G y 0nx (n. G y d = I nu N (5) (6) J uu 2 = and. J ud H1 H

Transcription

1 29 American Control Conference Hatt Regenc Riverfront, St Lois, MO, USA Jne 1-12, 29 WeC112 Conve initialization of the H 2 -optimal static otpt feeback problem Henrik Manm, Sigr Skogesta, an Johannes Jäschke Abstract Recentl we have establishe a link between invariants for qaratic optimization problems an linearqaratic (LQ) optimal control [1 The link is that for LQ control one invariant is c k = k K k, which iels zero loss from optimalit when controlle to a constant setpoint c = c s = In general there eists infinitel man sch invariants to a qaratic programming (QP) problem In [2 we show how the link can be se to generate otpt feeback control b sing crrent an ol measrements In this paper we eten this approach b consiering in more etail some interesting eamples, an the se of aitional (ol) measrements In particlar, we show that if the nmber of measrements is less than the nmber of istrbances (initial states) pls inepenent inpts, we can not with this metho fin a polic k = K k that minimizes the original problem, becase K is not optimall constant However, this metho ma be se to fin initial vales for H 2 -optimal static otpt feeback snthesis Ine Terms linear qaratic control, fie-strctre control I INTRODUCTION Consier a finite horizon LQ problem of the form min J(,()) = E{ T, 1,, N 1 NP N + + N 1 k= sbject to = () [ T k Q k + T kr k } k+1 = A k + B k, k k = C k + n k, where k R n are the states, k R n are the inpts an k R n are the measrements Frther P = P T >, Q, an R > are matrices of appropriate imensions, an E{ } is the epectation operator It is well-known that if C = I an n =, sch that k = k, the soltion to (1) is state feeback k = K k, where the gain matri K can be fon b solving an iterative Riccati eqation For the case with white noise assmption on an (n ), the optimal soltion is k = Kˆ k, where ˆ k is a state estimate from a Kalman filter [3, which in effect gives a namic compensator K lqg (from to ) of same orer n as the plant In this paper we consier the static otpt feeback problem, k = K k, where K is a static gain matri Note that the case with a fie-orer controller of orer less than n ma also be broght back to the static otpt Department of Chemical Engineering, Norwegian Universit of Science an Technolog, N-7491 Tronheim, Norwa Athor to whom corresponence shol be aresse skoge@ntnno (1) Fig 1 G G c = H( + n ) n Measrement combination (H) Notation for self-optimizing control feeback problem A particlar controller consiere in this paper is the mlti-inpt mlti-otpt proportional-integralerivative controller (MIMO-PID) where we have as man controlle otpts c as there are inpts The allowe measrements k in the formlation in (1) are the present vale of the controlle otpt k c (P), the integrate vale k i= c i (I) an the erivative c k δt (D) This optimal soltion to this problem is nsolve [4 so one cannot epect to fin an analtic or conve nmerical soltion The contribtion of this paper is therefore to propose a conve approach to fin a goo initial estimate for K, as a starting point for a nmerical search A Notation Notation aopte from self-optimizing control is smmarize in figre 1 Tpicall, = (, 1,, N 1 ), = an = (,) or = (, t,,), bt also other variables will be consiere II MAIN RESULTS A Reslts from self-optimizing control 1) Nllspace metho: From [5 we have the following theorem: Theorem 1: (Nllspace theorem = Linear invariants for qaratic optimization problem) Consier an nconstraine qaratic optimization problem in the variables (inpt vector of length n ) an (istrbance vector of length n ) min J(,) = [ T J J (2) J T J In aition, there are measrement variables = G + G If there eists n n + n inepenent measrements [ (where inepenent means that the matri G = G G has fll rank), then the optimal soltion to (2) has the propert that there eists n c = n linear variable combinations (constraints) c = H that are invariant to /9/$25 29 AACC 1724

2 the istrbances The optimal measrement combination matri H is fon b selecting H sch that HF =, (3) where F = opt is the optimal sensitivit matri which can be obtaine from F = (G J 1 J G ), (4) (That is, H is in the left nllspace of F ) 2) Generalization: Eact local metho: A generalization of Theorem 1 is the following: Theorem 2: (Eact local metho = Loss b introcing linear constraint for nois qaratic optimization problem [5) Consier the nconstraine optimization problem in Theorem 1, (2), an a set of nois measrements m = + n, where = G +G Assme that n c = n constraints c = H m = c s are ae to the problem, which will reslt in a non-optimal soltion with a loss L = J(,) J opt () Consier istrbances an noise n with magnites [ = W ; n = W n n ; 1 n Then for a given H, the worst-case loss introce b aing the constraint c = H is L wc = σ 2 (M)/2, where M is M [ M M n M = J 1/2 (HG ) 1 HFW M n = J 1/2 (HG ) 1 HW n, an σ( ) is the maimm singlar vale The optimal H that minimizes the loss can be fon b solving the conve optimization problem min H F F H sbject to HG = J 1/2 Here F = [FW W n The reason for sing the Frobenis norm is that minimization of this norm also minimizes σ(m) [6 Remark 1: From [5 we have that an optimal H premltiplie b a non-singlar matri n c n c D, ie H 1 = DH is still optimal One implication of this is that for a sqare plant, n c = n, we can write c = H 1 = H m 1 m +I To see this, assme = ( m,), so H = [H m H, where H is a nonsinglar n n matri Now, H 1 = (H ) 1 [H m H = [(H ) 1 H m I Remark 2: More generall, for the case when F F T is singlar, we can solve the conve problem (6) sing for eample CVX, a package for specifing an solving conve programs [7, with the following coe: cv_begin variable H(N*n,n+n*N); minimize norm(h*ftile, fro ) sbject to H*G == sqrtm(j); cv_en Remark 3: Noise will not be frther iscsse in this paper, bt is covere in [8 2 (5) (6) B Some special cases Some special cases will now be consiere where eplicit epressions can be fon 1) Fll information: No new reslts are represente here, bt we show the matrices G, G, J, an J for LQoptimal control Assme that noise-free measrements of all the states are available It is well known that the LQ problem (1) can be rewritten on the form in (2) (see for eample [9) b treating as the istrbance, an letting = (, 1,, N 1 ) Ths, from Theorem 1 we know that for the LQ problem there eists infinitel man invariants (bt onl one of these involves onl present states) Withot loss of generalit consier the case when the moel in (1) is stable Let = (,, 1,, N 1 ) = (,) Note that this incles also ftre inpts, bt we will se the normal trick in MPC of implementing onl the present (first) inpt change Since we have n = n + n an no noise, we can se Theorem 1 The open loop moel becomes: = G + G [ G n (n = N) R (n+nn) (nn) I nn G = I n N R (n+nn) n (nn) n Here I m is an m m ientit matri an m n is a m n matri of zeros The matrices J an J are the erivatives of the linear qaratic objective fnctionfor the objective an process moel in (1) we show in [1 that an J 2 = J 2 = B T PB+R B T A T PB B T (A N 1 ) T PB B T PAB B T PB+R B T (A N 2 ) T PB B T PA N 1 B B T PA N 2 B B T B T B T B T PB+R P PA PA N 1 (7) (8) A (9) The sensitivit matri (optimal change in when is pertrbe) becomes: F = opt T = (G J 1 J G ) = I n J 1 (1) J We can se Theorem 1 to get the combination matri H, ie fin an H sch that HF = : I H1 H n 2 J 1 = H J 1 H 2 (J 1 J ) = (11) To ensre a non-trivial soltion we can choose H 2 = I nn an get the following optimal combination of an : c = H = J 1 J +, (12) 1725

3 which can be interprete as: Invariant 1: = K Invariant 2: 1 = K 1 Invariant N: N 1 = K N 1 (13) From Theorem 1 implementation of (13) give zero loss from optimalit, ie the correspon to the optimal inpt trajector, 1,, N 1 from the soltion of (1) Moreover, since the states captre all information, we mst have that 1 = K 1 = K 1 (A + BK ) 1 1 (14) =K From this we ece that the soltion to (1) can be implemente as k = K k, k =,1, In [1 we prove that this gives the same reslt as conventional linear qaratic control, b conventional meaning for eample eqation (3) in Rawlings an Mske [9 2) Otpt feeback: In this section we will show how Theorem 2 can be se for the special (bt common) case when k = C k + k, k =,1,,N an we look for controllers on the form k = K k If C is fll colmn rank, then we have fll information (state feeback), bt we here consier the general case where C has fll row rank (inepenent measrements), bt not fll colmn rank Let = (,) an as before = (, 1,, N 1 ) The istrbance = The open loop moel is now = [ C = +, (15) I G G an the sensitivit matri F is F = (G J 1 J G ) = C J 1 (16) J Since we now have that n = nỹ + n < n + n, where nỹ are the nmber of measrements from the plant, nỹ < n, we cannot simpl set HF =, bt we nee to solve (6) Let s analze this problem For G T = [ I, HG = J 1/2 is eqivalent to H 2 = J 1/2, where H nc (nỹ+n ) = H 1nc nỹ H 2nc n (17) With this partitioning we get that H F = H[FW W n = [HFW HW n, (18) an for W n =, ie the noise-free case, H F = [HFW (19) We want to minimize the Frobenis-norm of this matri an we have that [HFW F = HFW F + F (2) Assme withot loss of generalit that W = I, an let J = J 1 J With F T = [C T J T we have that HF = H 1 C + H 2 J = H 1 C J 1/2 J (21) H2=J 1/2 We want to minimize H 1 C J 1/2 J, hence we look for a H 1 sch that Using the pseo-inverse, we fin that an we get that the optimal H is H 1 C = J 1/2 J (22) H 1 = J 1/2 J C, (23) H = [J 1/2 J C J 1/2 (24) In the final implementation we can ecople the invariants in the inpts b H = J 1/2 H = [ J 1 J C I (25) This means that the open-loop optimal otpt feeback is k = J 1 J C k = K k, (26) K state feeback that is, for an optimal state feeback K, the optimal otpt feeback is KC This means that for this case we have Invariant 1: = K C Invariant 2: 1 = K 1 C Invariant N: N 1 = K N 1 C (27) We have calle these variable combinations invariants in qotation marks becase the are not invariant to the soltion of the original problem, bt rather the variable combinations that minimize the (open-loop) loss Inee, the non-negative loss is HF = J 1/2 J C C J 1/2 J = J 1/2 J (C C I) J 1/2 J C C I For otpt feeback we have in the least sqares sense (28) 1 = K 1 C C(A + BK C C) 1 C 1 (29) K 1 Unfortnatel, in general K 1 K an hence the open loop soltion (27) cannot be implemente as a constant feeback k = K C k, as was the case for state feeback, see (14) III MAIN ALGORITHM We now propose an algorithm for fining otpt feeback controllers This is a two-step procere where we first fin initial vales sing Theorem 2 These initial vales correspon to a controller that in the open-loop sense is closest to the optimal state feeback LQ controller Thereafter we improve this controller b solving a close-loop optimization problem where the controller parameters are the egrees of freeom In the previos section we showe that if = (C,,, N 1 ) Theorem 1 gives = K = K state feeback C The algorithm presente here is more 1726

4 n 1/2 W W 1/2 n R 1/2 P z k = [1; B c (si A c ) 1 K C Q 1/2 m Fig 2 Interconnection strctre for close-loop optimization of K k 1 1: k = K k : k = K k = [;1 3: k = K 1, k K 2, k 1 4: k = K 1 k K 2 k 1 5: k = K lqr k general in the sense that it hanles measrements sch as = (, 1,, M,,, N 1 ) (In the latter case a casal controller is M = K [ T T M T ) Fig Time Simlation reslts for istrbances in initial conitions, eample Algorithm 1 Low-orer controller snthesis 1: Define a finite-horizon qaratic objective J(,) = T N P N + N 1 i= T i Q i T i R i + 2 T i N i 2: Calclate J an J as in (8), (9) 3: Define caniate variables = G + G, = (, 1,, N 1 ) 4: Decie weights W an Wn (Defalt: W = I, Wn = ) 5: Fin H b solving the conve optimization problem (6) 6: Optional: Improve control b close-loop optimization (Section III-A) A Relationship between LQ-control an H 2 optimal control It is well-known that the LQG problem ma be cast into the H 2 framework an that a class of H 2 optimal controllers ma be implemente in an LQG-scheme with a Kalman estimator an a constant feeback gain from the estimate states [11 In this paper we propose to improve the soltion from the open-loop control b minimizing the H 2 norm min K F l(p,k) 2 (3) In this contet min K means minimizing over the parameters in K The lower-fractional transform F l (P,K) = P 11 + P 12 (I P 22 K) 1 P 21 for a P = [ P 11 P 12 P 21 P 22 [12 The interconnection strctre we se for P is shown in figre 2 The last row of Algorithm 1 consists of solving (3) with initial vales as H on step 5 in algorithm 1, which is the soltion to (6) IV EXAMPLES In this section two eamples will be consiere First we iscss P-control of a secon orer plant, then MIMO-PID control of a moel of a istillation colmn Eample 1: (P-control of secon orer plant) Consier the 2 plant g(s) = s 2 +3s+2 The plant is sample with T s = 1 to get k+1 = k + 91 k k = [ 1 (31) k The objective is to erive two LQ-optimal controllers for this process, one P-controller on the form k = K k an a PD-controller k = (K 1 k + K 2 k 1) In the snthesis of the controllers we se algorithm 1 The open loop objective to be minimize is J(,) = i= T i Q i + T i R i with Q = [ 1 an R = 1 The infinite horizon objective can be approimate b the following objective: N 1 J(,) = T NP N + T i Q i + T i R i, (32) i= with P = [ an N = 1 (P is a soltion to the iscrete Lapnov eqation P = A T PA+Q) The objective is now on the form of step 1 in the algorithm P-control: For the P-controller, the variables to combine are 1 = (,,, N 1 ) The matrices J an J are the same as those reporte in eqations (8) an (9) Since we o not consier noise W = I an W n = We can now fin H either b solving the conve problem in (6), or we can simpl se the eplicit formla in (23), ie H 1 = J 1 J C As shown in section II-B2 (see eqation 27) we now get N = 1 invariants to the soltion to the original optimization problem in (32) The first one of these invariants is reporte as controller 1 in table I We observe that H F >, which is epecte from Theorem 1, as n < n + n in this case (n = 1 + 1,n = 1,n = 2) Using this P-controller ( k = 44 k ) as the initial estimate, the H 2 -optimal close-loop controller K in Figre 2 is obtaine nmericall Note from row 2 in Table I that the H 2 -norm is onl rece slightl (from 2993 to 2981), althogh K changes from -44 to -313 PD-control: For the snthesis of a PD controller we again se algorithm 1 The variables to combine are now 2 = (, 1,,, N 1 ) The objective fnction an the matrices J an J remain the same The open-loop moel 1727

5 TABLE I CONTROLLERS FOR EXAMPLE 1 Controller H F F F l (P, K) 2 1: k = 44 k First invariant sing Theorem 2 for 1 = (,,, N 1 ) 2: k = 313 k 2981 Close-loop optimal P-controller 3: k = (149 k 111 k 1 ) 3176 Secon invariant sing Theorem 2 for 2 = (, 1,,, N 1 ) 4: k = (416 k 19 k 1 ) 2979 Close-loop optimal controller PD-controller 5: k = [ k 2972 LQR = G + G is now: 2 = 1 = CB C I + CA (33) I For this particlar variable combination (23) cannot be se, as the variables occr on ifferent instances in time We therefore solve the optimization problem (6) sing cv, as shown in remark 2 The soltion is again on the form of (27), for which the secon invariant is reporte as controller 3 in table I The soltion (all the invariants) gives H F =, which is epecte from Theorem 1, as n = n + n an no noise is present Frther nmerical optimization reces the H 2 norm from 3176 to 2979 It can be verifie that the variable combination is inee optimal after one step with the following calclations: = K, 1 = K 1 1 K 2 1 = (K 1 C + K 2 C(A BK) 1 ) 1, (34) =K where K is the LQR controller For implementation some sb-optimalit mst be epecte since we are not starting the control with LQR, rather we se the PD controller at all time instances Simlations: From the close-loop norms reporte in table I the controllers are epecte to perform similarl in close loop This is confirme in the close loop simlations of istrbances in initial states, see figre 3 Eample 2: (Linear namic moel of istillation colmn) In this eample we consier MIMO-PI an -PID control of colmn A in [13 The moel is se as an eample for offset-free control in [14 The moel is base on the following assmptions: binar separation, 41 stages, incling reboiler an total conenser, each stage is at eqilibrim, with constant relative volatilit α = 15, linearize liqi flow namics, negligible vapor holp, constant pressre The fee enters on stage 21 = [ V L D an = [ B We here consier the LV-configration, where D an B are se to control the levels With level controllers implemente (P-control with K c = 1) the rest of the colmn is stable Balance rection is se to rece the nmber of states to 16 Then integrate otpts are ae to the moel, reslting in a moel with 18 states If we let the otpts of top bottom Inpts L,V LQR 5 PI 1 3 PID pper: V lower: L Time [min Fig 4 Simlation reslts for eample 2 At t = a step-change of 1 in F occrs, an at t = 7 z F is change from 5 to 6 the moel be P, I, an D, we get a moel with the following strctre: [ẋ a b = + σ c σ P c (35) I = I + D σ ca cb This moel is sample with T s = 1 to get a iscrete time moel Again[ we set p an infinite time objective fnction, with Q = C T I C, an R = 1 I, an for intermeiate calclations we approimate this b a finite horizon objective with N = 15 an P = Q We now look for controllers on the form k = ( K P k P + K I k I + K D k D ) (36) an we assme measrements of the compositions with a sample time of 1 minte is available Table II shows first-move PI (= first move invariant realize as feeback), close loop PI an PID controllers, 1728

6 TABLE II CONTROLLERS FOR EXAMPLE 2 Description Control eqation H F F F l (P, K) ( ) First-move PI k = k P k I ( ) Close-loop optimal PI k = k P k I 365 ( First-move PID k = k P k I ) k D ( Close-loop optimal PID k = k P k I ) k D LQR k = k (1 : 6) k (7 : 12) k (13 : 18) TABLE III ITERATION COUNT USING FMINUNC (MATLAB c R28A) WITH DIFFERENT INITIALIZATIONS Algorithm 1 K = SIMC-tne PI controllers PI PID an the LQR controller for reference In aition to the initialziation propose in this paper we trie to initialize the nmerical search with K = an two SIMC-tne [15 PI controllers with τ c = 1 mintes, leaing to KSIMC = As reporte in table III i K = not converge, whereas intializing with two SIMC-tne PI controllers converge in both cases (both for PI an PID esign), thogh with some more iterations than the metho propose in this paper Figre 4 shows simlation reslts where we at t = introce a step in the fee rate an at t = 7 a step in the fee composition PI an PID refers to the closeloop optimal controllers As one observes is the MIMO-PID controller qite close in performance to the LQR controller V CONCLUSIONS In this paper we have iscsse snthesis of H 2 -optimal static otpt feeback, an in particlar the MIMO-PID We have shown that initial conitions for close loop optimization can be fon b solving a conve program, an that the reslting close loop optimization problem converges for some interesting cases VI ACKNOWLEDGMENTS The athors gratefll acknowlege the comments from Bjarne Grimsta [2, Eplicit MPC with otpt feeback sing self-optimizing control, in Proceeings of IFAC Worl Conference, Seol, Korea, 28 [3 K J Åström an B Wittenmark, Compter Controlle Sstems, T Kailath, E Prentice-Hall, 1984 [4 V Srmos, C Aballah, an P Dorato, Static otpt feeback: a srve, Decision an Control, 1994, Proceeings of the 33r IEEE Conference on, vol 1, pp vol1, Dec 1994 [5 V Alsta, S Skogesta, an E Hori, Optimal measrement combinations as controlle variables, 28, in Press: Jornal of Process Control, oi:1116/jjprocont2812 [6 V Kariwala, Y Cao, an S Janarhanan, Local self-optimizing control with average loss minimization, In Eng Chem Res, vol 46, pp , 27 [7 M Grant an S Bo, CVX: Matlab software for iscipline conve programming (web page an software), Agst 28 [Online Available: bo/cv [8 H Manm an S Skogesta, Conve initialization of the H 2 -optimal static feeback problem with noise, in In preparation for Proceeings of CDC, 29 [9 J B Rawlings an K R Mske, The stabilit of constraine receing-horizon control, in IEEE Transactions on Atomatic Control, vol 38, 1993, pp [1 H Manm, S Narasimhan, an S Skogesta, A new approach to eplicit MPC sing self-optimizing control, 27, avaliable at: [11 J Dole, K Glover, P Khargonekar, an B Francis, State-space soltions to stanar H 2 an H control problems, IEEE Transactions on Atomatic Control, vol 34, no 8, pp , 1989 [12 S Skogesta an I Postlethwaite, Mltivariable Feeback Control Wile, 25 [13 S Skogesta, Dnamics an control of istillation colmns - a ttorial introction, Trans IChemE, Part A, vol 75, pp , September 1997 [14 K R Mske an T A Bagwell, Distrbance moeling for offsetfree linear moel preictive control, Jornal of Process Control, vol 12, pp , 22 [15 S Skogesta, Simple rles for moel rection an pi controller tning, Jornal of Process Control, 23 REFERENCES [1 H Manm, S Narasimhan, an S Skogesta, A new approach to eplicit MPC sing self-optimizing control, in Proceeings of American Control Conference, Seattle, USA,