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.     

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.     

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.     

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.     

Analysis and control of the jump modes behavior of 2-D singular systems—Part I: Structural stability

This paper considers the problem of structural stability of 2-D singular systems. Firstly, some properties of structural stability of 2-D general singular systems are presented. Sufficient and necessary conditions for the structural stability of the 2-D singular systems are given. Then, by extending the Lyapunov approach for the structural stability of 1-D continuous singular systems to the discrete case, a generalized Lyapunov equation approach to the analysis of the structural stability of 2-D singular Roesser models (2-D SRM) is proposed. The existence of a solution to the generalized Lyapunov equation gives a sufficient condition for the structural stability of the 2-D SRM.     

Isogeometric segmentation. Part II: On the segmentability of contractible solids with non-convex edges

Motivated by the discretization problem in isogeometric analysis, we consider the challenge of segmenting a contractible boundary-represented solid into a small number of topological hexahedra. A satisfactory segmentation of a solid must eliminate non-convex edges because they prevent regular parameterizations. Our method works by searching a sufficiently connected edge graph of the solid for a cycle of vertices, called a cutting loop, which can be used to decompose the solid into two new solids with fewer non-convex edges. This can require the addition of auxiliary vertices to the edge graph. We provide theoretical justification for our approach by characterizing the cutting loops that can be used to segment the solid, and proving that the algorithm terminates. We select the cutting loop using a cost function. For this cost function we propose terms which help to select geometrically and combinatorially favorable cutting loops. We demonstrate the effects of these terms using a suite of examples.     

Spatial stability of shear deformable curved beams with non-symmetric thin-walled sections. II: F. E. solutions and parametric study

In the companion paper, an improved formulation for spatial stability analysis of shear deformable thin-walled curved beams with non-symmetric cross-sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of semi-tangential rotations. Thus the elastic strain energy and the potential energy due to initial stress resultants are consistently derived. Also closed-form solutions for in-plane and lateral-torsional buckling of curved beams subjected to uniform compression and pure bending are newly derived for mono-symmetric thin-walled curved beams under simply supported and clamped end conditions. In this paper, F. E. procedures are developed by using curved and straight beam elements with non-symmetric cross-sections. Analytical and numerical solutions for spatial buckling of shear deformable thin-walled circular beams are presented and compared in order to illustrate the accuracy and the practical usefulness of this study. In addition, the extensive parametric studies are performed on spatial stability behavior of curved beams. Particularly transition and crossover phenomena of buckling mode shapes with change in curvature and length of beam on buckling for curved beams are investigated for the first time.     

Coupled model for the non-linear analysis of anisotropic sections subjected to general 3D loading. Part 1: Theoretical formulation

A numerical model for the coupled analysis of arbitrary shaped cross sections made of heterogeneous-anisotropic materials under 3D combined loading is formulated. The theory is derived entirely from equilibrium considerations and based on the superposition of the 3D section’s distortion and the traditional plane section hypothesis. 3D stresses and strains fields are obtained as well as a section stiffness matrix reflecting coupling effects between normal and tangential forces due to material anisotropy. Traditional generalized strains and stresses are maintained as input and output variables. The proposed model is suitable as a constitutive law for frame elements in the analysis of complete structures.