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Interaction and Structure Concepts for Large-Scale Systems
Published in Spyros G. Tzafestas, Keigo Watanabe, Stochastic Large-Scale Engineering Systems, 2020
P. V. S. Ponnapalli, Michael A. Johnson, Michael J. Grimble, David H. Owens
The polar decomposition of a matrix involves rewriting the matrix as a product of two matrices, one containing the magnitude part, and the other containing the phase part. It was shown by Postlethwaite et al. (1981) that for square systems σmin(s) and σmax(s) are the lower and upper bounds, respectively, for the moduli of the eigenvalues of G(s). Postlethwaite et al. (1981) also showed that principal phases, which are defined as the arguments of the eigenvalues of the polar decomposition matrix XV*, under the condition that their spread is less than 180°, are the bounds for the arguments of the eigenvalues of G(s). The usefulness of the polar representation lies in the clear characterization of phase information which can be used to study the stability of multivariable systems by constructing a principal region, which is a conic sector in the complex plane defined by the maximum and minimum principal gains and phases.
Application of Eigenvalues and Eigenvectors
Published in Timothy Bower, ®, 2023
There is another factoring of a matrix that uses the sub-matrices of the SVD. The polar decomposition splits the matrix up into a symmetric matrix and an orthogonal matrix. The factoring is A=RQ, which is intended to be a generalization to matrices of the polar representation of vectors on a complex plane, z=reiθ, where r is the scalar length of z and eiθ gives the direction of the vector according to Euler's complex exponential formula. In the polar decomposition, Q is a unitary rotation matrix, and R has the same |·|2 matrix norm as A. But with multiplication by a vector, the R matrix will both scale and rotate the vector. It can be found by simply inserting an identity matrix in the form of UTU into the SVD equation.
Procrustes based closed-form solution to the point-wise weighted rigid-body transformation in asymmetric and symmetric cases
Published in Journal of Spatial Science, 2021
Marcin Ligas, Dominik Prochniewicz
Derivations included in this paper are based on principles presented in Schönemann and Carroll (1970), i.e., the least squares optimality criterion for multivariate case. However, they are extended to the case of error-contaminated source system in contrast to the most common approach, i.e., error-affected target system. We also generalise the mentioned asymmetric cases to the symmetric one where both systems are subject to random errors. In addition, all derivations consider a point-wise (row-wise) weighting schema, i.e., weights are assigned to points not to individual coordinates, hence, e.g., positional errors may be used to construct weight matrices. The main solution for a rotation matrix is presented in the polar decomposition form which, on the other hand, may be expressed with the use of singular value decomposition. We also use a modification (Markley 1988, Umeyama 1991, Sjöberg 2013) to ensure the resulting orthogonal matrix to be a rotation one having the determinant equal to unity. All derivations contained in the paper are summarised in step-by-step algorithms covering all considered adjustment scenarios. The paper concludes with a discussion on similarities and differences between mentioned scenarios and weighting strategies. Also, a numerical example supporting all considerations is provided.
Necessary and sufficient stability condition for second-order switched systems: a phase function approach
Published in International Journal of Control, 2019
Xiaozhan Yang, H. K. Lam, Ligang Wu
To apply phase function into stability analysis, we need to know the properties of it. First, we want to check whether we can move the layout of phase function ϕ(A, θ) up, down, left and right by changing the parameters in matrix A. This shifting property can be analysed based on the polar decomposition of matrix A. Clearly for any matrix A, we can always find its polar decomposition: right polar decomposition, A = UrPr, and left polar decomposition, A = PlUl, with Ur and Ul being unitary matrices, Pr and Pl being negative semidefinite symmetric matrices. Moreover, for the obtained symmetric matrices Pr and Pl, we can make further decompositions and , where and are diagonal matrices, and Tr and Tl are unitary matrices. Overall, we have