Positive Definite Diagonal Matrix Research Articles

The Distributed Adaptive Bipartite Consensus Tracking Control of Networked Euler–Lagrange Systems with an Application to Quadrotor Drone Groups

Actuator faults and external disturbances, which are inevitable due to material fatigue, operational wear and tear, and unforeseen environmental impacts, cause significant threats to the control reliability and performance of networked systems. Therefore, this paper primarily focuses on the distributed adaptive bipartite consensus tracking control problem of networked Euler–Lagrange systems (ELSs) subject to actuator faults and external disturbances. A robust distributed control scheme is developed by combining the adaptive distributed observer and neural-network-based tracking controller. On the one hand, a new positive definite diagonal matrix associated with an asymmetric Laplacian matrix is constructed in the distributed observer, which can be used to estimate the leader’s information. On the other hand, neural networks are adopted to approximate the lumped uncertainties composed of unknown matrices and external disturbances in the follower model. The adaptive update laws are designed for the unknown parameters in neural networks and the actuator fault factors to ensure the boundedness of estimation errors. Finally, the proposed control scheme’s effectiveness is validated through numerical simulations using two types of typical ELS models: two-link robot manipulators and quadrotor drones. The simulation results demonstrate the robustness and reliability of the proposed control approach in the presence of actuator faults and external disturbances.

The Computational Solution of Generalized Inverse Eigenvalue Problem for Pseudo‐Jacobi Matrix

This paper is concerned with two generalized inverse eigenvalue problems for a kind of pseudo‐Jacobi matrix. From a non‐Hermite matrix, an r × r Jacobi matrix, two distinct real eigenvalues, and part of the corresponding eigenvectors, an n × n pseudo‐Jacobi matrix is constructed. Furthermore, an n × n pseudo‐Jacobi matrix can be made by two different eigenpairs and a positive definite diagonal matrix. It is shown that a unique pseudo‐Jacobi matrix can be recovered from partial eigenpairs and certain special mixed eigendata. Two algorithms are provided for the reconstruction of such a pseudo‐Jacobi matrix, and illustrative numerical examples are presented to verify the proposed algorithms.

Exact Solutions v.s. Perturbative Calculations of Finite Φ3-Φ4 Hybrid-Matrix-Model

There is a matrix model corresponding to a scalar field theory on noncommutative spaces called Grosse-Wulkenhaar model (Φ4 matrix model), which is renormalizable by adding a harmonic oscillator potential to scalar Φ4 theory on Moyal spaces. There are more unknowns in Φ4 matrix model than in Φ3 matrix model, for example, in terms of integrability. We then construct a one-matrix model (Φ3-Φ4 Hybrid-Matrix-Model) with multiple potentials, which is a combination of a 3-point interaction and a 4-point interaction, where the 3-point interaction of Φ3 is multiplied by some positive definite diagonal matrix M. This model is solvable due to the effect of this M. In particular, the connected ∑i=1BNi-point function G|aN11⋯aN11|⋯|a1B⋯aNBB| of Φ3-Φ4 Hybrid-Matrix-Model is studied in detail. This ∑i=1BNi-point function can be interpreted geometrically and corresponds to the sum over all Feynman diagrams (ribbon graphs) drawn on Riemann surfaces with B boundaries (punctures). Each |a1i⋯aNii| represents Ni external lines coming from the i-th boundary (puncture) in each Feynman diagram. First, we construct Feynman rules for Φ3-Φ4 Hybrid-Matrix-Model and calculate perturbative expansions of some multipoint functions in ordinary methods. Second, we calculate the path integral of the partition function Z[J] and use the result to compute exact solutions for 1-point function G|a| with 1-boundary, 2-point function G|ab| with 1-boundary, 2-point function G|a|b| with 2-boundaries, and n-point function G|a1|a2|⋯|an| with n-boundaries. They include contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams or higher genus surfaces.

Hierarchical Estimation Approach for RBF-AR Models With Regression Weights Based on the Increasing Data Length

In the radial basis function-based state-dependent autoregressive (RBF-AR) models with regression weights, the local linear models are included between the hidden layers and the output layers of the networks. The parameter estimation for the RBF-AR models with regression weights is studied in this brief. Considering the separable feature of the models, two criterion functions based on the increasing data length are defined to fit the observation data of the whole dynamical process. Two sub-algorithms are proposed by minimizing the criterion functions. Aiming to overcome the existence of the singular matrix during the Newton search and to make the algorithm more stable, a positive definite diagonal matrix is introduced to the algorithm. Based on the hierarchical principle, a hierarchical Newton recursive algorithm is proposed, which can realize the on-line parameter estimation. Simulation results verify the validity.

Attitude stabilization of rigid spacecraft with angular velocity and torque limitations

An adaptive backstepping control scheme has been put forward to investigate the attitude stabilization problem for a spacecraft with modeling uncertainties, external disturbances, actuator failures, angular velocity limitations and actuator limitations simultaneously. Based on the active disturbance rejection concept (ADRC), a positive definite diagonal matrix is selected by the designer to separate a control item from the system. Then, an extended state observer is constructed to deal with the total uncertain item including actuator failure. In order to deal with the limitation of torques, a saturation coefficient is introduced and its adaptive law is obtained. Meanwhile, a command filter and the compensation signal are constructed to handle the angular velocity limitations. Finally, the stability of the closed-loop system is analyzed. Simulation results are given to verify the effectiveness of the proposed scheme.

On the derivative-free quasi-Newton-type algorithm for separable systems of nonlinear equations

A derivative-free quasi-Newton-type algorithm in which its search direction is a product of a positive definite diagonal matrix and a residual vector is presented. The algorithm is simple to implement and has the ability to solve large-scale nonlinear systems of equations with separable functions. The diagonal matrix is simply obtained in a quasi-Newton manner at each iteration. Under some suitable conditions, the global and R-linear convergence result of the algorithm are presented. Numerical test on some benchmark separable nonlinear equations problems reveal the robustness and efficiency of the algorithm.

Extremal eigenvalues of sample covariance matrices with general population

We consider the eigenvalues of sample covariance matrices of the form Q=(Σ1/2X)(Σ1/2X)∗. The sample X is an M×N rectangular random matrix with real independent entries and the population covariance matrix Σ is a positive definite diagonal matrix independent of X. Assuming that the limiting spectral density of Σ exhibits convex decay at the right edge of the spectrum, in the limit M,N→∞ with N/M→d∈(0,∞), we find a certain threshold d+ such that for d>d+ the limiting spectral distribution of Q also exhibits convex decay at the right edge of the spectrum. In this case, the largest eigenvalues of Q are determined by the order statistics of the eigenvalues of Σ, and in particular, the limiting distribution of the largest eigenvalue of Q is given by a Weibull distribution. In case d<d+, we also prove that the limiting distribution of the largest eigenvalue of Q is Gaussian if the entries of Σ are i.i.d. random variables. While Σ is considered to be random mostly, the results also hold for deterministic Σ with some additional assumptions.

Interval Observer Design for One Class of Uncertain Linear Strictly Metzlerian Time-Delay Systems

The paper is concerned with design requirements when the problem of nonnegative state estimation for one class of uncertain linear Metzlerian time-delay systems with constant delays is tackled, while system states take nonnegative values whenever the initial conditions are nonnegative, the upper and lower system matrix bounds are strictly Metzler matrices, and the upper and lower output matrix bounds are nonnegative matrices. By defining positive definite diagonal matrix variables and introducing an associate structure of linear matrix inequalities, the design conditions are proven, guaranteeing if they are feasible, the resulting observer gain matrix is positive and the reflected observer system matrices are strictly Metzler and Hurwitz. A numerical example illustrates the solvability of the proposed design conditions.

Extension of diagonal stability and stabilization for continuous-time fractional positive linear systems

This paper considers an extension of diagonal stability for continuous fractional positive linear systems (FPLS) with the fractional order 0<α<1. Based on diagonal stability of Metzler matrix, an extension that involves the combination of a diagonal positive definite matrix and a skew-symmetric anti-diagonal matrix related a Hurwitz and Metzler matrix is established. Combining this extension and the linear matrix inequalities (LMIs) criteria of stability for fractional order systems (FOS), a necessary and sufficient condition for extension diagonal stability of FPLS is presented. A state feedback controller is given, which ensures the stabilization and positivity of the closed-loop systems. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods.

Optimization of formation for multi-agent systems based on LQR

In this paper, three optimal linear formation control algorithms are proposed for first-order linear multiagent systems from a linear quadratic regulator (LQR) perspective with cost functions consisting of both interaction energy cost and individual energy cost, because both the collective object (such as formation or consensus) and the individual goal of each agent are very important for the overall system. First, we propose the optimal formation algorithm for first-order multi-agent systems without initial physical couplings. The optimal control parameter matrix of the algorithm is the solution to an algebraic Riccati equation (ARE). It is shown that the matrix is the sum of a Laplacian matrix and a positive definite diagonal matrix. Next, for physically interconnected multi-agent systems, the optimal formation algorithm is presented, and the corresponding parameter matrix is given from the solution to a group of quadratic equations with one unknown. Finally, if the communication topology between agents is fixed, the local feedback gain is obtained from the solution to a quadratic equation with one unknown. The equation is derived from the derivative of the cost function with respect to the local feedback gain. Numerical examples are provided to validate the effectiveness of the proposed approaches and to illustrate the geometrical performances of multi-agent systems.

H ∞ control via dynamic output feedback for positive systems with multiple delays

This study is concerned with H∞ control problems for positive systems with time delays using dynamic output feedback. Novel bounded real lemma for continuous-time systems with time delays is first presented, which shows one of the remarkable properties of diagonal positive definite matrix solutions to linear matrix inequalities for positive systems. The proposed results reveal that H∞ norms of positive systems with time delays in state and output equations are equivalent to that of the corresponding systems with zero delays. The positivity preserving H∞ control problem by means of dynamic output feedback is also considered, and thus necessary and sufficient conditions in the forms of matrix equalities are established for the solvability of such problem. The desired controller gains can be obtained by means of cone complementarity linearisation techniques. A numerical example is finally given to illustrate the effectiveness of the obtained results.

Extension of Matic's results

Matic (2014) [6] proved two inequalities regarding the ratio det(A+D)/det(A), where A is positive definite, and D is a positive definite block diagonal matrix. In this paper, we extend these results to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector.

X−1-balance of some partially balanced experimental designs with particular emphasis on block and row-column designs

Summary This paper considers block designs and row-column designs where the information matrix C has two different nonzero eigenvalues, one of multiplicity 1 and the other of multiplicity h−1, where h is the rank of the matrix C. It was found that for each such design there exists a diagonal positive definite matrix X such that the design is X −1-balanced.

A modified quasi‐Newton diagonal update algorithm for total variation denoising problems and nonlinear monotone equations with applications in compressive sensing

SummaryIn this paper, we present a new algorithm to accelerate the Chambolle gradient projection method for total variation image restoration. The new proposed method considers an approximation of the Hessian based on the secant equation. Combined with the quasi‐Cauchy equations and diagonal updating, we can obtain a positive definite diagonal matrix. In the proposed minimization method model, we use the positive definite diagonal matrix instead of the constant time stepsize in Chambolle's method. The global convergence of the proposed scheme is proved. Some numerical results illustrate the efficiency of this method. Moreover, we also extend the quasi‐Newton diagonal updating method to solve nonlinear systems of monotone equations. Performance comparisons show that the proposed method is efficient. A practical application of the monotone equations is shown and tested on sparse signal reconstruction in compressed sensing. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Bounded real lemmas for positive descriptor systems

A well known result in the theory of linear positive systems is the existence of positive definite diagonal matrix (PDDM) solutions to some well known linear matrix inequalities (LMIs). In this paper, based on the positivity characterization, a novel bounded real lemma for continuous positive descriptor systems in terms of strict LMI is first established by the separating hyperplane theorem. The result developed here provides a necessary and sufficient condition for systems to possess H∞ norm less than γ and shows the existence of PDDM solution. Moreover, under certain condition, a simple model reduction method is introduced, which can preserve positivity, stability and H∞ norm of the original systems. An advantage of such method is that systems׳ matrices of the reduced order systems do not involve solving of LMIs conditions. Then, the obtained results are extended to discrete case. Finally, a numerical example is given to illustrate the effectiveness of the obtained results.

Regularized solution of LCP problems with application to rigid body dynamics

For Linear Complementarity Problems (LCP) with a positive semidefinite matrix M, iterative solvers can be derived by a process of regularization. In [3] the initial LCP is replaced by a sequence of positive definite ones, with the matrices M + ?I. Here we analyse a generalization of this method where the identity I is replaced by a positive definite diagonal matrix D. We prove that the sequence of approximations so defined converges to the minimal D-norm solution of the initial LCP. This extension opens the possibility for interesting applications in the field of rigid multibody dynamics.

THE BETA-MANOVA ENSEMBLE WITH GENERAL COVARIANCE

We consider adding arbitrary covariance to the β-Jacobi random matrix model. We recall that for β = 1 the Jacobi random matrix model may be thought of as the eigenvalues, λi, of YtY(XTX + YtY)-1 where X and Y are matrices whose elements are i.i.d. standard normals. Equivalently we can take the generalized cosine singular values of (Y, X), ci, and use [Formula: see text]. When β = 1 we add covariance by considering YtY(YtY + ΩXtXΩ)-1, for a positive definite diagonal matrix Ω. Equivalently, and preferably, we consider the generalized singular value decomposition (gsvd) of (Y, XΩ). We refer to Ω = I as the Jacobi case and the general Ω case as the MANOVA case. In this paper, we provide a matrix model for the general β-MANOVA ensemble. In particular, we provide an algorithm for the numerical sampling of eigenvalues or generalized cosine singular values. The β-MANOVA algorithm uses the β-Wishart algorithm of Forrester and Dubbs–Edelman–Koev–Venkataramana as a subroutine, perhaps making β-MANOVA the first "second-order" continuous-β random matrix algorithm. Our proofs make use of a conjecture of MacDonald (proven by Baker and Forrester), a theorem of Kaneko, and many identities from Forrester's Log-Gases and Random Matrices. We supply numerical evidence that our theorems are correct.

Bayes minimax estimation of the multivariate normal mean vector under quadratic loss functions

The problem of estimating the mean vector μ of a multivariate normal distribution with the covariance matrix σ2Ip is considered under the loss function, (δ−μ)′D(δ−μ)σ2, where σ2 is unknown and D is a known positive definite diagonal matrix. A large class of Bayes minimax estimators of μ is found. This class includes classes of estimators obtained by Lin and Mousa (1982) and Zinodiny et al. (2011).

Stability of Matrix Polytopes with a Dominant Vertex and Implications for System Dynamics

The paper considers the class of matrix polytopes with a dominant vertex and the class of uncertain dynamical systems defined in discrete time and continuous time, respectively, by such polytopes. We analyze the standard concept of stability in the sense of Schur—abbreviated as SS (resp., Hurwitz—abbreviated as HS), and we develop a general framework for the investigation of the diagonal stability relative to an arbitrary Hölderp-norm,1≤p≤∞, abbreviated asSDSp(resp.,HDSp). Our framework incorporates, as the particular case withp=2, the known condition of quadratic stability satisfied by a diagonal positive-definite matrix, i.e.SDS2(resp.,HDS2) means that the standard inequality of Stein (resp., Lyapunov) associated with all matrices of the polytope has a common diagonal solution. For the considered class of matrix polytopes, we prove the equivalence between SS andSDSp(resp., HS andHDSp),1≤p≤∞(fact which is not true for matrix polytopes with arbitrary structures). We show that the dominant vertex provides all the information needed for testing these stability properties and for computing the corresponding robustness indices. From the dynamical point of view, if an uncertain system is defined by a polytope with a dominant vertex, then the standard asymptotic stability ensures supplementary properties for the state-space trajectories, which refer to special types of Lyapunov functions and contractive invariant sets (characterized through vectorp-norms weighted by diagonal positive-definite matrices). The applicability of the main results is illustrated by two numerical examples that cover both discrete- and continuous-time cases for the class of uncertain dynamics studied in our paper.

H∞Control for Linear Positive Discrete-Time Systems

This paper is concerned withH∞control for linear positive discrete-time systems. Positive systems are characterized by nonnegative restriction on systems’ variables. This restriction results in some remarkable results which are available only for linear positive systems. One of them is the celebrated diagonal positive definite matrix solutions to some existed well-known results for linear systems without nonnegative restriction. We provide an alternative proof for criterion ofH∞norm by using separating hyperplane theorem and Perron-Frobenius theorem for nonnegative matrices. We also considerH∞control problem for linear positive discrete-time systems via state feedback. Necessary and sufficient conditions for such problem are presented under controller gain with and without nonnegative restriction, and then the desired controller gains can be obtained from the feasible solutions.