Input Vector Field Research Articles

Data-driven control of input-affine systems via approximate nonlinearity cancellation

We consider data-driven control of input-affine systems via approximate nonlinearity cancellation. Data-dependent semi-definite program is developed to characterize the stabilizer such that the linear dynamics of the closed-loop systems is stabilized and the influence of the nonlinear dynamics is decreased. Because of the additional nonlinearity brought by the state-dependent input vector field, nonlinearity cancellation is more difficult to achieve. We show that under some assumptions on the nonlinearity, the nonlinearity cancellation control approach can render the equilibrium locally asymptotically stable even if the additional nonlinearity is neglected. Data-based estimation of the region of the attraction is also presented.

Vortex Lens: Interactive Vortex Core Line Extraction using Observed Line Integral Convolution.

This paper describes a novel method for detecting and visualizing vortex structures in unsteady 2D fluid flows. The method is based on an interactive local reference frame estimation that minimizes the observed time derivative of the input flow field v(x, t). A locally optimal reference frame w(x, t) assists the user in the identification of physically observable vortex structures in Observed Line Integral Convolution (LIC) visualizations. The observed LIC visualizations are interactively computed and displayed in a user-steered vortex lens region, embedded in the context of a conventional LIC visualization outside the lens. The locally optimal reference frame is then used to detect observed critical points, where v=w, which are used to seed vortex core lines. Each vortex core line is computed as a solution of the ordinary differential equation (ODE) · w(t)=w(w(t), t), with an observed critical point as initial condition (w(t0), t0). During integration, we enforce a strict error bound on the difference between the extracted core line and the integration of a path line of the input vector field, i.e., a solution to the ODE · v(t)=v(v(t), t). We experimentally verify that this error depends on the step size of the core line integration. This ensures that our method extracts Lagrangian vortex core lines that are the simultaneous solution of both ODEs with a numerical error that is controllable by the integration step size. We show the usability of our method in the context of an interactive system using a lens metaphor, and evaluate the results in comparison to state-of-the-art vortex core line extraction methods.

Controller Reduction for Nonlinear Systems by Generalized Differential Balancing

In this article, we aim at developing computationally tractable methods for nonlinear model/controller reduction. Recently, model reduction by generalized differential (GD) balancing has been proposed for nonlinear systems with constant input-vector fields and linear output functions. First, we study incremental properties in the GD balancing framework. Next, based on these analyses, we provide GD linear quadratic Gaussian (LQG) balancing and GD <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> -balancing as controller reduction methods for nonlinear systems by focusing on linear feedback and observer gains. Especially for GD <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> -balancing, we clarify when the closed-loop system consisting of the full-order system and a reduced-order controller is exponentially stable. All provided methods for controller reduction can be relaxed to linear matrix inequalities (LMIs).

Reconstructing Unsteady Flow Data From Representative Streamlines via Diffusion and Deep-Learning-Based Denoising.

We propose VFR-UFD, a new deep learning framework that performs vector field reconstruction (VFR) for unsteady flow data (UFD). Given integral flow lines (i.e., streamlines), we first generate low-quality UFD via diffusion. VFR-UFD then leverages a convolutional neural network to reconstruct spatiotemporally coherent, high-quality UFD. The core of VFR-UFD lies in recurrent residual blocks that iteratively refine and denoise the input vector fields at different scales, both locally and globally. We take consecutive time steps as input to capture temporal coherence and apply streamline-based optimization to preserve spatial coherence. To show the effectiveness of VFR-UFD, we experiment with several vector field data sets to report quantitative and qualitative results and compare VFR-UFD with two VFR methods and one compression algorithm.

Empirical differential Gramians for nonlinear model reduction

In this paper, we present an empirical balanced truncation method for nonlinear systems whose input vector fields are constants. First, we define differential reachability and observability Gramians. They are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory), and it is difficult to find them as functions of the initial state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems.

Output tracking for a class of non-minimum phase nonlinear systems: A two-point boundary value problem formulation with a hybrid regulator

The asymptotic output tracking problem is studied for a class of non-minimum phase nonlinear systems without requiring the a priori knowledge, or even the existence, of a finite-dimensional exosystem that generates the prescribed periodic reference signal. The design strategy is illustrated in two steps. First, it is shown that the knowledge of a solution to a certain two-point boundary value problem involving the underlying zero-dynamics of the plant is instrumental and sufficient to construct a state feedback regulator that achieves boundedness of the trajectories and (exact) asymptotic tracking, hence completely circumventing the need for solving partial differential equations. Then, since the computation of the latter solution may be affected by numerical errors that are particularly detrimental in the presence of unstable zero-dynamics, the above architecture is robustified by means of an additional hybrid feedback loop whose trajectories converge to a solution of the two-point boundary value problem. Once the latter scheme has been established and discussed, the extension to the case of output feedback is presented, firstly in the specially structured case in which the input vector field depends only on the measured output and then extended to the generic case. The theory is then corroborated by means of a physically motivated numerical example involving an inverted pendulum on a cart. Interestingly, it is also shown that the solution provided by the hybrid scheme above coincides with the limiting solution of a suitably defined cheap optimal control problem.

5G Multimedia Precision Marketing Based on the Improved Multisensor Node Collaborative Filtering Recommendation Algorithm

The rapid development of web technology has brought new problems and challenges to the recommendation system: on the one hand, the traditional collaborative filtering recommendation algorithm has been difficult to meet the personalized recommendation needs of users; on the other hand, the massive data brought by web technology provides more useful information for recommendation algorithms. How to extract features from this information, alleviate sparsity and dynamic timeliness, and effectively improve recommendation quality is a hot issue in the research of recommendation system algorithms. In view of the lack of an effective multisource information fusion mechanism in the existing research, an improved 5G multimedia precision marketing based on an improved multisensor node collaborative filtering recommendation algorithm is proposed. By expanding the input vector field, the features of users’ social relations and comment information are extracted and fused, and the problem of collaborative modelling of these two kinds of important auxiliary information is solved. The objective function is improved, the social regularization term and the internal regularization term in the vector domain are analysed and added from the perspective of practical significance and vector structure, which alleviates the overfitting problem. Experiments on a large number of real datasets show that the proposed method has higher recommendation quality than the classical and mainstream baseline algorithm.

Lorenz chaotic system generated from Shimizu-Morioka system or Finance system: Differential geometric approach

The problem of how to generate the Lorenz attractor from several nonlinear control systems is investigated in this paper. To be more precise, the conversions from the controlled Shimizu-Morioka system and the controlled Finance system to the Lorenz system are achieved by using the differential geometric control theory. For each case a scalar control input and a state transformation are proposed. The main approach of this paper is to convert all of those three-order systems into so called lower triangular forms which all have the same first two equations. Thus converting the controlled Shimizu-Morioka system or the controlled Finance system into the Lorenz attractor is feasible by choosing an appropriate scalar control input in the third equation of each of the two control systems. To this end, firstly, in order to use the tools of the differential geometry we construct a controlled Lorenz system by treating the vector field of the Lorenz attractor as the drift vector field and treating a linear vector field with three parameters as an input vector field. When those parameters are selected in a special manner, the conditions under which the controlled Lorenz system can be equivalently transformed into the lower triangular form are satisfied. Secondly, a state transformation, through which the controlled Lorenz system can be described as a lower triangular form, is obtained by a method like Gaussian elimination instead of solving three complicated partial differential equations. Employing several partial state transformations, choosing those three parameters and setting a scalar control input, we can reduce the equations of the controlled Lorenz system into its simplest lower triangular form. Thirdly, through two state transformations designed for the controlled Shimizu-Morioka system and the controlled Finance system respectively, the two control systems are converted into their lower triangular forms which are both similar to that of the Lorenz system in a way aforementioned. A smooth scalar controller is given to achieve the anti-control from the controlled Shimizu-Morioka system to the Lorenz attractor while another non-smooth scalar controller is designed to realize the generalized synchronization from the controlled Finance system to the Lorenz system no matter what the initial values of the two systems are. Finally, two numerical simulations demonstrate the control schemes designed in this paper.

Adaptive tracking of switched nonlinear systems with prescribed performance using a reference-dependent reparametrisation approach

ABSTRACTIn this paper, adaptive tracking control of switched nonlinear systems in the parametric strict-feedback form is investigated. After defining a reparametrisation lemma in the presence of a non-zero reference signal, we propose a new adaptive backstepping design of the virtual controllers that can handle the extra terms arising from the reparametrisation (and that the state-of-the-art backstepping designs cannot dominate). The proposed adaptive design guarantees, under arbitrarily fast switching, an a priori bound for the steady-state performance of the tracking error and a tunable bound for the transient error. Finally, the proposed method, by overcoming the need for subsystems with common sign of the input vector field, enlarges the class of uncertain switched nonlinear systems for which the adaptive tracking problem can be solved. A numerical example is provided to illustrate the proposed control scheme.

Stability Analysis and Control of Nonholonomic Systems with Potential Fields

This paper presents a theoretical analysis on the stability of nonholonomic systems when a potential field method is applied to them, and shows a numerical example for a nonholonomic underwater vehicle among obstacles. Nonholonomic systems with a potential function have an infinite number of equilibrium points, because the motion of the systems cannot always be generated exactly along the gradient vector of the potential function. By utilizing the component of the input that does not increase or decrease the potential function, the equilibrium points other than the critical points of the function can be destabilized, if the controllability of the systems is satisfied with the first-order Lie brackets of input vector fields. A time-invariant controller is proposed based on the theoretical analysis on the stability of equilibrium points, and applied to an underwater vehicle among obstacles. When the potential function has saddles as its critical points, the potential function is modified to be time-varying near the saddles in order to prevent the system from being trapped in the saddles. Numerical simulation results demonstrate that the underwater vehicle with the proposed controller converges to the desired point without collision with obstacles.

Improved backstepping control for large angle maneuvers of spacecraft

For the control of large angle maneuvers of a spacecraft, variable gain backstepping control is proposed. The controller can make the states of the system converge to the commanded position along the input vector field orientation. The controller stabilizes the system with the amplitude of the commanded torques decreased. Considering the uncertainty of the disturbance torques and the estimation error of the inertia matrix, the control design is improved to provide strong self-adaptability and robustness of the system. Simulation is conducted, and the results show that the design has good tracking performance and convergence, consistent with the theoretical analysis.

Feature-aware streamline generation of planar vector fields via topological methods

Methods for vector field visualization strive to have a sparse representation of the field, while encoding all of its important features. Streamline visualization is one of the most popular such methods. Traditionally, a set of streamline methods have focused on capturing the salient features of the vector field such as sources, sinks, and vortices. However, not all features are created equal, and some features of the vector field are more important than others, which could simply be mere noise. It is this problem of characterizing feature importance through streamline visualization that we try to address in this paper. Specifically, a given 2D vector field can be decomposed into a rotation-free (gradient) component, divergence-free (curl) component and a harmonic component by the so-called Hodge decomposition. Features in the original vector field, in some sense, correspond to features in the first two components. Furthermore, the gradient and curl components are each induced by a scalar field. By analyzing these two corresponding scalar fields using topological methods (in particular the contour tree and the persistent homology), we develop a simple and novel algorithm whose streamline density tends to reflect the topological importance of the features in the input vector field. Such a feature-aware streamline sketch is more informative, yet still simple both visually and in terms of its generation. It enhances our understanding of the underlying vector field, which is demonstrated here by several experimental results.

Robust Morse Decompositions of Piecewise Constant Vector Fields

In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. To overcome the intrinsic difficulty in PC vector fields (in particular, discontinuity along mesh edges), we borrow results from the theory of differential inclusions. The input vector field and its PC variant have similar Morse decompositions. We introduce a robust and efficient algorithm to compute Morse decompositions of a PC vector field. Our approach provides subtriangle precision for Morse sets. In addition, we describe a Morse set classification framework which we use to color code the Morse sets in order to enhance the visualization. We demonstrate the benefits of our approach with three well-known simulation data sets, for which our method has produced Morse decompositions that are similar to or finer than those obtained using existing techniques, and is over an order of magnitude faster.

Young’s two-slit interference of vector light fields

We explore the peculiar interference behaviors of the vector fields in the Young's two-slit configuration. The interference patterns have a chessboard structure in the middle region and depend on the topological charge and the initial phase of the input vector field. The results have potential applications such as characterizing the topological properties of the arbitrary vector fields.

Unveiling locally linearly polarized vector fields with broken axial symmetry

We explore how the focusing of axial-symmetry-broken vector fields is associated with a sector aperture whose apex coincides with the field axis. Its focused field is azimuthally split into a pair of orthogonal right- and left-handed circularly polarized partial rings which are rotated by angles of $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ with respect to the input vector field carrying no orbital angular momentum (OAM) and no spin angular momentum (SAM), respectively. Each member of the pair of partial rings carries not only the opposite OAM but also the opposite SAM. We unveil the underlying physics behind such an effect.

Visual Reconstructability as a Quality Metric for Flow Visualization

AbstractWe present a novel approach for the evaluation of 2D flow visualizations based on the visual reconstructability of the input vector fields. According to this metric, a visualization has high quality if the underlying data can be reliably reconstructed from the image. This approach provides visualization creators with a cost‐effective means to assess the quality of visualization results objectively. We present a vision‐based reconstruction system for the three most commonly‐used visual representations of vector fields, namely streamlines, arrow glyphs, and line integral convolution. To demonstrate the use of visual reconstructability as a quality metric, we consider a selection of vector fields obtained from numerical simulations, containing typical flow features. We apply the three types of visualization to each dataset, and compare the visualization results based on their visual reconstructability of the original vector field.

Stable Morse Decompositions for Piecewise Constant Vector Fields on Surfaces

AbstractNumerical simulations and experimental observations are inherently imprecise. Therefore, most vector fields of interest in scientific visualization are known only up to an error. In such cases, some topological features, especially those not stable enough, may be artifacts of the imprecision of the input. This paper introduces a technique to compute topological features of user‐prescribed stability with respect to perturbation of the input vector field. In order to make our approach simple and efficient, we develop our algorithms for the case of piecewise constant (PC) vector fields. Our approach is based on a super‐transition graph, a common graph representation of all PC vector fields whose vector value in a mesh triangle is contained in a convex set of vectors associated with that triangle. The graph is used to compute a Morse decomposition that is coarse enough to be correct for all vector fields satisfying the constraint. Apart from computing stable Morse decompositions, our technique can also be used to estimate the stability of Morse sets with respect to perturbation of the vector field or to compute topological features of continuous vector fields using the PC framework.

Interactive Vector Field Feature Identification

We introduce a flexible technique for interactive exploration of vector field data through classification derived from user-specified feature templates. Our method is founded on the observation that, while similar features within the vector field may be spatially disparate, they share similar neighborhood characteristics. Users generate feature-based visualizations by interactively highlighting well-accepted and domain specific representative feature points. Feature exploration begins with the computation of attributes that describe the neighborhood of each sample within the input vector field. Compilation of these attributes forms a representation of the vector field samples in the attribute space. We project the attribute points onto the canonical 2D plane to enable interactive exploration of the vector field using a painting interface. The projection encodes the similarities between vector field points within the distances computed between their associated attribute points. The proposed method is performed at interactive rates for enhanced user experience and is completely flexible as showcased by the simultaneous identification of diverse feature types.

Design of Globally Adaptive Stabilizer for Underwater Navigator

In this paper, the problem of adaptive stabilization of an underwater navigator with unknown parameters both in the state vector-field and the input vector-field has been considered. By employing the control Lyapunov function method and parameter projection techniques, a direct adaptive controller is designed to complete the globally adaptive stability of the navigator. Simulations are provided to illustrate the effectiveness of our proposed method.

Animation of Orthogonal Texture Patterns for Vector Field Visualization

This paper introduces orthogonal vector field visualization on 2D manifolds: a representation by lines that are perpendicular to the input vector field. Line patterns are generated by line integral convolution (LIC). This visualization is combined with animation based on motion along the vector field. This decoupling of the line direction from the direction of animation allows us to choose the spatial frequencies along the direction of motion independently from the length scales along the LIC line patterns. Vision research indicates that local motion detectors are tuned to certain spatial frequencies of textures, and the above decoupling enables us to generate spatial frequencies optimized for motion perception. Furthermore, we introduce a combined visualization that employs orthogonal LIC patterns together with conventional, tangential streamline LIC patterns in order to benefit from the advantages of these two visualization approaches. In addition, a filtering process is described to achieve a consistent and temporally coherent animation of orthogonal vector field visualization. Different filter kernels and filter methods are compared and discussed in terms of visualization quality and speed. We present respective visualization algorithms for 2D planar vector fields and tangential vector fields on curved surfaces, and demonstrate that those algorithms lend themselves to efficient and interactive GPU implementations.