Local stability analysis and domain of attraction estimation for a class of uncertain nonlinear discrete‐time systems

This paper proposes a linear matrix inequality based method for the estimation of domain of attraction for a class of discrete‐time nonlinear systems subject to uncertain constant parameters. Recursive algebraic representations of the system dynamics and of the Lyapunov stability conditions are applied to obtain convex conditions which guarantee the system robust local stability while providing an estimate of the domain of attraction. A large class of discrete‐time nonlinear systems and of Lyapunov functions can be embedded in the proposed methodology including the whole class of regular rational functions of the system state variable and uncertain parameters. Numerical examples illustrate the application of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

On the estimation of the domain of attraction for discrete-time switched and hybrid nonlinear systems

This paper addresses the estimation of the domain of attraction for discrete-time nonlinear systems where the vector field is subject to changes. First, the paper considers the case of switched systems, where the vector field is allowed to arbitrarily switch among the elements of a finite family. Second, the paper considers the case of hybrid systems, where the state space is partitioned into several regions described by polynomial inequalities, and the vector field is defined on each region independently from the other ones. In both cases, the problem consists of computing the largest sublevel set of a Lyapunov function included in the domain of attraction. An approach is proposed for solving this problem based on convex programming, which provides a guaranteed inner estimate of the sought sublevel set. The conservatism of the provided estimate can be decreased by increasing the size of the optimisation problem. Some numerical examples illustrate the proposed approach.     

一类不确定脉冲型混杂系统的保代价控制

针对混合代价函数,研究了参数不确定脉冲型混杂系统的保代价控制问题,给出了混杂状态反馈保代价控制律的设计方法,由此得到的控制律既能使系统闭环鲁棒渐近稳定,又可使系统的闭环混合代价指标在对象参数摄动的范围内不超过确定的上界.本文提出的控制律不仅包含连续时间动态,也包含离散事件动态,而且其离散事件动态行为不需要与被控系统的离散事件动态行为一致,因此设计时不要求被控系统的每个连续时间子系统都具有可控性.仿真结果表明所提设计方法是可行有效的.     

On the domain of attraction and local stabilization of nonlinear parameter‐varying systems

In this paper, the domain of attraction (DA) and the local stabilization problem are investigated under the nonlinear parameter‐varying (NPV) framework. Specifically, to take into account the time‐varying parameter, we generalize the definition of robust DA (RDA) and propose the concept of parameter‐dependent DA (PDA) for NPV systems, together with the sum‐of‐squares (SOS) conditions for their estimations. Differently from the existing DA‐related works, the theoretical results in this paper can be applied to a large class of nonlinear polynomial systems including the time‐invariant, the parameter‐dependent, and the uncertain ones. Moreover, the commonly used iterative/coordinate‐wise search is avoided because we eliminate the bilinear product terms between the Lyapunov functional and the SOS multipliers. We also consider the local stabilization problem of NPV systems, in which the RDA can be specified as a control performance index. Finally, several examples are given for illustration purposes.     

Stability analysis and guaranteed domain of attraction for a class of hybrid systems: an LMI approach

This paper presents sufficient conditions for the regional stability problem for switched piecewise affine systems, a special class of Hybrid Systems. This class of systems are described by an affine differential equation of the type x˙=A(δ)x+b(δ), where x denotes the continuous state vector and δ is a vector of logical variables that modifies the local model of the system in accordance with the continuous dynamics. Using a Lyapunov function of the type v(x)=x′P(x)x, we present LMI conditions that, when feasible, guarantee local stability of the origin of the switched system. Examples of switched affine systems are used to illustrate the results. Copyright © 2003 John Wiley & Sons, Ltd.     

On the region of attraction of nonlinear quadratic systems

Quadratic systems play an important role in the modelling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems, it is of mandatory importance not only to determine whether the origin of the state space is locally asymptotically stable but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the region of attraction of the equilibrium. The proposed algorithm requires the solution of a suitable feasibility problem involving linear matrix inequalities constraints. An example illustrates the effectiveness of the proposed procedure by exploiting a population interaction model of three species.     

Estimating the domain of attraction for uncertain polynomial systems

Estimating the Domain of Attraction (DA) of equilibrium points is a problem of fundamental importance in systems engineering. Several approaches have been proposed for the case of known polynomial systems allowing one to find the Largest Estimate of the DA (LEDA) for a given Lyapunov Function (LF). However, the problem of estimating the Robust DA (RDA), that is the DA guaranteed for all possible uncertainties in an uncertain system, it is still an unsolved problem. In this paper, LMI methods are proposed for estimating the RDA in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope. Specifically, the issue of computing the Robust LEDA (RLEDA), that is the intersection of all LEDAs, is considered for common and parameter-dependent LFs, providing constant and parameter-dependent lower bounds. The computation of approximations with simple shape of the RLEDA in the case of parameter-dependent LFs is also discussed.     

On estimation of the domain of attraction for sliding mode control of underactuated nonlinear systems

A system is considered underactuated if the number of the actuator inputs is less than the number of degrees of freedom for the system. Sliding mode control for underactuated systems has been shown to be an effective way to achieve system stabilization. It involves exponentially stable sliding surfaces so that when the closed‐loop system trajectory reaches the surface, it moves along the surface while converging to the origin. In this paper, a general framework that provides sufficient conditions for asymptotic stabilization of underactuated nonlinear systems using sliding mode control in the presence of system uncertainties is presented. Specifically, it is shown that the closed‐loop system trajectories reach the sliding surface in finite time, and a constructive methodology to determine exponential stability of the closed‐loop system on the sliding surface is developed, which ensures asymptotic stability of the overall closed‐loop system. Furthermore, the aforementioned framework provides the basis to determine an estimate of the domain of attraction for the closed‐loop system with uncertainties. Finally, the results developed in the paper are experimentally validated using a linear inverted pendulum testbed to show a good match between the actual domain of attraction of the upward equilibrium state of the pendulum and its analytical estimate.Copyright © 2012 John Wiley & Sons, Ltd.     

Stability analysis for linear discrete-time systems subject to actuator saturation

In this paper, stability of discrete-time linear systems subject to actuator saturation is analyzed by combining the saturation-dependent Lyapunov function method with Finsler’s lemma. New stability test conditions are proposed in the enlarged space containing both the state and its time difference which allow extra degree of freedom and lead to less conservative estimation of the domain of attraction. Furthermore, based on this result, a useful lemma and an iterative LMI-based optimization algorithm are also developed to maximize an estimation of domain of attraction. A numerical example illustrates the effectiveness of the proposed methods.     

Enlarging the domain of attraction of MPC controllers

This paper presents a method for enlarging the domain of attraction of nonlinear model predictive control (MPC). The usual way of guaranteeing stability of nonlinear MPC is to add a terminal constraint and a terminal cost to the optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated Lyapunov function. The domain of attraction of the controller depends on the size of the terminal region and the control horizon. By increasing the control horizon, the domain of attraction is enlarged but at the expense of a greater computational burden, while increasing the terminal region produces an enlargement without an extra cost.In this paper, the MPC formulation with terminal cost and constraint is modified, replacing the terminal constraint by a contractive terminal constraint. This constraint is given by a sequence of sets computed off-line that is based on the positively invariant set. Each set of this sequence does not need to be an invariant set and can be computed by a procedure which provides an inner approximation to the one-step set. This property allows us to use one-step approximations with a trade off between accuracy and computational burden for the computation of the sequence. This strategy guarantees closed loop-stability ensuring the enlargement of the domain of attraction and the local optimality of the controller. Moreover, this idea can be directly translated to robust MPC.     

Guaranteed domains of attraction for multivariable Luré systems via open Lyapunov surfaces

In this paper we provide guaranteed stability regions for multivariable Luré-type systems. Specifically, using the Luré–Postnikov Lyapunov function a guaranteed subset of the domain of attraction for a feedback system whose forward path contains a dynamic linear time-invariant system and whose feedback path contains multiple sector-bounded time-invariant memoryless nonlinearities is constructed via open Lyapunov surfaces. It is shown that the use of open Lyapunov surfaces yields a considerable improvement over closed Lyapunov surfaces in estimating the domain of attraction of the zero solution of the nonlinear system. An immediate application of this result is the computation of transient stability regions for multimachine power systems and computation of stability regions of anti-windup controllers for systems subject to input saturation. © 1997 by John Wiley & Sons, Ltd.     

Resilient iterative learning control for a class of discrete-time nonlinear systems under hybrid attacks

The security control problem for a class of unknown nonlinear systems is considered in this paper. For the nonlinear system running in the network environment, the measurement channel is subjected to hybrid attacks. Intermittent denial of service attacks and false data injection attacks are modeled as the hybrid attacks. According to the characteristics of the repetitive system, a resilient iterative learning control (ILC) algorithm under hybrid attacks is devised. Subsequently, the stability of the system is proved by mathematical derivation and theoretical analysis in the sense of mathematical expectation. The theoretical analysis results indicate that the resilient ILC algorithm can ensure the stability of the system, and the tracking error converges with the increased number of iterations. Finally, the validity of the algorithm is illustrated by numerical simulation and mobile robot simulation.     

Enlarging the domain of attraction and maximising convergence rate for delta operator systems with actuator saturation

In this paper, we present some new approaches to improve the feedback property of delta operator systems with actuator saturation. Both enlarging the domain of attraction and maximising the convergence rate are the desired feedback properties. The lifting technique is used to enlarge the domain of attraction for the delta operator systems subject to actuator saturation. The maximisation of convergence rate is realised by designing control gain inside a given ellipsoid. A necessary and sufficient condition is proposed for the contractive invariance of the given ellipsoid. Simulation results are provided to demonstrate the effectiveness of the developed techniques.     

Stability analysis and control of a class of differential‐algebraic nonlinear systems

A new methodology based on the linear matrix inequality (LMI) framework is presented for the computation of polynomial Lyapunov functions for differential–algebraic nonlinear systems. The method is applied for estimating the domain of attraction of equilibria and also as a control design tool to increase the damping of the system. Numerical examples are used to demonstrate the approach. The results show the potential of the proposed approach as a tool for analysis and control design. Copyright © 2004 John Wiley & Sons, Ltd.     

Static anti‐windup design for a class of nonlinear systems

This paper focuses on the problem of static anti‐windup design for a class of multivariable nonlinear systems subject to actuator saturation. The considered class regards all systems that are rational on the states or that can be conveniently represented by a rational system with algebraic constraints considering some variable changes. More precisely, a method is proposed to compute a static anti‐windup gain which ensures regional stability for the closed‐loop system assuming that a dynamic output feedback controller is previously designed to stabilize the nonlinear system. The results are based on a differential algebraic representation of rational systems. The control saturation effects are taken into account by the application of a generalized sector bound condition. From these elements, LMI‐based conditions are devised to compute an anti‐windup gain with the aim of enlarging the closed‐loop region of attraction. Several numerical examples are provided to illustrate the application of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

基于系统输出的嵌套饱和系统抗饱和补偿器设计

针对嵌套输入饱和系统的吸引域扩大问题,本文提出了一种基于系统输出的抗饱和补偿器激发策略,将被控系统输出信号经性能补偿器馈入到抗饱和补偿器激发环节中,形成蕴含系统实时性能信息的抗饱和激发新机制,克服了传统抗饱和激发机制无法直接反映系统性能的缺点.基于上述抗饱和控制新框架,本文建立了抗饱和补偿器及性能补偿器存在的充分条件,并依此构建了优化问题求解最优补偿器增益以实现扩大闭环系统吸引域的目的.仿真结果表明本文方法的有效性.     

Fault estimation observer design for a class of nonlinear multiagent systems in finite‐frequency domain

This paper focuses on the fault estimation observer design problem in the finite‐frequency domain for a class of Lipschitz nonlinear multiagent systems subject to system components or actuator fault. First, the relative output estimation error is defined based on the directed communication topology of multiagent systems, and an observer error system is obtained by connecting adaptive fault estimation observer and the state equation of the original system. Then, sufficient conditions for the existence of the fault estimation observer are obtained by using a generalized Kalman‐Yakubovich‐Popov lemma and properties of the matrix trace, which guarantee that the observer error system satisfies robustness performance in the finite‐frequency domain. Meanwhile, the pole assignment method is used to configure the poles of the observer error system in a certain area. Finally, the simulation results are presented to illustrate the effectiveness of the proposed method.     

Estimating the domain of attraction for non-polynomial systems via LMI optimizations

This paper proposes a strategy for estimating the domain of attraction (DA) for non-polynomial systems via Lyapunov functions (LFs). The idea consists of converting the non-polynomial optimization arising for a chosen LF in a polynomial one, which can be solved via LMI optimizations. This is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst-case remainders in truncated Taylor expansions. Moreover, a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen LF, and another condition is provided for establishing whether such an estimate has been found. The proposed strategy can readily be exploited with variable LFs in order to search for optimal estimates. Lastly, it is worth remarking that no other method is available to estimate the DA for non-polynomial systems via LMIs.     

An analysis and design method for a class of nonlinear systems with nested saturations

This paper is concerned with the analysis and design of a class of nonlinear systems subject to nested saturations. The proposed controller incorporates both an extended state observer (ESO), which is utilised to estimate the nonlinear dynamics of the plant, as well as a set of observer-based feedbacks. We first present analysis results for systems with nonlinear ESOs and show that local stabilisation can be achieved in a region including the origin. Then, in the case that the ESO is in linear form (LESO), the conditions for determining the estimate of the domain of attraction of the resulting closed-loop system are formulated as a convex optimisation problem. A linear matrix inequality based algorithm is then established to design the feedback gains and the LESO gain. An illustrative example is given to show the effectiveness of the proposed approach.     

Inner approximations of domains of attraction for a class of switched systems by computing Lyapunov‐like functions

Domain of attraction plays an important role in control systems analysis, which is usually estimated by sublevel sets of Lyapunov functions. In this paper, based on the concept of common Lyapunov‐like functions, we propose an iteration method for estimating domains of attraction for a class of switched systems, where the state space is divided into several regions, each region is described by polynomial inequalities, and any region has no intersection among with each other. Starting with an initial inner estimate of domain of attraction, we first present a theoretical framework for obtaining a larger inner estimate by iteratively computing common Lyapunov‐like functions. Then, for obtaining a required initial inner estimate of domain of attraction, we propose a higher‐order truncation and linear semidefinite programming–based method for computing a common Lyapunov function. Successively, the theoretical framework is under‐approximatively realized by using S‐procedure and sum‐of‐squares programming, associated with a coordinatewise iteration idea. Finally, we implement our method and test it on some examples with comparisons. These computation and comparison results show the advantages of our method.