Left Half Of Complex Plane Research Articles

Exponential stability of a non-uniform Euler-Bernoulli beam with axial force

The aim of this paper is to investigate the boundary stabilization of a non-uniform Euler-Bernoulli beam with axial force generated by the equation ρ(x)utt+(σ(x)uxx)xx−(q(x)ux)x=0 defined in (0,1)×(0,∞), where the coefficients ρ(x)>0, σ(x)>0, and q(x)≥0. By adopting a new technique based on a qualitative theory of fourth-order linear differential equations, we prove that the spectrum of the system operator is confined to the open left-half complex plane and that all the associated eigenvalues are geometrically simple. Using this result and the Riesz basis approach, we obtain the exponential stability and the optimal decay rate of the system. This study extends the earlier paper by B.-Z. Guo [SIAM J. Control Optim., 40 (2002), pp. 1905–1923], where no axial force acting on the system (i.e., q≡0).

On the stability of parametric polynomials

This work introduces analytical tools for investigating stability regions of parametric polynomials. Our approach extends the classical D-decomposition method to accommodate any number of parameters, whether linearly or non-linearly involved in the coefficients of a polynomial ψ. Through Hermite's stability method, we establish the existence of stability regions within the parameter space, where all roots of ψ reside exclusively in the closed left half complex plane, defining the stability boundary. To identify points within the stability region, we employ the quotient and remainder from the Euclidean division of ψ by specific polynomials. Additionally, we introduce the ‘dynamics correlator’, a polynomial expressing ψ in its assignable roots space, whose zero set generalises the Root Locus. Confluence polynomials and loci, derived from the dynamics correlator, are used to realise stabilisability strategies. Extensive numerical examples demonstrate the application and effectiveness of the proposed methods in stabilising challenging systems and meeting additional performance criteria.

Spectral distribution and semigroup properties of a queueing model with exceptional service time

<p>In this paper, we studied the spectrum and semigroup properties of the M/G/1 queueing model with exceptional service time for the first customer in each busy period. First, we described the point spectrum of the system operator that corresponds to the model, and we prove that the system operator has an uncountable infinite number of eigenvalues on the left-half complex plane. Second, by using the spectrum analysis and semigroup theory, we obtained that the spectrum-determined growth (SDG) condition holds and the semigroup is not asymptotically stable, compact, eventually compact, or even quasi-compact. Finally, in order to clarify the results of spectral distribution, some numerical analysis were conducted.</p>

Effect of a Large Proton Exchange Membrane Electrolyser on Power System Small-Signal Angular Stability

The dynamics of electrical systems have changed significantly with the increasing penetration of non-conventional loads such as hydrogen electrolysers. As a result, detailed investigations are required to quantify and characterize these loads’ effects on the dynamic response of interconnected synchronous machines after being subjected to a disturbance. Many studies have focused on the effects of conventional static and dynamic loads. However, the impact of hydrogen electrolysers on the stability of power systems’ rotor angles is rarely studied. This paper assesses the effect of proton exchange membrane (PEM) electrolysers on small-disturbance rotor-angle stability. Dynamic modelling and the control of a PEM electrolyser as a load are first studied to achieve this. Then, the proposed electrolyser model is tested in the Amercoeur plant, which is part of the Belgian power system, to study its effect on the small-signal rotor-angle stability. Two approaches are considered to examine this impact: an analytical approach and time-domain simulations. The analytical approach consists of establishing a state-space model of the Belgian test system through linearisation around an operating point of the non-linear differential and the algebraic equations of the synchronous generators, the PEM electrolyser, the loads, and the network. The obtained state-space model allows for the determination of the eigenvalues, which are useful to evaluate the effect of the PEM electrolyser on the small-signal rotor-angle stability. This impact is investigated by examining the movement of the eigenvalues in the left complex half-plane. The obtained results show that the PEM electrolyser affects the electromechanical modes of synchronous machines by increasing their oscillation frequencies. The results also show that the effect of the electrolyser on these modes can be improved by adjusting the inertial constant and the damping coefficient of the synchronous machines. These results are consolidated through time-domain simulations using the software Matlab/Simscape from the version MatlabR2022a-academic use from Mathworks.

Delayed controllers for time-delay systems

In this paper, an analysis to design, tune and apply delayed controllers on a general class of first-order up to fourth-order LTI systems with or without dead-time is presented. This analysis is based on a study of the corresponding characteristic equation for obtaining σ-stabilization regions and to place dominant roots in the left-half complex plane. Obtaining σ-stabilization regions allows to characterize a parametric plane, which leads to obtain stabilizing conditions for LTI systems with two non-commensurable delays. The placement of dominant roots due to parametric variations (including delays) allows the tuning of delayed controllers. For this, analytic expressions to place three dominant real roots are given guaranteeing an exponential decay rate in the system response. The scope of the methodology is extended to nonlinear delayed controllers using recursive techniques for a class of nonlinear systems with dead-time in tracking tasks. To illustrate the obtained theoretical results, the following cases are considered: first, the tuning and application of a novel delayed controller with two delays are presented to stabilize a delay-free LTI model of fourth-order of the underactuated mechanical system called cart-pendulum using only pendulum position and cart position. Second, the design and tuning of a tracking delayed controller for a class of nonlinear systems with dead-time are proposed. Here, a delay is inherent to the system and another delay is artificially used to stabilize.

Static feedback stabilization of a rotating disk–cable system with measurements from each boundary

In this paper, we consider the stabilization of a unitized rotating disk–cable system, which consists of a uniform cable attached to a disk rotating at an angular velocity about an axis through the center of the disk and perpendicular to the disk. When the angular velocity is greater than π2, the system always has at least one positive eigenvalue, which makes the system cannot be stabilized only with the collocated static feedback control. This is why the existing literature usually considers the case where the admissible angular velocity is always less than π2. To overcome this obstacle, we design a static proportional feedback control with both collocated and anti-collocated measurements from each boundary to stabilize the system, where the admissible angular velocity is improved from less than π2 to less than π while ensuring that the system is minimum-phase. Through a detailed spectral analysis, the spectrum of the closed-loop system is proved to be in the left-half complex plane, and the asymptotic expressions of both eigenvalues and eigenfunctions are obtained. By the Riesz basis approach, the stability of the closed-loop system is proved. The effectiveness of the feedback control is verified by numerical simulation.

Eigenvalue Based Method to Maximize the Exponential Convergence Rate for Consensus Tracking with Heterogeneous Delays

This paper focuses on the consensus tracking problem for a group of single-integrator agents with heterogeneous communication and input delays, with the goal of achieving the maximum exponential convergence rate. Existing research discusses a tracking protocol consisting of a consensus-based estimator and a tracking controller for each agent, as well as stability analysis. In contrast, we examine the performance evolution of multi-agent systems for such tracking protocols in terms of their exponential convergence rate. We present a systematic frequency domain method for analyzing the rightmost eigenvalue in the open left half complex plane, which determines the exponential decay rate of the tracking error. The spectral characteristics of the linear time-invariant retarded delay system are investigated, leading to the optimization of the tracking controller and estimator gains. Simulations are given to demonstrate how the system performs when the design parameters are optimized.

Stabilization of two coupled wave equations with joint anti-damping and non-collocated control

In this paper, the non-collocated control problem of two connected wave equations with joint anti-damping is studied. The boundary dampings are introduced into the control at both ends of the boundary, and then the non-collocated observation is input into the right boundary to obtain the closed-loop system. The well-posedness of closed-loop system is proved by using the Riesz basis method. By using the Nyquist criterion, the admissible non-collocated feedback gain is concluded so that all eigenvalues of the closed-loop system are in the left-half complex plane, and the exponential stability of the closed-loop system is proved. The effectiveness of the proposed feedback control law is verified by numerical simulation.

On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.

On the Robust Multiple Objective Control with Simultaneous Pole Placement in LMI Regions

This article studies the problem of designing robust control laws to achieve multiple performance objectives for linear uncertain systems. Specifically, in this study we have selected one of the control objectives to be a closed-loop pole placement in specific regions of the left-half complex plane. As such, a guaranteed cost-based multi-objective control approach is proposed and compared with the H_2/H_∞ control by means of an application example.

Static boundary feedback stabilization of an anti-stable wave equation with both collocated and non-collocated measurements

In this paper, we consider the stabilization of a wave equation with an unknown anti-stable injection on the left boundary and the control input on the right boundary, where there are both collocated and non-collocated measurements. A static output feedback control law is designed to stabilize the wave equation. The value ranges of the feedback gains are given, such that all eigenvalues of the closed-loop system are shown to be inside the left-half complex plane by applying the Nyquist criterion for distributed parameter systems. Then the exponential stability of the closed-loop system is established. Numerical simulations are presented to verify the effectiveness of the proposed feedback control law.

Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

In this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

Dynamic feedback stabilization of an unstable wave equation

In this paper, we consider the stabilization of an unstable wave equation through a dynamic boundary compensator. The measurement, which is one internal point of the state of the wave equation, is fluxed into the compensator while the output of the compensator is forced into the boundary of the wave equation. The well-posedness of the closed-loop system is proved by using the semigroup method. By choosing appropriate controller parameters and adopting Nyquist criterion for distributed parameter systems, the eigenvalues of the closed-loop systems are shown to be inside the left-half complex plane. Then the closed-loop system is proved to be exponentially stable by using the Riesz basis approach. Numerical simulations are presented to verify the effectiveness of the proposed compensator.

Non-collocated boundary control for contact-force control of a one-link flexible arm

This paper deals with contact-force control of a one-link flexible arm whose tip is constrained to a rigid environment. To realize the contact-force control, a boundary controller is proposed based on a dynamic model represented by an infinite dimensional model. In particular, the proposed controller does not need the physical parameters in its implementation, and this results in the non-collocated boundary controller. The closed-loop system is analyzed in an appropriate Hilbert space, and it is shown that the exponential stability of the closed-loop system is obtained by setting the feedback gains to locate the eigenvalues of the closed-loop system on the complex left half-plane. In addition, in an attempt to realize the better control performance, another controller which is a modified version of our controller is proposed. Finally, the stability, robustness to the uncertainty in physical parameters, and disturbance response of the closed-loop system are investigated by numerical simulations.

Asymptotic Stability of Non-Autonomous Systems and a Generalization of Levinson’s Theorem

We study the asymptotic stability of non-autonomous linear systems with time dependent coefficient matrices { A ( t ) } t ∈ R . The classical theorem of Levinson has been an indispensable tool for the study of the asymptotic stability of non-autonomous linear systems. Contrary to constant coefficient system, having all eigenvalues in the left half complex plane does not imply asymptotic stability of the zero solution. Levinson’s theorem assumes that the coefficient matrix is a suitable perturbation of the diagonal matrix. Our objective is to prove a theorem similar to Levinson’s Theorem when the family of matrices merely admits an upper triangular factorization. In fact, in the presence of defective eigenvalues, Levinson’s Theorem does not apply. In our paper, we first investigate the asymptotic behavior of upper triangular systems and use the fixed point theory to draw a few conclusions. Unless stated otherwise, we aim to understand asymptotic behavior dimension by dimension, working with upper triangular with internal blocks adds flexibility to the analysis.

A CLASS OF S-STEP NON-LINEAR ITERATION SCHEME BASED ON PROJECTION METHOD FOR GAUSS METHOD

Various iteration schemes are proposed by various authors to solve nonlinear equations arising in the implementation of implicit Runge-Kutta methods. In this paper, a class of s-step non-linear scheme based on projection method is proposed to accelerate the convergence rate of those linear iteration schemes. In this scheme, sequence of numerical solutions is updated after each sub-step is completed. For 2-stage Gauss method, upper bound for the spectral radius of its iteration matrix was obtained in the left half complex plane. This result is extended to 3-stage and 4-stage Gauss methods by transforming the coefficient matrix and the iteration matrix to a block diagonal form. Finally, some numerical experiments are carried out to confirm the obtained theoretical results.

On the linear static output feedback problem: The annihilating polynomial approach

One of the fundamental open problems in control theory is that of the stabilization of a linear time invariant dynamical system through static output feedback. We are given a linear dynamical system defined throughw.=Aw+Buy=Cw. The problem is to find, if it exists, a feedback u=Ky such that the matrix A+BKC has all its eigenvalues in the complex left half plane and, if such a feedback does not exist, to prove that it does not. Substantial progress has not been made on the computational aspect of the solution to this problem.In this paper we consider instead ‘which annihilating polynomials can a matrix of the form A+BKC possess?’.We give a simple solution to this problem when the system has either a single input or a single output. For the multi-input - multi-output case, we use these ideas to characterize the annihilating polynomials when K has rank one, and suggest possible computational solutions for general K. We also present some numerical evidence for the plausibility of this approach for the general case as well as for the problem of shifting the eigenvalues of the system.

Minimum-Rank Dynamic Output Consensus Design for Heterogeneous Nonlinear Multi-Agent Systems

In this paper, we propose a new and systematic design framework for output consensus in heterogeneous Multi-Input Multi-Output (MIMO) general nonlinear Multi-Agent Systems (MASs) subjected to directed communication topology. First, the input-output feedback linearization method is utilized assuming that the internal dynamics is Input-to-State Stable (ISS) to obtain linearized subsystems of agents. Consequently, we propose local dynamic controllers for agents such that the linearized subsystems have an identical closed-loop dynamics which has a single pole at the origin whereas other poles are on the open left half complex plane. This allows us to deal with distinct agents having arbitrarily vector relative degrees and to derive rank-$1$ cooperative control inputs for those homogeneous linearized dynamics which results in a minimum rank distributed dynamic consensus controller for the initial nonlinear MAS. Moreover, we prove that the coupling strength in the consensus protocol can be arbitrarily small but positive and hence our consensus design is non-conservative. Next, our design approach is further strengthened by tackling the problem of randomly switching communication topologies among agents where we relax the assumption on the balance of each switched graph and derive a distributed rank-$1$ dynamic consensus controller. Lastly, a numerical example is introduced to illustrate the effectiveness of our proposed framework.

Laplace inversion for the solution of an abstract heat equation without the forward transform of the source term

AbstractWe consider the discretization in time of inhomogeneous parabolic equations, using the technique of Laplace inversion along a contour located in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a contour quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. A serious problem is how to treat the source term

Outer transfer functions of differential-algebraic systems

We consider differential-algebraic systems (DAEs) whose transfer function is outer : i . e ., it has full row rank and all transmission zeros lie in the closed left half complex plane. We characterize outer, with the aid of the Kronecker structure of the system pencil and the Smith–McMillan structure of the transfer function, as the following property of a behavioural stabilizable and detectable realization: each consistent initial value can be asymptotically controlled to zero while the output can be made arbitrarily small in the ℒ 2 -norm. The zero dynamics of systems with outer transfer functions are analyzed. We further show that our characterizations of outer provide a simple and very structured analysis of the linear-quadratic optimal control problem.