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Algorithms Used in Control Systems
Published in Arun G. Phadke, Handbook of Electrical Engineering Calculations, 2018
The properties of controllability and observability of a state space model were defined. Rank tests for each property were developed from the definitions; and, even though the developments were in the context of a DT model, the tests are valid for a CT model as well. As pointed out, one of the drawbacks of these tests is the binary nature of their results. As a means of examining to what extent these properties hold, the reciprocal of the condition number of the controllability (or observability) matrices can be useful. The condition number of a matrix is the ratio of its largest singular value to its smallest. Consequently, a “good” number is near unity, and the condition number of a singular matrix is infinite. To avoid extremely large numbers, the reciprocal is used, in which case very small values indicate a poor numerical “condition.”
Command, Control, and Communication (C3) Systems Engineering and Artificial Intelligence
Published in Anna M. Doro-on, Handbook of Systems Engineering and Risk Management in Control Systems, Communication, Space Technology, Missile, Security and Defense Operations, 2023
Reciprocal is defined as a mutual connection and integration between or among objects, systems, or elements. Regardless of the fact that an object or element is not exclusively dependent on the other, they rely on one another. Interaction is defined as an interchange or action among or between objects, systems, or elements. As a result of linking together, reciprocal interaction means a mutual exchange exists in which objects, systems, or elements perform complementary behavior simultaneously or alternately.
Errors in trigonometry
Published in Breach Mark, Essential Maths for Engineering and Construction, 2017
This is not so. The index of −1 associated with a trigonometric function has a special meaning. It is the inverse of the function concerned, not the reciprocal. For example, y = tan−1x means the value of the angle y such that its tangent is x. If y = tan−1x then x = tan y.
Statistical Modeling of Multivariate Destructive Degradation Tests With Blocking
Published in Technometrics, 2020
Qiuzhuang Sun, Zhi-Sheng Ye, Yili Hong
We also make two remarks on the numerical computation of the developed EM algorithm. First, Section S.1.1 of the supplementary materials shows that the E-step needs to invert two matrices of size M × M. One of them is a diagonal matrix. The inverse is computed by replacing each diagonal element with its reciprocal. The other matrix is block diagonal with each block of size d × d. The computational complexity for its inverse is with the singular value decomposition. It is significantly smaller than the general case where the computational complexity is of order . Therefore, the computational burden at each iteration is low, ensuring a fast implementation of the EM algorithm in reality. Second, when the data size is large, the likelihood function may have numerous local optima. Therefore, it is important to start the EM algorithm with an educated guess of the parameters. In Section S.1.2 of the supplementary materials, we propose a workable procedure to find such an educated guess.