Boundedness Problem Research Articles

Event-Triggered Extended Dissipative FTB for T-S Fuzzy Switched Systems With Mismatched Phenomena and Deception Attacks: A Multidomain Framework.

This article investigates the extended dissipative finite-time boundedness (ED-FTB) problem for fuzzy switched systems under deception attacks. To improve the network resource efficiency, a multidomain probabilistic event-triggered mechanism (MDPETM) is proposed. The mode mismatched phenomenon is modeled based on the switching delay information between the controller mode and the system mode. To extract the true signal generated by the MDPETM, a virtual delay concept is developed. The constraint that the controller and the system must have the same premise variables is removed. Based on the MDPETM, mismatched fuzzy state feedback controllers are first devised which may not share the same modes with the system. Then, by establishing fuzzy basis and controller mode-dependent Lyapunov functionals, sufficient criteria free of nonlinear terms existing in the literature are derived, which ensure the ED-FTB of the closed-loop system under admissible delays and deception attacks. Finally, an application-oriented one-link robotic arm system is utilized to validate the theoretical results.

Integral reinforcement learning‐based optimal tracking control for uncertain nonlinear systems under input constraint and specified performance constraints

AbstractThis article addresses the optimal tracking control problem with prescribed performance for uncertain nonlinear systems subject to input constraint and unknown disturbances. First, a fixed‐time monotonic convergence function is introduced to restrain tracking error, and a nonlinear mapping technique is employed to transform the constrained error into an unconstrained variable, then the fixed‐time output tracking issue is boiled down to the boundedness problem of the transformed variable. With the aid of a nonquadratic cost function, the input constraint is encoded into the optimization problem. To solve the unknown disturbances, an auxiliary system and an auxiliary disturbance policy are constructed, and the optimal control problem is formulated as a two‐player zero‐sum game. Moreover, a Hamilton–Jacobi–Isaacs (HJI) equation associated with this nonquadratic zero‐sum game is established to give the optimal control and the worst‐case disturbance policy solution. Subsequently, to avoid using knowledge of the system dynamics, three neural network approximators, namely, actor, critic, and disturbance, which are tuned online and simultaneously for approximating the solution of HJI, are constructed based on the integral reinforcement learning algorithm. Theoretical analysis shows that the reconstructed error system states and the weight estimation errors are semi‐globally uniformly ultimately bounded. Finally, the simulation study further tests the availability of the proposed control strategy.

Geometric prescribed‐time control of quadrotors with adaptive extended state observers

AbstractThis article studies an adaptive extended state observer (AESO) based prescribed‐time geometric attitude stabilization problem of quadrotors with endogenous uncertainties, exogenous disturbances, and actuator saturation, which can be estimated and compensated by AESOs. The employment of the geometric representation method for system attitude can effectively avoid ambiguity or singularity problems that may arise with other representations. The temporal and scale transformations are employed to solve the global boundedness problem on infinite time instead of the prescribed‐time convergence problem on finite time. By integrating the proposed controller with the backstepping technique, the system of quadrotors can be prescribed‐time stable. Further, a prescribed‐time geometric attitude control technique is devised such that the control signal is bounded and the state signals converge to a small domain near the equilibrium point in the prescribed time. The prescribed time is determined by the user and is independent of the initial conditions. The comparative simulations with different initial information are performed to demonstrate the effectiveness of the proposed design technique.

Finite-time boundedness of switched linear systems via periodically intermittent control

This paper focuses on finite-time boundedness problem of switched linear systems under periodically intermittent control. Sufficient conditions are established to guarantee finite-time boundedness or finite-time stability of switched linear systems under periodically intermittent control. The proof procedures are implemented successfully resorting to piecewise Lyapunov functions. Based on the linear matrix inequality technique, the conditions are casted into a convex optimisation problem and therefore, they can be efficiently handled. The method proposed in this paper is a much potentially useful one in practice because of its economical and efficient nature that is easy to employ. Finally, two examples are given to express the validity and efficiency of the proposed methods.

Differences between the geometric phase and propagation phase: clarifying the boundedness problem.

We show white light interferometer experiments that clearly demonstrate the basic differences between geometric and propagation phases. These experimental results also suggest a way to answer the "boundedness problem" in geometric phase-whether geometric phase is unbounded (i.e.,can take on any values without limit) or bounded (i.e.,limited to values between -π and +π). We show why the answer to this question is not as easy as it seems, from both a theoretical and an experimental perspective, and explain how the answer depends on one's choice of phase convention. We also hope that the videos provided will be pedagogically useful for explaining geometric phase.

Finite-region boundedness and [formula omitted]-dissipativity of 2-D singular Roesser systems with mixed time-varying delays

This paper is concerned with the problems of finite-region boundedness and dissipativity analysis of two-dimensional (2-D) singular Roesser systems with mixed directional time-varying delays. The concepts of singular finite-region boundedness (SFRB), which deduces the property of singular finite-region stability (SFRS) when exogenous disturbance is ignored, and finite-region (Q,S,R)-dissipativity are first proposed. Then, by using an energy function constructed as a weighted 2-D Lyapunov–Krasovskii functional, and by utilizing zero-type free matrix equations, new delay- and time-region-dependent conditions, which guarantee the properties of finite-region stability and dissipativity with respect to a quadratic supplied rate function, are derived. The obtained results are shown to encompass and extend existing works in the literature. The effectiveness of the analysis results is validated by numerical examples.

Cooperative Operation Control of Virtual Coupling High-Speed Trains With Input Saturation and Full-State Constraints

In this paper, the distributed cooperative control of virtual coupling high-speed trains (HSTs) subject to full-state constraints, actuator limitations, dynamical uncertainties and environmental disturbances is investigated. Targeting at the full-state constraints in cooperative operation of HSTs, a distributed nonlinear state-dependent function (DNSDF) is first proposed to convert the state-constrained problem of the leader-following consensus control to the boundedness problem of DNSDF. Then, the distributed control law of each train is designed by combining the command filtering backstepping method and the adaptive neural network approximation technique. Meanwhile, combined with the DNSDF, a novel auxiliary dynamical system (ADS) is designed to compensate for the adverse effects of actuator input saturation, and thus ensure the closed-loop stability of the HSTs system when the state constraints and the input saturation are considered simultaneously. By utilizing the Lyapunov theory, the convergence of the proposed controller is analyzed. Finally, the feasibility and effectiveness of the proposed control scheme are verified by simulations. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This work was motivated by the problem of cooperative control for virtual coupling HSTs with different initial states, actuator input saturation, full-state constraints, etc. The proposed approach addresses the train position and speed constraints by applying DNSDF, which can ensure train coordinated operation with relative braking distance and further reduce the tracking interval between trains. Specifically, the upper bound of the train position constraint varies with the real-time position of the preceding train and the relative speed of the adjacent trains, rather than a constant value. More importantly, an ADS is designed based on the DNSDF, which guarantees the stability of the controller when the input saturation and state constraints exist simultaneously, and avoids the chattering phenomenon of the train control input. The simulation results show that the trains can operate cooperatively at any initial speed within the speed limitations. In future, we will focus on the cooperative control of HSTs with communication delay, and the energy saving optimization cooperative control of HSTs.

Boundedness analysis of neutral stochastic systems driven by [formula omitted]-Brownian motion

This paper is concerned with exponential ultimate boundedness problem for a class of neutral stochastic differential systems driven by G-Brownian motion. By developing a method that combines the comparison principle and the reduction to absurdity, we provide sufficient conditions for exponential ultimate boundedness in the mean square of neutral stochastic differential systems driven by G-Brownian motion. The obtained results improve greatly some previous works given in the literature. Finally, some comparisons with existing results and examples are provided to show the effectiveness and superiority of the theoretical results.

Grounded L-Graphs Are Polynomially chi -Bounded

A grounded L-graph is the intersection graph of a collection of “L” shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number omega has chromatic number at most 17omega ^4. This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially chi -bounded. We also survey chi -boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.

Asymptotic Boundedness of Conformable Fractional Delay Differential Systems

This article focuses on the problem of asymptotic boundedness for non-autonomous conformable fractional delay differential systems. Firstly, by utilizing the definition of conformable fractional derivative and fractional comparison principle, we propose non-autonomous conformable fractional Halanay inequalities. And then, based on the Halanay inequalities, sufficient conditions for the asymptotic boundedness are constructed for the systems. Finally, a numerical example shows the applicability of the proposed technique.

Criteria for the boundedness of a certain class of matrix operators from lpv into lqu

One of the main aims in the theory of matrices is to find necessary and sufficient conditions for the elements of any matrix so that the corresponding matrix operator maps continuously from one normed space into another one. Thus, it is very important to find the norm of the matrix operator, at least, to find upper and lower estimates of it. This problem in Lebesgue spaces of sequences in the general case is still open. This paper deals with the problem of boundedness of matrix operators from lpv into lqu for 1 &lt; q &lt; p &lt; ∞, and we obtain necessary and sufficient conditions of this problem when matrix operators belong to the classes O2+- satisfying weaker conditions than Oinarov’s condition.

Resilient annular finite‐time synchronization for master‐slave systems under scaling attacks

AbstractThis paper investigates the resilient annular finite‐time synchronization and boundedness problems for master‐slave systems under dynamic event‐triggered scheme (ETS), actuator faults, and scaling attacks. A comprehensive model for characterizing scaling attacks is presented. The switched dynamic ETS is developed to reduce the bandwidth pressure. Two different internal variables are designed, each of which changes based on the switching manifold. Simultaneously, the resilient problem of switched master‐slave systems with respect to stochastic scaling attacks is considered, where there are limited malicious signals introduced by the adversary. Based on the Lyapunov‐like function involving the different internal dynamic variables, a set of criteria is established in the form of linear matrix inequalities (LMIs), which guarantee the annular finite‐time boundedness. Finally, two examples are presented to verify the theoretical findings.

Exponential stability and boundedness of nonlinear perturbed systems by unbounded perturbation terms

We study the exponential stability and boundedness problem for perturbed nonlinear time-varying systems using Lyapunov Functions with indefinite derivatives. As the limiting function for the perturbation term, we use different forms and give stability and boundedness conditions in terms of the coefficients in these bounds. Contrary to most of the available conditions, we allow the coefficients to be unbounded. But instead, we put forward a condition that requires a series produced by coefficients to be limited and exponentially decaying. We perform our results on Linear time-varying systems and generalize many of the available results.

Numerical treatments for large eddy simulations of liquid–liquid dispersions via population balance equation

In this work, we employ the two-fluid model under the large eddy simulations (LES) framework to investigate liquid–liquid dispersions in stirred tanks. The population balance equation was solved by the one primary and one secondary particle method, which was proven as identical as one-node quadrature method of moments. First, Aiyer's break-age kernel was investigated for its capability in the context of chemical stirred tank applications [Aiyer et al., “A population balance model for large eddy simulation of polydisperse droplet evolution,” J. Fluid Mech. 878, 700–739 (2019)]. Second, two new methods were proposed to handle the consistency problem and boundedness problem. These numerical problems were shown in our previous studies but had never been discussed in detail. Three test cases were launched, and results showed that our implementation ensures the moments' boundedness. The inconsistency problem was also treated properly. The predicted diameter also agrees well with experiments. Meanwhile, the phase segregation problem as observed in the unsteady Reynolds-Averaged Navier–Stokes simulations disappeared when a LES turbulence model was employed.

New conditions on finite-time stability of linear discrete-time system

This paper is concerned with the problems of set-based finite-time stability (SFTS) and set-based finite-time boundedness (SFTB) for both certain and uncertain linear time-varying systems. The concepts of SFTS and SFTB are defined. Different from existing results, sufficient conditions for SFTS and SFTB are directly derived from the basic definitions of finite-time stability (FTS) and finite-time boundedness (FTB) by using the convex hull technique rather than utilizing the weighted quadratic functions. Thus, more practical constraints on the system states can be dealt with. Furthermore, intervals, zonotopes and polytopes are employed to describe the typical compact convex sets. For linear uncertain systems, the uncertain time-varying state sets are assumed to be represented by interval matrices and matrix zonotopes, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the main results.

Soft Bornological Group Acts on Soft Bornological Set

In this paper, we introduce the notation of the soft bornological group to solve the problem of boundedness for the soft group. We combine soft set theory with bornology space to produce a new structure which is called soft bornological group. So that both the product and inverse maps are soft bounded. As well as, we study the actions of the soft bornological group on the soft bornological sets. The aim soft bornological set is to partition into orbital classes by acting soft bornological group on the soft bornological set. In addition, we explain the centralizer, normalizer, and stabilizer in details. The main important results are to prove that the product of soft bornological groups is soft bornological group and the action for different elements are the same actions.

Finite-Time H∞ Control for Time-Delay Markovian Jump Systems with Partially Unknown Transition Rate via General Controllers

This paper deals with the problems of finite-time boundedness (FTB) and H∞ FTB for time-delay Markovian jump systems with a partially unknown transition rate. First of all, sufficient conditions are provided, ensuring the FTB and H∞ FTB of systems given by linear matrix inequalities (LMIs). A new type of partially delay-dependent controller (PDDC) is designed so that the resulting closed-loop systems are finite-time bounded and satisfy a given H∞ disturbance attenuation level. The PDDC contains both non-time-delay and time-delay states, though not happening at the same time, which is related to the probability distribution of the Bernoulli variable. Furthermore, the PDDC is extended to two other cases; one does not contain the Bernoulli variable, and the other experiences a disordering phenomenon. Finally, three numerical examples are used to show the effectiveness of the proposed approaches.

Finite-time boundedness and finite-time stabilization boundedness for nonlinear interconnected systems with Atangana–Baleanu–Caputo fractional derivative

In this paper, we consider the finite-time boundedness (FTB) and finite-time stabilization boundedness (FTSB) problems for a class of nonlinear interconnected systems with Atangana–Baleanu–Caputo (ABC) fractional derivative. By employing the FTB theory and some properties of ABC fractional derivative, and some basic mathematical transformations, we derive new conditions for the FTB and FTSB problems of ABC fractional-order nonlinear interconnected systems. These conditions are formed in linear matrix inequalities (LMIs), which can be effectively solved using the MATLAB LMI Control Toolbox. Finally, two numerical examples with simulations are provided to illustrate the efficiency of the proposed method.

Finite‐time event‐triggered control for a class of cascade switched affine nonlinear systems

AbstractThis paper is concerned with the finite‐time boundedness (FTB) problem of a class of cascade switched affine nonlinear systems. In order to avoid unnecessary waste of network communication and Zeno behavior, a dynamical event‐triggered scheme is proposed, which is more general than some existing event‐triggered schemes with a common parameter matrix. By using the average dwell time method and multiple Lyapunov function technologies, sufficient conditions for the existence of controller gains are first proposed such that the closed‐loop system is finite‐time bounded with a finite‐time disturbance attenuation performance. It is also proved that the Zeno phenomenon is excluded. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the proposed method.

Strong and weak associativity of weighted Sobolev spaces of the first order

A brief overview of the recent results on the problem of characterization of associative and double associative spaces of function classes, including both ideal and non-ideal structures, is presented. The latter include two-weighted Sobolev spaces of the first order on the positive semi- axis. It is shown that, in contrast to the notion of duality, associativity can be ‘strong’ or ‘weak’. In addition, double associative spaces are further divided into three types. In this context it is established that a weighted Sobolev space of functions with compact support possesses weak associative reflexivity, while the strong associative space of a weak associative space is trivial. Weighted classes of Cesàro and Copson type have similar properties; for these classes the problem us fully investigated, and their connections with Sobolev spaces with power weights are established. As an application, the problem of boundedness of the Hilbert transform from a weighted Sobolev space to a weighted Lebesgue space is considered. Bibliography: 49 titles.