Minimum Eigenvalue Research Articles

Fuzzy Lyapunov-Based Model Predictive Sliding-Mode Control of Nonlinear Systems: An Ellipsoid Recursive Feasibility Approach

In this article, we introduce the design of an artificial fuzzy Lyapunov-based model predictive integral sliding-mode control to achieve the stability of the closed-loop Takagi–Sugeno fuzzy-model-based nonlinear systems. First, to reach the proper performance against the model uncertainties, a fuzzy integral sliding-mode controller (FISMC) is designed. Then, the fuzzy model predictive control (MPC) is developed based on a fuzzy Lyapunov function by considering a contractive constraint and an ellipsoidal terminal constraint. A systematic method is developed to reach the recursive feasibility of the MPC optimization problem based on an ellipsoidal terminal set. Also, a contractive fuzzy Lyapunov condition is imposed on the fuzzy MPC problem to guarantee the stability of closed-loop systems, which is led to a linear matrix inequality-based generalized eigenvalue minimization problem. In the proposed approach, FISMC greatly improves the robustness property of the fuzzy Lyapunov-based model predictive control and the asymptotic stability of the closed-loop system is achieved in comparison with the tube-based MPC. In addition, the proposed robust MPC has a less computational burden and improves the equilibrium point attractivity compared with the max–min MPC and the equality terminal constraint-based MPC. To illustrate the superiority of the proposed strategy, the suggested robust MPC is applied to a truck–trailer system and a numerical example. The simulation results show the capabilities of the proposed robust MPC.

Drag coefficient prediction of complex-shaped snow particles falling in air beyond the Stokes regime

This study considers complex ice particles falling in the atmosphere: predicting the drag of such particles is important for developing of climate models parameterizations. A Delayed-Detached Eddy Simulation model is developed to predict the drag coefficient of snowflakes falling at Reynolds number between 50 and 2200. We first consider the case where the orientation of the particle is known a posteriori, and evaluate our results against laboratory experiments using 3D-printed particles of the same shape, falling at the same Reynolds number. Close agreement is found in cases where the particles fall stably, while a more complex behavior is observed in cases where the flow is unsteady. The second objective of this study is to evaluate methods for estimating the drag coefficient when the orientation of the particles is not known a posteriori. We find that a suitable average of two orientations corresponding to the minimum and maximum eigenvalues of the inertia tensor provides a good estimate of the particle drag coefficient. Meanwhile, existing correlations for the drag on non-spherical particles produce large errors (≈ 50%). A new formula to estimate snow particles settling velocity is also proposed. Our approach provides a framework to investigate the aerodynamics of complex snowflakes and is relevant to other problems that involve the sedimentation of irregular particles in viscous fluids.

Modeling and Adaptive Fault Compensation for Two Physically Linked 2WD Mobile Robots

This article addresses the fault-tolerant control problem of two-wheel-drive (2WD) mobile robots under actuator faults. To improve the actuation redundancy, a physical link is employed to connect two 2WD mobile robots. The kinematics and dynamics of such a two-robot system are modeled. An adaptive fault-tolerant control scheme is developed to make the two-robot system asymptotic track a desired reference. To deal with the uncertainty of the control gain matrix caused by actuator faults, a new positive definite control gain matrix is constructed, and its minimum eigenvalue is utilized for the adaptive compensation design. Simulation results verify the effectiveness of the proposed fault-tolerant control scheme.

Relaxation dynamics of the three-dimensional Coulomb glass model.

In this paper, we analyze the dynamics of the Coulomb glass lattice model in three dimensions near a local equilibrium state by using mean-field approximations. We specifically focus on understanding the role of localization length (ξ) and the temperature (T) in the regime where the system is not far from equilibrium. We use the eigenvalue distribution of the dynamical matrix to characterize relaxation laws as a function of localization length at low temperatures. The variation of the minimum eigenvalue of the dynamical matrix with temperature and localization length is discussed numerically and analytically. Our results demonstrate the dominant role played by the localization length on the relaxation laws. For very small localization lengths, we find a crossover from exponential relaxation at long times to a logarithmic decay at intermediate times. No logarithmic decay at the intermediate times is observed for large localization lengths.

A novel Stewart-type parallel mechanism with topological reconfiguration: Design, kinematics and stiffness evaluation

A new approach is put forward to enhance the stiffness of parallel mechanisms by topological reconfiguration. First, a novel 6-DOF Stewart-type parallel mechanism is designed and analyzed. This mechanism can be reconfigured into three topological configurations, each permitting one rotational motion by means of lockable prismatic joints. Then, an overall rotational stiffness matrix is analytically deduced by relating the external loads exerted on the end-effector to the magnitude of the induced micro-angular displacements. It is proved that the minimum eigenvalue of this matrix can serve as a stiffness index of the parallel mechanism. Subsequently, an optimization objective function is developed for stiffness enhancement through topological reconfiguration, and a singularity-free path planning model for full mobility motion control is formulated. Finally, numerical simulations are provided to compare the stiffness index values of the unlocked and locked mechanisms. The results show that the stiffness of the latter is substantially larger than that of the former, thereby demonstrating the effectiveness of the proposed approach.

An accelerated first-order method with complexity analysis for solving cubic regularization subproblems

We propose a first-order method to solve the cubic regularization subproblem (CRS) based on a novel reformulation. The reformulation is a constrained convex optimization problem whose feasible region admits an easily computable projection. Our reformulation requires computing the minimum eigenvalue of the Hessian. To avoid the expensive computation of the exact minimum eigenvalue, we develop a surrogate problem to the reformulation where the exact minimum eigenvalue is replaced with an approximate one. We then apply first-order methods such as the Nesterov’s accelerated projected gradient method (APG) and projected Barzilai-Borwein method to solve the surrogate problem. As our main theoretical contribution, we show that when an $$\epsilon$$ -approximate minimum eigenvalue is computed by the Lanczos method and the surrogate problem is approximately solved by APG, our approach returns an $$\epsilon$$ -approximate solution to CRS in $${\tilde{O}}(\epsilon ^{-1/2})$$ matrix-vector multiplications (where $${\tilde{O}}(\cdot )$$ hides the logarithmic factors). Numerical experiments show that our methods are comparable to and outperform the Krylov subspace method in the easy and hard cases, respectively. We further implement our methods as subproblem solvers of adaptive cubic regularization methods, and numerical results show that our algorithms are comparable to the state-of-the-art algorithms.

The minimax inverse eigenvalue problem for matrices whose graph is a generalized star of depth 2

In this paper, we study an inverse eigenvalue problem, referred to as the minimax inverse eigenvalue problem, of constructing matrices whose graph is a special type of tree called a generalized star of depth 2. A scheme of labelling the vertices of such a tree is introduced in order to express the corresponding matrices in a number of special forms. We then solve the minimax inverse eigenvalue problem of constructing such matrices from given eigen data consisting of the minimal and maximal eigenvalues of each of the leading principal submatrices of the required matrices. We pose an open problem of solving the minimax inverse eigenvalue problem for matrices described by arbitrary trees.

Ricci curvature, graphs and eigenvalues

We express the discrete Ricci curvature of a graph as the minimal eigenvalue of a family of matrices, one for each vertex of a graph whose entries depend on the local adjacency structure of the graph. Using this method we compute or bound the Ricci curvature of Cayley graphs of finite Coxeter groups and affine Weyl groups. As an application we obtain an isoperimetric inequality that holds for all Cayley graphs of finite Coxeter groups.

Distributed diffusion fusion cooperative spectrum sensing based on reinforcement learning

The openness and complexity of wireless channels make collaborative spectrum sensing vulnerable to malicious users[1]. Therefore, it is very important to identify the attributes of malicious users before collaborative spectrum awareness networks make data fusion decisions. In this paper, a method combining reinforcement learning and cognitive user credit model is proposed, in which the maximum and minimum eigenvalues of signals are used as the initial information for exchange, and the whole sensing network tends to diffuse and fuse nodes with high credit. Finally, the convergence value of the whole network is compared with the decision threshold to complete collaborative spectrum sensing. By combining with consensus fusion network and traditional collaborative sensing algorithm, the proposed method can effectively improve the convergence speed of fusion network and shorten the sensing time on the premise of effectively identifying malicious users[2], so as to improve the spectrum sensing performance and make the collaborative sensing network more adaptive and stable.

Eigenvalues and chromatic number of a signed graph

For a signed graph Σ, let χ(Σ), λ1 and λn be the chromatic number, the maximum eigenvalue and the minimum eigenvalue of Σ, respectively. This paper proves that, for any nonempty signed graph Σ with n vertices,χ(Σ)≥max⁡{1+λ1|λn|,nn−λ1}. These two bounds extend the classical spectral lower bounds of Hoffman and Cvetković for an ordinary graph, respectively.

Sparse noncommutative polynomial optimization

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre [SIAM J. Optim. 17(3) (2006), pp. 822--843] and Waki et al. [SIAM J. Optim. 17(1) (2006), pp. 218--242]. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue and trace of noncommutative polynomials from the literature.

Optimization of nonlinear eigenvalues under measure or perimeter constraint

In this paper we recall some recent results about variational eigenvalues of the p-Laplacian, we show new applications and point out some open problems. We focus on the continuity properties of the eigenvalues under the gamma_p-convergence of capacitary measures, which are needed to prove existence results for the minimization of nonlinear eigenvalues in the class of p-quasi open sets contained in a box under a measure constraint. Finally, the new contribution of this paper is to show that these continuity results can be employed to prove existence of minimizers for nonlinear eigenvalues among measurable sets contained in a box and under a perimeter constraint.

Predicting the Maximum and Minimum Eigenvalues of a Random Matrix Filled with iid Normal Variates: A Study in Simulation

Predicting the Maximum and Minimum Eigenvalues of a Random Matrix Filled with iid Normal Variates: A Study in Simulation

First-order quantum phase transitions as condensations in the space of states

We demonstrate that a large class of first-order quantum phase transitions, namely, transitions in which the ground state energy per particle is continuous but its first order derivative has a jump discontinuity, can be described as a condensation in the space of states. Given a system having Hamiltonian H = K + gV, where K and V are two non commuting operators acting on the space of states , we may always write where is the subspace spanned by the eigenstates of V with minimal eigenvalue and . If, in the thermodynamic limit, M cond/M → 0, where M and M cond are, respectively, the dimensions of and , the above decomposition of becomes effective, in the sense that the ground state energy per particle of the system, ϵ, coincides with the smaller between ϵ cond and ϵ norm, the ground state energies per particle of the system restricted to the subspaces and , respectively: ϵ = min{ϵ cond, ϵ norm}. It may then happen that, as a function of the parameter g, the energies ϵ cond and ϵ norm cross at g = g c. In this case, a first-order quantum phase transition takes place between a condensed phase (system restricted to the small subspace ) and a normal phase (system spread over the large subspace ). Since, in the thermodynamic limit, M cond/M → 0, the confinement into is actually a condensation in which the system falls into a ground state orthogonal to that of the normal phase, something reminiscent of Anderson’s orthogonality catastrophe (Anderson 1967 Phys. Rev. Lett. 18 1049). The outlined mechanism is tested on a variety of benchmark lattice models, including spin systems, free fermions with non uniform fields, interacting fermions and interacting hard-core bosons.

Multi-Path Hybrid Spectrum Sensing in Cognitive Radio

Inefficient utilization of the authorized spectrum emerges cognitive radio (CR) as a hopeful technology for both present and future telecommunications. It is owing to the potency to leverage the obtainable bandwidth of other wireless communication networks and thereby increase its occupancy. The key feature for the cognitive radio system for distinguishing the blank spectrum is spectrum sensing. This paper is intended to establish a hybrid sensing model for spectrum detection in CR to enhance the sensing efficiency of traditional techniques of spectrum sensing, which consists of two parallel paths of hybrid detectors. The first path is formed from two sequential detector stages; in the first phase, energy detector is used to recognize the PU signal existence where the signal has not been identified. Maximum–Minimum Eigenvalue (MME) is used as a second stage to detect the PU signal presence. The second path consists of two parallel stage detectors employing separate ED and MME to detect the PU signal individually, the two results are gathered to make a decision, and then the final detection decision is determined based on the two paths’ detection combined results. The proposed hybrid sensing approach adopted for enhancing the sensing performance is validated with conventional methods. Simulation results show that the proposed approach outperforms various traditional and hybrid approaches in terms of maximizing the detection probability on the specified limitations on the false alarm probability, as it can increase the detection probability to 94% instead of 79% for the parallel detector at SNR = − 10 dB and Pfa = 0.1.

On the spectral radius of graphs without a star forest

Let F=∪i=1kSdi be the union of pairwise vertex-disjoint k stars of order d1+1,…,dk+1, respectively, where k≥2 and d1≥⋯≥dk≥1. In this paper, we present two sharp upper bounds for the spectral radius of F-free (bipartite) graphs and characterize all corresponding extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of an F-free graph and all extremal graphs are obtained.

Joint estimation of frequency and angle-of-arrival for two element antenna array using Eigen-approach

ABSTRACT An Eigen-approach method using a two-element antenna array for estimating the frequencies of source signals jointly with the angle-of-arrival (AOA) was proposed. The eigenvector corresponding to the minimum eigenvalue of the autocorrelation matrix was used initially to find the frequencies of the source signals. By moving the roots produced from the eigenvector one-by-one and finding the power of the output, the frequencies of the source signal were estimated. Finally, using the eigenvector of the autocorrelation for each output signals from the first stage produced the AOA for the source signal. The results presented validate the use of the Eigen-approach with the two-element antenna array method in finding an accurate estimation of frequencies and the AOA for the source signals at different noise power levels even when multiple source signals have the same AOA.

Effect of Objective Function on Data-Driven Greedy Sparse Sensor Optimization

The selection problem of an optimal set of sensors estimating the snapshot of high-dimensional data is considered. The objective functions based on various criteria of optimal design are adopted to the greedy method: D-optimality, A-optimality, and E-optimality, which maximizes the determinant, minimize the trace of inverse, and maximize the minimum eigenvalue of the Fisher information matrix, respectively. First, the Fisher information matrix is derived depending on the numbers of latent state variables and sensors. Then, the unified formulation of the objective function based on A-optimality is introduced and proved to be submodular, which provides the lower bound on the performance of the greedy method. Next, the greedy methods based on D-, A-, and E-optimality are applied to randomly generated systems and a practical data set of global climates. The sensors selected by the D-optimality objective function works better than those by A- and E-optimality with regard to the determinant, trace of the inverse, and reconstruction error, while those by A-optimality works the best with regard to the minimum eigenvalue. On the other hand, the performance of sensors selected by the E-optimality objective function is worse for all indices and reconstruction error. This might be because of the lack of submodularity as proved in the paper. The results indicate that the greedy method based on D-optimality is the most suitable for high accurate reconstruction with low computational cost.

Joint Frequency and DOA Estimation with Automatic Pairing Using the Rayleigh–Ritz Theorem

This paper presents a novel scheme for joint frequency and direction of arrival (DOA) estimation, that pairs frequencies and DOAs automatically without additional computations. First, when the property of the Kronecker product is used in the received array signal of the multiple-delay output model, the frequency-angle steering vector can be reconstructed as the product of the frequency steering vector and the angle steering vector. The frequency of the incoming signal is then obtained by searching for the minimal eigenvalue among the smallest eigenvalues that depend on the frequency parameters but are irrelevant to the DOAs. Subsequently, the DOA related to the selected frequency is acquired through some operations on the minimal eigenvector according to the Rayleigh–Ritz theorem, which realizes the natural pairing of frequencies and DOAs. Furthermore, the proposed method can not only distinguish multiple sources, but also effectively deal with other arrays. The effectiveness and superiority of the proposed algorithm are further analyzed by simulations.

Strict positivity and D-majorization

ABSTRACT Motivated by quantum thermodynamics, we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector d to majorization on square matrices with respect to a positive definite matrix D. For the two-dimensional case, we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of D has multiplicity one.