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Unformatted text preview: 2.160 Identification, Estimation, and Learning Lecture Notes No. 2 February 13, 2006 2. Parameter Estimation for Deterministic Systems 2.1 Least Squares Estimation y Deterministic System m u u u M 2 1 M w/parameter Linearly parameterized model Input-output y = u b 1 + u b 2 + K + b u 1 2 m m ] T m Parameters to estimate: = [ b K b R ] 1 m T m = [ u K u R m Observations: 1 T y = (1) ] T The problem is to find the parameters = [ b K b from observation data: 1 m ( ), 1 y (1) ( ), 2 y (2) M ( N ), y ( N ) The system may be a linear dynamic system, e.g. t y ) = t u b 1) + t u b 2) + + b t u m ) ( ( ( ( ( 1 2 m T m ( t ) = [ t u ), 1 t u ), 2 , t u m )] R ( ( or a nonlinear dynamic system, e.g. t y ) = t u b 1) + t u b 2) t u 1) ( ( ( ( 1 2 ( t ) = [ t u ), 1 t u 2) t u )] 1 ( ( ( T Note that the parameters, b 1 , b 2 , are linearly involved in the input-output equation. Using an estimated parameter vector , we can write a predictor that predicts the output T from inputs: t y ) = ( t ) (2) ( 1 We evaluate the predictors performance by the squared error given by N 1 V N ( ) = ( t y | ) t y )) 2 (3) ( ( N t = 1 Problem: Find the parameter vector that minimizes the squared error: = min arg V N ( ) (4) Differentiating V N ( ) and setting it to zero,...
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.160 taught by Professor Harryasada during the Spring '06 term at MIT.
- Spring '06