Derivation of the MUSMAR Dual Algorithm

For the sake of simplifying the notation, the mean conditioned on the observations E {-j Ok} is denoted by E {■}.

E. 1 Deduction of Equation (3.137)

image458

Since,

and

E {u2(k + i _ 1)} = E I ^i_1u(k) + f[_1s(k)j

= E |^2_1 u2(k) + 2u(k)/M-1p[_1s(k) + sT(k)p-1pT’_1s(k)^

= E {m2-1} u2(k) + 2u(k)E _1pJ_^ s(k) + sT(k)E |Фі_1 pf_^ s(k)

= [A2_1 + u2(k) + 2u(k) lM-1ФІ-1 + s(k)

+ sT(k)E [фі_1ф[_1} s(k) (E.2)

the minimization of the cost function according to Eq. (3.133) leads to

Подпись: (X [E (У2(k + i)} + PE [u2(k + i - 1)}]^T

2 9u(k) 2 9u(k)

T

Подпись: u(k)"УA2 + aei + pfa 2-1 + pa,

image461 image462

i = 1

Since the data vector, z(k) = [u (k) sT (k)] T, used to estimate the predictive models parameters is common to all models (actually, the vector used is z(k – T) so the T-steps ahead predictor can use the last output available to perform the estimation and that is y(k)), there is only one RLS covariance matrix, P(k) common to all models. To obtain the parameters of the true covariance matrix, matrix P (k) should be multiplied by the variance of the prediction error of each model. Therefore,

Подпись: (E.5)

Подпись: with Подпись: P= Подпись: Puu PuTs Pus Pss Подпись: (E.6)

a0i = Puuayi; a0pi = Pusayi; afa = Puua2; a^i = Pusali; ’

image468 Подпись: (E.7)

where the argument k is omitted for notation simplicity, which yields the control law

E. 2 Deduction of Equation (3.140)

Since is quadratic with negative second derivative, the minimum of this criteria coincides with one of the borders of the interval, and is given by

u(k) = Uc(k) + P(k)sgn { J(Uc(k) – P(k)) – J(Uc(k) + tf(k))} (E.8)

The expected prediction error for the output and input can be computed as

E j[y(k + i) – y(k + i)]2J = E j[(&i – §i)u(k) + (pT – pT )s(k)j J

= E {(0i – §i)2} u[3] [4](k) + 2u(k)E {(0i – §i)(pf – pf )} s(k) + sT(k)E [(Pi – Pi)(pf – pf )} s(k)

= aeiu2 (k) + 2u(k’)0Qpis (k)

+ sT(k)E [(Pi – Pi)(pf – pf )} s(k), (E.9)

E [[u(k + i – 1) – u(k + i – 1)]2J = E I[(m-1 – pi-{)u(k) + (pT-1 – p’T-1)s(k)[ J
= E [(pi-1 – pi_1)2J u2(k) + 2u(k)E [(pi-1 – pi-1)(pT-1 – Ф, Т_1^ s(k)

+ sT(k)E [(pi-1 – Фі-1)(p1T-1 – Ф-)] s(k)

Подпись: 22 = °piu2(k) + 2u(k)apfpis(k) + sT(k)E [(pi-1 – pi-1)(p1T-1 – s(k).

Подпись: (E.10)

Подпись: Replacing (E.9) and (E.10) in T T 4 =- E T y(k + і) – y(k + i)]2 + « [u(k + i – 1) – u(k + i – 1)]2

1 t

= -^X E[[y(k + i) – y(k + i)]2J

T i = 1 T

Подпись: a T ^e[[u(k + i – 1) – u(k + i – 1)]2[ (E.11)

Подпись: i=1

yields

ja(u)=-T °ві+aapPu2(k)+2u(k)[aepi+aaTpis(k)

i=1

image475
Подпись: puuu2 (k) + 2u(k) plss(k)
Подпись: (E.12)
Подпись: + c

and

J(uc(k) -&(k)) – J“(uc(k) + &(k))

= -у [puu(u2(k) – 2uc(k)$(k) + d(k)2) + 2(uc(k) – §(k))plss(k)j

+ у [puu(u;?(k) + 2uc(k)&(k) + &(k)2) + 2(uc(k) + &(k))plss(k)j

= 4y&(k) [puuuc(k) + plss(k(E.13) Equation (E.8) reduces then to Eq. (3.140). □

E.3 Local Models and Local Controllers for the Air Heating Fan Example

This appendix contains the numeric data for SMMAC control of the air heating fan laboratory plant, as described in Sect.4.3. The local models are described in Table E.1 and the local controllers, that match each of them, are described in Table E.2.

Подпись: Table E.1 Estimated local models of the air heating fan plant Low temperature Medium temperature Low flow A*(q-1) = 1 - 1.84q-1 + 0.8442q-2 A*(q-1) = 1 - 1.862q-1 + 0.8649q-2 B*(q-1) = 0.001866q-3 - 0.0003835q-4 B* (q-1) = 0.003823q-2 C *(q-1) = 1 - 1.389q-1 + 0.5118q-2 C * (q-1) = 1 - 1.15q-1 + 0.3074q-2 Medium flow A*(q-1) = 1 - 1.82q-1 + 0.8257q-2 A*(q-1) = 1 - 1.83q-1 + 0.8348q-2 B*(q-1) = 0.00143q-2 B* (q-1) = 0.003763q-2 C *(q-1) = 1 - 1.349q-1 + 0.4493q-2 C * (q-1) = 1 - 1.118q-1 + 0.3484q-2 High flow A*(q-1) = 1 - 1.783q-1 + 0.7902q-2 A*(q-1) = 1 - 1.8q-1 + 0.806q-2 B*(q-1) = 0.001222q-2 + 0.0002227q-3 B* (q-1) = 0.00399q-2 C *(q-1) = 1 - 1.334q-1 + 0.4904q-2 C * (q-1) = 1 - 1.193q-1 + 0.4593q-2

Table E.2 Local controllers for the different operating regions of the air heating fan plant Low temperature Medium temperature

Подпись:R*(q-1) = 1 – 1.696q-1 + 0.8719q-2 R*(q-1) = 1 – 1.156q-1 + 0.1556q-2

—0.1910q-3 + 0.01537q -4

S*(q-1) = 37.02 – 70.75q-1 + 33.84q-2 S*(q-1) = 41.41 – 76.24q-1 + 35.18q-2

T*(q-1) = 0.8677q-1 – 1.205q-2 T*(q-‘) = 2.176 – 2.503q-1 + 0.6690q-2

+0.4440q-3

Med – R*(q-‘) = 1 – 1.471q-1 + 0.4707q-2 R*(q-‘) = 1 – 1.170q-1 + 0.1698q-2

ium

flow S* (q-1) = 27.47 – 52.17q-1 + 24.81q-2 S*(q-1) = 51.96 – 96.19q-1 + 44.63q-2 T*(q-1) = 1.132 – 1.526q-1 + 0.5086q-2 T*(q-1) = 1.742 – 1.947q-1 + 0.6069q-2

Подпись:R* (q-1) = 1 – 1.479q-1 + 0.4935q-2 R*(q-1) = 1 – 1.291q-1 + 0.2912q-2

-0.01485q-3

S*(q-1) = 58.66 – 111.1q-1 + 52.68q-2 S*(q-1) = 51.48 – 96.22q-1 + 45.07q-2

T*(q-1) = 1.756 – 2.342q-1 + 0.8611q-2 T*(q-1) = 1.254 – 1.497q-1 + 0.5760q-2

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