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State-Variable Representation in Digital Control Systems
Published in Raymond G. Jacquot, Modern Digital Control Systems, 2019
This set of equations is a set of n linear equations in mn unknowns with the quantities on the left side being known. For there to exist a solution to these linear equations, the rank of the matrix of coefficients must be the same as the number of unknowns, or () rank[An−1BAn−2B⋯ABB]=n
Matrix Analysis
Published in Ramin S. Esfandiari, Bei Lu, Modeling and Analysis of Dynamic Systems, 2018
The rank of a matrix is the number of non-zero rows in the REF of that matrix. The determinant of An×n is a real scalar, calculated as |A|=∑k=1naik(−1)i+kMik,i=1,2,…,n
Matrices and linear transformations
Published in Alan Jeffrey, Mathematics, 2004
It is not proposed to offer more than a few general remarks about the solutions of m equations involving n variables. If the equations are consistent, but there are more equations than variables so that m >n, it is clear that there must be linear dependence between the equations. In the case that the rank of the coefficient matrix is equal to n there will obviously be a unique solution for, despite appearances, there will be only n linearly independent equations involving n variables. If, however, the rank is less than n we are in the situation of solving for r variables x1x2,. . . . in terms of the remaining n – r variables whose values may be assigned arbitrarily. In the remaining case where there are fewer equations than variables we have m < n. When this system is consistent it follows that at least n – m variables must be assigned arbitrary values.
Distributed estimation design for LTI systems: a linear quadratic approach
Published in International Journal of Systems Science, 2019
A. Rodríguez del Nozal, L. Orihuela, P. Millán
Let us transpose expression (17) in order to obtain several systems of linear equations with the structure Ax=b where the coefficient matrix A is , the searched variable vector x is given by the row vectors of and the matrix b corresponds to the row vectors of . Next, according to the Rouché-Capelli Theorem, the previous systems of equations are consistent if and only if the coefficient matrix A has full rank. Recall that this matrix is common for all the systems.
Development of linear-element boundary element method for inverse solution from induced far-field displacements to reservoir loading source
Published in European Journal of Environmental and Civil Engineering, 2023
Yu Zhao, Hong Li, Zhengzhao Liang, Hui Zhou, Bin Gong
The forwarding analysis of boundary value problems in engineering mostly refers to establishing the governing equation from boundary loading as source senders and to solve displacement, stress and strain at objective receivers through the constitutive relationship of rock mass under the premise of known properties. In contrasted with a forwarding problem, an inverse problem is to measure the responses to backwardly calculate their causes. The latter can usually not be directly monitored or explicitly represented by the forwarding problem (Maniatty et al., 1989; Tarantola, 2005).The solution to governing equations of inverse problem always falls into ill-posed. When the number of equations is less than that of unknowns, the coefficient matrix is rank deficient and the problem is called rank deficient problem. While the number of characteristic values of coefficient matrix is larger than that of unknown variable or the error of calculation or measured sample data are enough large, solution to the equations will change dramatically and lose its continuity between neighboring elements, which is called ill-posed problem (Cai, 2015).The calculation methods of inverse problems mainly include regularization methods, optimization methods, statistical methods, etc (Wang, 2007). Regularization method is very effective in treating ill-posed problems. Variational regularization method proposed by Tikhonov (Tikhonov, 1963), iterative regularization method put forward by Landweber (Landweber, 1951), and spatial regularization method founded by Zabaras (Zabaras et al., 1989), all of these can be used to deal with the ill-posed problem for inverse problems. With the continuous development of inverse problem theory and the increasingly maturity of numerical methods, it is a natural demand from practice that solution to inverse problems using numerical modelling should be applied to analyze engineering process. Maniatty et al. adopted diagonal regularization method to solve unknown stress boundary conditions with the help of finite element method (Maniatty et al., 1989). Schnur et al. combined spatial regularization with boundary element method to solve boundary pressure with analytical solution (Schnur & Zabaras, 1990). Zhang et al. used constrained least squares optimization method to reversely calculate residual stress and contact stress (Zhang et al., 1997). Combined with numerical modelling and solutions, the application scopes of inverse problem in engineering will be greatly extended, and the approximating accuracy in complicated shape of modelling will be also improved significantly through the corresponding boundary element method development.