Robust Model Predctve Control Formulatons of robust control [1] he robust control problem concerns to the control of plants that are only approxmately nown. Usually, t s assumed that the plant les n a set of possble plants and ths set can be quanttatvely characterzed. It s sought a control desgn that assures some nd of performance, whch ncludes stablty, for all the members of the famly of canddate plants. Robust control theory usually assumes that the controller s lnear and most of the avalable theory wll reman useful only when the operatng pont of the system s such that the system s unconstraned or has a fxed set of actve constrants. Norm-bounded uncertanty Assume that one has a nomnal model of the plant, for nstance represented by a transfer functon G ( z ), and that the real plant G(z) can descrbed by one of the followng models: Gz ( ) = G( z) + ( z) (1) [ ] Gz ( ) = G( z) 1 + ( z) (2) 1 [ ] [ ] Gz ( ) = D( z) + ( z) N( z) + ( z), where G ( z) = D ( z) N ( z) (3) D N 1 We do not now exactly what ( z) s, but we assume that t s stable and norm bounded. he most commonly used norm s the H nfnty norm the uncertanty model [2]., whch s the largest gan of Consder a lnear feedbac controller K(z) that stablzes the nomnal plant G( z ). hs controller wll not necessarly stablze the real plant G(z), and t s mportant to have some way of testng whether ths controller can stablze all plants allowed by the uncertanty descrpton. Consder ntally the case n whch uncertanty s represented by the addtve model defned n (1). he closed loop system s shown n Fg. 1 and the transfer functon of the bloc consttuted by the controller and the nomnal system s 1
( ) 1 M ( z) = K( z) I + G ( z) K( z) hs transfer functon M(z) s stable by desgn. Consequently, the loop transfer functon s M ( z) ( z). he small gan theorem (Chap. 1 of [2]) assures that the uncertan plant wth the proposed controller wll be stable f σ jω ( M e ) jω ( ) ( e ) < 1 (4) where σ (.) denotes the largest sngular value. he condton defned n (4) s only suffcent for robust stablty. If condton (4) s not satsfed, t does not mean that stablty wll not be attaned. Inequalty (4) may be tested for a gven controller, to verfy f stablty s guaranteed for any plant lyng n the set that defnes the uncertan plant. When tunng the controller, we may attempt to shape the frequency response of the nomnal closed loop system, such that (4) wll hold for the true system. Fgure 1 Closed loop wth addtve model uncertanty When, uncertanty s descrbed as n (2), the closed loop can be represented as n Fg. 2 below Fgure 2 Closed loop wth nput multplcatve uncertanty In ths case, the transfer functon of the bloc consttuted by the controller and plant s represented by 2
( ) 1 M( z) = K( z) G ( z) I + KG ( Z) and the followng suffcent condton for robust stablty s obtaned: ( ) 1 jω jω jω jω jω σ Ke ( ) G( e ) I+ Ke ( ) G( e ) ( e ) < 1 Analogously, the model uncertanty defned n (3) produces the followng condton for robustness: jω Ke ( ) jω jω 1 jω 1 jω jω σ I + G( e ) K( e ) N( e ) N( e ), D( e ) < 1 I (6) he uncertanty descrptons depcted n (4) to (6), n whch the only specfcaton on uncertanty s a norm-bound, are nown as unstructured uncertanty descrptons. (5) Robust Stablty of the Unconstraned DMC In the conventonal form of DMC, whch s based on the step response model, the state vector of the predctng model s updated as follows [3]: where { } [ y] A[ y] B u K y C[ y] = + ( 1) + F (7) / 1/ 1 / 1 [ ] [ ] y = A y + B u( 1) (8) / 1 1/ 1 s the present samplng tme y = y( ) y( + 1)! y( + n h ) n h s selected such that t s larger than the stablzng tme of the system Iny! I ny! A= " " " # ", A $! Iny I! ny (( nh+ 1). ny) (( nh+ 1). ny) S1 S 2 B= " B $ Sn h S nh + 1 (( nh + 1). ny) ny, 3
K F Iny I ny = " $ Iny I ny (( nh + 1). ny) ny, KF S1,1, S1,2,! S1, nu, S S S S! = " " # " Sny,1, Sny,2,! Sny, nu, 2,1, 2,2, 2, nu, ny nu, S $ S corresponds to the step response matrx related to tme step y ( ) = C[ y], [ ] / / y s the state vector of the true plant and y ( ) s the measurement of the plant output at tme. C = C = I ny! ny s the number of outputs and nu s the number of nputs. Combnng eqs. (7) and (8), we obtan: [ ] [ ] [ ] y = ( I K C) A y + K C y + ( I K C) B u( 1) (9) / F 1/ 1 F / F Let us defne the nput horzon as m, and let the control sequence to be appled at tme nu m nu u = u( ) u( + 1) u( + m 1)! $. hs step be gven [ ] (. ) control sequence, for the case of the unconstraned DMC, s obtaned as the least squares soluton to the problem of mnmzng the predcted error norm: [ ] ( ) 1 m m m sp { n n } ( n ) [ ] { } u = S Q QS + R R S Q QN y y (1) where N = I. ny n ( ny. n) ( ny.( nh 1) ), S m n / S1! S2 S1! = " " # " Sm Sm 1! S1 " " " Sn Sn 1! Sn m+ 1 Q and R are weghtng matrces and n s the output horzon. From Eq. (1), t s clear that the frst control move can be represented as 4
sp { [ ] } u ( ) = K y y (11) MPC where K MPC s constant. / We can assume that the true plant model s represented by [ ] [ ] y = A y + B u ( ) (12) / 1/ 1 Now, substtutng Eq. (12) n Eq. (9), we obtan [ ] F [ ] F [ ] ( F ) y = ( I K C) A y + K CA y + B+ K C( B B) u( 1) (13) / 1/ 1 1/ 1 Combnng equatons (12) and (13), the state space model of the system that encloses the predcton model and the true plant model s obtaned as follows y A y B u ( 1) y = K CA ( I K C) A y + B+ K C( B B) / F F 1/ 1 F (14) Wth the extended state defned above, the control move gven n (11) can be wrtten as follows sp sp y y y y u ( ) = K = K [ ] MPC sp sp y y y y / / Suppose now that we have an addtve uncertanty n the step response model of the system: B = B+ where S = S S1 S! 2 S nh + 1 hen, the state space model of the system consderng the model uncertanty as represented n (14) can be wrtten as follows: where [ ] φ[ ] β1 β2 (15) y % = y % + u( ) + u( ) (16) + 1/ + 1 / A φ = KFCA ( I KFC) A, β1 B = B, I ' β2 = S = β2 KFC S 5
herefore, the uncertan system n closed loop wth the DMC controller can be schematcally represented as n Fgure 3. Fgure 3 Addtve model uncertantes wth DMC Consequently, for the addtve uncertanty, we obtan 1 1 1 ' 1 2 M( z) = K I + ( zi φ) β K ( zi φ) β (17) Suppose now that we have a multplcatve uncertanty n the process gan. In ths case, the true gan nput matrx between nput u j and output y s represented as B ( 1, ) = B + (18) j j j Consequently, the uncertanty n the step response coeffcents can be wrtten as follows S S! S S S! S " " # " S S! S 1,1 1,1 1,2 1,2 1, nu 1, nu 2,1 2,1 2,2 2,2 2, nu 2, nu S = ny,1 ny,1 ny,2 ny,2 ny, nu ny, nu he above expresson can be wrtten n a more convenent form S1,1! S1,2!! S1, nu! S2,1 S2,2 S2, nu!!!! S = " " # " " " # "! " " # "! Sny,1! Sny,2!! Sny, nu 6
1! 1,1 " "! 1 #! " ny,1 1! # " "! " 1, nu 1! #! 1 ny, nu " "! "! 1 & '('') nu or = S % N S K u herefore, for ths nd of model uncertanty, Eq. (16) can stll be wrtten wth B% β = N = % β % N 2 K u 2 K u KFCB where S% 1 B% = " S % nh + 1 Wth ths nd of model uncertanty, Eq. (16) corresponds to the followng transfer functon 1 [ φ] ( β % 1 β2 K u) yz %( ) = zi + N uz ( ) (19) From Eq. (15), we can wrte sp uz ( ) = K y ( z) yz ( ) % % (2) Equatons (19) and (2) can be translated nto the dagram represented n Fg. 4. 7
Fgure 4. Multplcatve model uncertantes wth DMC hus, for the multplcatve uncertan represented n Eq. (18), we have 1 1 1 1 2 M( z) = Nu K I + ( zi φ) β K ( zi φ) % β (21) Example 1 Consder the followng 2 2 system 1.77 + 11 5.88 + 12 6s+ 1 5s+ 1 Gs () = 4.41+ 21 7.2 + 22 44s+ 1 19s+ 1 whch s based on the FCC heavy ol fractonator system proposed by Shell as a challenge MPC problem. Our purpose s to evaluate the robustness of a gven DMC controller wth respect to the addtve uncertanty n the process gan. he DMC to be tested s characterzed by the followng parameters: =5, n=12, m=5, Q=dag(1,1), R=dag(.3,.3). We wll ntally evaluate the robustness of ths controller for the general addtve uncertanty, whch maxmum bound s gven by ( ) 1 σ = mn S ω σ M( e ω ) where M(z) s gven by Eq. (17). Fg.5 shows 1 as a functon of the frequency ω. σ M( e ω ) We observe that the mnmum corresponds to σ ( S ) = 1.195 and t occurs at ω *.214 rad/mn. It s nterestng to evaluate the correspondng sze of ths uncertanty n terms of the step response coeffcents. Snce n ths case, t s assumed that uncertanty s unstructured and can appear n any coeffcent of the step response model, matrx S 8
has the same dmensons as the nput matrx B defned n Eq. (7). hs means that for ths partcular DMC, S s a matrx of dmensons 122 2. Wth the help of Matlab routne rand we observe that we can generate random matrces that have the maxmum sngular value larger than 1.195 and whose elements are lmted to approxmately.13. hs means that the method wll not guarantee robustness outsde the followng approxmate range S.13 S S +.13 = 1,, ny; j = 1,, nu; l = 1,, nh, j, l, j, l, j, l Fgure 5. Robust stablty of DMC General uncertanty We can also compute the model uncertanty that s tolerated by the DMC controller, usng M(z) gven by Eq. (21) where t s assumed that model uncertanty s lmted to the multplcatve uncertanty n the model gan. Fg. 6 shows the values of 1 σ M( e ω ) as a functon of the frequency for ths case. 9
Fgure 6. Robust stablty of DMC Gan uncertanty In ths case, we observe that the mnmum of the curve occurs at ω =.5 rad/mn and corresponds to σ ( K ) =.112, whch s smaller than the uncertanty bound for the case of general uncertanty. However, n ths case, matrx K s a 4 4 dagonal matrx whose sngular values correspond to the absolute values of the dagonal elements. hus, we may conclude that the followng relaton s vald: K (1.112) K K (1 +.112) = 1,, ny; j = 1,, nu, j, j, j Notes -From the above example t s clear that, when followng the small gan approach, the resultng ranges of general model uncertantes n whch DMC s guaranteed to be stable are usually qute conservatve. By smulaton, we can verfy that, for the studed system, DMC remans stable for a much larger range of gan uncertantes. Less conservatve uncertanty bounds can be obtaned by tang nto account the structure of the uncertanty n the search of the worst uncertanty. However, the results appear stll not useful for practcal applcatons. -he procedure presented so far has not taen nto account the exstence of constrants on the nput values or on the nput moves. In the presence of nput constrants, the controller becomes nonlnear and the methods presented here may not be general. In other words, when a constrant becomes actve, the closed loop system may become unstable, even f 1
uncertanty s smaller than the maxmum uncertanty tolerated by the unconstraned MPC controller. he Infnte Horzon MPC and the Robust Regulator Another class of robust MPC derves from the nfnte horzon model predctve control (IHMPC) proposed by Rawlngs and Muse [4]. hs controller s based on a state space model of the process n the conventonal form: x( + 1) = Ax( ) + Bu( ) (22) y ( ) = Cx ( ) (23) where nx x $ s the state vector, nu u $ s the nput vector, s the samplng nstant, ny y $ s the output vector, A, B and C are matrces of approprate dmenson. he control law of IHMPC s obtaned from the soluton to the followng optmzaton problem: mn u ( ), u ( + 1),, u ( + m 1) J m 1 (24) = j= J = x ( + ) Qx ( + ) + u ( + j) Ru ( + j) mn max u u( + j) u ; j =,1,!, m 1 (25) where m s the control horzon, x( + ) s the predcted state at samplng tme + ; nx nx Q $ and nu nu R $ are postve defnte matrces. For open loop stable systems, the nfnte horzon optmzaton problem defned above can be transformed nto a fnte horzon problem by defnng a termnal state weght P, whch can be obtaned from the soluton of the followng dscrete Lyapunov equaton: APA P= Q (26) Wth the termnal state weght, the IHMPC objectve functon defned n (24) can be wrtten n the form: 11
m m 1 J = x( + j) Q x( + j) + x( + m) Px( + m) + u( + j) Ru( + j) (27) j= j= At ths pont, t s mportant to emphasze that the IHMPC as presented above deals only wth the regulator problem, where the system steady state les at the orgn of the (u,x) plane. hs means that the model defned n (22) and (23) s wrtten n terms of the devaton varables u uss and y yss. In ther semnal paper, Rawlngs and Muse [4] show that ths controller s stable and stablty s ndependent of the tunng parameters of the controller. Badgwell [5] extended the nfnte horzon regulator presented above, to the case n whch we have uncertanty n the process model. o present the method of Badgwell, we assume that the true plant can be represented by the model defned n (22) and wth model parameters desgnated as θ = ( A, B ). We also assume that the plant s stable, or that all the egenvalues of A le nsde the unt crcle. In the presence of model uncertanty, θ s not nown, but t s supposed to be a member of a set Ω of L stable models wth the same dmensons as θ. hs means that {,,, } θ Ω, θ θ θ (28) 1 2 L hs nd of uncertanty s usually desgnated as multple-plant uncertanty where each plant may represent the real process at a partcular operatng condton. Wth ths notaton, for nstance, plant corresponds to θ = ( A, B ). For any of the plants defned n (28), the frst term on the rght hand sde of Eq. (27) can be wrtten as follows where ( ) m c c x( + j) θ Q x( + j) θ = x Q θ d xθ j= c xθ = Eu + Dx ( ) 12
c θ x xθ ( + 1) xθ ( + 2) = " xθ ( + m) ; E B! AB B! = ; " " # " m 1 m 2 A B A B! B D A 2 A = ; " m A u u ( ) u ( + 1) = " u ( + m 1) Qd = dag[ Q! Q] &'(') m Analogously, the second term on the rght hand sde of (27) can be wrtten as follows ( ) c c θ θ θ θ θ x( + m+ 1) Px( + m+ 1) = x( + m) A PAx( + m) = x C% A PACx % where C% =! Iny and P s obtaned from the soluton of Eq. (26) for the state matrx A. Consequently, the control objectve defned n (27), can be wrtten as follows [ ] [ ] J, θ = Eu ( ) ( ) + Dx Qd + C% A PAC% Eu + Dx + u Rdu (29) where R = dag [ R! R] d &'(') m Equaton (29) can now be wrtten n the general quadratc form, θ 2 θ f, θ θ J = u H u + c u + c (3) where Hθ = E Q + C% A P A C% E + R d D c, θ = x( ) D Q + C% A P A C% E f d cθ = x( ) D ( ) Qd + C% A P AC% Dx Let J, θ be the value of the objectve functon (3) correspondng to the nomnal plant, N 13
whch s represented by ( A, B ) θ =. hs s the most probable model, or the model N N N correspondng to the desgn condton of the process. Let also J, θ be the control objectve correspondng to the true plant. Remember that the true plant s unnown, but t s assumed to be one of the plants lyng n Ω. hus, assumng that the state s perfectly nown, the robust regulator proposed by Badgwell [5] can be formulated as the soluton of the followng problem: Problem P1 subject to mn u J J, θ N ˆ, θ J, θ θ Ω (31) mn max u u( + j) u ; j =,1,!, m 1 (25) Let the optmal soluton to the above problem be represented by the followng sequence of control actons * * * * u = u ( ) u ( + 1) u ( + m 1)! (32) In the nequalty defned n (31), Jˆ, θ ( = 1, 2,, L ) represents the numercal value of the control objectve for each of the plants lyng n Ω and obtaned wth the followng control sequence * * uˆ = u ( ) u ( + m 2)! (33) 1 whch s computed usng the optmal control sequence obtaned at samplng nstant -1 as follows: 14
uˆ I! " # # " = " # # " u!! I!! * 1 As the control sequence defned n (33) s nherted from the optmal control sequence at tme step -1, t s clear that t satsfes the nput constrants defned n (25). Also, t s easy to show that for the undsturbed true plant ˆ J = J x( 1) Qx( 1) u( 1) Ru( 1) *, θ 1, θ where J * 1, θ s the value of the objectve functon for the true plant wth the optmal control sequence obtaned at samplng tme -1, and Jˆ, θ s the value of the objectve functon of the true plant at samplng tme computed wth the control sequence defned n Eq. (33). hus, as the optmal soluton to Problem P1 at tme wll result n a objectve functon that s not larger than any feasble soluton, we can wrte J Jˆ J J J * * * *, θ, θ 1, θ, θ 1, θ hs means that the control objectve s a contractng functon, and snce t s postve defnte, t wll converge to zero. Consequently, as long as the model of the true plant les n set Ω, the MPC regulator defned n Problem P1 wll drve the true plant to the orgn. Note: In the development of the controller presented above, t s assumed that the term = x( + ) Q x( + ), n the control objectve functon, s bounded for all the plants lyng n set Ω. However, ths cannot happen when the plants have dfferent gans and a change of set pont s ntroduced nto the system. hs case s usually desgnated as servo or output tracng operaton. In ths case, as the soluton of Problem P1 wll produce a sngle optmal control sequence gven by (32), ths optmal control sequence cannot drve smultaneously the outputs of all the plants n Ω to the same set pont. herefore, the robust MPC presented above does not assure stablty for the case n whch there s a 15
change n the set pont of the system output. hs means that ths robust MPC cannot be mplemented n practce where the control system has to face both regulator and output tracng operatons. In the next secton we wll present a robust MPC [6] that extends the approach of Badgwell [5] to the practcal case. Extended IHMPC and Robust MPC o extend the controller presented n the prevous secton, such that the resultng controller can be appled to real systems, t s convenent that the state space model defned n (22) and (23) be wrtten n the followng ncremental form [7] where s ( 1) s s x + Iny x ( ) B = + u ( ) d ( 1) F d d x + x ( ) B s x ( ) y ( ) = Iny Ψ d x ( ) (34) (35) s ny x $, d nd x -, nd nd ny nd F -, Ψ $, u ( ) = u ( ) u ( 1) Suppose now that the objectve functon of the extended IHMPC s defned as follows: m 1 = (( + ) δ ) (( + ) δ ) + ( + ) ( + ) +δ δ j= j= (36) V e j Q e j u j R u j S sp where e ( + j) = y ( + j) y, y ( + j) s the predcted output at samplng step +j, m s the control horzon and ny δ $ s a vector of slac varables. ny ny Q $, R $ nu nu and ny ny S $ are postve defnte weghtng matrces. he vector of slac varables δ allows the nfnte horzon controller to be appled to the cases n whch there are not enough degrees of freedom to force all the system outputs to the desred set pont. It can be shown that the control objectve defned n (36) wll be bounded only f [6] 16
s sp x ( + m) δ y = (37) When the constrant defned n (37) s satsfed, the nfnte horzon control objectve can be wrtten as follows m ( ) ( ) d d j= V = e( + j) δ Q e( + j) δ + x ( + m) Qx ( + m) m-1 + u ( + j) R u ( + j) +δ Sδ j= (38) where Q ny ny $ s such that Q F QF = F Ψ QΨ F (39) Usng model equatons (34) and (35) to represent the output predcton as a functon of the future control actons and the current state, and (39) to compute the termnal weght, the control objectve represented n (38) can be wrtten n the quadratc form: where u u V = u δ H + 2cf + c δ δ s s s ( m +Ψ u) ( m +Ψ u) + u u + ( m +Ψ u) s 1( m 1 u) 1 B 1F Q1 B 1F F Q2F R1 B 1F QI 1 H = I Q B +Ψ F S+ I QI + Q c f s s d d ( Bm +Ψ 1Fu) Q1( Ie ( ) +Ψ 1Fxx ( ) ) + Fu Q2( Fxx ( ) ) s d I Q1( Ie ( ) +Ψ1Fx x ( ) ) Qe( ) = ( ) 1 x 1( 1 x ) ( x ) 2( x ) s d s d d d c= e( ) Qe( ) + Ie ( ) +Ψ F x ( ) Q Ie ( ) +Ψ F x ( ) + F x ( ) Q F x ( ) (4) (41) (42) (43) I Iny = ", Iny B s m s B! = " # ", s s B! B u ( ) u = " u ( + m 1) 17
F x d F B! 2 d d F m FB B, F # ".'/' = u =, Q 1 = dag[ Q! Q], " " " # " m m 1 d m 2 d d F F B F B! B m.''/'' m m.'/'.'/' Q2 = dag[! Q], R1 = dag[ R! R] and Ψ 1 = dag[ Ψ! Ψ], e ( ) = x ( ) y s s sp Analogously, the constrant defned n (37) can be wrtten as follows: s s e ( ) + B% u δ = (44) s s s where B% = B! B &''('') m Fnally, we can defne the extended nfnte horzon MPC [6] that s based on the soluton to the followng optmzaton problem: Problem P2 mn u, δ V subject to (4), (44) and mn max u u( + j) u max j max! = u u( 1) + u( + ) u ; j =,1,, m 1 It can be proved that f the system remans controllable along the trajectory from the present state to the steady state correspondng to the desred set pont, then the control law, resultng from the soluton to Problem P2 at successve tme steps, wll drve the system output asymptotcally to the desred set pont. If an open loop stable system s not controllable at the desred output, the closed loop wth the controller defned above s stll stable, but t wll not be able to drve the outputs to the desred set ponts. 18
We can now apply the approach of Badgwell [5] to nclude model uncertanty n the extended nfnte horzon MPC. A robust MPC, whch can be appled to both regulator and output tracng cases, s obtaned as the soluton to the followng optmzaton problem: Problem P3 mn V, θn, u δ, θ, = 1,..., L subject to V Vˆ = 1,..., L (45), θ, θ s e ( ) + B% u δ = = 1,..., L (46) mn max s, θ u u( + j) u max j max! = u u( 1) + u( + ) u ; j =,1,, m 1 In (45), V ˆ, θ s computed wth u ˆ and ˆ δ, θ. he control sequence uˆ s obtaned from the optmal control sequence computed at the prevous samplng tme as follows * * uˆ = u ( ) u ( m 2)! + 1 and ˆ δ, θ s such that s s e ( ) + B% uˆ δ ˆ = = 1,..., L (47), θ he ntroducton of ˆ δ, s necessary to accommodate the feedbac from the state of the θ true plant to the other models lyng wthn Ω. It s easy to show that, for the undsturbed system, we have δ ˆ =δ. *, θ 1, θ Suppose that at tme -1, Problem P2 was solved and the resultng optmal soluton s * u 1 * 1, θ, 1,..., L represented by (, δ = ). For the true plant, the correspondng cost s 19
( ) ( ) m 1 * * * * * 1, θ θ θ j= j= V = e( 1 + j) δ Q e( 1 + j) δ + u ( 1 + j) R u ( 1 + j) + * * 1, θ S 1, θ +δ δ Assume that the frst control acton u * ( 1) was njected nto the true plant and we move to the present samplng tme. At ths tme step, consder the value of the control objectve for the true plant wth the control sequence uˆ and slac vector equal to δ %, θ : ( ) ( ) Vˆ = V e( 1) δ Q e( 1) δ u ( 1) R u ( 1) * * * * *, θ 1, θ 1, θ 1, θ * * V( u, δ( θ), θ) hus, as Problem P3 s always feasble at tme, the control objectve for the true plant wll satsfy the relaton V * *, θ V 1, θ Consequently, V, θ, whch s postve and bounded below by zero, s also nonncreasng. hus, f the nputs do not become saturated and Inequalty (45) holds true for all the models lyng wthn Ω, the control objectve functon wll converge to zero for the true plant. However, smlarly to the nomnal case, for a square system, f one of the nputs becomes saturated, the resultng cost wll stll be bounded but t wll not converge to zero as there wll not be enough degrees of freedom to zero the error on all the outputs. he system outputs wll converge to an equlbrum pont, whch does not correspond to the set pont. Remar Problem P3 can be smplfed as for each θ, there s a slac vector δ, θ, whch becomes part of the set of decson varables of the control optmzaton problem. However, of the robust IHMPC optmzaton problem can be reduced because δ, θ s not constraned, and for each δ, θ there s an equalty constrant defned n (47). herefore, the slacs can be 2
elmnated from the optmzaton problem. After the substtuton of δ, θ, the left-hand sde of (46) can be wrtten as follows V% u H% u c% u c%, θ = 2 θ + f, θ + θ where H θ H11, θ H 12, θ = H21, θ H 22 =, cf, θ c f,1, θ c f,2, θ H% H H B% B% H B% H B% s s s s θ = 11, θ + 12, θ + 21, θ + 22 c% e H H B% c c B% s s s f, θ = ( ) 21, θ + 22 + f1, θ + f 2, θ c% e H e c e c s s s θ = ( ) 22 ( ) + 2 2, ( ) f θ + θ H θ, c f, θ and c θ can be defned as n (4), (41) and (42) respectvely. Wth ths reduced expresson for the objectve functon, the control law for the robust MPC can be obtaned as the soluton to the followng optmzaton problem Problem P4 mn u % V, θ N subject to V% Vˆ = 1,..., L, θ, θ mn max u u( + j) u max j max! = u u( 1) + u( + ) u ; j =,1,, m 1 We now llustrate the applcaton of the robust MPC wth a classcal system of the process control lterature. 21
Example 2 Ralhan and Badgwell (2) studed the control of a hgh purty dstllaton column and compared the performance of several regulators, whch are robust to model uncertanty. Here, we smulate ths system wth the controller defned by Problem P3. he model below relates the outputs: dstllate composton (y 1 ) and bottom stream composton (y 2 ), to the nputs: reflux flow rate (u 1 ) and reboler vapor flow rate (u 2 ). y ( s) 1.7868(1 +γ ).6147(1 +γ ) u ( s) 1 1 2 1 2( ) = 22.98 + 1.898(1 +γ1).982(1 +γ2) 2( ) y s s u s In ths model, γ 1 and γ 2 are parameters that defne the model uncertanty, and t s assumed that.4 γ1, γ2.4. Here, we consder an approxmated soluton n whch set Ω s dscretzed as follows Ω= [(, ) (.4,.4) (.4,.4) (.4,.4) (.4,.4) ] It s assumed that the nomnal model corresponds to (, ) (,) γ γ =. In the smulaton of Case 1, the nomnal model also represents the true plant. In the smulaton cases 2 and 3, the true plant s represented by ( γ, γ ) = (.4,.4) and ( γ, γ ) = (.4,.4) 1 2 respectvely. In these three cases, t s smulated the followng scenaro: at tme, the system starts at the orgn and the output reference s changed to ( y1, y 2 ) =(.1, ); at samplng tme 5, t s ntroduced an unmeasured dsturbance correspondng to ( u, u ) = (.5,.5). he tunng parameters of the controller are shown n able 1. 1 2 1 2 set 1 2 set able 1. unng parameters of the cost constranng robust MPC m Q R S u max u mn u max u mn 2 3 dag(1,1) dag(1-2,1-2 ) dag(1 2,1 2 ).25 -.25 1.5-1.5 Fgure 7 shows the system responses wth the robust controller for the three cases smulated. As expected, a better performance s obtaned when the plant model s the nomnal model. We may also note that the ntroducton of a dsturbance that moves the steady state of the nput to a dfferent value does not affect the convergence of the controller, nether t maes the control optmzaton problem to become unfeasble. 22
Fgure 7. Robust control of the dstllaton system. Case 1 ( ), Case 2 ( - - - -), Case 3 ( ) In the second smulaton scenaro, we verfy the effect of an nput saturaton on the convergence of the robust controller. It was consdered the same plant as n Case 2 but wthout the unmeasured dsturbance. he max bound on nput u 1 was reduced to.5, whch s low enough such that u 1 becomes saturated. Fg. 8 shows the responses of the system for ths case, and we can observe that the system outputs stll converge to a steady state, but, as a consequence of the nput saturaton, the outputs does not converge ther set ponts. Nether y 1 tends to ts reference, whch s.1, nor y 2 tends to zero. For the same case, Fg. 9 shows the cost functon (V ) for the true plant. hs cost s non-ncreasng ndcatng convergence, but, because of the nput saturaton, the cost cannot be reduced to zero. 23
Fgure 8. Responses of the dstllaton system wth nput saturaton Fg. 3. Cost for the true dstllaton system wth nput saturaton References [1] - Macejows J.M., Predctve Control wth Constrants, Prentce Hall, 22. [2] - Morar M., Zaphrou E., Robust Process Control, Prentce Hall Internatonal, 1989. [3] L S., Lm K. Y., Fsher D.G., A State Space formulaton for Model Predctve Control, AIChE Journal, 35 (2), 241-249 (1989). [4] Rawlngs J.B., Muse K. R., he stablty of constraned multvarable recedng horzon control, IEEE ransactons on Automatc Control, 38 (1), 1512-1516 (1993). [5] Badgwell. A., Robust model predctve control of stable lnear systems, Internatonal Journal of Control, 68, 797-818 (1997). 24
[6] Odloa D., Extended robust model predctve control, AIChE Journal, 5 (8), 1824-1836 (24). [7] Rodrgues M. A., Odloa D., MPC for stable lnear systems wth model uncertanty, Automatca, 39, 569-583 (23). Complementary bblography to Robust Model Predctve Control C1) Cuzzola F.A., Geromel J.C., Morar M., An mproved approach for constraned robust model predctve control, Automatca, 38 (7), 1183-1189 (22). C2) Kothare, M. V., Balarshnan V., Morar M., Robust constraned model predctve control usng lnear matrx nequaltes, Automatca, 32, 1361-1379 (1996). C3) Lee J. H., Cooley B.L., Mn-max predctve control technques for a lnear statespace system wth bounded set of nput matrces, Automatca, 36, 463-473 (2). C4) Lee J.H, Xao J., Use of two-stage optmzaton n model predctve control of stable and ntegratng systems, Computers and Chemcal Engneerng, 24, 1591-1596 (2). C5) Ralhan S., Badgwell.A., Robust control of stable lnear systems wth contnuous uncertanty, Computers and Chemcal Engneerng, 24, 2533-2544 (2). C6) Ralhan S., Badgwell.A.,, Robust model predctve control for ntegratng systems wth bounded parameters, Ind. Eng. Chem. Res., 39, 2981-2991 (2). C7) Scoaert P.O.M., Mayne D.O., Mn-max feedbac model predctve control for constraned lnear systems, IEEE ransactons on Automatc Control, 43 (8), 1136-1142 (1998). C8) Wan Z.Y., Kothare M.V., An effcent off-lne formulaton of robust model predctve control usng lnear matrx nequaltes, Automatca, 39 (5), 837-846 (23). C9) Wang Y. J., Rawlngs J.B., A new robust model predctve control method I: theory and computaton, Journal of Process Control, 14, 231-247 (24). 25
Exercses 1-Wth the help of Matlab routne PASI_Robust you should study, for the system of example 1, the effect of the followng parameters on the robustness of the unconstraned MPC controller: a) Control horzon m b) Input weght rr c) Weght q of one of the outputs whle the weght of the other output s ept constant. d) For the controller depcted n example 1 of the class notes, show by smulaton that the method based on the small gan theorem may be very conservatve. 2-Based on the results of exercse 1, propose a set of tunng parameters for the unconstraned MPC, such that the closed loop system wll be stable for gan uncertantes as large as ±12% of the nomnal gans. 3-Use Matlab routne PASI_MAIN_SIM to evaluate the performance of the MPC controller when appled to the nomnal system and to the worst case system wth ±12% uncertanty n the model gans. 4-For the system of example 2 (hgh purty dstllaton column), use Matlab routne PASI_RIHMPC to explore by smulaton to what extent the process gan uncertanty can be tolerated by the robust extended IHMPC before becomng unstable. ry to justfy why the controller becomes unstable. 5-For the dstllaton column, verfy by smulaton the effect of weght S. For the other tunng parameters of the controller eep the same values used n example 2. 26