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     P  o   l   i   t  e  c  n   i  c  o   d   i   T  o  r   i  n  o 14.1 Cosimo Greco THE OVERALL DEVELOPMENT SYSTEM Overall technological structure (13/14)     P  o   l   i   t  e  c  n   i  c  o   d   i   T  o  r   i  n  o 24.1 Cosimo Greco    n1n1 1i1i bbaa e)ni(u)1i(u)ni(y)1i(y herew)i(y      Model #1 can be rewritten as n2m    is defined “information set” (m  1)with  is the parameters vector  (m  1).The model tfis: n1n1nn1n1nn11nn11 azaz bzbzaza1 zbzb )z(G     ARXmodel (2/3)#2     P  o   l   i   t  e  c  n   i  c  o   d   i   T  o  r   i  n  o 34.1 Cosimo Greco iiii||1i2iNi|| y A wichfrom Ay)i(y )1i(y )1Ni(y        From #1 (or #2), by ordering N samples backward from i , we can obtain: #3 Nm 1 mN :conditionNecessary   ARXmodel (3/3)     P  o   l   i   t  e  c  n   i  c  o   d   i   T  o  r   i  n  o 44.1 Cosimo Greco  Problem: inversion of ; IMPORTANT: is it invertible?   A possible solution: use of the QRfactorization:it is possible to determine a matrix Q N  N such that:withR i upper triangularand Q ortonormal LS estimation(2/4)   ii  A A  0RQA ii1 QQ,IQQ    #5 QNmmN-m0 A i R i m
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