LESSONS OF EASTER ISLAND
March 17th, 2016
F. 1 Proof of the WARTICi/o Control Law
Hereafter we prove Eq. (5.16) that yields a closedform expression for the WARTIC – i/o control law. For that sake, consider the predictive model (5.15) and assume that u is constant and equal to u (k) over the prediction horizon, yielding
ni
70(k + i) = au(k) 22 R(k – 1 + j) + a22 R(k – p)u(k – p) + вTin(k + i – n). j=i p=i
(F.1)
Assume now that the future values of radiation at time k + 1 up to time k + T (that are unknown at time k) are equal to R(k). Equation (F.1) becomes
ni
T0(k + і) = au(k)R(k)i + y R(k – p)u(k – p) + вTin(k + і – n). (F.2)
p=i
Insert (F.2) in the definition of the cost Jk given by (5.14) to get
T
Jk = 22 [ATout(k + i) – YiR(k)u(k) – аїр(k, i)]2 , (F.3)
i = 1
where
AT*t(k) = T* (k) – вTin(k – n)
and
ni
П(k, i) = ^ R(k – p)u(k – p). (F.5)
p=1
= —aR(k) ^ i [ATo*ut(k + i) — atp(k, i)] + a2R2(k)u(k) ^ i2. (F.6)
i=1 i=1
Equating to zero the derivative (F.5) and solving with respect to u(k) yields (5.16).
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F. 2 Proof of the WARTICState Control Law
Hereafter we prove Eq. (5.42) that yields a closedform expression for the WARTIC – state control law.
From the assumptions (5.40) and (5.41) that both the manipulated variable and the reference are constant over the control horizon, the predictive model for the output (5.39) becomes
The matrices A, B and C are defined in (5.27). The cost function (5.33) can then be expanded as
Jk = (r(k) – CAx(k) – щ1u(k)) +———–
+ (r(k) – CATx(k) – щ1u(k)j2 + pTu2(k), (F.8)
where
ii
m = X cA – jB = £ CAjB = 1 + в + ••• + в1, (F.9)
j=0 j=0
with beta the parameter introduced in (5.13). Observing that
gives
Jk — {t?2 + ••• + ЇЇТ1 + pT }u(k)2
+ 2{t0Xn (k) + щв[5] Xn1(k) + ••• + tT1в T1 XnT+1(k)}u(k)
2r(k){щ0 + t1 + ••• + tT1}u(k) + terms independent of u(k),
and the minu(k) Jk is obtained by solving the equation




T1 i fT1 1
+ X в Xnj (k) + rn r(k) — 0
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