Cluster Treatment Of Characteristic Roots Research Articles

A general framework for the exact stability analysis of love dynamics with four independent time delays

We investigate the asymptotic stability of a linear love model with four independent time delays to mimic the complicated human emotions more precisely compared with the classic Strogatz model. These four delays stem from the perception of their own past emotions and their partner's past emotions for both individuals. The central idea is to exactly ascertain the stability switching boundaries where the emotions of these two individuals change in the domain of delays. For this, we propose a general stability analysis framework synthesizing the Dixon resultant theory, the frequency sweeping technique, and the cluster treatment of characteristic roots (CTCR) paradigm. We then present a detailed procedure of this framework to particularly study how one partner reacts to the other. Finally, we demonstrate the effectiveness of this framework through a case study and then provide some suggestions for the couple to achieve a sweet love relationship.

A Simple Procedure for the Creation of Stability Charts in Delay Parameter Space for a Class of LTI Systems With Two Delays

The central task of the stability robustness with respect to delay uncertainties lies in creating the whole stability chart (delay map) in the delay parameter space. In this paper, we present a very simple frequency-sweeping procedure for creating stability chart in the delay parameter plane for a class of linear time-invariant (LTI) two-delay systems with a delay crossing talk. Actually, the procedure based on using Rekasius pseudo-delay substitution and discriminant of quadratic polynomial to characterize the crossing frequency set. The exact and exhaustive determination of crossing frequency intervals involves only finding all positive real roots of a real-coefficient polynomial. With the availability of the entire crossing frequency set, the famous cluster treatment of characteristic roots (CTCR) paradigm is applied to construct complete stability chart in the delay plane. A by-product of the proposed procedure is the revelation of the non-zero finite frequencies corresponding to infinite pseudo-delays. These frequencies divide the crossing frequency intervals into subintervals over each of which the frequency-sweeping technique provides continuous stability crossing curves in the domain of pseudo delays. For illustration and validation, two examples are provided.

Complete stability assessment of LTI systems with multiple constant-coefficient distributed delays using the improved frequency sweeping framework

The stability problem of linear time-invariant systems with multiple constant-coefficient distributed delays is investigated from a new perspective. We present an equivalent system with multiple lumped delays and then apply the Dixon resultant to determine the exact upper bound of the projected imaginary spectra of the system. We then adopt the improved frequency sweeping framework over the obtained range to get the kernel and offspring hypersurfaces (KOH). Furthermore, the singularity at the zero characteristic root is scrutinised, leading to what we call the stationary root boundary (SRB) in the domain of the delays. With these, we resort to the Cluster Treatment of Characteristic Roots (CTCR) paradigm to determine the complete stability map of the system with the complete knowledge of the KOH and SRB. Finally, the effectiveness of our proposed procedure is shown by two case studies.

Improved frequency sweeping technique and stability analysis of the second-order consensus protocol with distributed delays

The consensus problem of a general class of second-order multi-agent system with distributed delays is investigated from a new perspective. We propose a consensus protocol with distributed delays to enhance the delay robustness of the system. First, we establish a spectral equivalence of such a system and the one with multiple discrete delays from the stability viewpoint. We then employ a factorization procedure to reduce the computational complexity. Next, we uniquely use the Dixon resultant theory to determine the exact upper bound of the imaginary spectra, leading to the kernel and offspring hypersurfaces (KOH) in the delay domain. Furthermore, the associated standing roots of the system are analyzed to determine the precise stationary root boundaries, which constitute the potential stability changing curves together with the KOH. Finally, we resort to the Cluster Treatment of Characteristic Roots (CTCR) paradigm to examine the exact and complete stability of the system.

A Novel Frequency-Domain Approach for the Exact Range of Imaginary Spectra and the Stability Analysis of LTI Systems With Two Delays

This paper presents a novel frequency-domain approach to reveal the exact range of the imaginary spectra and the stability of linear time-invariant systems with two delays. First, an exact relation, i.e., the Rekasius substitution, is used to replace the exponential term caused by the delays in order to transform the transcendental characteristic equation to a quasi-polynomial. Second, this quasi-polynomial is uniquely tackled by our proposed Dixon resultant and discriminant theory, leading to the elimination of delay-related elements and the revelation of the exact range of the frequency spectra of the original system of interest. Then, by sweeping the frequency over this obtained range, the stability switching curves are declared exhaustively. Last, we deploy the cluster treatment of characteristic roots (CTCR) paradigm to reveal the exact and complete stability map. The proposed methodologies are tested and verified by a numerical method called Quasi-Polynomial mapping-based Root finder (QPmR) over an example case.

Analysis of Composite Blade/Casing Rub Stability Through Delayed Differential Equations

Abstract To improve aerodynamic efficiencies, the clearances between blades and casings are becoming smaller and smaller in the aero-engine industry, which might lead to the interactions between these components. These unexpected interactions are known as the so-called blade/casing rubs. Abradable materials are implemented on the inner surface of the casings to reduce the potential damages caused by it. However, failures may still arise from blade/casing rubs according to experimental investigations and actual accidents. In this paper, a reduced-order delayed differential equations (DDEs) are used to simplify the rubbing process between composite blade and casing. It is assumed that the removal of the abradable material in blade/casing rubbing process shares a resemblance with machine tool chatters encountered in machining. The DDEs are established with centrifugal stiffness and the impacts of stacking sequences on the blade damping taking into consideration. Semidiscretization method (SDM) is used to study the stabilities of the simplified system, which is verified by cluster treatment of characteristic roots (CTCR) and direct integrations. The results show that the stacking sequences, rub positions, blade damping, and stiffness could have much impact on the relatively dangerous interaction regimes. With the help of this method, one can assist the design processes of the composite blade-casing interface in initial aero-engine structural designs.

Analysis and design aspects of delayed resonator absorber with position, velocity or acceleration feedback

A thorough analysis of six delayed resonator configurations is performed. Considering either lumped or distributed delays, feedback is taken from acceleration, velocity or position measurements. Next to presenting stability maps derived by the Cluster Treatment of Characteristic Roots, the robustness in the vibration suppression is analyzed. It is shown that the deterioration of vibration suppression caused by mismatch between the design and true excitation frequencies can be decreased by a proper selection of the delayed resonator configuration and the frequency operable range. Besides, the force and energy demands from the resonator feedback are studied with respect to the absorber physical parameters. All the results are derived for a dimensionless model form and as such they are universally valid. Thus, with respect to the important design specifications, the analysis is complete and can be used as guidelines for parametrization of the given delayed resonator set, with respect to the operable conditions.

Stability Analysis for the DC Microgrid of Chained Communication Network with Cluster Treatment of Characteristic Roots Paradigm

The chained communication network is a common topology for the application of control strategies, in which distributed generators (DGs) interact successively in DC microgrids. Through the chained communication network of a distributed control strategy, the power allocation and the voltage maintenance of DC microgrids can be achieved without central controllers. However, the time delay issues are unavoidable and critical herein, which might lead to stability problem and lack clear investigations. The fact that the stability and performance of the overall system depend on the time delay renders its analysis paramount. For this purpose, the cluster treatment of characteristic roots (CTCR) is employed to reveal the effect of time delay on the chained communication network. As a result, an unstable scenario of DC microgrids with the chained communication network is detected. Besides, the relationship between the power output and time delay is investigated, which is helpful to the stability maintenance of DC microgrid with the consideration of physics and information simultaneously.

Multiple Time-delay Stability Analysis for the DC-Microgrid Cluster with Distributed Control

With the increasing availability of DC microgrids (MGs), the DC-MG cluster will develop quickly as the connection of different DC MGs. To alleviate the burden of the central control, the distributed control strategy is attracting more and more interests among researchers. Besides, the communication cost is obviously reduced with only one receiver of the global system information. However, the bidirectional neighboring communication will make the communication network more complex, which makes DC-MG cluster as a time-delay system. Moreover, the different time-delay values among neighboring MGs further deteriorate the whole system to a multiple time-delay issue. To maintain the stable operation of DC-MG cluster, the cluster treatment of characteristic roots (CTCR) is proposed in the work for the stability analysis considering the multiple time delays.

Stability Robustness for Secondary Voltage Control in Autonomous Microgrids With Consideration of Communication Delays

The future extensive usage of open communication networks for secondary voltage control in autonomous microgrids (MGs) inherently introduce time delays, which would degrade the system performance or even cause instability in the worst case. However, existing literatures have largely ignored these communication delays between the microgrid central controller and primary local controllers. To fill the knowledge gap, this paper proposes an analytical method to determine the stability robustness of secondary voltage control in MGs by taking into account the communication latency effect. Based on static output feedback, a closed loop small-signal model of an MG installed with proportional-integral secondary voltage controllers is first derived. Delay-dependent criteria are then formulated to detect the stability posture of MG system by tracing critical eigenvalues and cluster treatment of characteristic roots, in the forms of delay margin for a single delay and stability tableau over the delay domain for multiple delays, respectively. By a series of trial declarations, the qualitative impact of secondary controller gains on system stability regions against uncertain delays is investigated, from which it can be observed that generally the increase of integral gains would yield to a less delay-dependent stability of control system and the results can be utilized to guide the controller design for a compromise between the dynamic performance and stability robustness. The effectiveness of the proposed methodology is verified by a simulation study.

An Algebraic Approach for a Stability Analysis Methodology for Multiple Time-delay Systems

This paper presents an improvement on Cluster Treatment of Characteristic Roots (CTCR), which is a well-known methodology for delay-dependent stability analysis of multiple time-delay systems (MTDS). We propose an algebraic approach to extract the stability switching hypersurfaces in spectral delay space, instead of a numerical procedure in CTCR with Extended Kronecker Sum (EKS) operation. The proposed algebraic approach is based on an efficient zero location test, and the deployment of this test to an auxiliary characteristic polynomial whose unique properties have recently been revealed. The achieved improvement is demonstrated by applying the new CTCR procedure to a system with three delays.

Analytical and Experimental Study on Passive Stabilization of Thermoacoustic Dynamics in a Rijke Tube

This paper deals with passive stabilization of thermoacoustic dynamics in a Rijke tube using a Helmholtz resonator. Thermoacoustic instabilities result from the dynamic coupling between the heat release and pressure in a chamber. Helmholtz resonators are used akin to vibration absorbers to suppress unwanted pressure oscillations in such structures and prevent instabilities. The first contribution of the paper is a state-space representation of the thermoacoustic dynamics for the resonator-mounted Rijke tube. This relationship happens to be in the class of linear time invariant, neutral multiple time delay systems (LTI-NMTDS). Then, benefiting from the cluster treatment of characteristic roots (CTCR) paradigm, we investigate the effect of resonator location on suppression of thermoacoustic instability. CTCR is a mathematical tool that determines the stability of LTI-NMTDS exhaustively and nonconservatively in the parameter space of the system. This capability provides a novel tool for the futuristic design concepts of combustors. These analytically obtained findings are also supported with experimental results from a laboratory-scale Rijke tube. In addition, a conceptual case study is presented where the stabilizing contributions of the resonator to the dynamics are investigated under strong thermoacoustic coupling.

Stability Analysis of a Human-in-the-Loop Telerobotics System with Two Independent Time-Delays

In this paper, stability of a human-in-the-loop telerobotics system with force feedback and communication delays is investigated. A general linear time-invariant time-delayed mathematical model of the human operator is incorporated into the system dynamics based on the interaction of the human operator with the rest of the telerobotic system. The resulting closed loop dynamics contains two independent time-delays mainly due to back and forth communication delay and human reaction time delay. Stability of this dynamics is characterized next on the plane of the two delays by rigorous mathematical investigation using Cluster Treatment of Characteristic Roots (CTCR). An illustrative numerical example is further provided in the results section along with interpretations.

Stability analysis for LTI systems with multiple time delays using the bounds of its imaginary spectra

The stability of linear time invariant (LTI) systems with multiple independent time delays and the cluster treatment of characteristic roots (CTCR) paradigm are investigated from a new perspective. Any delay composition that results in an imaginary characteristic root lies either on a small number of kernel hypersurfaces (KH) or their infinitely many offspring hypersurfaces (OH). The complete description of KH is the only prerequisite for the CTCR-based stability assessment procedure. As the number of delays increases, however, the determination of these KH becomes computationally costly. Instead, we present a practical procedure to extract the 2-D cross-sections of the KH set in the domain of the two arbitrarily selected delays. First, we determine the exact upper and lower bounds of the imaginary spectra in this 2-D cross-section of interest. This process starts with a half-angle tangent representation of the characteristic equation followed by the Dixon resultant operation. Based on the full knowledge of KH, the CTCR paradigm creates the complete stability map in the domain of these two arbitrarily selected delays. We demonstrate the effectiveness of this methodology over an example case study.

Delayed Feedback Control Laws for Rijke Tube Thermoacoustic Instability, Synthesis, and Experimental Validation

We approach the thermoacoustic instability problem in Rijke tubes from a mathematically novel perspective. In this benchmark experimental setting, the complex dynamic exchange between the unsteady heat release and the acoustic pressure variations creates the instability. When linearized, this behavior leads to a neutral class time-delayed dynamics with multiple independent delays, where our major contribution arrives. We aim to synthesize a series of stabilizing feedback control strategies for this dynamics. A recent mathematical paradigm, called the cluster treatment of characteristic roots (CTCR), is used to facilitate this objective. This paradigm, in essence, declares the necessary and sufficient stability conditions in the space of the system and controller parameters. The main contributions of this brief are in the first-time deployment of CTCR in the syntheses of various feedback control laws and validating experiments.

Delayed Resonator With Distributed Delay in Acceleration Feedback—Design and Experimental Verification

The main results of this paper are design, analysis, and experimental verification of a delayed resonator with a distributed-time-delay acceleration feedback. Analogously to the classical delayed resonator, the novel resonator is applied to suppress vibrations of a mechanical system. The theoretical part of this paper includes control parameter design and analysis of their ranges. Next, dynamical properties of the resonator are analyzed including the derivation of a complete stability picture for the combined resonator with the single-degree-of-freedom mechanical system. For the stability analysis, spectral methods are applied including the method of cluster treatment of characteristic roots. Utilizing the distributed delay instead of the lumped delay on the acceleration feedback, both the resonator and the overall system dynamics are converted from neutral to more convenient retarded character. Another positive aspect of the distributed delay is that it acts like a moving average filter on the acceleration measurements. Furthermore, we perform a computational analysis of spectral properties using a numerical root-finding QPmR algorithm. All these stability findings are also verified experimentally on a laboratory setup designed and implemented for this purpose.

Bounds of imaginary spectra of LTI systems in the domain of two of the multiple time delays

The stability of linear time invariant (LTI) systems with independent multiple time delays and the cluster treatment of characteristic roots (CTCR) paradigm are investigated from a new perspective. It is known that for such systems, all the imaginary characteristic roots can be detected completely on a small set of hypersurfaces in the domain of the delays (Sipahi and Olgac, 2006a). They are called kernel hypersurfaces (KH). The complete description of KH is the only prerequisite for the CTCR stability assessment procedure. As the number of delays increases, however, their evaluation becomes infeasible. Instead, we present a procedure to extract the 2-D cross-sections of these hypersurfaces in the domain of any two of the delays by fixing the remaining delays. In the 2-delay domain of interest, the exact upper and lower bounds of the imaginary spectra are determined. For this, a combination of half-angle tangent representation of the characteristic equation and the Dixon resultant theory is used as the main contributions of this paper. The complete KH are obtained by sweeping the root crossing frequency in this interval. Using this knowledge CTCR creates the cross-section of the stability map in the domain of the two arbitrarily selected delays. We demonstrate the effectiveness of this methodology over an example case study with three independent delays and two commensurate ones.

A study of Helmholtz resonators to stabilize thermoacoustically driven pressure oscillations.

This paper studies passive control of thermoacoustic instabilities from an unconventional mathematical perspective. These instabilities are notoriously known to result from the complex dynamic exchange between the unsteady heat release and the acoustic waves within a finite volume such as a combustor. One possible passive control strategy is to utilize Helmholtz resonators. Under certain simplifications, the ensemble combustion dynamics including the resonators reduces to a linear-time invariant-multiple time-delayed system (LTI-MTDS). As the main contribution of the paper, an exact analytical procedure is proposed to determine the placement of the resonators to avoid instabilities. A unique mathematical paradigm, called the cluster treatment of characteristic roots, is used to accomplish this task. It declares exactly the necessary and sufficient stability conditions for an LTI-MTDS in the space of the system parameters. This concept paper is written with the mindset that this analytical tool can invite yet unexplored design capabilities for similar noise control applications where acoustic dampers are used.

Stability and Performance Analysis of Time-Delayed Actuator Control Systems

Time delay is a common phenomenon in robotic systems due to computational requirements and communication properties between or within high-level and low-level controllers as well as the physical constraints of the actuator and sensor. It is widely believed that delays are harmful for robotic systems in terms of stability and performance; however, we propose a different view that the time delay of the system may in some cases benefit system stability and performance. Therefore, in this paper, we discuss the influences of the displacement-feedback delay (single delay) and both displacement and velocity feedback delays (double delays) on robotic actuator systems by using the cluster treatment of characteristic roots (CTCR) methodology. Hence, we can ascertain the exact stability interval for single-delay systems and the rigorous stability region for double-delay systems. The influences of controller gains and the filtering frequency on the stability of the system are discussed. Based on the stability information coupled with the dominant root distribution, we propose one nonconventional rule which suggests increasing time delay to certain time windows to obtain the optimal system performance. The computation results are also verified on an actuator testbed.

Passive suppression of thermoacoustic instability in a Rijke tube

This article investigates passive stabilization of thermoacoustic dynamics in a Rijke tube by using a Helmholtz resonator. Under certain assumptions, the mathematical model of the dynamics is shown to fall into the linear time-invariant multiple time delay systems class of neutral type. To study the thermoacoustic instability phenomenon and its passive suppression, the cluster treatment of characteristic roots (CTCR) paradigm is implemented. This effort reveals the placement of Helmholtz resonator along the Rijke tube for stabilizing the unstable dynamics. The analytically obtained results are compared with experiments conducted on a laboratory scale Rijke tube. In addition, the effect of resonator’s design parameters on its stabilization capability is investigated.