DIRECTIONAL CHANGE IN A PRIORI ANTI-WINDUP COMPENSATORS VS. PREDICTION HORIZON

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1 DIRECTIONAL CHANGE IN A PRIORI ANTI-WINDUP COMPENSATORS VS. PREDICTION HORIZON Dariusz Horla Poznan Universiy of Technology, Insiue of Conrol and Informaion Engineering Division of Conrol and Roboics, ul. Piorowo a, 6-965, Poland Dariusz.Horla@pu.poznan.pl Keywords: Absrac: Direcional change, windup phenomenon, opimal conrol, linear marix inequaliies, predicive conrol. The paper presens he correspondence in beween direcional change and ani-windup phenomenon wih respec o a priori ani-windup compensaor on he basis of MPC (simulaion resuls include plans wih no equal number of inpus and oupus). I shows wha is he excess of direcional change for consecuive predicions of conrol vecors for a given predicion horizons. 1 INTRODUCTION Taking conrol limis ino consideraion is necessary o achieve high performance of he designed conrol sysems (Horla, 6b). There are wo ways in which one can consider possible consrains a synhesis of conrollers. In he firs approach, imposing consrains during he design procedure of he conroller usually leads o difficulies wih obaining explici form of conrol laws, apar from very simple cases. The oher way is o assume he sysem is fully linear and, subsequenly, having designed he conroller for unconsrained sysem (by means of opimisaion, using Diophanine equaions, ec) impose consrains, wha would require addiional changes in conrol sysem due o presence of consrains (Horla, 7b; Öhr, ; Peng e al., 1998). The siuaion when because of consrains inernal conroller saes do no correspond o he acual signals presen in he conrol sysems is referred in he lieraure as windup phenomenon (Öhr, ). One can expec inferior performance because of infeasibiliy of compued (unconsrained) conrol signals when conrol limis are no aken ino accoun. A few mehods of compensaing he windup phenomenon from SISO framework work well enough in he case of mulivariable sysems (Öhr, ; Walgama and Sernby, 199). In such a case, apar from he windup phenomenon iself, one can also observe direcional change in he conrol vecor due o differen implemenaions of consrains, wha could affec direcion of he unconsrained conrol vecor (Horla, 4; Horla, 7a). The oher problem is, in general form, decoupling, wih respec o no equal number of conrol signals and oupu signals, when conrol direcion corresponds no only o inpu principal direcions or maximal direcional gain of he ransfer funcion marix, bu also o he degree of decoupling (Alberos and Sala, 4; Maciejowski, 1989). The problem of direcional change has been iniially discussed in (Walgama and Sernby, 199). The paper (Horla, 7a) defined he connecion of direcional change problem wih ani-windup compensaion (AWC) for sysems wih equal numer of inpus and oupus. The curren paper has been given rise by research carried ou in (Horla, 4; Horla, 7a; Horla, 7b) and exends he undersanding of aniwindup compensaion o non-square sysems wih imposed consrains, comparing conrol performance wih opimisaion-based approach, relaed o MPC (Camacho and Bordons, 1999; Doná e al., ; Maciejowski, ) ha is widely-spread and applied in he indusry. In he paper, he problem of direcional change has been sudied wih respec o opimal a priori ani-windup compensaion and differen predicion horizons. A PRIORI AWC One can perform ani-windup compensaion by incorporaing AWC implicily ino he conroller. In order o use all he advanages of such an approach (as opimaliy of he soluion, no need o design decoupling sages, ec.), le he opimal consrained conrol vecor 18

2 DIRECTIONAL CHANGE IN A PRIORI ANTI-WINDUP COMPENSATORS VS. PREDICTION HORIZON follow from wih consrains u : J (u ) = inf u D(J ) { } J (u ), (1) u j,i, α j, () where α j is an ampliude consrain of he i h elemen of conrol vecor, and u j,, D(J ) is he se of all conrol vecors such ha J has a finie value, u =. u m,, () and u j, (1 j m) comprises sequences of conrol acions applied o he j h inpu wih conrol horizon N u (Horla, 6a). The conroller is responsible for racking given reference (implici model) oupu vecor r M, wih y minimising J = p k=1 d+n u 1 (r M,k,+l y k,+l ), (4) l=d which can be presened in he sense of L norm as J = r M,+d ŷ ŷ, (5) +d +d where r M,+d = r M,1,+d r M,,+d. r M, p,+d (6) comprises vecors including reference signals known for d + N u 1 seps in advance. The vecor of predicion of plan response forced by he sough conrol sequence is compued in an ieraive manner ŷ +d = ŷ 1,+d ŷ,+d. ŷ p,+d, (7) where ŷ i,+d = Gu, and G comprises marices of plan impulse response samples. The conroller is o search for conrol vecors u j, wih 1 j m, each of hem being conrol sequences applied o he j h inpu {u j,, u j,+1,..., u j,+nu 1} in horizon N u >. Based on he superposiion rule, he decay-response vecor ŷ +d subjec o iniial condiions u k (k ) is compued ieraively alike. Having expressed (4) as ) T ) J = (Gu +ŷ r +d M,+d (Gu +ŷ r +d M,+d, (8) minimisaion subjec o consrains is equivalen o (Boyd e al., 1994; Boyd and Vandenberghe, 4) minγ s.. I u T (GT G) 1/ γ (r M,+d ŷ ) T +d (r M,+d ŷ )+ +d +(r M,+d ŷ ) T Gu +d } diag {F (1,1),..., F (1,m), } diag {F (,1),..., F (,m),, (9) where he las o LMIs define upper and lower bounds of u, and is a symmerical enry. SIMULATION STUDIES The following mulivariable CARMA plan model will be of ineres A(q 1 )y = B(q 1 )u d, (1) wih lef co-prime polynomial marices A(q 1 ), B(q 1 ), delay d = 1, wih y R p as he oupu vecor, u R m is he consrained conrol vecor (v R m will denoe unconsrained conrol vecor). The considered plans are assumed o be cross-coupled: P1 (m =, p = ) [ ] A(q ) = I+ q [ ] q.1.5, [ ] B(q ) =,.5.8 P (m =, p = ) [ A(q ) = I B(q 1 ) = [ [ ] q 1 + ], ] q, 19

3 ICINCO 8 - Inernaional Conference on Informaics in Conrol, Auomaion and Roboics P (m =, p = ) A(q 1 ) = I + B(q 1 ) = q, q 1 + The reference vecor is pre-filered by implici reference model wih characerisic polynomial marix A M (q 1 ) = (1.5q 1 )I p p, wha corresponds o closed-loop racking wih dynamics described by A M (q 1 ). Evaluaion of conrol performance conneced wih ani-windup compensaion qualiy requires following performance indices o be inroduced: J 1 = 1 N J = 1 N i = 1 N i = 1 N p N i=1=1 p N i=1=1 N =1 N =1 r i, y i,, (11) (r i, y i, ), (1) (v,i ) (u,i ) [ ], (1) ((v,i ) (u,i )), (14) where (11) corresponds o mean absolue racking error of p oupus, (1) is a mean absolue direcion change in beween compued and consrained conrol vecor, and (i) denoes angle measure of conrol vecor sequence in predicion horizon N u = i. 4 SIMULATION RESULTS For plans P1 and P he reference vecors comprise piecewise consan reference signals, whereas for P he hird oupu is o be kep a zero a all imes, wha is difficul when here is a inferior number of conrol inpus in comparison wih plan oupus. Numerical resuls of performed simulaions have been presened in Tables 1 and. The firs se of simulaions esed o wha excess he direcional change phenomenon will ake place for plans P1 P and differen predicion horizon. As i can be seen from Table 1a and Figures 1 and 4, for P1, he greaes direcional change (wih respec o unconsrained conrol vecor generaed a he same ime insan, bu no applied) akes place in he curren sample. The greaer he predicion horizon, he smaller he direcional change becomes. Since a mean angle deviaions is approx. 1 hen, one can say ha consrained conrol vecor is close o he compued unconsrained conrol vecor. This migh also ake place because of equal number of inpus and oupus, wha leads o easier decoupling. In he case of P (Tab. 1b, Fig., 5), mean angle deviaion is near he righ angle, wha corresponds o o normal vecors wih he hird componen unchanged, i.e. roaion wih respec o a fixed axis. This migh be be conneced wih plan principal direcions and wih he need o decouple oupus from inpus. Since he number of conrol inpus is greaer han plan oupus, he excessive change in direcion is needed, because one can obain beer racking performance han for m = p =. If he plan has insufficien number of conrol inpus (P, Tab. b, Fig., 6), i is impossible o assure high conrol performance and one has o cope wih poenial problem of unconrollable modes. As i can be seen, he speed of ransiens has been reduced, wha lead o beer decoupling, aiding aniwindup compensaion. In such a case, ofen direcional change is a resul of he need of decoupling. For he case of no direcional change requiremen (Tab. ), such a regime of work (presen in some applicaions in roboics, or e.g. in racking, (Öhr, )), resuls in inferior conrol performance. For P1 and increasing N u one obains performance degradaion, for P he closed-loop sysem becomes unsable (in order o decouple, he conroller would have o aler conrol direcion) he only improvemen can be observed in he case of P because of m < p (where some coupling is always presen and resuls in proporions beween conrol vecor componens ha conroller has o abide o). 5 SUMMARY As i has been shown in he paper, he problem of direcional change can be presened in a differen way for plans wih m p han in (Horla, 7a; Walgama and Sernby, 199). No allowing direcional change, may cause insabiliy in he case of unsable plans (see P), whereas for he oher cases i degrades conrol performance. Alering conrol direcion is relaed o decoupling, hus one can expecs problems wih performance for m > p and good conrol qualiy for m < p when componens of conrol vecor mus be kep in proporion (e.g., in a circular shape cuing ask) a all imes. 11

4 DIRECTIONAL CHANGE IN A PRIORI ANTI-WINDUP COMPENSATORS VS. PREDICTION HORIZON Table 1: a) p =, m =, b) p =, m =, c) p =, m =. a) N u = 1 N u = N u = N u = 4 N u = 5 J J b) N u = 1 N u = N u = N u = 4 N u = 5 J J c) N u = 1 N u = N u = N u = 4 N u = 5 J J Table : no direcional change, a) p =, m =, b) p =, m =, c) p =, m = ( denoes unsable closed-loop sysem). a) N u = 1 N u = N u = N u = 4 N u = 5 J J b) N u = 1 N u = N u = N u = 4 N u = 5 J J REFERENCES Alberos, P. and Sala, A. (4). Mulivariable Conrol Sysems. Springer-Verlag, London, Unied Kingdom. Boyd, S., Ghaoui, L. E., Feron, E., and Balakrishnan, V. (1994). Linear Marix Inequaliies in Sysem and Conrol Theory. Sociey for Indusrial and Applied Mahemaics, Philadelphia, Unied Saes of America, rd ediion. Boyd, S. and Vandenberghe, L. (4). Convex Opimizaion. Cambridge Universiy Press, Unied Kingdom. Camacho, E. and Bordons, C. (1999). Model Predicive Conrol. Springer-Verlag, Unied Kingdom. Doná, J. D., Goodwin, G., and Seron, M. (). Aniwindup and model predicive conrol: Reflecions and connecions. European Journal of Conrol, 6(5): Horla, D. (4). Direcional change and ani-windup compensaion for mulivariable sysems. Sudies in Auomaion and Informaion Technology, 8/9:5 68. Horla, D. (6a). LMI-based mulivariable adapive predicive conroller wih ani-windup compensaor. In Proceedings of he 1h IEEE Inernaional Conference MMAR, pages , Miedzyzdroje. Horla, D. (6b). Sandard vs. LMI approach o a convex opimisaion problem in mulivariable predicive conrol ask wih a priori ani-windup compensaor. In Proceedings of he 18h ICSS, pages , Covenry. Horla, D. (7a). Direcional change and windup phenomenon. In Proceedings of he 4h IFAC Inernaional Conference on Informaics in Conrol Auomaion and Roboics, pages CD ROM, Angers, France. Horla, D. (7b). Opimised condiioning echnique for a priori ani-windup compensaion. In Proceedings of he 16h Inernaional Conference on Sysems Science, pages 1 19, Wrocław, Poland. Maciejowski, J. (1989). Mulivariable Feedback Design. Addison-Wesley Publishing Company, Cambridge, Unied Kingdom. Maciejowski, J. (). Predicive Conrol wih Consrains. Pearson Educaion Limied, Unied Kingdom. Öhr, J. (). Ani-windup and Conrol of Sysems wih Muliple Inpu Sauraions: Tools, Soluions and Case Sudies. PhD hesis, Uppsala Universiy, Uppsala, Sweden. Peng, Y., Vrančić, D., Hanus, R., and Weller, S. (1998). Ani-windup designs for mulivariable conrollers. Auomaica, 4(1): Walgama, K. and Sernby, J. (199). Conidioning echnique for muliinpu mulioupu processes wih inpu sauraion. IEE Proceedings-D, 14(4):1 41. c) N u = 1 N u = N u = N u = 4 N u = 5 J J

5 ICINCO 8 - Inernaional Conference on Informaics in Conrol, Auomaion and Roboics a) b), c), d), e), 5, Figure 1: p =, m =, a) N u = 1, b) N u =, c) N u =, d) N u = 4, e) N u = 5. a) b) 1 5, c) , d) 1 5, , e) 1 5, , Figure : p =, m =, a) N u = 1, b) N u =, c) N u =, d) N u = 4, e) N u = 5. 11

6 DIRECTIONAL CHANGE IN A PRIORI ANTI-WINDUP COMPENSATORS VS. PREDICTION HORIZON a) 1, 5 5 b) 1, 5 5, 5 5 c) 1, y 1, y, Figure 4: p =, m =, N u =., d) 1, 5 5, y 1, y, u, Figure 5: p =, m =. e) 1, 5 5, , Figure : p =, m =, a) N u = 1, b) N u =, c) N u =, d) N u = 4, e) N u = 5. y 1, y, y, Figure 6: p =, m =. 11