By Cornelius T. Leondes
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Additional resources for Advances in Theory and Applications : System Identification and Adaptive Control, Part 2
0 •· 0 ' | ... 0 0 ... ρ21 Ρ 2 η ρ2 0 ο 0 ' αΊ , · ! ΐρΐ 0 ·· 0 | _| j ... j • • 0 ο ! • ο [ • | j ο | . , (1, 1) ··· k (1, s) ' n, ' Β (nxs) (45c) k 1 (p / 1) ··· k1(p, kn P s) (pf 1) ··· k n (p, s) P Note that A and C have a very specific structure while B is fully parametrized. In an identification context, once the structure indices n. have been estimated, the structure of A and C is completely specified by the 0 and 1 elements, while the a. , and k 7 (i, j) are free parameters. ,  or  for de tails).
If the input u(t) happens to be stepwise, 32 CHI-TSONG CHEN as in the case of digital computer control, then the G(s) in (27) will be an exact description of the system. Once Gj(z) = D (z)N(z) is identified, a discrete-time state- variable equation can also be readily obtained, as shown in the example in Section V. To find a continuous-time equation, we ~ AT ~ must solve A from A = e . We remark that if A is obtained from a continuous-time system, it is nonsingular or, equivalently, ~ AT its eigenvalues are all nonzero.
1318E-1S 28 CHI-TSONG CHEN Because of M = t~ N o : Do -N-. 25. 5)' This completes the identification. Note that the row degrees of the system are v, = 1 and v~ = 2. is v, + v« = 3. (20) The degree of the system It can be readily shown that the transfer ma trix of (17) is indeed equal to the one in (19) and (20). If a state variable description of the system is desired, we use the procedure in [7, p. 2 85] to write D(z) = N(z) = Γ z 0 1 ri 0' [o z 2J lo i. 95 . 7 . (21a) y(n) = "i · o o x(n) . o : o lj This equation is equivalent to the one in (17).
Advances in Theory and Applications : System Identification and Adaptive Control, Part 2 by Cornelius T. Leondes