A MODEL ORDER AND TIME-DELAY SELECTION METHOD FOR MIMO NON-LINEAR SYSTEMS AND IT S APPLICATION TO NEURAL MODELLING. D. W. Yu, J. B. GOMM AND D. L.

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Vome, Nmber, Pages Inse for Scenfc Comng and Informaon A MODEL ORDER AND TIME-DELAY SELECTION METHOD FOR MIMO NON-LINEAR SYSTEMS AND IT S APPLICATION TO NEURAL MODELLING D. W. Y, J. B. GOMM AND D. L. Y Absrac. A new mode order and me-dea seecon mehod for nera nework modeng of SISO non-near ssems has been recen roosed. The exenson of hs mehod o he MIMO case s deveoed n hs aer. The MIMO form of he NARX mode s consdered and he order and me-dea for each n are seeced b denfng nearsed modes of he ssem. Acaon of he mehod o a smaed connos srred ank reacor (CSTR) rocess s nvesgaed o demonsrae he seecon rocedre. Nera modes are sbseqen deveoed for he rocess based on he order and me-dea seeced sng he roosed mehod and are comared o oher nera modes wh dfferen srcres o demonsrae he effecveness of he mehod. Ke Words. Mode srcre seecon, non-near ssem denfcaon, nera nework modeng, MIMO ssems, CSTR rocess.. Inrodcon Acaon of nera neworks o modeng, conro and fa dagnoss for non-near ssems has been nensve sded n recen ears [5],[6],[9],[], [4], [6]. Nera neworks rovde a owerf modeng oo n non-near ssem denfcaon, eseca n conro-orened acaons. Comared wh he convenona onoma mode-based non-near denfcaon, on he mode order and he me-dea are needed n nera modeng, as a nera nework can reresen an non-near o an re-secfed accrac b s ooog and non-near ransformaon rovded ha here are enogh nerons n he hdden aers. Mode order and me-dea seecon mehods shod, herefore, be nvesgaed for se n nera modeng. Mode srcre for he near ARX mode of a ssem s sa chosen b checkng he rank of he nformaon or covarance marces or evaang he mode redcon error sng a gven creron sch as Akake's fna redcon error creron (FPE) []. These mehods are eas o memen and effcen for near ssems. However, s no sraghforward o a hose mehods o non-near mode srcre seecon. Genera, a non-near ssem descrbed b a NARMAX mode or a NARX mode needs o be arameerzed o rodce a near-n-he-arameers mode. Then, accordng o he rose of he mode, cod be frher aroxmaed b a se of secfc near-n-he-arameers modes ncdng a onoma mode, exonena mode, ec. However, deermnaon of each non-near fncon n he near-n-he-arameers mode s a ver comex ask. Even f on monomas are consdered for he non-near fncon, he nmber of erms can be ver arge f a he ossbe combnaons of he n and o are sed. Leonars and Bngs [] roosed a mehod o choose he sgnfcan erms from a ossbe combnaons b evaang he error redcon rao n he orhogona eas-sqares esmaon. B, he mehod needs o rea a hge nmber of Receved b he edors November 4, 004. Ths work s fnded b he EPSRC UK, nder Gran No GR/K

2 40 D. W. YU, J. B. GOMM AND D. L. YU erms n a f mode se, whch needs a arge amon of execon me and comer memor. Therefore, hs mehod s no economc for se n nera modeng. Nera nework modeng s ofen based on he ssem NARX mode srcre nsead of he near-n-he-arameers mode [],[5],[9]. Ths means ha on he eemens of he non-near fncon are necessar b no he fncon sef. Conseqen, he arameerzaon, whch s necessar n he non-near ssem denfcaon, s no exc reqred n nera modeng, on he mode order for he n and o as we as he me-dea are needed. Seecon of he mode order and me-dea for nera modeng s eqvaen o choosng he nework n node assgnmen. Research erare wde commens ha he nework n node assgnmen n nera modeng sa does no foow a se of secfc res. A common aroach s o r severa choces of he nework ns and seec he bes ones n erms of a rade-off beween mnmm redcon error and a ow nera mode comex [4],[7]. A comrehensve sd s ver me consmng and s ofen he case ha he nfence of some exermena facors, sch as he earnng rae n back-roagaon ranng or he wdh of he Gassan fncon n a RBF nework, cod ead o an narorae seecon. A sme mode order and me-dea seecon mehod for nera modeng has been roosed b he ahors n [0], based on denfng nearsed modes of a SISO non-near ssem arond severa oerang ons. The mehod s exended o he MIMO case n hs aer. An agorhm s deveoed n Secon o seec he orders of n and o and he me-deas for a MIMO non-near ssem mode over he oerang regon consdered. The acaon of hs mehod o a m-varabe CSTR rocess s resened n Secon 3 o demonsrae he acaon rocedre. The seeced order and me-dea are hen sed n nera modeng n Secon 4 o deveo a rada bass fncon (RBF) nework mode for he CSTR rocess. Ths mode s comared wh oher RBF modes whch have dfferen order and me-dea o show he effecveness of he mehod.. Mode order and me-dea seecon for MIMO non-near ssems The mode srcre seecon for nera modeng of a SISO, non-near ssem descrbed n[0] s exended o he MIMO case. Consderng he foowng MIMO form of a NARX mode () ( ) f ( ( ),..., ( n ), ( k),..., ( k n + )) e( ), where ( ) ( ) ( ) M, ( ), M ( ) ( ) m + e ( ) e( ) M e ( ) are he ssem o, n and nose resecve; and m are he nmber of he os and ns resecve; n and n are he maxmm ags n he o and n resecve; k s he maxmm me-dea n he ns; e () s assmed o be a whe nose seqence; and f ( ) s a vecor-vaed, connos non-near fncon. Ths mode s smar o he MIMO mode sed b Bngs and Chen [5] b wh he ncson of he me-dea.

3 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 4 The NARX mode () can be aroxmaed b a frs order Taor seres exanson of he non-near fncon f ( ) abo an oerang on, : [ n k,,, ) ( L + ; n ( ),,, L ] gvng an o resonse. Therefore, he o vae of he non-near fncon n eqaon () can be aroxmae exressed as () () $( ) () () () f e + + ψ, where (3) ψ ψ ψ () () (). ψ( ) s gven b (4) ψ( ) [ ( ),..., ( ), ( ),..., ( ) ] n k k n T T T T T +, and f s he vae a oerang on of he Jacoban marx of f ( ) wh resec o s varabes and can be reresened b (5) f a a b b n n,,...,,,..., Θ wh he eemen marces (6) f f f f f a ) ( ) ( ) ( ) ( ) ( L M L M L, n,, L. (7) k m f k f k m f k f k f b ) ( ) ( ) ( ) ( ) ( L M L M L, n,, L The regresson vecor ψ( ) n eqaon () s of he foowng form

4 4 D. W. YU, J. B. GOMM AND D. L. YU ( ) ( ) M ( n ) ( n ) (8) ψ( ) ( k) ( k) M ( k n + ) ( k n. + ) The nearsed mode can hen be wren as (9) () () Θ ψ () + e (). If he oerang on s chosen o be a ssem eqbrm on, whch s sa for he denfcaon of a near mode, where ( k) L ( k n + ), ( ) L ( n), hen he nearsed mode a he eqbrm on, :(, ), s of he form (0) () Θ ψ () + e (). Eqaon (0) has he form of a near MIMO ARX mode whch s vad oca for sma devaons arond he eqbrm on. I s moran o noce ha a he erms n he regresson vecor, ψ( ), of he nearsed ARX mode (0) are resen n he NARX mode eqaon (). I foows ha a erms denfed n a nearsed ARX mode arond an oerang on are ncded n a NARX mode of he ssem, whs a erms n a NARX mode are ncded n s nearsed mode rovded ha ever ara dfferenae s no zero. Ths s he reaonsh beween a NARX mode and s nearsed ARX mode. Hence, hs reaonsh can be sed o denf he mode order and he me-dea for a non-near ssem b sm denfng he mode order and he me-dea of s nearsed modes. I s ossbe ha a regresson erm n he NARX mode ma no aear n an denfed ARX mode f he corresondng eemen n f s zero or cose o zero. However, he kehood of hs occrrng can be redced b denfng ARX modes a more han one oerang on. To hs on, he mehod for MIMO ssems has been resened n vecor form. Eqaons () o (0) rovde a basc formaon for he mode order and me-dea seecon agorhm. However, he mehod can be memened more eas b decomosng he MIMO ssem mode () no MISO sb-ssem modes and reang each sb-ssem searae, () ( ) f ( ( ), L, ( n ), L, ( ), L, ( n ), ( k ), L

5 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 43 ( k n + ), L, m( k ), L, m( k n + )) + e ( ),, L, m m m where n and n are he orders for he n and o n he h h sb-ssem resecve; k s he me-dea of he n n he sb-ssem and f ( ) h h h s he non-near fncon n he sb-ssem. Corresondng, he nearsed mode eqaon (0) arond he sead-sae oerang on,, s aso decomosed no () () θ ψ () + e (), L,, h where θ s he row of he marx Θ, whch s gven n he dea θ f, L, ( ) f f, LL,, L, ) ( ) ( n ( n f, ) f f (, L, k ) ( k n, LL, + ) m f ( k m, L, ) m ( k f m n m + ) and ψ () s ψ L LL L ( ) ( ),, ( n ),, ( ),, ( n ), ( k ), L, ( k n + ), LL, ( k ), L, ( k n + ). m m m m m m From he defnon of ψ () above, s observed ha n each MISO sb-ssem he oher os of he ssem are he ns of he sb-ssem mode, Hence, he order and me-dea shod aso be seeced for hese o erms. In he MIMO ssem mode defnon of eqaon (), he noaons, m T and n are he maxmm orders of he mode n and o resecve, b he ndvda orders for dfferen ns and os ma be ess han hese wo vaes. Therefore, he mode orders ms be seeced wh resec o each n and o whn he regon of ess han or eqa o he maxmm orders consdered. Besdes, he me-deas ma aso be dfferen for dfferen ns and os, hs, he same consderaon for he mode orders shod aso be aed o he me-dea seecon. For MIMO modes he nmber of combnaons of dfferen orders and me-deas n

6 44 D. W. YU, J. B. GOMM AND D. L. YU for dfferen ns and os consdered drng he seecon hase can be excessve. Redcon of he members n he mode se herefore shod be nderaken o make he seecon easer. In hs aer, he seecon rocedre consss of wo ses: one for me-dea seecon and he oher for mode order seecon. Frs, he orders for a ns and os are fxed a vaes sffcen hgh sch ha a re orders for he ssem ns and os are covered. The frs mode se s hen formed nvovng a he combnaons of he me-deas for dfferen ns and os. An oma mode s seeced from hs mode se as a comromse beween mnmm redcon error and mode comex, based on he arsmon rnce. Second, he me-deas for he ns and os are fxed o hose chosen n he frs se and hen a second mode se s formed nvovng a he combnaons of orders for dfferen ns and os. An oma mode s hen seeced from he second mode se accordng o he same re as n he frs se. Ths rocedre grea redces he members n he mode ses hs, redcng he amon of comaon reqred. The mode orders and he me-deas chosen, sng he mehod descrbed above, are on vad arond he oerang on. The same rocedre shod be aed o a he oerang ons secfed. Fna, orders and me-deas whch cover hose seeced for he nearsed modes denfed a a he oerang ons are chosen for NARX modeng. Based on he above anass, he rocedre of seecng he mode order and he me-dea for a MIMO non-near ssem s oned beow. Seecon Procedre: Se. Choose NARX mode () for a gven MIMO non-near ssem. Choose he oerang ons dsrbed n he oerang regon of neres. Se. Decomose he MIMO ssem mode no MISO sb-ssem modes and choose he maxmm order for he n and o and he me-dea, n n k,,, for each MISO mode accordng o re-knowedge avaabe for he ssem. Se 3. Form a mode se for each nearzed sb-ssem, n whch each mode has fxed, maxmm orders b dfferen me-deas for dfferen ns and os, ha s, n n, n n and k k. Idenf he modes n he se sng a near denfcaon echnqe. Seec an oma mode from he se b comarng he Akake's FPE measre for each mode.

7 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 45 Ths s he seecon of he me-dea. Se 4. Form a mode se for each nearsed sb-ssem n whch each mode has he fxed me-dea seeced n Se 3 b dfferen orders for he n and o, ha s, n n, n n and fxed k. Idenf he modes n he se sng a near denfcaon echnqe. Seec an oma mode from he se b comarng he Akake's FPE measre for each mode. Ths s he seecon of he ssem order. Se 5. Reea Se 3 o Se 4 for a he oerang ons chosen. Comare he chosen near mode orders and he me-deas for a he oerang ons,. The NARX mode order and he me-dea shod be chosen o ncde a denfed erms n he seeced near ARX modes over he oerang regon consdered. I shod be noed ha he nearzaon dscsson n hs mehod has no assmed an arcar non-near denfcaon echnqe. Therefore, he mehod s aso acabe o an aroach sed o aroxmae he NARX mode, sch as dfferen es of nera nework, a fzz mode or onoma exanson. I s known ha, n nera nework modeng, a consan or near consan n can be reresened b a bas n. On he oher hand, f he ara dfferenaon of he non-near fncon, f () n eqaon (), wh resec o one of s varabes s zero, hen he fncon w no change aong he drecon of hs varabe. I foows ha even f he mode order and me-dea denfed sng hs mehod are no he re ones of he non-near ssem, becase of mssng erms cased b zero or near zero graden over he oerang regon consdered, he order and me-dea are s arorae for nera modeng as he effec of he mssng erms can be reazed b he bas n of he nera nework. 3. Mode order and me-dea seecon for a MIMO CSTR rocess The mode order and me-dea seecon mehod descrbed n Secon s aed o a non-near, mvarabe CSTR rocess o demonsrae he seecon rocedre and he effecveness of he mehod. I s foowed b he modeng of he rocess sng one of he ossbe negen mehods, nera neworks, accordng o he order and he me-dea seeced sng he roosed mehod. 3. The CSTR rocess The CSTR rocess nvesgaed here s a ca chemca rocess emoed as a es bed b man researchers. The rocess s shown n Fgre and s smar o ha descrbed b Foss and Johansen [8]. In Fg., (T), (A) and (C) denoe ransdcers, acaors and conro, resecve. The rocess consss of a smaed connos srred ank reacor n whch a second order endohermc chemca reacon A B akes ace. The reacor eve s mananed a a consan se on b a dga PI conroer. The rocess was smaed sng SIMULINK accordng o he foowng mass and energ baances,

8 46 D. W. YU, J. B. GOMM AND D. L. YU (3) A dh d h q, R v (4) Ah dc d A A A 0 0 A A r c q c q Ahk c ex( E / RT ), (5) r ρrcah dt r d ρ c ( q T q T ) HAhk c ex( E / RT ) + U ( T T ) U ( T T ), r r 0 r 0 A A r h r r x h (6) ρ h Vc dt h h d QU T T ( h r ). q(a) T(T) sr ca(t) h(t) h(c) ca(t) Rv(A) Th(T) Tr(T) Q(A) Fg. Smaed CSTR rocess Descrons of a qanes n eqaons (3)-(6) are gven n he Aendx. The man rocess dsrbances were smaed as zero-mean Gassan dsrbed fcaons on he nfow rae, nfow concenraon and he nfow emerare. Measremen nose was smaed as zero-mean Gassan whe nose added o he reacor qd eve, concenraon and emerare. Two ns and wo os are chosen for he rocess o be q Q c A,. Tr Snce he qd eve, h, s no coed wh he concenraon and emerare n eqaon (3) and s mananed consan, he qd eve dnamcs s sm a frs order ssem, and herefore he modeng of he qd eve s no consdered here. The rocess s decomosed no wo MISO sb-ssems of he concenraon and he emerare, whch can be reresened n he foowng form ca f ( q, Q, Tr, ), Tr f ( q, Q, ca, ).

9 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 47 From eqaons (4) and (5) s known ha non-near s nvoved n boh he dnamcs and sead saes of he reacor concenraon and emerare. In he foowng sb-secon, he roosed mehod w be sed o seec he order and me-dea for NARX modeng of he CSTR rocess. 3. Order and me-dea seecon The order and me-dea seecon mehod s aed o he CSTR rocess. The seecon foows he rocedre gven n Secon. Frs, wo oerang ons for each of he wo ns are chosen o be ( q ) 0.5, 3.5 /s for he oerang range (, 4) /s and ( ) 400, 600 kw for he oerang range (300, 700) kw. Q 0 Conseqen, for oerang ons are formed n he wo dmensona n sace from he combnaons of he oerang ons for each n varabe: U A each oerang on, n and o daa were coeced from he SIMULINK mode of he CSTR rocess. In order o mnmze he excaon of he rocess non-near, for near denfcaon, he excaon sgna for each oerang on s chosen o be a random bnar seqence (RBS) wh sma amde n he range where 0 0 ( 0. 0) ( ), 0 0 s he eemen of U a an oerang on. I was consdered ha an RBS wh arge amde wod exce he non-near of he rocess, so ha he denfcaon of he nearsed eqaon wod no be recse. On he oher hand, f he amde of he RBS excaon sgna was chosen oo sma, he sgna-o-nose rao wod be ow de o he dsrbances and nose sbeced b he rocess, so ha he denfcaon of he nearsed eqaon was aso no recse. Consderng he wo asecs of nqr, a comromse RBS amde was chosen as he above. The samng nerva was chosen as T 00sec accordng o he dnamcs of he rocess where a re of aroxmae /0 of he rse me o a se n was aed. Two hndred sames of rocess n and o daa were coeced a each oerang on and sed n he denfcaon of he nearsed modes. The MIMO CSTR rocess was decomosed no wo MISO sb-ssems, formang he foowng mode eqaons, (7) c () f ( c ( ), L, c ( n ), T( k ), L, T( k n + ), A A A ca r Tr r Tr ca q ( kq ), L, q ( kq nq + ), Q( kq), L, Q( kq nq + )) e ( ), + (8) T() f ( T( ), L, T( n ), c ( k ), L, c ( k n + ), r r r Tr A ca A ca ca

10 48 D. W. YU, J. B. GOMM AND D. L. YU q ( kq ), L, q ( kq nq + ), Q( kq), L, Q( kq nq + )) + e( ). Second, foowng Se of he seecon rocedre, he maxmm vaes of, n n and k were chosen o be (9) ( n ) max 3, ( n ) max 3, k max 3. Foowng Se 3, for he nearsed form of he MISO sb-ssem mode (7), a mode se, Se, was formed b fxng he order of n and o o be he maxmm vaes and he me-dea o be ess han or eqa o he maxmm, name, n ca (0) 3, n Tr 3, n q 3, n Q 3, k Tr 3, k q 3, k 3. Wh hs choce, he mode se for each oerang on hen has 7 modes. The bach eas sqares agorhm was sed o esmae he arameers for each mode n mode Se, and he Akake's FPE obaned for each mode s dsaed n Fg.. The for crves n Fg. are for he for oerang ons. I can be seen n Fg. ha for he for oerang ons, mode nmbers o gve he owes FPE vaes. Therefore, mode nmber was chosen whch has he me-dea as beow () k Tr, k q, k Q. Q. x 0-9 me dea seecon for reacor concenraon.9 Akakes FPE mode nmber n he mode se Fg. Akake's FPE for reacor concenraon modes n Se a for oerang ons A smar rocedre was aed o he nearsed form of he MISO sb-ssem mode (8). A mode se, Se, was formed conanng he same 7 modes as sed n mode Se. The Akake's FPE vaes for he dfferen modes n mode Se for he for oerang ons are dsaed n Fg.3. I can be seen ha mode nmbers o 6 have he owes FPE vaes and herefore, he me-dea n mode nmber was agan chosen () k ca, k q, k Q.

11 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 49 The nex se s o choose he mode orders for he n and o. Foowng Se 4, mode Se 3 was formed for he reacor concenraon b fxng k Tr k q,, k Q accordng o () and changng he order whn nc A 3, n Tr 3, n q 3, n Q 3. Ths mode se has 8 modes. 0. me dea seecon for reacor emerare 0. Akakes FPE mode nmber n he mode se Fg.3 Akake's FPE for reacor emerare modes n Se a for oerang ons.7 x 0-9 order seecon for reacor concenraon.65 Akakes FPE Mode nmber n he mode se 3 Fg.4 Akake's FPE for reacor concenraon modes n Se 3 a for oerang ons For each oerang on he arameers of hese modes were esmaed sng he same ses of n-o daa as for he me-dea seecon. The Akake's FPEs obaned are dsaed n Fg.4.

12 50 D. W. YU, J. B. GOMM AND D. L. YU.68 x magnfed vew of Fg Akakes FPE mode nmber n he mode se 3 Fg.5 Magnfed vew of he FPEs for mode se 3 For he for oerang ons, he FPEs obaned mnmm vaes on mode nmber 7. Ths can be seen more cear n Fg.5 whch s a magnfed grah. Mode nmber 7 has he order (3) n ca, n Tr, n q 3, n Q. Smar, based on he nearsed eqaon of he MISO sb-ssem mode of reacor emerare (8), he orders of n and o were seeced. Mode se 4 was formed b fxng he me-dea k c, k A q, k Q accordng o () and changng he n and he o orders whn n Tr 3, n ca 3, n q 3, nq 3. Therefore, mode se 4 aso has 8 modes. Akake's FPE for each esmaed near mode n Se 4 are dsaed n Fg.6 for he for oerang ons..9 Fg.7 Order seecon for ank emerare.8.7 Akakes FPE Mode nmber n he mode se 4 Fg.6 Akake's FPE for reacor emerare modes n Se 4 a for oerang ons. Fg.6 shows ha mode nmber 40 obans he mnmm FPE vae for he for oerang ons, whch has n and o orders (4) n Tr, n, n, n c A q Q.

13 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 5 Ths can be seen more cear n Fg.7. Combnng a he erms n he nearsed sb-ssem modes seeced a each oerang on gves he me-deas (), () and he orders (3), (4) for NARX modeng of he reacor concenraon and emerare resecve sng he roosed mehod. The me-deas seeced mach he me-deas n he orgna smaed mode. The orders chosen are aso reasonab conssen wh he dnamcs of he smaed mode. In he foowng secon, nera neworks are emoed o mode he MIMO non-near CSTR rocess based on he orders and me-deas chosen n hs secon..9 magnfed vew of Fg Akakes FPE mode nmber n he mode se 4 4. Nera nodeng of he CSTR rocess Fg.7 Magnfed vew of FPEs n mode se 4 In hs secon, nera modeng of he CSTR rocess n wo dfferen was s descrbed based on he orders and me-deas revos chosen. One wa s o se wo neworks o mode he wo MISO sb-ssems resecve (Secon 4. and 4.). The second wa s o se one nework o mode he enre MIMO ssem (Secon 4.4 ). The nework sed was a rada bass fncon (RBF) nework whch erforms a N non-near mang x R $ R va he ransformaon, T T (5) $ φ W, (6) φ φ( d) d d og( d ), (7) d x c, L, n h. nh where $ and x are he nework o and n vecors resecve; W R s he weghng marx wh eemen w denong he wegh connecng he h hdden node o o he h nework o; φ R n h s he o vecor of he

14 5 D. W. YU, J. B. GOMM AND D. L. YU non-near fncon n he hdden aer, denoes he eemen arra mng oeraon; c n R h h s he cenre vecor; n h s he nmber of nodes n he hdden aer; d s he eemen of vecor d. The weghng marx, W, s comed sng a recrsve eas-sqares agorhm (RLS) (Lng and Sodersrom [3]) sch ha he mode redcon error s mnmzed. In order o mrove he accrac of he nera mode, normazaon s made o he ranng daa as we as he es daa n he foowng wa, h x ( x x )/ σ ( ), nor where x ra and σ x( ra) are he mean vae and he sandard devaon of he ranng daa n, resecve. When he nework s sed afer ranng, x ra and ra x ra σ x( ra) are sed o resore he nera mode o sng a reverse forma $ $ σ ( ) + x. nor x ra ra The excaon sgna sed o generae he rocess n-o daa for non-near modeng was desgned o cover he whoe oerang sace. Here, a random se n seres wh he amde nform dsrbed n he regon of q 4( / s) 300 Q 700( kw ) and wh he hod on me nform dsrbed n he range of [,7] was sed. Ths excaon sgna was sed nsead of he more common random amde seres (RAS) wh a fxed hod on me becase a varng hod on me w exce more nformaon of he rocess dnamcs. The rocess dsrbances and measremen nose are smaed b he same whe nose as descrbed n Secon 3.. The excaon sgnas and os of he rocess were coeced for 900 sames wh he samng nerva 00 sec, whch s he same as ha sed for he order and me-dea seecon. The frs 800 daa sames were sed o ran he RBF neworks whe he as 00 sames of daa were sed for cross vadaon of he nera modes. A neworks had 40 ceners whch were chosen sng he k-means cserng mehod. 4. Modeng reacor concenraon Accordng o he order and me-dea chosen for he reacor concenraon n Secon 3, k Tr n q 3, nq, k q, k n eqaon () and n Q c A, n Tr,, ( 3 ) n eqaon (3), he n vecor o he nework s

15 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 53 corresondng formed as beow wh reference o eqaon (7), T x( ) [ c ( ), T ( ), q ( ), q ( ), q ( 3), Q( )]. A r In he remanng of he aer, he n vecors for he oher nera modes w be formed accordng o he seeced order and me-dea n he same wa as he above. Therefore, he formaon of he n vecors for he oher neworks w no be gven and on he order and me-dea w be secfed. The arge of he nework o was c A (). Therefore he RBF nework for he concenraon sb-ssem mode has he srcre 6:40: (6 ns, 40 hdden nodes, o). Afer ranng, he es daa s aed o he nera mode. The nera mode o and he rocess o for he reacor concenraon 'o' s he nera mode o. c A are dsaed n Fg.8, where he crve wh 0.09 measremen and RBF mode o of reacor concenraon reacor concenraon ca (mo/) samng me Fg.8 Nera mode o and rocess o for reacor concenraon The fness beween he wo crves s measred sng an ndex of he mean-sqare error (MSE) whch s defned as N MSE ( ) ( ( ) $( )) N where N s he nmber of daa ons. The MSE for he reacor concenraon s MSE(c )7.8846e-8. A,

16 54 D. W. YU, J. B. GOMM AND D. L. YU 380 measremen and RBF mode o of reacor emerare 370 reacor emerare Tr (K) samng me Fg.9 Nera mode o and rocess o for reacor emerare 4. Modeng reacor emerare The nera mode for he reacor emerare sb-ssem was se foowng he same rocedre for he reacor concenraon above. Accordng o he order and me-dea chosen sng he roosed mehod, k ca, k q k Q, n eqaon () and n Tr, n ca, n q, n Q, ( ) n eqaon (4), he n vecor o he nework was formed wh reference o eqaon (8). The arge of he nework o was Tr () gvng a nera mode srcre of 7:40:. The nera mode o and he rocess o for he reacor emerare are dsaed n Fg.9, where he crve wh "o" s he mode o. The MSE for he emerare s cacaed as MSE ( T r ) Vadaon of mode order Correc or oma mode order and me-dea s necessar for a nera mode o reab reresen a rocesses dnamcs. If he order chosen for nera modeng s ower han he oma one, he modeng error w be arger de o defcenc of he necessar nformaon. On he oher hand, f he order chosen s hgher han he oma one, ahogh he nformaon s sffcen, he redndan ns w ncrease he nmber of hdden nodes needed n he nework, so ha onger comng me and more comer memor ms be aken. To confrm ha he order chosen for he CSTR rocess s oma n erms of achevng a ow mode redcon error wh a mnmm nmber of ns, anoher wo nera sb-ssem modes were raned for each sb-ssem, one wh ower order, whch s referred o as he defcen order

17 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 55 mode, and he oher wh hgher order, whch s referred o as he redndan order mode. The me-deas for hese modes remaned he same as n he oma order modes (eqaons () and ()). For he reacor concenraonc A, he oma order chosen for c A, T r, q, Q was ( 3 ). The defcen order mode sed an order of ( ) and he redndan order mode sed an order of ( 3 ). For he reacor emerare, he oma order was chosen as ( ) for T r, c A,, Q. The defcen order mode sed an order of ( ) and he redndan order mode sed an order of ( 3 ). The same ses of ranng and es daa and he same nmber of ceners were sed for he wo addona modes of each varabe. The MSEs of he sb-ssem mode os for he wo varabes are dsaed n Tabe. q Tabe Comarson beween MSEs of dfferen nera sb-ssem modes Concenraon order MSE Temerare order MSE oma order mode e-8 oma order 3.0 mode defcen order.0055e-7 defcen order mode mode redndan order e-8 redndan order mode mode From he comarson of he oma order mode wh he defcen and he redndan order modes for boh he reacor concenraon and he reacor emerare (Tabe ), can be seen ha he oma order modes chosen sng he roosed mehod have mnmm MSE. The defcen order modes have maxmm MSE de o nsffcen nformaon. The redndan order modes have smaer MSE o hose of he corresondng defcen order modes b are s arger han he oma order modes. A ossbe reason for he arger MSEs s ha he redndan order ncreases he nmber of ns so ha he nmber of nework ceners ma no be enogh for he ncreased n sace. Ahogh ncreasng he nmber of hdden nodes of he redndan order modes can redce he modeng error; comng me, comer memor and eseca he dffc of he mode generazaon w corresondng ncrease (Pschogos and Ungar, [5]). 4.4 A m-o nera mode for he enre ssem In he revos sb-secons, wo neworks are emoed o mode he wo sb-ssems searae. Par of he nformaon n he n sace can be shared b boh of he neworks. Ths, modeng he enre ssem sng a m-o nera nework based on he oma order and me-dea seeced has aso been nvesgaed and s descrbed beow. In he m-o nera mode, he order for c A ( 3 ) and he order for ( ) were combned and a mnmm combnaon was chosen as ( 3 ) for T r

18 56 D. W. YU, J. B. GOMM AND D. L. YU c A, T r, q and Q. Ths mode s referred o as he oma order m-o nera mode. Becase he me-dea for he n and o are he same for he wo sb-ssems, hs me-dea was nara chosen for he m-o nera mode. The nmber of ceners was chosen as 60 and he arge of he nework o was T [ c ( ), T ( )]. So he nework srcre was 8:60:. The same daa ses for A r boh ranng and esng of he sb-ssem modes were sed. The mode os and he rocess os for he reacor concenraon s dsaed n Fg.0 and of he reacor emerare s dsaed n Fg., where crves wh "o" are mode os. The MSEs for he concenraon and emerare are resecve, MSE(c A )7.8846e-8 and MSE(T r ) The mean-sqare errors b he oma wo-o RBF mode are beer han hose b he defcen order and redndan order sb-ssem modes b s sgh worse for he reacor emerare han ha b he oma snge-o sb-ssem mode (Tabe ). The m-o mode has a arger hdden aer han he ndvda sb-ssem modes, de o he ncrease n he nework n sace. However, for modeng he enre CSTR rocess, he oma m-o mode acheves redcon accrac comarabe o he wo snge-o sb-ssem modes wh 60 fewer arameers han he combned sb-ssem modes measremen and RBF mode o of reacor concenraon reacor concenraon ca (mo/) samng me Fg.0 M-o nera mode o and rocess o for he concenraon

19 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS measremen and RBF mode o of reacor emerare 370 reacor emerare Tr (K) samng me 5. Concsons Fg. M-o nera mode o and rocess o for he emerare A mode order and me-dea seecon mehod for MIMO non-near ssems s deveoed. A reaonsh beween he NARX mode of a non-near ssem and s nearsed modes s nvesgaed and emoed o seec he oma mode order and me-dea. No arcar non-near aroxmaon s assmed n he seecon mehod hence, he mehod can be sed for an dnamc non-near fncon aroxmaon echnqe. The mehod was aed o he nera modeng of a MIMO non-near CSTR rocess based on a smaon of he rocess connos dfferena eqaons. A nera nework mode was deveoed wh he oma order and me-dea seeced b he roosed mehod, and was comared wh defcen order and redndan order nera nework modes raned and esed sng he same ses of n-o rocess daa. The MSE of he mode redcon errors ndcae he effecveness of he mehod for seecng oma order and me-dea n erms of achevng a ow redcon error wh ow mode comex. References [] Akake, H., 974. A new ook a he sasca mode denfcaon. IEEE Trans. Aomac Conro, Vo.9, [] Bha, N. and McAvo, T.J., 990. Use of nera nes for dnamc modeng and conro of chemca rocess ssems. Comers and Chemca Engneerng, Vo.4, No.4/5, [3] Bngs, S.A. and Chen, S., 989. Exended mode se, goba daa and hreshod mode denfcaon of severe non-near ssems. In. J. Conro, Vo.50, No.5, [4] Bngs, S.A., Janadden, H.B. and Chen, S., 99. Proeres of nera neworks wh acaons o modeng non-near dnamca ssems. In. J. Conro, Vo.55, No., [5] Chen, S., Bngs, S.A. and Gran, P.M., 990a. Non-near ssem denfcaon sng nera neworks. In. J. Conro, Vo.5, No.6, 9-4. [6] Chen, S., Bngs, S.A., Cowan, C.F.N. and Gran, P.M., 990b. Pracca denfcaon of NARMAX modes sng rada bass fncons. In. J. Conro, Vo.5, No.6, [7] Doher, S.K., Gomm, J.B. and Wams, D., 997. Exermen desgn consderaons for non-near ssem denfcaon sng nera neworks. Comers and Chemca Engneerng, Vo., No.3,

20 58 D. W. YU, J. B. GOMM AND D. L. YU [8] Foss, B.A. and Johansen, T.A., 99, An negraed aroach o on-ne fa deecon and dagnoss -- ncdng arfca nera neworks wh oca bass fncons. Proc. IFAC Smosm on On-ne Fa Deecon and Servson n he Chemca Process Indsres, Newark, USA, Ar -4, 07-. [9] Gomm, J.B., Wams, D., Evans, J.T., Doher, S.K. and Lsboa, P.J.G. 996a. Enhancng he non-near modeng caabes of MLP nera neworks sng sread encodng. Fzz Ses and Ssems, Vo.79, No., 3-6. [0] Gomm, J.B., Y, D.L. and Wams, D., 996b. A new mode srcre seecon mehod for non-near ssems n nera modeng. Proc. UKACC In. Conf. Conro'96, -5 Se., Exeer, UK, [] Hn, K.J., Sbarbaro, D., Zbkowsk, R. and Gawhro, P.J., 99. Nera neworks for conro ssems -- a srve. Aomaca, Vo.8, No.6, [] Leonars, I.J. and Bngs, S.A., 987. Mode seecon and vadaon mehods for non-near ssems. In. J. Conro, Vo.45, No., [3] Lng, L. and Sodersrom, T., 983. Theor and racce of recrsve denfcaon. MIT Press, Cambrdge MA. [4] Narendra, K.S. and Parhasarah, K., 990. Idenfcaon and Conro of dnamc ssems sng nera neworks. IEEE Trans. Nera Neworks, Vo., 4-7. [5] Pschogos, D.C. and Ungar, L.H., 994. SVD-NET: an agorhm ha aomaca seecs nework srcre. IEEE Trans. Nera Neworks, Vo.5, No.3, [6] Y, D.L., Sheds, D.N. and Dae, S., 996. A hbrd fa dagnoss aroach sng nera neworks. Nera Comng & Acaons, Vo.4, No., -6. Aendx The meanng of he qanes n he CSTR mode of eqaons (3) -- (6) are defned as foows. q reacor nfow h qd eve q 0 reacor ofow V h effecve vome of R v c A ofow e ressance de o conro vave concenraon of comonen A n reacor hea exchanger A reacor cross-secona area c A concenraon of comonen A n nfow k 0 freqenc facor E A acvaon energ R nversa gas consan T r reacor emerare ρ r mass dens n reacor c r secfc hea caac n reacor ρ h mass dens n hea exchanger c h secfc hea caac n hea exchanger T nfow emerare H reacon energ

21 A MODEL ORDER AND TIME-DELY SELECTION METHOD FOR MIMO SYSTEMS 59 U hea ransmsson coeffcen T h emerare of hea beween hea exchanger medm exchanger and reacor T x envronmen emerare U hea ransmsson coeffcen Q heang ower beween reacor and envronmen Dngwen Y receved he B. Eng degree n rocess conro and M.Sc degree n ssem engneerng from Beng Unvers of Chemca Technoog(BUCT), Chna n 98 and 987, and he PhD degree from Nongham Tren Unvers, U.K. n 000. Dr. Y was a ecrer a BUCT from 987 o 994 before he came o Unvers of Saford as a vsng researcher n 994. He worked a Lveroo John Moores Unvers as a os-docora researcher from 00 o 003. He s crren a rofessor n Norheasern Unvers a Qnhangdao, Chna. Hs research neress ncde adave and redcve conro, nera nework, omzaon, rocess modeng and smaon, and fa deecon. J. Barr Gomm receved he BEng frs cass degree n eecrca and eecronc engneerng n 987 and he PhD degree n rocess fa deecon n 99 from Lveroo John Moores Unvers (JMU), UK. He oned he academc saff n he Schoo of Engneerng a JMU n 99 and s a Reader n Inegen Conro Ssems. He was coedor of he book Acaon of Nera Neworks o Modeng and Conro (London, UK: Chaman and Ha, 993), Ges Edor for seca sses of he ornas Fzz Ses and Ssems (Amserdam, he Neherands: Esever, 996) and Transacons of he Inse of Measremen and Conro (London, UK: InsMC, 998). He has bshed more han 00 aers n nernaona ornas and conference roceedngs. Dr Gomm s a member of he IEE, IEEE and he IEE Technca Advsor Pane for he Conces for Aomaon and Conro Professona Nework. Hs crren research neress are n arfca negence mehods for rocess modeng, conro and fa dagnoss. Dng Y receved he B. Eng degree from Harbn Unvers of Cv Engneerng, Chna n 98, he M. Sc degree from Jn Unvers of Technoog (JUT), Chna n 986, and he PhD degree from Covenr Unvers, U.K. n 995, a n Eecrca Engneerng. Dr. Y was a ecrer a JUT from 986 o 990 before he came o Unvers of Saford as a vsng researcher n 99. He hen worked a Lveroo John Moores Unvers as a os-docora researcher snce 995 and became a ecrer n 998. He s crren a Reader n Process Conro. Hs crren research neress ncde fa deecon and fa oeran conro of bnear and nonnear ssems, adave nera neworks and her conro acaons, mode redcve conro for chemca rocesses and engne ssems.

22 60 D. W. YU, J. B. GOMM AND D. L. YU Dearmen of Aomaon, Norheasern Unvers a Qnhangdao, Qnhangdao, Hebe Provnce, P.R. Chna, ema: dw@ma.neq.ed.cn Dearmen of Engneerng, Lveroo John Moores Unvers, Brom sree, Lveroo, L3 3AF, UK ema: d.@vm.ac.k