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Journal articles on the topic 'Poles and zeros'

Author: Grafiati

Published: 4 June 2021

Last updated: 20 June 2025

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1

Wyman, Bostwick F. "Poles, zeros, and sheaf cohomology." Linear Algebra and its Applications 351-352 (August 2002): 799–807. http://dx.doi.org/10.1016/s0024-3795(01)00467-0.

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2

Sun, Yi, and Hugh G. A. Burton. "Complex analysis of divergent perturbation theory at finite temperature." Journal of Chemical Physics 156, no. 17 (2022): 171101. http://dx.doi.org/10.1063/5.0091442.

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We investigate the convergence properties of finite-temperature perturbation theory by considering the mathematical structure of thermodynamic potentials using complex analysis. We discover that zeros of the partition function lead to poles in the internal energy and logarithmic singularities in the Helmholtz free energy that create divergent expansions in the canonical ensemble. Analyzing these zeros reveals that the radius of convergence increases at higher temperatures. In contrast, when the reference state is degenerate, these poles in the internal energy create a zero radius of convergence in the zero-temperature limit. Finally, by showing that the poles in the internal energy reduce to exceptional points in the zero-temperature limit, we unify the two main mathematical representations of quantum phase transitions.

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3

Pishkoo, Amir, and Maslina Darus. "Translation, Creation and Annihilation of Poles and Zeros with the Biernacki and Ruscheweyh Operators, Acting on Meijer's G-Functions." Chinese Journal of Mathematics 2014 (February 12, 2014): 1–5. http://dx.doi.org/10.1155/2014/716718.

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Meijer's G-functions are studied by the Biernacki and Ruscheweyh operators. These operators are special cases of the Erdélyi-Kober operators (for m=1). The effect of operators on Meijer's G-functions can be shown as the change in the distribution of poles and zeros on the complex plane. These poles and zeros belong to the integrand, a ratio of gamma functions, defining the Meijer's G-function. Displacement in position and increasing or decreasing in number of poles and zeroes are expressed by the transporter, creator, and annihilator operators. With special glance, three basic univalent Meijer's G-functions, Koebe, and convex functions are considered.

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4

Langley, J. K. "On the zeros of the second derivative." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 2 (1997): 359–68. http://dx.doi.org/10.1017/s0308210500023672.

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Suppose that f is meromorphic of finite order in the plane, and that f″ has only finitely many zeros. We prove a strong estimate for the frequency of distinct poles of f. In particular, if the poles of f have bounded multiplicities, then f has only finitely many poles.

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5

Clark, R. L. "Accounting for Out-of-Bandwidth Modes in the Assumed Modes Approach: Implications on Colocated Output Feedback Control." Journal of Dynamic Systems, Measurement, and Control 119, no. 3 (1997): 390–95. http://dx.doi.org/10.1115/1.2801270.

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Colocated, output feedback is commonly used in the control of reverberant systems. More often than not, the system to be controlled displays high modal density at a moderate frequency, and thus the compliance of the out-of-bandwidth modes significantly influences the performance of the closed-loop system at low frequencies. In the assumed modes approach, the inclusion principle is used to demonstrate that the poles of the dynamic system converge from above when additional admissible functions are used to expand the solution. However, one can also interpret the convergence of the poles in terms of the zeros of the open-loop system. Since colocated inputs and outputs are known to have interlaced poles and zeros, the effect of a modification to the structural impedance locally serves to couple the modes of the system through feedback. The poles of the modified system follow loci defined by the relative location of the open-loop poles and zeros. Thus, as the number of admissible functions used in the series expansion is increased, the interlaced zeros of the colocated plant tend toward the open-loop poles, causing the closed-loop poles to converge from above as predicted by the inclusion principle. The analysis and results presented in this work indicate that the cumulative compliance of the out-of-bandwidth modes and not the modes themselves is required to converge the zeros of the open-loop system and the poles of the closed-loop system.

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6

Jedynak, R., and J. Gilewicz. "Distributions of zeros and poles of -point Padé approximants to complex-symmetric functions defined at complex points." Ukrains’kyi Matematychnyi Zhurnal 73, no. 8 (2021): 1034–55. http://dx.doi.org/10.37863/umzh.v73i8.333.

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UDC 517.5 The knowledge of the location of zeros and poles Padé and -point Padé approximations to a given function provides much valuable information about the function being studied.In general PAs reproduce the exact zeros and poles of considered function, but, unfortunately, some spurious zeros and poles appear randomly.Then, it is clear that the control of the position of poles and zeros becomes essential for applications of Padé approximation method.The numerical examples included in the paper show how necessary for the convergence of PA is the knowledge of the position of their zeros and poles.We relate our research of localization of poles and zeros of PA and NPA in the case of Stieltjes functions because we are interested in the efficiency of numerical application of these approximations. These functions belong to the class of complex-symmetric functions.The PA and NPA to the Stieltjes functions in different regions of the complex plane is also analyzed. It is expected that the appropriate selection of the complex point for the definition of approximant can improve it with respect to the traditional choice of All considered cases are graphically illustrated.Some unique numerical results presented in the paper, which are sufficiently regular should motivate the reader to reflect on them.

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7

Cordero, Raymundo, Matheus Caramalac, and Wisam Ali. "Generalized Predictive Control with Added Zeros and Poles in Its Augmented Model for Power Electronics Applications." Energies 17, no. 23 (2024): 6037. https://doi.org/10.3390/en17236037.

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Generalized predictive control (GPC) became one of the most popular and useful control strategies for academic and industry applications. An augmented model is applied to predict the future plant responses. This augmented model can be designed to embed the model of the plant reference, allowing its tracking by the controller according to the internal model principle (IMP). On the other hand, the performance of many controllers can be improved by adding zeros and poles in their structures (e.g., lead and lag compensators). However, according to the authors’ research, adding arbitrary poles or zeros to the GPC augmented model has not been explored yet. This paper presents a simple methodology to add arbitrary zeros and poles in the GPC augmented model. A new augmented model state variable is defined. The control law of the proposed approach embeds zeros and poles when zero-pole cancellation is avoided. Simulation results (considering a LCL filter controlled by a single-phase inverter of 500 W and a polynomial reference tracking controller) and experimental tests (using a third-order linear plant controlled by a resonant controller) prove that the proposed approach improves the transient response of different kinds of predictive tracking controllers applied to control different plants (including power electronics applications), without affecting the steady-state tracking capabilities of the control systems.

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8

Abbas, Ghulam, Jason Gu, Umar Farooq, et al. "Optimized Digital Controllers for Switching-Mode DC-DC Step-Down Converter." Electronics 7, no. 12 (2018): 412. http://dx.doi.org/10.3390/electronics7120412.

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In this paper, a nonlinear least squares optimization method is employed to optimize the performance of pole-zero-cancellation (PZC)-based digital controllers applied to a switching converter. An extensively used step-down converter operating at 1000 kHz is considered as a plant. In the PZC technique, the adverse effect of the (unwanted) poles of the buck converter power stage is diminished by the complex or real zeros of the compensator. Various combinations of the placement of the compensator zeros and poles can be considered. The compensator zeros and poles are nominally/roughly placed while attempting to cancel the converter poles. Although PZC techniques exhibit satisfactory performance to some extent, there is still room for improvement of the controller performance by readjusting its poles and zeros. The (nominal) digital controller coefficients thus obtained through PZC techniques are retuned intelligently through a nonlinear least squares (NLS) method using the Levenberg-Marquardt (LM) algorithm to ameliorate the static and dynamic performance while minimizing the sum of squares of the error in a quicker way. Effects of nonlinear components such as delay, ADC/DAC quantization error, and so forth contained in the digital control loop on performance and loop stability are also investigated. In order to validate the effectiveness of the optimized PZC techniques and show their supremacy over the traditional PZC techniques and the ones optimized by genetic algorithms (GAs), simulation results based on a MATLAB/Simulink environment are provided. For experimental validation, rapid hardware-in-the-loop (HiL) implementation of the compensated buck converter system is also performed.

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9

Filimonov, A. B., and N. B. Filimonov. "Control of Zeros and Poles in Problems of Synthesis of Regulation Systems. Part I. Compensation Approach." Mekhatronika, Avtomatizatsiya, Upravlenie 21, no. 8 (2020): 443–52. http://dx.doi.org/10.17587/mau.21.443-452.

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A highly important place in the theory and practice of automatic systems occupy the problems of synthesis of automatic regulation systems (ARS) based on the requirements for the dynamic quality of regulation processes. For class of linear stationary ARS such requirements are imposed upon the type and parameters of its transition characteristic which is uniquely determined by its transfer function (TF). In this connection the setting of problem of ARS synthesis with desired TF, corresponding to the given amplification coefficient, zeros and poles of the synthesized system is regularity. The given problem is worth while to call as the control problem of zeros and poles of ARS. In the present paper consists into two parts, the questions of control of zeros and poles ARS, which are important for engineering applications are considered. A critical analysis of the known compensation and compensation-modal regulation schemes is given. And also the new circuit solutions combining the functionality possibilities of compensation and modal approaches are proposed. The first part of the article the effect of compensation of zeros and poles of control objects in ARS is analyzed. The understanding of this effect gives the representation of the system of canonical structure of R. Kalman, according to which the compensation of zeros and poles of object does not mean their physical liquidation. As a result of compensation, they become the poles of its unobservable and unmanageable parts, which will be tell upon the regulation processes in the conditions of disturbances of the state of control object. The given effect and its negative results are clearly detected in the classical method of constructing of controllers on a priori given (desired, reference) TF of closed ARS. The influence of the factor of non-minimal phase zeros on the dynamics of control systems is studied. The effect of negative ejection in the transition characteristic of the system is described and its quantitative assessment is given for the case of a single real right zero. In the second part of the article the well-known methods for ARS synthesis with desired TF, based on use of polynomial calculus apparatus are considered and analyzed. New compensatory modal methods which may be of interest in engineering applications are proposed.

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10

Johnson, Timothy J., and Douglas E. Adams. "Transmissibility as a Differential Indicator of Structural Damage." Journal of Vibration and Acoustics 124, no. 4 (2002): 634–41. http://dx.doi.org/10.1115/1.1500744.

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This article discusses the use of frequency domain transmissibility functions for detecting, locating, and quantifying damage in linear and nonlinear structures. Structural damage affects both the system poles and zeros; however, zeros are much more sensitive than poles to localized damage. This is because zeros depend on the input and output locations whereas poles do not. It is demonstrated here that since transmissibility functions are determined solely by the system zeros, they are potentially better indicators of localized linear and nonlinear types of damage. Furthermore, excitation measurements are not required to compute transmissibility functions so damage indices can be calculated directly from response measurements. It is also demonstrated that sensor arrays can sometimes be used to yield mixed transmissibility functions that are differential in nature, that is, they are less sensitive to gross fluctuations in the dynamic loading or environmental variables.

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11

Peng, Lijuan, Jian Wang, Guicheng Yu, Zuoxue Wang, Aijun Yin, and Hongji Ren. "Active Vibration Control of PID Based on Receptance Method." Journal of Sensors 2020 (August 14, 2020): 1–8. http://dx.doi.org/10.1155/2020/8811448.

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Active vibration control approaches have been widely applied on improving reliability of robotic systems. For linear vibratory systems, the vibration features can be altered by modifying poles and zeros. To realize the arbitrary assignment of the closed-loop system poles and zeros of a linear vibratory system, in this paper, an active PID input feedback vibration control method is proposed based on the receptance method. The establishment and verification of the proposed method are demonstrated. The assignable poles during feedback control are calculated and attached with importance to expand the application of the integral control. Numerical simulations are conducted to verify the validity of the proposed method in terms of the assignment of closed-loop poles, zeros, and both. The results indicate that the proposed method can be used to realize the active vibration control of closed-loop system and obtain the desired damping ratio, modal frequency, and dynamic response.

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12

Foote, Richard, and V. Kumar Murty. "Zeros and poles of Artin L-series." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 1 (1989): 5–11. http://dx.doi.org/10.1017/s0305004100001316.

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Let E/F be a finite normal extension of number fields with Galois group G. For each virtual character χ of G, denote by L(s, χ) = L(s, χ, F) the Artin L-series attached to χ. It is defined for Re (s) > 1 by an Euler product which is absolutely convergent, making it holomorphic in this half plane. Artin's holomorphy conjecture asserts that, if χ is a character, L(s, χ) has a continuation to the entire s-plane, analytic except possibly for-a pole at s = 1 of multiplicity equal to 〈χ, 1〉, where 1 denotes the trivial character. A well-known group-theoretic result of Brauer implies that L(s, χ) has a meromorphic continuation for all s.

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13

Pommaret, J. F. "Poles and Zeros of Nonlinear Control Systems." IFAC Proceedings Volumes 34, no. 13 (2001): 397–400. http://dx.doi.org/10.1016/s1474-6670(17)39023-7.

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14

Frank, Günter, Xinhou Hua, and Rémi Vaillancourt. "Meromorphic Functions Sharing the Same Zeros and Poles." Canadian Journal of Mathematics 56, no. 6 (2004): 1190–227. http://dx.doi.org/10.4153/cjm-2004-052-6.

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AbstractIn this paper, Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function f is determined by its zeros and poles and the zeros of f(j) for j = 1, 2, 3, 4.

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15

MacCluer, C. R. "A Bound on the Impulse Response of Stable Rational Plants." Journal of Dynamic Systems, Measurement, and Control 113, no. 2 (1991): 315–16. http://dx.doi.org/10.1115/1.2896382.

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The magnitude of the impulse response of a strictly proper stable rational plant at time t is estimated to be of order 0(1/t) where the implied constant is explicitly given in terms of the spectral energy and the real parts of the zeros and poles. When exact location of poles and zeros is uncertain, this estimate can usefully replace the actual but uncertain exponential decay.

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16

Chen, Jun-Fan. "Nonexceptional functions and normal families of zero-free meromorphic functions." Filomat 31, no. 14 (2017): 4665–71. http://dx.doi.org/10.2298/fil1714665c.

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Let k be a positive integer, let F be a family of zero-free meromorphic functions in a domain D, all of whose poles are multiple, and let h be a meromorphic function in D, all of whose poles are simple, h . 0, ?. If for each f ? F, f(k)(z)- h(z) has at most k zeros in D, ignoring multiplicities, then F is normal in D. The examples are provided to show that the result is sharp.

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17

Knospe, C. R., and V. S. Lefante. "Understanding System Zeros." International Journal of Mechanical Engineering Education 21, no. 4 (1993): 316–26. http://dx.doi.org/10.1177/030641909302100402.

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An intuitive physical interpretation of a transfer function's poles is easily made: the natural frequencies of the system. A similar physical interpretation of a transfer function's zeros can also be made: the natural frequencies of special subsystems. In this paper, such an interpretation is advanced for certain spring-mass-damper systems using a rigorous and intuitive demonstration. This interpretation may serve as a useful paradigm for understanding the relationship between structural dynamics and transfer functions.

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18

Tomizuka, Masayoshi. "Zero Phase Error Tracking Algorithm for Digital Control." Journal of Dynamic Systems, Measurement, and Control 109, no. 1 (1987): 65–68. http://dx.doi.org/10.1115/1.3143822.

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A digital feedforward control algorithm for tracking desired time varying signals is presented. The feedforward controller cancels all the closed-loop poles and cancellable closed-loop zeros. For uncancellable zeros, which include zeros outside the unit circle, the feedforward controller cancels the phase shift induced by them. The phase cancellation assures that the frequency response between the desired output and actual output exhibits zero phase shift for all the frequencies. The algorithm is particularly suited to the general motion control problems including robotic arms and positioning tables. A typical motion control problem is used to show the effectiveness of the proposed feedforward controller.

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19

LANGLEY, J. K. "Non-real zeros of derivatives of real meromorphic functions of infinite order." Mathematical Proceedings of the Cambridge Philosophical Society 150, no. 2 (2010): 343–51. http://dx.doi.org/10.1017/s030500411000054x.

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20

Yan, Wei, Yong Chun Yang, and Xu Feng Guo. "Analyze the Transient Response of a Control Systems." Advanced Materials Research 706-708 (June 2013): 639–43. http://dx.doi.org/10.4028/www.scientific.net/amr.706-708.639.

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It is important for the control system analyst to understand the complete relationship of the complex-frequency representation of a linear system, the poles and zeros of its transfer function, and its time-domain response to step and other inputs. In such areas as signal processing and control, many of the analysis and designed collations are done in the complex-frequency plane, where a system model is represented in terms of the poles and zeros of its transfer functions.

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21

Lin, Jong-Lick. "On Transmission Zeros of Mass-Dashpot-Spring Systems." Journal of Dynamic Systems, Measurement, and Control 121, no. 2 (1999): 179–83. http://dx.doi.org/10.1115/1.2802452.

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For a noncollocated mass-dashpot-spring system with B=CTΓ, a novel approach is proposed to gain a better insight into the fact that none of its transmission zeros lie in the open right-half of the complex plane. In addition, the transmission zeros have physical meanings and will simply be the natural frequencies of a substructure constrained in the equivalently transformed system. Moreover, it is also shown that transmission zeros interlace with poles along the imaginary axis for a mass-spring system with B=CTΓ. They also interlace with poles along the negative real axis for a mass-dashpot system with B=CTΓ. Finally, two examples are used to illustrate the interlacing property.

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22

Andrievskii, V. V., H. P. Blatt, and R. K. Kovacheva. "On the Distribution of Zeros and Poles of Rational Approximants on Intervals." Abstract and Applied Analysis 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/961209.

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The distribution of zeros and poles of best rational approximants is well understood for the functions , . If is not holomorphic on , the distribution of the zeros of best rational approximants is governed by the equilibrium measure of under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, -values, and poles of best real rational approximants of degree at most to a function that is real-valued, but not holomorphic on . Generalizations to the lower half of the Walsh table are indicated.

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23

Hengartner, Nicolas W., and Oliver B. Linton. "Nonparametric regression estimation at design poles and zeros." Canadian Journal of Statistics 24, no. 4 (1996): 583–91. http://dx.doi.org/10.2307/3315335.

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24

Hribsek, M. "Simple allpass sections with complex poles and zeros." IEE Proceedings - Circuits, Devices and Systems 142, no. 5 (1995): 273. http://dx.doi.org/10.1049/ip-cds:19952108.

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25

HERSH, M. A. "The zeros and poles of delta operator systems." International Journal of Control 57, no. 3 (1993): 557–75. http://dx.doi.org/10.1080/00207179308934407.

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26

Bernal, Dionisio. "Fixed-base poles and eigenvectors from transmission zeros." Mechanical Systems and Signal Processing 45, no. 1 (2014): 68–79. http://dx.doi.org/10.1016/j.ymssp.2013.10.020.

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27

Reddy, A. R. "Rational approximation with real, negative zeros and poles." Journal of Approximation Theory 54, no. 2 (1988): 149–54. http://dx.doi.org/10.1016/0021-9045(88)90014-7.

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28

Wyman, Bostwick F., Michael K. Sain, Giuseppe Conte, and Anna Maria Perdon. "Poles and zeros of matrices of rational functions." Linear Algebra and its Applications 157 (November 1991): 113–39. http://dx.doi.org/10.1016/0024-3795(91)90107-8.

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29

Mårtensson, Jonas, and Håkan Hjalmarsson. "Variance-error quantification for identified poles and zeros." Automatica 45, no. 11 (2009): 2512–25. http://dx.doi.org/10.1016/j.automatica.2009.08.001.

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30

Ahmad, Irshad, and Ab Liman. "Inequalities pertaining to rational functions with prescribed poles." Filomat 31, no. 5 (2017): 1149–65. http://dx.doi.org/10.2298/fil1705149a.

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Let Rn be the space of rational functions with prescribed poles. If t1, t2, ..., tn are the zeros of B(z) + ? and s1, s2, ..., sn are zeros of B(z) - ?, where B(z) is the Blaschke product and ? ? T; then for z ? T |r'(z)| ? |B'(z)|/2 [(max 1?k?n |r(tk)|)2 + (max 1 ? k ? n |r(sk)|)2]. Let r, s ? Rn and assume s has all its n zeros in D- ? T and |r(z)| ? |s(z)| for z ? , then for any ? with |?| ? 1/2 and for z ? T |r'(z) + ?B'(z)r(z)|? |s'(z) + ?B'(z)s(z)|. In this paper, we consider a more general class of rational functions rof ? Rm*n, defined by (rof)(z) = r(f(z)), where f(z) is a polynomial of degree m* and prove some generalizations of the above inequalities.

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31

Shavala, O. "ON THE CONSTRUCTION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS ACCORDING TO GIVEN SEQUENCES OF ZEROS AND CRITICAL POINTS." Bukovinian Mathematical Journal 11, no. 1 (2023): 134–37. http://dx.doi.org/10.31861/bmj2023.01.12.

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A part of the theory of differential equations in the complex plane $\mathbb C$ is the study of their solutions. To obtain them sometimes researchers can use local expand of solution in the integer degrees of an independent variable. In more difficult cases received local expand in fractional degrees of an independent variable, on so-called Newton - Poiseux series. A row of mathematicians for integration of linear differential equations applied a method of so-called generalized degree series, where meets irrational, in general real degree of an independent variable. One of the directions of the theory of differential equations in the complex plane $\mathbb C$ is the construction a function $f$ according given sequence of zeros or poles, zeros of the derivative $f'$ and then find a differential equation for which this function be solution. Some authors studied sequences of zeros of solutions of the linear differential equation \begin{equation*} f''+Af=0, \end{equation*} where $A$ is entire or analytic function in a disk ${\rm \{ z:|z| < 1\} }$. In addition to the case when the above-mentioned differential equation has the non-trivial solution with given zero-sequences it is possible for consideration the case, when this equation has a solution with a given sequence of zeros (poles) and critical points. In this article we consider the question when the above-mentioned differential equation has the non-trivial solution $f$ such that $f^{1/\alpha}$, $\alpha \in {\mathbb R}\backslash \{ 0;-1\} $ is meromorphic function without zeros with poles in given sequence and the derivative of solution $f'$ has zeros in other given sequence, where $A$ is meromorphic function. Let's note, that representation of function by Weierstrass canonical product is the basic element for researches in the theory of the entire functions. Further we consider the question about construction of entire solution $f$ of the differential equation \begin{equation*} f^{(n)} +Af^{m} =0, \quad n,m\in {\mathbb N}, \end{equation*} where $A$ is meromorphic function such that $f$ has zeros in given sequence and the derivative of solution $f'$ has zeros in other given sequence.

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32

Antipov, Y. A. "Vector Riemann–Hilbert problem with almost periodic and meromorphic coefficients and applications." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2180 (2015): 20150262. http://dx.doi.org/10.1098/rspa.2015.0262.

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The vector Riemann–Hilbert problem is analysed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros (rational functions), (ii) periodic poles and zeros, and (iii) an infinite number of non-periodic zeros and poles, are considered. The first case is illustrated by the heat equation for a composite rod with a finite number of discontinuities and a system of convolution equations; both problems are solved explicitly. In the second case, a Wiener–Hopf factorization is found in terms of the hypergeometric functions, and the exact solution of a mixed boundary value problem for the Laplace equation in a wedge is derived. In the last case, the Riemann–Hilbert problem reduces to an infinite system of linear algebraic equations with the exponential rate of convergence. As an example, the Neumann boundary value problem for the Helmholtz equation in a strip with a slit is analysed.

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33

Langley, J. K. "Non-real zeros of real differential polynomials." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 3 (2011): 631–39. http://dx.doi.org/10.1017/s0308210510000284.

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The main results of the paper determine all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non-real poles and such that certain combinations of derivatives of f have few non-real zeros.

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34

Lin, Jong-Lick, Kuo-Chin Chan, Jyh-Jong Sheen, and Shin-Ju Chen. "Interlacing Properties for Mass-Dashpot-Spring Systems With Proportional Damping." Journal of Dynamic Systems, Measurement, and Control 126, no. 2 (2004): 426–30. http://dx.doi.org/10.1115/1.1650384.

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A mass-dashpot-spring system with proportional damping is considered in this paper. On the basis of an appropriate nonlinear mapping and the root-locus technique, the interlacing property of transmission zeros and poles is investigated if the columns of the input matrix are in the column space generated by the transpose of the output matrix. It is verified that transmission zeros interlace with poles on a specific circle and the nonpositive real axis segments for a proportional damping system. Finally, three examples are given to illustrate the property.

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35

Langley, J. K. "THE SECOND DERIVATIVE OF A MEROMORPHIC FUNCTION." Proceedings of the Edinburgh Mathematical Society 44, no. 3 (2001): 455–78. http://dx.doi.org/10.1017/s0013091599001029.

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AbstractLet $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or$f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.AMS 2000 Mathematics subject classification: Primary 30D35

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36

Dubinin, V. N. "Inequalities for derivatives of rational functions with critical values on an interval." Dal'nevostochnyi Matematicheskii Zhurnal 24, no. 2 (2024): 187–92. https://doi.org/10.47910/femj202417.

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37

Ahanger, Uzma M., and Wal M. Shah. "Inequalities for the derivative of rational functions with prescribed poles and restricted zeros." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 10 (68), no. 3 (2023): 554–67. http://dx.doi.org/10.21638/spbu01.2023.309.

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In this paper, instead of assuming that a rational function r(z) with prescribed poles has a zero of order s at origin, we suppose that it has a zero of multiplicity s at any point inside the unit circle, whereas the remaining zeros are within or outside a circle of radius k and prove some results which besides generalizing some inequalities for rational functions include refinements of some polynomial inequalities as special cases.

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38

Köhler, Lothar, and E. Mues. "Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros." Complex Variables, Theory and Application: An International Journal 11, no. 1 (1989): 39–48. http://dx.doi.org/10.1080/17476938908814322.

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39

Vlach, J., and E. Christen. "Poles, zeros, and their sensitivities in switched-capacitor networks." IEEE Transactions on Circuits and Systems 32, no. 3 (1985): 279–84. http://dx.doi.org/10.1109/tcs.1985.1085701.

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40

Kida, Takashi, Yoshiaki Ohkami, and Shigeo Sambongii. "Poles and transmission zeros of flexible spacecraft control systems." Journal of Guidance, Control, and Dynamics 8, no. 2 (1985): 208–13. http://dx.doi.org/10.2514/3.19961.

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Zenger, Kai, and Raimo Ylinen. "POLES AND ZEROS OF MULTIVARIABLE LINEAR TIME-VARYING SYSTEMS." IFAC Proceedings Volumes 35, no. 1 (2002): 261–66. http://dx.doi.org/10.3182/20020721-6-es-1901.00205.

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42

Srinivasan, B., and P. Myszkorowski. "Model reduction of systems with zeros interlacing the poles." Systems & Control Letters 30, no. 1 (1997): 19–24. http://dx.doi.org/10.1016/s0167-6911(96)00072-2.

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43

Gleizer, E. V. "Meromorphic functions with zeros and poles in small angles." Siberian Mathematical Journal 26, no. 4 (1986): 493–505. http://dx.doi.org/10.1007/bf00971296.

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44

Wyman, Bostwick F., Michael K. Sain, Giuseppe Conte, and Anna-Maria Perdon. "On the zeros and poles of a transfer function." Linear Algebra and its Applications 122-124 (September 1989): 123–44. http://dx.doi.org/10.1016/0024-3795(89)90650-2.

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45

Wang, Wen-June, and Tsu-Tian Lee. "Poles-zeros placement and decoupling in dicrete LQG systems." IEE Proceedings D Control Theory and Applications 134, no. 6 (1987): 388. http://dx.doi.org/10.1049/ip-d.1987.0061.

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46

Bourlès, H., and M. Fliess. "Poles and Zeros of Linear Systems: An Invariant Approach." IFAC Proceedings Volumes 28, no. 8 (1995): 383–88. http://dx.doi.org/10.1016/s1474-6670(17)45493-0.

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47

Zaris, P., J. Wood, and E. Rogers. "Controllable and Uncontrollable Poles and Zeros of nD Systems." Mathematics of Control, Signals, and Systems 14, no. 3 (2001): 281–98. http://dx.doi.org/10.1007/pl00009886.

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48

Stahl, H. "Poles and zeros of best rational approximants of |x|." Constructive Approximation 10, no. 4 (1994): 469–522. http://dx.doi.org/10.1007/bf01303523.

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49

Hippe, Peter. "Design of MIMO PI-compensators without exponentially unstable poles." at - Automatisierungstechnik 72, no. 4 (2024): 271–80. http://dx.doi.org/10.1515/auto-2023-0122.

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Abstract There is a class of systems that can only be stabilized by unstable compensators. However, for most practical systems suitable for use with linear regulators, this is possible with stable ones. To avoid exponentially unstable results in the design of pole placing PI-compensators it is of crucial importance to consider the position of the transmission zeros. This also has a favourable effect on the disturbance rejection. In multi-input multi-output (MIMO) systems, however, not only the zero position but also the zero direction is of importance. In this paper, a design method for multivariable PI-controllers in the time- and in the frequency-domains is presented which helps to avoid the occurrence of exponentially unstable controllers. A modified design method for multivariable PI-controllers, recently published in this journal, facilitates the procedure.

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50

Joelianto, Endra. "On Minimal Second-order IIR Bandpass Filters with Constrained Poles and Zeros." Journal of Engineering and Technological Sciences 53, no. 4 (2021): 210401. http://dx.doi.org/10.5614/j.eng.technol.sci.2021.53.4.1.

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In this paper, several forms of infinite impulse response (IIR) bandpass filters with constrained poles and zeros are presented and compared. The comparison includes the filter structure, the frequency ranges and a number of controlled parameters that affect computational efforts. Using the relationship between bandpass and notch filters, the two presented filters were originally developed for notch filters. This paper also proposes a second-order IIR bandpass filter structure that constrains poles and zeros and can be used as a minimal parameter adaptive digital second-order filter. The proposed filter has a wider frequency range and more flexibility in the range values of the adaptation parameters.

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