Advanced Process Identification & Control by Enso Ikonen

By Enso Ikonen

Complex strategy identity & regulate КНИГИ ;НАУКА и УЧЕБА Автор: Kaddour NajimНазвание: complicated strategy id & ControlИздательство: CRC PressГод: 2001Страниц: 632Формат: PDFISBN-10: 082470648XЯзык: АнглийскийРазмер: 11.64 MBPresents the latest options in complex method modeling, identity, prediction, and parameter estimation for the implementation and research of business platforms. obtain reflect zero 1 2 three four five

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78), is a rec~sive formul~ for the le~t squ~ ~timate. 84), the matrix R (k) needs to be inverted at each time step. 86) The t~get is to be able to update P (k) directly, without needing to matrix inversion. This can be done by ~ing the matrix inversion lemma. TLFeBOOK 30 CHAPTER 2. LINEAR REGRESSION Lemma1 (Matrix inversion lemma) Let A, B, C and D be matrices compatible dimensions so that A ÷ BCDexists. 64). 93) l+~T(k)P(k_ 1)~(k) Thusthe inversion of a square matrix of size dim0 is replaced by the inversion of a scalar.

The Kalmanfilter approach also showsthat optimal weightingfactor ak in the least squares criterion is the inverse of the variance of the noise term, a~ = 1/cr 2 (k), at least whenthe noise is white and Gaussian. e. 184) This prevents the covariance matrix from tending to zero. In fact P (k) --~ whenthe numberof iterations increases, and the algoritlmi remains alert to changes in model parameters. For example, in [23] the addition (regularization) of a constant scaled identity matrix at each sampleinterval is suggested, V ~- ~I.

99. horizon H = T=X WhenA is close to 1, the time constant for the exponential decay is approximately H. 99 corresponds, roughly, to rememberingthe 20 - 100 most recent data. TLFeBOOK 36 CHAPTER 2. tious. 84). 2 Algorithm Nowthe recursive least squares algorithm with exponential forgetting can be given. 5. 130) where 0 < A _< 1, and A = 1 gives the RLSalgorithm with no forgetting. The effect of the forgetting factor £ is that the P (k) and hence the gain L (k) are kept larger. With ~ < 1, the P (k) will not tend to zero and algorithm will always be alert to changesin 0.

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