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Design Methods for Discrete-Time, Linear Time-Invariant Systems
Published in William S. Levine, Control System Fundamentals, 2019
Mohammed S. Santina, Allen R. Stubberud, Gene H. Hostetter
where jn† is the transpose of the nth-unit coordinate vector jn†=[00…01],
Vector Spaces
Published in Lina Oliveira, Linear Algebra, 2022
where AB is the coordinate vector of matrix A relative to B. Find a basis BS for S.
Vector spaces
Published in Qingwen Hu, Concise Introduction to Linear Algebra, 2017
Rn+1 $ {\mathbb{R}}^{n + 1} $ the coordinate vector of f with respect to the basis S.
New formulation and application of the segment method for extensible risers
Published in Ships and Offshore Structures, 2022
Iwona Adamiec-Wójcik, Lucyna Brzozowska, Łukasz Drąg, Stanisław Wojciech
Transformation of coordinates from local system to the global system (Figure 2) is carried out as follows: where is the vector of coordinates of a chosen point of segment in global system is the coordinate vector of this point with respect to local coordinate system, is the rotation matrix, , , , , , .
Vibration control of a piezoelectric cantilever smart beam by ℒ 1 adaptive control system
Published in Systems Science & Control Engineering, 2021
A. Ebrahimi-Tirtashi, S. Mohajerin, M. R. Zakerzadeh, M. A. Nojoomian
Since the dynamic of FE model has large number of degrees of freedom, a truncated modal matrix is introduced to reduce the model order and as a result the computation cost is reduced. So, the nodal displacement vector can be transformed to the reduced vector as (Zhang, 2014): Therefore, the new states can be measured from the previous displacement vectors and the truncated Matrix which is defined as the first n modes shape matrix. By considering the first two bending modes' vibration of the beam the truncated Matrix is formed by the eigenvectors of the first two modes. Thus, the will be displacement vector, will be truncated Matrix and will be the modal coordinate vector.
Calculus of variations as a basic tool for modelling of reaction paths and localisation of stationary points on potential energy surfaces
Published in Molecular Physics, 2020
Josep Maria Bofill, Wolfgang Quapp
The basic definition of the RP is a curve located on a PES that monotonically increases from a stationary point (with the character of a minimum) to a first-index saddle point and from that point it monotonically decreases to a new stationary point, a minimum. The first index saddle point according to the previous definition is the highest energy point of the RP. The first and the second minimum are labelled as reactants and products, respectively, while the first-index saddle point is the transition state (TS). The parametrization of a curve, say by parameter t, satisfying the above RP requirements, is the reaction coordinate. More concisely, if is a coordinate vector of dimension N, then the RP is represented by . Normally, the parameter arc-length, t=s, of the curve is taken as the reaction coordinate; however, special values of the PES can also be taken as reaction coordinates.