Stabilizing decentralized model predictive control of nonlinear systems

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1 Automatca 42 (26) Techncal communque Stablzng decentralzed model predctve control of nonlnear systems L. Magn a,, R. Scattoln b a Dpartmento d Informatca e Sstemstca, Unverstà degl Stud d Pava, va Ferrata, 27 Pava, Italy b Dpartmento d Elettronca e Informazone, Poltecnco d Mlano, p.za Leonardo da Vnc, 32, 233 Mlano, Italy Receved 8 March 25; receved n revsed form 3 November 25; accepted 2 February 26 Avalable onlne 7 Aprl 26 Abstract Ths note presents a stablzng decentralzed model predctve control (MPC) algorthm for nonlnear dscrete tme systems. No nformaton s assumed to be exchanged between local control laws. The stablty proof reles on the ncluson of a contractve constrant n the formulaton of the MPC problem. 26 Elsever Ltd. All rghts reserved. Keywords: Nonlnear model predctve control; Decentralzed control; Dscrete tme systems; Stablzaton. Introducton The development of synthess methods for decentralzed control schemes has receved a great attenton for a long tme, many results are nowadays avalable, see e.g. Sljak (99) the references reported theren, Ioannou (986), Han Chen (995), Jang (22), just to cte a few. In fact, a dstrbuted control structure s often the most approprate one n many dfferent felds, such as n the power ndustry, n aerospace chemcal applcatons or n the manufacturng ndustry. At the same tme, the last two decades have seen the wdespread dffuson of model predctve control (MPC) technques, whch are now recognzed as the most useful approach to deal wth the control problems typcal of the process ndustry. Indeed, wth MPC t s possble to formulate the control problem as an optmzaton one, where many dfferent ( possbly conflctng) goals are easly formalzed state control constrants can be ncluded. Also for MPC, many Ths paper was not presented at any IFAC meetng. Ths paper was recommended for publcaton n revsed form by Assocate Edtor Jay H. Lee under the drecton of Edtor André Tts. The authors acknowledge the partal fnancal support by MURST Project New technques for the dentfcaton adaptve control of ndustral systems. Correspondng author. Tel.: ; fax: E-mal addresses: lalo.magn@unpv.t (L. Magn), rccardo.scattoln@elet.polm.t (R. Scattoln). results are nowadays avalable concernng stablty robustness, see e.g. Mayne, Rawlngs, Rao, Scokaert (2), so that t can now be seen as a well assessed methodology. In vew of the above consderatons, t s then natural to look for MPC algorthms to be mplemented accordng to a decentralzed structure. Indeed, the possblty to use MPC n a decentralzed fashon can also have the advantage to reduce an orgnal, large sze, optmzaton problem nto a number of smaller easly tractable ones. Decentralzed MPC methods have already been studed n Dunbar Murray (24), Camponogara, Ja, Krough, Talukdar (22) n a number of papers quoted there. In partcular, n Camponogara et al. (22), the system under control has been assumed to be composed by a number of lnear dscrete-tme subsystems, dfferent nformaton structures have been consdered, all of them guaranteeng the possblty to exchange some knd of nformaton between the dstrbuted controllers. Conversely, n ths paper, a stablzng decentralzed MPC algorthm s derved under the man assumptons that the overall system under control s nonlnear, dscrete-tme no nformaton can be exchanged between local control laws,.e. a fully decentralzed nformaton structure s consdered. The proposed method deeply reles on the MPC approach presented n de Olvera Kothare Morar (2), where the closedloop stablty property s acheved through the ncluson n the optmzaton problem of a contractve constrant. Wth respect to other methods often adopted n MPC to acheve stablty, see 5-98/$ - see front matter 26 Elsever Ltd. All rghts reserved. do:.6/j.automatca.26.2.

2 232 L. Magn, R. Scattoln / Automatca 42 (26) e.g. Mayne et al. (2) or Magn Scattoln (24), the use of a contractve control law s preferred here snce t does not requre the knowledge of an auxlary stablzng control law, whch could be dffcult to derve n vew of the dstrbuted nature of the problem. 2. Problem statement Let the system under control be composed by the nterconnecton of N local subsystems descrbed by the followng nonlnear, dscrete-tme models: x p (k + ) = f (x p (k), u (k)) + g (x p (k)) + d (k), x p () = xp, =,...,N, () where x p R ν s the state of the th subsystem, d R r s the dsturbance, whle u s the control whch s restrcted to fulfll the followng constrant: u (k) U, k, (2) where U s a compact subset of R m contanng the orgn as an nteror pont. In (), the mutual nfluence of the N subsystems s descrbed by the functons g, whch depend on the overall state [ x p (k) = x p (k) xp 2 ] (k)... xp N (k) R ν, ν = ν. Defne also the overall dsturbance vector d(k) = [ d (k) d 2 (k)... d N (k)] R r, r = = r. Concernng subsystems (), the followng assumptons are ntroduced: Assumpton. The functons f, =,...,N, are C functons of ther arguments, such that f (, ) = the followng Lpschtz condton s verfed: f (ξ,u ) f (ζ,u ) L f ξ ζ, =,...,N, ξ, ζ R ν. Assumpton 2. There exst postve Lpschtz constants L j,,j [, 2,...,N], such that g (x p ) L j x p j, j= =,...,N. As for the dsturbances d, lettng N p be a gven postve nteger, henceforth called the predcton horzon, t s assumed that the followng assumpton s fulflled. Assumpton 3. The dsturbances d, =,...,N, are asymptotcally decayng bounded, that s, = d (k) B ρd := {d R r : d ρ d [, )}, k Z +, where Z + s the set of nonnegatve ntegers. The problem here consdered can now be formally stated as the one of fndng a set of N local control laws u (k) = κ (x p (k)), =,...,N, (3) such that, under Assumptons 4, the orgn of the overall system composed by the N subsystems () control laws (3) s an asymptotcally stable fxed pont defned, accordng to Scokaert, Rawlngs, Meadows (997), as follows: Defnton. The orgn s an asymptotcally stable fxed pont of the perturbed system (), (3) f: () there exst strctly postve constants ρ, ρ ρ d, =,...,N, such that, f x p B ρ,=,...,n, d (k) B ρd,=,...,n, for all k, then the soluton of the th perturbed system (), (3), =,...,N, remans n a ball B ρ for all k ; () f x p B ρ, =,...,N, d (k) ask, =,...,N, then the soluton of the th perturbed system (), (3) converges asymptotcally to the orgn. 3. Decentralzed state-feedback MPC The contractve MPC algorthm can now be formally stated as n de Olvera Kothare Morar (2). To ths end, lettng ū (t) =[u (t) u (t + )... u (t + N p )], the th, =,...,N,decentralzed control law (3) s obtaned by (locally) mnmzng at any tme nstant t wth respect to ū (t) the followng performance ndex: J (x (t), N p ) = t+n p j=t subject to constrants (2) x (j) 2 Q + u (j) 2 R (4) x (k + ) = f (x (k), u (k)), x (t) = x p (t), k t, (5) x t (nn p + N p ) < α x p (nn p), α [, ), (6) where n = max λ Z+ λn p t, x t (k + ) = f ( x t (k), u (k)), k t, (7) { p x x t (t) := (t) f t = nn p, x t (8) (t) f t = nn p. In the defnton of J, the postve nteger N p s the predcton horzon assumed for smplcty equal for any subsystem, whle Q R are postve defnte matrces. Note that: () the mnmzaton s performed wth respect to the nomnal model concdng wth () when the system s decoupled the dsturbance s null; () the contractve constrant (6), whch s crucal for the closed-loop stablty, s modfed every N p tme steps.

3 L. Magn, R. Scattoln / Automatca 42 (26) Thus constrant s mposed on the trajectory x t defned through (7) (8). Accordng to the recedng horzon approach, for the th subsystem the state-feedback MPC control law s derved by solvng at any samplng tme t the optmzaton problem (4) (8) by applyng the control sgnal u (t). In so dong, one mplctly defnes the decentralzed state-feedback control laws u = κ RH (x p ), =,...,N. (9) In order to derve the man stablty result, the followng assumpton s ntroduced. Assumpton 4. For subsystems (), (9), =,...,N, wth d (k) = g (x p (k)) =, there exst β (, ) so that x p (k) B β x p (nn p), k [nn p,(n+ )N p ), n Z +. If (9) are Lpschtz state-feedback functons of x p, the values of β, =,...,N, can be easly computed. Lemma. Under Assumptons, 2, 4 let ρ be such that x p (nn p) B ρ there exsts a feasble soluton of the optmzaton problem (4) (8), =,...,N. Then, defnng k >, D(nN p,k) =[ d(nn p ) d(nn p + )... d(nn p + k ) ] R k X p (nn p ) =[ x p (nn p) x p 2 (nn p)... x p N (nn p) ] R N, there exst computable functons γ (X p (nn p ), D(nN p,k),k), =,...,N, such that for the closed-loop systems (), (9) (7), (9), k =,...,N p, =,...,N, the followng relaton holds: x p (nn p + k) x (nn p + k) γ (X p (nn p ), D(nN p,k),k). Defnng ρ =[ ρ ρ 2... ρ N ], α = max =,...,N α I(δ,j)=[δδ... δ] R j for any postve nteger j> real number δ, the man stablty result of the proposed approach can now be stated. Theorem. Under the assumptons of Lemma Assumpton 3 f () γ ( ρ,i( ρ d,n p ), N p )< ρ ( α ), (2) there exst α g <, α d, ε > such that ρ ρ, γ (ρ,i(ε,j),j)<(α g ρ M + α d ε)( α), ε ε, j >, wth ρ =[ρ ρ 2... ρ N ], ρ M = max =,...,N ρ. Then, () there exst ρ > such that xp B ρ, xp (nn p) B ρ, n ; () the orgn s an asymptotcally stable fxed pont of the perturbed closed-loop system (), (9) wth ρ := β ρ + max γ ( ρ,i( ρ d,j),j). j=,...,n p The result of the theorem s rather conservatve, due to the need to consder bounds on the mutual nfluence between subsystems, ther unmodelled dynamcs the effect of the dsturbances. However, these bounds could be relaxed when partal nformaton can be exchanged between subsystems. Moreover, n order to reduce the conservatveness nherent to any robust open-loop mnmzaton based MPC algorthm, one could resort to mn max closed-loop strateges. For a dscusson on ths pont see e.g. Magn Scattoln (25). 4. Example Consder the followng second order system composed of two subsystems S S 2 : S : x p (k + ) = x p (k)2 + + u (k) + η x p 2 (k) + d (k), x () = x, S 2 : x p 2 (k + ) = e sn(xp 2 (k)) + u 2 (k) + η 2 x p (k) + d 2 (k), x 2 () = x 2, where the nomnal part of S, S 2 s gven by the Lpschtz functons f (x p,u ) = x p2 + + u, f 2 (x p 2,u 2) = e sn(xp 2 ) + u 2, whle ther mutual nfluence s descrbed by g (x p ) = η x p 2, g 2 (x p ) = η 2 x p. As for the dsturbances, they are assumed to be the states of the followng asymptotcally stable frst order systems d (k + ) = γ d (k), d () = d, =, 2. Fnally, the control varables are requred to fulfll the followng constrants:.2 u (k).5, =, 2. The MPC algorthm descrbed n Secton 3 has been used n a number of smulaton experments wth ntal condtons x = d =,=, 2, wth performance ndces characterzed by N p = 5, Q = R =, =, 2. As for the dsturbance dynamcs, t has been defned by γ =.9,=, 2. Fnally, the contracton constrants have been chosen as α =.9, =, 2. In Fg. the transents of the state control varables are reported when η = η, =, 2, wth η ={,.2,.4}.

4 234 L. Magn, R. Scattoln / Automatca 42 (26) p x tme x 2 p tme u tme u 2 tme Fg.. Transents of the state control varables for η = (contnuous lne), η =.2 (dashed lne) η =.4 (dotted lne). Note that the stablty of the orgn of the closed-loop system s always acheved, whle the performances decrease when the nteracton ( η) ncreases. Note also that constrant on the control varable u s actve n the ntal nstants of the transents. Fnally wth η.5, startng from the same ntal condtons, the state transents dverge. 5. Conclusons The decentralzed predctve control algorthm presented n ths note can be extended n several drectons. Among them, the output feedback case ts modfcatons when partal nformaton can be exchanged between local control laws appear to be of nterest. Appendx. Proof of Lemma. In vew of Assumptons 4 one has x p (nn p + k) x (nn p + k) L f x p (nn p + k ) x (nn p + k ) + L j { x j (nn p + k ) j= + x p j (nn p + k ) x j (nn p + k ) } + d (nn p + k ) L f x p (nn p + k ) x (nn p + k ) + L j {β j x p j (nn p) j= + x p j (nn p + k ) x j (nn p + k ) } + d (nn p + k ). Fnally by teratng backwards the rght-h-sde of ths expresson by recallng (8) the result follows. Proof of Theorem. In vew of Lemma, for =,...,N, x p (nn p + N p ) x (nn p + N p ) x p (nn p + N p ) x (nn p + N p ) γ ( ρ,i( ρ d,n p ), N p ). () Hence, from (6) condton () of the theorem x p (nn p + N p ) α x p (nn p) +γ ( ρ,i( ρ d,n p ), N p ) () x p (nn p) α n ρ + γ ( ρ,i( ρ d,n p ), N p ) α ρ + γ ( ρ,i( ρ d,n p ), N p ). α So, n order to guarantee x p (nn p) < ρ, n, (2)

5 L. Magn, R. Scattoln / Automatca 42 (26) t s suffcent to have ρ + γ ( ρ,i( ρ d,n p ), N p ) < ρ α, ths s true f only f ρ < ρ γ ( ρ,i( ρ d,n p ), N p ), α but n vew of condton () ths holds true wth ρ >, then () s satsfed. We now prove that x p (k) B ρ, k>. To ths end wth the same arguments used to derve (), t s easy to show that, j [,N p ], x p (nn p + j) x (nn p + j) +γ ( ρ,i( ρ d,j),j), (3) but from Assumpton 4 recallng that x (nn p ) = x p (nn p) x (nn p + j) β x (nn p ) = β x p (nn p), j [,N p ]. Hence x p (nn p + j) β x p (nn p) +γ ( ρ,i( ρ d,j),j). (4) From (2) condton (2), n Z + j [,N p ] t follows that x p (nn p + j) β ρ + γ ( ρ,i( ρ d,j),j) β ρ + max γ ( ρ,i( ρ d,j),j)= ρ, j=,...,n p so that () of Defnton s satsfed. In order to prove () of Defnton frst we note that f d (k) ask, then for any ε > there exsts a fnte n ε Z + so that d (k) ε, k n ε N p. In vew of () one has x p ( n εn p + N p ) α x p ( n εn p ) +γ ( ρ,i(ε,n p ), N p ), l Z +, lettng ρ max ( n ε N p,l):= max k=,...,l xp (( n ε + k)n p ), ρ max ( n ε N p,l):= max ( n ε N p,l), =,...,N ρmax then x p (( n ε + h)n p ) < α h xp ( n εn p ) + γ (I ( ρ max ( n ε N p, h), N), I (ε,n p ), N p ). (5) α Then, n vew of (5) condton (2) there exsts ε such that ε < ε ρ max (( n ε + h)n p, ) < α h ρ max ( n ε N p, ) + α g ρ max ( n ε N p,h)+ α d ε (α h + α g ) ρ max ( n ε N p,h)+ α d ε. In vew of condton (2) there exsts a postve nteger l such that ᾱ g (h) := α h + α g <, h l, ρ max (( n ε + l)n p,l) = max φ=,...,l { ρ max (( n ε + l + φ)n p, )} max ᾱg(l) ρ max ( n ε N p,l+ φ) + α d ε. φ=,...,l Moreover, ρ max ( n ε N p,l+ φ) = max{ ρ max ( n ε N p,l+ φ ), ρ max (( n ε + l + φ )N p, )} max{ ρ max ( n ε N p,l+ φ ), ᾱ g (l + φ ) ρ max ( n ε N p,l+ φ ) + α d ε} so that lm ε ρmax ( n ε N p,l+ φ) lm ρ max ( n ε N p,l+ φ ) lm ρ max ( n ε N p,l), ε ε then lm ε ρmax (( n ε + l)n p,l) lm ᾱ g (l) ρ max ( n ε N p,l) ε lm ε ρmax (( n ε + ml)n p,l) lm ᾱ m g (l) ρmax ( n ε N p,l), ε so that lm ε Hence lm n lm m ρmax (( n ε + ml)n p,l)=. lm d xp (nn p) =, =,...,N. (6) Fnally, n vew of Lemma, j, =,...,N, x p (nn p + j) x p (nn p + j) x (nn p + j) + x (nn p + j) γ (X p (nn p ), D(nN p,j),j)+ β j x p (nn p) from (6) Assumpton 4 lm n lm d xp (nn p + j) =, j [,N p ], =,...,N, so that () of Defnton s satsfed the orgn s an asymptotcally stable fxed pont of the perturbed closed loop system () (9). References Camponogara, E., Ja, D., Krough, B. H., & Talukdar, S. (22). Dstrbuted model predctve control. IEEE Control Systems Magazne, de Olvera Kothare, S. L., & Morar, M. (2). Contractve model predctve control for constraned nonlnear systems. IEEE Transactons on Automatc Control, 53 7.

6 236 L. Magn, R. Scattoln / Automatca 42 (26) Dunbar, W. B., & Murray, R. M. (24). Recedng horzon control of multvehcle formatons: A dstrbuted mplementaton. In 43rd IEEE Conference on Decson Control (pp ). Han, M. C., & Chen, Y. H. (995). Decentralzed control desgn: Uncertan systems wth strong nterconnectons. Internatonal Journal of Control, 6(6), Ioannou, P. (986). Decentralzed adaptve control of nterconnected systems. IEEE Transactons on Automatc Control, 3(4), Jang, Z.-P. (22). Decentralzed dsturbance attenuatng output feedback trackers for large scale nonlnear systems. Automatca, 38, Magn, L., & Scattoln, R. (24). Stablzng model predctve control of nonlnear contnuous tme systems. Annual Revews n Control,. Magn, L., & Scattoln, R., 25. Robustness robust desgn of MPC for nonlnear dscrete-tme systems. In Internatonal workshop on assessment future drectons of NMPC, freudenstadt-lauterbad, Germany (pp. 3 46) [also to appear n Sprnger Book]. Mayne, D. Q., Rawlngs, J. B., Rao, C. V., & Scokaert, P. O. M. (2). Constraned model predctve control: Stablty optmalty. Automatca, 36, Scokaert, P. O. M., Rawlngs, J. B., & Meadows, E. S. (997). Dscretetme stablty wth perturbatons: Applcaton to model predctve control. Automatca, 33, Sljak, D. D. (99). Decentralzed control of complex systems. New York: Academc Press.