Convex Cone Research Articles

An MPC-DCM Control Method for a Forward-Bending Biped Robot Based on Force and Moment Control

For a forward-bending biped robot with 10 degrees of freedom on its legs, a new control framework of MPC-DCM based on force and moment is proposed in this paper. Specifically, the Diverging Component of Motion (DCM) is a stability criterion for biped robots based on linear inverted pendulum, and Model Predictive Control (MPC) is an optimization solution strategy using rolling optimization. In this paper, DCM theory is applied to the state transition matrix of the system, combined with simplified rigid body dynamics, the mathematical description of the biped robot system is established, the classical MPC method is used to optimize the control input, and DCM constraints are added to the constraints of MPC, making the real-time DCM approximate to a straight line in the walking single gait. At the same time, the linear angle and friction cone constraints are considered to enhance the stability of the robot during walking. In this paper, MATLAB/Simulink is used to simulate the robot. Under the control of this algorithm, the robot can reach a walking speed of 0.75 m/s and has a certain anti-disturbance ability and ground adaptability. In this paper, the Model-H16 robot is used to deploy the physical algorithm, and the linear walking and obstacle walking of the physical robot are realized.

The Orlicz–Pettis theorem for locally convex cones

The Orlicz–Pettis theorem for locally convex cones

Shape and stability of rotation sets for incompressible flows on the torus

AbstractIn this paper we introduce the concept of rotation cones, determined by rotation sets, and use these to describe the stability of rotation sets of incompressible and fixed-point free continuous flows on the torus $$\mathbb{T}^{d}$$ T d , d ⩾ 2. The previous concept is equivalent and extends the stability of rotation numbers (and rotation vectors) in the special case of fixed-point free continuous flows on $$\mathbb{T}^{2}$$ T 2 . We prove that incompressible Lipschitz vector fields with stable rotation sets have convex rotation cones with non-empty interior, and that all extremal edges are collinear with vectors in ℤd. If, in addition, the rotation cones are proper, then these are polyhedral with finitely many edges. Finally we prove that the set of vector fields with stable rotation sets is C0-open and dense among those having proper rotation cones.

Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces

Abstract Let Y be a real vector metric space and K ⊂ Y be a closed convex cone such that K ∩ (− K) = {0}. We prove that a convex compact-valued map F : ℝ → 2 Y ∖ {∅} is K-continuous and K-additive if and only if there are non-empty convex compact sets A, B ⊂ Y such that 0 ∈ A − B ⊂ K and F is equal “modulo K” to the continuous set-valued map G ( t ) = t A , t ≥ 0 , t B , t < 0. $$\begin{array}{} \displaystyle G(t)=\begin{cases} tA,&t\geq0,\\ tB,& t \lt 0. \end{cases} \end{array}$$ Next, we use this result to characterize convex compact-valued maps F : ℝ N → 2 Y ∖ {∅}.

Invariant cones for jump-diffusions in infinite dimensions

In this paper we provide sufficient conditions for stochastic invariance of closed convex cones for stochastic partial differential equations (SPDEs) of jump-diffusion type, and clarify when these conditions are necessary. We emphasize that the diffusion coefficient of the SPDE does not need to be smooth, an assumption which is frequently imposed when dealing with stochastic invariance problems. Our results apply to the positive cone of abstract L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-spaces. Furthermore, we present a series of applications, where we investigate SPDEs arising in natural sciences and economics.

Vector optimization problems in near vector space

ABSTRACT The conventional vector optimization problems intend to optimize the objective functions, which are defined on a vector space into another vector space. In this paper, we are going to study the vector optimization problems by minimizing the objective functions, which are defined on a vector space into another so-called near vector space. The concept of near vector space is axiomatically defined, and it does not necessarily contain the additive inverse element. A so-called pointed-like convex cone is introduced to propose the concept of minimizer of this new type vector optimization problem. Its corresponding scalar optimization problem is also formulated. Finally, we show that the optimal solution of the corresponding scalar optimization problem is also a minimizer of the original vector optimization problem based on the near vector space.

Early detection and lesion visualization of pear leaf anthracnose based on multi-source feature fusion of hyperspectral imaging.

Pear anthracnose, caused by Colletotrichum bacteria, is a severe infectious disease that significantly impacts the growth, development, and fruit yield of pear trees. Early detection of pear anthracnose before symptoms manifest is of great importance in preventing its spread and minimizing economic losses. This study utilized hyperspectral imaging (HSI) technology to investigate early detection of pear anthracnose through spectral features, vegetation indices (VIs), and texture features (TFs). Healthy and diseased pear leaves aged 1 to 5 days were selected as subjects for capturing hyperspectral images at various stages of health and disease. Characteristic wavelengths (OWs1 and OWs2) were extracted using the Successive Projection Algorithm (SPA) and Competitive Adaptive Reweighted Sampling (CARS) algorithm. Significant VIs were identified using the Random Forest (RF) algorithm, while effective TFs were derived from the Gray Level Co-occurrence Matrix (GLCM). A classification model for pear leaf early anthracnose disease was constructed by integrating different features using three machine learning algorithms: Support Vector Machine (SVM), Extreme Learning Machine (ELM), and Back Propagation Neural Network (BPNN). The results showed that: the classification identification model constructed based on the feature fusion performed better than that of single feature, with the OWs2-VIs-TFs-BPNN model achieving a highest accuracy of 98.61% in detection and identification of pear leaf early anthracnose disease. Additionally, to intuitively and effectively monitor the progression and severity of anthracnose in pear leaves, the visualization of anthracnose lesions was achieved using Successive Maximum Angle Convex Cone (SMACC) and Spectral Information Divergence (SID) techniques. According to our research results, the fusion of multi-source features based on hyperspectral imaging can be a reliable method to detect early asymptomatic infection of pear leaf anthracnose, and provide scientific theoretical support for early warning and prevention of pear leaf diseases.

Bionic Design of High-Performance Joints: Differences in Failure Mechanisms Caused by the Different Structures of Beetle Femur-Tibial Joints.

Beetle femur-tibial joints can bear large loads, and the joint structure plays a crucial role. Differences in living habits will lead to differences in femur-tibial joint structure, resulting in different mechanical properties. Here, we determined the structural characteristics of the femur-tibial joints of three species of beetles with different living habits. The tibia of Scarabaeidae Protaetia brevitarsis and Cetoniidae Torynorrhina fulvopilosa slide through cashew-shaped bumps on both sides of the femur in a guide rail consisting of a ring and a cone bump. The femur-tibial joint of Buprestidae Chrysodema radians is composed of a conical convex tibia and a circular concave femur. A bionic structure design was developed out based on the characteristics of the structure of the femur-tibial joints. Differences in the failure of different joint models were obtained through experiments and finite element analysis. The experimental results show that although the spherical connection model can bear low loads, it can maintain partial integrity of the structure and avoid complete failure. The cuboid connection model shows a higher load-bearing capacity, but its failure mode is irreversible deformation. As key parts of rotatable mechanisms, the bionic models have the potential for wide application in the high-load engineering field.

Discrete dynamics and supergeometry

We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and naturally incorporates laboratories. The latter are embedded symplectic submanifolds of an odd-dimensional symplectic structure. When suitably defined, symplectic geometry in odd dimensions is exactly the structure needed for covariance. A fundamentally probabilistic viewpoint allows classical supergeometries to describe discrete dynamics. We solve the problem of how to construct probabilistic measures on supermanifolds given a (possibly odd dimensional) supersymplectic structure. This relies on a superanalog of the Hodge star for differential forms and a description of probabilities by convex cones. We also show how stochastic processes such as Markov chains can be described by supergeometry.

Designing a switching law for Mittag-Leffler stability in nonlinear singular fractional-order systems and its applications in synchronization

In this work, the stability and synchronization issue of switched singular continuous-time fractional-order systems with nonlinear perturbation are examined. Using the fixed-point principle and S-procedure lemma, a sufficient condition for the existence and uniqueness of the solution to the switched singular fractional-order system is first stated. Next, using the Lyapunov functional method in combination with some techniques related to singular systems and fractional calculus, a switching rule for regularity, impulse-free, and Mittag-Leffler stability is developed based on the formation of a partition of the stability state regions in convex cones. For synchronizing switched fractional singular dynamical systems, we propose a state feedback controller that ensures regularity, impulse-free, and Mittag-Leffler stable in the error closed-loop system. Finally, the ease of use and computational convenience of our proposed methods are illustrated by two numerical examples and a practical example about DC motor controlling an inverted pendulum accompanied by simulation results.

C-Semigroups and Their Induced Order

Let C⊂Np be an integer polyhedral cone. An affine semigroup S⊂C is a C-semigroup if |C∖S|<+∞. This structure has always been studied using a monomial order. The main issue is that the choice of these orders is arbitrary. In the present work, we choose the order given by the semigroup itself, which is a more natural order. This allows us to generalise some of the definitions and results known from numerical semigroup theory to C-semigroups.

Stochastic Convex Cone Programming for Joint Optimal BESS Operation and Q-Placement in Net-Zero Microgrids

Microgrids have emerged as a pivotal solution in the quest for efficient, resilient, and sustainable energy systems. Comprising diverse distributed energy resources, microgrids present a compelling opportunity to revolutionize how we generate, store, and distribute electricity, while simultaneously reducing carbon footprints. This paper proposes an optimal battery energy storage system (BESS) management scheme, along with capacitor placement for reactive power (Q)-compensation, and scheduling for the purpose of a renewable-based microgrid’s loss minimization. The proposed model evaluates the impact of BESS management on energy efficiency and analyzes how optimal scheduling of BESS influences system losses. Furthermore, it investigates the coordinated planning and operation of active assets within the microgrid, such as controllable capacitor banks, in enhancing overall efficiency. The model is formulated as a mixed-integer second-order cone programming (MISOCP) problem which is solved for both deterministic and stochastic generation and consumption data. The proposed model is tested on a 21-bus microgrid comprising photovoltaic and hydropower energy resources, and the efficacy of the model is approved by several case studies. The simulation results show that the proposed method can reduce microgrid energy losses by approximately 12 percent using the deterministic approach and around 14 percent with the stochastic approach.

Interval-Valued Multiobjective Programming Problems Based on Convex Cones

The new solution concepts of interval-valued multiobjective optimization problems using ordering cone are proposed in this paper. An equivalence relation is introduced to divide the collection of all bounded closed intervals into the equivalence classes. The family of all equivalence classes is also called a quotient set. In this case, this quotient set can turn into a vector space under some suitable settings for vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. It means that an ordering in the quotient set can be defined to study the Pareto optimal solution in multiobjective optimization problems. In this paper, we consider the multiobjective optimization problem such that its coefficients are taken to be the bounded closed intervals. With the help of the convex cone, we can study the Pareto optimal solutions of the multiobjective optimization problem with interval-valued coefficients.

Methods for testing the random utility model

The Random Utility Model, central in stochastic choice theory, is equivalent to assume that a probability vector belongs to a convex cone. We investigate its underlying geometry, introduce two new testing procedures, and compare them by simulation.

Theorem of Alternative and Scalarization of Optimization Problems in Topological Vector Spaces

In this paper, we introduce the definitions of affinelike, preaffinelike, generalized affinelike, and generalized preaffinelike functions by use of "pointed convex cone", and prove that those definitions of generalized affine functions are all different. We discuss optimization problems in topological vector spaces and obtain a theorem of alternative and a scalarization theorem. Our inequalities are given by partial order relations. Our generalized affineness may be used for many discussions in mathematics or applied mathematics wherever the affineness is a condition.

Condition number of a convex cone

Condition number of a convex cone

The natural ordering cone of a polyhedral convex set-valued objective mapping

ABSTRACT For a given polyhedral convex set-valued mapping we define a polyhedral convex cone which we call the natural ordering cone. We show that the solution behaviour of a polyhedral convex set optimization problem can be characterized by this cone. Under appropriate assumptions, the natural ordering cone is proven to be the smallest ordering cone which makes a polyhedral convex set optimization problem solvable.

A new concept of semistrict quasiconvexity for vector functions

We establish a new concept of semistrict quasiconvexity for vector functions defined on a nonempty convex set in a real linear space X that take values in some real topological linear space Y, partially ordered by a proper solid convex cone C. The so-called semistrict C-quasiconvexity notion recovers the classical concept of semistrict quasiconvexity of scalar functions when Y = R and C = R + . Additionally, analogous to the scalar scenario, if the cone C is closed, a vector function is both semistrictly C-quasiconvex and C-quasiconvex (in the sense of Luc, 1989) if and only if it is explicitly C-quasiconvex (in the sense of Popovici, 2007). Finally, we convey a characterization of semistrictly C-quasiconvex functions by means of scalar semistrictly quasiconvex functions that are compositions of the nonlinear scalarization functions introduced by Gerstewitz (Tammer) in 1983 with the initial vector function. In light of this characterization, the new concept of semistrict C-quasiconvexity seems to be a natural vector counterpart for the scalar concept of semistrict quasiconvexity.

Decay rate of discrete-time homogeneous delayed systems with cone invariance

This paper is dedicated to analysing the decay rate of discrete-time homogeneous delayed systems that exhibit order preservation within a proper cone. First, a sufficient criterion ensuring the cone invariance of the system is introduced. By exploring the monotonicity of system trajectories, a condition that is both sufficient and necessary for guaranteeing the asymptotic stability of the system is presented. Importantly, under a weak assumption for the growth rate of time-varying delays, a universal framework is provided for characterising the decay rate of the system. Furthermore, typical examples of decay rate analysis are presented when delays are bounded or experience linear, sublinear, or logarithmic growth rates. Eventually, the rationality of the results is illustrated through a numerical simulation.

Optimal quantitative stability for a Serrin-type problem in convex cones

Optimal quantitative stability for a Serrin-type problem in convex cones