On Stochastic ISS of Time-Varying Switched Systems with Generic Lévy Switching Signals

Switched systems in which switching among subsystems occurs at random time instants find numerous applications in engineering. Stability analysis of such systems, however, is quite challenging. This paper investigates the stochastic input-to-state stability of this class of switched systems. The random switching instants are modeled by a non-decreasing, positive, and real-valued Levy process, which, at every time instant, selects the active subsystem from a family of deterministic systems. No assumption on the stability of subsystems is presumed; they can be stable or unstable. Stochastic properties of the switching signal are coupled with a family of Lyapunov-like functions to obtain a sufficient condition for stochastic input-to-state stability

Qualitative Studies of Nonlinear Hybrid Systems

A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. Hybrid systems arise in a wide variety of important applications in diverse areas, ranging from biology to computer science to air traffic dynamics. The interaction of continuous- and discrete-time dynamics in a hybrid system often leads to very rich dynamical behavior and phenomena that are not encountered in purely continuous- or discrete-time systems. Investigating the dynamical behavior of hybrid systems is of great theoretical and practical importance. The objectives of this thesis are to develop the qualitative theory of nonlinear hybrid systems with impulses, time-delay, switching modes, and stochastic disturbances, to develop algorithms and perform analysis for hybrid systems with an emphasis on stability and control, and to apply the theory and methods to real-world application problems. Switched nonlinear systems are formulated as a family of nonlinear differential equations, called subsystems, together with a switching signal that selects the continuous dynamics among the subsystems. Uniform stability is studied emphasizing the situation where both stable and unstable subsystems are present. Uniformity of stability refers to both the initial time and a family of switching signals. Stabilization of nonlinear systems via state-dependent switching signal is investigated. Based on assumptions on a convex linear combination of the nonlinear vector fields, a generalized minimal rule is proposed to generate stabilizing switching signals that are well-defined and do not exhibit chattering or Zeno behavior. Impulsive switched systems are hybrid systems exhibiting both impulse and switching effects, and are mathematically formulated as a switched nonlinear system coupled with a sequence of nonlinear difference equations that act on the switched system at discrete times. Impulsive switching signals integrate both impulsive and switching laws that specify when and how impulses and switching occur. Invariance principles can be used to investigate asymptotic stability in the absence of a strict Lyapunov function. An invariance principle is established for impulsive switched systems under weak dwell-time signals. Applications of this invariance principle provide several asymptotic stability criteria. Input-to-state stability notions are formulated in terms of two different measures, which not only unify various stability notions under the stability theory in two measures, but also bridge this theory with the existent input/output theories for nonlinear systems. Input-to-state stability results are obtained for impulsive switched systems under generalized dwell-time signals. Hybrid time-delay systems are hybrid systems with dependence on the past states of the systems. Switched delay systems and impulsive switched systems are special classes of hybrid time-delay systems. Both invariance property and input-to-state stability are extended to cover hybrid time-delay systems. Stochastic hybrid systems are hybrid systems subject to random disturbances, and are formulated using stochastic differential equations. Focused on stochastic hybrid systems with time-delay, a fundamental theory regarding existence and uniqueness of solutions is established. Stabilization schemes for stochastic delay systems using state-dependent switching and stabilizing impulses are proposed, both emphasizing the situation where all the subsystems are unstable. Concerning general stochastic hybrid systems with time-delay, the Razumikhin technique and multiple Lyapunov functions are combined to obtain several Razumikhin-type theorems on both moment and almost sure stability of stochastic hybrid systems with time-delay. Consensus problems in networked multi-agent systems and global convergence of artificial neural networks are related to qualitative studies of hybrid systems in the sense that dynamic switching, impulsive effects, communication time-delays, and random disturbances are ubiquitous in networked systems. Consensus protocols are proposed for reaching consensus among networked agents despite switching network topologies, communication time-delays, and measurement noises. Focused on neural networks with discontinuous neuron activation functions and mixed time-delays, sufficient conditions for existence and uniqueness of equilibrium and global convergence and stability are derived using both linear matrix inequalities and M-matrix type conditions. Numerical examples and simulations are presented throughout this thesis to illustrate the theoretical results

Stabilization of positive linear discrete-time systems by using a Brauer's theorem

The stabilization problem of positive linear discrete-time systems (PLDS) by linear state feedback is considered. A method based on a Brauer s theorem is proposed for solving the problem. It allows us to modify some eigenvalues of the system without hanging the rest of them. The problem is studied for the single-input single-output (SISO) and for multi-input multioutput (MIMO) cases and sufficient conditions for stability and positivity of the closed-loop system are proved.The results are illustrated by numerical examples and the proposed method is used in stochastic systems.This work is supported by the Spanish DGI Grant MTM2010-18228.Cantó Colomina, B.; Cantó Colomina, R.; Kostova, S. (2014). Stabilization of positive linear discrete-time systems by using a Brauer's theorem. Scientific World Journal. 2014:1-6. https://doi.org/10.1155/2014/856356S162014Caccetta, L., & Rumchev, V. G. (2000). Annals of Operations Research, 98(1/4), 101-122. doi:10.1023/a:1019244121533Allen, L. J. S., & van den Driessche, P. (2008). The basic reproduction number in some discrete-time epidemic models. Journal of Difference Equations and Applications, 14(10-11), 1127-1147. doi:10.1080/10236190802332308Delchamps, D. F. (1988). State Space and Input-Output Linear Systems. doi:10.1007/978-1-4612-3816-4Méndez-Barrios, C.-F., Niculescu, S.-I., Chen, J., & Maya-Méndez, M. (2013). Output feedback stabilisation of single-input single-output linear systems with I/O network-induced delays. An eigenvalue-based approach. International Journal of Control, 87(2), 346-362. doi:10.1080/00207179.2013.834075Anderson, B. D. O., Ilchmann, A., & Wirth, F. R. (2013). Stabilizability of linear time-varying systems. Systems & Control Letters, 62(9), 747-755. doi:10.1016/j.sysconle.2013.05.003De Leenheer, P., & Aeyels, D. (2001). Stabilization of positive linear systems. Systems & Control Letters, 44(4), 259-271. doi:10.1016/s0167-6911(01)00146-3Fornasini, E., & Valcher, M. E. (2012). Stability and Stabilizability Criteria for Discrete-Time Positive Switched Systems. IEEE Transactions on Automatic Control, 57(5), 1208-1221. doi:10.1109/tac.2011.2173416Bru, R., Cantó, R., Soto, R. L., & Urbano, A. M. (2011). A Brauer’s theorem and related results. Central European Journal of Mathematics, 10(1), 312-321. doi:10.2478/s11533-011-0113-0Soto, R. L., & Rojo, O. (2006). Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra and its Applications, 416(2-3), 844-856. doi:10.1016/j.laa.2005.12.026Silva, M. S., & de Lima, T. P. (2003). Looking for nonnegative solutions of a Leontief dynamic model. Linear Algebra and its Applications, 364, 281-316. doi:10.1016/s0024-3795(02)00569-4Mourad, B. (2013). Generalization of some results concerning eigenvalues of a certain class of matrices and some applications. Linear and Multilinear Algebra, 61(9), 1234-1243. doi:10.1080/03081087.2012.746330Pakshin, P. V., & Ugrinovskii, V. A. (2006). Stochastic problems of absolute stability. Automation and Remote Control, 67(11), 1811-1846. doi:10.1134/s0005117906110051Brauer, A. (1952). Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices. Duke Mathematical Journal, 19(1), 75-91. doi:10.1215/s0012-7094-52-01910-8Perfect, H. (1955). Methods of constructing certain stochastic matrices. II. Duke Mathematical Journal, 22(2), 305-311. doi:10.1215/s0012-7094-55-02232-8Berman, A., & Plemmons, R. J. (1994). Nonnegative Matrices in the Mathematical Sciences. doi:10.1137/1.9781611971262Cantó, B., Cardona, S. C., Coll, C., Navarro-Laboulais, J., & Sánchez, E. (2011). Dynamic optimization of a gas-liquid reactor. Journal of Mathematical Chemistry, 50(2), 381-393. doi:10.1007/s10910-011-9941-1Fieberg, J., & Ellner, S. P. (2001). Stochastic matrix models for conservation and management: a comparative review of methods. Ecology Letters, 4(3), 244-266. doi:10.1046/j.1461-0248.2001.00202.

On input-to-state stability of stochastic retarded systems with Markovian switching

This note develops a Razumikhin-type theorem on pth moment input-to-state stability of hybrid stochastic retarded systems (also known as stochastic retarded systems with Markovian switching), which is an improvement of an existing result. An application to hybrid stochastic delay systems verifies the effectiveness of the improved result

Robust filtering for a class of stochastic uncertain nonlinear time-delay systems via exponential state estimation

Copyright [2001] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.We investigate the robust filter design problem for a class of nonlinear time-delay stochastic systems. The system under study involves stochastics, unknown state time-delay, parameter uncertainties, and unknown nonlinear disturbances, which are all often encountered in practice and the sources of instability. The aim of this problem is to design a linear, delayless, uncertainty-independent state estimator such that for all admissible uncertainties as well as nonlinear disturbances, the dynamics of the estimation error is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are proposed to guarantee the existence of desired robust exponential filters, which are derived in terms of the solutions to algebraic Riccati inequalities. The developed theory is illustrated by numerical simulatio

Time-and event-driven communication process for networked control systems: A survey

Copyright © 2014 Lei Zou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In recent years, theoretical and practical research topics on networked control systems (NCSs) have gained an increasing interest from many researchers in a variety of disciplines owing to the extensive applications of NCSs in practice. In particular, an urgent need has arisen to understand the effects of communication processes on system performances. Sampling and protocol are two fundamental aspects of a communication process which have attracted a great deal of research attention. Most research focus has been on the analysis and control of dynamical behaviors under certain sampling procedures and communication protocols. In this paper, we aim to survey some recent advances on the analysis and synthesis issues of NCSs with different sampling procedures (time-and event-driven sampling) and protocols (static and dynamic protocols). First, these sampling procedures and protocols are introduced in detail according to their engineering backgrounds as well as dynamic natures. Then, the developments of the stabilization, control, and filtering problems are systematically reviewed and discussed in great detail. Finally, we conclude the paper by outlining future research challenges for analysis and synthesis problems of NCSs with different communication processes.This work was supported in part by the National Natural Science Foundation of China under Grants 61329301, 61374127, and 61374010, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany

On almost sure stability of hybrid stochastic systems with mode-dependent interval delays

This note develops a criterion for almost sure stability of hybrid stochastic systems with mode-dependent interval time delays, which improves an existing result by exploiting the relation between the bounds of the time delays and the generator of the continuous-time Markov chain. The improved result shows that the presence of Markovian switching is quite involved in the stability analysis of delay systems. Numerical examples are given to verify the effectiveness

SMC design for robust H∞ control of uncertain stochastic delay systems

Recently, sliding mode control method has been extended to accommodate stochastic systems. However, the existing results employ an assumption that may be too restrictive for many stochastic systems. This paper aims to remove this assumption and present in terms of LMIs a sliding mode control design method for stochastic systems with state delay. In some cases, the proposed method provides a control scheme for finite-time stabilization of stochastic delay systems