Stable Eigenvalues Research Articles

Adaptive Vibration Isolation for Flexible Structures

Vibration isolation control for flexible structures restricts the response resulting from external disturbances to areas not requiring high precision positioning and/or pointing. This paper introduces adaptive feedback isolation controllers, based on Lyapunov theory, that regulate and allow tracking of the undisturbed (controlled) coordinates in a flexible structure. Under assumptions of inertially decoupled controlled and uncontrolled coordinates, symmetric and positive definite mass matrix for the controlled subsystem, asymptotically stable eigenvalues for the uncontrolled subsystem, and bounded disturbances, an adaptive regulator asymptotically drives the controlled coordinates to zero. Under similar assumptions, an adaptive tracking algorithm drives the controlled coordinates to desired time trajectories. Experimental results on a three mass system compare the response of the adaptive isolation controllers with standard PID control. The adaptive regulator provides faster transient decay than PID control using the same control effort. The adaptive tracking controller has the same tracking error as PID using 30 percent less control effort.

Noise and O(1) amplitude effects on heteroclinic cycles.

The dynamics of structurally stable heteroclinic cycles connecting fixed points with one-dimensional unstable manifolds under the influence of noise is analyzed. Fokker-Planck equations for the evolution of the probability distribution of trajectories near heteroclinic cycles are solved. The influence of the magnitude of the stable and unstable eigenvalues at the fixed points and of the amplitude of the added noise on the location and shape of the probability distribution is determined. As a consequence, the jumping of solution trajectories in and out of invariant subspaces of the deterministic system can be explained. (c) 1999 American Institute of Physics.

The Solution of Singular Linear Difference Systems under Rational Expectations

Many linear rational expectations macroeconomic models can be cast in the first-order form, AE[subscript t]y[subscript t + 1] = By[subscript t] + CE[subscript t]x[subscript]t, if the matrix A is permitted to be singular. The authors show that there is a unique stable solution under two requirements: (1) the determinantal polynomial Az - B is not zero for some value of z, and (2) a rank condition. The unique solution is characterized using a familiar approach: a canonical variables transformation separating dynamics associated with stable and unstable eigenvalues. In singular models, however, there are new canonical variables associated with infinite eigenvalues. These arise from nonexpectational behavioral relations or dynamic identities present in the singular linear difference system. Copyright 1998 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Canonical formalism and quantization for a class of classical fields with application to radiative atomic decay in dielectrics

For the description of spectral and radiative decay properties of atoms or molecules, placed in a photonic material, the electromagnetic field in the material must be quantized and active research is taking place in this area. Here a unified account is given of such quantization procedures. Led by the Maxwell example, we consider the canonical formalism and its quantization for a class of linear evolution equations ${\ensuremath{\partial}}_{t}F=NF\ensuremath{-}G$, obeying a conservation law for $G=0$. If $N$ has a nonempty null space (zero is an eigenvalue with associated nonpropagating solutions), an abstract form of the gauge concept makes its appearance and generalizations of the familiar Coulomb and Lorentz gauges are obtained. A canonical formalism is set up and quantized. The application to spatially inhomogeneous nonconducting electrodynamical systems is immediate, including the interaction with matter. Next atomic decay in a medium is considered, in particular in the presence of band gaps. For a simple two-level model with transition frequency in the gap, single-photon decay is inhibited and also a different stable eigenvalue of the combined system is found. An open problem in connection with random dielectrics, showing Anderson localization, is discussed. Finally, a mass renormalization, by means of a Kramers transformation, is presented. In general, the renormalized mass is no longer a scalar quantity.

When is OGY Control More Than Just Pole Placement

Deterministic chaotic systems give rise to very complex dynamical behavior in spite of the fact that they are usually of very low dimension. The exceptionally high sensitivity to changes in initial conditions makes dealing with these systems non-trivial due to inherent measurement errors and limited numerical precision. Within a chaotic attractor it is usually possible to identify unstable point attractors and unstable periodic orbits. One of the widely accepted techniques for stabilizing unstable point attractors and periodic orbits is the so called OGY controller method developed by Ott, Grebogi, and Yorke. In the case of stabilizing an unstable point attractor within the chaotic map, the OGY controller effectively forces the system to move on the corresponding stable manifold in the state-space. On the other hand, it can be shown that this strategy coincides with the design of a full state feedback controller that places the unstable eigenvalues of the state matrix of the locally linearized system at zero while keeping the location of the stable eigenvalues unchanged. When the stabilization of the original, open-loop system is the only goal, the OGY approach does not offer any advantage over other stabilizing control actions such as pole placement to arbitrary but stable eigenvalue locations. In addition, neither OGY nor arbitrary pole placement explicitly deal with the problem of noise sensitivity. Hence, when the original chaotic system is noise corrupted, it is not clear if the OGY approach is even close to being optimal with respect to noise cancelation with "acceptable" control action. Although the answer to this question is negative in the general case, there are situations where the optimal system behavior is achieved. The contribution of this paper is twofold. The first contribution is the rigorous discussion of the relationship between the OGY approach and the pole placement and the extension of the OGY method to stabilization of systems with more than one unstable eigenvalue. The second contribution of the paper is the proof that the OGY control approach to the equilibrium point stabilization in the presence of persistent, magnitude bounded process noise is actually optimal when the system performance is measured by the l∞-norm of the control signal. The proof is based on properties of the optimal l1-norm controller design and on the parameterization of all stabilizing controllers for a given linear system.

A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow

A systems theory framework is presented for the linear stabilization of two-dimensional laminar plane Poiseuille flow. The governing linearized Navier-Stokes equations are converted to control-theoretic models using a numerical discretization scheme. Fluid system poles, which are closely related to Orr-Sommerfeld eigenvalues, and fluid system zeros are computed using the control-theoretic models. It is shown that the location of system zeros, in addition to the well-studied system eigenvalues, are important in linear stability control. The location of system zeros determines the effect of feedback control on both stable and unstable eigenvalues. In addition, system zeros can be used to determine sensor locations that lead to simple feedback control schemes. Feedback controllers are designed that make a new fluid-actuator-sensorcontroller system linearly stable. Feedback control is shown to be robust to a wide range of Reynolds numbers. The systems theory concepts of modal controllability and observability are used to show that feedback control can lead to short periods of highamplitude transients that are unseen at the output. These transients may invalidate the linear model, stimulate nonlinear effects, and/or form a path of ‘bypass’ transition in a controlled system. Numerical simulations are presented to validate the stabilization of both single-wavenumber and multiple-wavenumber instabilities. Finally, it is shown that a controller designed upon linear theory also has a strong stabilizing effect on two-dimensional finite-amplitude disturbances. As a result, secondary instabilities due to infinitesimal three-dimensional disturbances in the presence of a finite-amplitude two-dimensional disturbance cease to exist.

High-mode-number ballooning modes in a heliotron/torsatron system. II. Stability

As described in the companion paper [N. Nakajima, Phys. Plasmas 3, 4545 (1996)], in heliotron/torsatron systems that have a large Shafranov shift, the local magnetic shear is found to have no stabilizing effect on high-mode-number ballooning modes at the outer side of the torus, even in the region where the global shear is stellarator-like in nature. The disappearance of this stabilization, in combination with the compression of the flux surfaces at the outer side of the torus, leads at relatively low values of the plasma pressure to significant modifications of the stabilizing effect due to magnetic field-line bending on high-mode-number ballooning modes—specifically, that the field-line bending stabilization can be remarkably suppressed or enhanced. In an equilibrium that is slightly Mercier-unstable or completely Mercier-stable due to peaked pressure profiles, such as those used in standard stability calculations, high-mode-number ballooning modes are destabilized due to these modified stability effects, with their eigenfunctions highly localized along the field line. Highly localized mode structures such as these cause the ballooning mode eigenvalues ω2 to have a strong field line dependence (i.e., α-variation) through the strong dependence of the local magnetic curvature, such that the level surfaces of ω2(ψ,θk,α) (≤0) become spheroids in (ψ,θk,α) space, where ψ labels flux surfaces and θk is the radial wave number. Because the spheroidal level surfaces for unstable eigenvalues are surrounded by level surfaces for stable eigenvalues of high-mode-number toroidal Alfvén eigenmodes, those high-mode-number ballooning modes never lead to low-mode-number modes. In configuration space, these high-mode-number modes are localized in a single toroidal pitch of the helical coils, and hence they may experience substantial stabilization due to finite Larmor radius effects.

Smooth stabilization of nonlinear control systems

This paper provides the framework for a general approach for constructing smooth stabilizing feedback for smooth nonlinear systems when it is possible to construct such a feedback. Algorithms for determining the local smooth asymptotic stabilizability of two specific classes of smooth nonlinear control systems are developed. These two classes of systems are identified with systems whose linearizations possess either one zero eigenvalue or a pair of purely imaginary eigenvalues. The local stabilizability of any critical nonlinear system in either of these classes may be tested once the system's linearization is block diagonalized with respect to the critical and stable eigenvalues.

Sources of Error in Eigenvalue Spectra for Pipe-Poiseuille Flows

An operational Chebyshev Tau approach was used to establish temporal eigenvalue stability spectra for pipe-Poiseuille flows, using the Sexl equation which was transformed to remove the singularity at the origin. Substantial deviations were noted between the long-standing results of Davey and Drazin (1969) which are scattered and display branching patterns, and those of the present work which manifest a more orderly progression of the stable high-order eigenvalues. Formal distinctions are made between errors arising from round-off, basis truncation, and poor choice of basis function: the inaccuracies in Davey and Drazin’s data are thought to have arisen from the latter two effects.

Optimal entire eigenstructure assignment of discrete-time linear systems

An improved method which assigns the closed-loop eigenvalues of a discrete-time linear system in desired preselected stable locations and which simultaneously selects those eigenvectors which satisfy a quadratic cost criterion with suitable weighting matrices is presented. The optimal feedback gain-matrix is determined without solving the algebraic matrix Riccati equation. The proposed explicit solution of the Riccati equation is feasible for both real and complex closed-loop stable eigenvalues.

Limiting Exit Location Distributions in the Stochastic Exit Problem

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point S. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of S. We analyze the case when, as $\epsilon\to0$, the exit location on the boundary $\partial\Omega$ is increasingly concentrated near a saddle point H of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at H. If $\mu < 1$, then the exit location distribution is generically asymptotic as $\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the ${\cal O}(\epsilon^{\mu/2})$ lengthscale near H. If $\mu > 1$, it is generically asymptotic to a distribution on the ${\cal O}(\epsilon^{1/2})$ lengthscale, whose moments we compute. Our treatment employs both matched asymptotic expansions and stochastic analysis. As a byproduct of our treatment, we clarify the limitations of the traditional Eyring formula for the weak-noise exit time asymptotics.

On an eigenvalue problem arising in the study of the stability of ocean currents

We study a singular Sturm-Liouville eigenvalue problem which arises in studying the stability of rotating shallow-water shear flows on an equatorial β-plane. Earlier studies have shown that the problem has only stable eigenvalues if a certain parameter φ exceeds 3/4. On the other hand, numerical studies have found unstable eigenvalues if φ is sufficiently small. In this note, we give an analytical proof that the bound of 3/4 is indeed sharp, i.e. there are instabilities for any φ < 3/4.

Stable electronic energy levels in the presence of off‐diagonal disorder

AbstractIt is demonstrated that the topological arrangement of atoms can guarantee the existence of stable electronic eigenvalues. These levels are stable in the presence of off‐diagonal disorder. Graph theory is used to describe the topological structure of the Hamiltonian. The actual applications are presented in tight‐binding approximation. It was found that graph theory can be used even if the atoms have more than one atomic orbital. Also, the localization properties of the topologically determined eigenvalues are studied. © 1993 John Wiley &amp; Sons, Inc.

Stable and accurate boundary treatments for compact, high-order finite-difference schemes

The stability characteristics of various compact fourth- and sixth-order spatial operators sre used to assess the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semidiscrete initial boundary value problem (IBVP). In all cases, favorable comparisons are obtained between G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable (bounded, left half-plane (LHP) eigenvalues). It is shown that many of the higher-order schemes that are G-S-K stable are not asymptotically stable. A series of compact fourth- and sixth-order schemes is developed, all of which are asymptotically and G-K-S stable for the scalar case. A systematic technique is then presented for constructing stable and accurate boundary closures of various orders. The technique uses the semidescrete summation-by-parts energy norm to guarantee asymptotic and G-K-S stability of the resulting boundary closure. Various fourth-order explicit and implicit discretizations are presented, all of which satisfy the summation-by-parts energy norm.

Some properties and pitfalls of the stabilization method: a unified approach

Many recipes and techniques that have been proposed in the framework of the stabilization method can be better understood and systematized by considering the general behavior of the stable eigenvalues in the neighborhood of a resonance position. Such a unified approach underlines the fact that the method is founded both on standard resonance theory and on the properties of the basis sets employed in the calculations. As illustrations on the usefulness of the approach, the article focuses on the properties and pitfalls of the method with regards to the appearance of the classical stabilization pattern of plateaux and pseudocrossings; the determination of the resonance position, wave function, and width; the separation of closely lying resonances; and the case of coupled continua

Numerical solution of singularly perturbed initial value problems using fourth order cubic spline method

In this paper, we present a numerical method using a fourth order cubic spline for singularly-perturbed problems: ε y′ = f(x y ε)y(a) prescribeda ≤ x ≤ b, 0 < ε ≪ 1, where f y (x y, 0) is non-singular on (a b) and has stable eigen values for all x and y. The inner and outer regions are separated by a numerical cutting point technique and using the stretching of the independent variable x, the inner region problem is solved by a fourth order cubic spline. Finally, solutions of the inner and outer region problems are combined to give a global approximation to the original singularly perturbed initial value problem. The convergence analysis of the method giving rise to fourth order approximation of the spline solution is presented. Some well known examples have been solved to demonstrate the efficiency of the method.

Existence of stable electronic energy levels in the presence of vibrational and off-diagonal disorder.

We have found that the topological arrangement of atoms can guarantee the existence of stable electronic eigenvalues. These levels are stable in the presence of vibrational and off-diagonal disorder. Graph theory is used to describe the topological structure of the Hamiltonian. The actual applications are presented in tight-binding approximation. It was found that graph theory can be used even if the atoms have more than one atomic orbital.

Effect of paramagnetism on the global stability of straight screw pinches

An analytical expression for the eigenvalue of the global m = 1 kink mode in a straight screw pinch with arbitrary profiles of current density and axial magnetic field is derived in a long wavelength approximation of the incompressible ideal MHD equations. While the influence of the current profile is related to the inductance li of the plasma column, as has been shown previously (Nucl. Fusion 24 (1984) 1357), it is found here that the effect of the axial magnetic field profile can be expressed in terms of the poloidal beta value βp of the pinch. In turn, βp is related to a possible paramagnetism (βp < 1) or diamagnetism (βp > 1) of the plasma. Considering the range of stable eigen-values, the Kruskal-Shafranov condition q(a) > 1, independent of li and βp, is recovered in the tokamak regime Bz, ≫ Bϕ. In the regime Bz ⪅ Bϕ, on the other hand, the stability condition is strongly dependent on both li and βp. Optimal stability conditions in this regime are obtained for li, ≪ 1 (hollow current distribution), βp ≪ 1 (strong paramagnetism) and Bz(a) ≪ Bϕ(a) (low q(a)). In this case, wavelengths of up to fourteen times the plasma radius become stable, and also the growth rates of the unstable wavelengths are considerably reduced as compared to the situation with li, βp ∼ 1 and Bz ∼ Bϕ, or compared to the growth rates obtained in the tokamak regime. The results obtained are consistent with recent experimental observations in ‘ultra low q’ discharges.

Stable eigenvalue placement by constrained optimization

A pole placement algorithm is proposed which uses constrained nonlinear optimization techniques on a finite-dimensional model of a linear n degree of freedom system. Low-order feedback control is assumed where r poles may be assigned; r being the rank of the sensor coefficient matrix. It is shown that by combining feedback-control-theory methods with optimization techniques, one can ensure the stability characteristics of a system, and simultaneously improve its transient response

Discrete linear quadratic regulators with eigenvalue placement in specified regions

A new procedure is presented for the design of discrete linear quadratic regulators in the frequency domain with eigenvalue placement at exact locations and/or within specified regions of the complex z-plane. The method utilizes the frequency domain optimality conditions for achieving the desired pole placement. Also, the proposed procedure is sequential and enables the retention of some stable open-loop eigenvalues and associated eigenvectors in the closed-loop system. An illustrative example is provided to demonstrate the effectiveness of the proposed method.