Global Stability of Positive Discrete-time Time-varying Nonlinear Feedback Systems

The stability of time-invariant positive nonlinear systems is addressed. Necessary conditions for the stability of positive time-invariant continuous-time and discrete-time nonlinear systems are established. It is shown that the positive nonlinear systems are asymptotically stable only if the corresponding positive linear systems are asymptotically stable. Considerations are illustrated by numerical examples and simulations are performed in Matlab-Simulink environment.

The global stability of positive continuous-time standard and fractional order nonlinear feedback systems is investigated. New sufficient conditions for the global stability of these classes of positive nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple examples of positive nonlinear systems.

The paper presents new characterizations of the uniform global asymptotic stability (UGAS) for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability (WZSD) is equivalent to UGAS for globally uniformly stable systems. On the other hand, WZSD is further simplified by employing the notion of reduced limiting systems. Then, a second characterization of UGAS is proposed in terms of the detectability condition of the reduced limiting systems associated with the original system. As a result, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem but for time-varying, not necessarily periodic, systems.

Global asymptotic stability of nonlinear cascade systems

While the asymptotic stability of positive linear systems in the presence of bounded time delays has been thoroughly investigated, the theory for nonlinear positive systems is considerably less well-developed. This paper presents a set of conditions for establishing delay-independent stability and bounding the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, when the time delays have a known upper bound, we derive necessary and sufficient conditions for exponential stability of (a) continuous-time positive systems whose vector fields are homogeneous and cooperative, and (b) discrete-time positive systems whose vector fields are homogeneous and order preserving. We then present explicit expressions that allow us to quantify the impact of delays on the decay rate and show that the best decay rate of positive linear systems that our bounds provide can be found via convex optimization. Finally, we extend the results to general linear systems with time-varying delays.

This paper is concerned with practical asymptotic and exponential stability analysis of time-varying nonlinear systems. With the help of the new notion of practical stable functions, some differential Lyapunov inequalities based necessary and sufficient conditions are derived for testing global uniform practical asymptotic stability and practical exponential stability of general time-varying nonlinear systems. Therefore, we generalised some works which are already made in the literature. Furthermore, some illustrative examples are presented.

The object of this paper is to outline a stability theory for input-output problems using functional methods. More particularly, the aim is to derive open loop conditions for the boundedness and continuity of feedback systems, without, at the beginning, placing restrictions on linearity or time invariance. It will be recalled that, in the special case of a linear time invariant feedback system, stability can be assessed using Nyquist's criterion; roughly speaking, stability depends on the mounts by which signals are amplified and delayed in flowing around the loop. An attempt is made here to show that similar considerations govern the behavior of feedback systems in general-that stability of nonlinear time-varying feedback systems can often be assessed from certain gross features of input-output behavior, which are related to amplification and delay. This paper is divided into two parts: Part I contains general theorems, free of restrictions on linearity or time invariance; Part II, which will appear in a later issue, contains applications to a loop with one nonlinear element. There are three main results in Part I, which follow the introduction of concepts of gain, conicity, positivity, and strong positivity: THEOREM 1: If the open loop gain is less than one, then the closed loop is bounded. THEOREM 2: If the open loop can be factored into two, suitably proportioned, conic relations, then the closed loop is bounded. THEOREM 3: If the open loop can be factored into two positive relations, one of which is strongly positive and has finite gain, then the closed loop is bounded. Results analogous to Theorems I-3, but with boundedness replaced by continuity, are also obtained.

In this paper a new systematic switching control approach to adaptive stabilization of linear time-varying (LTV) discrete-time systems is presented.A feature of the localization-based method is its high model falsification capability, which in the case of LTI systems is manifested as the rapid convergence of the switching controller.We believe that the proposed method may help pave the way for design of practical adaptive switching controllers applicable to a wide range of linear time-invariant and timevarying systems.

This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time control systems is given. Numerical examples are included to illustrate the effectiveness of our results. MSC:15A09, 52A10, 74M05, 93D05.

In this note, we deal with the stability and stabilization problems for nonlinear quadratic discrete-time periodic systems. By using the quadratic Lyapunov function, and a so called periodic invariant set, delay-independent sufficient conditions for local stability and local stabilization for nonlinear quadratic discrete-time periodic systems are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

Adaptive stabilization of time-varying discrete-time linear systems

New stability criteria and stabilization methods on networked control systems (NCSs) with short time-varying delay (STVD) are proposed in this paper. An NCS with STVD is transformed into a time-varying discrete time system. And then, this discrete time system is converted into a equivalent time-invariant system with norm-bounded uncertainties by using robust control techniques. Using this method, the conservatism of the stability condition caused by STVD can be reduced. Based on that, a single norm-bounded uncertainty is replaced by N norm-bounded uncertainties to further reduce conservatism. Theoretical analysis shows that when N is increased, the stability condition becomes less conservative. For a fixed sampling period, the obtained stability conditions explicitly depend on the upper and lower bounds of the time delay. The existence condition and design method for the controllers are also presented. Finally, three numerical examples are provided to demonstrate the effectiveness of the proposed scheme.

The positivity and asymptotic stability of the discrete-time nonlinear systems are addressed. Necessary and sufficient conditions for the positivity of the systems and sufficient conditions for asymptotic stability of the positive systems are established. The proposed stability tests are based on Lyapunov method. The effectiveness of the tests are demonstrated on examples.

This paper investigates the stability of the nonlinear discrete-time strict-feedback system (NDSS) with Poisson noise. This work examines two n-dimensional systems: one with Poisson noise in the n-th state equation, and the other deals with Poisson noise in all state equations. First, a controller is designed for the discrete-time stochastic systems using the backstepping method. Then sufficient conditions are developed to ensure the exponential mean square stability of the system using Lyapunov theory. Finally, numerical examples are presented to demonstrate the usefulness of the proposed results.

Minimax control for discrete-time time-varying stochastic systems