Stable Eigenvectors Research Articles

Rigid motion tracking using moments of inertia in TOF-PET brain studies

A data-driven method is proposed for rigid motion estimation directly from time-of-flight (TOF)-positron emission tomography (PET) emission data. Rigid motion parameters (translations and rotations) are estimated from the first and second moments of the emission data masked in a spherical volume. The accuracy of the method is analyzed on 3D analytical simulations of the PET-SORTEO brain phantom, and subsequently tested on 18F-FDG as well as 11C-PIB brain datasets acquired on a TOF-PET/CT scanner. The estimated inertia-based motion is later compared to rigid motion parameters obtained by directly registering the short frame backprojections. We find that the method provides sub mm/degree accuracies for the estimated rigid motion parameters for counts corresponding to typical 0.5 s, 1 s, and 2 s 18F-FDG brain scans, with the current TOF resolutions clinically available. The method provides robust motion estimation for different types of patient motion, most notably for a continuous patient motion case where conventional frame-based approaches which rely on little to no intra-frame motion of short time intervals could fail. The method relies on the detection of stable eigenvectors for accurate motion estimation, and a monitoring of this condition can reveal time-frames where the motion estimation is less accurate, such as in dynamic PET studies.

Estimating the boundary of the asymptotic stability region of Lotka–Volterra system by using the trajectory reversing method

This paper proposes a topological approach for plotting the boundary of the region of asymptotic stability (RAS) of Lotka–Volterra predator-prey system. First, stability analysis was used to determine the specific saddle point that has eigenvalues with one positive and two negative real parts in a linearized Jacobian matrix. A set of initial states located around the saddle point on the specific eigenplane spanned by the two stable eigenvectors was then selected. Finally, the trajectory reversing method was used and the trajectories that had initial states on the eigenplane delineated the boundary of the asymptotic stability region. The trajectories of the initial states that started from the opposite sides of the RAS exhibited different dynamic behaviour. The numerical simulation are presented to demonstrate the effectiveness of the proposed approach.

Design of new fuzzy sliding mode controller based on parallel distributed compensation controller and using the scalar [formula omitted] function

This paper presents a new design of fuzzy sliding mode controller based on parallel distributed compensation and using a scalar sign function. The proposed fuzzy sliding mode controller (FSMC) uses the parallel distributed compensation (PDC) scheme to design the state feedback control law. The controller gains are determined in offline mode via linear quadratic regulator technique. Moreover, the fuzzy sliding surface of the system is designed using stable eigenvectors and the scalar sign function in order to overcome the discontinuous switching. This later is obtained by a sign function of the standard FSMC. The advantages of the proposed design are a minimum energy control effort, faster response and zero steady-state error. Finally, the validity of the proposed design strategy is demonstrated through the simulation of a flexible joint robot.

A new optimal sliding mode controller design using scalar sign function

This paper presents a new optimal sliding mode controller using the scalar sign function method. A smooth, continuous-time scalar sign function is used to replace the discontinuous switching function in the design of a sliding mode controller. The proposed sliding mode controller is designed using an optimal Linear Quadratic Regulator (LQR) approach. The sliding surface of the system is designed using stable eigenvectors and the scalar sign function. Controller simulations are compared with another existing optimal sliding mode controller. To test the effectiveness of the proposed controller, the controller is implemented on an aluminum beam with piezoceramic sensor and actuator for vibration control. This paper includes the control design and stability analysis of the new optimal sliding mode controller, followed by simulation and experimental results. The simulation and experimental results show that the proposed approach is very effective.

A multiplet analysis of spectra in the presence of broken symmetries

We introduce the notion of a generalised symmetry M of a hamiltonian H. It is a symmetry which has been broken in a very specific manner, involving ladder operators R and R†. In Theorem 1 these generalised symmetries are characterised in terms of repeated commutators of H with M. Breaking supersymmetry by adding a term linear in the supercharges is discussed as a motivating example.The complex parameter γ which appears in the definition of a generalised symmetry is necessarily real when the spectrum of M is discrete. Theorem 2 shows that γ must also be real when the spectrum of H is fully discrete and R and R† are bounded operators.Any generalised symmetry induces a partitioning of the spectrum of H in what we call M-multiplets. The hydrogen atom in the presence of a symmetry breaking external field is discussed as an example. The notion of stability of eigenvectors of H relative to the generalised symmetry M is discussed. A characterisation of stable eigenvectors is given in Theorem 3.

Modeling and Performance Evaluation of Multiple Mass Resonator Sensors for Electronic Nose Application

This paper presents development of a smart multiple mass resonator array (MMR) sensor based on a coupled resonator for the electronic nose application. The change of mass of each micro mass balance resonator is determined by measuring the eigenfrequencies of the coupled array using a single input and output electrode. To verify the uniqueness of each resonator, the changes in the eigenfrequencies of the coupled resonators must be distinctive and the structure eigenvectors are stable against resonator mass change. This paper highlights the sensor design concept, finite element analysis (FEA), lumped mass analysis, the sensor response measurement, and sensor mass change pattern analysis using Inverse Eigenvalue analysis. The FEA and the lumped mass analysis of the staggered coupled resonators produce the unique frequency response pattern for the sensor fingerprint with stable eigenvectors. Hence the overall mass change pattern of the sensor is well estimated.

Manifold based feedback linearization of a 1-DOF beam with one-sided spring

Vibration reduction in a harmonically excited 1-DOF beam with one-sided spring is realized by control-ling the system state from a stable large amplitude 1/2 subharmonic response towards a coexisting unstable small amplitude harmonic response using feedback linearization. At the unstable harmonic response no control effort is needed because the unstable harmonic response is a long term solution of the uncontrolled system. To reduce control effort when stabilizing the unstable harmonic response, the stable manifold can be used within the control design because at the stable manifold the system state approaches the unstable harmonic response without control effort. Unfortunately, the calculation of this manifold acquires much off-line computational effort while its usage complicates the on-line control design. Therefore, the stable manifold is approximated by the stable eigenvectors of the monodromy matrix. Due to the local validity of the approximation, a two-stage control approach is used. In the first stage, the system state is controlled towards the unstable harmonic response to reach the region where the stable manifold can be approximated accurately by the stable eigenvectors. In the second stage, the system state is controlled towards the stable eigenvectors and approaches the unstable harmonic response with hardly any control effort

Polarization of Astronomical Maser Radiation. II. Polarization Modes and Unsaturated Growth

Maser polarization is analyzed in the limit of overlapping Zeeman components (gν B Δν D , where gν B is the Zeeman splitting and Δν D is the Doppler width). All the discrepancies among the conflicting conclusions of previous studies that identified maser polarizations in this limit are fully resolved. In the case of m-independent pumping, proper application of the eigenvalue technique of Goldreich, Seeley, & Swan shows that the polarization eigenvectors are the same for saturated and unsaturated masers and are independent of spin for pure spin states, in agreement with the results of the first paper in this series. Stable eigenvectors correspond to the peak of the polarization mode distribution of self-amplified radiation at any degree of saturation

The Lorenz System: I. The Global Structure of its Stable Manifolds

The Lorenz system of equations has three fixed points when r > 1, two of which are stable if r < rt ≡ σ(σ + b + 3)/(σ - b - 1), denoted as C±. The unstable fixed point at the origin has two stable characteristic eigenvectors, s1, s2, and one unstable eigenvector, u. For r less than some r = rho < rt, the set of points which are asymptotic to the origin, as t → + ∞, (hence tangent to s1, s2) form a continuous two-dimensional surface (the stable manifold, Ws), which separates the basis of attraction (stable manifolds) of C+ and C-. The primary objective of the present study is to describe the global structure of these manifolds. When r = rho the one-dimensional unstable manifold (Wu), of the origin becomes a honoclinic orbit. The manner by which Wu becomes attached to (embedded in) Ws as r → rho is described. In particular it is shown that, not only does Ws become tangent to u as r → rho, but it must have a double tangency with one of the stable eigenvectors si, and two infinite portions of Ws are annihilated by coalescing.

Optimization of cascade blade mistuning. II - Global optimum and numerical optimization

The values of the mistuning which yield the most stable eigenvectors are analytically determined, using the simplified equations of motion which were developed in Part I of this work. It is shown that random mistunings, if large enough, may lead to the maximal stability, whereas the alternate mistunings cannot. The problem of obtaining maximum stability for minimal mistuning is formulated, based on numerical optimization techniques. Several local minima are obtained using different starting mistuning vectors. The starting vectors which lead to the global minimum are identified. It is analytically shown that all minima appear in multiplicities which are equal to the number of compressor blades. The effect of mistuning on the flutter speed is studied using both an optimum mistuning vector and an alternate mistuning vector. Effects of mistunings in elastic axis locations are shown to have a negligible effect on the eigenvalues. Finally, it is shown that any general two-dimensional bending-torsion system can be reduced to an equivalent uncoupled torsional system.