带有不稳定子系统的切换非线性系统的积分输入状态稳定

本文针对带有不稳定子系统的切换非线性系统研究了系统的积分输入状态稳定性问题. 应用导数不定的 类Lyapunov函数得出切换非线性系统的积分输入状态稳定. 导数不定的类Lyapunov函数方法比传统的导数正定 的Lyapunov函数的方法更具有一般性. 文中包含两种情况: 当所有子系统为积分输入状态稳定时, 切换非线性系统 是积分输入状态稳定的; 当部分子系统为非积分输入状态稳定时, 本文证明了切换非线性系统的积分输入状态稳 定. 最后应用一个仿真例子描述了所提结果的有效性.     

Output feedback for stochastic nonlinear systems with unmeasurable inverse dynamics

This paper considers a concrete stochastic nonlinear system with stochastic unmeasurable inverse dynamics. Motivated by the concept of integral input-to-state stability （iISS） in deterministic systems and stochastic input-to-state stability （SISS） in stochastic systems, a concept of stochastic integral input-to-state stability （SiISS） using Lyapunov functions is first introduced. A constructive strategy is proposed to design a dynamic output feedback control law, which drives the state to the origin almost surely while keeping all other closed-loop signals almost surely bounded. At last, a simulation is given to verify the effectiveness of the control law.     

Construction of Lyapunov–Krasovskii functionals for networks of iISS retarded systems in small-gain formulation

This paper tackles a problem of verifying stability of retarded dynamical networks in a dissipative formulation. Subsystems are assumed to be integral input-to-state stable (iISS). Time-delays are allowed to reside in both subsystems and interconnection channels, and may be both discrete and distributed. No assumption is made on the interconnection topology. A small-gain methodology is developed for constructing a Lyapunov–Krasovskii functional to establish iISS of such a network.     

Necessary and Sufficient Small Gain Conditions for Integral Input-to-State Stable Systems: A Lyapunov Perspective

This paper is concerned with conditions for the stability of interconnected nonlinear systems consisting of integral input-to-state stable (iISS) systems with external inputs. The treatment of iISS and input-to-state stable (ISS) systems is unified. Both necessary conditions and sufficient conditions are investigated using a Lyapunov formulation. In the presence of model uncertainty, this paper proves that, for the stability of the interconnected system, at least one subsystem is necessarily ISS which is a stronger stability property in the set of iISS. The necessity of a small-gain-type property is also demonstrated. This paper proposes a common form of smooth Lyapunov functions which can establish the iISS and the ISS of the interconnection comprising iISS and ISS subsystems whenever the small-gain-type condition is satisfied. The result covers situations more general than the earlier study and removes technical conditions assumed in the previous literature. Global asymptotic stability is discussed as a special case.     

A geometrical formulation to unify construction of Lyapunov functions for interconnected iISS systems

In recent years, the ability to accommodate various nonlinearities has become even more important to support systems design and analysis in a broad area of engineering and science. In this line of research, this paper discusses usefulness of the notion of integral input-to-state stability (iISS) in assessing and establishing system properties through interconnection of component systems. The focus is to construct Lyapunov functions which explain mechanism and provide estimate of stability and robustness of interconnected systems. Unique issues arising in dealing with iISS systems are reviewed in comparison with interconnections of input-to-state stable (ISS) systems. The max-separable Lyapunov function and the sum-separable Lyapunov function which are popular for ISS and iISS, respectively, are revisited. The max-separable function cannot be qualified as a Lyapunov function when component systems are not ISS. Level sets of the max-separable function are rectangles, and the rectangles cannot be expanded to encompass the entire state space in the presence of non-ISS components. The sum-separable function covers iISS components which are not ISS. However, it has practical limitations when stability margins are small. To overcome the limitations, this paper brings in a new idea emerged recently in the literature, and proposes a new type of construction looking at level sets of a Lyapunov function. It is shown how an implicit function allows us to draw chamfered rectangles based on fictitious gain functions of component systems so that they provide reasonable estimates of forward invariant sets producing a Lyapunov function applicable to both iISS and ISS systems equally.     

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.     

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.     

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.     

Discrete-time Lyapunov-based small-gain theorem for parameterized interconnected ISS systems

Input-to-state stability (ISS) of a feedback interconnection of two discrete-time ISS systems satisfying an appropriate small gain condition is investigated via the Lyapunov method. In particular, an ISS Lyapunov function for the overall system is constructed from the ISS Lyapunov functions of the two subsystems. We consider parameterized families of discrete-time systems that naturally arise when an approximate discrete-time model is used in controller design for a sampled-data system.     

具有状态和控制约束的受扰离散线性切换系统的反馈控制

本文的主要贡献是针对一类具有重置函数及由外部不能控事件决定动态的离散时间线性切换系统,给出一些稳定性综合结论. 当系统受到外部有界扰动, 及状态和控制量约束时, 在输入到状态稳定性理论框架下, 研究使得系统镇定的线性状态反馈控制器设计方法. 针对这类混杂系统, 本文引入了受控D不变性的概念, 并给出检测某一混杂区域具有受控D不变性的充要条件. 进而, 提出一种能够使得受扰的线性切换系统镇定, 同时保证状态和控制量满足其约束的反馈矩阵的计算方法. 最后, 通过一个由两个子系统构成的数值例子来说明本文技术的应用性.     

Strong iISS is preserved under cascade interconnection

In a recent paper, we have introduced the notion of Strong iISS as a compromise between the strength of input-to-state stability (ISS) and the generality of integral ISS (iISS). In this note, we continue the investigations around this property by studying its behavior in an interconnection context. In particular, we show that the cascade of Strongly iISS systems is itself Strongly iISS and we recall some useful tools to study Strongly iISS systems in feedback interconnection.     

Robust ISS of uncertain discrete-time singularly perturbed systems with disturbances

This paper is concerned with robustly input-to-state stable (ISS) and Robust ISS by feedback of uncertain discrete-time singularly perturbed systems (SPSs) with disturbances. Meanwhile, robust stability and stabilisation of uncertain discrete-time SPSs are also obtained as the particular cases of robust ISS and robust ISS by feedback. We first find a sufficient condition by using the fixed-point principle in terms of linear matrix inequalities (LMIs) to guarantee that the considered system is always standard discrete-time SPSs subject to uncertainty and disturbances. Then, the full systems could decompose into the continuous-time uncertain slow subsystem with disturbance and discrete-time uncertain fast subsystems with disturbance, respectively. Based on the two-time-scale decomposition technique, sufficient condition in terms of LMIs is given such that the full systems are uniformly standard and robust ISS simultaneously. In addition, a state feedback controller is constructed by using the LMI approach such that the resulting closed-loop systems are robust ISS. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.     

Input-to-state stability of switched nonlinear systems

The input-to-state stability （ISS） problem is studied for switched systems with infinite subsystems. By using multiple Lyapunov function method, a sufficient ISS condition is given based on a quantitative relation of the control and the values of the Lyapunov functions of the subsystems before and after the switching instants. In terms of the average dwell-time of the switching laws, some sufficient ISS conditions are obtained for switched nonlinear systems and switched linear systems, respectively.     

Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems

This paper addresses the problem of verifying stability of networks whose subsystems admit dissipation inequalities of integral input-to-state stability (iISS). We focus on two ways of constructing a Lyapunov function satisfying a dissipation inequality of a given network. Their difference from one another is elucidated from the viewpoint of formulation, relation, fundamental limitation and capability. One is referred to as the max-type construction resulting in a Lipschitz continuous Lyapunov function. The other is the sum-type construction resulting in a continuously differentiable Lyapunov function. This paper presents geometrical conditions under which the Lyapunov construction is possible for a network comprising $n\ge 2$ subsystems. Although the sum-type construction for general $n>2$ has not yet been reduced to a readily computable condition, we obtain a simple condition of iISS small gain in the case of $n=2$. It is demonstrated that the max-type construction fails to offer a Lyapunov function if the network contains subsystems which are not input-to-state stable (ISS).     

Input-to-state stability for discrete-time time-varying impulsive switched delayed systems with asynchronous phenomena

This paper studies the input-to-state stability (ISS) for a class of discrete-time time-varying impulsive switched delayed systems, in which the asynchronous phenomena are considered. Asynchronous phenomena here include two implications: asynchronous impulses and switches and asynchronous switching. The former means that the events, that is, impulses and switches, are not necessary to occur simultaneously, while the latter indicates that the actual switching modes and related controllers or specified modes do not coincide. Applying the mode-dependent concept to impulses and switches, both the improved Krasovskii-type and Razumikhin-type ISS criteria are provided. The time differences of Lyapunov functionals or functions here are sign-changing and time-varying, releasing from the traditional assumption that always requires negative definiteness. Meanwhile, in this paper, ISS or non-ISS subsystems and stabilizing or destabilizing impulses are taken into account. Finally, three numerical examples are given to illustrate the effectiveness of the proposed results.     

Local ISS of large-scale interconnections and estimates for stability regions

We consider interconnections of locally input-to-state stable (LISS) systems. The class of LISS systems is quite large, in particular it contains input-to-state stable (ISS) and integral input-to-state stable (iISS) systems.Local small-gain conditions both for LISS trajectory and Lyapunov formulations guaranteeing LISS of the composite system are provided in this paper. Notably, estimates for the resulting stability region of the composite system are also given. This in particular provides an advantage over the linearization approach, as will be discussed.     

Markov跳跃非线性系统逆最优增益设计

证明了一类严格反馈Markov跳跃系统是依概率输入–状态可稳定的.其次,证明了逆最优增益设计问题可解的一个充分条件是存在一组满足小控制量的依概率输入–状态稳定控制李雅普诺夫函数.最后,利用积分反推方法,给出了严格反馈Markov跳跃系统逆最优增益设计问题的一个构造性解.其中,为了克服由于Markov跳跃引起的耦合项所带来的困难,所设计的李雅普诺夫函数以及控制器是与模态无关的.     

一类前馈系统的鲁棒镇定

Two kinds of saturated controllers are designed for a class of feedforward systems and the closed-loop resulted is locally input-to-state stable and input-to-state stable, respectively. By the word "locally", it is meant that there are restrictions on the amplitude of inputs. At first, under the guidance of suitable energy functions, two kinds of saturated controllers are designed as locally input-to-state stabilizers for a class of perturbed linear systems, from which explicit gain estimations can be obtained for the subsequent design. Then under the conditions that two subsystems of the feedforward system are respectively of locally input-to-state stability and input-to-state stability, the small gain theory is used to determine saturated degrees for corresponding robust stabilizers. The stability proofs are given by using a new characterization of input-to-state stability that is based on the concept of ultimate boundedness. As an application, saturated controllers are designed for the partial dynamics of a certain inverted pendulum.     

Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation

We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed.     

具有 iISS 逆动态的非线性系统的输出反馈调节

研究了具有积分输入状态稳定 (Integral input-to-state stability, iISS)逆动态和未知控制方向的更一般的非线性系统的输出反馈调节问题. 利用自适应反推的方法, 所设计的输出反馈控制器使得闭环系统的输出调节到原点, 并且闭环系统的其他信号有界.