Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays

This paper is concerned with analyzing input-to-state stability (ISS) and integral-ISS (iISS) for nonlinear impulsive systems with delays. Razumikhin-type theorems are established which guarantee ISS/iISS for delayed impulsive systems with external input affecting both the continuous dynamics and the discrete dynamics. It is shown that when the delayed continuous dynamics are ISS/iISS but the discrete dynamics governing the impulses are not, the ISS/iISS property of the impulsive system can be retained if the length of the impulsive interval is large enough. Conversely, when the delayed continuous dynamics are not ISS/iISS but the discrete dynamics governing the impulses are, the impulsive system can achieve ISS/iISS if the sum of the length of the impulsive interval and the time delay is small enough. In particular, when one of the delayed continuous dynamics and the discrete dynamics are ISS/iISS and the others are stable for the zero input, the impulsive system can keep ISS/iISS no matter how often the impulses occur. Our proposed results are evaluated using two illustrative examples to show their effectiveness.     

A degree of flexibility in Lyapunov inequalities for establishing input-to-state stability of interconnected systems

In this paper, a novel approach to constructing flexible Lyapunov inequalities is developed for establishing Input-to-State Stability (ISS) of interconnection of nonlinear time-varying systems. It aims at a useful tool for using nonlinear small-gain conditions by allowing some flexibility in Lyapunov inequalities each subsystem is to satisfy. In the application of the ISS small-gain “theorem”, achieving a Lyapunov inequality conforming to a nonlinear small-gain “condition” is not a straightforward task. The proposed technique provides us with many Lyapunov inequalities with which a single trade-off condition between subsystems gains can establish the ISS property of the interconnected system. Proofs are based on explicit construction of Lyapunov functions.     

Analysis of input-to-state stability for discrete time nonlinear systems via dynamic programming

The input-to-state stability (ISS) property for systems with disturbances has received considerable attention over the past decade or so, with many applications and characterizations reported in the literature. The main purpose of this paper is to present analysis results for ISS that utilize dynamic programming techniques to characterize minimal ISS gains and transient bounds. These characterizations naturally lead to computable necessary and sufficient conditions for ISS. Our results make a connection between ISS and optimization problems in nonlinear dissipative systems theory (including L₂-gain analysis and nonlinear H_∞ theory). As such, the results presented address an obvious gap in the literature.     

Local input-to-state stability: Characterizations and counterexamples

We show that a nonlinear locally uniformly asymptotically stable infinite-dimensional system is automatically locally input-to-state stable (LISS) provided the nonlinearity possesses some sort of uniform continuity with respect to external inputs. Also we prove that LISS is equivalent to existence of a LISS Lyapunov function. We show by means of a counterexample that if this uniformity is not present, then the equivalence of local asymptotic stability and local ISS does not hold anymore. Using a modification of this counterexample we show that in infinite dimensions a uniformly globally asymptotically stable at zero, globally stable and locally ISS system possessing an asymptotic gain property does not have to be ISS (in contrast to finite dimensional case).     

Input-to-state stability for networked control systems via an improved impulsive system approach

This paper presents a novel impulsive system approach to input-to-state stability (ISS) analysis of networked control systems (NCSs) with time-varying sampling intervals and delays. This approach is based upon the new idea that an NCS can be viewed as an interconnected hybrid system composed of an impulsive subsystem and an input delay subsystem. A new type of time-varying discontinuous Lyapunov-Krasovskii functional, which makes full use of the information on the piecewise-constant input and the bounds of the network delays, is introduced to analyze the ISS property of NCSs. Linear matrix inequality based sufficient conditions are derived for ISS of NCSs with respect to external disturbances. When applied to the approximate tracking problem for NCSs, the derived ISS result provides bounds on the steady-state tracking error. Numerical examples are provided to show the efficiency of the proposed approach.     

Robust exponential input-to-state stability of impulsive systems with an application in micro-grids

This paper studies the robust exponential input-to-state stability (robust e-ISS) for impulsive systems. New notions of input-to-state exponent (IS-e) and e-property are proposed. Based on the established relation between IS-e and e-property, and the method of variation of constants formula, the equivalent conditions for robust e-ISS have been derived. Then the notion of robust event-e-ISS is defined. The sufficient conditions and the robust regions for robust e-ISS and robust event-e-ISS are also derived by using the IS-e of every subsystem. It shows the whole system may have robust event-e-ISS while every subsystem may have no ISS. It also shows the external disturbances may lead to relatively small robust regions. The results are then specialized to derive the equivalent conditions of interval e-ISS for interval impulsive systems. As an application, the result is used to test the ISS for a controlled micro-grid.     

Further results on input-to-state stability for nonlinear systems with delayed feedbacks

We consider a class of nonlinear control systems for which stabilizing feedbacks and corresponding Lyapunov functions for the closed-loop systems are available. In the presence of feedback delays and actuator errors, we explicitly construct input-to-state stability (ISS) Lyapunov-Krasovskii functionals for the resulting feedback delayed dynamics, in terms of the available Lyapunov functions for the original undelayed dynamics, which establishes that the closed-loop systems are input-to-state stable (ISS) with respect to actuator errors. We illustrate our results using a generalized system from identification theory and other examples.     

Stochastic input-to-state stability for impulsive switched stochastic nonlinear systems with multiple jumps

In this study, we investigate the stochastic input-to-state stability (SISS) of impulsive switched stochastic nonlinear systems. In this model, the impulse jumps are component multiple maps that depend on time. Thus the model differs from traditional impulsive systems with single impulse between two adjacent switching times. We provide sufficient conditions in three cases with the SISS system by using the Lyapunov function and average impulsive interval approach. The destabilising impulses cannot destroy the SISS properties if the impulses do not occur too frequently when all the subsystems that control the continuous dynamics are SISS. In other words, the average impulsive interval satisfies a lower bound restraint. Conversely, when all subsystems that control the continuous dynamics are not SISS, impulses can contribute to stabilising the system in the SISS sense when the average impulsive interval satisfies an upper bound. Then, we investigate the SISS property of impulsive switched stochastic nonlinear systems with some subsystems that are not SISS under certain conditions such that the property remains obtained. Finally, we show three examples to demonstrate the validity of the main result.     

Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems

We study the stability properties of a class of time-varying non-linear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.     

On input-to-state stability of systems with time-delay: A matrix inequalities approach

Nonlinear matrix inequalities (NLMIs) approach, which is known to be efficient for stability and L₂-gain analysis, is extended to input-to-state stability (ISS). We first obtain sufficient conditions for ISS of systems with time-varying delays via Lyapunov-Krasovskii method. NLMIs are derived then for a class of systems with delayed state-feedback by using the S-procedure. If NLMIs are feasible for all x, then the results are global. When NLMIs are feasible in a compact set containing the origin, bounds on the initial state and on the disturbance are given, which lead to bounded solutions. The numerical examples of sampled-data quantized stabilization illustrate the efficiency of the method.     

Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability

We consider nonlinear systems with input-to-output stable (IOS) unmodeled dynamics which are in the range of the input. Assuming the nominal system is globally asymptotically stabilizable and a nonlinear small-gain condition is satisfied, we propose a first control law such that all solutions of the perturbed system are bounded and the state of the nominal system is captured by an arbitrarily small neighborhood of the origin. The design of this controller is based on a gain assignment result which allows us to prove our statement via a Small-Gain Theorem [JTP, Theorem 2.1]. However, this control law exhibits a high-gain feature for all values. Since this may be undesirable, in a second stage we propose another controller with different characteristics in this respect. This controller requires morea priori knowledge on the unmodeled dynamics, as it is dynamic and incorporates a signal bounding the unmodeled effects. However, this is only possible by restraining the IOS property into the exp-IOS property. Nevertheless, we show that, in the case of input-to-state stability (ISS)—the output is the state itself-ISS and exp—ISS are in fact equivalent properties.Yuan Wang was supported in part by NSF Grant DMS-9403924 and by a scholarship from Université Lyon I, France.     

Limits of input-to-state stability

The relations between attractors, input-to-state-stability, and controllability properties are discussed. In particular it is shown that loss of the attractor property under perturbations is connected with a qualitative change in the controllability properties due to a ‘merger’ with a control set.     

Exponential stability of impulsive systems with application to uncertain sampled-data systems

We establish exponential stability of nonlinear time-varying impulsive systems by employing Lyapunov functions with discontinuity at the impulse times. Our stability conditions have the property that when specialized to linear impulsive systems, the stability tests can be formulated as Linear Matrix Inequalities (LMIs). Then we consider LTI uncertain sampled-data systems in which there are two sources of uncertainty: the values of the process parameters can be unknown while satisfying a polytopic condition and the sampling intervals can be uncertain and variable. We model such systems as linear impulsive systems and we apply our theorem to the analysis and state-feedback stabilization. We find a positive constant which determines an upper bound on the sampling intervals for which the stability of the closed loop is guaranteed. The control design LMIs also provide controller gains that can be used to stabilize the process. We also consider sampled-data systems with constant sampling intervals and provide results that are less conservative than the ones obtained for variable sampling intervals.     

基于ISS的非线性纯反馈系统的自适应动态面控制

研究一类具有未知死区的非线性纯反馈系统的自适应控制问题．基于输入状态稳定理论和小增益定理,提出一种自适应动态面控制方案．该方案有效地减少了可调参数的数目,避免了传统后推设计中由于需要对虚拟控制反复求导而导致的计算复杂性．理论分析证明了闭环系统是半全局一致终结有界的．     

Input to state stability of min-max MPC controllers for nonlinear systems with bounded uncertainties

Min-max model predictive control (MPC) is one of the control techniques capable of robustly stabilize uncertain nonlinear systems subject to constraints. In this paper we extend existing results on robust stability of min-max MPC to the case of systems with uncertainties which depend on the state and the input and not necessarily decaying, i.e. state and input dependent bounded uncertainties. This allows us to consider both plant uncertainties and external disturbances in a less conservative way.It is shown that the input-to-state practical stability (ISpS) notion is suitable to analyze the stability of worst-case based controllers. Thus, we provide Lyapunov-like sufficient conditions for ISpS. Based on this, it is proved that if the terminal cost is an ISpS-Lyapunov function then the optimal cost is also an ISpS-Lyapunov function for the system controlled by the min-max MPC and hence, the controlled system is ISpS. Moreover, we show that if the system controlled by the terminal control law locally admits certain stability margin, then the system controlled by the min-max MPC retains the stability margin in the feasibility region.     

Connection between cooperative positive systems and integral input-to-state stability of large-scale systems

We consider a class of continuous-time cooperative systems evolving on the positive orthant . We show that if the origin is globally attractive, then it is also globally stable and, furthermore, there exists an unbounded invariant manifold where trajectories strictly decay. This leads to a small-gain-type condition which is sufficient for global asymptotic stability (GAS) of the origin.We establish the following connection to large-scale interconnections of (integral) input-to-state stable (ISS) subsystems: If the cooperative system is (integral) ISS, and arises as a comparison system associated with a large-scale interconnection of (i)ISS systems, then the composite nominal system is also (i)ISS. We provide a criterion in terms of a Lyapunov function for (integral) input-to-state stability of the comparison system. Furthermore, we show that if a small-gain condition holds then the classes of systems participating in the large-scale interconnection are restricted in the sense that certain iISS systems cannot occur. Moreover, this small-gain condition is essentially the same as the one obtained previously by [Dashkovskiy et?al., 2007] and Dashkovskiy et al., in press for large-scale interconnections of ISS systems.     

Input-to-state stability of impulsive and switching hybrid systems with time-delay

This paper investigates input-to-state stability (ISS) and integral input-to-state stability (iISS) of impulsive and switching hybrid systems with time-delay, using the method of multiple Lyapunov–Krasovskii functionals. It is shown that, even if all the subsystems governing the continuous dynamics, in the absence of impulses, are not ISS/iISS, impulses can successfully stabilize the system in the ISS/iISS sense, provided that there are no overly long intervals between impulses, i.e., the impulsive and switching signal satisfies a dwell-time upper bound condition. Moreover, these impulsive ISS/iISS stabilization results can be applied to systems with arbitrarily large time-delays. Conversely, in the case when all the subsystems governing the continuous dynamics are ISS/iISS in the absence of impulses, the ISS/iISS properties can be retained if the impulses and switching do not occur too frequently, i.e., the impulsive and switching signal satisfies a dwell-time lower bound condition. Several illustrative examples are presented, with their numerical simulations, to demonstrate the main results.     

具有脉冲效应的切换随机系统的几乎必然稳定性

针对一类在切换时刻具有脉冲行为的Markov切换非线性随机系统,首先,应用切换的Lyapunov函数方法研究系统的稳定性,给出系统几乎必然稳定的充分条件,该条件不依赖于系统的矩稳定性;然后,进一步对线性系统的稳定化问题进行分析与设计,对随机子系统的控制结构同时出现在方程的位移部分与扩散部分,给出相应的状态反馈增益矩阵的求解方法;最后,数值算例说明了所设计方法的有效性.     

Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform global asymptotic, and input-to-state stability of fast time-varying dynamics. We also construct strict Lyapunov functions for our systems using a strictification approach. We illustrate our results using several examples.     

Input-to-state stability of switched systems and switching adaptive control

In this paper we prove that a switched nonlinear system has several useful input-to-state stable (ISS)-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included.