Stability Boundary Of Systems Research Articles

The kinematics and kinetics of pedestrians on a laterally swaying footbridge

Progress in understanding human-structure interaction (HSI) on footbridges has been hampered by the shortage of quality data collected in realistic environments. This paper reports a novel experiment conducted on a naturally-swaying 7m footbridge of frequency 0.67Hz and amplitudes up to 125mm. Subjects crossed the bridge while wearing infrared motion-tracking markers and pressure-sensing insoles. The bridge was fitted with bespoke force plates, allowing investigators to simultaneously record kinematic and kinetic reactions to the structure's motion, which was assumed simple harmonic. The bridge was naturally excited by the test subjects, who were allowed to walk at a comfortable self-chosen pace. The data show that the subjects adopted a fixed-in-space Centre of Mass (CoM) strategy but their Centres of Pressure (CoP) were highly correlated to the position of the bridge deck within its lateral oscillation cycle (henceforth ‘bridge phase’), allowing for the prediction of wide and crossed steps. Ground forces generally correlated to CoM-CoP separation, which reflected the phase of the bridge at the previous heel-strike. Instantaneous step width was correlated to the bridge phase and is proportional to the offset in the Medial-Lateral (M-L) ground force between consecutive steps. The Inverted Pendulum Model (IPM) was evaluated using the data, exhibiting a limited fit to the recorded ground forces. Finally, the pedestrian-induced work on the bridge and system stability boundaries are also calculated, revealing mechanisms for bridge instability to occur.

Impact of phase-locked loop on stability of active damped LCL-filter-based grid-connected inverters with capacitor voltage feedback

Active damped LCL-filter-based inverters have been widely used for grid-connected distributed generation (DG) systems. In weak grids, however, the phase-locked loop (PLL) dynamics may detrimentally affect the stability of grid-connected inverters due to interaction between the PLL and the controller. In order to solve the problem, the impact of PLL dynamics on small-signal stability is investigated for the active damped LCL-filtered grid-connected inverters with capacitor voltage feedback. The system closed-loop transfer function is established based on the Norton equivalent model by taking the PLL dynamics into account. Using an established model, the system stability boundary is identified from the viewpoint of PLL bandwidth and current regulator gain. The accuracy of the ranges of stability for the PLL bandwidth and current regulator gain is verified by both simulation and experimental results.

Saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems

ABSTRACTA complete characterization of the stability boundary of an asymptotically stable equilibrium point in the presence of type-k saddle-node non-hyperbolic equilibrium points, with k ≥ 0, on the stability boundary is developed in this paper. Under the transversality condition, it is shown that the stability boundary is composed of the stable manifolds of the hyperbolic equilibrium points on the stability boundary, the stable manifolds of type-0 saddle-node equilibrium points on the stability boundary and the stable centre and centre manifolds of the type-r saddle-node equilibrium points with r ≥ 1 on the stability boundary. This characterization is the first step to understanding the behaviour of stability regions and stability boundaries in the occurrence of saddle-node bifurcations on the stability boundary.

Small-signal discrete-time modeling of digitally controlled three-phase PWM boost rectifier under balanced voltage

PurposeThe model for digitally controlled three-phase pulse width modulation (PWM) boost rectifiers is a sampled data model, which is different from the continuous time domain models presented in previous studies. The controller, which is tuned according to the model in continuous time domain and discretized by approximation methods, may exhibit some unpredictable performances and even result in unstable systems under some extreme situations. Consequently, a small-signal discrete-time model of digitally controlled three-phase PWM boost rectifier is required. The purpose of this paper is to provide a simple but accurate small-signal discrete-time model of digital controlled three-phase PWM boost rectifier, which explains the effect of the sampling period, modulator and time delays on system dynamic and improves the control performance.Design/methodology/approachBased on the Laplace domain analysis and the waveforms of up-down-count modulator, the small signal model of digital pulse width modulation (DPWM) in the Laplace domain is presented. With a combination of state-space average and a discrete-time modeling technique, a simplified large signal discrete time model is developed. With rotation transformation and feed-forward decoupling, the large-signal model is decoupled into a single input single output system with rotation transformation. Then, an integrated small signal model in the Laplace domain is constructed that included the time delay and modulation effect. Implementing the modified z-transform, a small-signal discrete-time model is derived from the integrated small signal model.FindingsIn a digital control system, besides the circuit parameters, the location of pole of open-loop transfer function is also related to system sampling time, affecting the system stability, and the time delay determines the location of the zero of open-loop transfer function, affecting the system dynamic. In addition to the circuit parameters discussed in previous literature, the right half plane (RHP) zero is also determined by the sampling period and the time delay. Furthermore, the corner frequency of the RHP zero is mainly determined by the sampling period.Originality/valueThe model developed in this paper, accounting for the effect of the sampling period, modulator and time delays on the system dynamic, give a sufficient insight into the behavior of the digitally controlled three-phase PWM rectifier. It can also explain the effect of sampling period and control delay time on system dynamic, accurately predict the system stability boundary and determine the oscillation frequency of the current loop in critical stable. The experimental results verify that the model is a simple and accurate control-oriented small-signal discrete-time model for the digitally controlled three-phase PWM boost rectifier.

On the application of the asymptotic method of global instability in aeroelasticity problems

The asymptotic method of global instability developed by A.G. Kulikovskii is an effective tool for determining the eigenfrequencies and stability boundary of one-dimensional or multidimensional systems of sufficiently large finite length. The effectiveness of the method was demonstrated on a number of one-dimensional problems; and since the mid-2000s, this method has been used in aeroelasticity problems, which are not strictly one-dimensional: such is only the elastic part of the problem, while the gas flow occupies an unbounded domain. In the present study, the eigenfrequencies and stability boundaries predicted by the method of global instability are compared with the results of direct calculation of the spectra of the corresponding problems. The size of systems is determined starting from which the method makes a quantitatively correct prediction for the stability boundary.

Control of sub‐harmonic oscillation in peak current mode buck converter with dynamic resonant perturbation

SummaryIn this paper, a dynamic resonant perturbation (DRP) method is proposed to control sub‐harmonic oscillation in a peak current mode controlled buck converter. Different from the traditional non‐feedback resonant parametric perturbation method, the proposed DRP method extracts an approximate sinusoidal control signal from the output voltage. The self‐stabilizing condition for the designed control signal is presented, and its analog circuit implementation is given as well. Furthermore, the system stability boundary is obtained by investigating the system eigenvalues, and the control parameter is effectively determined. The presented simulation and experiment results show that the proposed DRP method eliminates sub‐harmonic oscillation without sacrificing the peak value of the inductor current and features with a self‐stabilizing characteristic for external condition changes. Copyright © 2014 John Wiley & Sons, Ltd.

Nonlinear stability boundary of journal bearing systems operating with non-Newtonian couple stress fluids

The non-Newtonian effects on the nonlinear stability boundary of short journal bearings are investigated through the transient nonlinear analysis. Two coupled nonlinear equations are solved by using the fourth-order Runge-Kutta method. According to the results, there exists a nonlinear stability boundary within the clearance circle. Any initial positions of the shaft center outside of this boundary would yield an unstable trajectory, even though the bearing should be stable in accordance with the linear stability theory. The non-Newtonian effects provide a larger stability boundary within the clearance circle as compared to the bearing lubricated with a Newtonian fluid.

Limit‐cycle stable control of current‐mode dc‐dc converter with zero‐perturbation dynamical compensation

SummaryRegarding the non‐limit‐cycle instabilities, which commonly exist in the feedback‐controlled switching power converters, a new zero‐perturbation dynamical compensation method is proposed based on a simplified self‐stable dynamical compensation condition in this paper. With a current‐mode Buck converter as the subject of investigation, the corresponding self‐stable perturbation control equation is given. At the same time, the system stability boundary is obtained based on the investigation of the system eigenvalues, and hence, the working range of control parameters is determined. Finally, the presented simulation and experiment results reveal that the new zero‐perturbation dynamical compensation controller is easily realized with an analog circuit and it will not sacrifice the working range of the original reference current compared with the traditional slope compensation. Copyright © 2013 John Wiley & Sons, Ltd.

Characterization of Stability Region for General Autonomous Nonlinear Dynamical Systems

The existing characterization of stability regions was developed under the assumption that limit sets on the stability boundary are exclusively composed of hyperbolic equilibrium points and closed orbits. The characterizations derived in this technical note are a generalization of existing results in the theory of stability regions. A characterization of the stability boundary of general autonomous nonlinear dynamical systems is developed under the assumption that limit sets on the stability boundary are composed of a countable number of disjoint and indecomposable components, which can be equilibrium points, closed orbits, quasi-periodic solutions and even chaotic invariant sets.

Solving Multi-Objective Voltage Stability Constrained Power Transfer Capability Problem using Evolutionary Computation

Competitive market forces and the ever-growing load demand are two of the key issues that cause power systems to operate closer to their system stability boundaries. Open access has since introduced competition and therefore promotes inter-regional electrical power trades. However, the economic flows of electrical energy between interconnected regions are usually constrained by system physical limits, e.g. transmission lines capacity and generation active/reactive power capability etc. As such, there is a limitation to the capacity of electrical power that regions can export or import. This maximum allowable electrical power transfer is normally referred to as Total Transfer Capability (TTC). It is critical to understand that TTC does not necessarily represent a safe and reliable amount of inter-regional power transfer as one or more operational limits are usually binding when quantifying TTC. Hence, it is of interest that power system stability issues are being considered during power transfer capability assessment in order to provide a more appropriate and secure power transfer level.The aim of this paper is to formulate an Optimal Power Flow (OPF) algorithm, which is capable of evaluating inter-area power transfer capability considering mathematically-complex voltage collapse margins. Through a multi-objective optimization setup, the proposed OPF-based approach can reveal the nonlinear relationships, i.e. the pareto-optimal front, between transfer capability and voltage stability margins. The feasibility of this approach has been intensively tested on a 3-machine 9-bus and the IEEE 118-bus systems.

Codimension-two singularities on the stability boundary in 2D Filippov systems

AbstractBifurcation theory provides powerful tools for the analysis of the dynamics of open-loop or closed-loop nonlinear control systems. These systems are Filippov (or piecewise smooth) when the dynamics depends discontinuously on the state, for example as a consequence relay feedback actions. In this paper we contribute to the analysis of codimension-two bifurcations in Filippov systems by reporting some results on the equilibrium bifurcations of 2D systems that involve a sliding limit cycle. There are only two such local bifurcations: a degenerate boundary focus that we call homoclinic boundary focus; and the boundary Hopf. We address both of them, and provide the complete set of curves that exist around such codimension-two bifurcation points. Existing numerical software can be used to exploit these results for the analysis of the stability boundaries of nonlinear piecewise smooth control systems. In the final part of this paper, we discuss a 2D Filippov system modelling an ecosystem subject to on-off harvesting control that exhibits both codimension-two bifurcations.

Bifurcation Analysis of Thermoacoustic Instability in a Horizontal Rijke Tube

A bifurcation analysis of the dynamical behavior of a horizontal Rijke tube model is performed in this paper. The method of numerical continuation is used to obtain the bifurcation plots, including the amplitude of the unstable limit cycles. Bifurcation plots for the variation of nondimensional heater power, damping coefficient and the heater location are obtained for different values of time lag in the system. Subcritical bifurcation was observed for variation of parameters and regions of global stability, global instability and bistability are characterized. Linear and nonlinear stability boundaries are obtained for the simultaneous variation of two parameters of the system. The validity of the small time lag assumption in the calculation of linear stability boundary has been shown to fail at typical values of time lag of system. Accurate calculation of the linear stability boundary in systems with explicit time delay models, must therefore, not assume a small time lag assumption. Interesting dynamical behavior such as co-existing multiple attractors, quasiperiodic behavior and period doubling route to chaos have been observed in the analysis of the model. Comparison of the linear stability boundaries and bifurcation behavior from this reduced order model are shown to display trends similar to experimental data.

Stability boundary characterization of nonlinear autonomous dynamical systems in the presence of a type-zero saddle-node equilibrium point

Under the assumption that all equilibrium points are hyperbolic, the stability boundary of nonlinear autonomous dynamical systems is characterized as the union of the stable manifolds of equilibrium points on the stability boundary. The existing characterization of the stability boundary is extended in this paper to consider the existence of non-hyperbolic equilibrium points on the stability boundary. In particular, a complete characterization of the stability boundary is presented when the system possesses a type-zero saddle-node equilibrium point on the stability boundary. It is shown that the stability boundary consists of the stable manifolds of all hyperbolic equilibrium points on the stability boundary and of the stable manifold of the type-zero saddle-node equilibrium point.

The Estimation of Stability Boundaries for an Anaerobic Digestion System

The operation of biochemical reaction systems requires substantial expertise and a good understanding of system dynamics. Due to various inhibition effects, these systems usually possess several equilibrium points. Selecting the initial state of the system is as important as selecting the system's inputs for a proper operation and for achieving the technological goal. This paper presents an anaerobic digestion system with two biochemical reactions, which possesses six equilibria for some ranges of system's inputs. The system's stability boundary separates the set of initial states from which the system converges to the desired operating point and the set of initial states which lead the system to a totally undesired condition, the acidification point. A methodology for estimating this stability boundary is described. Additionally, a procedure for visualizing the extent of the estimated boundary in a lower dimensional space is introduced. The algorithms are simple and provide accurate estimates. Moreover, the results may be displayed as two-dimensional diagrams that can be easily understood by the system's operator.

Optimality and stability in a class of bang–bang controlled biochemical reaction systems

This paper deals with the optimization of biochemical reaction systems of rank one. Two optimization problems are solved: the problem of optimal operation for maximum productivity in steady state and the problem of the start-up to the optimal steady state. Application of Pontryagin's maximum principle shows that the controller is of the bang–bang type, with no singular intervals. The determination of the optimal switching surface involves the solution of a two point boundary value problem. Solving such a difficult problem is avoided by choosing candidate switching surfaces on a heuristic basis. This study shows that switching on the stability boundary of the nominal operating point corresponding to the maximum dilution rate is the best choice. Here the value of the cost index is minimum amongst the various switching surfaces considered and the stability boundary satisfies the conditions imposed on a candidate switching surface for proper operation. Simple, robust algorithms are formulated for accurately estimating the system's stability boundary. The obtained results display the influence of feedback control on the stability of the set point. The bang–bang controller substantially increases the set point's region of attraction in state space as compared to the uncontrolled bioreactor.

Nonlinear random responses of a structure parametrically coupled with liquid sloshing in a cylindrical tank

The nonlinear random interaction of an elastic structure with liquid sloshing dynamics in a cylindrical tank is investigated in the neighborhood of 2:1 internal resonance. Such internal resonance takes place when the natural frequency of the elastic structure is close to twice the natural frequency of the anti-symmetric sloshing mode (1,1). The excitation is generated from the response of a linear shaping filter subjected to a Gaussian white noise. The analytical model involves three sloshing modes (1,1), (0,1) and (2,1). The system response statistics and stability boundaries are numerically estimated using Monte Carlo simulation. The influence of the excitation center frequency, its bandwidth, and the liquid level on the system responses is studied. It is found that there is an irregular energy exchange between the structure and the liquid free surface motion when the center frequency is close to the structure natural frequency. Depending on the excitation power spectral density, the liquid free surface experiences zero motion, uncertain motion (intermittency), partially developed motion, and fully developed random motion. The structure response probability density function is almost Gaussian, while the liquid elevation deviates from normality. The unstable region, where the liquid motion occurs, becomes wider as the excitation intensity increases or as the bandwidth decreases. As the liquid depth or the structure spring stiffness decreases, the region of nonlinear interaction shrinks and is associated with a shift of the peak of the structure mean square response toward the left side of the frequency axis.

Eliminating drift in inventory and order based production control systems

An Inventory and Order Based Production Control System lies at the heart of many commercial and bespoke ordering systems based on periodic review of stock and production targets. This simple and elegant control system works well, even when dealing with scenarios in which there are many competing value streams. However, such “interferences” inevitably cause some uncertainty in pipeline delivery times. We show via linear z-transform analysis that the consequences may include the possibility of inventory drift and instability. In this paper we establish the stability boundaries for such systems, and demonstrate an innovative method of eliminating inventory drift due to lead-time effect. This new principle is confirmed by simulation results.

Stability analysis of rotor-bearing systems via Routh-Hurwitz criterion

A method of analysis is developed for studying the whirl stability of rotor-bearing systems without the need to solve the governing differential-equations of motion of such systems. A mathematical model, comprised of an axially-symmetric appendage at the mid-span of a spinning shaft mounted on two dissimilar eight-coefficient bearings, is used to illustrate the method. Sufficient conditions for asymptotic stability of both the translational and rotational modes of motion of the system have been derived. The system's stability boundaries, presented graphically in terms of the various system non-dimensionalized parameters, afford a comprehensive demonstration of the effects of such parameters on the system's stability of motion.

Correction to "Computing small-signal stability boundaries for large-scale power systems"

Correction to "Computing small-signal stability boundaries for large-scale power systems"

Computing small-signal stability boundaries for large-scale power systems

This paper describes two algorithms for determining the value of a given system parameter that causes the crossing of a complex-conjugate eigenvalue pair through the small-signal stability boundary (Hopf bifurcation). A large-scale test system was utilized to validate the two proposed Hopf bifurcation algorithms. The results presented demonstrate the computational efficiency and numerical robustness of the algorithms.