Stability Arguments Research Articles

A Study of the Lyapunov Stability of an Self-Excited Induction Generator

The self-excited induction generators (SEIGs) can provide attractive renewable energy sources. Since it can be used in an open-loop fashion which includes no control closed-loop, its inherent stability can be counted on to allow operation over a wide range of operating conditions. Unlike classical arguments based on transient and steady-state analysis, it is proposed rigorous holistic analytical stability arguments based on the full nonlinear transient state-space model of a SEIG under two-phase stationary reference frame using dynamical systems theory. The conditions for critical transient stability of the unloaded SEIG are obtained in the sense of Lyapunov. Analytic formulas from the limit cycle conditions give the values of the range of capacitance, the range of rotor speed and the stator frequency for self-excitation. Analytic formulas from steady-state conditions give all the steady-state parameter and state values. Thus, the analytical results yield a set of simple and universal analytic formulas that can predict and evaluate the transient and steady-state stability characteristics of the SEIG. Good agreement between the experimental results and computed results validates the analytical results.

Regularity for $$C^{1,\alpha }$$ Interface Transmission Problems

We study existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with $C^{1,\alpha}$ interfaces. For this, we develop a novel geometric stability argument based on the mean value property.

Takagi–Sugeno Fuzzy Unknown Input Observers to Estimate Nonlinear Dynamics of Autonomous Ground Vehicles: Theory and Real-Time Verification

In this article, we address the simultaneous estimation problem of the lateral speed, the steering input, and the effective engine torque, which play a fundamental role in vehicle handling, stability control, and fault diagnosis of autonomous ground vehicles. Due to the involved longitudinal-lateral coupling dynamics and the presence of unknown inputs (UIs), a new nonlinear observer design technique is proposed to guarantee the asymptotic estimation performance. To this end, we make use of a specific Takagi-Sugeno (TS) fuzzy representation with nonlinear consequents to exactly model the nonlinear vehicle dynamics within a compact set of the vehicle state. This TS fuzzy modeling not only allows reducing significantly the real-time computational effort in estimating the vehicle variables but also enables an effective way to deal with unmeasured nonlinearities. Moreover, via a generalized Luenberger observer structure, the UI decoupling can be achieved without requiring a priori UI information. Using Lyapunov stability arguments, the UI observer design is reformulated as an optimization problem under linear matrix inequalities, which can be effectively solved with standard numerical solvers. The effectiveness of the proposed TS fuzzy UI observer design is demonstrated with real-time hardware-in-the-loop experiments.

New Analysis Framework of Lyapunov-Based Stability for Hybrid Wind Farm Equipped With FRT: A Case Study of Egyptian Grid Code

Due to the continuous increase of fuel prices and pollutions, the use of renewable energy especially wind has increased. In developing countries including Egypt, squirrel cage induction generator wind turbine (SCIG- WT) represents a considerable proportion of the total capacity of installed wind farm due to its qualities such as low cost and easy availability. However, its operation has a substantial effect on system stability. In contrast, doubly fed induction generator wind turbine (DFIG-WT) is broadly penetrated the electrical grid as it keeps the system stable. In this work, the ability of WT generators to continue operating rather than tripping at the time of faults is analyzed for proper stability investigation. The detailed control and stability of a grid-connected large scale SCIG and DFIG of Zafarana, Suez Gulf area, Egypt are discussed whereas the parameters of fault ride through (FRT) curve of Egypt grid code is utilized. Moreover, a precise analytical stability argument using a proposed integrated nonlinear dynamical model is presented. Conditions for global asymptotic stability of the SCIG in the sense of Lyapunov function (LF) are given and tested by time domain simulation. The eigenvalues of the matrices of LF and its derivative are determined by which the stability boundaries are determined depending on the positivity of these matrices. The dynamic behavior of the whole system is simulated in MATLAB/Simulink interface programming while the practical data are collected from an experimental model consisting of DFIG-WT to demonstrate the efficacy of the FRT control system.

Error estimation of the Besse Relaxation Scheme for a semilinear heat equation

The solution to the initial and Dirichlet boundary value problem for a semilinear, one dimensional heat equation is approximated by a numerical method that combines the Besse Relaxation Scheme in time [C. R. Acad. Sci. Paris Sér. I 326 (1998)] with a central finite difference method in space. A new, composite stability argument is developed, leading to an optimal, second-order error estimate in the discrete Lt∞(Hx2)-norm at the time-nodes and in the discrete Lt∞(Hx1)-norm at the intermediate time-nodes. It is the first time in the literature where the Besse Relaxation Scheme is applied and analysed in the context of parabolic equations.

Horizontal Nonlinear Path Following Guidance Law for a Small UAV With Parameter Optimized by NMPC

In this study, the problem of guiding a small fixed-wing unmanned aerial vehicle (UAV) toward a predefined horizontal path is studied. A stable nonlinear guidance law, which is a function of the inertial positions and velocities of the UAV and the predefined path, is designed using Lyapunov stability arguments. The concept of the nonlinear model predictive control (NMPC) technique was applied to optimize a key parameter of the guidance law to improve the performance of the controller (PFC_NMPC), where the stability of the relative nonlinear system is maintained. The proposed method was verified in the MATLAB/Simulink environment to realize following the straight-line, square and circular paths. The path- following performance of the proposed method is compared with those of the guidance laws with parameter fixed (PFC) or tuned by fuzzy logic (PFC_FL). With the predictive ability, the proposed method can make the UAV fly more on the desired square and circular paths than the other two methods, PFC and PFC_FL. The error overshoot by using PFC_NMPC is much smaller than those by using the PFC and PFC_FL methods in the presence of wind at 8m/s.

Fuzzy descriptor tracking control with guaranteed L∞ error-bound for robot manipulators

This paper is concerned with the nonlinear tracking control design for robot manipulators. In spite of the rich literature in the field, the problem has not yet been addressed adequately due to the lack of an effective control design. Using a descriptor fuzzy model-based framework, we propose a new approach to design a feedback-feedforward control scheme for robot manipulators in a general form. The goal is to guarantee a small level of an [Formula: see text] gain specification to improve the tracking performance while significantly reducing the numerical complexity for real-time implementation. Based on Lyapunov stability arguments, the control design is formulated as a convex optimization problem involving linear matrix inequalities. Numerical experiments performed with a high-fidelity manipulator benchmark model, embedded in the Simscape MultibodyTM environment, demonstrate the effectiveness of the proposed control solution over existing standard approaches.

Time-Accurate and highly-Stable Explicit operators for stiff differential equations

Unconditionally stable implicit time-marching methods are powerful in solving stiff differential equations efficiently. In this work, a novel framework to handle stiff physical terms implicitly is proposed. Both physical and numerical stiffness originating from convection, diffusion and source terms (typically related to reaction) can be handled by a set of predefined Time-Accurate and highly-Stable Explicit (TASE) operators in a unified framework. The proposed TASE operators act as preconditioners on the stiff terms and can be deployed to any existing explicit time-marching methods straightforwardly. The resulting time integration methods remain the original explicit time-marching schemes, yet with nearly unconditional stability. The TASE operators can be designed to be arbitrarily high-order accurate with Richardson extrapolation such that the accuracy order of original explicit time-marching method is preserved. Theoretical analyses and stability diagrams show that the s-stages sth-order explicit Runge-Kutta (RK) methods are unconditionally stable when preconditioned by the TASE operators with order p≤s and p≤2. On the other hand, the sth-order RK methods preconditioned by the TASE operators with order of p≤s and p>2 are nearly unconditionally stable. The only free parameter in TASE operators can be determined a priori based on stability arguments. Unlike classical implicit methods, the TASE methodology allows for solving non-linear problems with arbitrary order without requiring solving a nonlinear system of equations. A set of benchmark problems with strong stiffness is simulated to assess the performance of the TASE method. Numerical results suggest that the proposed framework preserves the high-order accuracy of the explicit time-marching methods with very-large time steps for all the considered cases. As an alternative to established implicit strategies, TASE method is promising for the efficient computation of stiff physical problems.

Region of attraction analysis of nonlinear stochastic systems using Polynomial Chaos Expansion

A method is presented to estimate the region of attraction (ROA) of stochastic systems with finite second moment and uncertainty-dependent equilibria. The approach employs Polynomial Chaos (PC) expansions to represent the stochastic system by a higher-dimensional set of deterministic equations. We first show how the equilibrium point of the deterministic formulation provides the stochastic moments of an uncertainty-dependent equilibrium point of the stochastic system. A connection between the boundedness of the moments of the stochastic system and the Lyapunov stability of its PC expansion is then derived. Defining corresponding notions of a ROA for both system representations, we show how this connection can be leveraged to recover an estimate of the ROA of the stochastic system from the ROA of the PC expanded system. Two optimization programs, obtained from sum-of-squares programming techniques, are provided to compute inner estimates of the ROA. The first optimization program uses the Lyapunov stability arguments to return an estimate of the ROA of the PC expansion. Based on this result and user specifications on the moments for the initial conditions, the second one employs the shown connection to provide the corresponding ROA of the stochastic system. The method is demonstrated by two examples.

Second-order bipartite consensus for networked robotic systems with quantized-data interactions and time-varying transmission delays

This paper investigates the second-order bipartite consensus (BC) problems for networked robotic systems (NRSs) subject to model uncertainties and external disturbances over signed directed graphs. We have newly presented two classes of hierarchical control algorithms (HCAs) to solve the BC problems for NRSs in a more practical and challenging case of quantized-data interactions (QDIs) and time-varying transmission delays (TVTDs). By using Hurwitz criterion and Lyapunov stability argument, several sufficient conditions on control parameters are obtained and the BC performance of the regulated system is analyzed. Three examples validate the presented theoretical findings.

Prediction-based control for a class of unstable time-delayed processes by using a modified sequential predictor

This work presents a prediction-based scheme to control a class of unstable delayed (bio)-chemical processes. The main characteristic of the control scheme is that it is based on a predicted future state that is estimated using a prediction-observation protocol that allows compensating large time-delays. The convergence of the prediction error is shown through standard stability arguments. The stability of the closed-loop system is shown by invoking the separation principle between the prediction error and the control based on estimated variables feedback. Four numerical examples are used to illustrate our results, including two time-delayed benchmark case studies taken from the control process literature.

Stable Solutions of $-\Delta u+\lambda u=|u|^{p-1}u $ in Strips

This paper is devoted to study the following equation $-\Delta u+\lambda u= |u|^{p-1}u \;\;\text{in}\;\Omega $ , with homogeneous Dirichlet or Neumann boundary conditions where $p>1$ , $\lambda >0$ , $\Omega =\mathbb{R}^{n-k}\times \omega $ , $n\geq 2$ , $k\geq 1$ , and $\omega $ is a smoothly bounded domain of $\mathbb{R}^{k}$ . We prove Liouville-type theorems for $C^{2}$ solutions which are stable or stable outside a compact set of $\Omega $ . We first provide an integral estimate using stability argument which combined with Pohozaev-type identity allow to obtain nonexistence results for $p\in [p_{s}(n), p_{s}(n-k)]$ , where $p_{s}(n)$ is the classical Sobolev exponent in dimension $n$ . Also, we establish monotonicity formula to prove the nonexistence of nontrivial solution which is stable or stable outside a compact set of $\Omega $ for all $p>1$ .

On the Construction of Approximate Solutions for the 1D Pollutant Transport Model

The purpose of this paper is to build sequences of suitably smooth approximate solutions to the 1D pollutant transport model that preserve the mathematical structure discovered in (Roamba, Zabsonré, Zongo, 2017). The stability arguments in this paper then apply to such sequences of approximate solutions, which leads to the global existence of weak solutions for this model. We show that when the Reynold number goes to infinity, we have always an existence of global weak solutions result for the corresponding model.

Non-linear Control of Inverted Pendulum

Presented is a study of non-linear control for an inverted pendulum system. The inverted pendulum system is a great example of an underactuated, non-minimum phase, and highly unstable system. The objective of this research paper is to derive non-linear control laws for an inverted pendulum system. First, dynamic equations of the inverted pendulum are derived by utilizing the Lagrange's equations and then it is linearized around an unstable upright position. Secondly, the corresponding analysis uses the standard linear stability arguments and the traditional Lyapunov method. The non-linear sliding mode control and feedback linearization control laws are then derived The feedback linearization control law is used to transform the non-linear system into an equivalent linear system such that a suitable feedback control law can be proposed. The stabilization of the initial condition and reference tracking is studied in this paper. I demonstrate the effectiveness of the proposed non-linear control strategies using MATLAB/Simulink software.

Homological stability for classical groups

We prove a slope 1 stability range for the homology of the symplectic, orthogonal, and unitary groups with respect to the hyperbolic form, over any fields other than F 2 \mathbb {F}_2 , improving the known range by a factor 2 in the case of finite fields. Our result more generally applies to the automorphism groups of vector spaces equipped with a possibly degenerate form (in the sense of Bak, Tits, and Wall). For finite fields of odd characteristic, and more generally fields in which − 1 -1 is a sum of two squares, we deduce a stability range for the orthogonal groups with respect to the Euclidean form, and a corresponding result for the unitary groups. In addition, we include an exposition of Quillen’s unpublished slope 1 stability argument for the general linear groups over fields other than F 2 \mathbb {F}_2 , and use it to recover also the improved range of Galatius–Kupers–Randal-Williams in the case of finite fields, at the characteristic.

Infinitesimal asphericity changes the universality of the jamming transition

The jamming transition of non-spherical particles is fundamentally different from the spherical case. Non-spherical particles are hypostatic at their jamming points, while isostaticity is ensured in the case of the jamming of spherical particles. This structural difference implies that the presence of asphericity affects the critical exponents related to the contact number and the vibrational density of states. Moreover, while the force and gap distributions of isostatic jamming present power-law behaviors, even an infinitesimal asphericity is enough to smooth out these singularities. In a recent work (Brito et al 2018 Proc. Natl Acad. Sci. 115 11736–41), we have used a combination of marginal stability arguments and the replica method to explain these observations. We argued that systems with internal degrees of freedom, like the rotations in ellipsoids, or the variation of the radii in the case of the breathing particles fall in the same universality class. In this paper, we review comprehensively the results about the jamming with internal degrees of freedom in addition to the translational degrees of freedom. We use a variational argument to derive the critical exponents of the contact number, shear modulus, and the characteristic frequencies of the density of states. Moreover, we present additional numerical data supporting the theoretical results, which were not shown in the previous work.

GLOBAL NEARLY-PLANE-SYMMETRIC SOLUTIONS TO THE MEMBRANE EQUATION

We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d\geqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^{2}$ by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.

Teoria parcial de justiça e estrutura política democrática na teoria de Nussbaum

Neste trabalho defendemos que o progresso da abordagem das capabilities está no desenvolvimento teórico da estrutura política democrática, para tanto, tomaremos como base o modelo normativo de Nussbaum. Primeiramente esta investigação classifica a teoria das capabilities em dois grupos: o modelo top down e o bottom up. Nussbaum tem um tipo de abordagem de “cima para baixo”, na qual inicia construindo uma teoria que parte de questões mais abstratas e somente depois desce para o nível de implementação institucional. Enquanto Amartya Sen, em contrapartida, possui um modelo “de baixo para cima”, iniciando das demandas de justiça que vem da esfera pública, depois subindo para questões normativas menos concretas (a sua “ideia de justiça”). No segundo momento, apresentamos a teoria parcial de justiça de Nussbaum argumentando que ela é composta de quatro estágios: (a) primeiro, a elaboração da lista de capabilities, (b) segundo, o processo de validade normativa, (c) terceiro, a estabilidade de uma sociedade justa (d) quarto, a implementação institucional do limiar básico de capabilities. Em relação a esse quarto estágio, no terceiro momento do artigo, desenvolvemos especificamente o papel da estrutura política como escopo teórico o qual a teoria das capabilities precisa investir como forma de progresso teórico. Em outras palavras, o futuro da teoria das capabilities está principalmente no entrelaçamento entre teoria das capabilities com uma forma de governo democrática. Finalmente, na última seção, propomos uma reelaboração do limiar básico de capabilities (a lista) de Nussbaum, de forma a torná-lo mais compatível com a estrutura política democrática.

Adaptive finite-time cluster synchronization of neutral-type coupled neural networks with mixed delays

This paper investigates the adaptive finite-time cluster synchronization problem of the neutral-type coupled neural networks (NCNNs) with mixed delays. Compared with the traditional neural networks, the model of NCNNs is more general in some sense, due to that it involves state delays, distributed delays and coupling delays. In this paper, a novel, adaptive and closed-loop control control algorithm is proposed to achieve the finite-time cluster synchronization of NCNNs with mixed delays. In addition, the sufficient conditions on the control parameters for stabilizing the closed-loop system are derived by leveraging the Lyapunov stability argument. Finally, the simulation results are carried out to illustrate the validity and feasibility of the proposed algorithm.

Unknown Input Observers for Simultaneous Estimation of Vehicle Dynamics and Driver Torque: Theoretical Design and Hardware Experiments

This article investigates a new observer design method to estimate simultaneously both the vehicle dynamics and the unknown driver's torque. To take into account the time-varying nature of the longitudinal speed, the vehicle system is transformed into a polytopic linear parameter-varying (LPV) model with a reduced level of numerical complexity. Based on Lyapunov stability arguments, we prove that the estimation errors of the system state and of the unknown input (UI) are norm-bounded, which can be made arbitrarily small by minimizing a guaranteed $\mathcal {L}_{\infty }-$ gain performance. The design of the LPV UI observer is reformulated as an linear matrix inequality-based optimization which can be effectively solved via semidefinite programming. Extensive hardware experiments are carried out under various driving test scenarios to confirm the effectiveness of the proposed observer design. In particular, a comparative study is performed with a widely adopted observer to emphasize the practical interests of the new estimation solution.