Residual Set Research Articles

Performance-Based Distributed Control of Multiagent Systems: A Dual Phase Approach.

In this article, we investigate the distributed tracking control problem for networked uncertain nonlinear strict-feedback systems with unknown time-varying gains under a directed interaction topology. A dual phase performance-guaranteed approach is established. In the first phase, a fully distributed robust filter is constructed for each agent to estimate the desired trajectory with prescribed performance such that the control directions of all agents are allowed to be nonidentical. In the second phase, by establishing a novel lemma regarding Nussbaum function, a new adaptive control protocol is developed for each agent based on backstepping technique, which not only steers the output to track the corresponding estimated signal asymptotically with arbitrarily prescribed transient response but also extends the application scope of the proposed control scheme largely since the unknown control gains are allowed to be time-varying and even state-dependent. In such a way, the underlying problem is tackled with the output tracking error converging into an arbitrarily preassigned residual set exhibiting an arbitrarily predefined convergence rate. Besides, all the internal signals are ensured to be semi-globally ultimately uniformly bounded (SGUUB). Finally, two examples are provided to illustrate the effectiveness of the co-designed scheme.

Event-Triggered-Based Adaptive NN Cooperative Control of Six-Rotor UAVs With Finite-Time Prescribed Performance

This paper studies the disturbance-observer-based adaptive neural network cooperative event-triggered control problem for six-rotor unmanned aerial vehicles with finite-time prescribed performance. Six-rotor unmanned aerial vehicle systems are divided into the position subsystem and the attitude subsystem. The switching threshold event-triggered mechanism that considers the influence of event triggering on the consensus control performance is designed of each six-rotor unmanned aerial vehicle to save the transmission resources, which excludes the Zeno behavior. Neural networks are introduced for estimating the uncertainties and solving the algebraic loop problem. In addition, for the position subsystem, by combining the prescribed performance with the velocity function, a finite-time control scheme is proposed, which can guarantee that the consensus errors converge to a prespecified neighborhood of the origin, and all closed-loop signals are bounded. Based on the designed control input signal of position subsystem, an adaptive neural network event-triggered control mechanism is designed to stabilize the attitude subsystem. Finally, some verification results are given to test the rationality of the proposed control strategy. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The purpose of this paper is to design an event-triggered cooperative control scheme to save the transmission resources for six-rotor unmanned aerial vehicle systems with performance limitations and lumped disturbances. In practical applications, if the impact of the event-triggered mechanism on the system performance is not taken into account, the system performance may be significantly degraded while the information transmission resource is saved. Thus, this paper designs an event-triggered mechanism that considers the system performance, which can effectively weigh the relationship between system performance and transmission resource consumption and better meet the practical requirements. Moreover, a finite-time specified performance control strategy is introduced to avoid the problem of difficult determination of residual set and improve the transient-state and steady-state performances of unmanned aerial vehicle systems, which is more in line with the actual demand.

The residual set dimension of a generalized apollonian packing

The residual set dimension of a generalized apollonian packing

Quantized adaptive practical fixed-time synchronization of stochastic complex networks with actuator faults

This paper investigates the practical fixed-time synchronization for stochastic complex networks (CNs) under quantized adaptive fault-tolerant control scheme. More accurate upper-bound estimates of stochastic settling time are obtained and novel residual sets are derived via proposed new lemmas for practical fixed-time stability in probability (PFTP). Quantized adaptive control scheme is designed based on the improved lemmas, which can deal with the unknown actuator faults and disturbances in CNs. Moreover, the proposed control strategy can make the stochastic CNs achieve practical fixed-time synchronization in probability (PFSP). In addition, the effectiveness of proposed method is confirmed by an example of Chua’s circuit.

Intestinal metabolites predict treatment resistance of patients with depression and anxiety

BackgroundThe impact of the gut microbiota on neuropsychiatric disorders has gained much attention in recent years; however, comprehensive data on the relationship between the gut microbiome and its metabolites and resistance to treatment for depression and anxiety is lacking. Here, we investigated intestinal metabolites in patients with depression and anxiety disorders, and their possible roles in treatment resistance.ResultsWe analyzed fecal metabolites and microbiomes in 34 participants with depression and anxiety disorders. Fecal samples were obtained three times for each participant during the treatment. Propensity score matching led us to analyze data from nine treatment responders and nine non-responders, and the results were validated in the residual sample sets. Using elastic net regression analysis, we identified several metabolites, including N-ε-acetyllysine; baseline levels of the former were low in responders (AUC = 0.86; 95% confidence interval, 0.69–1). In addition, fecal levels of N-ε-acetyllysine were negatively associated with the abundance of Odoribacter. N-ε-acetyllysine levels increased as symptoms improved with treatment.ConclusionFecal N-ε-acetyllysine levels before treatment may be a predictive biomarker of treatment-refractory depression and anxiety. Odoribacter may play a role in the homeostasis of intestinal L-lysine levels. More attention should be paid to the importance of L-lysine metabolism in those with depression and anxiety.

Fault-Tolerant Fuzzy Observer-Based Fixed-Time Tracking Control for Nonlinear Switched Systems

This paper focuses on developing an adaptive output feedback fault-tolerant fixed-time tracking control strategy for a class of switched nonlinear systems with non-affine faults. To approximate the unmeasurable states, a mode-dependent state switched fuzzy observer is generated using fuzzy logic systems and a Butterworth low-pass filter. Next, an adaptive fixed-time tracking control strategy is developed by combining dynamic surface technology with the backstepping method. An improved mode-dependent average dwell time switching rule is also introduced to ensure the stability of switched nonlinear systems with unstable subsystems. A stability analysis proves the effectiveness of the proposed control scheme in stabilizing the switched nonlinear system under mode-dependent average dwell time switching, ensuring that the tracking error reaches the residual set within a derived settling time. Finally, a switched resistor-inductor-capacitor circuit example is provided to demon- strate the validity of the proposed strategy.

Adaptive Fuzzy Decentralized Output-Feedback Control of Large-Scale Systems With Sensor Faults: A Multichannel Asynchronous Triggering Approach

This paper proposes an adaptive fuzzy decentralized output feedback control scheme for large-scale systems with multichannel asynchronous event-triggering. Uncertain system dynamics, unknown interconnections and time-varying sensor faults are handled simultaneously, even though only the corrupted measured local output is available for the design procedure. First, local state estimation filters are designed to estimate the unmeasured states. Then, considering that the resource-limited communication and computation may exist in all three types of channels including state estimation filter-to-controller channel, parameter estimator-to-controller channel and controller-to-actuator channel, an asynchronous event-triggering framework with time-varying triggering threshold is developed for multiple channels to release the communication and computation burden. Compared with the existing synchronous triggering mechanism, the proposed asynchronous one has several advantages: 1) the redundant signal transmission is reduced, 2) the triggering conditions for different channels are uncorrelated, which enables the designer to adjust the signal transmission frequency of each channel independently, and 3) the robustness could be improved in the face of noise as ideal synchronous triggering is not easy to be achieved in practice. In addition, both gain and bias faults, which are allowed to be unknown and time-varying, are covered in a unified sensor fault model. It is proved that, with the proposed control scheme, all the closed-loop signals are semi-globally uniformly ultimately bounded and the output tracking error converges into an adjustable residual set without Zeno behavior. Finally, simulation and comparison results illustrate the effectiveness and advantages of the proposed control scheme.

Adaptive Multi-Dimensional Taylor Network Tracking Control for a Class of Nonlinear Strict Feedback Systems

Nonlinear systems are very common in real life, but because they are not superposed and homogeneous, there are many difficulties in controlling nonlinear systems. Therefore, an adaptive control method based on a multi-dimensional Taylor network (MTN) is proposed for a class of nonlinear systems with strict feedback so that the output of the system can track the given signal. In order to achieve the control effect, we define a new state variable and transform the strict feedback system. After transformation, the original feedback system has a standard form, and two parameters to be identified are obtained. Then, the state observer is designed, and the two parameters are identified via the approximation of the MTN. On this basis, the controller design and a system stability analysis are completed. The lemma is introduced, and the stability condition is established by using this low-pass filter to ensure that all closed-loop signals are semi-globally uniform and finally bounded and the output tracking error converges to the residual set near zero. Finally, a numerical simulation of a hydraulic system is carried out to verify the effectiveness of the proposed method. Under the three indexes, the proposed method has obvious advantages.

Observer gain optimization for minimization of minimal robust positively invariant set and invariant set-based fault detection

This paper designs observer gains to optimize the minimal robust positively invariant set and uses this set for robust fault detection. First, the sizes of the minimal robust positively invariant set and corresponding minimal residual set are computed based on the Frobenius norm size (i.e., Frobenius radius) of zonotopes. Second, by formulating the size minimization problem as a solvable optimization problem, an observer gain is designed to minimize the minimal robust positively invariant set. Third, by maximizing the relative effect of faults to uncertainties, an observer gain is further designed to optimize robust fault detection performance. At the end, several examples are used to illustrate the effectiveness of the proposed methods.

Fault detection and identification for rolling mill main drive system based on integrated observer under iterative learning strategy

In this article, the problem of multiple fault detection, isolation and reconfiguration of the rolling mill main drive system containing external disturbances is investigated. Considering the nonlinear frictional damping between the rolls and the rolled parts, a nonlinear mathematical model of the main drive system of the mill is established. A comprehensive fault diagnosis scheme based on observer is addressed for this system subjected to unknown external interference. The proposed scheme is divided into two parts. In the first stage, a set of sliding mode observers is designed for system fault detection, and a fault isolation criterion is proposed based on observer redundancy and generalised residual set theory to reveal the fault source. In the second stage, combined with the iterative learning algorithm, an iterative learning-unknown input observer is constructed to realise the accurate estimation of the fault signal. Unlike the existing fault estimation methods, the iterative learning-unknown input observer designed in this article uses the state estimation error of the previous iteration to estimate the fault signal in the current iteration period. Using [Formula: see text] synthesis to design observers for the system will guarantee fault diagnosis robustness. The Lyapunov theory and linear matrix inequality are introduced to prove the convergence of the proposed observer. The simulation study of a 1780-mm hot strip mill evaluates the proposed scheme. Simulation results demonstrate that the sliding mode observer approach can detect faults in the main drive system and isolate faults accurately. In contrast, the iterative learning-unknown input observer method has the lowest fault reconfiguration error (99.87% smaller than the extended state observer, 99.77% smaller than the unknown input observer) and achieves accurate fault signal tracking.

Particle realization of Bondi-Metzner-Sachs symmetry in 2 + 1 space-time

We construct a Lorentz invariant massive particle model in (2+1) space-time with an enlarged set of symmetries which includes Bondi-Metzner-Sachs (BMS) translations (supertranslations), using the non-linear realization framework. The Hamiltonian formalism for the resulting Lagrangian is constructed, and the infinite phase-space constraints and the set of gauge transformations are analysed. We also compute the massless limit of the theory in phase-space. After eliminating the gauge degrees of freedom, the physical reduced space is left only with the degrees of freedom of a standard Poincaré particle but with a residual set of symmetries that we prove to be BMS. A similar result for the massless limit, including in this case superrotations, is pointed out.

Dynamics of a nonlinear laminated beam with thermodiffusion effects

A laminated beam system with thermodiffusion effects is considered. The main result is the long‐time dynamics of the system. By showing the system is gradient and asymptotic smoothness, the existence of a global attractor is obtained, which is characterized as unstable manifold of the set of stationary solutions. The quasi‐stability of the system and the finite fractal dimension of the global attractor are established by a stabilizability inequality. The continuity of global attractors regarding the parameter in a residual dense set is finally proved.

Existence and generic stability of open-loop Nash equilibria for noncooperative fuzzy differential games

Differential game is a critical area of study not only in theory but also in application. Based on the concept of the fuzzy process introduced by Liu, a fuzzy differential game model described using fuzzy differential equations is studied. Firstly, the existence theorem of open-loop Nash equilibrium for fuzzy differential games is given using Ky Fan inequality, and an example shows the applicability of the theorem. Secondly, the fuzzy differential game space Γ1 is constructed, and the stability of open-loop Nash equilibria of the fuzzy differential game γ∈Γ1 is studied. The conclusion shows that the fuzzy differential games whose all the open-loop Nash equilibria are stable form a dense residual set, i.e., most of the fuzzy differential games are stable in the sense of Baire classification.

Two-Phase Performance Adjustment Approach for Distributed Neuroadaptive Consensus Control of Strict-Feedback Multiagent Systems.

This article addresses the practical prescribed-time leaderless consensus problem for multiple networked strict-feedback systems under directed topology. Different from most existing protocols for finite-time consensus that rely on the signum function or fractional power state feedback (thus, the finite convergence time is contingent upon the initial positions of the agents or other design parameters), the proposed distributed neuroadaptive consensus solution is based on a two-phase performance adjustment approach, which exhibits several salient features: 1) the consensus error is ensured to converge to a preassigned arbitrarily small residual set within prescribed time; 2) the tunable transient behavior and desired steady-state control performance of the consensus error is maintained under any unknown initial conditions; and 3) the control scheme involves only one parameter estimation, significantly reducing the design complexity and online computation. Furthermore, we extend the result to practical prescribed-time leader-following consensus control under directed communication topology. Numerical simulation verifies the benefits and efficiency of the proposed method.

Cooperative Tracking Control of Unknown Discrete-Time Linear Multiagent Systems Subject to Unknown External Disturbances.

This article studies the tracking problem of a class of heterogeneous linear minimum-phase discrete-time multiagent systems (MASs) with unknown agent parameters in the presence of bounded disturbances. By introducing a distributed adaptive observer and a reference model, a novel framework is proposed to convert the complicated cooperative tracking problem of unknown heterogeneous MASs into a cooperative tracking problem of the reference models to the leader and a local robust model reference adaptive control problem. It is shown that under the adaptive controller designed based on the proposed framework, the tracking errors between the outputs of all the agents and the output of the leader converge to a residual set. It is also shown that the tracking errors will converge to zero asymptotically when the disturbances are absent. Compared with the existing related works, our main contribution is that the proposed framework could deal with the unknown MASs with arbitrary individual relative degrees and do not rely on any global graph information. Finally, the effectiveness of the proposed controller is illustrated by an example.

Event-triggered adaptive control for stochastic nonlinear systems with time-varying full state constraints and output dead-zone

This paper studies the problem of event-triggered adaptive control for a class of strictly-feedback stochastic nonlinear systems. The output dead zone and time-varying full-state constraints are considered in the controller. Firstly, time-varying barrier Lyapunov functions (TVBLFs) are introduced to address problems of time-varying state constraints. Then, the Nussbaum-type function is used to compensate for the influence of the output dead zone. The problem of state constraint is well solved when the system’s output signal is affected by the dead zone, which satisfies a wider application space. In addition, a new relative threshold of the event-triggered control (ETC) scheme is considered to save communication resources, and the Zeno behavior is excluded. This scheme makes the trigger interval more flexible and optimizes the performance of the controller. Moreover, the error compensation signals reduce the filtered errors and simplify the algorithm. Simultaneously, an adaptive tracking controller with the backstepping technique is designed so that all signals in the closed-loop system are bounded and the tracking error converges to a small residual set of the origin. Finally, two simulation examples are given to demonstrate the effectiveness of the control method.

Sets of Non-Lyapunov Behaviour for Scalar and Matrix Schrödinger Cocycles

Abstract We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schrödinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence $p_{n}$ there is a residual set of energies in the spectrum on which the middle singular value (the $W$-th out of $2W$) grows no faster than $p_{n}^{-1}$. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function $\rho (t)$ such that $\rho (t)/t$ is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each $k$, the average of the first $k$ Lyapunov exponents is the logarithmic potential of a probability measure.

Adaptive Containment Control for Heterogeneous MIMO Nonlinear Multiagent Systems With Unknown Direction Actuator Faults

In this article, a novel output-feedback adaptive containment control scheme is proposed for a class of heterogeneous multi-agent systems, where the agents are nonlinear multi-input multi-output (MIMO) systems whose relative degrees are allowed to be different. Unlike existing results, we only require the leading principal minors of agents' control gain matrices (CGMs) to be nonzero and take into account unknown direction actuator faults, which relaxes the restrictions on CGMs and enhances system reliability. The difficulties jointly caused by the unknown CGMs, unknown parameters, and unknown jumps introduced by the actuator faults are successfully overcome by a novel recursive contradiction argument based on some Nussbaum functions and a matrix similarity transformation. Moreover, an event-triggering mechanism is introduced to avoid continuous communication among agents and reduce the communication burden. It is shown that all closed-loop signals are globally uniformly bounded and the containment errors converge to a residual set that can be made arbitrarily small. Simulation results illustrate the effectiveness of the proposed scheme.

Fully distributed adaptive practical fixed-time consensus protocols for multi-agent systems

This paper investigates the practical fixed-time consensus problem for a multi-agent system in networks with an undirected topology. Both node-based and edge-based fully distributed adaptive protocols are proposed to achieve the practical fixed-time consensus for the single integrator-type multi-agent systems, i.e., fixed-time attractiveness of a residual set as well as asymptotic consensus. Most notably, the settling time estimate can be computed without using any global information, which is distinctive from the existing fixed-time protocols. Furthermore, a scaling technique is applied for the proposed protocol such that the fixed-time attraction region can be contracted by only tuning a scaling parameter and the original settling time estimate remains unchanged. Finally, a numerical simulation example is given to validate the proposed protocols.

Kato Chaos in Linear Dynamics

This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T¯. The main conclusions are as follows: (1) If a linear operator is accessible, then the collection of vectors whose orbit has a subsequence converging to zero is a residual set. (2) For a continuous linear operator defined on Fréchet space, Kato chaos is equivalent to dense Li–Yorke chaos. (3) Kato chaos is preserved under the iteration of linear operators. (4) A sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators T¯ are equivalent. (5) A continuous linear operator is sensitive if and only if its inducing operator T¯ is sensitive. It should be noted that this equivalence does not hold for nonlinear dynamics.