Nonsingular Surface Research Articles

Robust Control for Underactuated Fixed-Wing Unmanned Aerial Vehicles

Dynamic surface control (DSC) is a recognized nonlinear control approach for high-order systems. However, as the complexity of the system increases and the first-order filter (FOF) is introduced, there exists a singularity problem, i.e., the control input will reach infinity. This limits the application of the DSC algorithm to a class of real-world systems with complex dynamics. To address the problem of singularity, we present a novel DSC approach called nonsingular dynamic surface control (NDSC), which completely avoids the singularity problem and significantly improves the overall control performance. NDSC includes a nonsingular hypersurface, which is constructed by the error between system states and virtual control inputs. Then the nonsingular hypersurface will be applied to derive the corresponding control law with the aid of the DSC approach to ensure the output of the system can track arbitrary desired trajectories. NDSC has the following novel features: (1) finite time asymptotic stabilization can be guaranteed; (2) the performance of NDSC is insensitive to the FOF’s parameter variation once the maximum tracking error of FOF is bounded, which significantly reduces reliance on the control sampling frequency. We thoroughly evaluate the proposed NDSC algorithm in an unmanned aerial vehicle (UAV) system with an underactuated nature. Finally, the simulation results illustrate and highlight the effectiveness and superiority of the proposed control algorithm.

Estimator-based multi-target tracking of networked robotic systems via predefined-time hierarchical control method

In this paper, we study the multi-target tracking problem of networked robotic systems (NRSs) within predefined time under a directed interaction topology considering disturbances and physical parameter uncertainties. On the basis of non-singular sliding-mode surface, a new predefined-time estimator-based hierarchical control algorithm is put forward so as to address the tracking problem under the case of multiple moving targets. Using the time base generator (TBG), a distributed estimator algorithm is designed to ensure that the followers can estimate different target states within a predefined-time. Certain sufficient conditions of predefined-time multi-target tracking of NRSs are presented using the Lyapunov stability and neighbor information interaction principle. In the end, numerical simulation experiments are offered to indicate the availability and correctness of the theoretical results.

Approximation and homotopy in regulous geometry

Let $X$ , $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$ -regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$ , we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$ -regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$ -regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$ -dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$ -regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$ .

Output-feedback anti-disturbance consensus for multi-agent systems with application to secondary voltage regulation for microgrids

This paper investigates a distributed finite-time consensus problem for second-order multi-agent systems (MASs) with disturbances and unmeasurable states. Output-feedback anti-disturbance consensus protocols are designed by finite-time observer (FTO) methods and the nonsingular integral terminal sliding-mode control (NITSMC). Some FTOs are first introduced for the agents to estimate the disturbances and unmeasurable states together. Then, the nonsingular integral terminal sliding-mode surfaces are proposed, based on which consensus protocols are proposed. Moreover, the consensus methods are also applied to microgrids (MGs). Based on MG systems, output voltages achieve consensus through secondary regulation. Simulations verify the effectiveness.

On the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces

AbstractThe aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a nonsingular irreducible complex surface, and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;\,c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;\,c_1,c_2)$ to $Hilb_{\, \mathfrak{M}_{X,H}(n;\,c_1,c_2+m)}$ .

Fast fixed-time nonsingular terminal sliding-mode formation control for autonomous underwater vehicles based on a disturbance observer

In this paper, a novel disturbance observer-based fixed-time formation control method is proposed for autonomous underwater vehicles with actuator faults, model uncertainties and unknown disturbances. Firstly, a fast fixed-time stable system is improved, which has a faster convergence rate than existing stable system. Then, the leader-follower formation control strategy is combined with the path planning strategy based on the artificial potential field to obtain a collision-free formation configuration. Besides, a fast fixed-time disturbance observer is designed to compensate for unknown composite disturbances, which enhances the robustness of the system. Further, based on the disturbance estimation information and the improved nonsingular fast fixed-time terminal sliding-mode surface, a robust fast fixed-time formation control method is presented, which achieves the desired formation configuration. Finally, simulation and comparison results show the effectiveness of the proposed control strategy.

Continuous Nonsingular Terminal Sliding-Mode Control With Integral-Type Sliding Surface for Disturbed Systems: Application to Attitude Control for Quadrotor UAVs Under External Disturbances

This article proposes a continuous nonsingular terminal sliding-mode control with integral-type sliding surface (CNTSMC-ISS) framework for disturbed systems, in which we consider two types of finite-time controllers: the state feedback CNTSMC-ISS and the disturbance-observer-based CNTSMC-ISS. Compared with the existing sliding-mode controllers, the noteworthy contributions of two finite-time controllers in the proposed CNTSMC-ISS framework are the alleviation of the chattering phenomenon, the fast finite-time stability, and the singularity-free and ease-of-implementation characteristics. In the proposed CNTSMC-ISS framework, we first introduce a nonsingular integral terminal sliding-mode surface (NITSMS) such that the finite-time convergence of the system states to zero in the sliding phase is ensured and the singularity problem is avoided. Besides, a finite-time observer is developed to recover the external disturbance. Then, based on the constructed NITSMS with the supertwisting-like algorithm, the state feedback CNTSMC-ISS and the disturbance-observer-based CNTSMC-ISS are proposed, which generate the continuous control signals and guarantee the fast finite-time convergence of the system states to the designed sliding surface. Rigorous finite-time stability of the closed-loop system under two proposed controllers is provided. Finally, we apply two finite-time controllers in the proposed CNTSMC-ISS framework to the attitude control for quadrotor unmanned aerial vehicles under external disturbances. Extensive simulation and experimental results are illustrated to prove the effectiveness of the proposed attitude controllers.

Finite Time Trajectory Tracking with Full-State Feedback of Underactuated Unmanned Surface Vessel Based on Nonsingular Fast Terminal Sliding Mode

Marine transportation and operations have attracted the attention of more and more countries and scholars in recent years. A full-state finite time feedback control scheme is designed for the model parameters uncertainty, unknown ocean environment disturbances, and unmeasured system states in the underactuated Unmanned Surface Vessel (USV) trajectory tracking control. The external wind, wave and current environmental disturbances and model parameters perturbation are extended by Nonlinear Extended State Observer (NESO) to the state of the system, namely complex disturbances. The complex disturbances, positions and velocities of USV can be observed by NESO and feedback to USV control system. Next, the underactuated USV error model is obtained by operating the obtained feedback information and the virtual ship model. According to the error model, a Nonsingular Fast Terminal Sliding Model surface (NFTSM) is constructed to realize finite-time control. The control law is deduced through the Lyapunov stability theory to ensure the stability of the system. The results of MATLAB numerical simulations under different disturbances show that the trajectory tracking algorithm has fast responses, and a good convergence of the errors is observed, which verifies the effectiveness of the designed scheme.

Current-Limited Suboptimal Sliding Mode Control for Voltage-Regulated Synchronous Buck Converters

This article proposes a new sliding mode control based on a nonlinear switching surface that provides a suboptimal current-limited start-up response in step-down converters. The main idea behind the design consists in combining different switching surfaces. The first one forces the system trajectories to track a constant current path until the output voltage error is small. A nonsingular terminal switching surface then drives the trajectories to the vicinity of the steady-state operating point. Finally, a first-order switching surface regulates the steady state providing very good rates of load and line regulation. The article presents the design guidelines of the suboptimal sliding mode control and the experimental evaluation that confirms the effectiveness of the proposal.

Savage surfaces

Let G be the topological fundamental group of a given nonsingular complex projective surface. We prove that the Chern slopes c_1^2(S)/c_2(S) of minimal nonsingular surfaces of general type S with \pi_1(S) \simeq G are dense in the interval [1,3] .

Hartogs-type theorems in real algebraic geometry, I

Abstract Let f : X → ℝ {f:X\rightarrow\mathbb{R}} be a function defined on a connected nonsingular real algebraic set X in ℝ n {\mathbb{R}^{n}} . We prove that regularity of f can be detected by controlling the restrictions of f to either algebraic curves or algebraic surfaces in X. If dim ⁡ X ≥ 2 {\operatorname{dim}X\geq 2} and k is a positive integer, then f is a regular function whenever the restriction f | C {f|_{C}} is a regular function for every algebraic curve C in X that is a 𝒞 k {\mathcal{C}^{k}} submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of C is equivalent to the plane curve singularity defined by the equation x p = y q {x^{p}=y^{q}} for some primes p < q {p<q} . If dim ⁡ X ≥ 3 {\operatorname{dim}X\geq 3} , then f is a regular function whenever the restriction f | S {f|_{S}} is a regular function for every nonsingular algebraic surface S in X that is homeomorphic to the unit 2-sphere. We also have suitable versions of these results for X not necessarily connected.

Arnol’d’s type theorem on a neighborhood of a cycle of rational curves

Arnol’d showed the uniqueness of the complex analytic structure of a small neighborhood of a non-singular elliptic curve embedded in a non-singular surface whose normal bundle satisfies a Diophantine condition in the Picard variety. We show an analogue of this theorem for a neighborhood of a cycle of rational curves.

Nonlinear homogeneous sliding mode approach for fixed-time robust formation tracking control of networked quadrotors

A formation flight control algorithm for a multi-agent system composed of a group of perturbated quadrotors is thoroughly investigated in this paper. Both theoretical and practical aspects are addressed in detail to design and implement the control algorithm in a distributed fashion among the networked agents. By adopting the sliding mode framework: (i) A novel Nonlinear Homogeneous Nonsingular Terminal Sliding Surface (NHNTSS) is designed. To ensure Global Asymptotic Stability (GAS) as well as fixed-time convergence of the states, the sliding surface design skillfully employs the generalized weighted homogeneity theory. (ii) A novel Distributed Fixed-Time Continuous Nonsingular Control Protocol (DFCNCP) is proposed for the position-loop of each quadrotor agent by utilizing the designed NHNTSS. Moreover, the control scheme comprises a disturbance observer to compensate for the lumped disturbances affecting the position dynamics and estimate the unmeasurable linear velocities of the quadrotors. Overall, the synthesized controller can drive the formation tracking errors into close vicinity of the origin in fixed-time uniformly to the values of the Initial Conditions (ICs), i.e., initial positions of the quadrotors in the 3D state-space. The mathematical proofs based on the bi-limit approximation in the existing related works can only indicate that the system is guaranteed to be fixed-time stable without any estimation of the settling (convergence) time. In contrast, rigorous stability analyses are conducted in the present work by virtue of algebraic Lyapunov tools, i.e., Algebraic Lyapunov Equation (ALE) and Lyapunov Quadratic Function (LQF). Thus, an expression of the upper-bound on the settling-time is given explicitly. Numerical simulations and real experiments in the form of exhaustive comparative studies are carried out to validate the findings of this research work. Overall, the obtained results confirm the superiority of the proposed control strategy in terms of uniform fast fixed-time convergence, strong disturbance rejection, chattering alleviation, and control precision.

Linearization of transition functions of a semi-positive line bundle along a certain submanifold

Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. Assume that $L$ is semi-positive, namely $L$ admits a smooth Hermitian metric with semi-positive Chern curvature. Let $Y$ be a compact K\"ahler submanifold of $X$ such that the restriction of $L$ to $Y$ is topologically trivial. We investigate the obstruction for $L$ to be unitary flat on a neighborhood of $Y$ in $X$. As an application, for example, we show the existence of nef, big, and non semi-positive line bundle on a non-singular projective surface.

Three-dimensional adaptive fixed-time cooperative guidance law with impact time and angle constraints

This paper mainly addresses the simultaneous attack of multiple missiles against a maneuvering target in three-dimensional (3-D) space. Here, we show that a novel fixed-time fast nonsingular terminal slide mode surface (FNTSMS), which has the characteristics of nonsingularity, converges in fixed time. With the help of the fixed-time FNTSMS, fixed-time adaptive distributed guidance laws along and perpendicular to the line of sight (LOS) direction are proposed. This realizes the dynamic adjustment of the target parameters in guidance laws, reduces the energy consumption of the guidance process, and realizes the constraints of the impact time and angle simultaneously. Additionally, considering that the communication topology will cause topology switching failure, this paper proves through graph theory that the proposed adaptive guidance laws will not be affected by topology switching. Moreover, through relevant auxiliary transformations, guidance laws with control input delay are demonstrated. To close to the actual action process of the missile, the air resistance item is added to guidance laws and analyzed. In addition, the control input amplitude is restricted to meet the actual working conditions of the missile. Finally, simulations are conducted to demonstrate the performance of the guidance law and its robustness against the mutation of communication topology.

Donaldson–Thomas invariants, linear systems and punctual Hilbert schemes

We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain positivity conditions, we relate the DT invariants to Carlsson-Okounkov formulas for the twisted Euler's number of the punctual Hilbert schemes of nonsingular surfaces, and conclude they have a modular property.

Nonsingular cubic surfaces over F2k

We perform an opportunistic search for cubic surfaces over small fields of characteristic two. The starting point of our work is a list of surfaces complied by Dickson over the field with two elements. We consider the nonsingular ones arising in Dickson' s work for the fields of larger orders of characteristic two. We investigate the properties such as the number of lines, singularities and automorphism groups. The problem of determining the possible numbers of lines of a nonsingular cubic surface over the fields of $\mathbb{C}, \mathbb{R}, \mathbb{Q}, \mathbb{F}_q$ where q odd, $\mathbb{F}_2$ was considered by Cayley and Salmon, Schlafli, Segre, Rosati and Dickson, respectively. Our work contributes this problem over the larger fields of even characteristic. Besides that we investigate the structure of nonsingular surfaces with $15$ and $9$ lines. This work is a contribution to the study of nonsingular cubic surfaces with less than $27$ lines.

Algebraic surfaces determine analyticity of functions

Let f :X rightarrow mathbb {R} be a function defined on a nonsingular real algebraic set X of dimension at least 3. We prove that f is an analytic (resp. a Nash) function whenever the restriction f|_{S} is an analytic (resp. a Nash) function for every nonsingular algebraic surface S subset X whose each connected component is homeomorphic to the unit 2-sphere. Furthermore, the surfaces S can be replaced by compact nonsingular algebraic curves in X, provided that dimX ge 2 and f is of class mathcal {C}^{infty }.

Fixed-time formation control for spacecraft with prescribed performance guarantee under input saturation

This paper investigates the fixed-time stability of spacecraft formation flying by considering input saturation and prescribed performance constraints, including the safe distance constraint between members and control accuracy. By analyzing the relationship between the positional tracking error and the minimum safe distance between spacecrafts, a new form of the prescribed performance function with fixed-time convergence is developed to constrain the overshoot and accuracy of the tracking error. A nonsingular saturated terminal sliding-mode surface is developed to stabilize the system states in fixed time. A fixed-time coordinated tracking controller with input and rate saturation is finally developed by combining sliding-mode control, the back-stepping method, fixed-time observers, an anti-windup saturation compensator, and an auxiliary system. The stability of the closed-loop system is validated by Lyapunov stability theory. The results of simulation verified the effectiveness of the proposed control algorithm.

Higher rank Segre integrals over the Hilbert scheme of points

Let S be a nonsingular projective surface. Each vector bundle V on S of rank s induces a tautological vector bundle over the Hilbert scheme of n points of S . When s=1 , the top Segre classes of the tautological bundles are given by a recently proven formula conjectured in 1999 by M. Lehn. We calculate here the Segre classes of the tautological bundles for all ranks s over all K -trivial surfaces. Furthermore, in rank s=2 , the Segre integrals are determined for all surfaces, thus establishing a full analogue of Lehn's formula. We also give conjectural formulas for certain series of Verlinde Euler characteristics over the Hilbert schemes of points.