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02B Estimation of Sys Paras

# 02B Estimation of Sys Paras - SYS635 Adaptive Control...

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SYS635 Adaptive Control Systems KaCC 02B Estimation of Sys Paras 09/10/05 1 CHARACTERIZATION OF 1 st ORDER SYSTEM SYSTEM IDENTIFICATION - Parameter estimation Assumption. We will assume that the input u(k) and output y(k) for the system are accessible for measurements. That is we can acquire sample data {u(k) ,k = 1, 2, … N} {y(k), k = 1, 2, … N} where N = the # of samples taken or samples of interest to be considered. Batch Mode Parameter Estimation. From the ARMA model, we obtain a set of time series relationship ya y b u b u y b u b u y b u b u yN a buN buN () ( ) ( ) 21 2 1 32 3 2 43 4 3 11 01 =− + + + + + + + + M which can be vectorized as { 0 1 (2) (1) (2) (3) (2) (3) (2) (4) (3) (4) (3) ( 1 ) () ( 1 ) yy u u a u u b u u b uN uN YM θ   = −−  14243 1444442444443 which can be abbreviated as * = We can solve for the parameters using left pseudo inverse solution (Least Squared Error Estimate) = (' *) ' * MM MY 1

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SYS635 Adaptive Control Systems KaCC 02B Estimation of Sys Paras 09/10/05 2 Real-Time Parameter Estimation using Recursive Least Square Algorithm At time t = NT, we may compute the batch estimate for the parameters [] 1 ( ) '* covariance of estimation errors ( )= ( ) the estimate computed using N samples PN M M NP NMY θ == = The parameter estimate can also be updated at each sampling instance using a recursive algorithm. For instance, a new set of samples y(N+1) and u(N+1), obtained after N data has been collected, can be used to update the already computed parameter estimate as follows: [ ] () 3 (1 ) )(1 )( ) i n f o r m a t i o n ) ' ) 1(1 ) ( ) ' ) ( 1) ( ) ( 1)( ( ( 1) ( ) new estimate with the N+1 th samples ) ) ' ) ( ) / N y N u N u N newupdate KN N N N NN K N y N I N ϕ ϕϕ θθ +=− + + += + + + + + + +− + += − + + NOTE: The above solutions (batch and recursive) are mathematically solutions to the system of linear algebraic equation under precise modeling and measurement conditions.
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02B Estimation of Sys Paras - SYS635 Adaptive Control...

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