By Enso Ikonen
Complicated technique identity & keep an eye on КНИГИ ;НАУКА и УЧЕБА Автор: Kaddour NajimНазвание: complex procedure identity & ControlИздательство: CRC PressГод: 2001Страниц: 632Формат: PDFISBN-10: 082470648XЯзык: АнглийскийРазмер: 11.64 MBPresents the latest innovations in complex strategy modeling, id, prediction, and parameter estimation for the implementation and research of commercial structures. obtain replicate zero 1 2 three four five
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Additional info for Advanced Process Identification & Control
4 Properties Nex~we will be concerned with the properties of the least squares estimator 0. Owingto the fact that the measurementsare disturbed, the vector parameter estimation ~ is random. An estimator is said to be unbiased if the mathematical expectation of the parameter estimation is equal to the true parameters O. The least squares estimation is unbiased if the noise E has zero meanand if the noise and the data (I) are statistically independent. Notice, that the statistical independenceof the observations and a zero mean TLFeBOOK 26 CHAPTER 2.
The optimum (minimum or maximum) can be solved analytically by setting the derivative to zero and the examination of the second derivative shows whether a minimumor a maximumis in question. 18) k--~l aNote that this poses restrictions on the choice of ~ (k). 2. 29) k----1 TLFeBOOK 20 CHAPTER 2. 351 k=l For the optim~ to be a minim~, we req~re that the matrix is positive ~. definite Finally, the parameter vector ~ ~nimizing the c~t f~ction J is given by (if the inverse of the matrix exists): "~= o~k~(k)~T(k) Zo~k~(k)y(k) The optimumis a minimumif the second derivative is positive, matrix my ~ is positive definite.
40) k=l is invertible, then there is a uniquesolution. Theinverse exists if the matrix is positive definite. 41) was identified using sampledmeasurementsof the plant behavior, where ~ (k) is the output of the model (predicted output of the system) and 0 is a parameter estimate (based on K samples). 3 Matrix representation Often, it is moreconvenient to calculate the least squares estimate from a compact matrix form. (2) (K) -_~ ~p,(2) ~1 (2) ~ (2) ~o1 (K)2 (K) .. 42) TLFeBOOK 22 CHAPTER 2. 44) where E is a K × 1 columnvector of modeling errors.