Stability Region Of System Research Articles

Backward iteration approaches for the stability domain estimation of discrete non-linear polynomial systems

This paper deals with two evolutionary algorithms for calculating the asymptotic stability region for discrete non-linear polynomial systems. These algorithms rest on two new approaches of inverting discrete direct polynomial state equation. Based on some topological considerations, the systematic computational algorithms are then applied on the second-order polynomial model of a synchronous generator power system. Hence, a comparative study is presented to illustrate the benefits of the proposed algorithms so as to maximise the Region of Asymptotic Stability (RAS).

Estimate of Attractive Regions for Systems Satisfying Polytopic Uncertainties in Given Regions

To estimate stability region of systems satisfying polytopic uncertainties in given regions is very important since such systems are given as models of linear systems with saturating control or nonlinear systems with nonlinear elements which satisfy sector conditions in given regions. In this paper, we propose a method to estimate the maximal robust attractive region of such systems using polytope Lyapunov functions. To demonstrate the usefulness of the proposed method we show some numerical examples.

A reduced-order method for estimating the stability region of power systems with saturated controls

In a modern power system, there is often large difference in the decay speeds of transients. This could lead to numerical problems such as heavy simulation burden and singularity when the traditional methods are used to estimate the stability region of such a dynamic system with saturation nonlinearities. To overcome these problems, a reduced-order method, based on the singular perturbation theory, is suggested to estimate the stability region of a singular system with saturation nonlinearities. In the reduced-order method, a low-order linear dynamic system with saturation nonlinearities is constructed to estimate the stability region of the primary high-order system so that the singularity is eliminated and the estimation process is simplified. In addition, the analytical foundation of the reduction method is proven and the method is validated using a test power system with 3 buses and 5 machines.

An iterative water-filling algorithm for maximum weighted sum-rate of Gaussian MIMO-BC

We consider the maximization of weighted rate sum in Gaussian multiple-input-multiple-output broadcast channels. This problem is motivated by optimal adaptive resource allocation policies in wireless systems with multiple antenna at the base station. In fact, under random packet arrival and transmission queues, the system stability region is achieved by maximizing a weighted rate sum with suitable weights that depend on the queue buffer sizes. Our algorithm is a generalization of the well-known Iterative Multiuser Water-Filling that maximizes the rate sum under a total transmit power constraint and inherits from the latter its simplicity. We propose also a variation on the basic algorithm that makes convergence speed very fast and essentially independent of the number of users

Estimation of asymptotic stability regions via homogeneous polynomial Lyapunov functions

Asymptotic stability regions of non-linear dynamical systems are estimated by using homogeneous polynomial Lyapunov functions, which include quadratic Lyapunov functions as a special case. The non-linear system is represented as a convex hull of some linear systems in a specified region of the state space. It is shown that the estimation can be formulated as an LMI optimization problem in an augmented state space.

Stability region enlargement through anti-windup strategy for linear systems with dynamics restricted actuator

This paper addresses the problem of determination of stability regions for linear systems with amplitude and successive dynamics restricted actuator through anti-windup strategies. Considering a linear dynamic output feedback designed with respect to the linear system (without saturation), an anti-windup design method is investigated to guarantee both the stability of the closed-loop system and the respect of the controlled output constraints for a region of admissible initial states as large as possible. Based on the modelling of the closed-loop system resulting from the controller plus the anti-windup loop as a linear system with dead-zone and dynamics restricted nonlinearities, LMI stability conditions are formulated. Numerical optimization procedures are discussed.

Stability Analysis for Digital PD Control of Flexible Systems Including Damping

In this work we extent our recent results on the stability of single-rigid/single-flexible mode system to cases involving damping. We present closed form analytical expressions that describe the boundaries of the stability regions for digital PD control systems. This is obtained using a newly adopted approach based on the critical stability constraints of Jury test. The considered system simulates many practical systems such as antenna, space shuttle, and robot arm. It is found that, the stability regions for damped flexible systems have three identifiable and distinguished topologies corresponding to three classes of damped system. The three classes are separated from each other by two surfaces in the three-dimension-space of the system-parameters. The stability region for the first class is almost a right triangle in the gain space where the third boundary, the hypotenuse, is described by the lowest root of a cubic equation. The stability region for the second class is clearly larger in size and the three roots of the cubic equation participate in defining the third boundary. In the third class, the stability region is further larger in size and the highest root of the cubic equation defines the third boundary. A peculiar situation is found where a stable system of the second class is possible with negative derivative gain. Numerical simulation is presented to verify this peculiar situation.

Anti-windup Design for Time-delay Systems Subject to Input Saturation An LMI-based Approach

This paper focuses on the study and the characterization of stability regions for linear systems with delayed states and subject to input saturation through anti-windup strategies. In particular, the synthesis of anti-windup gains in order to guarantee the stability of the closedloop system for a region of admissible initial states as large as possible is addressed. Based on the modeling of the closed-loop system, resulting from the controller plus the anti-windup loop, as a linear time-delay system with a deadzone nonlinearity, stability conditions in an LMI form are stated, for both the delay independent and delay dependent contexts, by using quadratic functionals and a modified sector condition. LMI-based optimization schemes for computing the anti-windup gains that lead to the maximization of the size of the region of stability associated to the closed-loop system are then proposed. The application of the technique and the trade-off between the size of the delay and the region of stability are illustrated by means of numerical examples.

Stability Analysis for Digital PD Control of Flexible Systems

In this work we extent our recent results on the stability of rigid systems to cases involving flexible ones. We present closed form analytical expressions that describe the boundaries of the stability regions for digital PD control systems. This is obtained using a newly adopted approach based on the critical stability constraints of Jury test. The considered system consists of a single rigid and a single flexible mode. This simulates many practical systems such as antenna, space shuttle, and robot arm. The obtained closed-form stability criterion shows that the stability region is almost a right triangle in the gain space. The right-angle sides of the proposed triangle coincide exactly with two boundaries of the stability region. The hypotenuse of the triangle is found to well approximate the third boundary. An analytical expression is derived for the slope of the hypotenuse. This approximation results in defining the stability region using only two parameters which is convenient for practical applications. Case study is presented to show the practical usefulness of the proposed close-form stability criteria through investigating the control design of high speed flexible robot arm.

Anti-windup design with guaranteed regions of stability for discrete-time linear systems

The purpose of this paper is to study the determination of stability regions for discrete-time linear systems with saturating controls through anti-windup schemes. Considering that a linear dynamic output feedback has been designed to stabilize the linear discrete-time system (without saturation), a method is proposed for designing an anti-windup gain that maximizes an estimate of the basin of attraction of the closed-loop system in the presence of saturation. It is shown that the closed-loop system obtained from the controller plus the anti-windup gain can be locally modeled by a linear system with a deadzone nonlinearity. Then, based on the use of a new sector condition and quadratic Lyapunov functions, stability conditions in an LMI form are stated. These conditions are then considered in a convex optimization problem in order to compute an anti-windup gain that maximizes an estimate of the basin of attraction of the closed-loop system. Moreover, considering asymptotically stable open-loop systems, it is shown that the conditions can be slightly modified in order to determine an anti-windup gain that ensures global stability. An extension of the proposed results to the case of dynamic anti-windup synthesis is also presented in the paper.

크기가 제한된 입력을 갖는 가변구조제어 시스템을 위한 개선된 안정 영역 추정값

This paper deals with the problem of estimating the asymptotic stability region(ASR) of uncertain variable structure systems with bounded control. Using linear matrix inequalities(LMIs) we estimate the ASR and we show the exponential stability of the closed-loop control system in the estimated ASR. We show that our estimate is always better than the estimate of [3].

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.

An Analysis and Design Method for Uncertain Variable Structure Systems With Bounded Controllers

We present a method for estimating the asymptotic stability region(ASR) of uncertain variable structure systems with bounded switching feedback controllers. Using linear matrix inequalities(LMIs) we estimate the ASR and we show the exponential stability of the closed-loop control system in the estimated ASR. We also give a simple LMI-based method for designing switching surfaces that will make the estimated ASR big. Finally, we give numerical examples in order to show the effectiveness of our method.

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.

Anti-windup design with guaranteed regions of stability for discrete-time linear systems with saturating controls

The purpose of this paper is to study the determination of stability regions for discrete-time linear systems with saturating controls through anti-windup schemes. Considering that a linear dynamic output feedback has been designed to stabilize the linear discrete-time system (without saturation), a method is proposed for designing an anti-windup gain that maximizes an estimate of the basin of attraction of the closed-loop system in the presence of saturation. It is shown that the closed-loop system obtained from the controller plus the anti-windup gain can be modeled by a linear system connected to a deadzone nonlinearity. From this model, stability conditions based on quadratic Lyapunov functions are stated. Algorithms based on LMI schemes are proposed for computing both the anti-windup gain and an associated stability region.

Delay‐dependent anti‐windup strategy for linear systems with saturating inputs and delayed outputs

AbstractThis paper addresses the problem of the determination of stability regions for linear systems with delayed outputs and subject to input saturation, through anti‐windup strategies. A method for synthesizing anti‐windup gains aiming at maximizing a region of admissible states, for which the closed‐loop asymptotic stability and the given controlled output constraints are respected, is proposed. Based on the modelling of the closed‐loop system resulting from the controller plus the anti‐windup loop as a linear time‐delay system with a dead‐zone nonlinearity, constructive delay‐dependent stability conditions are formulated by using both quadratic and Lure Lyapunov–Krasovskii functionals. Numerical procedures based on the solution of some convex optimization problems with LMI constraints are proposed for computing the anti‐windup gain that leads to the maximization of an associated stability region. The effectiveness of the proposed technique is illustrated by some numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.

Design of Digital High-Gain PD Control Systems Using Analytical Approach

It is known that a continuous stable high-gain PD control system may become unstable when the controller is implemented digitally. Recent works consider this problem and determine the stability regions of such systems. In most cases the stability regions are obtained numerically. This work introduces a new analytical approach for obtaining the stability criteria for digital systems. The approach is based on the critical constraints of simplified version of Jury test and makes use of the capabilities of MATLAB software. The approach is applied to digital PD control systems. Here, the effect of computation time in addition to that of sampling period are considered. The resulting stability criteria are presented in closed forms that are suitable for design purposes and make it possible to map the stability region for various control design parameters. This versatile capability is illustrated via design examples. To the contrary of previous knowledge, the results show that, in certain cases, the stability is obtained through increasing the proportional gain or computation time rather than by decreasing them. A comparison with the literature shows that this approach is straightforward, versatile and even corrects some of the stability regions that have been reported.

Improving the stability region of saturated linear systems controlled by dynamic delayed output feedback through anti-windup strategy

This paper addresses the problem of determination of stability regions, in fact regions of admissible initial state, as large as possible, for linear systems with delayed outputs and subject to input saturation and bounded controlled outputs, through anti-windup strategies. The closed-loop system resulting from the open-loop system with delayed output, the dynamic output controller plus the anti-windup loop is modelled as a linear time-delay system with a dead-zone nonlinearity. Constructive delay-dependent stability condition is then formulated by using a quadratic Lyapunov-Krasovskii functional. The optimization problem for computing the anti-windup gain that maximizes the size of the associated region of stability is directly expressed as the optimization of a convex criterion under LMI constraints.

크기가 제한된 제어기를 갖는 가변구조제어 시스템의 점근 안정 영역 추정

크기가 제한된 제어기를 갖는 가변구조제어 시스템의 점근 안정 영역 추정

Delay-Dependent Anti-Windup Loops for Enlarging the Stability Region of Time Delay Systems With Saturating Inputs

This paper addresses the problem of the determination of regions of stability for linear systems with delayed inputs and subject to input saturation through anti-windup strategies. Differently of the most anti-windup techniques, where the design of the anti-windup loop is introduced with the objective of minimizing the performance degradation, we are particularly interested in the synthesis of anti-windup gains in order to guarantee the stability of the closed-loop system for regions of admissible initial states as large as possible. With this aim, due to the presence of delay in the input we propose delay dependent results.