Unstable Eigenvalues Research Articles

Shape Optimization of an Industrial Aeroengine Combustor to reduce Thermoacoustic Instability

Abstract We parameterize the geometry of an industrial aeroengine combustor using Free Form Deformation (FFD). We then define the thermoacoustic system parameters and impose the acoustic boundary conditions to calculate the thermoacoustic eigenmode of the model using an open source parallelized Helmholtz solver. We then use adjoint methods to calculate the shape derivatives of the unstable eigenvalue with respect to the shape parameters. First we calculate the sensitivities with respect to surface movements. Second, we calculate the sensitivities with respect to the FFD control points. We modify the FFD control point positions in order to reduce the thermoacoustic growth rate until the mode considered is stable. These findings show how, when combined with other constraints, this method could be used to reduce combustion instability in industrial annular combustors through geometric modifications.

Mechanism analysis on a type of limit induced oscillation for single‐machine infinite‐bus system of PMSG with positive damping

AbstractThe mechanism of oscillations that occur in the system connected to renewable energy sources is complex, generally including negative damping oscillation, limit induced oscillation etc. Combining with the dynamic characteristics of the low‐dimensional system after limit working, this paper analyses the mechanism of a type of limit induced oscillation that occurs when the output of the DC voltage outer loop continuously touches the limit for single‐machine infinite‐bus system of permanent magnet synchronous generator with positive damping under a large disturbance. Specific study as follows: firstly, a non‐smooth state space model of single‐machine infinite‐bus system of permanent magnet synchronous generator with nonlinear state limit (clarified as a Filippov system) is established. Secondly, the small disturbance characteristics of the system are analysed when the limit does not work. Thirdly, the piecewise dynamic description of the system is analysed with the limit working, and the reason that the final contraction of the system to an eight‐dimensional manifold is explained based on mathematical derivation and physical characteristics. Finally, the characteristics of equilibrium point of the system in a low‐dimensional manifold are analysed when the limit continuously working, revealing that the existence of a pair of conjugate unstable eigenvalues in the low‐dimensional system is the reason of system oscillation.

On the inviscid instability of the 2-D Taylor–Green vortex

We consider Euler flows on two-dimensional (2-D) periodic domain and are interested in the stability, both linear and nonlinear, of a simple equilibrium given by the 2-D Taylor–Green vortex. As the first main result, numerical evidence is provided for the fact that such flows possess unstable eigenvalues embedded in the band of the essential spectrum of the linearized operator. However, the unstable eigenfunction is discontinuous at the hyperbolic stagnation points of the base flow and its regularity is consistent with the prediction of Lin (Intl Math. Res. Not., vol. 2004, issue 41, 2004, pp. 2147–2178). This eigenfunction gives rise to an exponential transient growth with the rate given by the real part of the eigenvalue followed by passage to a nonlinear instability. As the second main result, we illustrate a fundamentally different, non-modal, growth mechanism involving a continuous family of uncorrelated functions, instead of an eigenfunction of the linearized operator. Constructed by solving a suitable partial differential equation (PDE) optimization problem, the resulting flows saturate the known estimates on the growth of the semigroup related to the essential spectrum of the linearized Euler operator as the numerical resolution is refined. These findings are contrasted with the results of earlier studies of a similar problem conducted in a slightly viscous setting where only the modal growth of instabilities was observed. This highlights the special stability properties of equilibria in inviscid flows.

Stages and main tasks of the century-long control theory and system identification development. Part VI. Model predictive control

The article is devoted to one of the relatively new areas in the field of control theory and practice, called model predictive control. The ideas underlying it, the connection with the theory of optimal control and numerical methods for solving optimization problems are described. It should be noted that currently the number of problems solved and control systems developed within the framework of this approach has come out on top in comparison with all others. This is evidenced by numerous publications concerning the theory and practical application of control methods and algorithms using a predictive model. The main distinctive feature of this method is that the control is not synthesized in advance, but is calculated directly during the operation of the system, i.e. at the current time. This made it possible, with such control, to implement original feedback that ensures stabilization of the process when the model is inaccurate and there is uncertainty associated with existing disturbances and measurements. It describes how such feedback is organized based on current measurement data. In this case, there is no need to interpret somehow the nature of the disturbances and errors acting on the system. Much attention in the article is paid to the possibilities that are admitted with this method of control. The model predictive control method seems especially attractive for systems with discrete impulse processes that are well described with using cognitive maps. It is considered an example of cognitive modeling of evolutionary demographic processes in Ukraine and the possibility of control them using a predictive model, when the model contains unstable eigenvalues. When solving problems of predictive control, it is proposed to use trajectory models, which directly follow from the Cauchy formula, in particular for discrete systems, instead of a local description in the state space. This makes it possible to take into account, when solving predictive control problems, how well or poorly controlled, observable or identifiable the system is. Non-standard capabilities of this method with a trajectory description appear during terminal control on a sliding interval with a finite horizon. A number of original formulations of terminal control problems are presented in the article and some of their properties are described.

Stabilization of SIMPLE-like RANS solvers for computing accurate gradients using the complex-step derivative method

The Reynolds-averaged Navier-Stokes (RANS) approach in Computational Fluid Dynamics (CFD) is increasingly vital for aerodynamic design across wind turbines, gas turbines, aircraft, and rotorcraft. Enhancing the process through CFD-based design optimization requires numerous design variables, favoring gradient-based methods for better scalability than gradient-free methods. This research uses numerical differentiation techniques, particularly the complex-step derivative method, to compute gradients. Challenges arise during aerodynamic shape optimization when unconventional shapes disrupt RANS solver assumptions, causing convergence failures. These failures undermine optimization, prompting the need to enhance solver convergence for robust optimization. The modified-Boostconv method is a residual recombination method bolstering unstable eigenvalues to stabilize convergence of iterative solvers for nonlinear systems of equations. This study extends the modified-Boostconv method to combine it with the complex-step derivative technique, creating robust optimizations via RANS-CFD with accurate gradients, even for these numerically unstable cases. The main issue encountered during the complexification of the modified-Boostconv method is how to correctly ensure that the model reduction leading to the least-squares problem satisfies the complex-step derivative method. We test whether the dot product operation involved in the model reduction should be a Hermitian or non-Hermitian inner product. The problem is first tested for a simple analytical case using the logistic equation in combination with the modified-Boostconv method. It shows that the non-Hermitian inner product should be used. This is also confirmed with a similar study using the RANS solver.

Symmetry and instability of marginally outer trapped surfaces

We consider an initial data set having a continuous symmetry and a marginally outer trapped surface (MOTS) that is not preserved by this symmetry. We show that such a MOTS is unstable except in an exceptional case. In non-rotating cases we provide a Courant-type lower bound on the number of unstable eigenvalues. These results are then used to prove the instability of a large class of exotic MOTSs that were recently observed in the Schwarzschild spacetime. We also discuss the implications for the apparent horizon in data sets with translational symmetry.

Exact delay range for the stabilization of linear systems with input delays

This paper is concerned with the exact delay range making input‐delay systems unstabilizable. The exact range means that the systems are unstabilizable if and only if the delay is within this range. Contributions of this paper are to characterize the exact range and to present a computation method to derive this range. It is shown that the above range is related to unstable eigenvalues of the system matrix. In the discrete‐time case, if none of the eigenvalues of the system matrix is a unit root, then the above range is a finite set. If there exist some eigenvalues which are unit roots, this range may be a finite set or may be composed of several arithmetic progressions. When this range contains finite elements, the number of these elements is bounded by the geometric multiplicities of eigenvalues. When this range contains arithmetic progressions, the number of such progressions is bounded by the above multiplicities. On the other hand, our results can provide an upper bound for the well‐known delay margin, which is the maximal delay value achievable by a robust controller to stabilize systems.

Flow and heat transfer in annuli owing to inner shrinking and outer stationary cylinder

The scope of the present work is to investigate the motion of the fluid flow and heat exchange within two concentric cylinders. It is assumed that the inner cylinder is moving (shrunk) and the outer cylinder is stationary. It derives the physical phenomenon, which can find industrial practical applications. The gap between the cylinders, curvature of cylinders, and Prandtl number are three physical parameters controlling the hydro-thermal flow motion. Asymptotic solutions are available when the gap size is moderately small. These and other solutions based on the numerical simulations clearly indicate that there are critical gap sizes beyond which either no solutions or dual solutions are possible. The common salient characteristic of solutions is that the axial fluid dragged into the cylinder as a result of surface shrinkage is thrown radially out through the outer cylinder. The non-existence of solutions after identifying critical gap sizes explains why the shrinking sheet and shrinking cylinder problems with an unbounded flow zone could not have been treated in the literature so far. This is the prime reason that with infinite gap size, other controlling mechanisms like surface transpiration, unsteadiness, velocity slip or magnetic field need to be included in the mathematical model to get a solvable problem under the surface shrinking process often studied among researchers. One of the solutions called branch I solution, has rational profiles with small pertinent physical quantities like skin friction, vertical/radial thrown velocity, and heat transfer rate. However, from a stability analysis, it can be shown that the branch II solution, possesses comparatively larger such quantities, making them questionable to appear in real physics with unstable eigenvalues.

Numerical spectral analysis of standing waves in quantum hydrodynamics with viscosity

<abstract><p>We study the spectrum of the linearization around standing wave profiles for two quantum hydrodynamics systems with linear and nonlinear viscosity. The essential spectrum for such profiles is stable; we investigate the point spectrum using an Evans function technique. For both systems we show numerically that there exists a real unstable eigenvalue, thus providing numerical evidence for spectral instability.</p></abstract>

Performance analysis for cyber–physical systems under two types of stealthy deception attacks

This paper focuses on the performance analysis of cyber–physical systems under the stealthy deception attack. Two types of attack models, strictly stealthy attack and ϵ-stealthy attack, are considered. The strictly stealthy deception attack model is firstly formulated based on the difference equations of the normal and compromised systems. We then present a necessary and sufficient condition for the existence of the strictly stealthy deception attack. The design method for the attacker to achieve the strictly stealthy deception attack is also provided. Considering that the strictly stealthy attack strategy is associated with the undetectable points of the difference equations, we propose a computationally efficient algorithm to find such points. It is shown that these points form a subspace which can be obtained in finite recursion steps. The corresponding defense strategy is further designed by avoiding the initial state falling into the subspace. For ϵ-stealthy attack, a necessary condition is provided for its existence. Furthermore, due to the fact that the difference system matrix does not have an unstable eigenvalue, it is proved that the ϵ-stealthy attack does not exist. Simulation examples are presented to illustrate the theoretical results.

Decentralized False-Data Injection Attacks Against State Omniscience: Existence and Security Analysis

This paper focuses on how false-data injection (FDI) attacks compromise state omniscience, which needs each node in a jointly detectable sensor network to estimate the entire plant state through distributed observers. To reveal the global vulnerability of state omniscience, we investigate decentralized false-data injection (DFDI) attacks that destabilize the estimation error dynamics but eliminate their influences on the residual in each sensor node. Firstly, the sufficiency and necessity for the existence of such attacks are studied from system eigenvalues and attackable sensors. Secondly, the self-generated DFDI attack sequences independent of system real-time data are designed to achieve the attack objective with elaborate parameters. Especially, the DFDI attack sequences are improved to maintain real values even if the system matrix only has unstable imaginary eigenvalues. Finally, we analyze the secure range for observer interaction weights and the sensor protection scheme to guarantee the security of state omniscience under DFDI attacks. The theoretical results for DFDI attacks are demonstrated with the linearized discrete-time model of an aircraft system.

Bifurcation analysis and control strategy for a car-following model considering jerk behavior

To further improve the adaptability of the optimal velocity model (OVM) in the actual traffic flow, this paper proposes a car-following model that considers both the jerk effect and time-delayed feedback control. To study the dynamic behavior of this model, the following analysis is carried out: A stability conditions and equilibrium points controlling OVM are obtained by stability analysis and Hopf bifurcation analysis; Using the intuitive and accurate definite integral stabilization method (DIS), by calculating the number of all unstable eigenvalues of the characteristic equation, we obtain a stable time-delayed interval, and a time-delayed feedback controller is designed to stabilize the traffic system. The results show that the time-delayed feedback controller expands stable time-delayed interval, suppresses the influence of unstable factors on the traffic system, and improves stability of vehicle traffic; Numerical simulations agree with the theoretical results, indicating that the time-delayed feedback controller can effectively suppress traffic congestion and control the managed traffic.

A new stability criterion and its application to robust stability analysis for linear systems with distributed delays

Recently, the possibility to verify positive definiteness of Lyapunov–Krasovskii functionals only on a specific Razumikhin-type set of functions to conclude on the stability of linear time delay systems was shown. In this paper, for a class of systems with multiple concentrated and distributed delays, we extend the result and prove that this specific set may be applied while verifying the negative definiteness of the functionals derivatives along the solutions as well. The result is applied in the robustness analysis with respect to uncertainties both in the system matrices and in the delays. It provides a simple way to prove the robust stability conditions due to obviating the need to work with indefinite derivatives of functionals along the solutions of a perturbed system. As a by-product, the estimates for an unstable eigenvalue of the system are derived, and the functionals defined on complex-valued initial functions are presented.

Results on the Spectral Stability of Standing Wave Solutions of the Soler Model in 1-D

We study the spectral stability of the nonlinear Dirac operator in dimension $1+1$, restricting our attention to nonlinearities of the form $f(\langle\psi,\beta \psi\rangle_{\mathbb{C}^2}) \beta$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-i\omega t} \phi_0$. For the case of power nonlinearities $f(s)= s |s|^{p-1}$, $p>0$, we obtain a range of frequencies $\omega$ such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition $\langle\phi_0,\beta \phi_0\rangle_{\mathbb{C}^2} > 0$ characterizes groundstates analogously to the Schr\"odinger case.

A Quasi-Exponential Transform for Unstable Modes Calculation of Large-Scale Power Systems

The calculation of the unstable eigenvalues without omission is vital in small signal stability analysis of power systems. In this letter, a quasi-exponential transform is proposed to compute all unstable eigenvalues with improved efficiency and accuracy. Numerical experiments on two public source systems, namely the Xingo6u and Juba5723 systems, show the effectiveness of the proposed method.

Gradient-Based Optimization of Chaotic Panel Flutter

This work considers the gradient-based optimization of an aeroelastic panel structure undergoing chaotic vibrations. A previously presented adaptive time-marching scheme is used to stabilize the design sensitivities that would normally grow unbounded in time, owing to intermittent unstable eigenvalues of the Jacobian. The design optimization entails forcing the root mean square (RMS) of the panel vibration to some specified value, in a way that minimizes the deviation of the structural thickness variables from their uniform baseline values. Depending on the location of the RMS constraint, the output physics may be chaotic, limit cycle oscillations, or statically buckled.

Stability of the aneurysm-type solution in a membrane tube with localized wall thinning filled with a fluid with a non-constant velocity profile

We perform the stability analysis of bulging localized structures on the wall of a fluid-filled axisymmetric membrane elastic tube. The wall of the tube is assumed to be subjected to localized thinning. The problem has no translational invariance anymore, hence the stability of a bulging wave centered in the point of the localization of imperfection is usual, and not orbital stability up to a shift as in the case of translationally invariant governing equations. Motionless wave solutions of the governing equations having the form of a localized bulging are called throughout the paper aneurysm solutions. We assume that the fluid is subjected to the power law for viscous friction of a non-Newtonian fluid, and also that the viscosity of the fluid does not play a significant role and it can be neglected. The velocity profile remains not constant along the cross section of the tube (even in the absence of the viscosity) because no-slip boundary conditions are performed on the tube walls. Stability is established by demonstrating the non-existence of the unstable eigenvalues with a positive real part of the linearized problem. This is achieved by constructing the Evans function depending only on the spectral parameter, analytic in the right half of the complex plane Ω+ and which zeroes in Ω+ coincide with the unstable eigenvalues of the problem. The non-existence of zeroes of the Evans function is performed using the argument principle from the analysis of complex variable. Finally, we discuss the possibility of applying of the results of the present analysis to the aneurysm formation in damaged human vessels under the action of the internal pressure.

Event-Triggered Stabilizing Bit Rate Conditions for an n-Dimensional Linear System With i.i.d. Feedback Dropouts.

This article aims to stabilize an n -dimensional linear time-invariant (LTI) system, whose feedback packets are transmitted through a digital communication network. The digital network suffers from network delay and independent and identically distributed (i.i.d.) feedback dropouts, which may destabilize the system. The coupling among multiple state variables may further harm the stability of the system. In order to deal with these issues and save the occupied bandwidth of the feedback network, we propose a periodic event-triggering strategy. In our strategy, the state is measured periodically, but only quantized and transmitted when a certain condition is triggered. By well balancing the state coupling and making full use of both the information inside transmitted feedback packets and the one carried by sampling time instants, our strategy can maintain the desired mean square stability at a lower bit rate than conventional periodic sampling policies. The obtained stabilizing bit rate conditions are determined by the processing and network delays, the dropout rate, and the unstable eigenvalues of the system matrix, but independent of the process noise. Moreover, the lack of the direct state access does not incur any additional stabilizing bit rate. Simulations are done to confirm the effectiveness of the obtained stabilizing bit rate conditions.

Damping control strategy of MMC-HVDC connecting to the PMSM-based wind farm for subsynchronous resonance suppression

It is a typical power electronics based system when flexible high voltage direct current (HVDC) converter station connects to the permanent magnet synchronous machine (PMSM) based wind farm. This system may have the risk of subsynchronous resonance needed to be studied. Firstly, the linear models of modular multilevel converter (MMC) considering the inner dynamics and the grid side voltage source converter (GSVSC) considering the dynamics of phase-locked loop (PLL) are established. Secondly, the participation variables and participation factors are used to identify the key parts affecting the unstable eigenvalues. Thirdly, a damping control strategy simulating the performance of paralleled impedance at the points of common coupling (PCC) is proposed to suppress the subsynchronous resonance. And then, the parameter selecting ranges of damping controller are analytically calculated. Finally, the effectiveness of proposed damping control strategy and the correctness of parameter selection are verified by the electromagnetic transient simulation models

Stabilization Strategy of a Novel Car-Following Model with Time Delay and Memory Effect of the Driver

In this paper, a novel car-following model is established integrating the drivers’ memory of previous information. The drivers’ memory of the vehicle ahead is introduced as an influencing factor on the drivers’ expected behavior. The time delay feedback control term is added to the model to increase the stability interval of the system. By comparing the stability intervals of the controlled and uncontrolled models, the necessity of adding a delay feedback control item is demonstrated. The validity and feasibility of the time delay feedback control strategy are proved by numerical simulations. In this paper, the stability interval of the system is determined by the definite integral stability method (DISM) and the Hopf bifurcation analysis method. According to the number of unstable eigenvalues derived from the system eigenvalue equation, the appropriate time delay feedback control parameters are set. By choosing the optimal parameters, the new model can optimize the traffic flow to the maximum extent, eliminate the stop-and-go of vehicles, and make the traffic stable. Numerical examples close to actual traffic conditions are given to verify the feasibility of the control strategy using the verified design steps. Next generation simulation (NGSIM) measurements are used to conduct parameter calibration of the new model.