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Recursive Least Squares Parameter Estimation for Linear Models

Learn about recursive least squares method for parameter estimation in steady state and dynamic linear models with sequential and non-sequential forms, estimation of multiple parameters, and practical considerations. Explore practical guidelines and applications in digital models and feedback control.

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Recursive Least Squares Parameter Estimation for Linear Models

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  1. Recursive Least Squares Parameter Estimation for Linear Steady State and Dynamic Models Thomas F. Edgar Department of Chemical Engineering University of Texas Austin, TX 78712 1

  2. Outline • Static model, sequential estimation • Multivariate sequential estimation • Example • Dynamic discrete-time model • Closed-loop estimation 2

  3. Least Squares Parameter Estimation Linear Time Series Models ref: PC Young, Control Engr., p. 119, Oct, 1969 scalar example (no dynamics) model y = ax data least squares estimate of a: (1) 3

  4. Simple Example The analytical solution for the minimum (least squares) estimate is (2) pk, bk are functions of the number of samples This is the non-sequential form or non-recursive form 4

  5. Sequential or Recursive Form To update based on a new data pt. (yi , xi), in Eq. (2) let (3) and (4) 5

  6. Recursive Form for Parameter Estimation (5) where (6) To start the algorithm, need initial estimates and p0. To update p, (7) (Set p0 = large positive number) Eqn. (7) shows pkis decreasing with k 6

  7. Estimating Multiple Parameters (Steady State Model) (8) (non-sequential solution requires n x n inverse) To obtain a recursive form for (9) (10) 7

  8. Recursive Solution (11) (12) need to assume 8

  9. Simple Example (Estimate Slope & Intercept) Linear parametric model input, u 0 1 2 3 4 10 12 18 output, y 5.71 9 15 19 20 45 55 78 y = a1 + a2u 9

  10. Sequential vs. Non-sequential Estimation of a2 Only (a1 = 0) Covariance Matrix vs. Number of Samples Using Eq. (12) 10

  11. Sequential Estimation Using Eq. (11) 11

  12. Application to Digital Model and Feedback Control Linear Discrete Model with Time Delay: (13) y: output u: input d: disturbance N: time delay 12

  13. Recursive Least Squares Solution (14) where (15) (16) 14

  14. Recursive Least Squares Solution (17) (18) (19) (20) K: Kalman filter gain 15

  15. Closed-Loop RLS Estimation • There are three practical considerations in implementation of parameter estimation algorithms - covariance resetting - variable forgetting factor - use of perturbation signal 16

  16. Enhance sensitivity of least squares estimation algorithms with forgetting factor l (21) (22) (23) l prevents elements of P from becoming too small (improves sensitivity) but noise may lead to incorrect parameter estimates 0 <l< 1.0 l→1.0 all data weighted equally l ~ 0.98 typical 17

  17. Closed-Loop Estimation (RLS) Perturbation signal is added to process input (via set point) to excite process dynamics large signal: good parameter estimates but large errors in process output small signal: better control but more sensitivity to noise Guidelines: Vogel and Edgar, Comp. Chem. Engr., Vol. 12, pp. 15-26 (1988) 1. set forgetting factor l = 1.0 2. use covariance resetting (add diagonal matrix D to P when tr (P) becomes small) 18

  18. 3. use PRBS perturbation signal only when estimation error is large and P is not small. Vary PRBS amplitude with size of elements proportional totr (P). 4. P(0) = 104I 5. filter new parameter estimates r: tuning parameter (qc used by controller) • Use other diagnostic checks such as the sign of the • computed process gain 19

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