In this appendix we explain how the predictive models used by MUSMAR, (3.62) are obtained from the ARX model (3.9). As explained in Sect. 3.2.4 the MUSMAR adaptive control algorithm restricts the future control samples (with respect to the present discrete time denoted k), from time k + 1 up to t + T — 1, to be given by a constant feedback of the pseudo-state, leaving u(k) free. In order to see how the predictive model (3.47) is modified by this assumption, start by observing that the pseudostate s(k) defined in (3.95) satisfies the dynamic state equation
s(k + 1) = 0ss (k) + Vsu (k) + eTe(k), (C.1)
in which
eT = [10 … 0], (C.2)
PT "
!n — 0(n— 1)xn (C 3)
01x(n+m)
0(m—1)xn Im — 1 — (m—1)x1
rs = [bo 0 … 010 … 0]T, (C.4)
and
PT = [—ax… — anbx… bm]. (C.5)
The matrix entries ai and bj above are the coefficients of the ARX model (3.9) and e is the innovations sequence of the same model.
Assume that a constant feedback law written as
u(k + i) = F0T s(k + i) (C.6)
will be acting on the plant from time к + 1 up to time t + T — 1, where T is the optimization horizon (do not confuse the horizon T with similar symbol used to refer the transpose of a matrix; the context in which they are used should be sufficient to distinguish both). Under this assumption, from time к + 1 up to time к + T — 1, the pseudo-state verifies
s (к + i + 1) — Ф Fs(k + i) + вв(к + i), (C.7)
with
ФР — Фs + VsFT. (C.8)
Combining (C.1) and (C.7), it is concluded by making a comparison with Kalman predictors Goodwin and Sin (1984) that the optimal predictor of s(к + i), given information up to time к is
s(к + i |к) — Фі—1 ^Ss (к) + Гы (к)]. (C.9)
The optimal predictor for ы(к + i) is
ы (к + i |к) — F^s^ + i |к), (C.10)
which implies that
ы(к + i|к) — F0TФі—1 [Фss(к) + Гы(к)]. (C.11)
Defining
№ — F0 Ф tF~1rs (C.12)
and
Фі — F0T ФF— Фs, (C.13)
the expression (C.11) reduces to the MUSMAR input predictor (3.63).
Furthermore, since by the construction of the pseudo-state the optimal predictor for the output у(к + i), given observations up to time к, is
у(к + i к) — ё$(к + i |к), (C.14)
it follows that
у(к + iк) — eTФ1—1 ^(к) + Гы(к)]. (C.15)
Defining and
Pi = ёФ1-1 Ф*, (C.17)
it is also recognized that the expression (C.15) reduces to the MUSMAR output predictor (3.62). □
