Finite-time Input-to-state Stability Research Articles

A Lyapunov-Based Small-Gain Theorem for a Network of Finite-Time Input-to-State Stable Systems

This paper presents a Lyapunov formulation of the small-gain theorem for the finite-time input-to-state stability (FTISS) of an interconnected nonlinear system composed of two or more FTISS subsystems. In addition, an FTISS-Lyapunov function for the interconnected system is constructed from the FTISS-Lyapunov functions of the subsystems. With respect to the previously developed nonlinear, Lyapunov-based small-gain theorem restricted to input-to-state stability, a new power-function-based scaling technique is proposed to deal with the challenge that a nonlinearly scaled FTISS-Lyapunov function may not retain a decreasing rate as a power function with a positive power less than one.

A Lyapunov-Based Small-Gain Theorem for Interconnected Finite-Time Input-to-State Stable Systems

This paper presents a Lyapunov formulation of the small-gain theorem for the finite-time input-to-state stability (FTISS) of an interconnected system composed of FTISS subsystems. In addition, an FTISS-Lyapunov function for the entire system is constructed from the FTISS-Lyapunov functions of the subsystems. With respect to the previously developed nonlinear, Lyapunov-based small-gain theorem restricted to input-to-state stability, a new power-function-based scaling technique is proposed to deal with the challenge that a nonlinearly scaled FTISS-Lyapunov function may not retain a decreasing rate as a power function with a positive power less than one.

Finite-time input-to-state stability of nonlinear systems: the discrete-time case

This paper extends the finite-time input-to-state stability (FTISS) issue to the discrete-time dynamical systems. Based on the Lyapunov method, sufficient conditions are derived to check the FTISS property of the discrete-time nonlinear systems with inputs. In addition, the generalised class- functions and the settling-time function capturing the finite settling-time behaviour of the dynamical system are constructed explicitly and well studied. In the end of the paper, a numerical example is provided to illustrate the feasibility/effectiveness of the obtained results.

Finite-time input-to-state stability of switched stochastic time-varying nonlinear systems with time delays

This article extends the finite-time input-to-state stability (FTISS) to switched stochastic time-varying nonlinear systems with time delays. By using the methods of Razumikhin theorem, comparison principle combined with average dwell-time (ADT) approach, several sufficient conditions of nonlinear switched stochastic delayed systems are analyzed to obtain ISS property in the framework of finite time. Compared with the existing results, we allow the upper bound estimation for the derivative of Lyapunov function to be time-varying, which takes both positive and negative values. Finally, an example is given to illustrate the effectiveness of the proposed results.

Input-to-state stability and sliding mode control of the nonlinear singularly perturbed systems via trajectory-based small-gain theorem

This paper presents the trajectory-based input-to-state stability (ISS) and input-to-output stability (IOS) small-gain theorem, and the finite-time ISS (FTISS) and finite-time IOS (FTIOS) of nonlinear singularly perturbed systems. The contribution of this paper is threefold. Firstly, a novel idea is proposed to analyze the stability of the nonlinear singularly perturbed system, which is regarded as an interconnected system by using two-time-scale decomposition. Secondly, the trajectory-based approach is applied to establish ISS and IOS small-gain theorem for singularly perturbed systems and the FTISS and FTIOS properties are proposed. Thirdly, a novel sliding mode controller is developed for a class of nonlinear singularly perturbed systems. Finally, the effectiveness of proposed method is illustrated by using a numerical example, a DC motor simulation and a multi-agent singularly perturbed system.

Adaptive finite-time stabilisation of output-constrained low-order uncertain nonlinear systems with time-varying powers

This paper investigates adaptive finite-time stabilisation for a class of output-constrained low-order nonlinear systems with unknown time-varying powers, parametric and dynamic uncertainties. Due to low-order nonlinear systems being merely continuous but not smooth, and the presence of time-varying powers and multiple unknowns, all the existing control methods of output-constrained nonlinear systems are inapplicable. By characterising unmeasured dynamic uncertainty with finite-time input-to-state stability (FTISS), applying finite-time stability theory and introducing the tan-type barrier Lyapunov function, low-power and high-power terms into control design, a unified adaptive state-feedback controller, which can deal with both constrained and unconstrained systems, is constructed. It is proved that all the closed-loop signals are uniformly bounded, the output constraint is not violated, and system states converge to zero in a finite time. Finally, a simulation example is given to illustrate the effectiveness of this control scheme.

A small gain theorem for finite-time input-to-state stability of infinite networks and its applications

We prove a small-gain sufficient condition for (global) finite-time input-to-state stability (FTISS) of infinite networks. The network under consideration is composed of a countable set of finite-dimensional subsystems of ordinary differential equations, each of which is interconnected with a finite number of its “neighbors” only and is affected by some external disturbances. We assume that each node (subsystem) of our network is finite-time input-to-state stable (FTISS) with respect to its finite-dimensional inputs produced by this finite set of the neighbors and with respect to the corresponding external disturbance. As an application we obtain a new theorem on decentralized finite-time input-to-state stabilization with respect to external disturbances for infinite networks composed of a countable set of strict-feedback form systems of ordinary differential equations. For this we combine our small-gain theorem proposed in the current work with the controllers design developed by S. Pavlichkov and C. K. Pang (NOLCOS-2016) for the gain assignment of the strict-feedback form systems in the case of finite networks. The current results address the finite-time input-to-state stability and decentralized finite-time input-to-state stabilization and redesign the technique proposed in recent work S. Dashkovskiy and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization, Automatica. – 2020. – 112. – 108643, in which the case of $\ell_{\infty}$-ISS of infinite networks was investigated. The current paper extends and generalizes its conference predecessor to the case of finite-time ISS stability and decentralized stabilization in presence of external disturbance inputs and with respect to these disturbance inputs. In the special case when all these external disturbances are zeroes (i.e. are abscent), we just obtain finite-time stability and finite-time decentralized stabilization of infinite networks accordingly.

Finite-time input-to-state stability of nonlinear impulsive systems

This paper extends the finite-time input-to-state stability (FTISS) property to nonlinear impulsive systems. By the Lyapunov method, several sufficient conditions are provided to obtain ISS property of nonlinear impulsive systems in the framework of finite time, where external inputs are considered in both the continuous dynamics and impulsive dynamics. Considering destabilizing impulsive actions, a relation of the impulsive frequency, the system structure, and exogenous disturbances is given to guarantee the FTISS and the uniform FTISS of impulsive systems. Examples are given to illustrate the effectiveness of the proposed results.

Sufficient conditions on finite-time input-to-state stability of nonlinear impulsive systems: a relaxed Lyapunov function method

This article presents sufficient conditions on Finite-Time Input-to-State Stability (FTISS) of nonlinear impulsive systems, which gathers the analysis of Input-to-State stability (ISS) and finite-time convergence rate. By utilising the Finite-Time Stable Function Pair (FTSFP) and Generalised () function, we established an unified FTISS criterion for nonlinear impulsive systems, in which the impulses maybe beneficial to stabilisation or detrimental to stabilisation. Moreover, a relaxed Lyapunov Function (LF) method is used to investigate FTISS. Compared with the traditional LF method, the main advantage is that the constructed LF is permitted to own indefinite-derivative. As a special case of FTISS, a corollary on Finite-Time Stability (FTS) is established, which is applicable to investigate the realisation of FTS for nonlinear system via impulsive control strategy. Finally, the proposed results are supported by two practical numerical examples.

Input-to-State Stability Analysis of Subhomogeneous Cooperative Systems

This paper analyzes the input-to-state stability (ISS) properties of subhomogeneous cooperative systems for the first time. A new definition of finite-time ISS (FISS) for nonlinear positive systems is given. For stability analysis, the max-separable Lyapunov function is used, and the min-separable Lyapunov-like function is presented for the first time in this paper. Using these functions, we obtain our main results on solution estimates for subhomogeneous positive systems. According to the proposed method of analysis, the ISS properties of these systems can be easily judged only on the basis of their differential and algebraic characteristics. Simulation examples verify the validity of our results.

Input-to-state stability of continuous-time systems via finite-time Lyapunov functions

In this paper, input-to-state stability (ISS) of continuous-time systems is analyzed via finite-time Lyapunov functions. ISS of a continuous-time system is first proved via finite-time robust Lyapunov functions for an introduced auxiliary system of the considered system. It is then obtained that the existence of a finite-time ISS Lyapunov function implies that the continuous-time system is ISS. The converse finite-time ISS Lyapunov theorem is proposed. Furthermore, we explore the properties of finite-time ISS Lyapunov functions for the continuous-time system on a bounded and compact set without a small neighborhood of the origin. The effectiveness of our results is illustrated by four examples.

Continuous Sliding-Mode Control Strategies for Quadrotor Robust Tracking: Real-Time Application

The design of robust tracking control for quadrotors is an important and challenging problem nowadays. In this paper, a robust tracking output-control strategy is proposed for a quadrotor under the influence of external disturbances and uncertainties. Such a strategy is composed of a finite-time sliding-mode observer, which estimates the full state from the measurable output and identifies some types of disturbances. It is also composed of a combination of PID controllers and continuous sliding-modes controllers that robustly track a desired time-varying trajectory with exponential convergence despite the influence of external disturbances and uncertainties. The closed-loop stability is provided based on the input-to-state stability (ISS) and finite-time ISS properties. Finally, experimental results in real time show the performance of the proposed control strategy.

Adaptive finite-time control of uncertain nonlinear systems with the powers of odd rational numbers

ABSTRACTThis paper studies adaptive finite-time control problem of a class of nonlinear systems with dynamic and parametric uncertainties. The power of systems in this paper is any positive odd rational number but not necessary equal to or greater than one. To solve this problem, finite-time input-to-state stability (FTISS) is used to characterise the unmeasured dynamic uncertainty. By skillfully combining Lyapunov function, sign function, adding a power integrator, adaptive control and FTISS methods, an adaptive state feedback controller is designed to drive the states to the origin in finite time while maintaining all the closed-loop signals bounded.

Adaptive finite-time stabilisation of high-order uncertain nonlinear systems

ABSTRACTThis paper studies the problem of finite-time stabilisation of more general high-order nonlinear systems with dynamic and parametric uncertainties. By characterising the unmeasured dynamic uncertainty with finite-time input-to-state stability (FTISS), skillfully combining Lyapunov function, sign function, backstepping, adaptive control and FTISS approaches, and using finite-time stability theory, an adaptive state feedback controller is designed to guarantee high-order uncertain nonlinear systems globally finite-time stable.

Adaptive Finite-Time Stabilization of High-Order Nonlinear Systems with Dynamic and Parametric Uncertainties

Under the weaker assumption on nonlinear functions, the adaptive finite-time stabilization of more general high-order nonlinear systems with dynamic and parametric uncertainties is solved in this paper. To solve this problem, finite-time input-to-state stability (FTISS) is used to characterize the unmeasured dynamic uncertainty. By skillfully combining Lyapunov function, sign function, backstepping, and finite-time input-to-state stability approaches, an adaptive state feedback controller is designed to guarantee high-order nonlinear systems are globally finite-time stable.

A three-dimensional robust nonlinear terminal guidance law with ISS finite-time convergence

ABSTRACTThis paper presents a novel three-dimensional nonlinear terminal guidance law with finite-time convergence for intercepting manoeuvring targets. Different from the usual method of decoupling the missile-target relative motion into two-dimensional planes, this law is designed via using the coupled dynamics. The guidance law is derived based on the theory of finite-time input-to-state stability (ISS), which needs no assumption of the linearisation and the estimation of target accelerations. Under this law, the line-of-sight angular rates can be stabilised to a small domain of convergence around zero in finite time. The convergence rate and convergence domain can be adjusted by changing the guidance parameters. First, a sufficient condition on finite-time ISS of the guidance system is given, and is subsequently used to design the guidance law. Finally, simulation results are provided to show that the proposed guidance law possesses fast convergence rate and strong robustness to target manoeuvres.

Finite-time stochastic input-to-state stability of impulsive switched stochastic nonlinear systems

This paper mainly tends to investigate the finite-time stochastic input-to-state stability (FTSISS) problem for a class of impulsive switched stochastic nonlinear systems. An efficient lemma is firstly established to construct a generalized class KL function. Further, a set of Lyapunov-based sufficient conditions are derived to check the finite-time stability in probability (FTSiP) and finite-time input-to-state stability (FTISS) properties of switched stochastic nonlinear systems with or without impulses. In particular, some useful results in favor of simulations are obtained in terms of average dwell-time technique. Finally, a numerical example is provided to illustrate the effectiveness of proposed results.

Finite-Time Input-to-State Stability and Applications to Finite-Time Control Design

This paper extends the well-known concept, Sontag's input-to-state stability (ISS), to finite-time control problems. In other words, a new concept, finite-time input-to-state stability (FTISS), is proposed and then is applied to both the analysis of finite-time stability and the design of finite-time stabilizing feedback laws of control systems. With finite-time stability, nonsmoothness has to be considered, and serious technical challenges arise in the design of finite-time controllers and the stability analysis of the closed-loop system. It is found that FTISS plays an important role as the conventional ISS in the context of asymptotic stability analysis and smooth feedback stabilization. Moreover, a robust adaptive controller is proposed to handle nonlinear systems with parametric and dynamic uncertainties by virtue of FTISS and related arguments.

Finite-Time Input-to-State Stability and Applications to Finite-Time Control

This paper extends the well-known concept, input-to-state stability (ISS), to finite-time control problems. In other words, a new concept, finite-time input-to-state stability (FTISS), along with related concepts such as finite-time input-output stability and finite-time small-gain theorems, is discussed, and then is applied to both the finite-time stability analysis and the finite-time stabilizing feedback design. With finite-time convergence, non-smoothness has to be considered, which poses serious technical challenges in the analysis and synthesis of closed-loop finite-time systems. It is found that FTISS plays a key role in the study of finite-time stability and stabilization of nonlinear systems.