AN EXTENDED MPC CONVERGENCE CONDITION

Transcription

1 Latn Amercan Aled esearch 36:57-6 (6 AN EXENDED MPC CONVEGENCE CONDIION A. H. GONZÁLEZ and. L. MACHEI Insttuto de Desarrollo ecnológco ara la Industra uímca, INEC (UNL - CONICE alegon@cerde.gov.ar lmarch@cerde.gov.ar Abstract Nomnal convergence of Constraned Model Predctve Control has been extensvely analyzed n the last ffteen years. he ncluson of a termnal constrant nto the otmzaton roblem and the exanson of the redcton horzon u to nfnty are the man strateges already roosed n order to acheve the desred stablty. However, when a model s used n whch the nuts are n the ncremental form, these strateges tend to be nfeasble. hs aer extends the contractng constrant dea by ncludng a smle-to-aly and less restrctve new set of constrants nto the otmzaton roblem, to allow nomnal convergence. Keywords Predctve Control, State Sace Model, ecedng Horzon, Pseudo Cost Functon, Infnte Constrant. I. INODUCION he ecedng Horzon dea uses an on-lne otmzaton that udates the manulated varable at each samle tme. In the tracng roblem, the dfference between the redcted future oututs and the set ont s the cost functon of a mnmzaton, and nomnal stablty reduces tself to ensure the convergence of the successve otmal costs. Snce consecutve otmzaton roblems are n essence dfferent, t s not smle to comare two consecutve cost functons (the redcton horzon recedes n tme so the successve cost functons dffer from each other n ther locaton. When an nfnte horzon (IHMPC s used, the end of the consecutve horzons does not vary whle the begnnng ncreases. Mang use of the Bellman s rncle of otmalty, whch states that the tal of any otmal traectory s tself the otmal traectory from ts started ont, the convergence can be guaranteed. See Maceows (, However, when an ncremental form of a model s used, the effect of the ntegratng modes at the end of the control horzon must be zeroed n order to mae the nfnte oen-loo cost bounded (odrgues and Odloa, 3. Smlar to the case of termnal constrants, ths roblem tends to be nfeasble and slac varables must be added (odrguez and Odloa 3, Odloa 4. Followng the strategy develoed by González, et al. (4, ths aer extends the dea of ncludng a set of contractng constrants to acheve outut convergence. In the mentoned wor, a relmnary study of convergence condtons that s dfferent from the 57 classcal method has been made. However, convergence could not be roerly roved. Now, two mrovements are made: the convergence of the method s roved, and the whole formulaton s translated nto a State Sace Model n order to tae advantage of ts well-nown benefts. II. BASIC FOMULAION OF MPC Consder a system wth nu nuts and ny oututs and consder an otmzaton cost functon as the sum of the future errors nsde the redcton horzon lus a manulated varable enalzaton, namely e + / e + / +Δu Δu ( where e( + / y( + / ( Δ u (, (,, ( Δu Δ u + Δ u + m ny and y( + / s the redcted outut, nu Δ u( + are the manulated varable ncrements, s the redcton horzon, m s the control horzon, ny ny nu nu and dag(,, are ostve defnte weghtng matrces, and r s the setont value. If Δu s the otmal nut ncrement vector, then / / e + e + +Δu Δu (3 s the otmal cost functon value at tme. In the same way, the otmal cost functon value at tme + wll be + e + / + e + / + +Δu Δu (4 Now, followng the dea used by awlng and Muse (993, an auxlary seudo cost functon at tme + s defned usng the otmal values of the nut changes calculated at tme : + + e( + / + e( + / + (5 +Δu Δu where e( + / s the error at tme +, calculated at tme +, usng Δu +, and he form of the seudo cost obeys to the fact that, n the nfnte horzon case, f no new control ncrements s ntroduced at +, then the otmzaton remans exactly the same from tme to tme +, excet for the startng ont.

2 Latn Amercan Aled esearch 36:57-6 (6 ( Δ u + Δ u + Δ u + Δ u + m (6 III. SAE SPACE MODEL he ntal model has the form : ( + ( + ( ( Cx( x Ax Bu y nu nx where u( and x(. Now the comonent errors e( / + + are wrtten as functons of Δ u n the case that state sace models are used. y ( + / y ( + / CdAsx( + CdAu ( ast y ( + / Δu ( / (8 Δ u ( + / + Cd A Δ u ( + m / Δu future where: A m A C As, m+. ny. nx A Cd C A B B AB B + m m AB A AB B, A m AB AB AB (7 In the same way, at tme +, the oututs are gven by ( + / + ( + 3/ + y y CAx d s ( + + y ( + + / + Note that the ntal model s wrtten n term of u. 58 ( / ( / Δ u + + Δ u CAu d ( + CA d Δ u ( + / + Δu + future where the nut u ( u( +Δ u ( was calculated wth the last otmal value Δ u ( / comuted at tme. hs corresonds to the recedng horzon dea of alyng only the frst calculated element of each otmzaton roblem. hen, a new otmzaton roblem s solved at the next tme. It can be roved that, for the undsturbed system, f Δ s used nstead of Δ, then u + u + ( + / + ( + / ( + 3/ + ( + 3/ e e e e e e ( + / + ( + / (9 therefore, the dfference of two successve cost functon values and + s gven by ( / ( / e e where e + Δu ( / e e. IV. HE NEW INFINIE CONACING CONSAIN ( An nequalty constrant that forces the cost functon to be non-ncreasng s gven by 3 + c : e + / Δu / ( + e + + / + or, tang nto account that s ostve defnte 4, : + / / + c e e ( In order to mae the deendence of the last errors on otmzaton varables exlct, t s useful to exress them n terms of Δ u : e( + / y( + / r Crow ( As x( ( + Crow A u + Crow A Δuˆ ( 3 he formulaton resented here s the same used n Maceowy. 4 Note that the constrant c n ( s more conservatve than the former.

3 A. H. GONZÁLEZ,. L. MACHEI ( + + / + ( + + / + Crow ( ˆ( ( ˆ As x + + Crow A u( + Crow ( A Δu e y r + (3 where row and row are the frst and the last bloc of matrx A s, A and A : row( As A, row( A B row ( A B [ ] row ( A A, row ( A A B s m row ( A AB AB and the hat means that the varables are otmzaton varables. Another way to fnd a general exresson for the successve contractng constrant s by movng the rows through the matrx nstead of the udatng nut and the state vector. hat s, c : Crow A x + Crow A u ( ( ( ˆ s + Crow A Δu r + Crow A x + + s ( Crow A u + Crow A Δuˆ + + (4 (5 where Δ uˆ s the vector of otmzaton varables and both, x ( and u(, are feedbac measurements. Here, t can be seen that all the necessary nformaton to wrte the nequalty constrants s n matrces A, B and C. Note that n the last exresson, matrces A s, A and A were extended n tme ( rows were added to the end of the orgnal defnton n (8. hen, the otmzaton roblem that ncludes the contractng constrant can be wrtten as: mn ˆ ˆ e + / e + / +Δu Δu Δuˆ st..: [ Δu ] ˆ Δu [ ] mn Δu max c ( + (6 Fnally, t must be shown that, f Δu + s a feasble soluton of the otmzaton roblem at tme +, then: + + (7 hus, the (global otmum obtaned n (6 wll not be worse than the one obtaned from ths artcular soluton. But the requred feasblty s not easy to obtan and, as shown n González, et al. (4, t demands an nfnte seres of (forward constrants of the followng form: ( where + are the successve seudo cost functons; that s, the seudo cost functons obtaned by the alcaton of Δu+, Δu +,, Δu +,, where Δ u + Δ u ( + Δ u ( + m ] elements he comlete MPC otmzaton roblem s then: P: mn ( / ( / ˆ ˆ e + e + +Δu Δu Δuˆ st..: [ Δu ] Δuˆ [ Δu ] mn max ( + c ( + + c c + + (9 V. DEALING WIH AN INFINIY NUMBE OF CONSAINS In the formulaton resented above, the number of constrants s nfnte, but consderng that the oen-loo system s stable, only a fnte number of constrants must be added snce the lant, wth null nut moves (from +m on, converges to a fnte value 5. Snce nut ncrements are null from +m on, the constrants at a generc tme +m+ tae the form c ˆ( ˆ m+ : C A x + m + C A Bu( + m CA xˆ( + m + C A Buˆ( + m ( whch s an exresson n terms of ˆx ( + m and uˆ ( + m only. Now, when, we have c ˆ m+ : C A Bu( + m, ( + C A Buˆ ( + m r 5 Note that, as ooste to the nfnte horzon case, the requrement s not u ( + u ra for m, but Δ u ( + ss ss for m (control horzon hyothess. In the nfnte horzon case, ths requrement s only stated to mae ossble the comutaton of the nfnte cost functon.

4 Latn Amercan Aled esearch 36:57-6 (6 snce matrx A s stable (has all ts egenvalues lower than unty, and then A s a null matrx. hs means that at tme aroachng nfnty, all the constrants are satsfed. Note that to satsfy the last equalty constrant, t s not necessary that r u ( + m uss C A B ( ass where the matrx C lant and u ( m A B s clearly the gan of the + s the last nut nected n the oen-loo redcton. It s mortant to notce that, n ste of a otental large number of contractng constrants, ths strategy s less restrctve than others, whch need the accomlshment of (. On the other hand, the weaness of ths strategy s that the mnmum number of constrants to ensure the overall convergence (n case to exst remans to be determned. In order to strengthen that the nfnte number of constrant resent n (9 stll holds when a fnte number of them s used, t s ossble to requre that the last nut n the control horzon be such that the steady state value of the oen-loo redcton be no greater than the set ont. hs condton added to a number of contractng constrants coverng the comlete dynamc order of the lant would hel to the accomlshment of the constrants for all future tme. Note that ths strategy may be a bt conservatve, but t s always feasble. VI. CONVEGENCE OF HE MEHOD Now, a matrx condton must be derved to guarantee that both, the outut error and the nut ncrements converge to zero. Suose that the roosed otmzaton roblem has a feasble soluton at each samlng tme, that s, a Δ u( / move exsts that mnmzes the cost functon. he nfnte contractng constrant ensures that the cost s non-ncreasng, + + Δu( / (3 + e ( + / e ( + + / byconstrant hen, because of ts ostve (quadratc condton, the seres of cost functons { } s bounded below by zero and therefore converges. From (3 both, the term ( / Δ u and the term ( + ( e / e /, (4 converge to zero for large. From the last one, two alternatves arse: e e for large, or + / + + / + (5 e e e c + / + + / + (6 where e c reresents a redcted outut offset. Note that the condton Δ u / (7 ( for large (whch mly, n turn, that u( / s also bounded, added to the oen-loo stablty hyothess, guarantees that the outut error converges to a bounded value. heorem If the system to be controlled s stable and roblem P s feasble at each tme ste, then, there exst matrces and, and horzon, such that the offset e c converges to zero as tends to nfnty. Proof: Suose that when - large enough- the outut error tends to ec. hen, tang nto account that the nut ncrements necessarly converge to zero, the corresondng cost (consderng a SISO case and m for smlcty s ( / ( / e + e + +Δu Δ u e A Δu ( / ( / f f ( / ( / + Δ u e U Ue + Ue c c c (8 where, U, Af C A and d e f Cd As x( + Cd Au ( reresents the vector of future errors n absence of new nut ncrements. In ths way, the (no otmal nut ncrement that gudes the system to the set onts (n steady state s gven by ( / ( ( ( u ( u +Δ u / uss G r Δ u G r u G r Gu G e c (9 where G s the lant gan, and r s the set ont. Wth ths nut ncrement (whch may not be the otmal the corresondng cost s gven by 6 6 It s suosed that the on-lne otmzaton roblem s always feasble, and then, the steady state usng the roosed nut ncrement exhbts a null cost (. + 6

5 A. H. GONZÁLEZ,. L. MACHEI ( / ( c ef AfG ec ( G ec Ue c AfG ec e + + G e + ( G ec ( U AfG ec ( G ec (( ( e U AG U AG G G e c f f c c mn e Note that e f. mn (3 Ue because t reresents the future c error n absence of new nut ncrements. hus, the condton that guarantees the convergence of the error to zero s < mn < (3 Now, let s remar de followng fact: equaton (3 can be exressed as U U> U AG U AG + G G where A f s ( f ( f Af ( CB ( C( AB+ B C A B. If matrces and are adoted such that U U G G > then, as ncreases, the last element of vector A f tends to G, and the last art of the term U Af G tends to zero. herefore, snce the left-hand sde ncreases wth whle the rght hand sde tends to a fnte value, a redcton horzon does exst that guarantees (3. hen, matrces and, and horzon exst that guarantee the convergence of the error to zero, and the theorem s roved. o summarze, f (3 s true and e c s dfferent from zero, then, n the oen-loo otmzaton roblem there are no reasons to revent Δu( / from movng to a non null value n order to elmnate e c. herefore, the null condton of Δ u (whch, n turn, s gven by the convergence of the cost functon mles, ndrectly (by means of the model, that e c s null. In other words, there are no reasons, out of nfeasblty, to mede the closed-loo otmzaton to fnd a value of u (u ss that taes the outut to ts desred value r. ecall that ths value of u s comuted tang nto account the outut feedbac and then, t wll be accurate to the actual lant., G ( s ( 9s ( 3s+ ( 5s+ Fgure shows the closed-loo resonse of the second order system for a ste change n the set ont when the constrants, defned n (9, are not ncluded n the control roblem. On the other hand, Fg. shows the ndexes, and + +, whch corresond to ths case. Due to the small outut horzon that was adoted n ths case, the closed-loo system becomes unstable. It can be seen that the ndex values volate n ths case the roosed constrant. Fgures 3 and 4 show the ste resonse and the ndexes when the nfnte constrant s used. It can be seen that ncludng the constrants defned n (9 turns the closed-loo system stable. he arameter values of the MPC for both cases (wth and wthout nfnte constrant are shown n able. able. Parameter values used n the Constraned MPC controller (n both cases: wth and wthout nfnte constrant Control Horzon (m 3 Predcton Horzon ( 8 Weght on Δ u (.5 Weght on errors ( Nº of Constrants Fgure : Outut resonse wthout nfnte constrant. + + VII. SIMULAION ESULS A numercal examle taen from Marchett, et al., (986 s used to llustrate some results; ths s a second order system wth a rght half lane zero: Fgure : Indexes wthout nfnte constrant. Only three successve ndexes are shown. 6

6 Latn Amercan Aled esearch 36:57-6 (6 Fgure 3: Outut resonse usng nfnte constrant. Marchett,. L., Mellcham, D. A. and Seborg, D. E. Predctve Control Based on Dscrete Convoluton Models, Ind. Eng. Chem. Process Des. Dev. (3, (983. Odloa D. Extended obust Model Predctve Control. AIChE ournal. 5 (8, (4. awlng,.b., & Muse, K.. he Stablty of Constraned ecedng Horzon Control. IEEE ransacton on Automatc Control 38, 5 (993. odrgues, M.A., Odloa, D. MPC for Stable Lnear Systems wth Model Uncertanty. Automatca 39, ( Fgure 4: Indexes usng nfnte constrant. Only three successve ndexes are shown. VIII. CONCLUSIONS A new MPC convergence condton s roosed when State Sace Models are used. he strategy conssts of the ncluson of a set of constrants that forces the MPC cost functon to decrease, lus an arorate selecton of cost weghtng matrces. he man contrbuton of ths aer conssts of the develoment of a detaled roof of convergence of the successve otmzng solutons, and a smle-to-aly relatonsh among the weghtng matrces, whch allows the outut offset elmnaton. Even though the theoretcal number of contractng constrants should be nfnty, the smulaton results show that, when the controlled lant s oen-loo stable, only a fnte number of them are necessary to accomlsh the desred convergence. However, the recse fnte number of constrants that guarantees the convergence and would wor as a suffcent condton was not determned. hs roblem and the robustness of the roosed rocedure n the resence of model uncertantes are consdered matter of future wors. EFEENCES González, A.H. Odloa D. and Marchett.L. An MPC Stablty Condton usng Ste esonse Model, AADECA (4. CD Verson. Maceows,.M. Predctve Control wth Constrants, England: Prentce Hall (. eceved: Setember, 5. Acceted for ublcaton: February 6, 6. ecommended by Guest Edtors C. De Angelo,. Fgueroa, G. García and. Solsona. 6