INCLUDING MODEL UNCERTAINTY IN THE MODEL PREDICTIVE CONTROL WITH OUTPUT FEEDBACK

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1 Brazilian Journal of Cheical Engineering ISSN Printe in Brazil Vol. 9, No. 04, pp , October - Deceber 2002 INCLUDING MODEL UNCERAINY IN HE MODEL PREDICIVE CONROL WIH OUPU FEEDBACK M.A.Rorigues an D.Oloak * Departent of Cheical Engineering, University of São Paulo C.P. 6548, , São Paulo, Brazil. E-ail: oloak@usp.br (Receive: March 5, 2002 ; Accepte: June 3, 2002) Abstract - his paper aresses the evelopent of an efficient nuerical output feeback robust oel preictive controller for open-loop stable systes. Stability of the close loop is guarantee by using an infinite horizon preictive controller an a stable state observer. he perforance an the coputational buren of this approach are copare to a robust preictive controller fro the literature. he case use for this stuy is base on an inustrial gasoline ebutanizer colun. Keywors: Preictive Control, Robust Stability, Infinite Horizon. INRODUCION Moel preictive control (MPC) is the ost popular avance control technique in the process inustry. For ore etails about the wie acceptance of MPC see the survey paper by Garcia et al. (989). In inustrial applications, prouction of a stable response is require espite changes in the operating conitions an oel uncertainties. his issue is referre to as robust stability an has receive consierable attention in acaeic research, as can be verifie in the reviews by Mayne et al. (2000) an Morari an Lee (999). Fro the point of view of inustrial application, the review papers by Qin an Bagwell (997, 2000) ephasize the lack of a practical inustrial solution to the robust MPC proble. In the robust MPC literature, the work of Bagwell (997), which propose extension of the noinal infinite horizon MPC (IHMPC) of Rawlings an Muske (993) to the robust case, can be cite. Robustness is achieve by the inclusion of contracting constraints in the objective function corresponing to each oel of the set of possible process oels consiere by the controller. Another contribution to the robust esign of MPCs was ae by Lee an Cooley (2000), who propose a robust infinite horizon MPC for stable an integrating systes with oel uncertainties on the input istribution atrix. In this case, the infinite preiction horizon cost becoes equal to the output error at the en of the control horizon. he paper by Kothare et al. (996), which presente a robust MPC base on the LMI fraework, can be also cite. An infinite preiction horizon is use, an the authors propose a etho that iniizes an upper boun for the objective function. At this point, it shoul be stresse that all these approaches can guarantee robust stability only for the regulator proble, since these authors assue that the state tens to zero when tie tens to infinity. Here, a new robust infinite horizon MPC that was recently presente in the literature (Rorigues an Oloak, 200) is extene to the output feeback case. he controller is ore general than the existing controllers qualifie as robust, since *o who corresponence shoul be aresse

2 476 M.A.Rorigues an D.Oloak it applies to both the regulator an the target tracking probles. he approach is base on a particular realization of the syste state space oel (Rorigues, 200; vrzská e Gouvêa an Oloak, 997). his oel realization allows the irect integration of the square error of the syste output so that the cost function of the controller can be efine in a ore suitable for. Following this strategy, equality constraints, which ensure that the infinite horizon cost function is boune, naturally arise. Moel uncertainties are approxiate by a finite set of isjunct linear oels. his paper is organize as follows: In Section 2 the state space oel use in this paper is briefly introuce. In Section 3 the evelopent of a oifie noinal IHMPC is presente an in Section 4 these results are extene to the robust case. Siulation results are presente in Section 5, the propose controller is copare to a robust MPC presente in a previous paper. In Section 6 soe concluing rearks are offere. MODEL SRUCURE In this paper, it is assue that the syste is represente by the following linear iscrete-tie oel: [x] k+ = A[x] k + B uk () y(k + t) = C(t)[x] k (2) nx x is the vector of states, u is the vector of inputs, uk = uk uk is the process input increent, k is the present tie step an y is the vector of outputs. Here the oel representation esignate output preiction oriente oel (OPOM), which was propose by Rorigues (200), is aopte. Using OPOM, the oel atrices involve in eqs. () an (2) have the following structure: s x [x] =, x I 0 A = 0 F, nu 0 D B =, DFN C(t) = [ I Ψ (t)], nx = ( + nu.na) (3) he state vector, x was partitione into two auxiliary vectors, x s an x. For stable systes, s coponent x correspons to the preicte value for the syste output at steay state. Vector n x is relate to the ynaic oes of the syste an na is the orer of the respective transfer function oel. Matrix D 0 contains the steay state gains of the process an atrix Ψ(t) is shown below. More etails on the OPOM forulation can be seen in Rorigues (200) an Rorigues an Oloak (2002). Φ(t) (t) 0 (t) Φ Ψ = 0 0 Φ (t) ri,, t ri,,nat e e Φ i () t = i=, 2,, ri,nu,t ri,nu,nat e Ψ(t) n, are the poles of the syste, an n (4) Φ n i, r i =,,n =.nu.na. INFINIE HORIZON MPC (IHMPC) An iportant feature of OPOM is that tie appears continuously in the output preiction apping. his peculiarity was use by Rorigues (200) to propose a oifie infinite horizon MPC, whose ain steps are escribe in the sequel. Consier a oifie expression for the objective function of MPC as given by (5): n J k, = {e (n ) k+ t +δ k} Q{e k+ t +δ k}t+ + {e k+ t +δ k} Q{e k+ t +δ k}t + + u R u+ δ k Sδk sp k+ t = +, sp (5) e y(k t) y y is the vector of output reference values; is the control horizon; is the length of the sapling tie, an Q, Brazilian Journal of Cheical Engineering

3 Incluing Moel Uncertainty 477 nu nu R an S are positive efinite weighting atrices. δk is the vector of slack variables that was introuce into the IHMPC objective function to allow for processes with > nu. Insie the control horizon, the output error can be represente by the following expression: s 0 k+ t k k n n e = [e ] +Ψ (t)[x ] + D u +Ψ(t)W Z u (6) (n ) t < n, n s s sp k = k [e ] [x ] y D [D D D 0 0], n n = 0.nu D n n (n ) W = [I F F 0 0], W n n n. DN D N 0 Z =, Z 0 0 D N u = u u u k k+ k+.n.nu Substituting (6) into the first ter on the righthan sie of (5), one obtains, n (n ) k+ t k k+ t k {e +δ } Q{e +δ }t = n n n n n 2 2 n u {D QD + 2D Q(G (n) G (n ))W Z + Z W (G (n) G (n ))W Z} u s 0 k n n + 2 {[e +δ ]Q(D + (G(n) G(n ))WZ)}u (7) 0 k n 2 2 n + {[x ][(G(n) G(n )) D + (G (n) G (n ))WZ]} u n G(n) = Ψ(t)tan 2 0 n. 0 G (n) = Ψ(t) Q Ψ(t)t When the tie t is within the interval t <, the error of the process output can be escribe by s 0 k t k k k [e + ] = [e ] +Ψ (t)[x ] + D u +Ψ(t)W Z u (8) Substituting (8) into the infinite integral on the righthan sie of (5), one obtains: s 0 s 0 k+ t k k+ t k k k k k [e +δ ] Q[e +δ ]t = {[e ] +δ + D u} Q{[e ] +δ + D u} t s 0 k k k + 2{[e ] +δ + D u} Q Ψ (t){[x ] + W Z u}t (9) k k + {x ] + W Z u} Ψ(t) Q Ψ (t){[x ] + W Z u}t Brazilian Journal of Cheical Engineering, Vol. 9, No. 04, pp , October - Deceber 2002

4 478 M.A.Rorigues an D.Oloak For stable systes li Ψ (t) = 0 an t consequently the secon an thir ters on the righthan sie of (9) are boune. However, the first ter on the right-han sie of (9) will be boune only if the equality below hols: s 0 k k [e ] +δ + D u = 0 (0) Introucing (0) into (9), results in 996). Briefly, the separation principle consists in splitting the feeback control esign into two steps. In the first step, a full-state feeback controller is esigne assuing that all the oel states can be easure, an the control sequence is calculate using these oel states. In the secon step, an observer is introuce to estiate the state of the syste base upon the syste outputs. his fraework was eploye here to esign the robust output feeback MPC. In the siulations presente here the following sequence is aopte: k+ t k k+ t k [e +δ ] Q[e +δ ]t = = {[x ] + W Z u} k [G( ) G()]{[x ] + W Z u} k () a) One oel of Ω (the ost probable) is assue to be the true plant. he output of this oel is esignate [y p] k. b) he states of the reaining oels are estiate as follows: [x ] + + = A [x ] + B u + K {[y ] i k k i i k k i k F p k ROBUS INFINIE HORIZON MPC (RIHMPC) C (A [x ] + B u )} = (I K C )A [x ] + i i i k k i k F i i i k k (3) In this section, the oifie infinite horizon MPC presente in the previous section is extene to the case of oel uncertainty. When the stabilizing control law takes into account oel uncertainties, the resulting controller is esignate robust. It is well known that a single linear oel can not accurately represent the actual plant uner ifferent operating conitions. Recently, the ultioel representation of uncertain processes has been introuce in the robust MPC literature (see Bagwell, 997 an Rorigues, 200). his oel isatch representation consiers a finite set of isjunct linear systes (Ω), which are obtaine at ifferent points of the operating oain, or better: ( A,B ) : {(A,B ) (A,B )} Ω Ω = (2) L L which eans that the atrices of the oel escribe by () an (2) are not exactly known, but belong to Ω. Consiering this kin of uncertainty an the evelopents presente in Section 3, Rorigues (200) propose a robust infinite horizon MPC for the case all state variables of the process can be easure (state feeback). However, uring actual plant operation only a subset of the oel states, the outputs, is easure. One way to exten the referre robust IHMPC to the output feeback case is base on the separation principle (Levine, + (I K C )B u + K [y ] i =,...,L F i i k F p k c) he observer use here has a static gain given by: F F n K = [I 0], K (4) his observer is base on the assuption that the ifference between oel output an plant easureent is prouce by a step isturbance in the syste output. his assuption is aopte by ost MPC packages but is restricte to stable systes, which is the case consiere here. herefore, assuing that the uncertain process belongs to Ω, one estiates the states of all the linear oels lying in Ω using (3) an (4). Substituting (7) an () into (5), one can write the following optiization proble for RIHMPC: Proble P) in u, δi,, δl, γ subject to I HiY i > 0 Yi Hi 2cf,iY γ i γ, i=,...,l (5) Brazilian Journal of Cheical Engineering

5 k+ j Incluing Moel Uncertainty 479 u U, j 0 (6) s 0 i k i k,i [e ] +δ [ ] + D u = 0, i =,..., L (7) Yi = u [ i] δ k, Hi + H, i + R, Hi 2 Hi = Hi Q S 2 + c c c c fi = fi + fi fiδ i =, n,i n,i + n,i,i,i n,i i + i n,i 2,i 2,i n,i i H {D QD 2D Q[G (n) G (n )]W Z } {Z W [G (n) G (n )]W Z } i, i,i 2,i 2,i,i i H = Z W [G ( ) G ()]W Z 0 i = 2 n,i + i n,i,i,i H {D Q Z W [G (n) G (n )] Q} s 0 0 fi = i k n,i +,i,i n,i i + i k,i,i n,i + 2,i 2,i n,i i n = n = c {[e ] Q[D (G (n) G (n ))W Z ]} {[x ] [G (n) G (n )] QD [G (n) G (n )]W Z } fi i k 2,i 2,i,i i c = [x ] [G ( ) G ()]W Z, s fi = i k + i k,i δ,i c [e ] Q [x ] [G (n) G (n )] Q, R = iag(r R) U efines the set of feasible solutions relate to the input constraints. Proble P can be solve by the available ethos of linear atrix inequalities (LMI). he theore below suarizes the robust stability of the RIHMPC with output feeback efine by Proble P). heore: Consier an open-loop stable process the true plant oel belongs to Ω an whose states are estiate by a stable observer. If there is a feasible solution to Proble P), then the resulting control law stabilizes all the plants belonging to Ω. Proof: he proof for the state feeback case can be foun in Rorigues an Oloak (2002). It reains to be prove that when a stable state observer is use the resulting close loop reains stable. It is easy to verify that the observer gain efine by (4) correspons to λ(i KFC i)ai <, i =,,L. hus, as a consequence of the separation principle, when one uses IHMPC, which stabilizes all the oels belonging to set Ω when the states are perfectly known, an a stable state observer, the resulting close-loop will also be stable. Brazilian Journal of Cheical Engineering, Vol. 9, No. 04, pp , October - Deceber 2002

6 480 M.A.Rorigues an D.Oloak COMPARING RIHMPC O A ROBUS MPC FROM HE LIERAURE [ ] N = I 0, s nu N R s [nu] x[ nu.] Rorigues an Oloak (2000) propose a finite horizon robust output feeback MPC, which was nae RSMPC. he approach is base on the paraeterization of the control oves by an output feeback schee siilar to the conventional linear quaratic regulator. In this way, it was possible to use the quaratic Lyapunov stability conition that was inclue in MPC as an aitional constraint. In their MPC, these authors consiere a polytopic representation of the uncertain oels. he ultioel version of the Rorigues an Oloak (2000) robust preictive controller is escribe below: Proble P2) in γ γ,k,p,,pl subject to ( + ) BQB R K [e] k k γ+ [e k] ka QBK [e 0 k] k+ > [e k] kk + [e k] kkb QA[e k] k Pi P(A i i + BiNsKC) > 0, (A P i BiNsKC) P + i i i=,,l γ> 0, B i i (8) (9) P = P > 0, i=,,l (20) S S S3 S 2 S Sn S p np = Sn S e ne Sne D D D DFN DFN DFN, S i is the step response coefficient atrix at sapling step i K is the feeback gain of the controller Rorigues an Oloak (2000) propose an iterative algorith base upon LMI tools to solve Proble P2). It is iportant to observe that atrices A an B of RSMPC are relate to the augente syste (paraeters + outputs at preiction steps). Consequently, the iension of the state x of this controller is ifferent fro the iension of the state consiere by RIHMPC. SIMULAION RESULS he process siulate in this work is a ebutanizer colun of an oil refinery, an it was borrowe fro Rorigues an Oloak (2000). his colun prouces LPG (liquefie petroleu gas) as the top strea an stabilize gasoline as the botto strea. he controlle outputs are the concentration of C5 + in the top strea (y ) an the Rei vapor pressure of the botto strea (y 2 ). he anipulate inputs are the top reflux flow rate (u ) an the reboiler heat uty (u 2 ). he set of uncertain oels, Ω, was built by linear oels ientifie at three ifferent operating points of the colun. hese oels can be seen in able. he ost probable oel is assue to be G(s). RIHMPC was siulate using G(s) as the true plant, which together with the other two oels constitutes the set Ω. he tuning paraeters of RIHMPC are = ; = 2 ; Q = iag(,) ; 2 2 R = iag(0,0 ); S = iag(00,00) ; uj,ax = 5; uj,in = 2 an uj,ax = 5. RSMPC was siulate using the sae set of oels as RIHMPC an the following tuning paraeters: = ; = 2 ; Q = iag(,) ; 2 2 R = iag(0,0 ); uj,ax = 5; uj,in = 2; Brazilian Journal of Cheical Engineering

7 Incluing Moel Uncertainty 48 uj,ax = 5, n p =3; n e = 75; ε = 0. n p is the nuber of equally space syste output preiction steps, n e is the extra preiction step inclue in the coputation of output error an ε is the convergence paraeter of RSMPC. Figure shows the results of RIHMPC an RSMPC for the oels an tuning paraeters escribe above. he case of output tracking the reference values of the two outputs were siultaneously increase is consiere. It can be observe that for controlle output y (C5 + in the LPG strea), RIHMPC has a better perforance than RSMPC. For y 2 the perforance of RIHMPC is quite siilar to that of RSMPC. he ratio between the coputational tie expene to run one saple step an the sapling tie length (CPU ie/) was for RIHMPC an for RSMPC at a COMPAQ XP000 workstation (52 MB RAM). able : Moels of the ebutanizer colun. G(s): G 2(s): G 3(s) s s+ 64s s s + 6.2s+ 70s + 20s s s+ 50.2s s s +.32s+ 20 s + 5s s s s s s s s s+ Figure : Input an output profiles for RIHMPC an RSMPC for the ebutanizer colun. Brazilian Journal of Cheical Engineering, Vol. 9, No. 04, pp , October - Deceber 2002

8 482 M.A.Rorigues an D.Oloak CONCLUSION In this paper an extension of a highly nuerical efficient robust infinite horizon MPC was evelope for the output feeback case. A case stuy using an inustrial process was carrie out in orer to copare the propose controller to another robust MPC presente earlier in the MPC literature. he siulation results showe that RIHMPC has a very low coputational buren copare to RSMPC. herefore, it can be conclue that RIHMPC is appealing for inustrial applications. Although RIHMPC is a controller a in-ax proble is solve, an consequently the worst case perforance is consiere, its close loop perforance is better then the perforance of RSMPC whose control law is base on the ost probable syste oel. Consequently, inclusion of the Lyapunov stability conition in the MPC proble can ake the controller ore conservative than the controller base on the in-ax approach. he ain avantage of RSMPC over RIHMPC is that Proble P2) ay be convex in the oel atrices if the Lyapunov atrix P i is the sae for all the oels lying in Ω. In this case, stability is guarantee for a plant whose oel is a convex cobination of the oels of Ω. his is not true for RIHMPC, since Proble P) is not convex an stability is only guarantee for the plants lying in Ω. ACKNOWLEDGEMENS Support for this research was provie by FAPESP uner grant 0/008-8 an CNPq uner grant /97-9. REFERENCES Bagwell,.A. (997). Robust Moel Preictive Control of Stable Linear Systes. International Journal of Control, 68, Garcia, C.E.; Prett, D.M. an Morari, M. (989). Moel Preictive Control: heory an Practice A Survey. Autoatica, 25, 3, Kothare, M. V.; Balakrishnan, V. an Morari, M. (996). Robust Constraine Moel Preictive Control Using Linear Matrix Inequalities. Autoatica, 32, 0, Lee, J.H. an Cooley, B.L. (2000). Min-ax Preictive Control echniques for a Linear State Space Syste with a Boune Set of Input Matrices. Autoatica, 36, Levine, W.S. (996). he Control Canbook, Chapter 37. CRC Press Inc. Mayne, D.Q.; Rawlings, J.B.; Rao C.V. an Scokaert, P.O.M. (2000). Constraine Moel Preictive Control: Stability an Optiality. Autoatica, 36, Morari, M. an Lee, J. H. (999). Moel Preictive Control: Past, Present an Future. Coputers & Cheical Engineering, 23, Qin S.J. an Bagwell,.A. (997). An Overview of Inustrial Moel Preictive Control echnology. Fifth International Conference on Cheical Process Control, pp , AIChE an CACHE, Eitors: J.C. Kantor, C. Garcia an B. Carnahan. Qin S.J. an Bagwell,.A. (2000). An Overview of Nonlinear Inustrial Moel Preictive Control Applications. In: Nonlinear Moel Preictive Control. Allgöwer, F., A. Zheng (Es.). Progress in Systes an Control heory Series, v. 26, Switzerlan: Birkhauser Boston. Rawlings, J.B. an Muske, K.R. (993). he Stability of Constraine Multivariable Receing Horizon Control. IEEE rans. Auto. Cont., 38, Rorigues, M.A. (200). Robust Stability of Moel Preictive Controllers. Ph.D. hesis, University of São Paulo (in Portuguese). Rorigues, M.A. an Oloak, D. (2000). Output Feeback MPC with Guarantee Robust Stability. Journal of Process Control, 0, Rorigues, M.A. an Oloak, D. (2002). Robust MPC for Stable Linear Systes, Braz. Jour. Che. Engng., 9: () -23. vrzská e Gouvêa, M. an Oloak, D. (997). ROSSMPC: A New Way of Representing an Analyzing Preictive Controllers. Che. Engng. Res. Des., 75, Brazilian Journal of Cheical Engineering