Showing papers on "Uniform boundedness published in 2020"

Adaptive Neural Network Learning Controller Design for a Class of Nonlinear Systems With Time-Varying State Constraints

Abstract: This paper studies an adaptive neural network (NN) tracking control method for a class of uncertain nonlinear strict-feedback systems with time-varying full-state constraints. As we all know, the states are inevitably constrained in the actual systems because of the safety and performance factors. The main contributions of this paper are that: 1) in order to ensure that the states do not violate the asymmetric time-varying constraint regions, an adaptive NN controller is constructed by introducing the asymmetric time-varying barrier Lyapunov function (TVBLF) and 2) the amount of the learning parameters is reduced by introducing a TVBLF at each step of the backstepping. Based on the Lyapunov stability analysis, it can be proven that all the signals in the closed-loop system are the semiglobal ultimately uniformly bounded and the time-varying full-state constraints are never violated. Finally, a numerical simulation is given, and the effectiveness of this adaptive control method can be verified.

Prespecified-Time Cluster Synchronization of Complex Networks via a Smooth Control Approach

Abstract: Most existing finite-/fixed-time synchronization control schemes are nonsmooth or discontinuous, and the settling time is estimated with conservatism. It is due to the utilization of signum function or fraction power state feedback. This brief considers the problem of prespecified-time cluster synchronization of complex networks with a smooth control protocol. The synchronization time is independent of any control parameters or any systems’ initial conditions, which is actually uniformly prescribed according to task requirements without any estimations. Moreover, the cluster synchronization can maintain after the specified time, and the smooth control input can always keep uniformly bounded in an infinite time interval as well. Finally, one numerical example is provided to illustrate the effectiveness of the proposed protocol and design method.

Stochastic Gradient Descent for Nonconvex Learning Without Bounded Gradient Assumptions

Abstract: Stochastic gradient descent (SGD) is a popular and efficient method with wide applications in training deep neural nets and other nonconvex models. While the behavior of SGD is well understood in the convex learning setting, the existing theoretical results for SGD applied to nonconvex objective functions are far from mature. For example, existing results require to impose a nontrivial assumption on the uniform boundedness of gradients for all iterates encountered in the learning process, which is hard to verify in practical implementations. In this article, we establish a rigorous theoretical foundation for SGD in nonconvex learning by showing that this boundedness assumption can be removed without affecting convergence rates, and relaxing the standard smoothness assumption to Holder continuity of gradients. In particular, we establish sufficient conditions for almost sure convergence as well as optimal convergence rates for SGD applied to both general nonconvex and gradient-dominated objective functions. A linear convergence is further derived in the case with zero variances.

Reservoir Computing Universality With Stochastic Inputs

Abstract: The universal approximation properties with respect to $L ^{p} $ -type criteria of three important families of reservoir computers with stochastic discrete-time semi-infinite inputs are shown. First, it is proven that linear reservoir systems with either polynomial or neural network readout maps are universal. More importantly, it is proven that the same property holds for two families with linear readouts, namely, trigonometric state-affine systems and echo state networks, which are the most widely used reservoir systems in applications. The linearity in the readouts is a key feature in supervised machine learning applications. It guarantees that these systems can be used in high-dimensional situations and in the presence of large data sets. The $L ^{p} $ criteria used in this paper allow the formulation of universality results that do not necessarily impose almost sure uniform boundedness in the inputs or the fading memory property in the filter that needs to be approximated.

Event-Triggered Adaptive Control for a Class of Nonlinear Systems With Unknown Control Direction and Sensor Faults

Abstract: In this note, a novel event-triggered adaptive control scheme is proposed for a class of nonlinear systems with unknown control direction and unknown sensor faults. The effects of the network-induced error and sensor faults are compressed by introducing some auxiliary filters and a bound estimation approach. Additionally, by introducing some differentiable auxiliary functions and high-order Lyapunov functions, we successfully circumvent the obstacle caused by unknown control direction and completely avoid Zeno phenomenon. The proposed scheme is able to ensure that all closed-loop signals are globally uniformly bounded and the tracking error converges to a residual set. Simulation results are presented to illustrate the effectiveness of the proposed scheme.

Finite-time trajectory tracking control of unmanned surface vessel with error constraints and input saturations

Abstract: In this paper, a finite-time trajectory tracking control for unmanned surface vessel with error constraints and input saturations is proposed. We take the limited actuator capability into account and employ the hyperbolic tangent function to address the saturation problem, which is converted into the unknown input gains. A tan-type barrier Lyapunov function tackles the error constraints, where error variables remain within the predefined bounds. In addition, the neural networks are utilized for model uncertainties and external disturbances. We present a semi-globally uniformly bounded control method, and develop a finite-time control approach subsequently. Finally, the effectiveness of the proposed strategies is verified by the simulation studies.

Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime

Abstract: This work deals with a parabolic chemotaxis model with nonlinear diffusion and nonlocal reaction source. The problem is formulated on the whole space and, depending on a specific interplay between the coefficients associated to such diffusion and reaction, we establish that all given solutions are uniformly bounded in time.

Adaptive fault-tolerant time-varying formation tracking for multi-agent systems under actuator failure and input saturation.

Abstract: This paper studies the time-varying formation tracking problem for general linear multi-agent systems with multiple leaders in the presence of both actuator failure and input saturation. The followers are required to uniquely determine and track the convex combination of the states of leaders, while maintaining a predefined time-varying formation. A hyperbolic tangent function is firstly introduced to modify the actuator model with input saturation constraint. Then, an augmented plant for dynamics of each follower is constructed to derive the control protocol by exploiting the dynamic surface control technique. The proposed control protocol deals with faults of bias and unknown bounded loss of effectiveness by means of adaptive fault-tolerant strategies, while a formation feasible condition should be satisfied. With the control signal generated by the augmented plant, the time-varying formation error is proved to be semi-globally uniformly bounded under the faults and input saturation, based on standard Lyapunov theory. Finally, a numerical simulation is implemented to demonstrate the effectiveness of the proposed algorithm.

Stability Analysis of Covariance Intersection-Based Kalman Consensus Filtering for Time-Varying Systems

Abstract: The phenomena of unknown correlations are ubiquitously existing in general distributed filtering problems over sensor networks. And the covariance intersection (CI) fusion rule is an effective tool to tackle with this phenomena. During the recent years, the related CI-based Kalman consensus filters (CIKCFs) have been adopted to deal with unknown correlations in sensor networks. However, a systematic stability analysis result for the general CIKCF in the time-varying system setting remains to be established. This paper is written for this purpose. First, a general CIKCF with full features of CI is presented. Accordingly, the conditions for CIKCF to reach consensus with varying weights are investigated. Furthermore, a novel detectability condition, i.e., collectively uniform detectability , is proposed to ensure the error covariances of the CIKCF are uniformly bounded. Based on this condition, the estimation errors are further proven to be exponentially bounded in mean square with the aid of the stochastic stability lemma . Finally, an example is given to validate the effectiveness of the theoretical results.

Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

Abstract: In this paper we are concerned with resolvent estimates for the Laplacian $$\Delta $$ in Euclidean spaces. Uniform resolvent estimates for $$\Delta $$ were shown by Kenig et al. (Duke Math J 55(2):329–347, 1987) who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp $$L^p$$–$$L^q$$ bounds depending on z has not been considered in a general framework which admits all possible p, q. In this paper, we present a complete picture of sharp $$L^p$$–$$L^q$$ resolvent estimates, which may depend on z. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner–Riesz operators of negative index.

Designing Robust Control for Mechanical Systems: Constraint Following and Multivariable Optimization

Abstract: This article proposes a novel robust control design for mechanical systems based on constraint following and multivariable optimization. The state of the concerned system is affected by (possibly fast) time-varying and bounded uncertainty. The objective is to drive the system to obey a set of prescribed constraints. A $\beta$ -measure is defined to gauge the constraint-following error; based on which, a feedback robust control scheme, which invokes design parameters, is proposed. For the seeking of optimal design parameters, a multivariable constrained optimization problem is formulated. The problem is successfully solved: with the existence, uniqueness , and analytical expression (i.e., closed form) of the optimal design parameters demonstrated. With the optimal parameters, the proposed robust control can render dual performance: guaranteed and optimal. As the guaranteed performance, the $\beta$ -measure is assured to be uniform boundedness and uniform ultimate boundedness. As the optimal performance, the performance index is globally minimized. This article is the first ever endeavour to cast both the constraint following and multivariable optimization into the control framework for uncertain mechanical systems.

On stability of a class of second alpha-order fractal differential equations

Abstract: In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions that are not differentiable or integrable on totally disconnected fractal sets such as middle-μ Cantor sets. Analogues of the Lyapunov functions and their features are given for asymptotic behaviors of fractal differential equations. The stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presented and proved. We present examples and graphs for more details of the results.

Solvability of a Keller-Segel system with signal-dependent sensitivity and essentially sublinear production

Abstract: In this paper, we consider the zero-flux chemotaxis system ut0=Δu−∇⋅(uχ(v)∇v) in Ω×(0,∞),=Δv−v+g(u) in Ω×(0,∞), in a smooth and bounded domain Ω of R2. The chemotactic sensitivity χ is a general nonnegative function from C1((0,∞)) while g, the production of the chemical signal v, belongs to C1([0,∞)) and satisfies λ1≤g(s)≤λ2(1+s)β, for all s≥ 0, 0≤β<1 and 0<λ1≤λ2. It is established that no chemotactic collapse for the cell distribution u occurs in the sense that any arbitrary nonnegative and sufficiently regular initial data u(x,0) emanates a unique pair of global and uniformly bounded functions (u,v) which classically solve the corresponding initial boundary value problem. Finally, we illustrate the range of dynamics present within the chemotaxis system by means of numerical simulations.

Wave Propagation on Microstate Geometries

Abstract: Supersymmetric microstate geometries were recently conjectured (Eperon et al. in JHEP 10:031, 2016. https://doi.org/10.1007/JHEP10(2016)031) to be nonlinearly unstable due to numerical and heuristic evidence, based on the existence of very slowly decaying solutions to the linear wave equation on these backgrounds. In this paper, we give a thorough mathematical treatment of the linear wave equation on both two- and three-charge supersymmetric microstate geometries, finding a number of surprising results. In both cases, we prove that solutions to the wave equation have uniformly bounded local energy, despite the fact that three-charge microstates possess an ergoregion; these geometries therefore avoid Friedman’s “ergosphere instability” (Friedman in Commun Math Phys 63(3):243–255, 1978). In fact, in the three-charge case we are able to construct solutions to the wave equation with local energy that neither grows nor decays, although these data must have non-trivial dependence on the Kaluza–Klein coordinate. In the two-charge case, we construct quasimodes and use these to bound the uniform decay rate, showing that the only possible uniform decay statements on these backgrounds have very slow decay rates. We find that these decay rates are sublogarithmic, verifying the numerical results of Eperon et al. (2016). The same construction can be made in the three-charge case, and in both cases the data for the quasimodes can be chosen to have trivial dependence on the Kaluza–Klein coordinates.

Stackelberg-Theoretic Approach for Performance Improvement in Fuzzy Systems

Abstract: This paper investigates the robust control for dynamical systems subject to uncertainty. The uncertainty is assumed to be (possibly fast) time varying and bounded. The bound is unknown but lies within a prescribed fuzzy set (hence the fuzzy dynamical system). We propose an approach for the robust control design which is implemented in two steps. First, a class of robust controls is proposed based on tunable parameters. The proposed controls are deterministic and are not conventionally IF-THEN rules based. By the Lyapunov minimax approach, we prove that the proposed controls are able to guarantee deterministic system performance, namely, uniform boundedness and ultimate uniform boundedness. Second, optima seeking from the proposed controls is considered to improve fuzzy system performance. We formulate the optima-seeking problem as a two-player (one leader and one follower) Stackelberg game by developing two cost functions, each of which is in charge of one tunable parameter (i.e., the player). Each cost function consists of an average fuzzy system performance index and the associated player’s control effort. We show that the solution of the optimal design problem (i.e., the optima of the tunable parameters), which is called the Stackelberg strategy, always exists and how to obtain the backwards-induction outcome is provided. Simulation results on the walking control of a biped robot model are presented for demonstration.

Improved regularity assumptions for partial outer convexification of mixed-integer PDE-constrained optimization problems

Abstract: Partial outer convexification is a relaxation technique for MIOCPs being constrained by time-dependent differential equations. Sum-Up-Rounding algorithms allow to approximate feasible points of the relaxed, convexified continuous problem with binary ones that are feasible up to an arbitrarily small δ > 0. We show that this approximation property holds for ODEs and semilinear PDEs under mild regularity assumptions on the nonlinearity and the solution trajectory of the PDE. In particular, requirements of differentiability and uniformly bounded derivatives on the involved functions from previous work are not necessary to show convergence of the method.

A Novel Model-Based Robust Control for Position Tracking of Permanent Magnet Linear Motor

Abstract: The main objective of this article is to address the position tracking control problem for permanent magnet linear motor by using a novel model-based robust control method. Specifically, based on the assumed upper bound of the lumped model uncertainties and external disturbances, we design a novel nonlinear control algorithm, which is characterized by error-based and model-based inheriting from traditional proportional-integral-derivative (PID) control and robust control. The proposed control can be considered as an improved PID control or a redesigned robust control, with simple implementation and practical effectiveness. Theoretical analysis is provided to demonstrate that the controller can guarantee the uniform boundedness and uniform ultimate boundedness of the system. Moreover, on the experimental platform, rapid controller prototyping cSPACE is designed to avoid long time programming and debugging, and provides great convenience for practical operation. Numerical simulations and real-time experimental results are finally presented to illustrate the effectiveness and the achievable control performance of the control design.

Fading memory echo state networks are universal

Abstract: Echo state networks (ESNs) have been recently proved to be universal approximants for input/output systems with respect to various $L ^p$-type criteria. When $1\leq p< \infty$, only $p$-integrability hypotheses need to be imposed, while in the case $p=\infty$ a uniform boundedness hypotheses on the inputs is required. This note shows that, in the last case, a universal family of ESNs can be constructed that contains exclusively elements that have the echo state and the fading memory properties. This conclusion could not be drawn with the results and methods available so far in the literature.

Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space

Abstract: We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.