# Warped Time Optimization

In this appendix the underlying control laws (i. e., the control law assuming that the plant parameters are known) used in the WARTIC-i/o and WARTIC-state control algorithms, described in Chap. 5, are deduced.

F. 1 Proof of the WARTIC-i/o Control Law

Hereafter we prove Eq. (5.16) that yields a closed-form 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

n-i

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

n-i

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).

F. 2 Proof of the WARTIC-State Control Law

Hereafter we prove Eq. (5.42) that yields a closed-form 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) – щ-1u(k)) +———–

+ (r(k) – CATx(k) – щ-1u(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 + ••• + ЇЇТ-1 + pT }u(k)2

+ 2{t0Xn (k) + щв Xn-1(k) + ••• + tT-1в T-1 Xn-T+1(k)}u(k) 2r(k){щ0 + t1 + ••• + tT-1}u(k) + terms independent of u(k),

and the minu(k) Jk is obtained by solving the equation 1 9 Jk 2 9u(k)

 + p T u (k)   T-1 i fT-1 1

+ X в Xn-j (k) + rn r(k) — 0-