PQI 5780 Chemical Processes Control I. (Ogata, K., Modern Control Engineering, Prentice Hall, 4 th Ed. 2002)

Transcription

1 PQI 578 Chemcal Processes Control I I-State-Space Models for Process Control (Ogata, K., Modern Control Engneerng, Prentce Hall, 4 th Ed. 22) Some basc defntons: State: he state of a dynamc system s a set of varables (called state varables) such that the nowledge of these varables at t t, together wth the nput for t t completely determnes the behavor of the system for any tme t t. When dealng wth tme nvarant systems, we usually choose the reference tme t to be zero. State varables: he state varables are the varables of the set that defnes the state of the system. If at least n varables x 1 (t), x 2 (t),., x n (t) are needed to completely descrbe the behavor of a dynamc system (such that once the nput s gven for t t and the ntal state at t t s ecfed, the future state of the system s completely determned), then such n varables are a set of state varables. State vector: he state vector s the vector consttuted by the state varables. x() t x () t x () t x () t 1 2 n State-ace: he state ace s the n-dmensonal ace whose coordnates axes consst of the x 1 axs, x 2 axs,, x n axs. 1

2 State-ace representaton of systems of lnear dfferental equatons: Let the system be represented by the followng dfferental equatons ( n) ( n1) (1) ( n) ( n1) (1) ya y a ya y b ub u b ub u (I.1) 1 n1 n 1 n1 n where () dy y. hen, consder the followng set of state varables: dt x y u 1 (1) (1) (1) x y uu x u (2) (2) (1) (1) x y u u x u ( n1) ( n1) ( n2) (1) (1) x y u u u u x u n 1 n2 n1 n1 n1 where, 1,..., n are determned from b 1 b1a1 2 b2 a11a2 b a a a b a a a n n 1 n1 n1 1 n (I.2) (I.3) hen, wth ths choce of state varables, the followng state equaton and output equaton are obtaned: x 1 1 x1 1 x 1 x x 1 x n1 n1 n1 x n an an 1 an2 a 1 x n n y 1 u x x 1 2 x x n1 n u 2

3 he above system can be represented n the general expresson for the lnear tme nvarant (LI) system n the state-ace form: x Ax Bu (I.4) y Cx Du Note that the dfferental equaton represented n (I.1) s equvalent to the followng transfer functon: Y() s bs bs b sb U() s s as a sa n n1 1 n1 n n1 1 n1 n n We now prove that (I.3) s true for the second order system: Y() s bs bs b U() s s as a he transfer functon defned n (I.5) s equvalent to the followng dfferental equaton: (2) (1) (2) (1) (I.5) y a ya y b ub u b u (I.6) hen, applyng (I.3) to ths system, we have x y u (I.7) 1 (1) (1) (1) x y uu x u (I.8) (1) (2) (2) (1) x y u u (I.9) 2 1 Substtutng (I.6) n (I.9) results (1) (1) (2) (1) (2) (1) x2 a1 ya2yb ub1ub2u u1u hen, substtutng (I.7) and (I.8) n (I.1), we obtan (I.1) (1) (2) (1) x axax ( b ) u( b a ) u( a a b) u (I.11) Suppose now that b b a b a a then, (I.11) becomes 3

4 (1) x axax u (I.12) he state ace model correonds to equatons (I.8), (I.12) and (I.7) wrtten as follows x x u x ax ax u y x u 1 or n the vector form x 1 1 x1 1 u x a a x y Example I.1: 1 x x 1 2 u Y( s) s U s s s 2 ( ) hen, a1 a2 b b1 b ; 35.89; ; 1; Consequently b b a b a a and, the state ace model becomes x 1 1 x1 1 u x x x1 y 1 x 2 Remar: Smlar results can be obtaned wth Matlab functon tf2ss [A,B,C,D] = F2SS(NUM,DEN) calculates the state-ace representaton:. x = Ax + Bu y = Cx + Du of the system: NUM(s) H(s) = DEN(s) 4

5 Soluton of the state-ace model x Ax Bu (I.13) gven x () and ut () It s easy to show that for the homogeneous case ( ut () ) the soluton s 1 1 x t I At A t A t x 2!! 2 2 () () At hen, f we defne e I At A t A t the soluton to the homogeneous equaton can be wrtten 2!! as At x() t e x() In the non-homogeneous case, model (I.13) can be wrtten as x Ax Bu and, f we multply both sdes of the above equaton by At d At At e ( x Ax) e x( t) e Bu dt Integratng the above equaton between and t gves or At A e x() t x() e Bu( ) d t At e we have t At A( t ) () () ( ) x t e x e Bu d (I.13a) State-ace model of dscrete-tme system When t,,1, 2,... and s the samplng perod, the state-ace model of the LI system s wrtten n the followng form x( 1) Gx( ) Hu( ) y ( ) Cx ( ) Du ( ) (I.14) he dscrete transfer functon model Y( z) bz bz b zb U( z) z a z a za n n1 1 n1 n n1 1 n1 n n 5

6 that s equvalent to the followng dfference equaton y ( n) ay ( n1) ay ( n2)... a y ( 1) ay ( ) 1 2 n1 bu( n) bu( n1) b u( n2)... b u( 1) b u( ) 1 2 n1 n n (I.15) can be wrtten n the state-ace form defned n (I.14). For ths purpose defne the followng state varables: x ( ) y( ) hu( ) 1 x ( ) x ( 1) hu( ) x ( ) x ( 1) h u( ) x ( ) x ( 1) h u( ) n n1 n1 where h, h1,, hn are determned as follows h b h b ah h b ah a h h b ah a h a h n b 1 n1 n1 1 n (I.16) Wth ths choce of state varables, we obtan the followng dscrete-tme state equaton and output equaton for the system defned n (I.15) x1( 1) 1 x1( ) h1 x ( 1) x ( ) h x ( 1) 1 x ( ) h n1 n1 n1 xn( 1) an an 1 a2 a 1 xn( ) h n x1 ( ) x2( ) y( ) 1 hu( ) xn1( ) xn ( ) u ( ) Example I.2: We can use a numercal example to show that the expressons gven n (16) are true. Let the system be represented by the followng dscrete transfer functon y ( z ) HGp ( z) z z uz ( ) z.8385 z

7 that correonds to the dfference equaton y( 2) y( 1).8385 y( ).5967 u( 1).1114 u( ) (I.17) hen, defne the varables x ( ) y( ) hu( ) (I.18) 1 x ( ) x ( 1) hu( ) (I.19) Applyng (I.19) to tme step +1, we have x ( 1) x ( 2) hu( 1) he, consderng (I.18) at tme +2 and substtutng n the above equaton, we obtan x ( 1) y( 2) hu( 2) hu( 1) 2 1 Substtutng (I.17) n the above equaton results x( 1) y ( 1).8385 y ( ).5967 u ( 1).1114 u ( ) hu ( 2) hu ( 1) 2 1 (I.2) Applyng (I.18) at tme +1 we have y ( 1) x( 1) hu ( 1) (I.21) 1 Substtutng (I.19) n (I.21) results y ( 1) x( ) hu ( ) hu ( 1) (I.22) 2 1 Fnally, substtutng (I.22) and (I.18) n (I.2) produces x ( 1) x ( ).8385 x ( ) hu( 2) (1.5191h.5967 h) u( 1) (1.5191h.8385h.1114) u( ) 1 (I.23) Assume now that h h.5967 h h Consequently: h h1.8385h and (23) becomes x ( 1) x ( ).8385 x ( ).28 u( ) (I.24) Fnally, combnng (I.19), (I.24) and (I.18) we obtan the state ace model of the system 7

8 x1( 1) 1 x1( ).5967 u ( ) x2( 1) x2( ).28 x1 ( ) y ( ) 1 x2( ) Solvng the dscrete-tme state equaton Consder the dscrete tme state ace model defned n (I.14) x( 1) Gx( ) Hu( ) (I.25) y( ) Cx( ) Du( ) (I.26) Suppose, we start at tme = wth ntal condton x() and we apply the nput sequence u(), u(1), u(2),. hen, applyng (I.25) at the successve tme steps, we obtan x(1) Gx() Hu() 2 x(2) Gx(1) Hu(1) G x() GHu() Hu(1) 3 2 x(3) Gx(2) Hu(2) G x() G Hu() GHu(1) Hu(2) By repeatng ths procedure, we obtan 1 j1 x( ) G x() G Hu( j), =1,2,3,. j Dscretzaton of the contnuous-tme state equaton Suppose we have the state-ace model of the system n the contnuous tme form: x Ax Bu (I.13) and we want to obtan the correondng dscrete tme model, whch wll have the form x( 1) G( ) x( ) H( ) u( ) (I.25) where s the samplng perod. he soluton of (I.13) s gven by equaton (I.13a) that s repeated below At t A( t ) x() t e x() e Bu( ) d (I.13a) 8

9 Suppose that (I.13a) s used to compute x(+1) startng from the state and nput at tme step A A x( 1) e x( ) e Bd u( ) (I.26) hen, comparng (I.25) and (I.26), we conclude that G ( ) e A H ( ) e At dt B II. Controlablty, Reachablty and Observablty of Systems he purpose of ths secton s to answer the followng questons: -Is t possble to drve a system from a gven ntal state to any other state? -How to determne the system state by observng the system nput and output? Controllablty and Reachablty Consder the system represented by the followng dscrete tme state-ace model x( 1) Gx( ) Hu( ) (II.1) y ( ) Cx ( ) (II.2) Assumng that the system starts from a gven ntal state x(), then, the state at tme step n (where n s the order of the system) s gven by: n n1 xn ( ) G x() G Hu() Hun ( 1) (II.3) n x( n) G x() WU (II.4) where Wc H GH G H c n 1 and ( 1) () U u n u If the ran of W c s equal to n, then t s possble to fnd a set of n equatons that can be used to determne the control sequence (u(), u(1),..., u(n-1)) that s capable of drvng the system from the ntal state to the desred fnal state x(n). Observe that ths soluton may not be unque. 9

10 Defnton of Controllablty: he system represented n (II.1) and (II.2) s controllable f, for any ntal state, t s possble to fnd a control sequence such that the orgn (x=) wll be reached n fnte tme. Defnton of Reachablty: A system s reachable f, for any ntal state, t s possble to fnd a control sequence that wll be able to drve the system to an arbtrary state n fnte tme. Controllablty does not mply reachablty, whch can be seen easly from (II.3). If n G x(), then the orgn can be reached applyng the nput u=, however the system s not necessarly reachable. he two concepts are equvalent when G s of full ran. heorem: he system defned n (II.1) s reachable f and only f matrx Example II.1: he system 1 1 x( 1) x( ) u( ) 1 1 W c has ran n. 1 1 s not reachable because Wc 1 1 has ran equal to 1 and the system s of order 2. It can be shown that all the states that are reachable from the orgn can be obtaned through the combnaton of the columns of W c. Example II.2: Gven the system x( 1) x( ) u( ).25.5 wth 2 x() 2 t s ased f t s possble to fnd a control sequence that drves the system to.5 x(2) 1. Soluton: From (II.3), we have 1

11 2 x(2) Gx() GHu() Hu(1) or u() u(1) u() u(1) that correonds to a sngle equaton:.5 u() u(1) 4 A possble control sequence s u() 2 and u(1) 3..5 Consder now that x(2) 1. In ths case we obtan the followng system of equatons u() u(1) that has no soluton. he reason s that the system s not reachable. he controllablty matrx, whch s 1.5 Wc.5.25 has ran equal to 1, whle the system s of order 2. Remar II.1 Consder the change of varables defned by z ( ) x ( ), where s a non-sngular matrx. In the new state varables the model defned n (II.1) becomes 1 z( 1) G z( ) Hu( ) Gz ( ) Hu ( ) In the new coordnate system, the controllablty matrx becomes n 1 W c H GH G H 1 n 1 1 W c H G H G H W Consequently, f W c s of full ran, then c W c wll also have ran equal to n. hs means that the reachablty of the system s ndependent of the system coordnates. 11

12 Observablty In order to solve the problem of determnng the state of system based on the observaton of the output (and nput) of the system, we frst ntroduce the concept of non-obsrvable states. Defnton: he state 1 when x s non-observable f there s a fnte 1 wth 1 n 1 such that y ( ) for x() x and u ( ) for 1. he system represented n (II.1) and (II.2) s observable f there s a fnte such that nowng the nputs u(), u(1),, u(-1) and the outputs y(), y(1),, y(-1) s suffcent to determne the ntal state of the system. o smplfy, assume that u ( ) and that y(), y(1),, y(n-1) are nown. hus, we can wrte the followng set of equatons: y() Cx() y(1) Cx(1) CGx() n1 y( n1) CG x() Usng vector notaton, the above set of equatons can be wrtten as follows C y() CG y(1) x() n 1 CG yn ( 1) (II.5) he state x() can be obtaned from (II.5) f and only f the observablty matrx W C CG n 1 CG has ran equal to n. he state x() s non-observable f x() les n the null ace of W. Example II.2 Consder the system 12

13 1.1,3 x( 1) x( ) 1 y ( ) 1.5 x ( ) he correondng observablty matrx s C 1.5 W CG.6.3 he ran of W s equal to 1 and the non-observable states le n the null ace of W, whch s defned by the drecton.5 1. In the fgure below, t s shown the system output for the followng ntal states: (a).5 1, (b) 1.5.5, (c) 2.5 (d) 1.5 State observers Usually, we have a mathematcal model of the process and we want to compute the states based on the nowledge of the nputs and outputs. Here, we dscuss several methods of computng the state. Suppose the system s descrbed by equatons (II.1) and (II.2) x( 1) Gx( ) Hu( ) (II.1) y( ) Cx( ) (II.2) 13

14 and we want to calculate the state x( ) based on the sequence of outputs and nputs y(), y(-1),, u(), u(-1),. he calculaton of the state wll be possble f the system s completely observable. Drect method to compute the state varables Consder the case of a SISO system. From (II.1) and (II.2) we have y ( n1) Cx ( n 1) y ( n2) CGx ( n1) CHu ( n 1) n1 n2 y ( ) CG x ( n1) CG Hu ( n1) CHu ( 1) he above set of equatons can be wrtten as follows y ( n1) u ( n1) y ( n 2) u ( n 2) Wx ( n1) y ( ) u ( 1) where C CG W and n 1 CG CH n2 n3 CG H CG H CH If the system s observable, the state x ( n 1) can be calculated as follows y ( n1) u ( n1) 1 y ( n2) 1 u ( n2) x ˆ( n1) W W y ( ) u ( 1) Usng (II.1), we can estmate the state at the present tme step n1 1 x ˆ( ) G W where y ( n1) u ( n1) y ( n 2) u ( n 2) y ( ) u ( 1) (II.6) 14

15 n2 n3 n1 1 G H G H H G W State reconstructon usng a dynamc system he drect calculaton method of the state has the advantage that the correct state wll be obtaned after at most n measurements. he dsadvantage s that the method may be senstve to dsturbances. hus, t s useful to have other alternatve methods. For the system represented (II.1) and (II.2), consder that the state s approxmated by the state ˆx of the model xˆ( 1) Gxˆ( ) Hu( ) (II.7) whch has the same nput as n (II.1). If the model used n (II.7) s perfect, or t has the same parameters as model (II.1) and the ntal condtons for models (II.1) and (II.7) are also the same, then the state ˆx of (II.7) wll be dentcal to state x of the real system (II.1). If the ntal condtons of (II.1) and (II.7) are dfferent, then the state ˆx wll converge to x only f the system s asymptotcally stable. he state reconstructon through (II.7) can be mproved ntroducng a correcton wth the predcton error y Cxˆ : x ˆ( 1/ ) Gx ˆ( / 1) Hu ( ) K y ( ) Cx ˆ( / 1) (II.8) where K s a gan matrx that has to be properly selected. he notaton xˆ( 1/ ) s used to ndcate that t s a predcton of x(+1) based on measurements untl tme nstant. Let us ntroduce the reconstructon error x xxˆ (II.9) and subtractng (II.8) from (II.1), we obtan x ( 1/ ) Gx ( / 1) K y ( ) Cx ˆ( / 1) x ( 1/ ) GKC x ( / 1) (II.1) hen, f K s selected such that (II.1) s asymptotcally stable, the reconstructon error wll always converge to zero. hus, ntroducng a feedbac from the measurements n the state reconstructon maes t possble to the state to converge to zero, even when system (II.1) s unstable. System (II.8) s desgnated as the state observer of system (II.1), because t estmates the states of the system based on measurements of the nputs and outputs. Another form of the state observer, n whch there s no measurement delay s gven by x ˆ( / ) Gx ˆ( 1/ 1) Hu ( 1) K y ( ) C Gx ˆ( 1/ 1) Hu ( 1) (II.11) 15

16 ˆ xˆ( / ) I KC Gx( 1/ 1) Hu( 1) Ky( ) (II.12) he reconstructon error of (II.12) s gven by x ( / ) x ( ) x ˆ( / ) GKCG x ( 1/ 1) When there s a gan K such that the egenvalues of G-KC can be postoned arbtrarly n the complex plane, the system s sad to be detectable. It can be shown that a system that s observable s also detectable. he same can be sad n relaton to matrx G-KCG. Matlab provdes the routne place to the computaton of matrx K that wll correond to a gven set of egenvalues for matrx G-KC or G-KCG. III. Model Predctve Control Suppose the process beng controlled s represented by the followng state-ace model where x( 1) Ax( ) Bu( ) (III.1) y( ) Cx( ) (III.2) nx x, nu u and ny y. It should be noted that the model defned n (III.1) and (III.2), t s assumed that the process s operatng around a steady-state that could be represented by ss ss u, y. Consequently, the nput and output vectors consdered n (III.1) and (III.2) can be consdered as ss ss devaton varables related to the steady-state y ( ) y ( ) y and u ( ) u ( ) u, where y ( ) and u ( ) would be the true system output and nput reectvely. Assume now that, for a tme step the state x() s nown and the controller computes a control sequence u ( ), u ( 1 ),..., u ( m 1 ) that mnmzes the followng objectve functon: p m1 J y( j ) y Q y( j ) y u( j ) Ru( j ) j1 j where (III.3) 16

17 p s the output predcton horzon; m s the nput horzon; y s the desred value of the output; u ( j) u ( j) u ( j 1 ) ; Q and R are matrces of approprate dmensons. o develop the functon defned n (III.3), one can use the model defned n (III.1) and (III.2) as follows y( 1 ) Cx( 1 ) CAx( ) CBu( ) y ( 2 ) CAx ( 1 ) CBu ( 1 ) ( ) ( ) ( 1 ) 2 CA x CABu CBu 3 2 y( 2 ) CA x( ) CA Bu( ) CABu( 1 ) CBu( 2 ) y j CA x CA Bu CA Bu CBu j j j1 j2 ( ) ( ) ( ) ( 1 ) ( 1 ) Suppose now that u ( m) u ( m1 ) u ( m1 ) (III.4) or the control nput remans constant after the tme nstant +m. Wth the assumpton defned n (III.4), the output predctons can be wrtten as follows: y m CA x CA Bu CA Bu CAB CB u m m 1 1 ( 1 ) m m ( ) ( ) ( 1 ) ( 1 ) y m CA x CA Bu CA Bu CA BCABCB u m m 2 m 1 m 2 ( 2 ) ( ) ( ) ( 1 ) ( 1 ) y p CA x CA Bu CA Bu CA Bu m p p1 p2 pm1 ( ) ( ) ( ) ( 1 ) ( 2 ) pm pm1 CA B CA B CB u( m 1 ) hus, the vector of output predctons can be wrtten as follows: y ( 1 ) CA CB 2 y ( 2 ) CA CAB CB u ( ) m m1 m2 y ( m) CA x ( ) CA B CA B CB u ( 1 ) m1 m m1 y ( m1 ) CA CA B CA B CA 1B u ( m1 ) p p1 p2 y ( p) CA CA B CA B CApmB 17

18 where A AI; A A AI; A A A I p m p m 1 pm (III.5) hen, the output predcton equaton (III.5) can be represented n the form y( ) x( ) u Now, defnng the set pont vector y y y p and the weght matrx Q dagq Q, the frst term of the rght hand sde of (III.3) becomes p j1 y j y Qy j y x u y Qx u y p ( ) ( ) ( ) ( ) Now, to develop the second term of the rght hand sde of (III.3), observe that u ( ) u ( ) u ( 1) u ( 1 ) u ( 1 ) u ( ) u Mu Iu( 1) u ( m1 ) u ( m1 ) u ( m2 ) where I M I I nu nu nu nu nu nu nu nu nu nu nu nu nu nu nu nu, M ( mnu. ) ( mnu. ) hen, the second term n (III.3) can be wrtten as follows m1 j Inu and I, I nu nu ( mnu. ) nu nu. m nu. m u ( j) Ru ( j) ( I Mu ) Iu ( 1) R( I Mu ) Iu ( 1) and the control cost functon can be wrtten as follows ( ) ( ) ( 1) ( 1) J x u y Q x u y I u Iu R I u Iu M M where IM Inu. m M and R dag R R. m he objectve can be reduced to a quadratc functon of the form 18

19 f J u Hu 2c u c where H Q IM RIM c x( ) y Qu( 1) I RI f c x( ) y Q x( ) y u( 1) I RIu( 1) Fnally, the control law of the MPC results from the soluton to the followng QP: M mn u Hu 2c u u st.. u u( j ) u, j,1,, m1 mn f max max u u( j ) u( j1 ) u, j,1,, m1 max (III.6) IV. Dynamc Matrx Control (DMC) hs was the frst predctve controller reported n the lterature and was developed n the 197s n ndustry (Cutler, C. R. & Ramaer B. L., Dynamc matrx control a computer control algorthm, Jont Automatc Control conference, San Francsco, CA, 198). It s based on the step reonse model of the process, whch for a system of ny outputs and nu nputs correonds to the followng non-mnmal state-ace model: where x( 1) Ax( ) B u( ) (IV.1) x ( ) y ( 1 ) y ( 2 ) y ( n) y ( j) s the output predcton at tme +j computed at tme n s the stablzng horzon of the system I I A I I ny ny ny ny ny ny ny ny ny ny ny ny ny ny ny ny, A ( ny. n) ( ny. n)) ; S S B S 2 3 ( ny. n) nu n1, B u ( j) u ( j) u ( j 1 ), nu u 19

20 S j s the matrx of coeffcents of the step reonse at tme j, S j nynu Remars -In the model representaton defned above, the state correonds to the trajectory of the system output f no control moves are njected nto the system. -he state of the model defned above s non-mnmal, whch means that a state wth dmenson smaller that the above state could be obtaned, for nstance by startng from the transfer functons of the system. Wth the model defned n (IV.1), the output predcton vector, tang nto account all the future control moves, becomes y 1 c( 1 ) S nynu nynu ( 2 ) S2 S1 yc nynu u( ) u ( 1 ) Nx( ) yc ( m ) Sm Sm 1 S1 u ( m1 ) yc ( p ) Sp Sp 1 Spm 1 (IV.2) Equaton (IV.2) can also be wrtten n the form y( ) Nx( ) DM u (IV.3) where D M s defned as the dynamc matrx of the system and N I horzon becomes ny. p ( ny. p) ( ny.( n p)) y y y p. Now, defnng the vector of desred values along the output ( ) ( ) and usng (IV.3), the control objectve defned n (III.3) J Nx D u y Q Nx D u y u R u M M 2 ( ) J u DMQDM R u Nx y QDMu Nx( ) y QNx( ) y hen, defnng H DMQDM R, cf Nx( ) y QDM, the QDMC controller defned as the soluton to the problem: 2

21 mn u Hu 2c u u st.. u u( j/ ) u, j,1,, m1 mn max f max u u( j/ ) u, j,1,, m1 max (IV.4) Remars 1) Observe that the model defned n (IV.1) s ncremental n the nput. Consequently, the decson varables of the problem defned n (IV.4) s the vector of nput ncrements u. o vsualze the advantage of usng a model of the form (IV.1) over the conventonal state-ace model defned n (III.1), consder the soluton to problem (IV.2) when no constrants are ncluded. In ths case, we have the least squares soluton of (IV.4): u NH D Q Nx y 1 ( / ) 1 M ( ) where hen N1 I nu nu nu u u NH D Q Nx y 1 ( / ) ( 1) 1 M ( ) 1 u( / ) u( 1) K Nx( ) y (IV.5) where K 1 s a constant gan and Nx( ) y s the output predcted error along the predcton horzon. hen, the controller defned n (IV.5) s an ntegral controller and capable of elmnatng the output offset. Now, consder the same unconstraned soluton to problem (III.6) u NH Q x y I RIu I 1 ( / ) 1 ( ) M ( 1) u ( / ) K 1 x ( ) y Ku 2 ( 1) (IV.6) Snce K2 Iny, t s clear that n ths case, the controller defned n (IV.6) s a pure proportonal controller and wll not be capable of elmnatng the output offset for changes n the output set pont. 2) o nclude the effect of measured dsturbances, the model defned n (IV.1) can be modfed as follows: x( 1) Ax( ) Bu( ) B d( ) d 21

22 where B d S S S d,2 d,3 dn, 1 V. MPC wth Output Feedbac In the MPC controllers defned n sectons (III) and (IV), t s assumed that the state of the model utlzed to buld up the output predcted trajectory s perfectly nown, whch does not usually happen n practce. In most applcatons, only the system controlled outputs are measured and the state must be estmated. For nstance, consder the case n whch the model s defned as n (IV.1). In ths case, at each samplng step, the state has to be updated wth the last output measurement followng the usual state observer strategy: x( ) Ax( 1) Bu( 1) K y ( ) C Ax( 1) Bu( 1) (V.1) F where y ( ) s the real plant output measurement at tme, K F s the observer gan and C s the output matrx. In the model defned n (IV.1), we have C Iny ny ny and KF ny ny ny I I I (V.2) n Equaton (V.1) can be developed as follows x ( 1) IK C Ax ( ) IK C Bu ( ) K y ( 1) F F F F Remar As dscussed n secton (II), matrx I K C must be stable n order to the state estmaton to converge to the true state. Wth matrces A and K F correondng to the DMC controller, we have ny ny ny ny ny ny ny Iny I ny ny ny ny ny Iny ny Iny ny ny I KFCA ny Iny ny ny ny I ny ny Iny ny ny ny Iny 22

23 whch has all the egenvalues at and consequently s stable. Systems wth Integratng Poles he model defned n (IV.1) nvolves the assumpton that f n s the system stablzng tme, then y( n1/ ) y( n/ ) hs means that, we are consderng only open-loop stable systems, or that the step reonse of the system has the shape below When we deal wth ntegratng system (e.g. the level of lqud n a drum) the above assumpton cannot be appled. For nstance, suppose we have a system whose step reonse s represented n the fgure below: 23

24 In ths case, we can consder a lnear extrapolaton of the step reonse curve, such that y( n1/ ) y( n/ ) y( n/ ) y( n1/ ) 2 y ( n/ ) y ( n1/ ) Wth ths consderaton, the state matrx of the model defned n (IV.1) becomes I I A I I 2I ny ny ny ny ny ny ny ny ny ny ny ny ny ny ny ny ny ny ny ny hen, usng an observer wth the gan defned n (V.2), the state matrx of the observer error model becomes ny ny ny ny ny ny ny Iny I ny ny ny ny ny Iny ny Iny ny ny I KFCA ny Iny ny ny ny I ny ny Iny ny ny Iny 2Iny It can be shown that the above matrx has ny egenvectors equal to 1 and all the remanng egenvalues are equal to. Consequently, the observer defned n (V.2), whch s the observer used by DMC, s not stable. 24

25 Remar An observer wth gan KF Iny Iny Iny 1.1Iny would stablze the output feedbac of the ntegratng system. VI. State-ace models n the ncremental form Besdes the state-ace model derved from the step reonse model that naturally produces a model n the ncremental form as shown n (IV.1), we can bult up an ncremental model startng from the usual form of the state-ace model defned n (III.1). Suppose we have the model x( 1) Ax( ) Bu( ) (VI.1) y( ) Cx( ) (VI.2) and to prevent offset we want a model n the form x( 1) Ax ( ) B u( ) (VI.3) y ( ) Cx ( ) (VI.4) Here, we present three new forms of constructng the ncremental model startng from the model defned n (VI.1) and (VI.2). 1) Consder model (VI.1) wrtten for tme nstant x ( ) Ax ( 1) Bu ( 1) (VI.5) y( ) Cx( ) hen, subtractng (VI.5) from (VI.1), we have x( 1) ( I A) x( ) Ax( 1) B u( ) (VI.6) As we saw n secton (I), that equaton (VI.6) s a dfference equaton of the form x( 2) ax ( 1) ax ( ) bu ( 2) bu ( 1) bu ( ) (VI.7) whch, through a change of varables, can be converted nto a state-ace model of the form where ( 1) 1 ( ) x( ) x ( ) h u( ) x1 x1 h1 u x2( 1) a2 a1 x2( ) h2 1 ( ) 25

26 h b h1 b1ab 1 h b ah a h Eq. (VI.6) has the form of eq. (VI.7) wth: a I A h, h1 B, h I AB and the resultng system s 2 x ( 1) I x ( ) B u ( ) A ( I A) ( I A) B 1 1 x2( 1) x2( ) y( ) C x ( ) 1 ; b ; b 1 B and b2. hus, It can be shown that the state matrx of the above system has the egenvalues of A and n x egenvalues equal to 1, where n x s the dmenson of the state of the system (VI.5). 2) Another way to obtan a state-ace model n the ncremental form also starts from the model defned n (VI.5) and we substtute the followng equaton u ( ) u ( 1) u ( ) (VI.7) nto (VI.5) to produce x( 1) Ax( ) B u( 1) u( ) hen, we can defne the followng system x ( 1) A B x ( ) B u ( ) u ( ) I nu u ( 1) I x ( ) y ( ) C u ( 1) In ths case, the state vector has dmenson equal to n x +n u and the state matrx has the same egenvalues as the orgnal state matrx A and n u egenvalues located at +1. 3) he thrd way to buld an ncremental model s to wrte (VI.5) at tme nstants +1 and, and subtract the two equatons. he result s x( 1) Ax( ) B ( ) (VI.8) hen, the system output can be wrtten as follows y ( 1) Cx ( 1) C x ( 1) x ( ) CAx( ) CBu( ) y( ) (VI.9) 26

27 Now combnng (VI.8) and (VI.9) we obtan x ( 1) A x ( ) B u ( ) y ( 1) CA I ny y ( ) CB x ( ) y ( ) Iny y ( ) (VI.1) In ths case, the state of the model defned n (VI.1) has dmenson n x +n y and the state matrx has the same egenvalues as A besdes n y egenvalues at +1. Plant Model plus State Observer In a MPC applcaton, the state-ace model s combned wth the state observer as follows Plant model: x( 1) Ax( ) Bu( ) y( ) Cx( ) State observer: xˆ( 1) Axˆ( ) Bu( ) K F y( 1) CAxˆ( ) Bu( ) Combnng the above equatons we have x ˆ( 1) I K ˆ FC A KFCAx ( ) BKFC( BB) u ( ) x ( 1) A x ( ) B Observe that the system model AB, used by the observer may be not exactly the same as the true plant system model AB,. We have already seen that the observer gan K F should be such that I K CA s stable otherwse the closed-loop system wll be unstable. F A non-mnmal state ace model where the states are the nput and output he realgnment model Intally, consder the case of a pure ntegrator process (e.g. the level of lqud n a drum) and let us use the followng model 1 y ( 1) y ( ) y ( ) y ( 1) S u ( ) y ( 1) 2 y ( ) y ( 1) S u ( ) (VI.11) Consder now the followng state 1 27

28 y ( ) x ( ) y ( 1) y ( 2) and obvously y ( ) 1 x( ). hen, the model defned n (VI.11) s equvalent to the followng non-mnmal state-ace model: y ( 1) 2 1 y ( ) S1 y( ) 1 y( 1) u( ) y ( 1) 1 y ( 2) x ( 1) A x ( ) B Consder also a state observer wth the followng gan the state observer error wll be the followng I K CA F K F 1, then the state matrx of I KFCA 1 that has all the egenvalues at and consequently s stable. he 1 model correondng to equaton (VI.11) wth the observer defned above, smply extrapolates the plant reonse. In the general case, consder the followng transfer functon model na nb 1 1 (VI.12) y( ) a y( ) bu( ) where na correonds to the order of the model and nb s equal to number of past tme nstants consdered for the nput and Suppose now, we defne the followng state a nyny, b nynu, y ny1, u nu1. x ( ) y ( ) y ( 1) y ( na1) u ( 1) u ( 2) u ( nb1) then, the correondng state-ace model can be wrtten as follows 28

29 a1 a ( ) 2 ana 1 ana b2 bnb 2 bnb 1 b y nb y ( 1) b1 I y ( 1) ny ny ny ny nynu nynu nynu nynu y ( 2) nynu y ( 2) ny Iny ny ny nynu nynu nynu nynu nynu y ( na1) y ( na1) ny ny Iny ny nynu nynu nynu nynu y ( na) u ( 1) nynu u ( 1) nuny nuny nuny nuny nu nu nu nu u ( 2) Inu u ( 2) nuny nuny nuny nuny Inu nu nu nu nu u ( nb 2) u ( nb2) nuny nuny nuny nuny nu Inu nu u ( nb1) nu nu u ( nb1) nuny nuny nuny nuny nu nu I u ( nb) nu nu nu or x ( ) Ax ( 1) Bu ( 1) (VI.13) y ( ) Cx ( ) where C [ Iny ny... ny nynu nynu... nynu ] na nb1 na. nynb1. nu 1 x ; A na ny nb nu na ny nb nu ; B na ny nb nu nu ; C ny na ny nb nu ; he state observer can be characterzed by the followng gan matrx K I F ny ny ny ny nu ny nu ny nu Wth ths state observer, the resultng state matrx of the model plus observer system becomes Z I K f CA J where Z A A B B and 2 na 2 nb Iny J Iny hs state matrx has all the egenvectors equal to zero. hs means that the observer stablzes any stable or unstable process. he realgnment model n the ncremental form Wrtng the model equaton represented n (VI.12) at tmes and -1 we have 29

30 na y a y a y a y a y na bu bu bu b u nb na y a y a y a y a y na bu b u bu b u nb hen, subtractng one equaton from the other results y I a y 1 a a y 2 a a y 3... ny ana ana1 y na ana y na 1 b u 1 b u 2 b u 3... b u nb nb nb nb Followng the same procedure adopted for the model defned n (VI.12), t can be shown that the above model s equvalent to a1iny a2 a1 a3 a2 a4 a3... ana ana 1 ana b2 b3 b4... b nb Iny ny ny ny... ny ny nynu nynu nynu... nynu ny Iny ny ny... ny ny ny nu ny nu ny nu... nynu ny ny Iny ny... ny ny nynu nynu nynu... nynu x 1 ny ny ny ny Iny ny nynu nynu nynu nynu nynu x nuny nuny nuny nuny nu ny nu ny nu nu nu nu nu nuny nuny nuny nuny nuny nuny Inu nu nu nu nu nuny nuny nuny nuny nu ny nu ny nu Inu nu nu nu nuny nuny nuny nuny nuny nuny nu nu Inu nu nu nuny nuny nuny nuny nuny nuny nu nu nu Inu nu b1 nynu nynu nynu... nynu u I nu nu nu nu... nu Example Consder the system represented by the followng dfference equaton model y ( ) ay ( 1) ay ( 2) bu ( 1) bu ( 2) he correondng realgnment model s gven by 3

31 y ( ) (1 a1) ( a2 a1) a2 b2 y ( 1) b1 y ( 1) 1 y ( 2) u ( 1) y ( 2) 1 y ( 3) u ( 1) u ( 2) 1 Now consderng a state observer wth gan K 1 and C 1 F We have 1 I KFCA, whch has all the egenvalues at zero. Consequently, the 1 observer stablzes all the poles (stable, ntegratng and unstable) of the system. VII. Model Predctve Control wth the state-ace model n the ncremental form Here, we assume that the process beng controlled s represented by the followng stateace model where x( 1) Ax( ) Bu( ) (VII.1) y ( ) Cx ( ) (VII.2) nx x, nu u, ny y and u ( ) u ( ) u ( 1). Now, for a tme step the state x() s supposed to be nown and the controller computes a control sequence u ( ), u ( 1 ),..., u ( m 1 ) that mnmzes the same objectve functon of the MPC controller developed n Secton 3: p m1 J y( j ) y Q y( j ) y u( j ) Ru( j ) j1 j where p s the output predcton horzon; m s the nput horzon; (III.3) y s the desred value of the output; u ( j) u ( j) u ( j 1 ) ; Q and R are matrces of approprate dmensons. 31

32 Usng the model defned n (VII.1), we can represent the vector of output predctons as follows y ( 1 ) Cx ( 1 ) CAx ( ) CBu ( ) y ( 2 ) CAx ( 1 ) CBu ( 1 ) 2 CA x( ) CABu( ) CBu( 1 ) y CA x CA B u CAB u CB u 3 2 ( 2 ) ( ) ( ) ( 1 ) ( 2 ) y j CA x CA B u CA B u CB u j j j 1 2 ( ) ( ) j ( ) ( 1 ) ( 1 ) Suppose now that u ( m) u ( m1 ) (VII.3) or the control nput remans constant after the tme nstant +m. Wth the assumpton defned n (VII.3), the output predctons can be wrtten as follows: y ( 1 ) CA CB 2 y ( 2 ) CA CAB CB u ( ) m m1 m2 y ( m) CA x ( ) CA B CA B CB u ( 1 ) m1 m m1 y ( m 1 ) CA CA B CA B CAB u ( m1 ) p p1 p2 pm y ( p) CA CA B CA B CA B hen, the output predcton equaton (III.5) can be represented n the form y ( ) x ( ) u (VII.4) he set pont vector s defned as n Secton III y y y p, as well as the weght matrces Q dagq Q and R dag R R. p m hen, the control cost functon can be wrtten as follows 32

33 ( ) ( ) J x u y Q x u y u R u where u u( / ) u( 1/ ) u( m1/ ) he objectve can also be reduced to a quadratc form where f J u Hu 2c u c H Q R c x( ) y Q f ( ) ( ) c x y Q x y he control law of the MPC based on ncremental state-ace mode results from the soluton to the followng QP: mn u Hu 2c u u st.. u u( j/ ) u, j,1,, m1 mn max f max u u( j/ ) u, j,1,, m1 max (VII.5) Remar Observe that, n the problem defned n (VII.5), the vector of nputs can be related wth the vector of nput moves, whch are the decson varables of the problem, as follows: u M u Iu ( 1) where Inu Inu Inu Inu I nu M ; I I nu Inu Inu Inu 33

34 VIII. he Output Predcton Orented Model (OPOM) Consder a MIMO nonntegratng system wth nu nputs and ny outputs, where the Laplace transfer functon relatng nput u j to output y s nb b, j, b, j,1 s b, j, nbs G, j() s na 1a, j,1 sa, j, nas where na, nb nb na (VIII.1). he correondng step reonse at tme step can be calculated, for a samplng perod, by the followng expresson: na d r, j, l, j( ), j, j, l l1 S d d e (VIII.2) where coeffcents d d, j,, j,1,,, j, na d d d are obtaned by partal fractons expanson and r, r,, r are the poles, whch are assumed to be dstnct. For the sequel of ths, j, j,1, j, na wor, t s convenent to defne the followng coeffcent matrces d1,1 d 1, nu D, dny,1 dny, nu D nynu d d d d d d d d d dag 1,1,1 1,1, na 1, nu,1 1, nu, na ny,1,1 ny,1, n1 ny, nu,1 ny, nu, na D d d d d d d d d D d ndnd For ths system, a dscrete tme state ace model of the form represented n (1) and (2) can be wrtten as follows where s s x ( 1) Iny x ( ) D u ( ) d ( 1) F d d x x ( ) D FN s x ( ) y ( ) Iny d x ( ) (VIII.3) (VIII.4) 34

35 s x x1 x ny, s ny d x, x xny 1 xny2 xny( nu na1), d nd x, nd nu na ny r1,1,1 r1,1, na r1, nu,1 r1, nu, na rny,1,1 rny,1, na rny, nu,1 rny, nu, na F dag e e e e e e e e F ndnd J N J 1 ny, ndnu N, Φ Ψ, Ψ Φ J 1 1, 1 1 nynd, Φ 1 1 nu nanu J, Φ nu na he Infnte Horzon MPC wth OPOM he nfnte horzon MPC s based on the followng cost or control objectve: m1 (VIII.5) j1 j J e ( j) Qe ( j) u ( j) Ru ( j) where e ( j) y ( j) y, y( j ) s the output predcton tang nto account the effects of the future control actons, y s the output reference, nyny Q, R nunu and nyny S are assumed postve defnte. he control objectve defned n (VIII.5) can be wrtten as follows m J e( j ) Qe( j ) j1 m1 Cx( j ) y Q Cx( j ) y u( j ) Ru( j ) jm1 j (VIII.6) Wth the proposed model formulaton, the second term on the rght hand sde of (VIII.6) can be developed as follows 35

36 jm1 Cx( j) y QCx( j) y j1 s j d s j d x ( m ) y F x ( m ) Qx ( m ) y F x ( m ) (VIII.7) s In ths nfnte sum, the term x ( m ) y does not depend on ndex j and, therefore, must be made equal to zero to eep the control objectve bounded. hus, the followng constrant has to be ncluded n the control optmzaton problem: s x ( m ) y (VIII.8) Wth ths condton, Equaton (VIII.7) can be smplfed as follows: jm1 Cx( j) y QCx( j) y d j j d x ( m ) ( F ) QF x ( m ) j1 (VIII.9) j For stable systems, lm F and the nfnte sum n the rght hand sde of (VIII.9) can j be reduced to a sngle term: j1 d j j d d d x ( m) ( F ) QF x ( m) x ( m) Qx ( m) where Q ndnd s such that QF QF F Q F (VIII.1) Equaton (VIII.1) s the Lyapunov equaton for the dscrete tme system represented n (3), and (4). Hence, the control objectve defned n (VIII.5) becomes: m m-1 d d J e( j ) Qe( j ) x ( m ) Qx ( m ) u( j ) Ru( j ) j1 j (VIII.11) o obtan a smpler expresson for the above control objectve, equatons (VIII.3) and (VIII.4) can be used to evaluate the state and output at future samplng nstants as follows s d y( j ) x ( j ) x ( j ) (VIII.12) 36

37 s s x ( j ) x ( ) D u( ) D u( j1 ) (VIII.13) d j d j1 d j2 d x ( j ) F x ( ) F D FNu( ) F D FNu( 1 ) d DFNu ( m1 ) (VIII.14) Applyng (VIII.13), for the tme steps nsde the control horzon, we obtan the followng expresson s x Ix s ( ) D u (VIII.15) where x D m s x ( 1 ) Iny mnyny, x, I, I ; s x ( m ) Iny s s mny D, D D D u ( ) u, u u ( m1 ) mny. mnu. mnu. m m, For the same tme steps, (VIII.14) produces d d x ( 1 ) F I DFN d 2 d x ( 2 ) F F I DFN d x ( ) u d m m1 m2 d x ( m ) F F F I DFN d d x F x ( ) F u (VIII.16) x u Wth the varables defned above, the control objectve represented n (VIII.5) can be wrtten as J u Hu 2c u c (VIII.17) where f m 1 u 1 m 1 u u 2 u 1 H D F Q D F F Q F R (VIII.18) s 1 1 ( ) d 1 ( ) d 2 ( ) c D F Q Ie F x F Q F x (VIII.19) f m u x u x 1 x 1 1 x x 2 x s d s d d d ce( ) Qe( ) Ie ( ) F x ( ) Q Ie ( ) F x ( ) F x ( ) Q F x ( ) (VIII.2) 37

38 m m m Q1 dag[ Q Q], Q2 dag[ Q], R1 dag[ R R] and m 1 dag[ ] e ( ) x ( ) y s s Usng (VIII.13), the constrant represented n (VIII.8) can be wrtten as follows: s e ( ) D u (VIII.21) where D D D Fnally, the control optmzaton problem of the nfnte horzon MPC (IHMPC) can be formulated as follows: mn u J subject to (VIII.22) s x ( ) y D u (VIII.23) u ( j) U, j (VIII.24) max max u u( j) u u ( j) u ( j) ; jm j mn max u u( 1) u( ) u ; j,1,, m1 he stablty of the closed loop systems wth the controller defned trough the soluton to the problem defned through (VIII.22) to (VIII.24) can be proved as long as ths problem remans feasble. hs result can be summarzed n the followng theorem: heorem: For an undsturbed stable system, f the problem defned through (VIII.22) to (VIII.24) s feasble, t wll reman feasble at the subsequent tme steps and, f the desred steady state s achevable, the controller produced by the soluton of ths problem s asymptotcaly stable. 38

39 Proof: Observe that H and the problem defned through (VIII.22) to (VIII.24) s convex and so t has an unque soluton at each tme step. Now, suppose that * * * * u u ( ) u ( 1 ) u ( m1 ) correonds to the optmal soluton to ths problem at tme step. hen, the frst control move u * ( ) s njected n the real system and at tme step +1 consder the followng control sequence * * 1 ( 1 ) ( 1 ) u u u. It s clear that u 1 satsfes (VIII.24), then, to show that ths control sequence s feasble, we need to show that t also satsfes (VIII.23). At tme +1, the left hand sde of (VIII.23) wth u 1 can be wrtten as follows: s Z x ( 1) y D u 1 Now, usng (VIII.3), the above expresson can be wrtten as follows Z x D u y D u D u m D s * * * ( ) ( ) ( 1 ) ( 1 ) Z x y D u s * ( ) hen, u 1 s a feasble soluton and recursve feasblty of the controller s proved. o prove the stablty of the system wth the proposed controller let the value of the objectve functon correondng to u 1 be desgnated J 1. It s straghtforward to show that * * * d J 1 J e ( 1 ) Qe ( 1 ) x ( m ) F Q Fx ( m ) u ( ) R u ( ) d If one of the terms e ( 1 ), x ( m ) or u ( ) s not equal to zero, then J 1 J * and consequently J J. If ths stuaton s repeated at the subsequent tme steps +1, * * 1 +2,, the objectve functon wll converge to zero. Consequently, the system output wll converge asymptotcaly to the set pont as tends to nfnte and convergence s proved. For large changes on x s () or y or f y correonds to an unreachable steady state, then the optmzaton problem defned through (VIII.22) (VIII.24) may become nfeasble 39

40 because of a conflct between constrants (VIII.23) and (VIII.24). Consequently, the IHMPC as defned above cannot be mplemented n practce. he extended Infnte Horzon MPC wth OPOM o produce an nfnte horzon MPC, whch s mplementable n practce, the objectve functon of nfnte horzon MPC s re-defned as follows: m1 ( ( ) ) ( ( ) ) ( ) ( ) j1 j J e j Q e j u j R u j S where ny s a vector of slac varables and nyny (VIII.25) S s a assumed postve defnte. Observe that each slac varable refers to a gven controlled output. Weght matrx S should be selected such that the controller tends to zero the slacs or at least mnmze them dependng on the number of nputs, whch are not constraned. Analogously to the control objectve defned n (VIII.5), t sconvenent to wrte (VIII.25) as follows m ( ) ( ) J e j Q e j j1 jm1 m1 j Cx( j ) y QCx( j ) y u ( j) Ru ( j) S (VIII.26) Analogously to the case wthout slac, the second term on the rght hand sde of (VIII.26) can be developed as follows jm1 j1 Cx( j ) y QCx( j ) y s j d s j d x ( m ) y F x ( m ) Qx ( m ) y F x ( m ) (VIII.27) 4

41 s In ths nfnte sum, the term x ( m ) y must be made equal to zero to eep the control objectve bounded. hus, nstead of (VIII.8), the followng constrant has to be ncluded n the control optmzaton problem: s x ( m ) y (VIII.28) Wth ths condton, Equaton (VIII.27) becomes: jm1 Cx( j ) y QCx( j ) y d j j d d d x ( m ) ( F ) QF x ( m ) x ( m ) Qx ( m ) j1 (VIII.29) where Q ndnd s obtaned by solvng (VIII.1) Hence, the control objectve defned n (VIII.25) becomes: m d d j1 J e( j ) Q e( j ) x ( m ) Qx ( m ) m-1 u ( j ) Ru ( j ) S j (VIII.3) Equatons (VIII.15) and (VIII.16) can now be used to wrte the objectve functon n a more compact form and t results n the followng quadratc expresson: where H c f u u J u H 2cf c m u m u u u m u 1 m 1 u 1 (VIII.31) D 1F Q1 D 1F F Q2F R1 D 1F Q1I (VIII.32) I Q D F S I Q I Q s d d Dm 1Fu Q1Ie ( ) 1Fxx ( ) Fu Q2Fxx ( ) s d I Q1Ie ( ) 1Fx x ( ) Qe( ) 1 x 1 1 x x 2 x s d s d d d ce( ) Qe( ) Ie ( ) F x ( ) Q Ie ( ) F x ( ) F x ( ) Q F x ( ) Usng the model equaton (VII.3), the constrant represented n (VIII.28) can be wrtten as follows: s x ( ) y D u (VIII.33) 41

42 Fnally, the control optmzaton problem of the extended nfnte horzon MPC (IHMPC) can be formulated as: u u J u H c c mn 2 f u, subject to (VIII.34) Equatons (VIII.33) and (VIII.24) he convergence of the extended nfnte horzon MPC resultng from (VIII.33) can be summarzed n the followng theorem: heorem: For a stable system, f n the control objectve defned n (VIII.25), the slac weght matrx S s postve defnte, then, the control law produced by the soluton to the problem defned n (VIII.34) drves the system output to the desred set pont f t correonds to a reachable state state. If the desred set pont s not reachable, the controller wll stablze the system n a steady state such that the dstance between ths steady state and the desred steady state s mnmzed. Proof: Suppose that * * u correonds to the optmal soluton of (VIII.34) at tme * step. It s easy to show that u 1 s a feasble soluton to (VIII.34) at +1. he correondng value of the objectve functon s d d J J e( 1 ) Q e( 1 ) x ( m ) F QFx ( m ) * * * 1 * * u Ru ( ) ( ) d If one of the terms e ( 1 ), x ( m ) or u ( ) s not equal to zero, then J J and consequently 1 * J J. If ths stuaton s repeated n the subsequent tme * * 1 steps +1, +2,, the cost wll converge ether to zero or to (wth ). If the objectve converges to zero, the system output wll converge to the reference as tends to nfnte and convergence s proved. If the desred steady state s not reachable, the closed loop system wll converge toa stedy state where S S s mnmzed. 42

43 Example: Consder a 22 system represented by the followng transfer functon: () 9.5s 1 9s 1 Gs s1 1.2s1 Intally (Case 1) consder the followng tunng parameters for the IHMPC =1; y = [.5 -.1]'; dumax = [.1.1]'; umax = [1 1]'; umn = [-1-1]'; Q=dag([1 1]); R=dag([1 1]); S=dag([1 1]); he system reonses can be seen n fgures 1 to 3. he outputs are drven to ther desred set ponts and the control cost tends to zero. y y me Fgure 1. Outputs of the example system (Case 1) me u me u me Fgure 2. Inputs of the example system (Case 1) 43

44 J me Fgure 3. Control cost for Case 1 In the second case, the outputs set ponts are changed to y = [1. -.1]'; he system reonses are represented n fgures 4 to 6. From fgure 4, we can see that n ths case the desred set ponts are not reched by the system outputs. hs s so because nput u 2 reaches ts mnmum bound (= -1) and consequently the desred set pont s not reachable. However, the system s stll stable and fgure 6 shows that the control cost tends to a mnnum value that s not equal to zero y y me Fgure 4. Outputs of the example system (Case 4) me u u me Fgure 5. Inputs of the example system (Case 2) me 44

45 J me Fgure 6. Control cost for Case 2 he Infnte Horzon MPC wth OPOM for ntegratng Systems Agan, t s assumed that we have a multvarable system wth nu nputs and ny outputs, and for each par (y, u j ) there s a transfer functon model: 1 2 nb b, j,1 z b, j,2 z b, j, nbz G, j( z) (VIII.35) 1 2 na 1 (1 a, j,1 z a, j,2 z a, j, naz )(1 z ) where na, nb. When the poles of the system are non-repeated, the th coeffcent of step reonse can be wrtten as follows: na d, j( ), j, j, l l, j l1 S d d r d (VIII.36) where r l, l =1, 2,..., na are the non-ntegratng poles of the system, s the samplng perod and coeffcents d d d, d,, d, na d are obtaned by partal fractons expanson of the transfer functon G,j. An equvalent state ace model, whch produces an offset free MPC, can be wrtten n the followng form x( 1) Ax( ) B u( ) (VIII.37) y( ) Cx( ) where s x d [ x] x, x x nx ny nd, nd ny. nu. na, nx, 2 s ny x, d ny. nu. na x, ny x 45

46 Iny Iny nxnx A F, A, Iny D D d B DFN, D B nxnu y y1, C I ny ny y ny s s s s x x1 x2 x ny, x x1 x2 x ny d d d d d d d d d 1,1,1 1,1, na 1,2,1 1,2, na 2,1,1 2,1, na ny,1, na ny, nu, na x x x x x x x x x D d d d d 1,1 1, nu ny,1 ny, nu, D ; nynu D d d d d 1,1 1, nu ny,1 ny, nu F r r r r r r r r dag 1,1,1 1,1, na 1, nu,1 1, nu, na ny,1,1 ny,1, na ny, nu,1 ny, nu, na D dag d d d d d d d d, D d d d d d d d d d 1,1,1 1,1, na 1, nu,1 1, nu, na ny,1,1 ny,1, na ny, nu,1 ny, nu, na nynu F ndnd D d ndnd N J J J 1 2 ny, N nd nu ; J 1 1, 1 1 J nu nanu, 1, 2,, ny Φ Ψ nynd nu na, Ψ, Φ 1 1, Φ Φ In such model representaton, x s correonds to the ntegratng states ntroduced by the ncremental form of the nput, x d correonds to the stable states and x correonds to the true ntegratng states of the system. In the state matrx A, the system poles appear n ts man dagonal. he frst ny components of the man dagonal are the ntegratng poles created by the ncremental form of the model. Matrx F s also dagonal and contans the stable poles of the system. Fnally, the last ny components of the man dagonal correond to the true ntegratng poles of the system. 46

47 Example: he system represented by the followng transfer matrx z 1 z -.95 Gz ( ) z z 1 Usng a samplng perod =1, ths model can be translated nto the model formulaton represented n (VIII.37) as follows: s s x1( 1) x 1( ) s s x2( 1) 1 x 2( ) d d x1 ( 1).95 x1 ( ) d.969 d u ( ) x2( 1) x2( ) x1( 1) x1( ) x2( 1) x 2( ).235 s x1 ( ) s x2 ( ) d y1( ) 1 1 x1 ( ) y2( ) 1 1 d x2 ( ) x1 ( ) x2( ) Now, consder agan the control objectve functon of the nfnte horzon MPC: m1 s s s s ( ( ) ) ( ( ) ) ( ) ( ) 1 j1 j J e j Q e j u j R u j S (VIII.38) For the model defned n (VIII.37) n order to eep the control objectve bounded, the followng equalty constrants should be obeyed by the control sequence: where x y D D u (VIII.39) s s ( ) ( 2, m m) x ( ) D u (VIII.4) 1m m D [ D D D ], m m, D1 [ D D D ], nym. nu D m m D1 m nym. nu 47

48 D2 m D ( m1) D, D 2 m nym. nu u ( ) u, u u ( m1 ) mnu. Substtutng Eqs. (VIII.39) and (VIII.4) nto (VIII.38), the control cost becomes m j1 d d m1 s s u ( j ) Ru ( j ) S1 j J e( j ) Q e( j ) x ( m ) Qx ( m ) where Q s obtaned from the soluton of the followng dscrete Lyapunov equaton QF QF F Q F Wrtng model equatons (VIII.37) for future tme nstants, we can obtan y Lx( ) G u where y y( 1 ) y( 2 ) y( m ) L L(1) L(2) L( m) j, L( j) Iny F j I ny G1,1 G2,1 G2,2 G G G G m,1 m,2 m, m It can also be shown that, jl d Gjl, D D F D FN ( j l) D d x ( m ) F x( ) F u where F x u m F, x ndny ndny d m d m1 d Fu D F N D F N D FN Fnally, the control cost can be wrtten as follows s u u J u H 2C s f c s (VIII.41) where 48

49 H G QG Fu QFu R G QI I QG I QI S1 Cf ( I y L x( )) QG x( ) FxQF u [ I y Lx( )] QI c[ I y L x( )] Q [ I y L x( )] x( ) F QF x( ) x ( ) x( ) x ( ) x( ) Iny I I I ny s d ( mny. ) ny m m Q dag[ Q Q], R dag [ R R ] hus, the nfnte horzon MPC for the ntegratng system can be wrtten as the followng optmzaton problem: x x mn J s u, subject to s s ( ) ( 2, m m) (VIII.42) e D D u (VIII.42a) x ( ) D u (VIII.43) 1m u ( j) U, j (VIII.44) max max u u( j) u u ( j) u ( j) ; jm j mn max u u( 1) u( ) u ; j,1,, m1 If at a gven tme step the problem defned n (VIII.42) s feasble and at steady state, the nput that correonds to y les nsde the set, the convergence of the system output n closed loop to the reference value s assured for the undsturbed system by the followng theorem: 49

50 heorem: For the undsturbed system wth stable and ntegratng poles and the slac weght S 1 s postve defnte, f Problem (VIII.42) s feasble at samplng step, then t wll be feasble at tme steps +1, +2,, and the output of the closed-loop system wth the control law defned by Problem (VIII.42) wll converge to the desred steady state. Proof: he proof of ths theorem follows the same steps as the controller defned by Problem (VIII.34) for stable systems. It can be proved that f * * u correonds to the * optmal soluton of (VIII.42) at tme step then u 1 s a feasble soluton to (VIII.42) at +1. he correondng value of the objectve functon s also smaller than at step and converges to zero f the desred steady state s reachable. However, dependng on the sze of the dsturbances or set pont changes, t may stll arse a conflct between constrants (VIII.43) and (VIII.44). In such case, Problem (VIII.42) becomes nfeasble and the controller can no longer be mplemented for the ntegratng system. o crcumvent such lmtaton, we propose a new extenson of the nfnte horzon controller appled to systems wth ntegratng poles. he extended proposed controller s based on the followng cost functon s s j1 J (( e j ) j ) Q(( e j ) j ) m1 j s s 1 2 u ( j ) Ru ( j ) S S (VIII.45) nyny where S2 s postve defnte and s a vector of addtonal decson varables of the control optmzaton problem. As t wll be shown n the sequel, these new slac varables provde the necessary degrees of freedom n order to the control problem to be feasble to a larger set of ntal states and external dsturbances. We can show that, for the new cost functon to be bounded, constrant (VIII.43) s substtuted by the followng equaton x ( ) D u (VIII.46) 1m 5