Stability radii of positive linear systems under fractional perturbations

In this paper we study stability radii of positive linear discrete‐time systems under fractional perturbations. It is shown that real and complex stability radii coincide and can be computed by a simple formula. From the obtained results, we apply to derive estimates and computable formulae for the stability radii of positive linear delay systems. Finally, a simple example is given to illustrate the obtained results. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust stability of Metzler operators under parameter perturbations

In this paper we study how the spectral bound of Metzler operators changes under parameter perturbations. Characterizations of the stability radii of Metzler operator with respect to this type of disturbances are established. The results generalize those obtained in (Vietnam J. Math. 2006; 34 :357–368; Vietnam J. Math. 1998; 26 :147–163). Copyright © 2009 John Wiley & Sons, Ltd.     

Strong stability radii of positive linear time‐delay systems

In this paper, we study robustness of the strong delay‐independent stability of linear time‐delay systems under multi‐perturbation and affine perturbation of coefficient matrices via the concept of strong delay‐independent stability radius (shortly, strong stability radius). We prove that for class of positive time‐delay systems, complex and real strong stability radii of positive linear time‐delay systems under multi‐perturbations (or affine perturbations) coincide and they are computed via simple formulae. Apart from that, we derive solution of a global optimization problem associated with the problem of computing of the strong stability radii of a positive linear time‐delay system. An example is given to illustrate the obtained results. Copyright © 2005 John Wiley & Sons, Ltd.     

Robustness of stability of nonlinear systems with stochastic delay perturbations

Suppose there exist random disturbances to a given exponentially stable system and the stochastically perturbed system is described by a stochastic differential-functional equation. In this paper a sufficient condition is given so that the perturbed system remains exponentially stable. In the case where the perturbation depends only on several states of the past we obtain a condition under which the perturbed system is absolutely exponentially stable.     

Stability radii of positive discrete-time systems under affine parameter perturbations

In this paper we study stability radii of positive linear discrete-time systems under affine parameter perturbations. It is shown that real and complex stability radii of positive systems coincide for arbitrary perturbation structures, in particular, for blockdiagonal disturbances as considered in μ-analysis. Estimates and computable formulae are derived for these stability radii. The results are derived for arbitrary perturbation norms induced by monotonic vector norms (e.g. p-norms, 1⩽p⩽∞). © 1998 John Wiley & Sons, Ltd.     

Robust stability analysis of uncertain multiorder fractional systems: Young and Jensen inequalities approach

Robust stability analysis of multiorder fractional linear time‐invariant systems is studied in this paper. In the present study, first, conservative stability boundaries with respect to the eigenvalues of a dynamic matrix for this kind of systems are found by using Young and Jensen inequalities. Then, considering uncertainty on the dynamic matrix, fractional orders, and fractional derivative coefficients, some sufficient conditions are derived for the stability analysis of uncertain multiorder fractional systems. Numerical examples are presented to confirm the obtained analytical results.     

New criteria for exponential stability of nonlinear difference systems with time-varying delay

General nonlinear time-varying difference systems with time-varying delay are considered. Some new explicit criteria for global exponential stability are given. Two examples are given to illustrate the obtained results.     

Stability radii of infinite-dimensional systems subjected to unbounded stochastic perturbations

In this paper, we use the framework of stability radii to study the robust stability of linear deterministic systems on real Hilbert spaces which are subjected to unbounded stochastic perturbations. First, we establish an existence and uniqueness theorem of the solution of the abstract equation describing the system. Then we characterize the stability radius in terms of a Lyapunov equation or equivalently in terms of the norm of an input-output operator.     

Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces

We study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space. The results are obtained by using semigroup theory, fractional calculus and Schauder’s fixed point theorem. Multi-delay controls and a fractional nonlocal condition are introduced. Furthermore, we present an appropriate set of sufficient conditions for the considered fractional nonlocal multi-delay control system to be approximately controllable. An example to illustrate the abstract results is given.     

Robust FOPID controller design for fractional‐order delay systems using positive stability region analysis

In this paper, a robust fractional‐order PID (FOPID) controller design method for fractional‐order delay systems is proposed based on positive stability region (PSR) analysis. Firstly, the PSR is presented to improve the existing stability region (SR) in D‐decomposition method. Then, the optimal fractional orders λ and μ of FOPID controller are achieved at the biggest three‐dimensional PSR, which means the best robustness. Given the optimal λ and μ, the other FOPID controller parameters k_p, k_i, k_d can be solved under the control specifications, including gain crossover frequency, phase margin, and an extended flat phase constraint. In addition, the steps of the proposed robust FOPID controller design process are listed at length, and an example is given to illustrate the corresponding steps. At last, the control performances of the obtained robust FOPID controller are compared with some other controllers (PID and FOPI). The simulation results illustrate the superior robustness as well as the transient performance of the proposed control algorithm.     

BIBO stability of some classes of delay systems and fractional systems

A new test is presented for the BIBO stability of delay systems of neutral type with a single delay, specified in terms of their transfer functions, enabling us to decide on some cases that were previously open. Next, a class of fractional systems is considered, and a method is given for determining the stability intervals for such systems.     

分数阶时滞非线性多智能体系统的自适应控制

针对分数阶多智能体系统中存在时滞和非线性特性, 时滞往往会引起控制系统的性能下降甚至出现系统 不稳定等问题, 提出了一种含时滞非线性的分数阶多智能体系统自适应控制方法. 对于多智能体系统的控制协议, 设计了基于领导者和相邻智能体状态信息的自适应控制协议, 减小了过大常数控制增益带来的能源浪费. 对于一 致性, 利用图论基础、分数阶Halanay不等式稳定性定理、Kronecker积和Schur补引理, 获得了分数阶时滞非线性多 智能体系统的LMI一致性条件. 仿真结果验证了本文算法的正确性和有效性. 由于整数阶系统是分数阶系统的特殊 形式, 本文结论可以直接推广到整数阶多智能体系统中.     

Analysis of stability of stochastic dynamic systems under Poisson perturbations. II. Stability of solutions in quadratic mean of systems of linear stochastic differential equations under Poisson perturbations

Necessary and sufficient conditions of asymptotic stability in quadratic mean are obtained for trivial solutions of systems of linear stochastic differential equations under Poisson perturbations. Model problems are analyzed. Part I of this article is published in No. 4 (2005). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 50–66, November–December 2005.     

Robust exponential stability conditions for retarded systems with Lipschitz nonlinear stochastic perturbations

This paper investigates robust mean‐square exponential stability of a class of uncertain stochastic state‐delayed systems with Lipschitz nonlinear stochastic perturbation. Based on Lyapunov–Krasovskii functional (LKF) method and free‐weighting matrix technique, some new delay‐dependent stability conditions are established in terms of linear matrix inequalities (LMIs). In order to reduce the conservatism, (1) the delay is divided into several segments, i.e. the delay decomposition method is applied; (2) cross terms estimation is avoided; (3) some information of the cross terms relationships which has not been involved in Reference (IET Control Theory Appl. 2008; 2(11):966–973) is considered. Moreover, from the mathematical point of view, the results obtained by free‐weighting matrix technique can be equivalently re‐formulated by simpler ones without involving any additional free matrix variables. The effectiveness of the method is demonstrated by numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust stability and stabilization of discrete singular systems with interval time-varying delay and linear fractional uncertainty

The robust stability and robust stabilization problems for discrete singular systems with interval time-varying delay and linear fractional uncertainty are discussed. A new delay-dependent criterion is established for the nominal discrete singular delay systems to be regular, causal and stable by employing the linear matrix inequality (LMI) approach. It is shown that the newly proposed criterion can provide less conservative results than some existing ones. Then, with this criterion, the problems of robust stability and robust stabilization for uncertain discrete singular delay systems are solved, and the delay-dependent LMI conditions are obtained. Finally, numerical examples are given to illustrate the e?ectiveness of the proposed approach.     

Analysis of fractional delay systems of retarded and neutral type

This paper analyses several properties linked to the robust control of fractional differential systems with delays. The BIBO stability of both retarded and neutral fractional delay systems is related to the location of their poles. In the particular case of retarded systems, we give some properties of the poles of the system and the singular values of its Hankel operator.     

Robust non-fragile fractional order PID controller for linear time invariant fractional delay systems

A fractional order PID controller is designed to stabilize fractional delay systems with commensurate orders and multiple commensurate delays, where the time delays in the system may belong to several distinct intervals. Moreover, the controller parameters should belong to given intervals. In order to stabilize the system, the D-subdivision method is employed to choose the stabilizing set of the controller parameters from their available values. Furthermore, the nearest values of the obtained stabilizing set to their mean values are selected as the controller parameters so that a non-fragile controller is concluded. Two numerical examples evaluate the proposed control design method.     

Exact bounds for robust stability of output feedback controlled fractional‐order systems with single parameter perturbations

This article investigates exact robust stability bounds of output feedback controlled fractional‐order systems with the commensurate order and single parameter perturbations in all system coefficient matrices. First, a sufficient and necessary condition for robust asymptotical stability of such systems is obtained by using the Kronecker product. Then the maximal upper bounds and minimum lower bounds for robust asymptotical stability are established, respectively, without conservatism by transforming such problems into checking whether the matrix with single parameter perturbations is nonsingular or not. Finally, two numerical examples are given to show the effectiveness of the proposed results.     

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.

     

Robust stability of non-standard nonlinear singularly perturbed discrete systems with uncertainties

In the past several decades, the singularly perturbed discrete systems have received much attention for the stability analysis and controller design. Recently, there are some results about the nonlinear singularly perturbed discrete systems. Compared with the existing result, we consider the robust stability of the uncertain nonlinear singularly perturbed discrete systems with the less conservative assumption via the Lyapunov function method. Moreover, the previous results of the singularly perturbed discrete system are only applied to the system, which is composed of the slow part and the fast part, separately. However, we consider the non-standard nonlinear singularly perturbed discrete system in which the slow part and the fast part coexist, that is, a general case of the nonlinear singularly perturbed discrete systems. Then, by using the lower-order subsystems from two standard systems, we present the robust stability of the non-standard nonlinear singularly perturbed discrete system with uncertainties.