1. تنظيم پارامترهاي يك سيستم با روش کمترين مربعات خطا Parameter Identification of a System with LSE … Vali Derhami Yazd University, Computer Department vderhami@yazduni.ac.ir

2. Author: Vali Derhami  Structure Identification: Priori knowledge  Parameter Identification (Jang’s book, Ch.5) 2/13  Aim is finding θ such that can be desired.  This chapter we discuss LSE

3. Author: Vali Derhami 3/13 Parameter Identification

4. Author: Vali Derhami 4/13 Parameter Identification Training Data, [y i ; u i ] i=1,2, …N

5. Author: Vali Derhami 5/13 Least square methods

6. Author: Vali Derhami Basics of matrix manipulation & calculus  From Ch 5.2 of Jang’s book 6/13

7. Author: Vali Derhami Least -squares estimator  f 1, f 2,..,f n known functions of u  θ 1, θ 2,…, θ n : unknown parameters  Training Data, [y i ; u i ] i=1,2, m  m linear equations: 7/13

8. Author: Vali Derhami Least -squares estimator 8/13

9. Author: Vali Derhami Least -squares estimator 9/13 Usually m> n: Modified as e=y-Aθ Find so minimize sum of squared errors:

10. Author: Vali Derhami Least -squares estimator (Theorem) 10/13 If W is defined as the desired weighting matrix (symmetric, and W>0) Ex 5.2

11. Author: Vali Derhami Recursive Least -Squares Estimator  If after tuning with k pairs, new data pair become available as k+1 entry : (a T,y) 11/13

12. Author: Vali Derhami Recursive Least -Squares Estimator  A T y=A T A θ, P k =(A T A) -1 Hence: 12/13  Lemma 5.6:  P 0 =αI where α >>0 for Ex. α=10 8

13. Author: Vali Derhami Recursive Least -Squares Estimator 13/13