On linear co-positive Lyapunov functions for sets of linear positive systems

In this paper we derive necessary and sufficient conditions for the existence of a common linear co-positive Lyapunov function for a finite set of linear positive systems. Both the state dependent and arbitrary switching cases are considered. Our results reveal an interesting characterisation of “linear” stability for the arbitrary switching case; namely, the existence of such a linear Lyapunov function can be related to the requirement that a number of extreme systems are Metzler and Hurwitz stable. Examples are given to illustrate the implications of our results.     

Discrete-time stochastic control systems: A continuous Lyapunov function implies robustness to strictly causal perturbations

Discrete-time stochastic systems employing possibly discontinuous state-feedback control laws are addressed. Allowing discontinuous feedbacks is fundamental for stochastic systems regulated, for instance, by optimization-based control laws. We introduce generalized random solutions for discontinuous stochastic systems to guarantee the existence of solutions and to generate enough solutions to get an accurate picture of robustness with respect to strictly causal perturbations. Under basic regularity conditions, the existence of a continuous stochastic Lyapunov function is sufficient to establish that asymptotic stability in probability for the closed-loop system is robust to sufficiently small, state-dependent, strictly causal, worst-case perturbations. Robustness of a weaker stochastic stability property called recurrence is also shown in a global sense in the case of state-dependent perturbations, and in a semiglobal practical sense in the case of persistent perturbations. An example shows that a continuous stochastic Lyapunov function is not sufficient for robustness to arbitrarily small worst-case disturbances that are not strictly causal. Our positive results are also illustrated by examples.     

Stability robustness of linear normal distributed parameter systems

This paper considers the stability robustness analysis problem for linear distributed parameter systems containing known perturbation operators multiplied by uncertain parameters. The nominal system operators are assumed to be normal, but allowed to be unbounded. The perturbation operators are confined to some relative bounded set, but may be unbounded also. By using the Lyapunov stability criterion, simple bounds on uncertain parameters are derived to ensure the stability of the perturbed systems. Examples are provided to illustrate the usage of the theoretical results.     

The stability robustness determination of state space models with real unstructured perturbations

This paper considers the robust stability of a linear time-invariant state space model subject to real parameter perturbations. The problem is to find the distance of a given stable matrix from the set of unstable matrices. A new method, based on the properties of the Kronecker sum and two other composite matrices, is developed to study this problem; this new method makes it possible to distinguish real perturbations from complex ones. Although a procedure to find the exact value of the distance is still not available, some explicit lower bounds on the distance are obtained. The bounds are applicable only for the case of real plant perturbations, and are easy to compute numerically; if the matrix is large in size, an iterative procedure is given to compute the bounds. Various examples including a 46th-order spacecraft system are given to illustrate the results obtained. The examples show that the new bounds obtained can have an arbitrary degree of improvement over previously reported ones. This work has been supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A4396.     

Piecewise Lyapunov functions for robust stability of linear time-varying systems

In this paper, we investigate the use of two-term piecewise quadratic Lyapunov functions for robust stability of linear time-varying systems. By using the so-called S-procedure and a special variable reduction method, we provide numerically efficient conditions for the robust asymptotic stability of the linear time-varying systems involving the convex combinations of two matrices. An example is included to demonstrate the usefulness of our results.     

Improved measures of stability robustness for linear state space models

In this paper, the aspect of "stability robustness" of linear systems is analyzed in the time domain. A bound on the structured perturbation of an asymptotically stable linear system is obtained to maintain stability using a Lyapunov matrix equation solution. The resulting bound is shown to be an improved bound over the ones recently reported in the literature. Also, special cases of the nominal system matrix are considered, for which the bound is given in terms of the nominal matrix, thereby, avoiding the solution of the Lyapunov matrix equation. Examples given include comparison of the proposed approach with the recently reported results.     

A Lyapunov theorem for semigroups of operators

The well known condition for the existence of a unique solution to finite dimensional Lyapunov equations is extended to a family of Hilbert space operators.     

Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls

This paper is concerned with piecewise-affine (PWA) functions as Lyapunov function candidates for stability analysis of time-invariant discrete-time linear systems with saturating closed-loop control inputs. Using a PWA model of saturating closed-loop system, new necessary and sufficient conditions for a PWA function be a Lyapunov function are presented. Based on linear programming formulation of these conditions, an effective algorithm is proposed for construction of such Lyapunov functions for estimation of the region of local asymptotic stability. Compared to piecewise-linear functions, like Minkowski functions, PWA functions are more adequate to capture the dynamical effects of saturation nonlinearities, giving strictly less conservative results. The complexity of the proposed approach is polynomial in state dimension and exponential in saturating control dimension, being hence appropriate for problems with large state dimension but with few saturating inputs.     

On the Lyapunov matrices for integral delay systems

We study Lyapunov matrices for the class of integral delay systems with constant kernel and one delay. The uniqueness and computational issues of these Lyapunov matrices for exponentially stable systems are investigated.     

On the simultaneous diagonal stability of linear discrete-time systems

This paper studies the set of time-invariant linear discrete-time systems in which each system has a diagonal quadratic Lyapunov function. First, it is shown that there is generally no common diagonal quadratic Lyapunov function for such a set of systems even if the set is assumed to be commutative. It is also shown that the commutativity assures the existence of a common diagonal quadratic Lyapunov function inside the set of 2×2 systems or the set of nonnegative systems. Then, two simple topological results are presented concerning the simultaneous diagonal stability on the set of nonnegative systems. The first is a measure of the difference of matrices that assures the simultaneous diagonal stability. The second is a measure of the commutativity of matrices.     

Lyapunov matrices: Existence and uniqueness issues

A necessary and sufficient condition for the existence and uniqueness of Lyapunov matrices for neutral type single delay systems is presented.     

Instability conditions for linear time delay systems: a Lyapunov matrix function approach

Instability conditions for linear time delay systems of retarded type, with distributed delay, and of neutral type are given. The approach is based on using the converse results on the existence of special quadratics lower bounds for the Lyapunov–Krasovskii functional of complete type associated to these systems.     

Stability Analysis of Systems With Time-varying Delay via a Novel Lyapunov Functional

This paper investigates the stability problem for time-varying delay systems. To obtain a larger delay bound, this paper uses the second-order canonical Bessel-Legendre (B-L) inequality. Secondly, using four couples of integral terms in the augmented Lyapunov-Krasovskii function (LKF) to enhance the relationship between integral functionals and other vectors. Furthermore, unlike the construction of the traditional LKF, a novel augmented LKF is constructed with two new delay-product-type terms, which adds more state information and leads to less conservative results. Finally, two numerical examples are provided to demonstrate the effectiveness and the significant improvement of the proposed stability criteria.      

2-D奇异线性离散系统渐近稳定性的一类Lyapunov 方法

引入2－D奇异一般离散状态空间模型的Lyapunov方程,探讨了该模型的渐近稳定性、特征多项式的根式以及2－D Lyapunov矩阵方程间的关系,给出了系统渐近稳定性的充分条件。     

采用多Lyapunov函数的混杂系统稳定性研究

针对一类由离散事件监控的连续动态子系统组成的混杂动态系统, 首先分析利用多Lyapunov函数方法已有成果, 指出切换超平面为滑动模时, 利用这种方法不能确保混杂系统的稳定. 基于Filipov理论给出了能活稳定性结果. 对于混杂系统的连续动态子系统为线性时不变情况下, 研究了混杂系统二次镇定条件. 最后给出一个例子来说明本文方法.     

Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.     

离散广义系统稳定性分析与控制的Lyapunov方法

利用Lyapunov方法,研究离散广义系统稳定性分析与控制问题.得到了离散广义 系统正则、具有因果关系且渐近稳定的等价条件;还给出了相关的鲁棒稳定性分析与镇定方 法.     

Stability of uncertain piecewise affine systems with time delay: delay-dependent Lyapunov approach

This article addresses the problem of robust stability of piecewise affine (PWA) uncertain systems with unknown time-varying delay in the state. It is assumed that the uncertainty is norm bounded and that upper bounds on the state delay and its rate of change are available. A set of linear matrix inequalities (LMIs) is derived providing sufficient conditions for the stability of the system. These conditions depend on the upper bound of the delay. The main contributions of the article are as follows. First, new delay-dependent LMI conditions are derived for the stability of PWA time-delay systems. Second, the stability conditions are extended to the case of uncertain PWA time delay systems. Numerical examples are presented to show the effectiveness of the approach.     

Stability analysis of discrete-time switched systems: a switched homogeneous Lyapunov function method

This paper addresses the stability issue of discrete-time switched systems with guaranteed dwell-time. The approach of switched homogeneous Lyapunov function of higher order is formally proposed. By means of this approach, a necessary and sufficient condition is established to check the exponential stability of the considered system. With the observation that switching signal is actually arbitrary if the dwell time is one sample time, a necessary and sufficient condition is also presented to verify the exponential stability of switched systems under arbitrary switching signals. Using the augmented argument, a necessary and sufficient exponential stability criterion is given for discrete-time switched systems with delays. A numerical example is provided to show the advantages of the theoretical results.     

Stability analysis for discrete linear systems with state saturation by a saturation-dependent Lyapunov functional

This paper concerns the stability analysis problem of discrete linear systems with state saturation using a saturation-dependent Lyapunov functional.We introduce a free matrix characterized by the sum of the absolute value of each elements for each row less than 1,which makes the state with saturation constraint reside in a convex polyhedron.A saturation-dependent Lyapunov functional is then designed to obtain a sufficient condition for such systems to be globally asymptotically stable.Based on this stability criterion,the state feedback control law synthesis problem is also studied.The obtained results are formulated in terms of bilinear matrix inequalities that can be solved by the presented iterative linear matrix inequality algorithm.Two numerical examples are used to demonstrate the effectiveness of the proposed method.