LMI stability conditions for uncertain rational nonlinear systems

This paper presents LMI conditions for local, regional, and global robust asymptotic stability of rational uncertain nonlinear systems. The uncertainties are modeled as real time varying parameters with magnitude and rate of variation bounded by given polytopes and the system vector field is a rational function of the states and uncertain parameters. Sufficient LMI conditions for asymptotic stability of the origin are given through a rational Lyapunov function of the states and uncertain parameters. The case where the time derivative of the Lyapunov function is negative semidefinite is also considered and connections with the well known LaSalle's invariance conditions are established. In regional stability problems, an algorithm to maximize the estimate of the region of attraction is proposed. The algorithm consists of maximizing the estimate for a given target region of initial states. The size and shape of the target region are recursively modified in the directions where the estimate can be enlarged. The target region can be taken as a polytope (convex set) or union of polytopes (non‐convex set). The estimates of the region of attraction are robust with respect to the uncertain parameters and their rate of change. The case of global and orthant stability problems are also considered. Connections with some results found in sum of squares based methods and other related methods found in the literature are established. The LMIs in this paper are obtained by using the Finsler's Lemma and the notion of annihilators. The LMIs are characterized by affine functions of the state and uncertain parameters, and they are tested at the vertices of a polytopic region. It is also shown that, with some additional conservatism, the use of the vertices can be avoided by modifying the LMIs with the S‐Procedure. Several numerical examples found in the literature are used to compare the results and illustrate the advantages of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Hierarchy of LMI conditions for the stability analysis of time-delay systems

Assessing stability of time-delay systems based on the Lyapunov–Krasovskii functionals has been the subject of many contributions. Most of the results are based, first, on an a priori design of functionals and, finally, on the use of the famous Jensen’s inequality. In contrast with this design process, the present paper aims at providing a generic set of integral inequalities which are asymptotically non conservative and then to design functionals driven by these inequalities. The resulting stability conditions form a hierarchy of LMI which is competitive with the most efficient existing methods (delay-partitioning, discretization and sum of squares), in terms of conservatism and of complexity. Finally, some examples show the efficiency of the method.     

Lyapunov stability analysis for nonlinear nabla tempered fractional order systems

Due to the restriction of practical systems in time or space, tempered fractional calculus becomes more reasonable than the traditional fractional calculus. It is known that stability analysis is a crucial issue for control systems. This paper concerns the stability analysis issue of nabla tempered fractional order systems for the first time. The (discrete time) tempered Mittag–Leffler stability is defined firstly and then a stability criterion is derived via Lyapunov method. Besides, boundedness and attractiveness are also investigated.     

Mittag-Leffler stability of fractional order nonlinear dynamic systems

In this paper, we propose the definition of Mittag-Leffler stability and introduce the fractional Lyapunov direct method. Fractional comparison principle is introduced and the application of Riemann-Liouville fractional order systems is extended by using Caputo fractional order systems. Two illustrative examples are provided to illustrate the proposed stability notion.     

分数阶线性定常系统的稳定性及其判据

介绍了分数阶微分方程和分数阶系统 ,给出分数阶线性定常系统的传递函数描述和状态空间描述 .给出了分数阶线性定常系统的稳定性条件 ,并结合分数阶状态方程给出定理的证明 .直接从复分析中的辐角原理出发 ,推导出分数阶线性定常系统 2个有效的稳定性判据 :分数阶系统奈奎斯特判据和分数阶系统对数频率判据 .通过实例验证了其有效性     

Tuning fractional order proportional integral controllers for fractional order systems

In this paper, two fractional order proportional integral controllers are proposed and designed for a class of fractional order systems. For fair comparison, the proposed fractional order proportional integral (FOPI), fractional order [proportional integral] (FO[PI]) and the traditional integer order PID (IOPID) controllers are all designed following the same set of the imposed tuning constraints, which can guarantee the desired control performance and the robustness of the designed controllers to the loop gain variations. This proposed design scheme offers a practical and systematic way of the controllers design for the considered class of fractional order plants. From the simulation and experimental results presented, both of the two designed fractional order controllers work efficiently, with improved performance comparing with the designed stabilizing integer order PID controller by the observation. Moreover, it is interesting to observe that the designed FO[PI] controller outperforms the designed FOPI controller following the proposed design schemes for the class of fractional order systems considered.     

不确定分数阶时滞系统的鲁棒稳定性判定准则

基于退化分析方法提出一种判定准则, 用于分析不确定分数阶时滞系统的稳定性. 介绍一种分数阶积分算子的有理逼近方法, 在此基础上采用整数阶系统逼近分数阶系统, 从而将难以判定的分数阶系统稳定性问题转化为由逼近偏差作为不确定项的整数阶系统稳定性问题进行处理. 利用积分不等式法研究逼近系统稳定性, 得到LMI 形式的稳定性判据. 仿真结果表明, 所提出方法能够有效分析这类系统的稳定性.

     

A note on the stability of fractional order systems

In this paper, a new approach is suggested to investigate stability in a family of fractional order linear time invariant systems with order between 1 and 2. The proposed method relies on finding a linear ordinary system that possesses the same stability property as the fractional order system. In this way, instead of performing the stability analysis on the fractional order systems, the analysis is converted into the domain of ordinary systems which is well established and well understood. As a useful consequence, we have extended two general tests for robust stability check of ordinary systems to fractional order systems.     

Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems

A Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems is presented via an auxiliary integer degree polynomial. The presented Routh test is a classical Routh table test on the auxiliary integer degree polynomial derived from and for the commensurate fractional degree polynomial. The theoretical proof of this proposed approach is provided by utilizing Argument principle and Cauchy index. Illustrative examples show efficiency of the presented approach for stability test of commensurate fractional degree polynomials and commensurate fractional order systems. So far, only one Routh-type test approach [1] is available for the commensurate fractional degree polynomials in the literature. Thus, this classical Routh-type test approach and the one in [1] both can be applied to stability analysis and design for the fractional order systems, while the one presented in this paper is easy for peoples, who are familiar with the classical Routh table test, to use.     

Improved LMI representations for delay-independent and delay-dependent stability conditions

Some new linear matrix inequality (LMI) representations for delay-independent and delay-dependent stability conditions are obtained by introducing additional matrices and eliminating the product coupling of the system matrices and the Lya-punov matrices. The results improve conservativeness of the given conditions for the analysis and the design of tune-delay systems with polytopic-type uncertainty.     

Finite‐time stability of fractional order impulsive switched systems

This paper considers the finite‐time stability of fractional order impulsive switched systems. First, by using the fractional order Lyapunov function, Mittag–Leffler function, and Gronwall–Bellman lemma, two sufficient conditions are given to verify the finite‐time stability of fractional order nonlinear systems. Then, the concept of finite‐time stability is extended to fractional order impulsive switched systems. A sufficient condition is given to verify the finite‐time stability of fractional order impulsive switched systems by combining the method of average dwell time with fractional order Lyapunov function. Finally, two numerical examples are provided to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

LMI stability criterion with less variables for time-delay systems

Slack variables approach is an important technique for tackling the delay-dependent stability problem for systems with time-varying delay. In this paper, a new delay-dependent stability criterion is presented without introducing any slack variable. The technique is based on a simply integral inequality. The result is shown to be equivalent to some existing ones but includes the least number of variables. Thus, redundant selection and computation can be avoided so that the computational burden can be largely reduced. Numerical examples are given to illustrate the effectiveness of the proposed stability conditions. Recommended by Editorial Board member Young Soo Suh under the direction of Editor Jae Weon Choi. The authors would like to thank the Associate Editor and the Reviewers for their very helpful comments and suggestions. This work was supported in part by the Funds for Creative Research Groups of China under Grant 60821063, by the State Key Program of National Natural Science of China under Grant 60534010, by the Funds of National Science of China under Grant 60674021, 60774013, 60774047, National 973 Program of China under Grant No. 2009CB320604, and by the Funds of Ph.D. program of MOE, China under Grant 20060145019 and the 111 Project B08015. Xun-Lin Zhu received the B.S. degree in Applied Mathematics from Information Engineering Institute, Zhengzhou, China, in 1986, the M.S. degree in basic mathematics from Zhengzhou University, Zhengzhou, China, in 1989, and the Ph.D. degree in Control Theory and Engineer-ing from Northeastern University, Shenyang, China, in 2008. Currently, he is an Associate Professor at Zhengzhou University of Light Industry, Zhengzhou, China. His research interests include neural networks and networked control systems. Guang-Hong Yang received the B.S. and M.S. degrees in Northeast University of Technology, China, in 1983 and 1986, respectively, and the Ph.D. degree in Control Engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, non-fragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Control, Automation, and Systems (IJCAS), and an Associate Editor of the Conference Editorial Board of the IEEE Control Systems Society. Tao Li was born in 1979. He is now pursuing a Ph.D. degree in Research Institute of Automation Southeast University, China. His current research interests include time-delay systems, neural networks, robust control, fault detection and diagnosis. Chong Lin received the B.Sci and M.Sci in Applied Mathematics from the Northeastern University, China, in 1989 and 1992, respectively, and the Ph.D in Electrical and Electronic Engineering from the Nanyang Technological University, Singapore, in 1999. He was a Research Associate with the University of Hong Kong in 1999. From 2000 to 2006, he was a Research Fellow with the National University of Singapore. He is currently a Profesor with the Institute of Complexity Science, Qingdao University, China. His current research interests are mainly in the area of systems analysis and control. Lei Guo was born in 1966. He received the Ph.D. degree in Control Engineering from Southeast University (SEU), PR China, in 1997. From 1999 to 2004, he has worked at Hong Kong University, IRCCyN (France), Glasgow University, Loughborough University and UMIST, UK. Now he is a Professor in School of Instrument Science and Opto-Electronics Engineering, Beihang University. He also holds a Visiting Professor position in the University of Manchester, UK and an invitation fellowship in Okayama University, Japan. His research interests include robust control, stochastic systems, fault detection, filter design, and nonlinear control with their applications.     

LMI conditions for time-varying uncertain systems can be non-conservative

Establishing robust asymptotic stability of uncertain systems affected by time-varying uncertainty is a key problem. LMI sufficient conditions have been proposed in the literature for addressing this problem based on homogeneous polynomial Lyapunov functions. Unfortunately, till now it has been unclear whether these conditions are also necessary. This paper proposes a proof in order to show that one of these conditions is not only sufficient but also necessary for a sufficiently large degree of the Lyapunov function.     

分数阶线性系统的内部和外部稳定性研究

介绍了分数阶线性定常系统的状态方程描述和传递函数描述．运用拉普拉斯变换和留数定理，给出并证明了分数阶线性定常系统的内部和外部稳定性条件，并讨论了其相互关系．以一个粘弹性系统的实例验证了上述方法的正确性．     

Rational multiplier IQCs for uncertain time-delays and LMI stability conditions

Describes a set of delay-dependent integral quadratic constraint (IQC) stability conditions for time-delay uncertainty. The IQCs are linearly parameterized in terms of a pair of rational stability multipliers, each active over one of a pair of complementary frequency intervals. Using the finite-frequency positive real lemma, each of these finite-frequency IQC conditions are shown to be equivalent to a frequency-independent linear matrix inequality condition, thereby dispensing with the need for frequency-sweeping.     

Arnoldi-based model reduction for fractional order linear systems

In this paper, the Arnoldi-based model reduction methods are employed to fractional order linear time-invariant systems. The resulting model has a smaller dimension, while its fractional order is the same as that of the original system. The error and stability of the reduced model are discussed. And to overcome the local convergence of Padé approximation, the multi-point Arnoldi algorithm, which can recursively generate a reduced-order orthonormal basis from the corresponding Krylov subspace, is used. Numerical examples are given to illustrate the accuracy and efficiency of the proposed methods.     

LMI conditions for quadratic and robust D-stability of interval systems

Sufficient conditions for the quadratic D-stability and further robust D-stability of interval systems are presented in this paper. This robust D-stability condition is based on a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities （LMIs） defined at a series of partial-vertex-based interval matrices other than the total vertex matrices as in previous results. The results contain the usual quadratic and robust stability of continuous-time and discrete-time interval systems as particular cases. The illustrative example shows that this method is effective and less conservative for checking the quadratic and robust D-stability of interval systems.     

On LMI conditions to design observers for Lipschitz nonlinear systems

This paper addresses the problem of observer design for Lipschitz nonlinear systems via LMI. The goal of this note is to clarify some recent results and to propose a new design methodology. This is based on the reformulation of the Lipschitz property using some mathematical tools. This reformulation is a relevant and useful Lipschitz condition, which leads to less restrictive LMI conditions. To show the superiority of this latter, two numerical examples are presented.