Input-to-state Stability Property Research Articles

A De Giorgi Iteration-Based Approach for the Establishment of ISS Properties for Burgers’ Equation With Boundary and In-domain Disturbances

This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De~Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in $L^2$-norm for Burgers' equation have been established using this method. Moreover, as an application of De~Giorgi iteration, ISS in $L^\infty$-norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-$D$ {linear} {unstable reaction-diffusion equation} have also been established. It is the first time that the method of De~Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear {partial differential equations (PDEs)

Input-to-State Stability of Perturbed Nonlinear Systems With Event-Triggered Receding Horizon Control Scheme

In this paper, input-to-state stability (ISS) properties of perturbed systems with event-triggered receding horizon control (RHC) schemes are studied. Two event-triggered control schemes, which are the event-triggered quasi-infinite RHC (EQRHC) and the event-triggered dual-mode RHC (EDRHC) strategies, respectively, are considered. In the EQRHC scheme, an optimal control problem (OCP) should be considered at triggering time and the event is triggered if the error between the actual system state and the optimal system state violating a threshold. While in the EDRHC strategy, an OCP is only solved outside the terminal region and a local control law will be used inside the terminal region. The corresponding event condition is redesigned based on if the system state enters the terminal region or not. The event-triggering condition outside the terminal region is the same with that of the EQRHC scheme and the event-triggering condition inside the terminal region is proposed based on the difference between the actual system state and the predictive system state. Sufficient conditions of feasibility are proposed and ISS properties of both event-triggered control schemes are studied, respectively. At last, numerical simulations show the validity of the proposed methods.

ISS implies iISS even for switched and time-varying systems (if you are careful enough)

For time-invariant systems, the property of input-to-state stability (ISS) is known to be strictly stronger than integral-ISS (iISS). Known proofs of the fact that ISS implies iISS employ Lyapunov characterizations of both properties. For time-varying and switched systems, such Lyapunov characterizations may not exist, and hence establishing the exact relationship between ISS and iISS remained an open problem, until now. In this paper, we solve this problem by providing a direct proof, i.e. without requiring Lyapunov characterizations, of the fact that ISS implies iISS, in a very general time-varying and switched-system context. In addition, we show how to construct suitable iISS gains based on the comparison functions that characterize the ISS property, and on bounds on the function f defining the system dynamics. When particularized to time-invariant systems, our assumptions are even weaker than existing ones. Another contribution is to show that for time-varying systems, local Lipschitz continuity of f in all variables is not sufficient to guarantee that ISS implies iISS. We illustrate application of our results on an example that does not admit an iISS-Lyapunov function.

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.

ISS with respect to boundary and in-domain disturbances for a coupled beam-string system

This paper addresses the robust stability of a boundary controlled system coupling two partial differential equations (PDEs), namely beam and string equations, in the presence of boundary and in-domain disturbances under the framework of input-to-state stability (ISS) theory. Well-posedness assessment is first carried out to determine the regularity of the disturbances required for guaranteeing the unique existence of the solution to the considered problem. Then, the method of Lyapunov functionals is applied in stability analysis, which results in the establishment of some ISS properties {}{with respect to} disturbances. As the analysis is based on the \emph{a priori} estimates of the solution to the PDEs, it allows avoiding the invocation of unbounded operators while obtaining the ISS gains in their original expression without involving the derivatives of boundary disturbances.

Pth moment exponential input‐to‐state stability of nonlinear discrete‐time impulsive stochastic delay systems

SummaryThis paper investigates the pth moment exponential input‐to‐state stability (ISS) of nonlinear discrete‐time impulsive stochastic delay systems. By employing Lyapunov functionals, some pth moment exponential ISS criteria are provided. The obtained results show that a discrete‐time stochastic delay system can become pth moment exponential input‐to‐state stable by impulsive controls even if it may be not input‐to‐state stable itself. On the other hand, the original system without impulses can retain its ISS property with appropriate destabilizing impulses. As an application, the theoretical results are used to test the ISS for a class of recurrent neural networks under stochastic perturbations. Finally, a numerical example is presented to illustrate the effectiveness of the results.

Input-to-State Stability of Discrete-Time Delay Systems with Delayed Impulses

This paper investigates the input-to-state stability (ISS) of discrete-time delay systems with delayed impulses. By employing Lyapunov functions together with Razumikhin technique, a number of ISS criteria are obtained. It is shown that if the original impulse-free system has no ISS property, it can become input-to-state stable by adding certain impulsive controls which are stabilizing. Correspondingly, the original system without impulses can retain its ISS property with appropriate impulsive perturbations which are destabilizing. Then, robust ISS property for a class of uncertain discrete-time delay systems with delayed impulses is considered. Finally, some numerical examples are given to illustrate the effectiveness and the superiority of the results.

On input‐to‐state stability of rigid‐body attitude control with quaternion representation

SummaryThe concept of input‐to‐state stability (ISS) is important in robust control, as the state of an ISS system subject to disturbances can be stably regulated to a small region around the origin. In this study, the ISS property of the rigid‐body attitude system with quaternion representation is thoroughly investigated. It has been known that the closed loop with continuous controllers is not ISS with respect to arbitrarily small external disturbances. To deal with this problem, hybrid proportional‐derivative controllers with hysteresis are proposed to render the attitude system ISS. The controller is far from new, but it is investigated in a new aspect. To illustrate the applications of the results about ISS, 2 new robust hybrid controllers are designed. In the case of large bounded time‐varying disturbances, the hybrid proportional‐derivative controller is designed to incorporate a saturated high‐gain feedback term, and arbitrarily small ultimate bounds of the state can be obtained; in the case of constant disturbances, a hybrid adaptive controller is proposed, which is robust against small estimate error of inertia matrix. Finally, simulations are conducted to illustrate the effectiveness of the proposed control strategies.

Multiple Lyapunov Functions-Based Small-Gain Theorems for Switched Interconnected Nonlinear Systems

Multiple Lyapunov functions (MLFs)-based small-gain theorems are presented for switched interconnected nonlinear systems with unstable subsystems, which extend the small-gain technique from its original non-switched nonlinear version to a switched nonlinear version. Each low dimensional subsystem does not necessarily have the input-to-state stability (ISS) property in the whole state space, and it only has individual ISS property in some subregions of the state space. The novelty of this paper is that integral-type MLFs and small-gain techniques are utilized to establish some MLFs-based small-gain theorems for switched interconnected nonlinear systems, which derive various stability results under some novel switching laws designed and construct integral-type MLFs. The small-gain theorems proposed cover several recent results as special cases, which also permit removal of a common restriction in which all low dimensional subsystems in switched interconnected systems are ISS or only some are ISS and others are not. Finally, two illustrative examples are presented to demonstrate the effectiveness of the results provided.

Input-to-state stability of impulsive systems and their networks

This paper considers input-to-state stability (ISS) characterization for a class of impulsive systems which jump map depends on time. We provide sufficient conditions in terms of exponential ISS-Lyapunov functions equipped with an appropriate dwell-time condition for establishing ISS property. Some modifications of dwell times which are more conservative, but easy to be verified are being introduced. We also show that impulsive system with multiple jump maps can naturally represent an interconnection of several impulsive systems with different impulse time sequences. Then we present a procedure to verify ISS of such networks.

Trajectory‐based small gain theorems for ISpS and ISS of large‐scale networks of switched systems with arbitrary switchings

The authors consider a large-scale switched non-linear system which is composed as an interconnection of N non-linear switched systems. Having assumed that every subsystem is stable uniformly with respect to the unobservable switching signals, it can happen that the entire interconnection is unstable. This study provides a sufficient condition for the uniform stability of the entire large-scale interconnection. The framework of input-to-state stability (ISS) is used for this purpose and the stability condition is of the small-gain type which is given in terms of trajectory-based ISS gains. To prove this result we extend the well-known result by Sontag and Wang on characterisations of the ISS property in terms of simpler stability notions to the case of switched systems with arbitrary switchings.

ISS with Respect to Boundary Disturbances for 1-D Parabolic PDEs

Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-to-state stability (ISS) properties and finite gains cannot be established with respect to disturbances at the boundary. This intuition has been reinforced by many unsuccessful attempts, as well as by the success in establishing ISS only with respect to the derivative of the disturbance. Contrary to this intuition, we establish such a result for parabolic PDEs. Our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time-varying subsets of the state space. The obtained results are used for the comparison of the gain coefficients of transport PDEs with respect to inlet disturbances and for the establishment of the ISS property with respect to control actuator errors for parabolic systems under boundary feedback control.

Robust observer-based absolute stabilization for Lur’e singularly perturbed systems with state delay

This paper is concerned with the problem of robust observer-based absolute stabilization for Lur’e singularly perturbed time-delay systems. The aim is to design a suitable observer-based feedback control law such that the resulting closed-loop system is absolutely stable. First, a full-order state observer is constructed. Based on the linear matrix inequality (LMI) technique, a delay-dependent sufficient condition is presented such that the observer error system is absolutely stable. Then, for observer-based feedback control, by introducing some slack matrices, a sufficient condition for input-to-state stability (ISS) of the closed-loop system with regard to the observer error is presented. Thus, the absolute stabilization of the closed-loop system can be guaranteed based on the ISS property. In addition, the criteria presented are both independent of the small parameter and the upper bound for the absolute stability can be obtained in a workable algorithm. Finally, two numerical examples are provided to illustrate the effectiveness of the developed methods.

Robustness of delayed multistable systems with application to droop-controlled inverter-based microgrids

ABSTRACTMotivated by the problem of phase-locking in droop-controlled inverter-based microgrids with delays, the recently developed theory of input-to-state stability (ISS) for multistable systems is extended to the case of multistable systems with delayed dynamics. Sufficient conditions for ISS of delayed systems are presented using Lyapunov–Razumikhin functions. It is shown that ISS multistable systems are robust with respect to delays in a feedback. The derived theory is applied to two examples. First, the ISS property is established for the model of a nonlinear pendulum and delay-dependent robustness conditions are derived. Second, it is shown that, under certain assumptions, the problem of phase-locking analysis in droop-controlled inverter-based microgrids with delays can be reduced to the stability investigation of the nonlinear pendulum. For this case, corresponding delay-dependent conditions for asymptotic phase-locking are given.

Input-to-state stability for networked control system with predictive scheme

This paper studies the input-to-state stability (ISS) problem for a class of networked control systems (NCSs). The improved varying controller gain predictive scheme is used to compensate the effects of network-induced delays and packet dropouts. By introducing a new auxiliary variable and analyzing the relationship of the network-induced delays and packet dropouts between two consecutive sampling instants, the considered networked control system (NCS) is converted into a coupled switched system. Based on the small gain theorem, a novel stability criterion is proposed to guarantee the ISS property of the considered system. Compared with existing works in literature, the stability criterion obtained here is easier to test and is much less conservative. A numerical simulation is presented to show the effectiveness of the proposed method.

Integral reinforcement learning for continuous-time input-affine nonlinear systems with simultaneous invariant explorations.

This paper focuses on a class of reinforcement learning (RL) algorithms, named integral RL (I-RL), that solve continuous-time (CT) nonlinear optimal control problems with input-affine system dynamics. First, we extend the concepts of exploration, integral temporal difference, and invariant admissibility to the target CT nonlinear system that is governed by a control policy plus a probing signal called an exploration. Then, we show input-to-state stability (ISS) and invariant admissibility of the closed-loop systems with the policies generated by integral policy iteration (I-PI) or invariantly admissible PI (IA-PI) method. Based on these, three online I-RL algorithms named explorized I-PI and integral Q -learning I, II are proposed, all of which generate the same convergent sequences as I-PI and IA-PI under the required excitation condition on the exploration. All the proposed methods are partially or completely model free, and can simultaneously explore the state space in a stable manner during the online learning processes. ISS, invariant admissibility, and convergence properties of the proposed methods are also investigated, and related with these, we show the design principles of the exploration for safe learning. Neural-network-based implementation methods for the proposed schemes are also presented in this paper. Finally, several numerical simulations are carried out to verify the effectiveness of the proposed methods.

Input-to-State Stability for Networked Predictive Control With Random Delays in Both Feedback and Forward Channels

The input-to-state stability (ISS) for a class of networked control systems with random delays and packet dropouts appearing simultaneously in both feedback and forward channels is thoroughly investigated in this paper. A new network predictive controller scheme is introduced in order to compensate the effect of transmission delays and packet dropouts. By making use of the small gain theorem, the stability criteria of the considered new system are derived. The proposed stability conditions are fairly easy to check and considerably less conservative than the existing ones. These criteria reveal that, if the original linear systems are controllable and observable, then, by adopting the proposed networked-predictive-control scheme, the ISS properties can be guaranteed for the overall system despite the effects of networking such as transmission bounded delays, packet dropouts, and possible disturbance inputs. When no disturbance inputs occur, the system stability can be guaranteed for random delays with a certain bound or else for any large constant delays. Results for two illustrative examples are given to validate the proposed control scheme, the second one being a laboratory-scale dc-motor rig.

Input-to-State Stability for Nonlinear Systems With Large Delay Periods Based on Switching Techniques

The paper is concerned with the problem of input-to-state stability (ISS) for nonlinear delay systems with large delay periods (LDP). The concepts of the length rate of LDP and the frequency of LDP are introduced. First the considered system is converted into a switched delay system which may include an unstable subsystem. Then based on a piecewise Lyapunov functional, the ISS properties of the systems are developed under the constraints of the length rate and the frequency of LDP. ISS for a special kind of nonlinear delay systems is also considered and the conditions of nonlinear matrix inequalities (NLMIs) are proposed to guarantee the ISS properties of such a system. A numerical example and an example of a series DC motor are given to show the effectiveness of the proposed methods.

Input‐to‐state stability of non‐linear impulsive and switched delay systems

This study addresses the input-to-state stability (ISS) of non-linear impulsive and switched delay systems. Based on the piecewise Lyapunov-Krasovskii functional method, the authors show that when only some of the constituent subsystems are ISS, the ISS property still can be retained for the non-linear impulsive and switched delay system, if the dwell time of the ISS subsystems satisfies a lower bound condition and the activation time of the non-ISS subsystems satisfies an upper bound condition, respectively. In common parlance, some of the intervals must be overly long, whereas the others must be short enough between impulses. Compared with existing results on related problems, the theory can be applied to a larger class of systems. The proposed approach also enables us to give the analysis of systems involving actuator failure, controller failure or temporary uncertain switching signal. As an example, the authors employ the method to analyse the problem of reliable control for a mechanical rotational cutting process.

A Strict Control Lyapunov Function for a Diffusion Equation With Time-Varying Distributed Coefficients

In this paper, a strict Lyapunov function is developed in order to show the exponential stability and input-to-state stability (ISS) properties of a diffusion equation for nonhomogeneous media. Such media can involve rapidly time-varying distributed diffusivity coefficients. Based on this Lyapunov function, a control law is derived to preserve the ISS properties of the system and improve its performance. A robustness analysis with respect to disturbances and estimation errors in the distributed parameters is performed on the system, precisely showing the impact of the controller on the rate of convergence and ISS gains. This is important in light of a possible implementation of the control since, in most cases, diffusion coefficient estimates involve a high degree of uncertainty. An application to the safety factor profile control for the Tore Supra tokamak illustrates and motivates the theoretical results. A constrained control law (incorporating nonlinear shape constraints in the actuation profiles) is designed to behave as close as possible to the unconstrained version, albeit with the equivalent of a variable gain. Finally, the proposed control laws are tested under simulation, first in the nominal case and then using a model of Tore Supra dynamics, where they show adequate performance and robustness with respect to disturbances.