Integral Input-to-state Stability Research Articles

Event-Triggered Control of Switched Nonlinear Time-Delay Systems With Asynchronous Switching.

This article investigates the event-triggered switching control (ETSC) of switched nonlinear time-delay systems (SNTDSs) with asynchronous switching. First, we study the input-to-state stability (ISS) and integral ISS (iISS) for SNTDSs with asynchronous switching, where switching instants are generated based on the designed event-triggered mechanism. Among existing works on the ETSC, systems behavior at event-triggered instants is neglected. In fact, whenever an event is triggered, the systems mode will jump suddenly such that a switch is imposed to the systems, leading to the change of subsystems. Moreover, asynchronous switching behavior may occur between the actual subsystem and its corresponding controller. These facts bring great challenges for the event-triggered mechanism design and ISS analysis. To tackle these problems, a new Lyapunov-based event-triggered mechanism with adjustable parameters is designed to establish the relationship between systems switches and event triggers, and exclude the Zeno phenomenon. The analysis of the asynchronous switching behavior can be implemented and some ISS and iISS criteria of SNTDSs are derived utilizing the merging switching technique. Finally, two numerical examples, including a practical stirred tank reactor system, are presented to show the validity of the proposed methods.

The ISS framework for time-delay systems: a survey

At the occasion of Eduardo D. Sontag’s 70th birthday, we provide here an overview of the tools available to study input-to-state stability (ISS) and related notions for time-delay systems. After a hopefully pedagogical presentation of the main differences with respect to the finite-dimensional theory, we review basic stability concepts for input-free time-delay systems, as well as instruments to guarantee them in practice, including the Lyapunov–Krasovskii, Lyapunov–Razumikhin, and Halanay approaches. We then consider the influence of inputs through the notions of ISS, integral ISS, and input-to-output stability and provide both Lyapunov-like and solutions-based characterizations of these properties. We also show how these notions can be helpful for the stability analysis of interconnected systems, whether in cascade or in feedback form. We finally provide a list of questions which remain open until now.

Input-to-State Stability of Switched Nonlinear Time-Delay Systems With Asynchronous Switching: Event-Triggered Switching Control

This letter addresses the input-to-state stability (ISS) and integral ISS (iISS) problems for switched nonlinear time-delay systems (SNTDSs) with asynchronous switching by the event-triggered switching control (ETSC) technique, where the switching times are generated by a novel Lyapunov-based event-triggered mechanism (ETM). Based on the relationship between the new ETM and the jump strength of adjacent Lyapunov functions, sufficient conditions of ISS and iISS for SNTDSs with asynchronous switching are presented, which can exclude the Zeno behavior. Since switches only occur when the event is triggered, the burden of information transmission for SNTDSs can plummet. Finally, two practical examples are introduced to verify the effectiveness of our proposed methods.

On characterisations of the input to state stability properties for conformable fractional order bilinear systems

This paper proposes for the first time the theoretical requirements that a fractional-order bilinear system with conformable derivative has to fulfil in order to satisfy different input-to-state stability (ISS) properties. Variants of ISS, namely ISS itself, integral ISS, exponential integral ISS, small-gain ISS, and strong integral ISS for the general class of conformable fractional-order bilinear systems are investigated providing a set of necessary and sufficient conditions for their existence and then compared. Finally, the correctness of the obtained theoretical results is verified by numerical example.

Newton’s method: sufficient conditions for practical and input-to-state stability

Newton’s method is a classical iterative algorithm for the numerical computation of isolated roots of algebraic equations and stationary points of functions. While its application is ubiquitous in a plethora of fields, questions concerning its robust stability to uncertainties in problem data and numerical accuracy often arise in practice. This paper seeks to provide sufficient conditions for practical stability, input-to-state-stability (ISS), integral ISS (iISS) and incremental ISS (δISS) of Newton’s method in the presence of such uncertainties, and provide illustrative examples of their application.

Input-to-State Stability of Time-Varying Switched Systems With Time Delays

This paper considers the input-to-state stability (ISS) of time-varying switched systems with time delays, where the upper bound estimation for the operator of Lyapunov function (UBEOL) is assumed to be time varying and mode dependent. The ISS and integral ISS are investigated for time-varying switched systems with time delays by using the Lyapunov–Razumikhin and comparison theorem methods. Since the coefficient in the UBEOL is time varying and takes a positive/negative value, the subsystems consist of both ISS and non-ISS subsystems, simultaneously. It is shown that our presented results have wider applications than some existing works. Two examples, including one of the consensus for time-varying multiagent systems with cooperative and competitive protocols, are presented to demonstrate the effectiveness of the proposed results.

Indefinite Lyapunov functions for input-to-state stability of impulsive systems

This paper presents a new Lyapunov method for the input-to-state stability (ISS) and integral ISS (iISS) of impulsive systems. The approach is proposed on the basis of an indefinite Lyapunov function rather than a negative definite one. When the continuous dynamics have a Lyapunov function with an indefinite time derivative, and the discrete dynamics are destabilizing, the impulsive system is ISS if an average dwell-time (ADT) condition is satisfying. Numerical examples show their effectiveness and advantages.

Qualitative equivalences of ISS and [formula omitted]-gain stability properties for discrete-time nonlinear systems

Input-to-state stability (ISS) and input–output L2-gain are two important robust stability concepts in analyzing the interconnections of nonlinear dynamical systems. In this paper, we demonstrate several qualitative equivalences between different ISS related properties and lp-gain properties for discrete-time systems via nonlinear changes of coordinates. Prior to that, certain summation-to-summation estimates are shown to be equivalent to the standard definitions of ISS and integral ISS (iISS) which lead to sufficient Lyapunov function conditions to verify lp-gain properties. Combined with known results on the equivalence of 0-input global asymptotic stability and iISS, and a superposition principle, we subsequently outline interesting implicative relationships between various discrete-time robust stability properties.

Input‐to‐state stability of non‐linear systems with distributed‐delayed impulses

This study addresses the input-to-state stability (ISS) and integral ISS (iISS) of non-linear systems with distributed-delayed impulses . Based on Lyapunov method and the analysis technique proposed by Hespanha et al. , the authors derive some sufficient condition for ISS/iISS, which depend on the distributed delays existing in impulses. They prove that the given system is uniformly ISS/iISS under the average dwell time conditions. Finally, the effectiveness of the proposed results is evidenced by two illustrative examples.

Input-to-State Stability, Integral Input-to-State Stability, and <inline-formula> <tex-math notation="LaTeX">${\cal L}_{2} $</tex-math></inline-formula>-Gain Properties: Qualitative Equivalences and Interconnected Systems

Input-to-state stability (ISS) and ${\cal L}_{2} $ -gain are well-known robust stability properties that continue to find wide application in the analysis and control of nonlinear dynamical systems and their interconnections. We investigate the relationship between ISS-type and ${\cal L}_{2} $ -gain properties, demonstrating several qualitative equivalences between these two properties via coordinate transformations. In particular, we investigate and clearly specify the relationship between ISS and linear ${\cal L}_{2}$ -gain, and between integral ISS and nonlinear ${\cal L}_{2} $ -gain. This latter relationship raises an interesting question as it is known that cascade and feedback interconnections of integral ISS systems are not generally well-behaved, whereas we demonstrate that such interconnections for systems with nonlinear ${\cal L}_{2} $ -gain are well-behaved. We subsequently present several new sufficient conditions for the stability of interconnected systems derived by exploiting the derived qualitative equivalences.

On input-to-state stability for stochastic coupled control systems on networks

This paper investigates the input-to-state stability (ISS) properties for stochastic coupled control systems on networks (SCCSNs). With the help of Lyapunov method and graph theory, some inequality techniques and stochastic analysis skills, some sufficient criteria are obtained for eλt-weighted integral ISS in mean and almost sure exponential ISS of SCCSNs. These sufficient criteria are closely related to the topological property of the network anatomy. Finally, an example is given to illustrate the results.

Equivalences of Stability Properties for Discrete-Time Nonlinear Systems

Several qualitative equivalences are demonstrated between various robust stability properties for discrete-time nonlinear systems. In particular, two different summation-to-summation estimates are shown to be equivalent to the standard definitions of Input-to-State Stability (ISS) and integral ISS (iISS), respectively. Furthermore, ISS and iISS are shown to be qualitatively equivalent, via a nonlinear change of coordinates, to linear and nonlinear l2-gain properties, respectively. These equivalences, together with a previous result on the equivalence of 0-input global asymptotic stability and iISS provide interesting relationships between discrete-time robust stability properties that do not hold in continuous-time.

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.

Lyapunov Technique and Backstepping for Nonlinear Neutral Systems

For nonlinear systems with delay of neutral type, we propose a new technique of stability and robustness analysis. It relies on the construction of functionals which make it possible to establish estimates of the solutions different from, but very similar to, estimates of input-to-state stability (ISS) or integral ISS (iISS) type. These functionals are themselves different from, but very similar to, ISS or iISS Lyapunov-Krasovskii functionals. The approach applies to systems which do not have a globally Lipschitz vector field and are not necessarily locally exponentially stable. We apply this technique to carry out a backstepping design of stabilizing control laws for a family of neutral nonlinear systems.

Input-to-state stability of nonlinear systems based on an indefinite Lyapunov function

This study investigated the input-to-state stability (ISS) and integral ISS (iISS) of nonlinear systems and discovered new sufficient conditions for them. These conditions are more relaxed than others in that they employ an indefinite Lyapunov function rather than a negative definite one. Thus, a nonlinear time-varying system satisfying them has uniform asymptotic stability. Two numerical examples show their effectiveness and advantages.

On Saturation, Discontinuities, and Delays, in iISS and ISS Feedback Control Redesign

This paper addresses input-to-state stability (ISS) and integral ISS (iISS) redesign to enhance robustness of feedback control systems in the face of actuator disturbances. The idea of the redesign is to modify control laws available for disturbance-free systems a priori in order to achieve robustness against the disturbance. In practice, control system design often encounters time-delays, discontinuities and actuator limitations at the same time. The purpose of this paper is to propose a unified treatment of these issues in the redesign for actuator disturbances. For the new class of systems, we pursue LgV-type formulas which are widely used for ordinary differential equations. In addition, we incorporate saturation into the LgV-type control for implementation in the presence of actuator limitations. To derive such a redesign formula, we develop a framework of ISS/iISS Lyapunov-Krasovskii functionals allowing for the Krasovskii solutions. This paper demonstrates that the notion of invariantly differentiable functionals is useful for deriving LgV-type control laws and dealing with discontinuities directly. The iISS formulation which includes the ISS as a special case allows us to address actuator limitations in the redesign problem in a flexible manner.

On input-to-state stability of stochastic nonlinear systems with Markovian jumping parameters

In this article, the input-to-state stability (ISS) properties of stochastic nonlinear systems with Markovian jump parameters are considered. Firstly, the notions of stochastic ISS, e λt -weighted (integral) ISS in mean and in the pth mean are introduced. Then by the Lyapunov-like function method, some corresponding criteria are provided. Lastly, a simulation example is provided to illustrate the effectiveness of our results.

Separation Principles for Input-Output and Integral-Input-to-State Stability

We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS (input-to-state stability) framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of separation principles for input-output-to-state stability---characterizations of the property as conjunctions of weaker stability notions. When applied to the notion of integral ISS, these characterizations yield analogous results.

Power characterizations of input-to-state stability and integral input-to-state stability

New notions of external stability for nonlinear systems are introduced, making use of average powers as signal norms and comparison functions as in the input-to-state stability (ISS) framework. Several new characterizations of ISS and integral ISS are presented in terms of the new notions. An example is discussed to illustrate differences and similarities of the newly introduced properties.