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.     

General solution of terminal control and observation problems

Mathematical problems of construction of general solutions for terminal control and observation problems are considered. The conditions of existence of a general solution of these problems for linear dynamic systems with a continuous and discrete argument are given. The study was carried out with the support of the Ukrainian Scientific and Technical Center (project No. 545) and the State Fund for Fundamental Research of Ukraine (project No. 1.4/369). Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 80–89, March–April, 2000.     

Direct methods of solving systems of linear algebraic ef ions with complex a-matrices

Algorithms of computer algebra are proposed for solving systems of linear algebraic equations with complex á- matrices. An analysis of roundoff errors for the computational schemes considered is given. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 144–156, March–April, 2000.     

Remarks on stability of solutions of the property domination problem

A class of combinatorial problems is considered whose investigation and solution require the notions of the theory of fuzzy sets. The necessary and sufficient conditions of stability are given. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 36–40, July–August, 1999.     

Implicit-system approach to the robust stability for a class of singularly perturbed linear systems

Asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear differential-algebraic equations (DAEs) are studied. The asymptotic behavior of the stability radius for a singularly perturbed implicit system is characterized as the parameter in the leading term tends to zero. The main results are obtained in direct and short ways which involve some basic results in linear algebra and classical analysis, only. Our results can be extended to other singular perturbation problems for DAEs of more general form.     

Modeling and Control of Linear Two-time Scale Systems: Applied to Single-Link Flexible Manipulator

This paper deals with the problem of H _∞ control of linear two-time scale systems. The authors’ attention is focused on the robust regulation of the system based on a new modeling approach under the assumption of norm-boundedness of the fast dynamics. In the proposed approach, the fast dynamics are treated as a norm-bounded disturbance (dynamic uncertainty). In this view, the synthesis is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. It should be noted, however, that this scheme is significantly different from the conventional approaches of order reduction for linear two-time scale systems. Specifically, in the present work, explicitly or implicitly, all the dynamics of the system are taken into consideration. In other words, the portion that is treated as a perturbation is incorporated in the design by its maximum possible gain – in the L ₂ sense – over different values of the inputs. One of the advantagesof this approach is that – unlike in the conventional approaches of the order reduction – the reduced-order system still keeps some information of the ‘deleted’ subsystem. Also, we consider the robust stability analysis and stability bound improvement of perturbed parameter (ɛ) in the two-time scale systems by using linear fractional transformations and structured singular values (μ) approach. In this direction, by introducing the parametric uncertainty and dynamic uncertainty in the two-time scale systems, we represent the system as a standard μ-interconnection framework by using linear fractional transformations, and derive a set of new stability conditions for the system in the frequency domain. The exact solution of ɛ-bound is characterized. It is shown that, in spite of the coupling between the dynamic uncertainties and certain dynamics, the designed H _∞ controller stabilizes the overall closed-loop system, in the presence of norm-bounded disturbances. To show the effectiveness of the approach, the modeling of the single-link flexible manipulator and control of the Tip-position of the manipulator by utilizing the mentioned method are presented in the case study.     

Solution of systems of linear algebraic equations with preliminary flattening

A method of improving computing properties of matrices of systems of linear algebraic equations is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 144–149, September–October, 1999.     

LP-oriented upper bounds for the weighted stability number of a graph

Upper bounds for the weighted stability number of a graph are considered that are based on the approximation of its stable set polytope by linear inequalities for odd cycles and p-wheels in the graph. Algorithms are developed for finding upper bounds on the basis of solution of LP problems with a finite number of inequalities produced by the shortest path algorithm for a special graph. The results of test experiments are given for graphs with several hundred or thousand vertices. This work was partially financially supported by the CRDF Cooperative Grants Program under grant UKM2-2812-KV-06. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 157–170, January–February 2009.     

The Regular Linear Systems Associated with the Shift Semigroups and Application to Control Linear Systems with Delay

We investigate the infinite dimensional control linear systems with delays in the state and input. We give a new variation of constants formula when the state and control delay operators are unbounded. We prove the existence of mild and classical solutions of such systems. Our approach is based on the theory of abstract and regular linear systems introduced by Salamon (Math Syst Theor 21:147–164, 1989) and Weiss (Isr J Math 65:17–43, 1989). Finally, we apply our abstract framework to an example from population dynamics.     

Cycles in economic systems with open labor markets

A dynamic macromodel of an economic system with a monopsonic labor market and variable labor resources is considered. The initiation conditions for classical business cycles are investigated for this model. Bifurcation analysis of the model is performed to estimate the effect of the minimum wage on the way such cycles form. The results of theoretical analysis are supplemented with numerical experiments. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 48–72, July–August 2008.     

离散区间系统的二次稳定性及其稳定裕度分析

研究了动态离散区间系统的鲁棒稳定性问题,提出了系统二次稳定的充分与必要条件,以及相应的稳定裕度的计算方法,并推广到离散参数不确定系统.所有结论以线性矩阵不等式(LMI)的形式给出.利用功能强大的LMI工具,求解非常方便.所给实例表明,该方法用于确定动态离散区间系统的鲁棒稳定性及其稳定裕度,非常有效.     

Lyapunov function method for investigation of stability of stochastic Ito random-structure systems with impulse Markov switchings. I. General theorems on the stability of stochastic impulse systems

The asymptotic stochastic stability on the whole and asymptotic p-stability on the whole are investigated with the help of Lyapunov functions and the concept of an infinitesimal operator over solutions to a system (to calculate this operator, it suffices to know only the coefficients of the system). The stability under constant perturbations is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 135–145, March–April 2009.     

Regularization of a lexicographic vector problem of integer programming

A stability criterion for a vector integer linear problem of lexicographic optimization is obtained. A regularization method is proposed that allows us to reduce a possible unstable output problem to a sequence of perturbed stable equivalent problems. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 125–130, November–December, 1999.     

Boolean problem of sequential minimization of moduli of linear functions and stability theorems

A multicriterion Boolean problem of lexicographic optimization with partial criteria represented by moduli (absolute values) of linear functions is considered. Five types of stability of a set of lexicographic optima under “small” variations in the parameters of a vector criterion are investigated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 178–187, September–October 2007.     

An Implicit Transition Matrix Approach to Stability Analysis of Flexible Multi-Body Systems

The stability of linear systems defined by ordinarydifferential equations with constant or periodic coefficients can beassessed from the spectral radius of their transition matrix. Inclassical applications of this theory, the transition matrix isexplicitly computed first, then its eigenvalues are evaluated; if thelargest eigenvalue is larger than unity, the system is unstable. Theproposed implicit transition matrix approach extracts the dominanteigenvalues of the transition matrix using the Arnoldi algorithm,without the explicit computation of this matrix. As a result, theproposed implicit method yields stability information at a far lowercomputational cost than that of the classical approach, and is ideallysuited for stability computations of systems involving a large number ofdegrees of freedom. Examples of application of the proposed methodologyto flexible multi-body systems are presented that demonstrate itsaccuracy and computational efficiency.     

General principles of construction of knowledge computer systems

The architecture of knowledge of computer systems and some problems of artificial intelligence (AI) are considered such as knowledge acquisition, knowledge processing, knowledge formation, etc. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 81–101, July–August 2006.     

Stability, pseudostability, and quasistability of a multicriterion problem on a system of subsets

Stability, pseudostability, and quasistability, and the stability kernel of a multicriterion problem are considered on a system of subsets with Σ-minimax and Σ-minimin criteria for perturbations of the parameters of a vector criterion. Lower attainable estimates are found for the radii of stability, quasi stability, and pseudostability and the stability kernel of such a problem in the space of perturbed parameters for the case of the Chebyshev norm. This work has been carried out with financial support from the Fundamental Research Fund of the Republic of Belorussia and the German Fund of Academic Exchanges (DAAD), FRG. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 111–124, September–October, 1999.     

参数摄动混杂离散系统的鲁棒稳定性

采用线性矩阵不等式(LMI)方法研究离散事件状态转移条件为状态依赖的参数摄动线性混杂离散系统的鲁棒稳定性问题, 提出此类系统全局鲁棒渐近稳定性判定定理, 基于分段Lyapunov函数给出了一般混杂离散系统在Lyapunov意义下局部稳定的判定定理, 该定理可将线性混杂离散系统的稳定性问题转化为LMI问题, 在此基础上提出了参数摄动线性混杂离散系统在Lyapunov意义下局部鲁棒稳定的充分条件.     

广义不确定周期时变系统的鲁棒稳定性分析

研究广义不确定周期时变系统的鲁棒稳定性问题.基于广义周期时变系统Lyapunov不等式,提出了广义不确定周期时变系统鲁棒稳定的概念,采用矩阵不等式(LMI)方法,得到了该类系统鲁棒稳定的充分必要条件;然后,进一步研究了在状态反馈控制下保证闭环系统鲁棒稳定的条件,给出了一族状态反馈鲁棒稳定器的设计方法;最后,引入了广义周期时变系统二次稳定的概念,并讨论了二次稳定性与鲁棒稳定性之间的关系.     

Stabilization of nonlinear sandwich systems via state feedback—Discrete‐time systems

A recent paper (IEEE Trans. Aut. Contr. 2010; 55 (9):2156–2160) considered stabilization of a class of continuous‐time nonlinear sandwich systems via state feedback. This paper is a discrete‐time counterpart of it. The class of nonlinear sandwich systems consists of saturation elements sandwiched between linear systems. We focus first on single‐layer sandwich systems, which consist of a single saturation sandwiched between two linear systems. For such systems, we present necessary and sufficient conditions for semi‐global and global stabilization problems by state feedback, and develop design methodologies to achieve the prescribed stabilization. We extend the results to single‐layer sandwich systems subject to additional actuator saturation. Finally, we discuss further extension to general multi‐layer sandwich systems with an arbitrary number of saturations sandwiched between linear systems, both with and without actuator saturation. The design methodologies can be viewed as extensions of classical low‐gain design methodologies developed during the 1990s in the context of stabilizing linear systems subject to actuator saturation. Copyright © 2010 John Wiley & Sons, Ltd.