Recurrent Neural Network Optimization for Model ... - IEEE Xplore

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he optimal VROYLQJ VWULFW FRQYH[ TXDGUDWLF SURJUDPPLQJ SUREOHPV DQG solution relies on a dynamic model of the process, satisfies UHODWHG OLQHDU SLHFHZLVH HTXDWLRQV 7KH SURSRVHG QHXUDO given input and state constraints, and minimizes a quadratic QHWZRUN LV VKRZQ WR KDYH D ILQLWHWLPH FRQYHUJHQFH DQG performance. Generally, 4XDGUDWLF SURJUDPPLQJ 43 H[SRQHQWLDO FRQYHUJHQFH > @ ,Q >@ D VLPSOLILHG GXDO DOJRULWKP LV XVHG WR VROYH WKLV RSWLPDO SUREOHP %XW the QHXUDO QHWZRUN ZDV SURSRVHG ,Q SURSRVHG QHXUDO QHWZRUN optimization problem depends on the current state, the QXPEHU RI QHXURQV LV HTXDO WR WKH QXPEHU RI LQHTXDOLW\ implementation of MPC requires the online solution of a QP FRQVWUDLQWV +RZHYHU WKHVH 511 PHWKRGV DUH DGRSWHG WR at each time step. Although efficient QP solvers based, for VROYH RSWLPL]DWLRQ SUREOHPV DQG QRW DSSO\ WR VROYH FRQWURO example, on active-set methods are available, computing the SUREOHPV,Q>@DVWUXFWXUHGQHXUDOQHWZRUNZDVSURSRVHG control output demands significant online computation LPSOHPHQWLQJWKHJUDGLHQW SURMHFWLRQ DOJRULWKP WR VROYH WKH effort. For this reason, the application of MPC has been TXDGUDWLFSURJUDPPLQJSUREOHPLQFRQVWUDLQHG03&%XWWKH restricted for a long time to rather slow processes. Recently, VWUXFWXUH RI SURSRVHG QHXUDO QHWZRUN LV FRPSOH[ DQG QRW a number of methods have been proposed to reduce the VXLWDEOHIRUKDUGZDUHLPSOHPHQWDWLRQ7KLVSDSHUSUHVHQWVD computational burden of solving the QP problem in QHZ VFKHPH RI VROYLQJ 03& SUREOHP EDVHG RQ RQH OD\HU constrained MPC. A popular technique is to solve the QP 511 approximately. In [1], an approximation algorithm was 7KLV SDSHU LV RUJDQL]HG DV IROORZV ,Q 6HFWLRQ ZH proposed and derives a quantitative condition under which LQWURGXFH 03& DQG WUDQVIRUP 30& LQWR 43 SUREOHPV ,Q MPC can be approximated effectively. In [2], an approach to 6HFWLRQ 511 LV SUHVHQWWR VROYH 03& SUREOHP DQG JLYH shift the computational effort from online to offline was VFKHPH GLDJUDP 7ZR QXPHULFDO H[DPSOHV DUH VLPXODWHG WR proposed. This approach relies on treating the cost function VKRZ WKH JRRG SHUIRUPDQFH RI WKH SURSRVHG PHWKRGV LQ as a multiparametric quadratic program (mp-QP), solving it VHFWLRQ6HFWLRQFRQFOXGHVWKLVSDSHU for every possible initial state, and thus, resulting in a control law. These approximation and offline strategy reduce ,, )2508/$7,212)03& the computational burden, but the control performance is 6XSSRVHDOLQHDUGLVFUHWHWLPHVWDWHVSDFHPRGHORIWKH sacrificed. SODQWLVJLYHQLQWKHIRUP [ N + = $[ N + %X N

$EVWUDFW²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

0

7KLVZRUNLVIXQGHGE\WKH+LWHFK5HVHDUFKDQG'HYHORSPHQW3URJUDPRI &KLQD XQGHU FRQWUDFW $$ DQG E\ WKH 1DWLRQDO 1DWXUDO 6FLHQFH )RXQGDWLRQRI&KLQDXQGHUFRQWUDFW The authors are with the School of automation, Wuhan University of Technology. (E_mail: ]O\ZKXW#ZKXWHGXFQ)

\ N = &[ N

ZKHUH [ ∈ 5 Q LV Q GLPHQVLRQDO VWDWH YHFWRU X ∈ 5 P LV P GLPHQVLRQDO PDQLSXODWHG YHFWRU \ ∈ 5 O LV O GLPHQVLRQDO RXWSXWYHFWRU

751 c 978-1-4244-1821-3/08/$25.002008 IEEE

)RU WKH EDVLF IRUPXODWLRQ RI SUHGLFWLYH FRQWURO ZH VKDOO DVVXPHWKDWWKHSODQWPRGHOLVOLQHDUWKDWWKHFRVWIXQFWLRQLV TXDGUDWLF DQG WKDW FRQVWUDLQWV DUH LQ WKH IRUP RI OLQHDU LQHTXDOLWLHV6LJQDOV X N + M _ N ZLOOGHQRWHDIXWXUHYDOXHDW WLPH N + M RI WKH PDQLSXODWHG YDULDEOH X DW WLPH N Signals [ N + M _ N and \ N + M _ N will denote the predictions,madeattimek,ofthevariablesxψandyψat time N + M , on the assumption that some sequence of inputs X N + L _ N L = " M − hasbeenapplied.These predictions will be made consistently with the system model(1). :HFRQVLGHUWKHSUREOHPRIKDYLQJDQLQLWLDOVWDWH[N DW WLPHNDQGILQGLQJDFRQWUROVHTXHQFH X N + M N1=− ZKLFKZLOO PLQLPL]HWKHILQLWHKRUL]RQ03&SUREOHP 1

PLQ - = ¦ ( \ N + M _ N − \ U ) 4( \ N + M _ N − \ U ) X

7

M =

1 F −

+ ¦ (X N + M _ N − X U ) 3(X N + M _ N − X U ) 7

M=

VW \PLQ ≤ \N + M _ N ≤ \PD[ M = " 1 XPLQ ≤ X N + M _ N ≤ X PD[ M = " 1 F − ZKHUH 1 LV SUHGLFWLRQKRUL]RQ 1 F LV FRQWUROKRUL]RQ \U DQG X U DUHUHIHUHQFHWUDMHFWRU\RIRXWSXWYDULDEOHDQGPDQLSXODWHG YDULDEOH UHVSHFWLYHO\ 4 ∈ 5 O ×O DQG 3 ∈ 5 P×P DUH ZHLJKW PDWULFHV ,WLVZHOONQRZQWKDW03&VROYHVWKLVSUREOHPDWHDFKWLPH VWHS N FRPSXWLQJ DOO QH[W 1 YDOXHV RI WKH PDQLSXODWHG YDULDEOHVXNM_N $FFRUGLQJWRV\VWHPPRGHO ZHFDQJHW

[

\ N + M _ N = &$ M [ N + & $ M −

ª XN _ N º » # $ M− " , % « » « «¬X N + M − _ N ¼»

]

M = " 1 ,Q WKH ILUVW OLQH ZH KDYH XVHG XN_N UDWKHU WKDQ XN EHFDXVHDWWKHWLPHZKHQZHQHHGFRPSXWHWKHSUHGLFWLRQVZH GRQRW\HWNQRZZKDWXN ZLOOEH 'HILQH ª \ N + _ N º « \ N + _ N » » < N = « « » # « » ¬ \ N + 1 _ N ¼ ª &% ª &$ º « &$% &% « &$ » « » [N + « &$ % =« &$% &% « # » « # # « 1» « # ¬&$ ¼ «¬&$ 1F − % &$ 1F − % &$ 1F − %

= Φ[ N + Γ8 N

ZKHUH ª &$ º « &$ » » Φ=« « # » « 1» ¬&$ ¼

752

:HFDQUHZULWHWKHREMHFWLYHIXQFWLRQLQ DV - = < N − \ U 7 4< N − \ U + 8 N − X U 7 38 N − X U = Φ[ N + Γ 8 N − \ U 7 4 Φ[ N + Γ 8 N − \ U + 8 N − X U 7 3 8 N − X U

= 8 N 7 Γ 7 4Γ + 3 8 N + [N 7 Φ 7 4Γ − \ 7U 4Γ − XU7 3 8 N

+ \ U7 4\ U + [ N 7 Φ 7 4Φ[N − \ U7 4Φ[ N − X U7 3X U 7KLVKDVWKHIRUP - = 8 N 7 +8 N + I 7 8 N + FRQ ZKHUH + = Γ7 4Γ + 5 I = Γ7 4Φ[ N − Γ 7 4\ U − 5X U = V[N + W V = Γ7 4Φ W = −Γ7 4\U − 5X U FRQ = \U7 4\U + [N 7 Φ7 4Φ[N − \U7 4Φ[N − XU7 3XU

+DQGIGRQ¶WGHSHQGRQ8N IGHSHQGVRQWKHLQLWLDOVWDWH [N &RQVWUDLQWV DQG FDQUHZULWWHQDV ªΓ 7 (XPLQ " XPLQ @7 DQG 8 PD[ = >X PD[ " XPD[ @7 DUH P

hPDWULFHV 'HILQH

8 PLQ = PD[^Γ 7 (@WRVROYHWKLV43SUREOHP &RQVLGHUWKH/DJUDQJLDQIXQFWLRQRIWKHSUREOHP /8 [η = 8 7 +8 + I 7 8 − [ 7 8 − η

ZKHUH [ ∈ 5 P1 F LVUHIHUUHGWRDVWKH/DJUDQJHPXOWLSOLHUDQG η ∈ ; = {[ ∈ 5 P1 F _ 8 PLQ ≤ [ ≤ 8 PD[ } $FFRUGLQJ WR 6DGGOH SRLQWWKHRUHP >@ ZH DVVXPH WKDW WKH YDOXH RI [ DQGη DUH

λ DQGη ZKHQ 8 LVDQRSWLPDOVROXWLRQRIWKHSUREOHP ,Q>@DQDSSURDFKZDVSURSRVHGWRWUDQVIRUP43SUREOHP LQWR 511 RSWLPL]DWLRQ ZKRVH G\QDPLF HTXDWLRQLV GHILQHG DVIROORZV 6WDWHHTXDWLRQ GY = λ{3[ :Y + T − Y − :Y − T} GW

2XWSXWHTXDWLRQ 8 W = 5YW + D ZKHUH : = 5 = + − LV PDWUL[ T = D = − + − I DUH YHFWRU Y ∈ 5 P1 LV WKH VWDWH YDULDEOH RI 511 λ > LV D VFDOLQJ FRQVWDQW 7 3; Y = [3; Y 3; Y " 3; YP1 F ] DQGIRU L = " P 1F F

8 PLQ L YL < 8 PLQ L ° 3; YL = ®YL 8 PLQ L ≤ YL ≤ 8 PD[ L °8 L Y > 8 L L PD[ ¯ PD[ 7KHSUREOHP FDQEHLPSOHPHQWHGE\DQDQDORJFLUFXLW ZLWKDVLQJOHOD\HUVWUXFWXUHDQGWKHDUFKLWHFWXUHRISURSRVHG 03& EDVHG RQ 511 LV VKRZQ LQ )LJ ZKHUH S = P 1F

7KH FLUFXLW FRQVLVWV RI S + S + VXPPHUV S LQWHJUDWRUV DQG S ZHLJKWHG FRQQHFWLRQV 7KH SURMHFWLRQ RSHUDWRU 3; YL LV LPSOHPHQWHG E\ XVLQJ D OLQHDU SLHFHZLVH IXQFWLRQ 0X[ GHQRWHVWKDWPXOWLSOHVLJQDOV YL DUHWUDQVIRUPHGLQWRDEXV9 'HPX[GHQRWHVWKDWDEXVVLJQDOLVVSOLWLQWRPXOWLSOHVLJQDOV 6HOHFWRU GHQRWHV WKDW JHW ILUVW P YDOXH IURP D EXV 8N $FFRUGLQJWRWKHUHVXOWVLQ>@WKHSURSRVHGDUFKLWHFWXUH KDVORZDUFKLWHFWXUHFRPSOH[LW\WKDQRWKHU511PHWKRGV

% $OJRULWKPRI03&EDVHGRQ511 7KH DOJRULWKP RI 03& EDVHG RQ 511 PHWKRG FDQ EH VXPPDUL]HGDVIROORZV ,QLWLDOL]DWLRQ /HW N 6HW WHUPLQDO WLPH 7HQG 6HW WKH YDOXH RI VWDWH YDULDEOH [N LQ SODQW PRGHO SUHGLFWLYH KRUL]RQ 1 FRQWURO KRUL]RQ 1 F VDPSOH WLPH W ZHLJKW PDWUL[ 4 DQG 3 FRPSXWDWLRQ WLPH ΔW RI 511 DW HDFK VDPSOH WLPH $FFRUGLQJ WR WKH SODQW PRGHO FDOFXODWH PDWUL[ Φ Γ +VWDQG: = 5 = + − &DOFXODWH 511 SDUDPHWHUV &DOFXODWH I = V ⋅ [ N + W T = D = − + − I $FFRUGLQJWR(T FDOFXODWHXSSHUDQG ORZOLPLWRIPDQLSXODWHGYDULDEOH 8 PD[ DQG8 PLQ 511G\QDPLFRSWLPL]DWLRQ$FFRUGLQJWR:5TDQGD

)LJ$UFKLWHFWXUHRIWKHSURSRVHGQHXUDOQHWZRUN

511VWDUWWRUXQDIWHUWLPH ΔW ZHJHWVWDEOHYDOXH8 3UHGLFWLRQRISODQWPRGHO*HWILUVWPYDOXHLQ 8 DVWKH YDOXH RI PDQLSXODWHG YDULDEOH DQG FDOFXODWLRQQH[W VWHS YDOXH[W RIVWDWHYDULDEOH ,I N < 7HQG VHWN NUHWXUQVWHSRWKHUZLVHSURJUDP HQG & &RQYHUJHQFHDQDO\VLV 7R VWXG\ RSWLPL]DWLRQ FDSDELOLW\ RI WKH SURSRVHG 03& EDVHGRQ511FRQYHUJHQFHSURSHUW\RI511DWHDFKVDPSOH VWHSPXVWEHLQYHVWLJDWHG 'HILQLWLRQ $ 511 LV VDLG WR KDYH D ILQLWHWLPH FRQYHUJHQFHWRWKHRSWLPDOVROXWLRQ Y RIWKHSUREOHP DW HDFKVDPSOHVWHSLIWKHUHH[LVWVDWLPH τ VXFKWKDWWKHRXWSXW WUDMHFWRU\YW RIWKLVQHWZRUNUHDFKHV Y IRU W ≥ τ $511LV VDLG WR EH H[SRQHQWLDOO\ FRQYHUJHQW WR WKH RSWLPDO VROXWLRQ Y RIWKHSUREOHP LIWKHRXWSXWWUDMHFWRU\YW RI WKLVQHWZRUNVDWLVILHV Y W − Y ≤ F H −η W −W ∀W > W

ZKHUHη LVDSRVLWLYHFRQVWDQWLQGHSHQGHQWRQWKHLQLWLDOSRLQW DQG F LVDSRVLWLYHFRQVWDQWGHSHQGHQWRQWKHLQLWLDOSRLQW 2XUPDLQUHVXOWVRQ511RIWKHSURSRVHGDSSURDFKDUHWKH IROORZLQJ 7KHRUHP7KHVWDWHWUDMHFWRU\RISURSRVHG511LVJOREDOO\ FRQYHUJHQWWRDVROXWLRQRISUREOHP DWHDFKVDPSOHVWHS 3URRI/HWYW EHWKHWUDMHFWRU\RIWKHVWDWHHTXDWLRQGHILQHG LQ(T ZLWKDQ\JLYHQLQLWLDOSRLQW YW = Y &RQVLGHUWKHIROORZLQJ/\DSXQRYIXQFWLRQ 9 Y =

* Y − Y

ZKHUH Y LV DQ HTXLOLEULXP SRLQW RI (T DQG * LV D V\PPHWULF DQG SRVLWLYH GHILQH PDWUL[ VDWLVI\LQJ * = , + : 7KHQ 7

G § G9 · GY 9 Y = ¨ ¸ GW © GY ¹ GW

= λ Y − Y * ^3; :Y + T − Y − :Y − T` $FFRUGLQJWRWKHUHVXOWVLQ>@ZHKDYH

2008 International Joint Conference on Neural Networks (IJCNN 2008)

753

G λμ 9 Y ≤ −λμ Y − Y ≤ − 9 Y α GW ,WIROORZVWKDW

^3; :Y − T − Y − :X − T`7 ^Y − Y − 3; :Y − T − Y

− :Y − T` ≥ 7KHQ Y − Y , + : ^3; :Y + T − Y − :Y − T`

Y − Y ≤ −9 YW H

≤ −Y − Y 7 : Y − Y − 3; :Y + T − Y − :Y − T

7KXV G 9 Y ≤ −λ Y − Y : Y − Y − λ 3; :Y + T − Y − :Y − T ≤ GW 7KHUHIRUH WKH VWDWH WUDMHFWRU\ RI WKH SURSRVHG 511 LV JOREDOO\FRQYHUJHQWWRDVROXWLRQRIWKHSUREOHP DWHDFK VDPSOHVWHS 7KHRUHP 7KHFRQYHUJHQFHWLPHRIWKHSURSRVHG511DW HDFK VDPSOH VWHSLQ SUREOHP LV ILQLWH SURYLGHG WKDW λ LV ODUJHHQRXJK 3URRI/HW I YW = 3; :Y + T − Y − :Y − T 7KHQ I YW ≥ DQG I YW ≥ 6LQFH I YW LV FRQWLQXRXV >@ WKHUH H[LVW τ > DQG δ > VXFK WKDW IRU DQ\ W ∈ >W W + τ I YW ≥ δ $FFRUGLQJWR7KHRUHP G 9 Y ≤ −λ Y − Y : Y − Y − λ 3; :Y + T − Y − :Y − T GW ≤ −λ 3; :Y + T − Y − :Y − T ,WIROORZVWKDW

W

9 YW ≤ 9 YW − λ ³ 3; :Y V + T − Y V − :Y V − T GV

W

τ +W

≤ 9 YW − λ ³

W

I YV GV ≤ 9 YW − λδτ

%\WDNLQJ 9 YW λ= δτ :HJHW 9 YW ≤ 9 YW − λδτ = ZKHQ W ≥ W + τ 7KHUHIRUH WKH FRQYHUJHQFH WLPH RI WKH SURSRVHG511DWHDFKVDPSOHVWHSLQSUREOHP LVILQLWH SURYLGHGWKDW λ LVODUJHHQRXJK 7KHRUHP ,I : = + − LVV\PPHWULFDQGVHPLGHILQLWHWKHQ WKH VWDWH WUDMHFWRU\ RI WKH SURSRVHG 511 LV JOREDOO\ H[SRQHQWLDOO\ FRQYHUJHQW WR D VROXWLRQ RI SUREOHP DW HDFKVDPSOHVWHS 3URRI ,Q 3UREOHP : = + − LV V\PPHWULF DQG

VHPLGHILQLWH 6R 9 Y ≥ α Y − Y ZKHUH α > LV WKH PLQLPDO HLJHQYDOXH RI ,: >@ 0RUHRYHU WKHUH H[LVWV D μ > VXFKWKDW YW − Y : YW − Y ≥ μ YW − Y

7

−ψ W −W

ZKHUHψ = λμ α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γ DQG γ DV T = γ TD T = γ TE T = − γ TE T = − γ TD ZKLFK DUH WKH ZDWHU IORZV WR HDFK WDQNWKH IORZ SDUDPHWHUV γ DQG γ FDQ YDU\ EHWZHHQ DQG 7KH GLIIHUHQWLDO HTXDWLRQV WKDW GHVFULEH WKH V\VWHP G\QDPLFVDUHDVIROORZV GK GW = −D $ ⋅ JK + D $ ⋅ JK + γ $ ⋅ TD GK GW = −D $ ⋅ JK + D $ ⋅ JK + γ $ ⋅ TE GK GW = −D $ ⋅ JK + − γ $ ⋅ TE GK GW = −D $ ⋅ JK + − γ $ ⋅ TD ZKHUH $L LVWKHFURVVVHFWLRQRI7DQNL DL LVWKHFURVVVHFWLRQ

RI WKH RXWOHW KROH $L LV WKH ZDWHU OHYHO 7KH HVWLPDWHG SDUDPHWHUYDOXHVRIWKHUHDOSODQWDUHVKRZQLQ7DEOH 7KHREMHFWLYHRIWKLVSUREOHPLVWRFRQWUROWKHZDWHUOHYHOLQ WKHWZRORZHUWDQNV7KLVLVGRQHE\FRQWUROOLQJWKHSXPS IORZ TD DQG TE 7R ZRUN ZLWK D OLQHDU GHVFULSWLRQ WKH PRGHO KDV WR EH OLQHDUL]HGDURXQGDQRSHUDWLQJSRLQW'HILQLQJWKHRSHUDWLQJ SRLQW DV KL = > @ WKH YDULDEOHV DV [L = KL − KL DQG [ M = T M − T M ZKHUHM DEDQG L = " OHDGVWR

6LQFH G 9 Y ≤ −λ Y − Y : Y − Y − λ 3; :Y + T − Y − :Y − T GW ≤ −λ Y − Y : Y − Y 7KHQ

754


$ $7 º ª γ $ º ª−7 « −7 » « » $ $ 7 γ G[ « $ » » = [+« X » « −γ $ » −7 GW « « » « » ¼ −7 ¼ ¬−γ $ ¬

ª º \=« »[ ¬ ¼

)LJ7KHSODQWRIWKHIRXUWDQNV

7$%/(, 3DUDPHWHUYDOXHV

1DPH $ $ $ $ D D D D γ γ J

8QLW FP FP FP FP FP FP FP FP FPV

9DOXH

WKHVHIRXUFDVHV7KLVVKRZVWKDW511KDVWKHDGYDQWDJHRI SDUDOOHOFRPSXWDWLRQ

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

D

E


755

F )LJ6LPXODWLRQUHVXOWZKHQSUHGLFWLYHKRUL]RQ1

D

E

D

E



D

756


E


>@ --+RSILHOG':7DQN&RPSXWLQJ ZLWK QHXUDOFLUFXLWV $ PRGHO 6FLHQFH ± >@ @ @ /;:DQJ):DQ6WUXFWXUHGQHXUDOQHWZRUNVIRUFRQVWUDLQHGPRGHO SUHGLFWLYHFRQWURO$XWRPDWLFD [13] M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear programming-----theory and algorithms (QG ed.) (New York, Wiley, 1993). [14] -63DQJ-&@ K.H. Johansson. The quadruple-tank process: a multivariable laboratory process with an adjustable aero, IEEE transactions on control systems technology, 2000, 8(3): 456-465.

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

[2] >@ >@ >@

$ =KHQJ 5HGXFLQJ 2QOLQH FRPSXWDWLRQDO GHPDQGV LQ PRGHO SUHGLFWLYHFRQWUROE\DSSUR[LPDWLQJ43FRQVWUDLQWV-RXUQDORI3URFHVV &RQWURO $ %HPSRUDG 0 0RUDUL 9 'XD (1 3LVWLNRSRXORV 7KH H[SOLFLW OLQHDUTXDGUDWLFUHJXODWRUIRUFRQVWUDLQHGV\VWHPV$XWRPDWLFDYRO ± 03 .HQQHG\ /2 &KXD 1HXUDO QHWZRUNV IRU QRQOLQHDU SURJUDPPLQJ,(((7UDQVDFWLRQRQ&LUFXLWVDQG6\VWHPV