Global Stability Region Research Articles

Parameter effects on high-speed UAV ground directional stability using bifurcation analysis

A loss of ground directional stability can trigger a high-speed Unmanned Aerial Vehicle (UAV) to veer off the runway. In order to investigate the combined effects of the key structural and operational parameters on the UAV ground directional stability from a global perspective, a fully parameterized mathematical high-speed UAV ground nonlinear dynamic model is developed considering several nonlinear factors. The bifurcation analysis procedure of a UAV ground steering system is introduced, following which the simulation efficiency is greatly improved comparing with the time-domain simulation method. Then the numerical continuation method is employed to investigate the influence of the nose wheel steering angle and the global stability region is obtained. The bifurcation parameter plane is divided into several parts with different stability properties by the saddle nodes and the Hopf bifurcation points. We find that the UAV motion states will never cross the bifurcation curve in the nonlinear system. Also, the dual-parameter bifurcation analyses are presented to give a complete description of the possible steering performance. It is also found that BT bifurcation appears when the UAV initial rectilinear velocity and the tire frictional coefficient vary. In addition, results indicate that the influence of tire frictional coefficient has an opposite trend to the influence of initial rectilinear velocity. Overall, using bifurcation analysis method to identify the parameter regions of a UAV nonlinear ground dynamic system helps to improve the development efficiency and quality during UAV designing phase.

Stability analysis of an ensemble of simple harmonic oscillators

In this paper, we investigate the stability of the configurations of harmonic oscillator potential that are directly proportional to the square of the displacement. We derive expressions for fluctuations in partition function due to variations of the parameters, viz. the mass, temperature and the frequency of oscillators. Here, we introduce the Hessian matrix of the partition function as the model embedding function from the space of parameters to the set of real numbers. In this framework, we classify the regions in the parameter space of the harmonic oscillator fluctuations where they yield a stable statistical configuration. The mechanism of stability follows from the notion of the fluctuation theory. In Secs. 7 and 8, we provide the nature of local and global correlations and stability regions where the system yields a stable or unstable statistical basis, or it undergoes into geometric phase transitions. Finally, in Sec. 9, the comparison of results is provided with reference to other existing research.

Interplay of Kelvin–Helmholtz instability with acoustics in a viscous potential flow

Among the hydrodynamic instabilities influencing the evolution, stabilization, and control of flows, the Kelvin–Helmholtz (KH) instability mode is a profound trigger to induce unsteadiness and turbulence—either within a single fluid, by means of a velocity shear, or along the interface of multiple fluids. This mechanism has been analytically studied by Funada and Joseph [“Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel,” J. Fluid Mech. 445, 263 (2001)], for the surface separating two fluids within the approximation of inviscid and viscous potential flows. The present investigation extends the Funada–Joseph formulation to incorporate the effect of imposed acoustic waves on the system under consideration. Specifically, the KH–acoustic interaction is studied by employing a modification of the Bychkov approach [V. Bychkov, “Analytical scalings for flame interaction with sound waves,” Phys. Fluids 11, 3168 (1999)], which has been originally derived for the acoustic coupling to the combustion instability. The analytic formulae for the dispersion relations, growth rates, and neutral curves describing the perturbed interface of the KH instability/acoustic region are derived. Specifically, the limits for stable/unstable regimes as a function of hydrodynamic and acoustic parameters are identified. Two interacting modes are of particular interest: resonant and parametric modes, characterized by acoustic fields having the same frequency (resonant) and twice the frequency (parametric) of the instability oscillations. It is shown that while relatively weak acoustics provide a promising contribution to stabilize the KH instability, those of higher strength can excite the parametric instability. Overall, a comprehensive parametric study of the KH–acoustic coupling and stability limits shows that a global stability region may exist between that of the resonant and parametrically unstable regimes.

Global Frequency Synchronization of Complex Power Networks Via Coordinating Switching Control

It is well known that the frequency synchronization of complex power networks is a prerequisite for stable operation. This paper investigates the global frequency synchronization in power networks via coordinating novel controllers on both the generator-side and the load-side. The designed controllers compute the monitored information from the generator-side and load-side simultaneously, and transmit the processed signals to steer the generators and load buses for maintaining frequency stability. In particular, the generator (load)-side buses can communicate with the load (generator)-side buses, not limited to the same side. Furthermore, nonlinear switching control protocols are proposed to analyze the frequency stability for overcoming the deficiency that the conventional linearized methods can only guarantee the local stability in or near an equilibrium state. The global stability region is rigorously proven, in which the sufficient conditions are derived for achieving global frequency synchronization of power systems. Finally, a prevailing modified Western System Coordinating Council (WSCC) 9-bus system and the widely-used IEEE 39-bus system are selected to validate the effectiveness and feasibility of the theoretical results.

Axial mixing and vortex stability to in situ radial injection in Taylor–Couette laminar and turbulent flows

Taylor–Couette flows have been widely studied in part due to the enhanced mixing performance from the variety of hydrodynamic flow states accessible. These process improvements have been demonstrated despite the traditionally limited injection mechanisms from the complexity of the Taylor–Couette geometry. In this study, using a newly designed, modified Taylor–Couette cell, axial mass transport behaviour is experimentally determined over two orders of magnitude of Reynolds number. Four different flow states, including laminar and turbulent Taylor vortex flows and laminar and turbulent wavy vortex flows, were studied. Using flow visualization techniques, the measured dispersion coefficient was found to increase with increasing $Re$, and a single, unified regression is found for all vortices studied. In addition to mass transport, the vortex structures’ stability to radial injection is also quantified. A dimensionless stability criterion, the ratio of injection to diffusion time scales, was found to capture the conditions under which vortex structures are stable to injection. Using the stability criterion, global and transitional stability regions are identified as a function of Reynolds number, $Re$.

Asymptotics of Solutions of the Generalized Hutchinson’s Equation

We discuss the dynamics of the Hutchinson’s equation and its generalizations. An estimate of the global stability region of a positive steady state is obtained. The main results refer to existence, stability and asymptotics of a slow oscillating solution. New asymptotic methods are applied to a problem of dynamical properties of ODE system describing Belousov — Zhabotinsky reaction.

A New Tuning Method for Stabilization Time Delay Systems Using PIλDμ Controllers

AbstractThis paper presents a new design procedure to tune the fractional order PIλDμ controller that stabilizes a first order plant with time delay. The procedure is based on a suitable version of the Hermite–Biehler Theorem and the Pontryagin Theorem. A Theorem and a Lemma are developed to compute the global stability region of the PIλDμ controller in the (kp,ki,kd) space. Hence, this Theorem and Lemma allow us to develop an algorithm for solving the PIλDμ stabilization problem of the closed loop plant. The proposed approach has been verified by numerical simulation that confirms the effectiveness of the procedure.

PI and PID stabilization of neutral and retarded systems with time delay

Stabilization using fixed-order controllers is one of the key topics in control system design for the linear time-invariant (LTI) systems. In this paper, the stabilization of LTI neutral and retarded time-delay systems by means of PI and PID controllers is investigated in detail. The basic theme of the presented approach is to obtain a global stability region in the 2-dimensional (2-D) (k_p, k_i)-plane for the PI controller and in the 3-dimensional (3-D) (k_p, k_i, k_d)-space for the PID controllers. This region is formed by stability boundaries that are defined as the real root boundary, infinite root boundary, and complex root boundary. The 3-D global stability region for the PID case is made up of the set of 2-D stability regions that are obtained in the (k_p, k_i)-plane, (k_p, k_d)-plane, or (k_i, k_d)-plane by changing the other third parameter. To achieve this, the stability boundary locus approach is incorporated into the D-decomposition method. Thus, the complete set of stabilizing PI and PID controllers for the system is obtained. The simulation results indicate that the presented stabilization method is effective and practically useful in the analysis and control of the neutral and retarded time-delay systems.

Global Stability Lobes of Turning Processes with State-Dependent Delay

We obtain global stability lobes of two models of turning processes with inherit nonsmoothness due to the presence of state-dependent delays. In the process, we transform the models with state-dependent delays into systems of differential equations with both discrete and distributed delays and develop a procedure to determine analytically the global stability regions with respect to parameters. We find that the spindle speed control strategy that we investigated in [SIAM J. Appl. Math., 72 (2012), pp. 1--24] can provide essential improvement on the stability of turning processes with state-dependent delay, and furthermore we show the existence of a proper subset of the stability region which is independent of system damping. Numerical simulations are presented to illustrate the general results.

DESIGN AND REAL TIME IMPLEMENTATION OF FRACTIONAL ORDER PROPORTIONAL-INTEGRAL CONTROLLER ( PI ? ) IN A LIQUID LEVEL SYSTEM.

This research article deals with the design and real time implementation of a fractional order Proportional-Integral controller (PI?) for a Liquid Level System (LLS). The system is approximated as a First Order Plus Time Delay (FOPTD) model. The equivalent transfer function of this system in polynomial format is considered here for controller design. Expressions for controller parameters (KP and KI) in terms of frequency (?) and fractional order (?) are derived from the Fractional Order Characteristic Polynomial (FOCP) of the closed loop system. The global stability region based on K P and K I for each ? is constructed. Average values of KP and KI, for each ?, are taken. Among these values, the best fit of KP average and KI average and corresponding ? are identified by means of optimization techniques. The real time implementation of PI? controller with the identified controller settings in LLS is done. The PI? controller performances are analyzed in terms of ISE and IAE. A comparison of this control strategy with other conventional based controller techniques is made. PI? controller outperforms the conventional PI controllers. In addition the load disturbance studies are also carried out and it justifies the supremacy of PI? controller.

Global stability for non-monotone delay equations (with application to a model of blood cell production)

A set of sufficient conditions for the global stability of the positive equilibrium is established for a large class of non-monotone time-delayed differential equations by the method of fluctuations and the exponential ordering approach. Then these analytic results are applied to a mathematical model of white blood cell production. Numerical simulations are also presented to illustrate the global stability regions of parameters and the long-term behavior of solutions.

An Algorithm for Stabilization of Fractional-Order Time Delay Systems Using Fractional-Order PID Controllers

This technical note presents a solution to the problem of stabilizing a given fractional-order system with time delay using fractional-order PllambdaDmu controllers. It is based on determining a set of global stability regions in the (kp, Ki, Kd)-space corresponding to the fractional orders lambda and mu in the range of (0, 2) and then choosing the biggest global stability region in this set. This method can be also used to find the set of stabilizing controllers that guarantees prespecified gain and phase margin requirements. The algorithm is simple and has reliable result which is illustrated by an example, and, hence, is practically useful in the analysis and design of fractional-order control systems.

Time-delayed feedback control of friction-induced instability

Time-delayed feedback control of different types of friction-induced instabilities is discussed. Mainly, three well-known mechanisms of friction-induced instabilities are considered: (1) Stribeck instability governed by velocity-weakening characteristics of friction force, (2) mode-coupling instability, and (3) sprag-slip instability. Control force is constructed utilizing the delayed difference of a single measurable state (displacement here) as proposed by Pyragas. Regions of local stability of the equilibrium are constructed in the plane of control parameters (gain and delay). Global stability regions are also constructed for the Stribeck type instability. A comprehensive methodology for selecting the control parameters is discussed. Analytical results are substantiated by direct numerical simulation.

Stabilization using fractional-order PI and PID controllers

This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PIλDμ controllers. It is based on plotting the global stability region in the (kp, ki)-plane for the PIλ controller and in the (kp, ki, kd)-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.

An iterative numerical method for dispersion-managed solitons

We present a new numerical method for finding regimes of stable transmission of pulses in nonlinear dispersion-managed (DM) systems. Unlike the known averaging method, whose iterations require finding the pulses with the minimum and maximum width, our method merely starts with an initial guess for the pulse. Even if the initial guess is far from the correct shape, the method rapidly converges to a DM soliton after several iterations. We show that this method is especially relevant for the application to realistic models including filtering and amplification, where the soliton plays the role of an attractor. A new result obtained by means of the iterative method and reported in this work is a global stability region for the solitons in the DM system with lumped filters and amplifiers. We also demonstrate the utility of the method, applying it to a more sophisticated three-step DM system.

A model for numerical evaluation of the global stability region in power systems determined by a nonlinear limit cycle

AbstractA power system is a large dynamic system, which includes many nonlinear elements. According to the nonlinear analyses using Hopf bifurcation theory, it can be detected that a limit cycle exists around an operating point, which may affect the global stability of a power system significantly. The authors have presented a numerical method to analyze the nonlinear characteristics in power systems by observing the power swing after some perturbation where the coefficients of nonlinear terms are determined by the least squares method. In this paper the method is modified for the application to a longitudinally interconnected power system including an excitation system, and the influence of the excitation voltage limiter on the nonlinear phenomena of the whole power system can be detected by some numerical analyses. © 2003 Wiley Periodicals, Inc. Electr Eng Jpn, 144(3): 17–27, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.10193

非線形周期軌道によって定まる電力系統大域的安定領域の特性評価のための数値的解析モデル

非線形周期軌道によって定まる電力系統大域的安定領域の特性評価のための数値的解析モデル

The Stability of Two-Station Multitype Fluid Networks

This paper studies the fluid models of two-station multiclass queueing networks with deterministic routing. A fluid model is globally stable if the fluid network eventually empties under each nonidling dispatch policy. We explicitly characterize the global stability region in terms of the arrival and service rates. We show that the global stability region is defined by the nominal workload conditions and the “virtual workload conditions,” and we introduce two intuitively appealing phenomena—virtual stations and push starts—that explain the virtual workload conditions. When any of the workload conditions is violated, we construct a fluid solution that cycles to infinity, showing that the fluid network is unstable. When all the workload conditions are satisfied, we solve a network flow problem to find the coefficients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero, proving that the fluid level eventually reaches zero under any nonidling dispatch policy. Under certain assumptions on the inter arrival and service time distributions, a queueing network is stable or positive Harris recurrent if the corresponding fluid network is stable. Thus, the workload conditions are sufficient to ensure the global stability of two-station multiclass queueing networks with deterministic routing.

Stability of a three-station fluid network

This paper studies the stability of a three-station fluid network. We show that, unlike the two-station networks in Dai and Vande Vate l18r, the global stability region of our three-station network is not the intersection of its stability regions under static buffer priority disciplines. Thus, the “worst” or extremal disciplines are not static buffer priority disciplines. We also prove that the global stability region of our three-station network is not monotone in the service times and so, we may move a service time vector out of the global stability region by reducing the service time for a class. We introduce the monotone global stability region and show that a linear program (LP) related to a piecewise linear Lyapunov function characterizes this largest monotone subset of the global stability region for our three-station network. We also show that the LP proposed by Bertsimas et al. l1r does not characterize either the global stability region or even the monotone global stability region of our three-station network. Further, we demonstrate that the LP related to the linear Lyapunov function proposed by Chen and Zhang l11r does not characterize the stability region of our three-station network under a static buffer priority discipline.

Stability and Instability of Fluid Models for Reentrant Lines

Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai (Dai, J. G. 1995. On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid models. Ann. Appl. Probab. 5 49–77.) which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served (FBFS) and Last-Buffer-First-Served (LBFS) disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such networks; however, all work-conserving policies are stable in a ring network.