Frequency-domain Criterion Research Articles

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

Connected systems with switching between three linear discrete-time subsystems are considered, and a new frequency-domain criterion for the existence of a quadratic Lyapunov function ensuring the stability of such systems under arbitrary switching is proposed. The application of this criterion is demonstrated on an example of a third-order system.

Fracture mechanical properties and identification of principal stress direction from the rock frequency domain

The characterization of rock instability precursors and the identification of principal stress direction are essential for the safety construction of underground engineering. This study explores the characterization of rock instability precursors from the frequency domain and identification of principal stress direction by the anisotropy of frequency variation. The results show that the dense distribution of peak frequency and frequency centroid in the high frequency domain, along with an increase in peak frequency bandwidth and a decrease in mean frequency centroid value, can be used as frequency domain criteria for rockburst prediction and early warning. Both peak frequency and frequency centroid are effective monitoring parameters for rock instability. The anisotropic frequency centroid variation can be used to identify the principal stress direction, and a process is proposed. This paper not only offers a valuable complement to existing methods for rock instability precursor and principal stress direction identification, but also provides a new analytical method for stability monitoring in rock engineering.

Frequency criteria for exponential stability

Abstract We discuss some frequency-domain criteria for the exponential stability of nonlinear feedback systems based on Popov’s hyperstability theorem. The main results are new bounds on convergence rates for perturbations of the damped harmonic oscillator, including the Hessian-damped case, when the perturbations satisfy certain sectoriality or convexity hypotheses. We also include a simplified proof of Popov’s theorem in its general form, which has the virtue that it often applies at the margins of stability when other criteria typically degenerate.

Identification and Estimation of Structural VARMA Models Using Higher Order Dynamics

We use information from higher order moments to achieve identification of non-Gaussian structural vector autoregressive moving average (SVARMA) models, possibly nonfundamental or noncausal, through a frequency domain criterion based on higher order spectral densities. This allows us to identify the location of the roots of the determinantal lag matrix polynomials and to identify the rotation of the model errors leading to the structural shocks up to sign and permutation. We describe sufficient conditions for global and local parameter identification that rely on simple rank assumptions on the linear dynamics and on finite order serial and component independence conditions for the non-Gaussian structural innovations. We generalize previous univariate analysis to develop asymptotically normal and efficient estimates exploiting second and higher order cumulant dynamics given a particular structural shocks ordering without assumptions on causality or invertibility. Finite sample properties of estimates are explored with real and simulated data.

Variable shearing holography with applications to phase imaging and metrology

We report variable shear interferometers employing liquid-crystal-based geometric-phase (GP) gratings. Conventional grating-based shear interferometers require two lateral shifts of the gratings to enable phase-shifting capabilities in x- and y- direction and two axial shifts of the gratings to adjust the shear amount in x- and y- direction, i.e., these systems need control of four axes mechanically. Here we show that the GP gratings combined with a pixelated polarization camera give instantaneous-phase shifting so that no mechanical movement is necessary to obtain phase shifts. Furthermore, we show that a single fixed shear with a rotational shear axis provides a more robust selection of shears while requiring the control of only one mechanical axis. We verify this statement using spatial domain and frequency domain criteria. We further show that the resolution of the reconstructed wavefield depends not only on the numerical aperture of the imaging system, the pixel size of the detector, or the spatial coherence of the source but also on the ability to determine the shear amount accurately. To improve this, we report a methodology to accurately detect the shear amounts using the second derivative of the autocorrelation function of the measured holograms, which further relaxes the requirements on mechanical stability.

Revisiting stability analysis of Phase-Locked Loops with the Popov Stability Criterion

This paper revisits the problem of stability analysis of Phase-Locked Loops (PLLs), focusing specifically on stability conditions derived using the Popov Stability Criterion. A new form of the Popov based frequency domain criterion is derived in which the size of the stability region, the PLL locking range, appears independently of the loop transfer-function. This enables one to maximize the stability region graphically and directly on the Popov plot, rather than iteratively. Various numerical and analytic results available in the literature are shown to be particular cases of the proposed new stability test. It is also shown that for PLLs of type r, in which r denotes the number of integrators in the loop, it is not possible to achieve full locking range if r is larger or equal than three.

A frequency-domain analysis of inexact gradient methods

We study robustness properties of some iterative gradient-based methods for strongly convex functions, as well as for the larger class of functions with sector-bounded gradients, under a relative error model. Proofs of the corresponding convergence rates are based on frequency-domain criteria for the stability of nonlinear systems. Applications are given to inexact versions of gradient descent and the Triple Momentum Method. To further emphasize the usefulness of frequency-domain methods, we derive improved analytic bounds for the convergence rate of Nesterov’s accelerated method (in the exact setting) on strongly convex functions.

Multiaxial fatigue assessment of jacket-supported offshore wind turbines considering multiple random correlated loads

Fatigue analysis is an important part of the design process for offshore wind turbines (OWTs). The aerodynamic wind forces are the main fatigue loads, and are generally transformed into six load components: Fx, Fy, Fz, Mx, My, and Mz, which may lead to hotspots of concentrated multiaxial stress. In addition, the six wind load components may be correlated, which makes fatigue analysis complex. To address these issues, this paper derives and presents two new formulae to account for load correlation in the determination of stress power spectral density (PSD) from multiple random loads based on the interaction equation approach and the first principle stress approach. These two formulae form two frequency domain fatigue criteria to evaluate fatigue life of OWT support structures. Two frequency domain criteria are validated through comparison with full time domain analysis results.

Power shaping control of DC–DC converters with constant power loads

This paper deals with the stabilization of DC–DC converters feeding constant power loads (CPLs). It is shown that power/energy dynamics are easily computed via the image of the input/output voltages/currents of the converter. Such image is represented by a high-order differential operator, obtained from first principles, acting on the variables of a stabilizing interconnected controller. This approach is based upon the premise that the destabilizing effect of CPLs, is caused due to a power imbalance in a dissipation equality, which prevents passivity. The proposed stabilizing mechanism shapes the power dynamics via an interconnected controller, inducing a feasible power flow and thus passivity. To do so, we developed a stability test, as well as control design tools in terms of linear matrix inequalities (LMIs), constructed from coefficient matrices of the system dynamics. As a practical advantage, the proposed power shaping approach permits to link stability conditions with the nominal power rate specification of power converters. Finally, we show that the proposed framework can also accommodate other traditional frequency domain, eigenvalue and immittance criteria. The main results are validated via a theoretical analysis and experimental results.

Some issues about conditional stability in LTI plants

Controlled plant stability always depends on some plant and controller parameters. If the dependency is not monotone, the system is denoted as conditionally stable. In this paper some basic parameters, such as the gain or the time delay, are considered to determine the linear time invariant plant stability by using frequency domain criteria. Also, a plant performance such as the phase margin is used for the same purpose. Even the sampling period, in some sampled data controlled plants, may characterize the stability condition. In this paper, the requirements to present a conditional stability in controlled plants are outlined. Some examples and a practical application related to a networked control of a resonant flexible manipulator are used to illustrate the results.

Receptance-based robust stability criteria for second-order linear systems with time-varying delay and unstructured uncertainties

This paper presents new frequency domain criteria for the robust stability analysis of closed-loop second-order systems under time-varying internal delay. The proposed approach is derived from the small gain theorem for systems described by the versatile system receptance to guarantee robust stability despite time-varying delays and model uncertainties. Robust stability of systems with time-varying delays is challenging to cope because time-invariant paradigm does not hold, but the proposed approach is directly based on the frequency response analysis of the closed-loop system with unstructured uncertainty bound. The criteria can be applied to either single or multiple inputs systems, and no constraint is imposed on the delay derivative. A simple receptance-based Internal Model Control strategy is proposed to improve robust stability margins while nominal state-feedback design is preserved. Thus, a controller obtained from the delay-free nominal model can be used to ensure the robust stability of the uncertain second-order system with time-varying delay. Simulation case studies are used to illustrate the benefits of the proposed criteria.

Formulation of multiaxial fatigue failure criteria for spectral method

Fatigue life assessment under multiaxial loading requires the use of multiaxial fatigue failure criteria which are defined in time and frequency domain. Frequency domain criteria are created via formulations using former criteria in the time domain. Therefore, there is a need for a general approach to formulate criteria in the frequency domain based on existing proposals. We discuss methods to determine parameters appearing in the frequency domain criteria; in particular, we present the critical plane approach proposed by Macha and other proposals based on stress invariants. An attempt to generalize the problem is presented and discussed in the section of conclusions.

A Study of Periodic Motions in Homogeneous Nonlinear Multivariable Systems Written in the Polynomial Vector-Matrix Representation

In this paper, we suggest a method for estimating the parameters of periodic motions and also a frequency-domain criterion of their stability for nonlinear time-invariant homogeneous (identical) multivariable automatic control systems with structurally identical subsystems. This method can be used to simply determine the parameters of periodic motions and their stability in the class of systems mentioned above based on the structural decomposition and the method of harmonic linearization. The results are confirmed by mathematical modeling.

Volterra Equations with Periodic Nonlinearities: Multistability, Oscillations and Cycle Slipping

In this paper, systems of nonlinear integro-differential Volterra equations are examined that can be represented as feedback interconnections of linear time-invariant block and periodic nonlinearities. The interest in such systems is motivated by their numerous applications in mechanical, electrical and communication engineering; examples include, but are not limited to, models of phase-locked loops, pendulum-like mechanical systems, coupled vibrational units and electric machines. Systems with periodic nonlinearities are usually featured by multistability and have infinite sequences of (locally) stable and unstable equilibria; their trajectories may exhibit nontrivial (e.g. chaotic) behavior. We offer frequency-domain criteria, ensuring convergence of any solution to one of the equilibria points, and this property is referred to as the gradient-like behavior and corresponds to phase locking in synchronization systems. Although it is hard to find explicitly the equilibrium, attracting a given trajectory, we give a constructive estimate for the distance between this limit equilibrium and the initial condition. The relevant estimates are closely related to the analysis of cycle slipping in synchronization systems. In the case where the criterion of gradient-type behavior fails, a natural question arises — which nonconverging solutions may exist in the system and, in particular, how many periodic solutions it has. We show that a relaxation of the frequency-domain convergence criterion ensures the absence of high-frequency periodic orbits. The results obtained in the paper are based on the method of integral quadratic constraints that has arisen in absolute stability theory and stems from Popov’s techniques of “a priori integral indices”. We illustrate the analytic results by numerical simulations.

When is it reasonable to implement the discontinuous sliding‐mode controllers instead of the continuous ones? Frequency domain criteria

SummaryProfessor Utkin proposed an example showing that the amplitude of chattering caused by the presence of parasitic dynamics (stable actuators) in some systems governed by the First‐Order Sliding‐Mode Controller is lower than that produced by the Super‐Twisting Algorithm. This example served to motivate this paper reconsidering the problem of comparison of chattering in systems with stable actuators, and driven by Discontinuous Sliding‐Mode Controllers (DSMCs) and Continuous Sliding‐Mode Controllers (CSMCs). Comparison of chattering produced by DSMC and CSMC taking into account their amplitudes, frequencies, and average power (AP) needed to maintain the system into real‐sliding modes, allowing to conclude the following: (i) for systems with slow actuators, the amplitude of oscillations and AP produced by DSMC be smaller than those caused by CSMC; (ii) for bounded disturbances with fixed Lipschitz constant, there exist sufficiently fast actuators for which the amplitude of oscillations and AP produced by CSMC be smaller than those caused by DSMC.

Tsypkin and Jury–Lee Criteria for Synchronization and Stability of Discrete-Time Multiagent Systems

Dynamics of complex networked systems consisting of a large number of interconnected components (agents) has attracted considerable attention of researchers in control theory and dynamical systems. One of the most interesting questions in this area concerns the emergence of a coherent joint behavior of nodes caused by couplings between them, such as, e.g., their synchronization. We study a network where each node is a linear dynamical system of arbitrary order in discrete time, and couplings between the nodes are nonlinear and unknown but satisfy the sector inequality with known bounds. We obtain frequency-domain criteria of synchronization and stability of the network, which are generalizations of the Tsypkin criterion and the Jury–Lee criterion for discrete-time Lurie systems.

Stability Analysis of Dynamic Systems in the Polynomial Vector-Matrix Representation

This paper suggests frequency-domain stability criteria for homogeneous multiconnected automatic control systems that allow an easy stability analysis of high-order systems of a large dimension. The results are confirmed by mathematical modeling.

Frequency-domain estimates of the sampling interval in multirate nonlinear systems by time-delay approach

ABSTRACTA class of nonlinear hybrid systems consisting of a continuous plant and a multirate sampled-data controller is considered. Applying the time-delay approach, the system is transformed to the system with a time-varying sawtooth delay. Then, the value of the sampling intervals ensuring stability of the closed-loop system is estimated based on the frequency-domain circle-like criterion for systems with time-varying delay. Such a criterion for multirate MIMO systems is established based on the previous result for SISO systems proposed by M.Churilova in 1995. Efficiency and advantages of the method are illustrated by numerical examples. The key advantage of the proposed approach is that it is applicable to multirate systems nonaffine in control. It allows one to use a nonlinear feedback for stabilisation. Particularly, the case of nonsmooth or discontinuous nonlinear systems can be treated.

Frequency-Domain Criterion for the Global Stability of Dynamical Systems with Prandtl Hysteresis Operator

In the present paper, dynamical systems with Prandtl hysteresis operator are considered. For the class of dynamical systems under consideration, a frequency-domain global stability criterion is formulated and proved. For a second-order dynamical system with Prandtl operator, we demonstrate the advantage of the obtained criterion as compared to the well-known criterion derived by Logemann and Ryan.