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Harmonic Analysis on Groups
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
Another test for irreducibility of compact Lie group representations is based solely on their continuous nature and is not simply a direct extension of concepts originating from the representation theory of finite groups. For a Lie group, we can evaluate any given representation U (g) at a one-parameter subgroup generated by exponentiating an element of the Lie algebra as U(exp(tXi)). Expanding this matrix function of t in a Taylor series, the linear term in t will have the constant coefficient matrix u(Xi)=dU(exp(tXi))dt|t=0.
Dynamical Systems and Linear Algebra
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Fritz Colonius, Wolfgang Kliemann
A periodic linear differential equation ẋ = A(θt)x is given by a matrix function A : ℝ → gl (d, ℝ) that is continuous and periodic (of period t^>0). As above, the solutions define a dynamical system via Φ:ℝ×S1×ℝd→S1×ℝd, if we identify ℝ mod t^ with the circle S1.
Dual-Manipulator Testing Technique
Published in Chunguang Xu, Robotic Nondestructive Testing Technology, 2022
In algebra, matrix exponential is a matrix function, similar to a general exponential function. The exponent of a matrix determines the mapping relation between the Lie algebra of the matrix and the region of its Lie group. On the other hand, matrix logarithm is the inverse function of matrix exponential [5]. Not all matrices have a logarithm. Those matrices with logarithm may have more than one logarithm. The study of logarithmic matrices has been accompanied by the development of the theory of Lie group, because a matrix with logarithm must be in a Lie group, and its logarithm must be the corresponding element of Lie algebra.
Double tracking control for the directed complex dynamic network via the state observer of outgoing links
Published in International Journal of Systems Science, 2023
Peitao Gao, Yinhe Wang, Lizhi Liu, LiLi Zhang, Shengping Li
In this paper, we consider the N continuous chaotic Chua's circuits in Thamilmaran et al. (2004) as the isolate nodes, which are coupled in the form of (2), the dynamics of which is as follows where is the state vector of the ith Chua's circuit, denotes the nonlinear vector function, is a constant matrix, , , , , a = −0.04725 and b = 3.15, stands the diagonal matrix, where , , and are randomly generated in the interval , the control input is given as (9)–(10); Furthermore, denotes the inner connection matrix function, where . Let the common coupling strength c be randomly generated in the interval .
Modelling and analysis of sustainable manufacturing system using a digraph-based approach
Published in International Journal of Sustainable Engineering, 2018
Vimal K. E. K., Vinodh S., Anand Gurumurthy
Using the values obtained from experts, matrices are formulated. After the series of interaction sessions, the consensus opinion has been fetched from the experts. For Interdependency, five-point scale (Table 4) used by Anand and Kodali (2009) has been used. Inheritance (Table 5) is evaluated using Saaty scale (Saaty 1980). With the assigned score, the permanent function of matrix is computed. This permanent function is a standard matrix function and is used and defined in combinatorial mathematics. The permanent value is computed as similar to determinant with all positive signs. Application of permanent concept will lead to a better appreciation as no information is lost. Jurkat and Ryser (1996) proposed an approach for computing permanent of matrix S = (sij) with order n > 1. Equation (2) is the general form used to compute permanent function of matrix S (Jangra et al. 2011).
State estimation and decoupling of unknown inputs in uncertain LPV systems using interval observers
Published in International Journal of Control, 2018
D. Rotondo, A. Cristofaro, T. A. Johansen, F. Nejjari, V. Puig
Given an invertible matrix function partitioned as in (16) and such that (17)–(18) hold, and a matrix for which the following holds: and provided that (20) and Assumptions 1–3 hold, determine an LPV interval UIO which, in addition to solve Problem 1, satisfies: where and are evaluable quantities that can be used as unknown input isolation signals. In particular, in this paper, it is shown that a valid choice for these signals is the following: