Positive Definiteness Requirement Research Articles

Sampled-Data Control for Exponential Synchronization of Delayed Inertial Neural Networks With Aperiodic Sampling and State Quantization.

This article is devoted to dealing with exponential synchronization for inertial neural networks (INNs) with heterogeneous time-varying delays (HTVDs) under the framework of aperiodic sampling and state quantization. First, by taking the effect of aperiodic sampling and state quantization into consideration, a novel quantized sampled-data (QSD) controller with time-varying control gain is designed to tackle the exponential synchronization of INNs. Second, considering the available information of the lower and upper bounds of each HTVD, a refined Lyapunov-Krasovskii functional (LKF) is proposed. Meanwhile, an improved looped-functional method is utilized to fully capture the characteristic of practical sampling patterns and further relax the positive definiteness requirement for LKF. Consequently, less conservative exponential synchronization conditions with extra flexibility are derived. Finally, a numerical example is employed to demonstrate the effectiveness and advantages of the proposed synchronization method.

A Switching Memory-Based Event-Trigger Scheme for Synchronization of Lur'e Systems With Actuator Saturation: A Hybrid Lyapunov Method.

This article is concerned with the event-triggered synchronization of Lur'e systems subject to actuator saturation. Aiming at reducing control costs, a switching-memory-based event-trigger (SMBET) scheme, which allows a switching between the sleeping interval and the memory-based event-trigger (MBET) interval, is first presented. In consideration of the characteristics of SMBET, a piecewise-defined but continuous looped-functional is newly constructed, under which the requirement of positive definiteness and symmetry on some Lyapunov matrices is dropped within the sleeping interval. Then, a hybrid Lyapunov method (HLM), which bridges the gap between the continuous-time Lyapunov theory (CTLT) and the discrete-time Lyapunov theory (DTLT), is used to make the local stability analysis of the closed-loop system. Meanwhile, using a combination of inequality estimation techniques and the generalized sector condition, two sufficient local synchronization criteria and a codesign algorithm for the controller gain and triggering matrix are developed. Furthermore, two optimization strategies are, respectively, put forward to enlarge the estimated domain of attraction (DoA) and the allowable upper bound of sleeping intervals on the premise of ensuring local synchronization. Finally, a three-neuron neural network and the classical Chua's circuit are used to carry out some comparison analyses and to display the advantages of the designed SMBET strategy and the constructed HLM, respectively. Also, an application to image encryption is provided to substantiate the feasibility of the obtained local synchronization results.

Aperiodic Sampled-Data Control for Anti-Synchronization of Chaotic Nonlinear Systems Subject to Input Saturation

This paper studies the aperiodic sampled-data (SD) control anti-synchronization issue of chaotic nonlinear systems under the effects of input saturation. At first, to describe the simultaneous existence of the aperiodic SD pattern and the input saturation, a nonlinear closed-loop system model is established. Then, to make the anti-synchronization analysis, a relaxed sampling-interval-dependent Lyapunov functional (RSIDLF) is constructed for the resulting closed-loop system. Thereinto, the positive definiteness requirement of the RSIDLF is abandoned. Due to the indefiniteness of RSIDLF, the discrete-time Lyapunov method (DTLM) then is used to guarantee the local stability of the trivial solutions of the modeled nonlinear system. Furthermore, two convex optimization schemes are proposed to expand the allowable initial area (AIA) and maximize the upper bound of the sampling period (UBSP). Finally, two examples of nonlinear systems are provided to illustrate the superiority of the RSIDLF method over the previous methods in expanding the AIA and enlarging the UBSP.

Coordinated Disturbance Observer-Based Flight Control of Fixed-Wing UAV

In this brief, a novel coordinated disturbance observer (CDO) is proposed to tackle the disturbance which affects the stable flight of a fixed-wing unmanned aerial vehicle (UAV). The above CDO is composed of multiple disturbance observers which have a basic form and depend on the designed coordinated variables. Due to the above special structure of the CDO, for one thing, the requirement of positive definiteness of the observer gain posed on a single disturbance observer is removed in the CDO. For another, the performance of the CDO can be enhanced by weighting multiple small disturbance observer gains. Via the Lyapunov function method, not only the decay rate and stable region of disturbance estimation error (DEE) of the CDO for the fixed-wing UAV are analyzed, but also the stability criteria of DEE and tracking error are given. Finally, the effectiveness of the proposed CDO is verified through the simulation of flight control of a fixed-wing UAV.

Distributed Control of DC Microgrids With Improved ZIP Load Adaptability

This article presents a distributed consensus-based controller for dc microgrids to achieve proportional current sharing and weighted average voltage regulation in the presence of ZIP [constant impedance (Z), constant current (I), and constant power (P)] loads. The proposed algorithm allows the regulation of the global weighted average voltage in a distributed manner. The precondition on initial bus voltages is relaxed. Furthermore, this study investigates the negative conductance introduced by constant power loads. Based on the properties of Laplacian matrices, the positive definiteness requirement on the conductance matrix is relaxed. A sufficient stability condition on ZIP loads is obtained with improved adaptability. By using the Lyapunov method, large-signal stability is analyzed rigorously for a wide range of loading conditions. The current sharing and voltage regulation errors are proved to converge to zero exponentially. Finally, simulations based on a switch-level dc microgrid model illustrate the advantages of the designed control algorithm.

Stability Analysis for Delayed Neural Networks via a Novel Negative-Definiteness Determination Method.

The stability of neural networks with a time-varying delay is studied in this article. First, a relaxed Lyapunov-Krasovskii functional (LKF) is presented, in which the positive-definiteness requirement of the augmented quadratic term and the delay-product-type terms are set free, and two double integral states are augmented into the single integral terms at the same time. Second, a new negative-definiteness determination method is put forward for quadratic functions by utilizing Taylor's formula and the interval-decomposition approach. This method encompasses the previous negative-definiteness determination approaches and has less conservatism. Finally, the proposed LKF and the negative-definiteness determination method are applied to the stability analysis of neural networks with a time-varying delay, whose advantages are shown by two numerical examples.

On the banded Toeplitz structured distance to symmetric positive semidefiniteness

This paper is concerned with the determination of a close real banded positive definite Toeplitz matrix in the Frobenius norm to a given square real banded matrix. While it is straightforward to determine the closest banded Toeplitz matrix to a given square matrix, the additional requirement of positive definiteness makes the problem difficult. We review available theoretical results and provide a simple approach to determine a banded positive definite Toeplitz matrix.

Optimizing and Factorizing the Wilson Matrix

The Wilson matrix, W, is a 4 × 4 unimodular symmetric positive definite matrix of integers that has been used as a test matrix since the 1940’s, owing to its mild ill-conditioning. We ask how close W is to being the most ill-conditioned matrix in its class, with or without the requirement of positive definiteness. By exploiting the matrix adjugate and applying various matrix norm bounds from the literature we derive bounds on the condition numbers for the two cases and we compare them with the optimal condition numbers found by exhaustive search. We also investigate the existence of factorizations with Z having integer or rational entries. Drawing on recent research that links the existence of these factorizations to number-theoretic considerations of quadratic forms, we show that W has an integer factor Z and two rational factors, up to signed permutations. This little matrix continues to be a useful example on which to apply existing matrix theory as well as being capable of raising challenging questions that lead to new results.

PSEUDO-RIEMANNIAN SPACES WITH A SPECIAL RIEMANN TENSOR

The paper considers pseudo-Riemannian spaces, the Riemann tensor of which has a special structure. The structure of the Riemann tensor is given as a combination of special symmetric and obliquely symmetric tensors. Tensors are selected so that the results can be applied in the theory of geodetic mappings, the theory of holomorphic-projective mappings of Kähler spaces, as well as other problems arising in differential geometry and its application in general relativity, mechanics and other fields. Through the internal objects of pseudo-Riemannian space, others are determined, which are studied depending on what problems are solved in the study of pseudo-Riemannian spaces. By imposing algebraic or differential constraints on internal objects, we obtain special spaces. In particular, if constraints are imposed on the metric we will have equidistant spaces. If on the Ricci tensor, we obtain spaces that allow φ (Ric)-vector fields, and if on the Einstein tensor, we have almost Einstein spaces. The paper studies pseudo-Riemannian spaces with a special structure of the curvature tensor, which were introduced into consideration in I. Mulin paper. Note that in his work these spaces were studied only with the requirement of positive definiteness of the metric. The proposed approach to the specialization of pseudo-Riemannian spaces is interesting by combining algebraic requirements for the Riemann tensor with differential requirements for its components. In this paper, the research is conducted in tensor form, without restrictions on the sign of the metric. Depending on the structure of the Riemann tensor, there are three special types of pseudo-Riemannian spaces. The properties which, if necessary, satisfy the Richie tensors of pseudoriman space and the tensors which determine the structure of the curvature tensor are studied. In all cases, it is proved that special tensors satisfy the commutation conditions together with the Ricci tensor. The importance and usefulness of such conditions for the study of pseudo-Riemannian spaces is widely known. Obviously, the results can be extended to Einstein tensors. Proven theorems allow us to effectively investigate spaces with constraints on the Ricci tensor.

Square roots of H-nonnegative matrices

Roots of matrices are well-studied. The conditions for their existence are understood: The block sizes of nilpotent Jordan blocks, arranged in pairs, have to satisfy some simple algebraic property.More interesting are structured roots of structured matrices. Probably the best known example is the existence and uniqueness of positive definite square roots of a positive definite matrix. If one drops the requirement of positive definiteness of the square root, it turns out that there exists an abundance of square roots. Here a description of all canonical forms of all square roots is possible and is straightforward.H-nonnegative matrices are H-selfadjoint and are nonnegative with respect to an indefinite inner product with Gramian H. An H-nonnegative matrix B allows a decomposition in a negative definite, a nilpotent H-nonnegative, and a positive definite matrix, B=B−⊕B0⊕B+. The interesting part is B0, as only Jordan blocks of size one and two occur. Determining a square root of B reduces to determining a square root of each of B−, B0, and B+. Here we investigate for an H-nonnegative matrix: its square roots without additional structure, as well as its structured square roots that are H-nonnegative or H-selfadjoint.For these three classes of square roots of H-nonnegative matrices we show a simple criterion for their existence and describe all possible canonical forms. This is based mainly on known results but an important new part is that in all three cases we describe all possible square roots of the nilpotent H-nonnegative matrix B0 explicitly. Moreover, we show how our results can be applied to the conditional and unconditional stability of H-nonnegative square roots of H-nonnegative matrices, where the explicit description of the square roots of B0 is used.

The indefinite proximal point algorithms for maximal monotone operators

The proximal point algorithm (PPA) has been widely used in convex optimization. Many algorithms fall into the framework of PPA. To guarantee the convergence of PPA, however, existing results conventionally need to ensure the positive definiteness of the corresponding proximal measure. In some senses, this essentially results in tiny step sizes (or over regularization) for the subproblems and thus inevitably decelerates the overall convergence speed of PPA. In this paper, we investigate the possibility of relaxing the positive definiteness requirement of the proximal measure in PPA. An indefinite PPA for finding a zero of maximal monotone operator is thus proposed via choosing an indefinite proximal regularization term, resulting in larger step sizes. Under some suitable conditions, we prove the global convergence of the proposed algorithm and its extension. To make our method more practical, we suggest to solve the subproblem in an approximate manner and propose two flexible inexact criteria. We show that the condition which guarantees the convergence of the proposed indefinite PPA is tight by a simple example. In addition, we show how to apply the indefinite PPA to some convex models. We report some preliminary numerical results, which show the efficiencies of the proposed algorithms.

Aperiodic Sampled-Data Control for Exponential Stabilization of Delayed Neural Networks: A Refined Two-Sided Looped-Functional Approach

This brief addresses the exponential stabilization of a class of delayed neural networks under the framework of aperiodic sampled-data control. Firstly, a two-sided looped-functional is precisely constructed to relax the stabilization conditions and to enlarge the maximum sampling period. It drops the common positive definiteness requirement and only requires it at the sampling instants. Combining the Gronwall-Bellman inequality with the reciprocally convex approach, a less conservative exponential stabilization criterion in terms of LMIs with fewer decision variables is presented. Meanwhile, an effective design algorithm for the feedback gain matrix is proposed. Finally, a simulation example is provided to illustrate the effectiveness and superiority of the main results over some popular ones.

Nonstationary cross-covariance functions for multivariate spatio-temporal random fields

In multivariate spatio-temporal analysis, we are faced with the formidable challenge of specifying a valid spatio-temporal cross-covariance function, either directly or through the construction of processes. This task is difficult as these functions should yield positive definite covariance matrices. In recent years, we have seen a flourishing of methods and theories on constructing spatio-temporal cross-covariance functions satisfying the positive definiteness requirement. A subset of those techniques produced spatio-temporal cross-covariance functions possessing the additional feature of nonstationarity. Here we provide a review of the state-of-the-art methods and technical progress regarding model construction. In addition, we introduce a rich class of multivariate spatio-temporal asymmetric nonstationary models stemming from the Lagrangian framework. We demonstrate the capabilities of the proposed models on a bivariate reanalysis climate model output dataset previously analyzed using purely spatial models. Furthermore, we carry out a cross-validation study to examine the advantages of using spatio-temporal models over purely spatial models. Finally, we outline future research directions and open problems.

Transition-metal-element dependence of ideal shear strength and elastic behaviors of γ′-Ni3Al: ab initio study to guide rational alloy design

When doped with transition-metal (TM) elements, modern commercial Ni-based superalloys achieve an extraordinary mechanical performance, including elastic behaviors and shear strength. By using density functional theory, the TM-element (3d:Sc–Zn, 4d:Y–Cd, 5d:Hf–Au) dependence of the elastic properties and the ideal shear strength of γ′-Ni3Al were determined. According to the positive definiteness requirement of the elastic stiffness coefficients matrix, we present the necessary and sufficient mechanical stability in terms of Born criteria for a monoclinic crystal system. By examining the mechanical stability and calculating the stress–strain curve at every single strain, the shear behavior of pure Ni3Al in the weakest shear-slip system was investigated in detail. Our results show that the d-orbital occupancy of TM elements can affect the mechanical properties of the γ′ phase significantly when they occupy the Al site in Ni3Al, where the elements toward the center part of each series (e.g. V, Cr, Mn, Nb, Mo, Ru, Ta, W, Re, and Os) can enhance the elastic constants, elastic moduli, and shear strength. Elements at the beginnings and ends of the series (such as Cu, Zn, Y, Ag, Cd, and Au) will decrease the elastic properties and shear strength of γ′-Ni3Al.

Nyström subsampling method for coefficient-based regularized regression**The work described in this paper is supported partially by the National Natural Science Foundation of China (Grants No.11571078, 11631015) and NSFC/RGC Joint Research Fund (Project No.11461161006 and Project No. CityU 104012). Lei Shi is also sponsored by Program of Shanghai Subject Chief Scientist (Project No.18XD1400700).

Kernel methods are attractive in data analysis as they can model nonlinear similarities between observations and provide means to rich representations, both of which are useful for the regression problems in general domains. Despite their popularity, they suffer from two primary inherent drawbacks. One drawback is the positive definiteness requirement of the kernel functions, which greatly restricts their applications to some real data analysis. The other drawback is their poor scalability in massive data scenarios. In this paper, we aim to address these two problems by considering the Nyström subsampling approach for coefficient-based regularized regression (or Nyström CRR for short). Nyström subsampling is an effective approach to reduce the space and time complexity by constructing a low-rank approximation of the original kernel matrix through column subsampling. Coefficient-based regularized regression can provide a simple paradigm for designing indefinite kernel methods. We show that a combination of these two schemes is not only computationally efficient but also statistically consistent with a mini-max optimal rates of convergence. We explicitly determine the lower bound of subsampling level as a function of the sample size such that the mini-max optimal convergence rates can be preserved. From our analysis, the subsampling level plays a role as a trade-off between computational and asymptotic behaviors of Nyström CRR, and hence is pivotal for algorithmic performances. In order to choose an appropriate subsampling level, we develop an incremental Nyström CRR for nested subsampling sets. The proposed algorithm can greatly reduce the cost of cross-validation, and even allows to compute the whole solution path through all possible subsampling levels. Based on our empirical studies, the incremental Nyström CRR can perform effective model selections and achieve the state-of-the-art results on both synthetic and real data sets.

What really happens if the positive definiteness requirement on the covariance matrix of returns is relaxed in efficient portfolio selection?

The Markowitz critical line method for mean–variance portfolio construction has remained highly influential today, since its introduction to the finance world six decades ago. The Markowitz algorithm is so versatile and computationally efficient that it can accommodate any number of linear constraints in addition to full allocations of investment funds and disallowance of short sales. For the Markowitz algorithm to work, the covariance matrix of returns, which is positive semi-definite, need not be positive definite. As a positive semi-definite matrix may not be invertible, it is intriguing that the Markowitz algorithm always works, although matrix inversion is required in each step of the iterative procedure involved. By examining some relevant algebraic features in the Markowitz algorithm, this paper is able to identify and explain intuitively the consequences of relaxing the positive definiteness requirement, as well as drawing some implications from the perspective of portfolio diversification. For the examination, the sample covariance matrix is based on insufficient return observations and is thus positive semi-definite but not positive definite. The results of the examination can facilitate a better understanding of the inner workings of the highly sophisticated Markowitz approach by the many investors who use it as a tool to assist portfolio decisions and by the many students who are introduced pedagogically to its special cases.

Observer-based output-feedback control of large-scale networked fuzzy systems with two-channel event-triggering

This paper concerns data transmissions for large-scale T–S fuzzy systems with event-triggering control, where each subsystem communicates its information via a two-channel network. We propose an event-triggering scheme in which two event-triggering mechanisms are used to verify the data transmissions. At first, a novel model transformation is presented, where the event-triggered control system is reconstructed as a constant-delay system with extra inputs and outputs. By using a relaxed Lyapunov–Krasovskii functional (LKF) without the requirement of positive definiteness for all Lyapunov matrices, and the scaled small gain (SSG) theorem, the co-design problem of desired observer and controller gains, event-triggering parameters, and the sampling period is resolved in the form of linear matrix inequalities (LMIs). It will be shown that the solution guarantees the stability of closed-loop fuzzy control system and the reductions of data communications in both the sensor-to-controller and controller-to-actuator channels. The proposed method is validated by using a numerical example.

Split quaternions and particles in (2+1)-space

It is well known that quaternions represent rotations in 3D Euclidean and Minkowski spaces. However, the product by a quaternion gives rotation in two independent planes at once and to obtain single-plane rotations one has to apply half-angle quaternions twice from the left and on the right (with inverse). This ‘double-cover’ property is a potential problem in the geometrical application of split quaternions, since the (2+2)-signature of their norms should not be changed for each product. If split quaternions form a proper algebraic structure for microphysics, the representation of boosts in (2+1)-space leads to the interpretation of the scalar part of quaternions as the wavelengths of particles. The invariance of space-time intervals and some quantum behaviors, like noncommutativity and the fundamental spinor representation, probably also are algebraic properties. In our approach the Dirac equation represents the Cauchy–Riemann analyticity condition and two fundamental physical parameters (the speed of light and Planck’s constant) emerge from the requirement of positive definiteness of the quaternionic norms.

Smooth support vector learning for fuzzy rule-based classification systems

This paper extends the previous work in the connection between fuzzy classifiers and kernel machines [2] to a general case. In [2], all membership functions for the same input variable are generated from location transformation of a reference function. A translation invariant kernel is constructed from reference functions. The kernel is a Mercer kernel if the reference functions are positive definite. A support vector learning approach for positive definite fuzzy classifiers PDFCs was proposed. In this paper, a smooth support vector learning algorithm for fuzzy rule-based classification systems is proposed. The smooth support vector machine SSVM is capable of generating nonlinear separating surfaces using arbitrary kernels. The positive definiteness requirement on reference functions is relaxed. A fuzzy classifier using arbitrary reference functions can be built from the training samples based on an SSVM. The resulting fuzzy classifier is called standard binary fuzzy classifier SBFC. Fuzzy rules are extracted with each rule given by a training sample. The reduced kernel technique is introduced to simplify the decision function of the SBFC and to reduce the number of fuzzy rules. Finally, the performance of SBFCs with different reference functions is illustrated by experimental results. Comparisons with PDFCs are also provided.

Kernel Discriminant Analysis for Positive Definite and Indefinite Kernels

Kernel methods are a class of well established and successful algorithms for pattern analysis thanks to their mathematical elegance and good performance. Numerous nonlinear extensions of pattern recognition techniques have been proposed so far based on the so-called kernel trick. The objective of this paper is twofold. First, we derive an additional kernel tool that is still missing, namely kernel quadratic discriminant (KQD). We discuss different formulations of KQD based on the regularized kernel Mahalanobis distance in both complete and class-related subspaces. Secondly, we propose suitable extensions of kernel linear and quadratic discriminants to indefinite kernels. We provide classifiers that are applicable to kernels defined by any symmetric similarity measure. This is important in practice because problem-suited proximity measures often violate the requirement of positive definiteness. As in the traditional case, KQD can be advantageous for data with unequal class spreads in the kernel-induced spaces, which cannot be well separated by a linear discriminant. We illustrate this on artificial and real data for both positive definite and indefinite kernels.