Hermite Biehler Research Articles

A generalized Hermite–Biehler theorem and non-Hermitian perturbations of Jacobi matrices

The classical Hermite–Biehler theorem describes the zero configuration of a complex linear combination of two real polynomials whose zeros are real, simple, and strictly interlace. We provide the full characterization of the zero configuration for the case when this interlacing is broken at exactly one location. We apply this result to solve the direct and inverse spectral problem for non-Hermitian rank-one multiplicative perturbations and rank-two additive perturbations of finite Hermitian and Jacobi matrices.

On the relation between Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions

In this paper, we elaborate on the relationship between the Hurwitz stability of matrix polynomials and matrix-valued Stieltjes functions. Our strategy is that, for a monic matrix polynomial, we associate a rational matrix-valued function with its even–odd split and then check the Hurwitz stability of the matrix polynomial by testing the Stieltjes property of the related rational matrix-valued function. On the basis of this relationship, we present matrix generalizations of a classical stability criterion by Gantmacher, Chebotarev theorem, Grommer theorem and some aspects of the modified Hermite–Biehler theorem. Our work is motivated by one of the authors’ recent stability studies linked with matricial Markov parameters.

Descent generating polynomials and the Hermite–Biehler theorem

Descent generating polynomials and the Hermite–Biehler theorem

Robust Model Predictive Control Paradigm for Automatic Voltage Regulators against Uncertainty Based on Optimization Algorithms

This paper introduces a robust model predictive controller (MPC) to operate an automatic voltage regulator (AVR). The design strategy tends to handle the uncertainty issue of the AVR parameters. Frequency domain conditions are derived from the Hermite–Biehler theorem to maintain the stability of the perturbed system. The tuning of the MPC parameters is performed based on a new evolutionary algorithm named arithmetic optimization algorithm (AOA), while the expert designers use trial and error methods to achieve this target. The stability constraints are handled during the tuning process. An effective time-domain objective is formulated to guarantee good performance for the AVR by minimizing the voltage maximum overshoot and the response settling time simultaneously. The results of the suggested AOA-based robust MPC are compared with various techniques in the literature. The system response demonstrates the effectiveness and robustness of the proposed strategy with low control effort against the voltage variations and the parameters’ uncertainty compared with other techniques.

Stabilising PID controller for time-delay systems with guaranteed gain and phase margins

A new analytical-graphical method is proposed for computing the region of stability for proportional-integral-derivative (PID) controllers based on the Hermite–Biehler theorem. By this method, a PID controller is designed to ensure the Hurwitz stability of a time-delay system with any order of transfer function. First, the possible range of the derivative part that makes the system stable is obtained. Then, the stability region is found by applying the Hermite–Biehler theorem. It is shown that this theorem can be extended to quasi-polynomials functions in the form of to find the stability region of systems with time delay under certain conditions. Using this, the proposed approach can guarantee specified gain and phase margins for time-delay systems; therefore, it is very beneficial and advantageous for the control of practical plants. Five different examples including two practical systems are studied throughout the paper to illustrate the applicability, performance, and efficiency of the proposed control approach.

Generic Noninteger Order Controller for Time-Delay Systems

This paper focuses on the problem of fractional controller P I stabilization for a first-order time-delay systems. For this reason, we utilize the Hermite–Biehler and Pontryagin theorems to compute the complete set of the stabilizing P I λ parameters. The widespread industrial utilization of PID controllers and the potentiality of their noninteger order representation justify a timely interest in P I λ tuning techniques. Step responses are calculated through K p , K i , l a m b d a parameters inside and outside stability region to prove the method efficiency.

Stabilizing and Robust Fractional PID Controller Synthesis for Uncertain First-Order plus Time-Delay Systems

In this paper, by using the noninteger P I λ D μ controllers, we conduct an investigation into the subject of robust stability area of time-delay interval process. Our method is based on setting up of the noninteger interval closed-loop characteristic equation using the inferior and superior bounds of uncertain parameters into several vertexes. We have combined the composition of the value set of vertex with the zero exclusion principle to analyse the stability of the uncertain process. A generalized version of the Hermite–Biehler theorem, applicable to fractional quasipolynomials, is exploited to determine the stability region of each vertex. The robust stability region of the noninteger regulator can be given by the crossing of the stability area of all the vertex characteristic noninteger quasipolynomials. By using the value set method and zero exclusion theory, the effectiveness of the stability region can be tested. Also, we propose a suitable procedure to determine the whole of stabilizing parameters for an interval process. An explicative example is given to point out the advantage and reliability of the approach.

Robust Optimal Proportional–Integral Controller for an Uncertain Unstable Delay System: Wind Process Application

In industrial practice, certain processes are unstable, such as different types of reactors, distillation columns, and combustion systems. To ensure greater maneuverability and improve the speed of response command, certain systems in the military and aviation fields are purposely configured to be unstable. These systems are often more difficult to control than stable systems and are of particular interest to designers and control engineers. Despite all advances in process control over the past six decades, the proportional–integral–derivative (PID) controller is still the most common. The main reasons are the simplicity, robustness, and successful applications provided by PID-based control structures. The design of proportional–integral (PI) controller for time-delay (TD) systems is a traditional and classical problem. On one hand, PI controllers are used in more than 95% of industrial processes. On the other hand, there are TD phenomena in almost all practical control systems. In this paper, we study the robustness of a PI controller in stabilizing systems containing uncertain parameters and delays. A robust controller for an unknown unstable second-order with margin TD is constructed using a generalized Kharitonov theorem for quasi-polynomials. A constructive method based on the Hermite–Biehler theorem is used to obtain all PI gains, which stabilize an uncertain and unstable second-order delay system. Genetic algorithms (GAs) and optimization methods are used to obtain optimal system and control parameters for providing the best PI control that makes the system robust and stable under uncertainty. By minimizing performance criteria such as the integral of square error, integral of absolute error, time-weighted integral of absolute error, and integral of time weighted square error, GAs and particle swarm optimization are used to optimize the PI parameters and system parameters that provide the best control under uncertainties.

An inverse problem for a class of canonical systems having Hamiltonians of determinant one

We study the inverse problem of canonical systems recovering the Hamiltonian from a given function E of the Hermite–Biehler class, and solve the inverse problem under some special assumptions on E.

A note on the zeros of approximations of the Ramanujan Xi -function

In this paper we review the study of the distribution of the zeros of certain approximations for the Ramanujan Xi -function given by Ki (Ramanujan J 17(1):123–143, 2008), and we provide new proofs of his results. Our approach is motivated by the ideas of Velásquez (J Anal Math 110:67–127, 2010) in the study of the zeros of certain sums of entire functions with some condition of stability related to the Hermite–Biehler theorem.

Real-Time Implementation of a Stable 2 DOF PID Controller for Unstable Second-Order Magnetic Levitation System with Time Delay

The design of a two degree of freedom (DOF) proportional–integral–derivative (PID) controller is proposed in this paper to stabilize an unstable second-order plant having time delay. The lower and upper bounds of the controller gains for which the compensated system will exhibit a stable response, is determined using Hermite–Biehler theorem. In order to determine the bounds of the controller gains, this work proposes an extension of the Hermite–Biehler theorem, suitable for the transfer function of the maglev plant, which contains a negative dc-gain and a missing ‘s’ term in the denominator polynomial. The optimal values of the controller gains, from the range thus estimated, are found by a recent bio-inspired algorithm, namely symbiotic organisms search (SOS). The obtained results have been experimentally verified, and it has been validated that the 2-DOF PID controller exhibits better response. The robustness of the compensated system for 1-DOF and 2-DOF configurations, in the presence of external disturbance and measurement noise, has also been validated in real time, and it is observed that the 2-DOF PID controller improves the robustness of the system, as compared to its 1-DOF counterpart.

Consensus of fractional-order double-integrator multi-agent systems

In this paper, several consensus problems are studied for fractional-order double-integrator multi-agent systems, where the order of fractional derivative belongs to (0, 1). Both continuous-time and sampled-data based consensus protocols are proposed for absolute and relative damping cases, respectively. For the continuous-time protocols, consensus is achieved by using the connection between integer-order and fractional-order multi-agent systems. For the sampled-data based protocols, convergence analysis is provided by using the Hermite–Biehler theorem, which is used to deal with the stability of a polynomial with complex coefficients. It is shown that the consensus value is a function of the initial conditions for the absolute damping case, while the consensus value is related to the initial conditions and the time for the relative damping case. In addition, some examples are provided to validate the correctness of the main theorems.

Ant Colony Optimization based on Pareto optimality: application to a congested router controlled by PID regulation

ABSTRACTThe subject of this research work is to stabilize the network TCP (transmission control protocol) as well as the queue of the router congestion by designing an Active Queue Management scheme able to ensure this role. The problem is dealt with under the theory of the command by using a tuned PID (proportional–integral–derivative) controller based on an extension of the Hermite–Biehler theorem applied to quasi-polynomials. This tuning approach uses Hurwitz stability concept, that is to say, a sufficient and necessary condition must be given so the roots of the quasi-polynomial lie in the left half plane. Since this stabilization method gives rise to a set of values for parameters ‘P’, ‘I’ and ‘D’, it turns out relevant to optimize the results achieved within this stability region. To achieve this purpose, a multi-criterion Ant Colony Optimization based on Pareto optimality is used, given the conflictual character of the closed-loop system performance parameters. The set of optimal solutions of the problem is given by determining the Pareto front of objective functions. The effectiveness of the proposed control scheme is evaluated via a series of numerical simulations in MATLAB and SIMULINK. The results are compared with those of the genetic algorithm and Ziegler–Nichols methods.

Linear finite difference operators preserving the Laguerre–Pólya class

ABSTRACTWe prove that the difference operator of the following form: where and are some complex functions and h is a non-zero complex number, preserves the Laguerre–Pólya class of entire functions if and only if, the step h is pure imaginary, , and the function is entire of order at most 2 and belongs to a subclass of the Hermite–Biehler class of entire functions.

Stability region of a simplified multirotor motor–rotor model with time delay and fractional-order PD controller

ABSTRACTThe main aim of this paper is to present stability region analysis for a closed-loop system with the second-order model with a time delay and continuous-time fractional-order proportional-derivative (PD) controller. The model of the plant used in the paper approximates the dynamics of a simplified motor–rotor model of multirotor's propulsion system. The controller tuning method is based on Hermite–Biehler and Pontryagin theorems. The tracking performance is also analysed in the paper by observing the integral of absolute error and integral of squared error indices. The presented results are expected to be useful in future when comparing simulation with experimental results.

De Branges functions of Schroedinger equations

We characterize the Hermite–Biehler (de Branges) functions E which correspond to Schroedinger operators with \(L^2\) potential on the finite interval. From this characterization one can easily deduce a recent theorem by Horvath. We also obtain a result about location of resonances.

New result on PID controller design of LTI systems via dominant eigenvalue assignment

This note considers the problem of assigning the dominant eigenvalues of a linear time-invariant (LTI) system to the desired positions by using proportional–integral–derivative (PID) controllers. The procedure is based on first setting some rightmost eigenvalues of the system at the desired positions and then guaranteeing the dominance of those rightmost eigenvalues by using the generalization of the Hermite–Biehler Theorem. It is worth pointing out that this work aims to ascertain the gains of PID controllers in a straightforwardly computational way which plays an important role in practical applications.

Controller design for delay systems via eigenvalue assignment – on a new result in the distribution of quasi-polynomial roots

This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment. First, a new result on the root distribution for a class of quasi-polynomials is developed based on the extension of the Hermite–Biehler theorem. Then, such result is applied to proportional–integral (PI) controller parameter design for a first-order plant with time delay through pole placement. The complete region of PI gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed-loop system with delay plant are assigned to desired positions in the complex plane. Furthermore, on the basis of the previous result, this paper also extended the PI control to the proportional–integral–derivative (PID) control. It is worth pointing out that this work aims to improve the performance of the closed-loop system on the premise of guaranteeing the stability.

The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem

Based on the Hermite–Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach. Basé sur le théorème de Hermite–Biehler, nous prouvons simultanément les polynômes eulériens de type $D$ et les polynômes eulériens affine de type $B$ ont seulement racines réelle, qui sont d’abord obtenue par Savage et Visontai en utilisant le théorie des polynômes $s$-eulériens. Nous confirmons aussi les conjectures de Hyatt sur la propriété entrelacement de polynômes mi-eulériens. Le travail de Borcea et Brändén sur la caractérisation des opérateurs linéaires préservant la stabilité Hurwitz est essentielle à cette approche.

Design of PID controllers for interval plants with time delay

First of all, the box theorem is extended to the interval plants with the fixed delay. An approach is presented to design the PID controller for interval plants with the fixed delay, which can obtain all of the stabilizing PID controllers. Then, using Hermite–Biehler theorem, extreme point results are provided by the virtual quasi-polynomials. When two virtual and two vertex quasi-polynomials corresponding to a Kharitonov-like segment plant are stable under a particular PID controller, it is sufficient that the same PID controller can stabilize this Kharitonov-like segment plant. The virtual quasi-polynomials are obtained in a simple way, and they are expressed in terms of the controller and the Kharitonov polynomials of the interval plants. A PID controller stabilizes interval plants with the fixed delay if it simultaneously stabilizes thirty-two quasi-polynomials. The example is given to illustrate the proposed method.