Robust stability analysis with cycling-based LPTV scaling: part I. Fundamental results on its relationship with lifting-based LPTV scaling

This paper is concerned with robust stability analysis of discrete-time linear periodically time-varying (LPTV) systems using the cycling-based LPTV scaling approach. To study the properties of this approach in comparison with the lifting-based LPTV scaling approach, we consider exploiting the framework of representing the associated robust stability conditions with infinite matrices. Since it serves as a common framework for comparing the two different LPTV scaling approaches, it provides us with new insights into the relationship between the cycling-based and lifting-based scaling approaches. In particular, we derive fundamental results that enable us to reduce the comparison, with respect to conservativeness in robust stability analysis, of the two scaling approaches with restricted and tractable classes of separators to a modified comparison of the associated classes of what we call infinite-dimensional separators arising in the above infinite matrix framework.     

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.     

Robust stability analysis of singular linear system with delay and parameter uncertainty

1Introduction It is well known that the existence of a delay in adynamical system may induce instability or poorperformances in various systems suchas electric,pneumatic,and hydraulic networks,chemical processes,longtransmission lines,etc.For a survey of time_delay systems,the reader can refer to a recent overview paper[1].Control of singular systems has been extensively studied inthe past years due to the fact that singular systems betterdescribe physical systems than regular ones.Agreat numb…     

Robust stability analysis of singular linear system with delay and parameter uncertainty

This paper deals with th e problem of robust stability for continuous-time sin gular systems with state delay and parameter uncerta inty.The uncertain singular systems with delay consi dered in this paper are assumed to be regular and impulse free.By decomposing the systems into slow and fast subsystems,a robust delay-dependent asymptotic stability criteria based on linear matrix inequality is proposed,which is derived by using Lyapunov-Krasovskii functionals,neither model transformatio n nor bounding for cross terms is required in the derivation of our delay-dependent result.The r obust delay-dependent stability criterion proposed in th is paper is a sufficient condition.Finally,numerical examples and Matlab simulation are provided to illustrate the effectiveness of the proposed method.     

Robust stability analysis of Smith predictor-based congestion control algorithms for computer networks

Congestion control is a fundamental building block in packet switching networks such as the Internet due to the fact that communication resources are shared. It has been shown that the plant dynamics is essentially made up of an integrator plus time delay and that a proportional controller plus a Smith predictor defines a simple and effective controller. It has also been shown that the TCP congestion control can be modelled using a Smith predictor plus a proportional controller. Due to the importance of this control structure in the field of data network congestion control, we analyse the robust stability of the closed-loop system in the face of delay uncertainties that are present in data networks due to queuing. In particular, by applying a geometric approach, we derive a bound on the proportional controller gain which is necessary and sufficient to guarantee the closed-loop stability for a given bound on the delay uncertainty.     

Robust stability of impulsive Takagi-Sugeno fuzzy systems with parametric uncertainties

This paper presents a kind of time-varying impulsive Takagi-Sugeno (T-S) fuzzy model with parametric uncertainties in which each subsystem of the model is time-varying. Several robust stabilities of time-varying systems with parametric uncertainties, such as general robust stability, robustly asymptotical stability and exponential stability, are studied using uniformly positive definite matrix functions and the Lyapunov method. Specifically, robust stability conditions of time-invariant impulsive T-S fuzzy systems are also derived in the formulation of quasi-linear matrix inequalities (QLMIs) and an iterative LMIs algorithm is designed for solving QLMIs. Finally, a unified chaotic system with continuous periodic switch and a unified time-invariant chaotic system are used for demonstrating the effectiveness of our respective results.     

Explicit construction of quadratic lyapunov functions for the small gain,positivity, circle,and popov theorems and their application to robust stability. part II: Discrete-time theory

In a companion paper (‘Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. Part I: Continuous-time theory’), Lyapunov functions were constructed in a unified framework to prove sufficiency in the small gain, positivity, circle, and Popov theorems. In this Part II, analogous results are developed for the discrete-time case. As in the continuous-time case, each result is based upon a suitable Riccati-like matrix equation that is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time-invariant system and a memoryless nonlinearity. Multivariable versions of the discrete-time circle and Popov criteria are obtained as extensions of known results. Each result is specialized to the linear uncertainty case and connections with robust stability for state-space systems is explored.     

Robust stability analysis of systems with real parametric uncertainty: A global optimization approach

This paper presents an optimization framework for the robustness analysis of linear and nonlinear systems with real parameter uncertainty. For linear systems, a nonlinear programming formulation for the exact calculation of the stability margin is presented. The potential of decomposition-based global optimization methods for the solution of this nonconvex problem is discussed. Next the concept of the stability margin is extended to a class of nonlinear systems. A nonlinear stability margin and a uniqueness margin are defined to address the effect of parametric uncertainty on the stability of a particular steady state, as well as on the number of steady states of the system. This analysis allows for the derivation of necessary and sufficient conditions for robust stability and robust uniqueness of the steady state of the system in the presence of parametric uncertainty.     

Noncausal linear periodically time-varying scaling for robust stability analysis of discrete-time systems: Frequency-dependent scaling induced by static separators

This article is concerned with robust stability analysis of discrete-time systems and introduces a novel and powerful technique that we call noncausal linear periodically time-varying (LPTV) scaling. Based on the discrete-time lifting together with the conventional but general scaling approach, we are led to the notion of noncausal LPTV scaling for LPTV systems, and its effectiveness is demonstrated with a numerical example. To separate the effect of noncausal and LPTV characteristics of noncausal LPTV scaling to see which is a more important source leading to the effectiveness, we then consider the case of LTI systems as a special case. Then, we show that even static noncausal LPTV scaling has an ability of inducing frequency-dependent scaling when viewed in the context of the conventional LTI scaling, while causal LPTV scaling fails to do so. It is further discussed that the effectiveness of noncausal characteristics leading to the frequency-domain interpretation can be exploited even for LPTV systems by considering the νN-lifted transfer matrices of N-periodic systems.     

Robust stability of Markovian jump discrete-time neural networks with partly unknown transition probabilities and mixed mode-dependent delays

This article discusses the robust stability problem for a class of uncertain Markovian jump discrete-time neural networks with partly unknown transition probabilities and mixed mode-dependent time delays. The transition probabilities of the mode jumps are considered to be partly unknown, which relax the traditional assumption in Markovian jump systems that all of them must be completely known a priori. The mixed time delays consist of both discrete and distributed delays that are dependent on the Markovian jump modes. By employing the Lyapunov functional and linear matrix inequality approach, some sufficient criteria are derived for the robust stability of the underlying systems. A numerical example is exploited to illustrate the developed theory.     

Robust stability analysis and stabilisation of uncertain impulsive positive systems with time delay

This paper considers the robust stability and stabilisation of uncertain impulsive positive systems with delay in state. The robust global exponential stability criterion under ranged dwell-time is first established by employing an impulse-time-dependent copositive Lyapunov function. Extensions to the exponential stability of the considered system under constant, arbitrary, maximal and minimal dwell-time are then derived. To positively stabilise the considered system under ranged dwell-time, a method of state-feedback controller design is presented based on the derived stability condition. Several numerical examples illustrate the reduced conservatism and effectiveness of the obtained theoretical results.     

A heat equation for freezing processes with phase change: stability analysis and applications

In this work, the stability properties as well as possible applications of a partial differential equation (PDE) with state-dependent parameters are investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (potential) Burgers’ equation. We show that for certain forms of coefficient functions, the PDE converges to a stationary solution given by (fixed) boundary conditions that make physical sense. These boundary conditions are either symmetric or asymmetric of Dirichlet type. Furthermore, we present an observer design based on the PDE model for estimation of inner-domain temperatures in block-frozen fish and for monitoring freezing time. We illustrate the results with numerical simulations.     

Robust stability analysis of switched uncertain systems with multiple time-varying delays and actuator failures

In this paper,the robust stability issue of switched uncertain multidelay systems resulting from actuator failures is considered.Based on the average dwell time approach,a set of suitable switching signals is designed by using the total activation time ratio between the stable subsystem and the unstable one.It is first proven that the resulting closed-loop system is robustly exponentially stable for some allowable upper bound of delays if the nominal system with zero delay is exponentially stable under these switching laws.Particularly,the maximal upper bound of delays can be obtained from the linear matrix inequalities.At last,the effectiveness of the proposed method is demonstrated by a simulation example.     

Linear stability analysis in fluid-structure interaction with transpiration. Part II: Numerical analysis and applications

This paper constitutes the numerical counterpart of the mathematical framework introduced in Part I. We address the problem of flutter analysis of a coupled fluid-structure system involving an incompressible Newtonian fluid and a reduced structure. We use the Linearization Principle approach developed in Part I, particularly suited for fluid-structure problems involving moving boundaries. Thus, the stability analysis is reduced to the computation of the leftmost eigenvalues of a coupled eigenproblem of minimal complexity. This eigenproblem involves the linearized incompressible Navier-Stokes equations and those of a reduced linear structure. The coupling is realized through specific transpiration interface conditions. The eigenproblem is discretized using a finite element approximation and its smallest real part eigenvalues are computed by combining a generalized Cayley transform and an implicit restarted Arnoldi method. Finally, we report three numerical experiments: a structure immersed in a fluid at rest, a cantilever pipe conveying a fluid flow and a rectangular bridge deck profile under wind effects. The numerical results are compared to former approaches and experimental data. The quality of these numerical results is very satisfactory and promising.     

On the transfer matrix of a neutral system: Characterizations of exponential stability in input—output terms

This paper investigates the stability of linear autonomous multivariable neutral systems from an input—output viewpoint. Several frequency-domain and input—output characterizations for exponential stability of neutral systems are given. We provide two examples which illustrate that the behaviour of neutral systems may be quite different from that of retarded systems. Moreover we give necessary and sufficient conditions for the transfer functions of a neutral system to belong to certain algebras of meromorphic functions introduced in this paper.     

Robust stability and control of uncertain linear discrete-time periodic systems with time-delay

This paper deals with the problems of robust stability analysis and robust control of linear discrete-time periodic systems with a delayed state and subject to polytopic-type parameter uncertainty in the state-space matrices. A robust stability criterion independent of the time-delay length as well as a delay-dependent criterion is proposed, where the former applies to the case of a constant time-delay and the latter allows for a time-varying delay lying in a given interval. The developed robust stability criteria are based on affinely uncertainty-dependent Lyapunov–Krasovskii functionals and are given in terms of linear matrix inequalities. These stability conditions are then applied to solve the problems of robust stabilization and robust _H∞$H_{\infty }$  control via static periodic state feedback. Numerical examples illustrate the potentials of the proposed robust stability and control methods.     

收益共享控制下闭环供应链系统稳定性研究

为了减少环境污染和提高资源的利用效率,产品再制造闭环供应链系统的稳定有序运行非常重要。为探索使系统达到稳定的方法,以供应链参与主体均为风险中性的两阶闭环供应链系统为背景,研究了收益共享控制模型对闭环供应链协调性和稳定性的影响,证明了收益共享控制能够使产品再制造闭环供应链稳定,提升供应链的绩效。最后,通过算例研究和收益共享控制模型的灵敏度分析,验证了该模型的有效性及实用性。     

On the degree of polynomial parameter-dependent Lyapunov functions for robust stability of single parameter-dependent LTI systems: A counter-example to Barmish's conjecture

In this paper, we consider the robust Hurwitz stability analysis problems of a single parameter-dependent matrix A(θ)?A₀+θA₁ over θ∈[-1,1], where A₀,A₁∈R^n×n with A₀ being Hurwitz stable. In particular, we are interested in the degree N of the polynomial parameter-dependent Lyapunov matrix (PPDLM) of the form that ensures the robust Hurwitz stability of A(θ) via . On the degree of PPDLMs, Barmish conjectured in early 90s that if there exists such P(θ), then there always exists a first-degree PPDLM P(θ)=P₀+θP₁ that meets the desired conditions, regardless of the size or rank of A₀ and A₁. The goal of this paper is to falsify this conjecture. More precisely, we will show a pair of the matrices A₀,A₁∈R^3×3 with A₀+θA₁ being Hurwitz stable for all θ∈[-1,1] and prove rigorously that the desired first-degree PPDLM does not exist for this particular pair. The proof is based on the recently developed techniques to deal with parametrized LMIs in an exact fashion and related duality arguments. From this counter-example, we can conclude that the conjecture posed by Barmish is not valid when n?3 in general.     

Sufficient and necessary conditions for the stability of second-order switched linear systems under arbitrary switching

Many practical systems can be modelled as switched systems, whose stability problem is challenging even for linear subsystems. In this article, the stability problem of second-order switched linear systems with a finite number of subsystems under arbitrary switching is investigated. Sufficient and necessary stability conditions are derived based on the worst-case analysis approach in polar coordinates. The key idea of this article is to partition the whole state space into several regions and reduce the stability analysis of all the subsystems to analysing one or two worst subsystems in each region. This article is an extension of the work for stability analysis of second-order switched linear systems with two subsystems under arbitrary switching.