Identification of MIMO State Space Model based on MISO High-order ARX Model: Design and Application

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1 Korean Chem. Eng. Res., Vol. 45, No. 1, February, 2007, pp MISO y ARX m MIMO m m : j nk kjiçlslçm Çm * ne e 1 *LS r l kke k ( o 20p r, o 21p }ˆ) Idenificaion of MIMO Sae Space Model based on MISO High-order ARX Model: Design and Applicaion Wangyun Won, Jieun Yoon, Kwang Soon Lee and Bongkook Lee* Deparmen of Chemical and Biomolecular Engineering, Sogang Universiy, 1, Shinsoo-dong, Mapo-gu, Seoul , Korea *LS Indusrial Sysem Co., Ld., Hogye-dong, Dongan-gu, Anyang, Gyeonggi , Korea (Received 20 November 2006; acceped 21 December 2006) k qd, q realizaion, q runcaionp p f, MIMO ˆ p pe p o rp p l. l MIMO edšp ARX MISO e dšp. p, ARX p Ž qd l p r., realizaionp p MISO ARX r l MISO ˆ p lv, MIMO ˆ. srp, q realizaion q runcaionp p q MIMO ˆ p llv. rk p k CO 2 n r e q p m rl o m rlp l rn l. h Absrac An efficien mehod for idenificaion of MIMO sae space model has been developed by combining parial leas squares (PLS) regression, balanced realizaion, and balanced runcaion. In he developed mehod, a MIMO sysem is decomposed ino muliple MISO sysems each of which is represened by a high-order ARX model and he parameers of he ARX models are esimaed by PLS. Then, MISO sae space models for respecive MISO ARX ransfer funcion are found hrough realizaion and combined o a MIMO sae space model. Finally, a minimal balanced MIMO sae space model is obained hrough balanced realizaion and runcaion. The proposed mehod was applied o he design of model predicive conrol for emperaure conrol of a high pressure CO 2 solubiliy measuremen sysem. Keywords: Idenificaion, PLS, Balanced Realizaion, Balanced Truncaion, Model Predicive Conrol 1. p rl rl p p e ˆ p ˆq p r p r l p r. v p rp rp q s l rp v p p pn l dv k. rp rl l e p pe rp r p r p p rp. e rp r p l o rp p 1970 l Ljungl p p m m (predicion error mehod) To whom correspondence should be addressed. kslee@sogang.ac.kr 1990 l rk pe (subspace idenificaion mehod) p [1]. p m m p parameric p p. rp p p (SISO)( p p p (MISO)) p p m m Ž r l rp kp r p rp r Ž p qrp v., pe p nonparameric l p, r p p n l, QR p p n l rp kr q p (MIMO) ˆ p }k vp [2, 3]. Van Overscheem DeMoorl p N4SID q o, p p l p pe p p rnl p l l [4-8]. pe p p 67

2 68 omlovpp p pn n p(biased) p r l, kr l l ˆ pe e p l o rl pep l. r l p r p k l p r p p l o m m l p p n rp v. r p er rp m p n, r p p rp l eˆv (collineariy)l p illcondiioningp edš Ž r rl q p. MIMO rp e r rp l eˆ l l l p r p kv., r p er r p n, PRBS(pseudo random binary sequence)m p k Ž p r e p r p pp p l p r. l p r rp kp p rp r p l o, l l sl l p s l lrp e p pe edšp m. p vp qd (parial leas square regression) p l (muli-collineariy) l p Ž r r, q (balanced) realizaion q runcaionp l p(minimal) q MIMO ˆ p ll. 2. m m rl rl p r p r l p p r, rp q p pp n n. p o pp p rp pe p m. Ì 1: p r}. Ì 2: MIMO rp MISO rp. Ì 3: 2 l r MISO rl ARX r qd n l Ž r. Ì 4: MISO ARX r ˆ. Ì 5: MISO ˆ p p(non-minimal) MIMO ˆ. Ì 6: q realizaion q runcaionp p q ˆ. op p np p m nz o sp pe l o p r} rp n. p p (oulier) r l o Ž mlp s p f, k vv kp p l p r p. p o, l l e (1) p p p p l p r, Ž r s p eˆ o l p p r (normalizaion) m. o45 o k y () l, y m m u p r r m p p, y m u p σ y m σ u p. p, MIMO rp p l MISO rp m y ARX }o ARX p Ž r k ARMAX, p Ž r p p, ARMAXm p q p n v k er rp q p qrp p [1]. l l MISO rp e (2)m p ARX m. l, y r, u rp, e n p. np, n u p p, n y p p. e (2) 1,...,N kp p k e e (3). e (3)l Φ i m Y i p r rl ppp ˆ rl m p ˆ v k m. Y i Φ i Θ i + E i l i1,..., n y Φ i y m y u m u , u() σ y σ u n n u a ik ( k) + b jk u j ( k) + e i k 1 j 1 l i1,..., n y (2) ( n + 1) Y i ( N) n () () 1 u 1 () n u 1 () 1 u nu () n u nu () 1 ( ) ( N n) u 1 ( N 1) u 1 ( N n) u nu ( N 1) u nu ( N n) N 1 ARX p r p mis-specificaionp p l p r ill-condiioning l, Ž r rl r p. p l p r o p m qd o. p e (4)m p p llv Φ i p p Y i p n p f, p v p pn l l p r r p. Φ i [ U Ũ ] D 0 V T UDV T UQ T 0 D Ṽ T e (3)p rl e (4)m p rp Φ i m Yi p p l p pn l, m p rp v p Φ i p Y i mp q l., qd e (5)m p Φ i m Y i el l, pp p (1) (3) (4)

3 p v p r } l, Φ i k Y i m r. pm p Φ i m Y i l p n l Ž r r p kr r vp prp p. l Abdi[9]p qd k vp pn l e (2)p Ž a ik, b jk (i 1,...,n y, j1,...,n u, k 1,...,n) r m. MISO ARX p MIMO ˆ p pe: m rn 69 Φ i ΤΡ Τ Y i TS T (5) l Sm P (loading marix)p, T qq (laen vecors) l r (score marix)p Realizaion MIMO m e (2)p MISO ARX r pp ˆ. x ( + 1) A i x () + B ij u j A i C i x () a i1 1 0 n u j 1 l i1,...,n y, j1,...,n u a i 2 0 0,, C i [ ] (6) B b ij2 ij 0 0 a in b ij1 b ijn (similariy ransformaion)p q realizaionp [10]. p p ˆ, ˆm pp l p e (8)p rl L c (x)m L o (x)p l p. L c ( x) 1 --x T 1 G C x 2 l G c C C T, C [ B AB A n 1 B] 1 L o ( x) --x T G o x 2 l G o Õ T Õ, Õ [ ( CT CA) T ( CA n 1 ) T ] T (8) l C rl, Õ, G c rl gramian, G o gramian, np p. l l e (7)p MIMO ˆ p XMZl q MIMO ˆ p m. p p e (9) p, e (10)p s. l Mp [11]p k vp l m. Z ( + 1) AZ + BU() Y () CZ() l A M 1 AM, B M 1 B, C CM (9) l MISO ˆ p e (7) p MIMO ˆ. X ( + 1) AX + BU() l Y () CX() Y () y 1, y ny A 1 0 U () 2-4. m realizaion m runcaion r p p rl l edšp ˆ pp sr p edšp rl, r p ed Šp ˆ r p edšp [12]. e (6)p MISO ˆ p rl ˆ p p realizaionp. v p e (7) p MIMO ˆ rl ˆ l p realizaionp. MIMO ˆ p rl rl p p p v k ˆ p. l q p ˆ, ˆm pp l p p p rp kr ˆ, rl p p p p u 1 u ny B 11 B 1nu A, B, C 0 A ny B ny B 1 ny n u C C ny (8) G c G o Σ 0 σ 2 0, σ 1 > σ 2 >... > σ n (10) o el q MIMO ˆ p rl gramian gramianp Hankel p p o p. l 1/σ i p i w ˆl p l vp l ˆ, σ i p l vl i w ˆp l ˆ. q realizaionp p vp pn l q runcaionp m. q runcaionp e (9)p q MIMO ˆ l rl ˆ r p f p q MIMO ˆ p l. p o r e (10)l Hankel p p l p rp Σ p p r m. Σ e (9)p Σ 1 0, Σ 1» Σ 2 0 Σ 2 ( A, B, C) p. (11) e (11)p Σl p, r A 11 A 12 A A 11, B B 1, C [ C 1 C 2 ] C 1 A 21 A 22 B 2 (12) e (12)l Σ 2 0l A 11 kr ( A 11, B 1, C 1 ) p p realizaionp [14]. σ σ n B 1 Korean Chem. Eng. Res., Vol. 45, No. 1, February, 2007

70 omlovpp p 3. m sm Fig. 1l pe l p C++ ll pe Žˆv p m. Fig. 1(a)l *.csv(d ) Žp rq p p m pe l p, qe Fig. 1(b)m p qq ˆ p ˆ. p, p Φm Y q rp qq ˆ. Fig. 1(c)m p q runcaionp o Hankel p p ˆ p ˆ, p rr Hankel p p l s rp llv p q MIMO ˆ p A.

4 70 omlovpp p 3. m sm Fig. 1l pe l p C++ ll pe Žˆv p m. Fig. 1(a)l *.csv(d ) Žp rq p p m pe l p, qe Fig. 1(b)m p qq ˆ p ˆ. p, p Φm Y q rp qq ˆ. Fig. 1(c)m p q runcaionp o Hankel p p ˆ p ˆ, p rr Hankel p p l s rp llv p q MIMO ˆ p A.csv, B.csv, C.csv Žpl rq. er p m p m p Y_Simulaion.csv Žpl rq, p pe llv r p r p. 4. Fig. 2l l l p k CO 2 n r e q ˆ l. lv k l l CO 2 ~m pm k~ l, k l p p p k r l ~p k~l n. k l p k l ms p m s r p f m r pr ov p rl rp. p msm p ql. rp rl D/A l 4-20 map r e sq, m rl, n m n p SISO rp. k CO 2 n r e q lv m l r n p rp o, r m rl n. e p p o m r l k r ov lk. v 60 bar p k CO 2 ~ n l p p, p p msm pp lr l p sq. rep edšp l, ˆ p o q r p. p vp sp PID rl rl l rm, qld p m p v rl Fig. 1. Human-machine inerface (HMI) of he developed idenificaion package program; (a) an iniial screen, (b) variance able, (c) Hankel singular value able. o45 o k Fig. 2. Process flow diagram of he concerned solubiliy measuremen experimenal sysem.

5 MISO ARX p MIMO ˆ p pe: m rn 71 Fig. 3. Resuls of emperaure predicion in he solubiliy measuremen sysem; (a) mean cenered and normalized emperaure, (b) inpu exciaion signal. rn o. m rl r p r l p p r, er rp q sp p pp n n. r p pe p rq lp, p o pe e p l. m ˆp p l Fig. 3(b)p 7-10 ma p e l - ˆl 57e k p m p, e p 1 p l 3,420 p m p m. p, pe p r p rq l m rl l rn m. r p 5 r l Ž r mp, m rl Won [13]p rl p p l m., PID rl r n l m rlp nr m m. p PID r l Ziegler-Nichols sp p sp, spp r sp m. Fig. 4. Closed-loop response agains a se poin change under MPC and PID conrol; (a) rajecory of emperaure, (b) process inpu of MPC, (c) process inpu of PID conrol. Korean Chem. Eng. Res., Vol. 45, No. 1, February, 2007

6 72 omlovpp p Table 1. Variance of Φ and Y explained by he laen vecors Laen Vecor Percenage of Explained Cumulaive Percenage of Percenage of Explained Cumulaive Percenage of Variance for Φ Explained Variance for Φ Variance for Y Explained Variance for Y m rk pe p rq r p rr m rl l rn l k CO 2 n r e q p m rl m. Fig. 3(a)l p ep r m r m pe p llv m m p m. p l pe p m m p er m r p q p p. p m m p Table 1 l 2 qq p ˆ l T p, e (5) o Ŷ TS pn l T l. Fig. 4l PID rlm pe p rq r p ˆq m rl k CO 2 n r q p m rll rn ˆ l. p, rl 1 p r m. PID rlp n m rlm l m r l rp rl qp p. PID rl p m p l l rl m l rl qp m p r, m r l l kr m e p. rp msm pp lr l p p v rp, 2 qo v PID rl rl l p. pe l p rq 5 r p ˆq m rl m r l sp rl p l. m rlp m p p rl p rk s p s e q l, p p m p p. p f m r l kr. rp n, p k rp p e sp rl p lp pl. 6. l qd pn l ARX p r, q realizaion, q runcaionp p q p e ˆ p rq p e p m. p rr m rl n r rp m rll rn p f, pe p lv r p p v m. l lqo l e r ll lp p p ld. 2 p v q p qrrp vol. y 1. Ljung, L., Sysem Idenificaion Theory for he user, 2nd ed., Prenice-Hall, New Jersey, NJ(1999). 2. Favoreel, W., Moor, B. D. and Overschee, P. V., Subspace Sae Space Sysem Idenificaion for Indusrial Processes, J. Process Conrol, 10(2), (2000). 3. Overschee, P. V. and Moor, B. D., N4SID: Subspace Algorihms for he Idenificaion of combined deerminisic-sochasic Sysems, Auomaica, 30(1), 75-93(1994). 4. Qin, S. J., An Overview of Subspace Idenificaion, Comp. Chem. Eng., 30(10), (2006). 5. Chiuso, A. and Picci, G., The Asympoic Variance of Subspace Esimaes, J. Economerics, 118(1), (2004). 6. Jia, C., Rohani, S. and Juan, A., FCC Uni Modeling, Idenificaion and Model Predicive Conrol, a Simulaion Sudy, Chem. Eng. Proc., 42(4), (2003). 7. Lee, K. S., Eom, Y., Chung, J. W., Choi, J. and Yang, D., A Conrol-relevan Model Reducion Technique for Nonlinear Sysems, Comp. Chem. Eng., 24(2), (2000). 8. Sima, V., Sima, D. M. and Huffle, S. V., High-Performance Numerical Algorihms and Sofware for Subspace-based Linear Mulivariable Sysem Idenificaion, J. Comp. Appl. Mah., 170(2), (2004). 9. Abdi, H., Parial Leas Squares (PLS) Regression, Enc. Soc. Sci. Res. Meh., Thousand Oaks (CA), TO(2003). 10. Helmke, U. and Hüper, K., A Jacobi-ype Mehod for Compuing Balanced Realizaion, Sys. Con. Le., 39(1), 19-30(2000). 11. Chen, C. T., Linear Sysem Theory and Design, 3rd ed., OXFORD, New York, NY(1999). 12. Åsröm, K. J. and Wienmark, B., Compuer-conrolled sysems heory and design, 3rd ed., Prenice-Hall, New Jersey, NJ(1997). 13. Won, W., Lee, K. S., Lee, B., Lee, S. and Lee, S., Model Predicive Conrol of Condensae Recycle Process in a Cogeneraion Power Saion: I. Conroller Design and Numerical Applicaion, Proceedings of SICE-ICASE Inernaional Join Conference 2006, Busan, Korea, 178(2006). 14. Zhou, K., Doyle, J. C. and Glover, K., Robus and Opimal Conrol, Prenice-Hall, New Jersey, NJ(1996). o45 o k

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