Recursive Least Squares Parameter Estimation for Linear Steady

Outline • • • Static model, sequential estimation Multivariate sequential estimation Example Dynamic discrete-time

Least Squares Parameter Estimation Linear Time Series Models ref: PC Young, Control Engr. ,

Simple Example The analytical solution for the minimum (least squares) estimate is (2) pk,

Sequential or Recursive Form To update based on a new data pt. (yi ,

Recursive Form for Parameter Estimation (5) where (6) To start the algorithm, need initial

Estimating Multiple Parameters (Steady State Model) (8) (non-sequential solution requires n x n inverse)

Recursive Solution (11) (12) need to assume Thomas F. Edgar (UT-Austin) RLS – Linear

Simple Example (Estimate Slope & Intercept) Linear parametric model input, u 0 1 output,

Sequential vs. Non-sequential Estimation of a 2 Only (a 1 = 0) Thomas F.

Sequential Estimation Thomas F. Edgar (UT-Austin) RLS – Linear Models Using Eq. (11) Virtual

Application to Digital Model and Feedback Control Linear Discrete Model with Time Delay: (13)

Recursive Least Squares Solution (14) where (15) (16) Thomas F. Edgar (UT-Austin) RLS –

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

Closed-Loop RLS Estimation • There are three practical considerations in implementation of parameter estimation

Enhance sensitivity of least squares estimation algorithms with forgetting factor l (21) (22) (23)

Closed-Loop Estimation (RLS) Perturbation signal is added to process input (via set point) to

3. use PRBS perturbation signal only when estimation error is large and P is

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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 Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 1

Outline • • • Static model, sequential estimation Multivariate sequential estimation Example Dynamic discrete-time model Closed-loop estimation Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 2

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) Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 3

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 Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 4

Sequential or Recursive Form To update based on a new data pt. (yi , xi), in Eq. (2) let (3) and (4) Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 5

Recursive Form for Parameter Estimation (5) where (6) To start the algorithm, need initial estimates and p 0. To update p, (7) (Set p 0 = large positive number) Eqn. (7) shows pk is decreasing with k Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 6

Estimating Multiple Parameters (Steady State Model) (8) (non-sequential solution requires n x n inverse) To obtain a recursive form for (9) (10) Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 7

Recursive Solution (11) (12) need to assume Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 8

Simple Example (Estimate Slope & Intercept) Linear parametric model input, u 0 1 output, y 5. 71 9 2 15 3 19 4 20 10 45 12 55 18 78 y = a 1 + a 2 u Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 9

Sequential vs. Non-sequential Estimation of a 2 Only (a 1 = 0) Thomas F. Edgar (UT-Austin) Covariance Matrix vs. Number of Samples Using Eq. (12) RLS – Linear Models Virtual Control Book 12/06 10

Sequential Estimation Thomas F. Edgar (UT-Austin) RLS – Linear Models Using Eq. (11) Virtual Control Book 12/06 11

Application to Digital Model and Feedback Control Linear Discrete Model with Time Delay: (13) y: output u: input Thomas F. Edgar (UT-Austin) d: disturbance RLS – Linear Models N: time delay Virtual Control Book 12/06 12

Recursive Least Squares Solution (14) where (15) (16) Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 14

Recursive Least Squares Solution (17) (18) (19) (20) K: Kalman filter gain Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 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 Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 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 Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 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) Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 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 to tr (P). 4. P(0) = 104 I 5. filter new parameter estimates r: tuning parameter (qc used by controller) 6. Use other diagnostic checks such as the sign of the computed process gain Thomas F. Edgar (UT-Austin) RLS – Linear Models Virtual Control Book 12/06 19