Matrix Congruence Research Articles

H∞ observer-based controller synthesis for fractional order systems over finite frequency range

This paper focuses on the H∞ observer-based control problem for fractional order systems (FOSs) over finite frequency range. Firstly, as an extension to the KYP lemma, the structure singular value–μ and its upper bound are analyzed and determined. The necessary and sufficient conditions of the H∞ performance for FOSs over finite frequency range are given through the generalized μ analysis method. This method is different from the traditional μ-analysis method, which is only suitable for integer order systems. Secondly, according to these obtained LMI-based criteria for H∞ performance index and a key projection lemma, we design an H∞ observer-based controller to stabilize the estimation error and decrease the H∞ norm for the FOS simultaneously. With the help of the matrix congruence transformation, the feedback gain matrix is parameterized by a scalar matrix. Thirdly, to deal with the nonlinear terms in matrix inequalities, an iterative linear matrix inequality (ILMI) algorithm is developed. Finally, two numerical examples are provided to explain the correctness of the established results.

Congruence of rational matrices defined by an integer matrix

We study algorithms that construct an invertible matrix B∈Mn(Z) that defines congruence Btr·X·B=Y of given square rational matrices X,Y∈Mn(Q). We describe a general algorithm and discuss special cases of positive-definite and singular matrices. In the first case, the algorithm solves the decision problem and unequivocally determines if such an integer matrix exists.Since the problem of isomorphism of various finite structures can be viewed as a special case of matrix congruence problem, our results are applicable in this setting. We discuss this topic and show that the presented algorithms solve this problem in the case of partially ordered sets, vertex and edge labeled directed and undirected graphs. Moreover, they allow computing the automorphism group Aut(G) of a given (di)graph G.The results of this work are applicable in the general framework of Coxeter spectral analysis of signed graphs Δ introduced in (SIAM J. Discrete Math. 27:827–854, 2013) as they make it possible to calculate strong and weak Gram Z-congruences. Moreover, they can be used to compute isotropy groups and thus are useful in the study of matrix morsifications in the sense of (J. Pure Appl. Algebra 215:3–34, 2011).

Connectome spatial smoothing (CSS): Concepts, methods, and evaluation

Structural connectomes are increasingly mapped at high spatial resolutions comprising many hundreds—if not thousands—of network nodes. However, high-resolution connectomes are particularly susceptible to image registration misalignment, tractography artifacts, and noise, all of which can lead to reductions in connectome accuracy and test-retest reliability. We investigate a network analogue of image smoothing to address these key challenges. Connectome Spatial Smoothing (CSS) involves jointly applying a carefully chosen smoothing kernel to the two endpoints of each tractography streamline, yielding a spatially smoothed connectivity matrix. We develop computationally efficient methods to perform CSS using a matrix congruence transformation and evaluate a range of different smoothing kernel choices on CSS performance. We find that smoothing substantially improves the identifiability, sensitivity, and test-retest reliability of high-resolution connectivity maps, though at a cost of increasing storage burden. For atlas-based connectomes (i.e. low-resolution connectivity maps), we show that CSS marginally improves the statistical power to detect associations between connectivity and cognitive performance, particularly for connectomes mapped using probabilistic tractography. CSS was also found to enable more reliable statistical inference compared to connectomes without any smoothing. We provide recommendations for optimal smoothing kernel parameters for connectomes mapped using both deterministic and probabilistic tractography. We conclude that spatial smoothing is particularly important for the reliability of high-resolution connectomes, but can also provide benefits at lower parcellation resolutions. We hope that our work enables computationally efficient integration of spatial smoothing into established structural connectome mapping pipelines.

Congruence of Unitary Matrices

C. R. Johnson and S. Furtado have shown that if two unitary matrices are *-congruent, then they are unitarily similar. A counterpart of this statement concerning another type of matrix congruence, namely, the T-congruence is established. In addition, the problem of checking the T-congruence of given matrices using a finite number of arithmetic operations is discussed.

Non-fragile synchronization for chaotic time-delay neural networks with semi-Markovian jump parameters

The issue of the non-fragile synchronization for chaotic time-delay neural networks subject to semi-Markovian jump parameters is addressed in this paper. Unlike the Markovian jump process, the sojourn time of the semi-Markovian jump process allows to be non-exponential distributed and the transition rate allows to be time-varying. By utilizing the discretized Lyapunov–Krasovskii functional method and introducing two free-weighting matrices, a sufficient condition is proposed to ensure the synchronization error system to be stochastically stable with an performance index. Then, by means of a matrix congruence transformation and some inequality techniques, an approach to the non-fragile controller design is developed. Finally, two illustrative examples are employed to show the usefulness of the proposed results.

Learning Paths from Signature Tensors

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry, and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.

Congruence method for global Darboux reduction of finite-dimensional Poisson systems

A new procedure for the global construction of the Casimir invariants and Darboux canonical form for finite-dimensional Poisson systems is developed. This approach is based on the concept of matrix congruence and can be applied without the previous determination of the Casimir invariants (recall that their prior knowledge is unavoidable for the standard reduction methods, thus requiring either the integration of a system of PDEs or solving some equivalent problem). Well the opposite, in the new congruence method, both the Darboux coordinates and the Casimir invariants arise simultaneously as the outcome of the reduction algorithm. In fact, the congruence algorithm proceeds only in terms of matrix-algebraic transformations and direct quadratures, thus avoiding the need of previously integrating a system of PDEs and therefore improving previously known approaches. Physical examples illustrating different aspects of the theory are provided.

The association between patient–therapist MATRIX congruence and treatment outcome

Objective: The present study aimed to examine the association between patient–therapist micro-level congruence/incongruence ratio and psychotherapeutic outcome. Method: Nine good- and nine poor-outcome psychodynamic treatments (segregated by comparing pre- and post-treatment BDI-II) were analyzed (N = 18) moment by moment using the MATRIX (total number of MATRIX codes analyzed = 11,125). MATRIX congruence was defined as similar adjacent MATRIX codes. Results: the congruence/incongruence ratio tended to increase as the treatment progressed only in good-outcome treatments. Conclusion: Progression of MATRIX codes’ congruence/incongruence ratio is associated with good outcome of psychotherapy.

H∞ output feedback control of linear time-invariant fractional-order systems over finite frequency range

This paper focuses on the H∞ output feedback control problem of linear time-invariant fractional-order systems over finite frequency range. Based on the generalized Kalman-Yakubovic-Popov (KYP) Lemma and a key projection lemma, a necessary and sufficient condition is established to ensure the existence of the H∞ output feedback controller over finite frequency range, a desirable property in control engineering practice. By using the matrix congruence transformation, the feedback control gain matrix is decoupled and further parameterized by a scalar matrix. Two iterative linear matrix inequality algorithms are developed to solve this problem. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.

Two-Generated Algebras and Standard-Form Congruence

Matrix congruence can be used to mimic linear maps between homogeneous quadratic polynomials in n variables. We introduce a generalization, called standard-form congruence, which mimics affine maps between non-homogeneous quadratic polynomials. Canonical forms under standard-form congruence for three-by-three matrices are derived. This is then used to give a classification of algebras defined by two generators and one degree two relation. We also apply standard-form congruence to classify homogenizations of these algebras.

Fault detection for a class of uncertain nonlinear Markovian jump stochastic systems with mode-dependent time delays and sensor saturation

This paper considers the problem of robust H∞ fault detection for a class of uncertain nonlinear Markovian jump stochastic systems with mode-dependent time delays and sensor saturation. We aim to design a linear mode-dependent H∞ fault detection filter that ensures, the fault detection system is not only stochastically asymptotically stable in the large, but also satisfies a prescribed H∞-norm level for all admissible uncertainties. By using the Lyapunov stability theory and generalised Itô formula, some novel delay-dependent sufficient conditions in terms of linear matrix inequality are proposed to guarantee the existence of the desired fault detection filter. Explicit expression of the desired mode-dependent linear filter parameters is characterised by matrix decomposition, congruence transformation and convex optimisation technique. Sector condition method is utilised to deal with sensor saturation, a definite relation of sector condition parameters with fault detection system robustness against disturbances and sensitivity to faults is put forward, and weighting fault signal approach is employed to improve the performance of the fault detection system. A simulation example and an industrial nonisothermal continuous stirred tank reactor system are utilised to verify the effectiveness and usefulness of the proposed method.

Rectangular group congruences on a semigroup

We study rectangular group congruences on an arbitrary semigroup. Some of our results are an extension of the results obtained by Masat (Proc. Am. Math. Soc. 50:107–114, 1975). We show that each rectangular group congruence on a semigroup S is the intersection of a group congruence and a matrix congruence and vice versa, and this expression is unique, when S is E-inversive. Finally, we prove that every rectangular group congruence on an E-inversive semigroup is uniquely determined by its kernel and trace.

Study on robust H ∞ filtering in networked environments

This paper is concerned with the robust H∞ filter problem for networked environments, which are subject to both transmission delay and packet dropouts randomly. By employing random series which have Bernoulli distributions taking value of 0 or 1, the data transmission model is obtained. Based on state augmentation and stochastic theory, the sufficient condition for robust stability with H∞ constraints is derived for the filtering error system. The robust filter is designed in terms of feasibility of one certain linear matrix inequality (LMI), which is formed by adopting matrix congruence transformations. A numerical example is given to show the effectiveness of the proposed filtering method.

Canonical forms for complex matrix congruence and ∗congruence

Canonical forms for congruence and ∗congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347–353], based on Sergeichuk’s paper [Math. USSR, Izvestiya 31 (3) (1988) 481–501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous ∗congruence of pairs of complex Hermitian matrices.

A note on canonical forms for matrix congruence

Canonical forms for matrix congruence for general matrices are exhibited as an easy consequence of results presented in a recent paper by R. C. Thompson on pencils of symmetric and skew symmetric matrices. We also show how to exhibit standard equivalence class representatives in the orbits of the semiorthogonal groups acting naturally on skew symmetric forms.

THE CONGRUENCE OF MORPHOMETRIC SHAPE IN RELATION TO GENETIC DIVERGENCE IN FOUR RACES OF MORABINE GRASSHOPPERS (ORTHOPTERA: EUMASTACIDAE).

The purpose of this study is to examine, by factor analysis and matrix congruence procedures, the relationship between genetic divergence and morphometric patterns among four chromosomal races of morabine grasshoppers. Determination of the amount of change in body form or shape that has occurred as a function of evolutionary divergence among closely related taxa is of considerable importance to many disciplines of biology including systematics, evolutionary genetics and developmental biology. Unfortunately, shape is a rather nebulous concept as evidenced by the wide variety of coefficients, indices and statistical procedures that have been proposed to describe variation (Pearson, 1926; Penrose, 1947, 1954; Jolicoeur and Mosimann, 1960; Jolicoeur, 1963; Rao, 1964; Burnaby, 1966; Vasicek and Jicin, 1971; Sprent, 1972; Spielmann, 1973). Quite obviously, the shape of an organism is the result of complex interactions of numerous morphometric variables and, as a result, the analysis of can only be accomplished using techniques that simultaneously consider the variation in large numbers of variables. Two interrelated problems, however, have hindered the multivariate analysis of shape. First, a workable definition had been lacking as to what constitutes and shape components of variation. Secondly, considerable difference of opinion exists among biologists as to the best quantitative methods for partitioning out the respective size and patterns. Mosimann (1970) partially resolved the first problem when he provided a generalized definition of size and shape. Assuming a random vector of body measurements (X1 ...., XmQ) measured in the same scale and associated with each individual in the population, Mosimann defined a variable as a dimensioned variable such as :X, (IIX)/l or simply any single Xi. A shape variable on the other hand, he defined as a dimensionless variable such as (Xi/Xm,. .. Xm i/Xm) or (Xi/:X, .... Xm/:X). Before considering what we believe to be the appropriate quantitative model for analyzing shape, consider the fundamental questions that must be answered in a comparative analysis of variation. These include: 1) What are the major patterns of variation within each taxon? 2) To what extent have these patterns become altered among closely related taxa? Within the framework of this question, we are interested not only in the relative contribution of the various individual variables to each pattern but also the relationships among the major patterns themselves. As noted above, a number of quantitative procedures have been proposed to study variation; however, some of these, such as canonical variate analysis and distance statistics, are incapable of answering the aforementioned two questions. Canonical variate analysis, as employed by Blackith and Blackith (1969), is inappropriate because one can not determine the patterns of variation within the various groups nor do the canonical variates repre-

The Matrix Congruence X 2 ≡I (mod p a )

The Matrix Congruence X 2 ≡I (mod p a )