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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.”
Multivariable Control of Polymerization Reactors
Published in F. Joseph Schurk, Pradeep B. Deshpande, Kenneth W. leffew, Vikas M. Nadkarni, Control of Polymerization Reactors, 2017
Schurk F. Joseph, Deshpande Pradeep B.
SVD analysis gives an important index of controllability, namely, the condition number of the system. The condition number is the ratio of the largest singular value, amu, to the smallest singular value, amin. Condition numbers are used to determine the condition of a set of equations; the larger the condition number, the more difficult it is to compute the matrix inverse that is free of computational errors. In tenns of process control, for systems having large condition numbers, it may be impossible to satisfy the entire set of control objectives.
Fundamentals
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
The condition of a problem is inherent to the problem whichever algorithm is used to solve it. The condition number of a numerical problem measures the asymptotically worst case of how much the outcome can change in proportion to small perturbations in the input data. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. The condition number is a property of the problem and not of the different algorithms that can be used to solve the problem.
A finite-difference and Haar wavelets hybrid collocation technique for non-linear inverse Cauchy problems
Published in Applied Mathematics in Science and Engineering, 2022
Muhammad Ahsan, Iltaf Hussain, Masood Ahmad
The condition number of a matrix measures the sensitivity of the matrix, like how much error be there in the output results for a small error in the input data. A matrix with a high condition number means that the determinant of the matrix is close to zero and is difficult to find its inverse accurately and hence the numerical results are unstable. The condition number of a can be mathematically defined as where and are maximal and minimal singular values of respectively and can be easily calculated by using the built-in command cond(A) in MATLAB.
A Multilevel in Space and Energy Solver for 3-D Multigroup Diffusion and Coarse-Mesh Finite Difference Eigenvalue Problems
Published in Nuclear Science and Engineering, 2019
Ben C. Yee, Brendan Kochunas, Edward W. Larsen
The primary deficiency of all Krylov methods is the dependence of their convergence rate on the condition number of the problem . As , the convergence rates of Krylov methods approach 1 (Ref. 23). This is particularly problematic for the CMFD problem because its condition number goes to infinity as the number of mesh elements goes to infinity. Thus, Krylov methods are prohibitively slow for larger problems of interest in MPACT. To remedy this issue, preconditioners are typically used to reduce the condition number of the linear system. That is, a preconditioned Krylov method will solve a system of the formcEquation (8) is an example of left-preconditioning, but right-preconditioning is also commonly used.
Analytical, Semi-Analytical, and Numerical Heavy-Gas Verification Benchmarks of the Effective Multiplication Factor and Temperature Coefficient
Published in Nuclear Science and Engineering, 2018
Matthew A. Gonzales, Brian C. Kiedrowski, Anil K. Prinja, Forrest B. Brown
Table IV also reports the condition number of the matrix formed for the linear solver. The condition number is the ratio of the largest singular value of a matrix to its smallest, and provides a measure of the numerical precision of the solution of the linear system. A large condition number implies the system is ill conditioned and there may be a significant loss of accuracy from numerical round-off. For this problem, the condition number varies by several orders of magnitude, from over at K to around at K. The reason for this is that as the temperature increases, the absorption becomes less peaked because of Doppler broadening, leading to a smaller variation in the flux spectrum and a smaller condition number. This implies that for lower temperatures, the accuracy of the results may be limited by their numerical precision given that double precision numbers are used in the computation; in practice, at least for the cases studied, this appears to not have a significant impact. For higher temperatures, the results should be accurate to several decimal digits if double precision numbers are used.