Estimation Of Stability Region Research Articles

Observer-basedH∞control for uncertain singular time-delay systems with actuator saturation

Summary In this paper, the observer-based H∞ control problem for uncertain singular time-delay systems with actuator saturation is concerned. First, a delay-dependent linear matrix inequality (LMI) condition is obtained which, guarantees that the uncertain singular time-delay systems with actuator saturation are regular, impulse free, and asymptotically stable with H∞ performance condition. Then, with this condition, the estimation of stability region and the design method of observer-based H∞ controller are given by solving LMIs and convex optimization problem. Finally, some numerical examples are provided to demonstrate the merit of the obtained results. Copyright © 2015 John Wiley & Sons, Ltd.

Estimation of Attractive Stability Region via Homogeneous Parameter-Dependent Quadratic Lyapunov Function for Impulsive Switched Linear System with Saturated Control Input

A class of homogeneous polynomial functions is used to estimate the attractive stability region for an impulsive switched linear system with saturated control input. Under the arbitrary switching rule and the dwell time switching rule, local stabilization conditions are obtained in terms of linear matrix inequalities. The corresponding optimization problems are formulated to obtain a larger attractive stability region. Using the simple ellipsoid method to estimate the attractive stability region produces very conservative results, because the attractive stability region is normally irregular. To solve this problem, a polyhedron constructed using a level set of the homogeneous parameter-dependent quadratic Lyapunov function is used to estimate the attractive stability region. The polyhedron is closer to the attractive stability region than the simple ellipsoid. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.

A Hybrid System Approach to the Analysis and Design of Power Grid Dynamic Performance

In this paper, we describe an approach to the analysis and design of power grid dynamic performance based on hybrid systems theory. Power grid is a large-scale cyber-physical system for transmission of electrical energy. The joint dynamics of physical processes and cyber elements in power grids are typical of a mixture of continuous and discrete behaviors, that is, hybrid dynamics. We address problems on stability that are basic concerns in the performance of current and future power grids with the hybrid dynamics. Measures for stability of power grids are interpreted as safety specifications in hybrid system models and are translated into restrictions on the systems' reachable sets of states. Algorithmic reachability analysis of hybrid systems enables analysis of safe initial states and hence quantitative estimation of stability regions. Also it contributes to synthesis of safe initial states as well as switching conditions in order to satisfy safety specifications in a power grid. We demonstrate the approach for two problems on transient stability of the single machine/infinite bus (SMIB) system and on fault release control of a multimachine power grid.

On expansion of estimated stability region: Theory, methodology, and application to power systems

The expansion of the estimated stability region plays an important role in the stability analysis of nonlinear systems. However, current literatures have not provided a complete mathematical description for this problem. This paper reveals that essentially the enlargement or the compression of the estimated stability region results directly from the diffeomorphism map, which is induced by the flow contained in the stability region. By proving that any integration algorithm with an order higher than one can approximately trace the flow of the system, a generalized methodology is proposed to construct various algorithms to realize the enlargement or the compression of the estimated stability region. With this methodology, two new algorithms based on symbolic calculation are suggested to reduce the computational burden. Furthermore, this methodology is applied to construct a scalable numerical algorithm to calculate the critical clearing time (CCT) of the power system for given faults. Tests on the IEEE 10-machine 39-bus system show that the computational results coincide well with the step-by-step simulation with high accuracy.

A Nonlinear-System Approach to Analysis and Design of Power-Electronic Converters With Saturation and Bilinear Terms

Power-electronic converters are intrinsically nonlinear. This paper proposes a Lyapunov approach to analysis and design of a class of nonlinear systems arising from power-electronic converters. The system has a bilinear term as the product of the state and the input-the duty cycle, which is subject to strict constraint (or saturation). The nonlinearities and the input saturation are considered in this paper by using piecewise-quadratic Lyapunov functions and by describing the system with a piecewise-linear differential inclusion. The problems considered include controller design for robust stability, and estimation of stability region and tracking domain. These analysis and design problems are converted into numerically efficient optimization algorithms involving linear-matrix inequalities (LMIs). A buck-boost dc-dc converter is used to demonstrate the proposed methods. The optimization results show that a simple state-feedback law can be constructed to achieve practically global stabilization and tracking, which is theoretically confirmed by the Lyapunov approach. An experimental buck-boost converter is constructed to verify the tracking of a square reference varying almost between the upper and the lower limit.

Equivalence of Three Quadratic Approximation Methods of Stability Boundaries of Nonlinear Dynamic Systems

Approximation of stable manifolds of unstable equilibrium points is a key problem in stability region estimation of nonlinear dynamic systems. Different quadratic approximation methods have been proposed in past literature. Then what is their relationship and which has better performance are of concern. The mathematical equivalence of three quadratic approximation methods is proved in this note. Their computation burdens of numerical implementation are compared, and the one with the least computation time is recommended for application.

New method for computing unstable equilibrium points of power systems with induction motors

The computation of the unstable equilibrium point (UEP) is a key step involved in stability region estimation of nonlinear dynamic systems. A new continuation-based method to compute the UEPs of a power system with induction motors is proposed. The mechanical torques of motors are changed to form a parameterized equation set. Then the solution curve of the equation set is traced by the continuation method from the stable equilibrium point to a UEP. The direction of mechanical torque change is varied to get multiple UEPs. The obtained UEPs are mostly type-1. Then fast assessment of induction motor stability is studied by approximating the stable manifolds of the UEPs. The method is tested in several systems and satisfactory results are obtained.

Decentralized coordinated attitude control for satellite formation flying via the state-dependent Riccati equation technique

The goal of the present study is to develop a decentralized coordinated attitude control algorithm for satellite formation flying. To handle the non-linearity of the dynamic system, the problems of absolute and relative attitude dynamics are formulated for the state-dependent Riccati equation (SDRE) technique. The SDRE technique is for the first time utilized as a non-linear controller of the relative attitude control problem for satellite formation flying, and then the results are compared to those from linear control methods, mainly the PD and LQR controllers. The stability region for the SDRE-controlled system was obtained using a numerical method. This estimated stability region demonstrates that the SDRE controller developed in the present paper guarantees the globally asymptotic stability for both the absolute and relative attitude controls. Moreover, in order to complement a non-selective control strategy for relative attitude error in formation flying, a selective control strategy is suggested. This strategy guarantees not only a reduction in unnecessary calculation, but also the mission-failure safety of the attitude control algorithm for satellite formation. The attitude control algorithm of the formation flying was tested in the orbital-reference coordinate system for the sake of applying the control algorithms to Earth-observing missions. The simulation results illustrate that the attitude control algorithm based on the SDRE technique can robustly drive the attitude errors to converge to zero.

A Method for Estimating the Stability Region of Singular Perturbation Systems with Saturation Nonlinearities

摘要: 针对奇异摄动饱和系统, 提出了一种估计其稳定域的降阶方法. 结合饱和函数的特殊性质, 证明了此类系统的稳定域可分解为伴随系统的不变集与一个足够大球体的笛卡尔积. 将原系统稳定域估计问题转化为低阶伴随系统稳定域的估计问题, 利用线性矩阵不等式(Linear matrix inequality, LMI)优化方法估计伴随系统的稳定域以减少保守性. 本方法不仅可以克服奇异摄动饱和系统的奇异性, 还可以一定程度克服系统的``维数灾''等问题. 关键词: 稳定域 / 奇异摄动系统 / 饱和非线性 / 线性矩阵不等式

Systems with a switch and estimation of absolute stability region

Two methods are proposed for estimation of absolute stability region in the space of parameters. The first method uses nontrivial necessary conditions, which follow from stability of switching systems. The second method is a modification of the harmonic balance approach; it is used for approximate determination of existence of periodic modes that arise when stability is lost.

Função energia generalizada de controle para estabilização de sistemas não lineares

Neste trabalho, o conceito de função de Lyapunov de controle é estendido em duas direções principais. Na primeira direção, o conceito de função energia de controle (FEC) é proposto com o objetivo de não somente projetar leis de controle estabilizantes para sistemas não lineares mas também com o objetivo de fornecer uma estimativa da região de estabilidade do sistema em malha fechada. A FLC garante estabilidade local de um certo ponto de equilíbrio do sistema em malha fechada, mas usualmente não fornece estimativasótimas da região de estabilidade. O conceito de FEC é mais exigente que o conceito de FLC. Além de garantir estabilidade local do equilíbrio, a FEC fornece informações globais a respeito dos conjuntos limites e estimativas ótimas das regiões de estabilidade. A segunda direção está relacionada com o fato de que, em geral, é difícil encontrar FLC ou FEC para muitos sistemas não lineares. De forma a minimizar este problema e aplicar õ conceito de CEF em uma classe maior de sistemas, propõe-se neste artigo o conceito de função energia generalizada de controle (FEGC). A FEGC permite o projeto de controladores mesmo quando a derivada da função energia é positiva em algumas regiões limitadas do espaço de estados.

Numerical algorithm for constructing Lyapunov functions of polynomial differential system

In this paper, a numerical algorithm for constructing Laypunov functions of polynomial system is described. The algorithm is described for second-dimension system although extension to higher dimension systems would appear to be possible. Using the algorithm, we can obtain the estimation of asymptotic stability regions of nonlinear autonomous dynamical systems. Numerical examples further illustrate the theoretical work in this paper.

A Technique of a Stability Domain Determination for Nonlinear Discrete Polynomial Systems

This paper is devoted to the asymptotic stability region estimation for nonlinear discrete polynomial systems. An algebraic method is derived for the enlargement of a guaranteed stability region in which the asymptotic stability is ensured. The advantages of the proposed method are the accuracy of determination of the largest stability boundary, its numerical and theoretical robustness and its applicability to wide classes of dynamical discrete systems. A numerical example illustrates the proposed method.

Impact of saturation nonlinearities/disturbances on the small-signal stability of power systems: An analytical approach

In this paper, a multi-objective optimization model is presented to estimate the practical stability region and maximum endurable disturbance rejection for a small-signal power system dynamic model with saturation nonlinearities and disturbance rejection. Iterative algorithms are developed to solve for Pareto optimized solutions (POS) of this optimization. Furthermore, as an application of this approach to power systems, a method to analyze the impact of saturation nonlinearities and disturbance rejection on power system small-signal stability is introduced based on the estimated stability region and maximum endurable disturbance rejection. Numerical results of a test power system with detailed saturated PSS controllers are described, indicating the reliability and simplicity of the method.

A method for evaluating the performance of PSS with saturated input

In this paper, we report upon a method for estimating an ellipsoid, which resides entirely in the stability region of a linear system with saturation nonlinearities. To reduce the conservatism in the estimation, a procedure to transform the estimation problem into a simple convex optimization with linear matrix inequality (LMI) constraints is presented. As an application of this development, a method is introduced to estimate the stability region of a multi-machine power system with power system stabilizers (PSS) subject to saturated feedback. The idea is to check if the system state after a disturbance resides inside the estimated stability region. Thereby a sufficient condition is derived to conclude the effectiveness of saturated PSS controls. Numerical results of a test power system with three generators and five buses are described, indicating the reliability and simplicity of this approach.

Methods for estimating stability regions with applications to power systems

AbstractA specialized method for constructing a hyper‐ellipse that resides inside the stability regions of a class of nonlinear autonomous systems such as electric power systems is provided. This method is further generalized to estimate the stability region of a fairly general class of high dimension nonlinear autonomous systems. Applications of the introduced results to power system transient stability analysis are described, together with numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.

Parametric Optimization of Digitally Controlled Nonlinear Reactor Dynamics using Zubov-like Functional Equations

The present work aims at the development of a systematic method to optimally choose the parameters of digitally controlled nonlinear reactor dynamics. In addition to traditional performance requirements for the controlled reactor dynamics such as stability, fast and smooth regulation, disturbance rejection, etc., optimality is requested with respect to a physically meaningful performance. The value of the performance index is analytically calculated via the solution of a Zubov-like functional equation and becomes explicitly parameterized by the digital controller parameters. A standard static optimization algorithm yields subsequently the optimal values of the above parameters. Within the proposed framework, stability region estimates are also provided through the solution of the above functional equation. Finally, a nonlinear chemical reactor example following Van de Vusse kinetics is used in order to illustrate the proposed parametric optimization method.

On the estimation of robust stability regions for nonlinear systems with saturation

This paper addresses the problem of determining robust stability regions for a class of nonlinear systems with time-invariant uncertainties subject to actuator saturation. The unforced nonlinear system is represented by differential-algebraic equations where the system matrices are allowed to be rational functions of the state and uncertain parameters, and the saturation nonlinearity is modelled by a sector bound condition. For this class of systems, local stability conditions in terms of linear matrix inequalities are derived based on polynomial Lyapunov functions in which the Lyapunov matrix is a quadratic function of the state and uncertain parameters. To estimate a robust stability region is considered the largest level surface of the Lyapunov function belonging to a given polytopic region of state. A numerical example is used to demonstrate the approach.

A Singular Fixed-Point Homotopy MethodtoLocate the Closest Unstable Equilibrium Point for Transient Stability Region Estimate

The closest unstable equilibrium point (UEP) method is a well-known direct method of the Lyapunov type for optimally estimating stability regions of nonlinear dynamical systems. One key step involved in the closest UEP methodology is the computation of the closest unstable equilibrium point that has the lowest Lyapunov function value on the stability boundary. In this paper, a new computational algorithm to compute the closest UEP is presented. The proposed algorithm is based on a homotopy-continuation method combined with the singular fixed-point strategy. Numerical simulation results show that the algorithm outperforms previously reported existing techniques.

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