Homotopy Continuation Solution Method in Nonlinear Model Predictive Control Applications

Transcription

1 Ian Davd Lockhart Bogle and Mchael Farweather (Edtors Proceedngs of the nd European Symposum on Computer Aded Process Engneerng 7 - June London. Elsever B.V. All rghts reserved Homotopy Contnuaton Soluton Method n onlnear Model Predctve Control Applcatons Panos Seferls ab Ioanns Stavraks a Athanasos I. Papadopoulos b a Department of Mechancal Engneerng Arstotle Unversty of Thessalonk P.O. Bo Thessalonk Greece b Chemcal Process Engneerng Research Insttute Centre for Research and Technology-Hellas 6th klm Charlaou-Therm 57 Therm Thessalonk Greece Abstract A new fast and effcent algorthm for the soluton of the dynamc optmzaton problem resulted from the mplementaton of a model predctve control (MPC framework n hghly nonlnear dynamc systems s presented. The sequence of the optmal control actons s obtaned through the soluton of the parameterzed set of the Karush-Kuhn- Tucker (KKT optmalty condtons for the nonlnear program as resulted from a constructed homotopy wth respect to the ntal pont of the dynamc optmzaton problem. A predctor-corrector contnuaton method talored for large scale sparse systems enables the quck calculaton of the optmal soluton as ehbted n challengng engneerng problems such as the control of a cart wth a double pendulum. Keywords: onlnear model predctve control Homotopy Contnuaton method. Introducton Model predctve control s a powerful control algorthm that enables the satsfacton of comple control objectves through the systematc consderaton of all process dynamc nteractons for multvarable control systems. The mplementaton of nonlnear model predctve control (MPC n dynamc systems that possess fast dynamcs s prohbtve as the soluton to the assocated dynamc optmzaton problem s comparable to the duraton of the control nterval sutable for adequate control performance. The computatonal delay appears to be a consderable factor for the deteroraton of the MPC performance []. Therefore ether a delayed control acton s appled to the system or a smplfed model usually a lnear realzaton of the nonlnear dynamc model s used n the model predctve control algorthm. In both cases however the performance of the controller dmnshes as crucal process dynamcs may be gnored through the utlzaton of a smplfed (lnear model or the adopton of a delayed control acton. Over the years a large number of effcent real-tme strateges have been proposed such as eplct MPC [] ewton-type controllers [3 4] and nonlnear program (LP senstvty-based controllers. Recently the advanced-step MPC [5] acheved sgnfcant mprovement n the speed for the soluton of the optmzaton problem as the control nterval s used n order to gude the soluton of the problem towards the real optmum usng the predcton model before the measurements become avalable. Despte all efforts the need for fast and effcent MPC soluton technques s stll a challengng problem.. onlnear model predcton control problem formulaton The MPC problem can be formulated as follows:

2 P. Seferls et al. Mn u s.t. h g J = Φ ( y + φ ( y ( & y ( y u = p = - = = K ( Vector s the real-valued state vector of dmenson n y s the real-valued output vector of dmenson l u s the real-valued nput (manpulated varables pecewse constant vector of dmenson m and s the ntal values vector for the state varables of dmenson n. Functonal J represents the objectve of the control problem conssted of a termnal term Φ( and a tme varyng term epressed usng nonlnear functons φ(. Vector h represents the set of nonlnear modelng equatons descrbng the dynamc behavor of the system whereas g represents the set of nequalty constrants. The model predctons etend nto the future for tme ntervals whereas the optmal sequence of manpulated varables etends to M tme ntervals wth M<. Accordng to the MPC algorthm gven an ntal pont for the system the sequence of optmal control actons s calculated from (. After mplementaton of the frst control acton a new set of measurements s acqured for the output varables. Provded that the system s observable the set of measurements s then used to update the model states/parameters through a sutable state estmator method. The updated vector of states thus becomes the ntal condtons for problem ( n the net tme nterval. Therefore the soluton of problem ( can be vewed as a parameterzed optmzaton problem wth respect to the ntal condtons vector p. Such property has been eploted by Zavala and Begler [5] n order to mprove the effcency of MPC soluton strateges... Dynamc model dscretzaton In the present work a drect soluton method s appled through the dscretzaton of the dfferental equatons usng an orthogonal collocaton on fnte elements (OCFE appromaton technque. The predcton horzon s parttoned n E fnte elements and wthn each fnte element the state profles are appromated by Lagrange t or order n. The state profles and ther tme dervatves are then polynomals ( W j epressed as functons of the state varables defned at specfed tme nstances; namely the collocaton ponts as follows: n ( t = W ( t ( t n ( t W j ( t = ( t t t t... E d j j j end = dt dt j= j= ( The collocaton ponts are selected as the roots of orthogonal Legendre polynomals of order equal to the number of collocaton ponts wthn each fnte element. aturally the element boundares concde wth the tme ntervals n the dgtal mplementaton of the MPC algorthm but also nteger multples of fnte elements can form one tme (control nterval. Constrants on the state varable at varous tme nstances can be best accommodated wthn the appromaton scheme by ether placng a collocaton pont or a fnte element breakpont at the specfc tme nstance of the constrant. OCFE appromaton requres that the modelng equaton resduals vansh only at the collocaton ponts. Zero-order contnuty for the state profles s mposed at the element boundares. The dscretzaton of the dynamc equatons usng OCFE transforms the MPC problem to the followng nonlnear program (the tlde denotes appromated

3 Homotopy Contnuaton Soluton Method n onlnear Model Predctve Control 3 varable profles. Mn u s.t. h g J = Φ ( y ( y u = p ( y + φ ( y = - = = K.. Parameterzed MPC algorthm The dscretzed MPC problem (3 can be solved usng conventonal LP soluton algorthms that may suffer from long soluton tmes and occasonally from convergence falures. An alternatve method for the soluton of (3 reles on the formulaton of the problem as a set of nonlnear equatons that correspond to the Karush-Kuhn-Tucker optmalty condtons parameterzed wth respect to the ntal condtons [6]. ( T T ( ( y + λ h y + µ A g A y h( y g ( y u = J F = (4 A p + p+ θζ p+ The frst entry n equaton (4 corresponds to the gradent of the Lagrangan wth respect to y and u of problem (3. Vectors λ and μ A denote the equalty and actve nequalty constrants Lagrange multplers respectvely. The second and thrd entres correspond to the feasblty condton under the assumpton that the lnear ndependence constrant qualfcaton (LICQ holds (.e. the gradents of the equalty and actve nequalty constrants are lnearly ndependent. The strct complementarty condton s ensured by the postveness of the Lagrange multplers correspondng to the actve nequaltes. Fnally the last entry determnes the transton from the ntal pont p (at current tme nterval to the target ntal pont p + (tme nterval + along the drecton θ n the mult-dmensonal space defned by the system states. Symbol ζ denotes the ndependent contnuaton parameter for the problem. Problem (4 can be vewed as a homotopy connectng the known optmal soluton at ntal vector p for ζ equal to zero to the unknown soluton at ntal vector p + for ζ equal to unty. Equaton set (4 s an under-determned set of nonlnear equatons as the overall number of varables eceeds by one the overall number of equatons. Provded that model constrants are twce contnuously dfferentable the equaton set s solved usng a predctor-corrector contnuaton method as mplemented n PITCO [7]. The sparse numercal solver UMFPACK [8] has been employed for mproved soluton speed. The Jacoban of equaton (4 nvolves the Hessan for the modelng equatons h and g whch has been analytcally evaluated for enhanced accuracy and computatonal performance. The soluton method evaluates a seres of contnuaton ponts for ncreasng values of the contnuaton parameter ζ. Αt every contnuaton pont the algorthm performs rgorous checks for possble volaton of nactve nequalty constrants ncludng state and control varable bounds. Addtonally the satsfacton of the strct complementarty condton µ > s ensured through the nspecton of the sgn of the Lagrange A (3

4 4 P. Seferls et al. multplers that correspond to the actve nequaltes. Any volaton of nequalty constrants and varable bounds as well as sgn changes for Lagrange multpler correspondng to actve nequaltes results n an actve set change for the problem. In such a case equaton set (4 s sutably modfed by ether ncludng a new actve nequalty or removng an nequalty that ceased to be actve n the equaton set. In addton nonlnear effects such as turnng or bfurcaton ponts n the optmal soluton path may result to optmalty loss volaton of the LICQ and multple soluton branches. The soluton of the parameterzed KKT condtons for varyng ntal pont condtons enables the robust and accurate dentfcaton of the optmal soluton of the assocated dynamc optmzaton problem. 3. Case Study 3.. Background The poston control wth mnmum sway of a cart wth a double pendulum as shown n Fgure usng MPC s consdered. Usng Lagrange s method the followng nd order nonlnear dfferental equatons of moton are derved for the system: F M ( M + m + m + ( m + m l [&& cos & && θ θ -θ snθ ] + m l && cos & θ θ -θ snθ + b& = (5 l m θ l θ m Fgure : Cart wth double pendulum. [ ] F ( m + m l θ&& + ( m + m l [&& cosθ ] + mll θ&& cos( θ -θ + θ& sn( θ -θ + ( m + m gl snθ = + m l l + m gl [ ] m l && cosθ + m l θ&& [ θ&& cos( θ -θ + θ& sn( θ -θ ] snθ The performance crteron for the system nvolves a penalty term for the devaton of the fnal poston and velocty of the cart from the desred values n addton to tme varyng terms penalzng devatons for the sway angles θ and θ and the control effort. = ( ( w - & + w ( θ -θ + w ( θ { + w u } J = w & θ (8 f f 3 t 4 - = The predcton horzon s selected equal to one second. The dscretzaton of the modelng equatons s acheved through twenty fnte elements utlzng 5 th order Lagrange nterpolatng polynomals therefore each fnte element represents a control nterval wth duraton of 5ms. The OCFE dscretzaton model ehbted a mamum absolute error of on the state vector profles for a gven control sequence when compared wth results from numercal ntegraton of the full order system. The soluton of the control problem as defned n problem (3 has been performed usng the augmented Lagrangan solver MIOS 5.5 [9] and a sparse verson of PITCO for t 5 (6 (7

5 Homotopy Contnuaton Soluton Method n onlnear Model Predctve Control 5 the parameterzed KKT condtons of equaton (4. The two problem formulatons nvolve 7 (equaton 3 and 45 (equaton 4 varables respectvely. Two hundred smulaton tme steps have been calculated wth actual control nterval equal to 5ms thus leadng to an overall smulaton tme span of s. o computatonal delay has been consdered n the mplementaton of the MPC algorthm. However the computatonal tme has been recorded n order to compare the performance of the two procedures n terms of soluton effort. umercal ntegraton of a plant model s used to smulate the plant behavor whereas a model wth msmatch has been utlzed n the MPC algorthm. The update of the state vector s based on the estmaton of a bas term equvalent to an ntegraton dsturbance model. The control profles and the state profles for the MPC are shown n Fgure. The MPC algorthm successfully placed the cart close to the new target poston whle mantaned the cart velocty and the two suspended masses swayng wthn acceptable lmts. The computatonal tme requred for the soluton of the MPC problem wth MIOS 5.5 was equal to 6.5ms on average for each control nterval relatvely large compared to the nterval duraton whereas the respectve soluton tme for the soluton of problem (4 wth a sparse mplementaton of PITCO was on average 4.4ms (4GHz Intel processor. The proposed soluton approach ehbted sgnfcantly better robustness as t fully converged at all smulaton steps. On the contrary the LP solver MIOS 5.5 faled at a small number of smulaton steps to converge to an optmal soluton. Force Tme s Cart poston (m - velocty (m/s cart poston cart velocty Tme s P angle poston (rad - velocty (rad/s P poston P velocty Tme s (a (b (c Fgure : Control actons (a and state profles (b-c usng the MPC algorthm. 4. Conclusons Ths paper presents a fast and effcent soluton approach of the dynamc optmzaton problem for nonlnear model predctve control applcatons. The soluton s obtaned from the parameterzed optmalty condtons resulted from a homotopy wth respect to the control problem ntal pont. The speed and robustness of the proposed method enables the effcent mplementaton of MPC n challengng engneerng problems. References [] Santos L.O. Afonso P.A.F..A. Castro J.A.A.M. Olvera.M.C. Begler L.T. Control Eng. Practce [] Bemporad A. Morar M. Dua V. Pstkopoulos E.. Automatca [3] De Olvera. M. C. Begler L. T. 995 Automatca 3 8. [4] Dehl M. Bock H.G. Schlöder J.P. 5 SIAM J. on Control and Optm [5] Zavala V.M. Begler L.T. 9 Automatca [6] Seferls P. Hrymak A Comput. & Chem. Eng. 77. [7] Rhenboldt W.C. Burkardt J.V. 983 ACM Trans on Math. Software [8] Davs T. A. UMFPACK User Gude. [9] Murtagh B. A. Saunders M. A. MIOS 5.5 User's Gude SOL 83-R. 99.