non-renormalizable effective field theories (EFTs) is developed. The method is applied to the O(N)-symmetric EFT, which is a two-dimensional sibling of the four dimensional O(N + 1)/O(N) sigma-model. For the first time the exact all-order summation of the (E2ln(1/E))n contributions (chiral logarithms) for the 2 → 2 scattering amplitudes is performed in closed analytical form. The cases when the resulting amplitudes turn to be meromorphic functions with an infinite number of poles (Landau poles) are identified. This provides the first explicit example of quasi-renormalizable field theories.

Asymptotic behaviour of the fifth Painlevé transcendents in the space of initial values

We study the asymptotic behaviour of the solutions of the fifth Painlevé equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with an essential singularity at zero has an infinite number of poles and zeroes, and any solution with an essential singularity at infinity has infinite number of poles and, moreover, takes the value unity infinitely many times.

Tracking of periodic reference signal: A parameterized finite dimensional repetitive control approach

This paper is concerned with the development of a Modified Finite Dimensional Repetitive Control (MFDRC) system. Conventional FDRC is modified through parameterization of the controller using co-prime factorization. The proposed method of controller design specifies the input output characteristics beforehand, ensuring highly accurate tracking and disturbance rejection property of the system subjected to periodic reference input. Another problem is the stabilization issues in the input–output and the disturbance rejection characteristics of a conventional repetitive control system with respect to the periodic reference input owing to have infinite number of poles in the transfer function of it. It is desirable that the transfer functions from both the reference input and the disturbance to the output have a finite number of poles and that will be taken care of in the proposed scheme of MFDRC design depicted in this paper. The performance of designed controller has been validated in steps applied to a laboratory based servo system. Comparisons are done using conventional PID controller, FDRC-based PID controller and MFDRC to show advantage, disadvantage or limitation of applied control scheme.

Graviton propagator, renormalization scale and black-hole like states

We study the analytic structure of the resummed graviton propagator, inspired by the possible existence of black hole precursors in its spectrum. We find an infinite number of poles with positive mass, but both positive and negative effective width, and studied their asymptotic behaviour in the infinite sheet Riemann surface. We find that the stability of these precursors depend crucially on the value of the normalisation point scale.

Asymptotic Behavior of the Fourth Painlevé Transcendents in the Space of Initial Values

We study the asymptotic behavior of solutions of the fourth Painleve equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any solution that is not rational has an infinite number of poles and infinite number of zeros.

Dimensional analysis approach to dominant three-pole placement in delayed PID control loops

PID control loops with time delay are characterized by infinite number of poles but the pole assignment technique for adjusting the controller parameters can be applied to placing three poles only. The dominance of these poles is therefore an essential condition for this application. A novel approach to this problem involves applying dimensional analysis theory to obtain a generalized model of the control loop and then to perform a parameter tuning for its dimensionless representation. A one-row dimensional matrix results from the assumption of the usual dimensionless interpretation of both control error and actuating signals of the controller. Dimensionless similarity numbers of the so-called swingability and laggardness are introduced to specify the plant dynamics in the controller synthesis. A trio of numbers is assigned to become the dominant zeros of the characteristic quasi-polynomial of the control loop and the corresponding PID parameter adjustment is derived in the form of uniform formulae. The tuning of the proper damping and the real pole position ratios is provided by means of an IAE optimization technique. A dominance degree notion is introduced and an argument increment criterion is proposed to check the dominance of any of the pole placement cases. The quality of the disturbance rejection response is taken as the general criterion in the design of the time delay plant control.

Analytical closed‐form solution for transient analysis of boost DC‐DC converters

PurposeThe purpose of this paper is to suggest a new analytical methodology for transient analysis of DC‐DC power converters. The closed‐form solution obtained following this methodology is suitable both for design of passive elements of the converter and for the development of control techniques.Design/methodology/approachThe methodology is based on a mixed use of Laplace transform and z‐transform. The expressions of variables of the set of equations, characteristic of a DC‐DC converter, are first evaluated in the Laplace domain for the generic switching interval. The solutions obtained are then z‐transformed in order that they match in each contiguous time interval, to form the complete transient response.FindingsThe new solution methodology allows the analytical determination of time constants of DC‐DC converters, also in presence of large duty‐cycle variations. Moreover, it is possible to evaluate easily the influence of passive elements on converter's behaviour, without several numerical simulations.Originality/valueThe analytical solution of linear systems is well known both in transient and in steady‐state conditions. However, when there is an infinite number of poles in the Laplace transform of the input signals, such as the case of switching power converters, the inversion in a closed form of the Laplace transform of the solution can be cumbersome. The methodology presented tries to overcome this problem by using an approach based on the z‐transform. Operating in this way, a closed‐form solution can be obtained both in transient and in steady‐state conditions, for all the main topologies of switching power converters. The procedure has been explained in detail for the sample case of boost DC‐DC converters.

A Design Method for Simple Repetitive Controllers for Time-Delay Plants

The modified repetitive control system is a type of servomechanism for the periodic reference input. That is, the modi¯ed repetitive control system follows the periodic reference input with small steady state error, even if a periodic disturbance or uncertainty exists in the plant. Using previously proposed modified repetitive controllers, even if the plants does not includes time-delay, the transfer function from the periodic reference input to the output and that from the disturbance to the output have an infinite number of poles. When the transfer function from the periodic reference input to the output and that from the disturbance to the output have an infinite number of poles, it is dificult to specify the input output characteristic and the disturbance attenuation characteristic. From the practical point of view, it is desirable that the input-output characteristic and the disturbance attenuation characteristic are easily specified. In order to specify the input-output characteristic and the disturbance attenuation characteristic easily, the transfer function from the periodic reference input to the output and that from the disturbance to the output are desirable to have a finite number of poles. Yamada et al. proposed the concept of simple repetitive control systems such that the controller works as a modified repetitive controller and the transfer function from the periodic reference input to the output and that from the disturbance to the output have a finite number of poles. In addition, Yamada et al. clarified the parametrization of all stabilizing simple repetitive controllers. However the method by Yamada et al. cannot be applied for time-delay plants. The purpose of this paper is to propose the parametrization of all stabilizing simple repetitive controllers for time-delay plants.

A design method of robust stabilizing simple multi-period repetitive controllers

A multi-period repetitive control system is a type of servomechanism for a periodic reference input. Even if a plant does not include time-delay, the transfer function from the periodic reference input to the output and that from the disturbance to the output of the multi-period repetitive control system generally have an infinite number of poles. In order to specify the input-output characteristic and the disturbance attenuation characteristic easily, Yamada et al. propose the concept of simple multi-period repetitive control systems such that the controller works as a stabilizing multi-period repetitive controller and the transfer function from the periodic reference input to the output and that from the disturbance to the output have a finite number of poles. However, the method by Yamada et al. cannot apply for the plant with uncertainty. The purpose of this paper is to propose the concept of robust stabilizing simple multi-period repetitive controllers for the plant with uncertainty and to clarify the parametrization of all robust stabilizing simple multi-period repetitive controllers.

Adaptive stabilization of Kirchhoff's non‐linear strings by boundary displacement feedback

AbstractThis paper is concerned with adaptive global stabilization of an undamped non‐linear string in the case where any velocity feedback is not available. The linearized system may have an infinite number of poles and zeros on the imaginary axis. In the case where any velocity feedback is not available, a parallel compensator is effective. The adaptive stabilizer is constructed for the augmented system which consists of the controlled system and a parallel compensator. It is proved that the string can be stabilized by adaptive boundary control. Copyright © 2007 John Wiley & Sons, Ltd.

Jahn-Teller systems at half filling: Crossover from Heisenberg to Ising behavior

The Jahn-Teller model with $E\ensuremath{\bigotimes}\ensuremath{\beta}$ electron-phonon coupling and local (Hubbard-like) Coulomb interaction is considered to describe a lattice system with two orbitals per site at half filling. Starting from a state with one electron per site, we follow the tunneling of the electrons and the associated creation of an arbitrary number of phonons due to electron-phonon interaction. For this purpose we apply a recursive method which allows us to organize systematically the number of pairs of empty/doubly occupied sites and to include infinitely many phonons which are induced by electronic hopping. In lowest order of the recursion (i.e., for all processes with only one pair of empty/doubly occupied sites) we obtain an effective anisotropic pseudospin $1∕2$ Heisenberg Hamiltonian ${H}_{\mathit{eff}}$ as a description of the orbital degrees of freedom. The pseudospin coupling depends on the physical parameters and the energy. This implies that the resulting resolvent ${[E\ensuremath{-}{H}_{\mathit{eff}}(E)]}^{\ensuremath{-}1}$ has an infinite number of poles, even for a single site. ${H}_{\mathit{eff}}$ is subject to a crossover from an isotropic Heisenberg model (weak electron-phonon coupling and isotropic hopping) to an Ising model (strong electron-phonon coupling).

Robust regulation of stable systems in the H ∞-algebra

In this paper, robust regulation of stable infinite-dimensional plants with transfer functions in the H ∞-algebra is considered. The reference and disturbance signals are allowed to have an infinite number of poles on the imaginary axis, which makes it possible to consider e.g. arbitrary periodic signals. A controller depending on a positive scalar ε and design parameters Kk is proposed. The design parameters are matrices over H ∞, and by the main result of the paper, if the design parameters satisfy a certain condition, the proposed controller is a robust regulator for sufficiently small values of ε. The only knowledge needed to verify this condition is the so-called -stability of the plant and values of the plant transfer function at the poles of the reference and disturbance signals. The analysis in this paper is in the frequency domain and it is based on the fractional representation approach.

Model Approximation for Dead-Time Recycling Systems

Chemical processes with recycling commonly contain delays in both the forward and the backward paths. The characteristic equation of such systems is a quasipolynomial function, so that the corresponding transfer function contains an infinite number of poles. This feature precludes the use of classical stability analysis and control design techniques. In this work, a simple and effective methodology to derive an approximate discrete-time model for continuous-time recycling processes with delay is proposed. The method is based on the discretization, via a fictitious sampler and hold device, of the internal delayed signal, resulting in a finite-dimensional discrete-time version of the original continuous model. In this way, standard analysis methods, such as root locus and stability margin techniques, can be easily applied to the approximate model to obtain some conclusions on the stability of the original recycling process. Illustrative examples are used to show that some stability measures (e.g., stability margin) obtained with the approximate discrete-time model closely describe the behavior of the original recycling process.

Stability results for two classes of linear time‐delay and hybrid systems

The stability of linear time‐delay systems with point internal delays is difficult to deal with in practice because of the fact that their characteristic equation is usually of transcendent type rather than of polynomial type. This feature causes usually the system to possess an infinite number of poles. In this paper, stability tests for this class of systems are obtained either based on extensions of classical tests applicable to delay‐free systems or on approaches within the framework of two‐dimensional digital filters. Some of those two‐dimensional stability tests are also proved to be useful for stability testing of a common class of linear hybrid systems which involve coupled continuous and digital substates after a slight “ad‐hoc” adaptation of the tests for that situation.

Low‐gain adaptive stabilization of semilinear second‐order hyperbolic systems

AbstractIn this paper low‐gain adaptive stabilization of undamped semilinear second‐order hyperbolic systems is considered in the case where the input and output operators are collocated. The linearized systems have an infinite number of poles and zeros on the imaginary axis. The adaptive stabilizer is constructed by a low‐gain adaptive velocity feedback. The closed‐loop system is governed by a non‐linear evolution equation. First, the well‐posedness of the closed‐loop system is shown. Next, an energy‐like function and a multiplier function are introduced and the exponential stability of the closed‐loop system is analysed. Some examples are given to illustrate the theory. Copyright © 2004 John Wiley & Sons, Ltd.

Resolution of the pion puzzle: The QCD string in Nambu-Goldstone mesons

Pions and kaons have double nature: the chiral dynamics of Nambu-Goldstone bosons together with the usual string dynamics common to all mesons. To uncover the interplay of both dynamics the effective chiral Lagrangian is derived from the QCD Lagrangian using the Field Correlator Method and the pion self-energy (mass) operator is written explicitly. The latter contains an infinite number of poles, but normalized to zero at zero momentum because of spontaneous chiral symmetry breaking. As a result one obtains the Gell-Mann-Oakes-Renner relation for the ground state pion and (slightly shifted by chiral dynamics) the usual spectrum of radially excited pions starting with $\pi(1300)$.

Stabilization of infinite-dimensional undamped second order systems by using a parallel compensator

In this paper stabilization of infinite‐dimensional undamped second‐order systems is considered in the case where the input and output operators are collocated. The systems have an infinite number of poles and zeros on the imaginary axis. In the case where only position feedback is available, a parallel compensator is effective. The stabilizer is constructed by a P‐controller for the augmented system which consists of the controlled system and a parallel compensator. The asymptotic stability of the augmented system is proved by LaSalle's invariance principle under compactness of the resolvent.