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EEG inverse problem I
Published in Munsif Ali Jatoi, Nidal Kamel, Brain Source Localization Using EEG Signal Analysis, 2017
Hence, the solution is provided as: J′=TY,withT=(WBTBW)−1LT{L(WBTBW)−1LT}+ where A+ denotes the Moore–Penrose pseudoinverse of matrix A.
Fault-tolerant sensor reconciliation schemes based on unknown input observers
Published in International Journal of Control, 2020
Hamid Behzad, Alessandro Casavola, Francesco Tedesco, Mohammad Ali Sadrnia
Let denote the set of real numbers whereas that of natural numbers. Let denote the transpose of a vector , the 2-norm of a vector (i.e. ) and the -norm of a signal (i.e. ). Given a matrix , the i-th row of M is denoted as . For a matrix having linearly independent rows, the Moore-Penrose Pseudoinverse is defined as and is computed as . LFTs are extensively used in the paper. For properly sized matrices N and the lower LFT is defined as
Dimension-reduced cross-section adjustment method based on minimum variance unbiased estimation
Published in Journal of Nuclear Science and Technology, 2018
Kenji Yokoyama, Akio Yamamoto, Takanori Kitada
Now, let us derive a new cross-section adjustment method under these assumptions. If the dimension-reduced cross-section set of is denoted as , and that of T0 as , then these are written as and where Equation (14) has been used. Similarly, if the dimension-reduced cross-section set of Tt is denoted as , then it is expressed by Using Equation (28), we obtain and Now, Equation (35) is rewritten as By using the Moore–Penrose pseudoinverse, the minimum-norm least-squares solution of the above equation can be written as Substituting Equations (37) and (38) into this equation yields On the other hand, premultiplying to both sides of Equation (36), we obtain Substituting Equation (44) into the above equation yields Furthermore, introducing Equation (30) into the above equation, we have Substituting Equations (37) and (39) into the above equation, we obtain
Multiplicative fault estimation based on the energetic approach for linear discrete-time systems
Published in International Journal of Control, 2022
Hamid Behzad, Mohammad Ali Sadrnia, Alessandro Casavola, Amin Ramezani, Ahmad Darabi
Let R denote the set of real numbers and N those of natural numbers. Let the transpose of a vector v. For a matrix , the Moore–Penrose Pseudoinverse is defined as and is computed as . Moreover, denotes the norm of a discrete-time signal z, which is defined as