Guaranteed asymptotic stability for a class of uncertain linear dynamical systems

We consider a class of linear dynamical systems with bounded, Lebesgue-measurable uncertainties in the system and input matrices as well as in the input itself. A state feedback control is derived, which guarantees global, uniform asymptotic stability of the zero state; this control is continuous, except at the zero state.This paper is based in part on research supported by the National Science Foundation.     

Asymptotic stability of switched linear time-varying systems based on top-floor function

This paper considers the problem of asymptotic stability for switched linear time-varying (SLTV) systems. First, some stability conditions for SLTV systems are given by using infinite integrals. Then, based on the results obtained, two stability conditions are proposed by combining the methods of top-floor function and average dwell time. Moreover, using strict top-floor function, a stability condition is also provided when some subsystems are unstable. With the help of top-floor function, the stability problem of SLTV systems can be simplified and solved by using the technique of linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the theoretical results.     

Hyperplane method for reachable state estimation for linear time-invariant systems

A numerical algorithm is presented for generating inner and outer approximations for the set of reachable states for linear time-invariant systems. The algorithm is based on analytical results characterizing the solutions to a class of optimization problems which determine supporting hyperplanes for the reachable set. Explicit bounds on the truncation error for the finite-time case yield a set of so-called -supporting hyperplanes which can be generated to approximate the infinite-time reachable set within an arbitrary degree of accuracy. At the same time, an inner approximation is generated as the convex hull of points on the boundary of the finite-time reachable set. Numerical results are presented to illustrate the hyperplane method. The concluding section discusses directions for future work and applications of the method to problems in trajectory planning in servo systems.This research was supported in part by Digital Equipment Corporation through the American Electronics Association Fellowship Loan Program and by the National Science Foundation under Grant No. ECS-84-04607.     

Equivalence of several characteristics of dual switched linear systems

This paper is concerned with the equivalence of several dynamic characteristics between mutually dual switched linear systems, both discrete- and continuous-time cases are considered. Two systems are mutually dual means that their system matrices are transpose of each other. The dynamic properties considered in this paper include four types: exponential stability, global uniform asymptotic stability, attractivity, and weak attractivity. It is shown that every one of the four dynamic properties of a switched linear system implies each of the four dynamic properties of its dual system, and vice versa. The main results enable us to investigate these dynamic properties of switched system by virtue of study related to the corresponding properties of its dual system, thus providing an alternative way to explore the dynamics of switched systems. A numerical example is provided to illustrate the obtained theoretical conclusions.     

Topological stability of linear semi-infinite inequality systems

In this note, we analyze the relationship between the lower semicontinuity of the feasible set mapping for linear semi-infinite inequality systems and the so-called topological stability, which is held when the solution sets of all the systems obtained by sufficiently small perturbations of the data are homeomorphic to each other. This topological stability and its relation with the Mangasarian-Fromovitz constraints qualification have been studied deeply by Jongen et al. in Ref. 1. The main difference of our approach is that we are not assuming any kind of structure for the index set and, consequently, any particular property for the functional dependence between the inequalities and the associated indices. In addition, we deal with systems whose solution sets are not necessarily bounded.This work has been supported partially by the DGICYT of Spain, Grant PB93-0943, by Generalitat Valenciana, Grant GV-2219/94, and by IVEI, Grant 003/026.The authors would like to thank J. E. Martínez Legaz for his valuable comments.     

Stochastic stability for Markovian jump linear systems associated with a finite number of jump times

This paper deals with a stochastic stability concept for discrete-time Markovian jump linear systems. The random jump parameter is associated to changes between the system operation modes due to failures or repairs, which can be well described by an underlying finite-state Markov chain. In the model studied, a fixed number of failures or repairs is allowed, after which, the system is brought to a halt for maintenance or for replacement. The usual concepts of stochastic stability are related to pure infinite horizon problems, and are not appropriate in this scenario. A new stability concept is introduced, named stochastic τ-stability that is tailored to the present setting. Necessary and sufficient conditions to ensure the stochastic τ-stability are provided, and the almost sure stability concept associated with this class of processes is also addressed. The paper also develops equivalences among second order concepts that parallels the results for infinite horizon problems.     

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.      

Some criteria for the existence of invariant measures and asymptotic stability for random dynamical systems on polish spaces

Tomasz Szarek presented interesting criteria for the existence of invariant measures and asymptotic stability of Markov operators on Polish spaces. Hans Crauel in his book presented the theory of random probabilistic measures on Polish spaces showing that notions of compactness and tightness for such measures are in one-to-one correspondence with such notions for non-random measures on Polish spaces, in addition to the criteria under which the space of random measures is itself a Polish space. This result allowed the transfer of results of Szarek to the case of random dynamical systems in the sense of Arnold. These criteria are interesting because they allow to use the existence of simple deterministic Lyapunov type function together with additional conditions to show the existence of invariant measures and asymptotic stability of random dynamical systems on general Polish spaces.     

On guaranteed stability of uncertain linear systems via linear control

This paper addresses the problem of stabilizing an uncertain linear system. The uncertaintyq(·) which enters the dynamics is nonstistical in nature. That is, noa priori statistics forq(·) are assumed; only boundsQ on the admissible variations ofq(·) are taken as given. The results given here applied to so-called matched systems differ from previous results in two ways. Firstly, the stabilizing control in this paper is linear; for this same class of problems, many of the existing results would require a nonlinear control. Furthermore, those results which do in fact yield linear controls are only valid when a certain matrix (q) (formed using the given data) is negative definite for allq Q. In contrast, the theory given here only requires compactness of the bounding setQ. Secondly, we show that the so-called matching conditions (used in earlier work) can be generalized so as to encompass a larger class of dynamical systems.This research was supported by the US Department of Energy under Contract No. ET-78-S-01-3390.     

受控马尔可夫跳线性系统的随机鲁棒稳定性

本文讨论了受控连续和离散时间马尔可夫跳线性系统的随机鲁棒稳定性,并且给出了此时该系统发生马尔可夫跳的转移速率的一个界.     

Uniform asymptotic stability for damped linear oscillators with variable parameters

In this paper, sufficient conditions for uniform asymptotic stability of the damped linear oscillators with variable coefficients are presented. The result of the present study can be applied to even the case of negative damping. The conditions presented herein are shown to be essential for uniform asymptotic stability.     

A quantitative study on the stability of delay discrete singular systems

For quadratic delay discrete singular systems, an algebraic criterion on the stability is established, and the size of the uniform stability region and asymptotic stability region around zero is estimated. Hence, the criterion is both qualitative and quantitative. With the computer techniques, the criterion dependent of delay is easy test and applies to the application in the practice. An illustrative simulation is given to illustrate the application of the obtained result.     

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

We consider the asymptotic stability of the time-varying dynamicsystem : = A(t)x, A(t) R^{n x n}, A(t) A= {A₁, ..., A_m}, where A_i is Hurwitz and where a set of non-singularmatrices T_{i j} exist such that any pair of matrices {T_{i j} A_iT_{i j}^–1, T_{i j} A_j T_{i j}^–1}, i, j {1, ..., m}, areupper triangular. Switching systems of this form are referredto as pairwise triangularizable switching systems. It can beestablished that (a) pairwise triangularizability is not sufficientto guarantee the existence of a common quadratic Lyapunov functionfor the linear time-invariant dynamic systems A_i : = A_i x; (b) additional conditions can be specified which guaranteeasymptotic stability of the switching system . In this paperwe also show that pairwise triangularizability is not even sufficientto guarantee asymptotic stability of the switching system .We also show that the method of proof of stability in (b), whichdoes not assume the existence of a common quadratic Lyapunovfunction, can be used to prove the asymptotic stability of moregeneral switching systems (systems that are not pairwise triangularizable).Finally, we show that our results can be used as the basis forthe design of practical control systems; namely, for the designof an automobile speed switched controller with guaranteed stabilityproperties.     

On almost sure stability conditions of linear switching stochastic differential systems

In modeling practical systems, it can be efficient to apply Poisson process and Wiener process to represent the abrupt changes and the environmental noise, respectively. Therefore, we consider the systems affected by these random processes and investigate their joint effects on stability. In order to apply Lyapunov stability method, we formulate the action of the infinitesimal generator corresponding to such a system. Then, we derive the almost sure stability conditions by using some fundamental convergence theorem. To illustrate the theoretical results, we construct an example to show that it is possible to achieve stabilization by using random perturbations.     

Improved results on the stability analysis of linear neutral systems with delay decay approach

In this paper, the investigation of the asymptotical stability of linear neutral systems with time-varying delay has been presented. In order to achieve the desired results, the integral inequality approach was used to express relationships between terms of Newton-Leibniz formula technique and was constructed an appropriate Lyapunov-Krasovskii functional. By improving a delay decay approach, the stability criteria for the zero solution of system were formulated as linear matrix inequalities (LMIs) which can be easily solved. Two numerical examples have been given to show the applicability of established assumptions and the effectiveness of proposed method by MATLAB-Simulink.     

On stability of discrete-time switched systems

This article deals with stability of discrete-time switched systems. Given a family of nonlinear systems and the admissible switches among the systems in the family, we first propose a class of switching signals under which the resulting switched system is globally asymptotically stable. We allow unstable systems in the family and our stability condition depends solely on asymptotic behaviour of the switching signals. We then discuss algorithmic construction of the above class of switching signals, and show that in the presence of exogenous inputs and outputs, a switching signal so constructed also ensures input/output-to-state stability for discrete-time switched nonlinear systems. We finally show that our class of switching signals that ensures global asymptotic stability also extends to the continuous-time setting with minor modifications under standard assumptions. We employ multiple Lyapunov-like functions and graph theoretic tools as the main apparatuses for our analysis.     

On the simultaneous diagonal stability of a pair of positive linear systems

In this paper we derive a necessary and sufficient condition for the existence of a diagonal common quadratic Lyapunov function (CQLF) for a pair of positive linear time-invariant (LTI) systems.     

时滞依赖型中立系统的鲁棒稳定性分析

研究了一类不确定中立型系统的时滞依赖稳定性问题.利用基于LMI正定解的存在性,给出了一个新时滞依赖型稳定性判据,相比于已有文献本文具有低的保守性.最后通过数值算例验证了本文得到的结论的正确性和有效性.