An Efficient Formulation of Off-line Model Predictive Control for Nonlinear Systems Using Polyhedral Invariant Sets

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1 04 International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies. International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies ttp://tuenr.co n Efficient Forulation of Off-line Model Predictive Control for Nonlinear Systes Usin Polyedral Invariant Sets Porncai Buroonsri a*, Pornpun rundecacai b, and Sooratep Keawo b a Departent of Ceical Enineerin Faculty of Enineerin, Maidol University, THIND b Departent of Ceical Enineerin Faculty of Enineerin, Culalonkorn University, THIND R T I C E I N F O rticle istory: Received 0 uust 03 ccepted 06 Deceber 03 vailable online 09 Deceber 03 Keywords: control law; real-tie interpolation; scedulin paraeter; MPC alorit B S T R C T In tis researc, an efficient forulation of off-line odel predictive control for nonlinear systes is presented. Te nonlinear systes are reforulated as linear paraeter varyin systes so teir coplexity is reduced witout any loss of enerality. Te on-line coputational burdens are decreased by pre-coputin off-line te sequences of explicit control laws correspondin to te sequences of polyedral invariant sets. t eac saplin tie, te current state and te scedulin paraeter are easured. Te real-tie control law is ten calculated by linear interpolation between te pre-coputed control laws. Te results indicate tat te proposed alorit can acieve better control perforance copared to te previously developed off-line robust odel predictive control alorit because te scedulin paraeter is incorporated into te controller desin. 04 INT TRNS J ENG MNG SCI TECH... Introduction Ceical processes are ultivariable processes tat cane one or ore ceical copounds to te desired products. Ceical processes are usually involved wit any coplex ceical reactions wic are nonlinear. In order to efficiently control nonlinear ceical processes, a ultivariable nonlinear control alorit needs to be developed Qin and Badwell, 003; Raes, et al., 009; Manent 0. *Correspondin autor P.Buroonsri. Tel: Ext.60. E-ail address: porncai.bu@aidol.ac.t. 04. International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies. Volue 5 No. ISSN eissn Online available at ttp://tuenr.co/v05/005.pdf. 5

2 Model predictive control MPC is an advanced control alorit for ultivariable processes. MPC is widely used in any ceical processes because input and output constraints are considered in a systeatic anner Morari and ee, 999; Mayne, et al., 000; ee, 0. linear odel is typically used in MPC forulation because te on-line optiization proble can be easily solved. However, ost of te ceical processes are nonlinear. Wen te operatin conditions undero canes, te perforance of linear MPC can sinificantly deteriorate Buroonsri and Keawo, 0a; Yu, et al., 0; Suzuki and Suie, 007. In order to deal wit nonlinear ceical processes, nonlinear MPC was developed by Man et al. 00. full nonlinear odel was used in MPC forulation. Te coplicated nonlinear control proble ad to be solved at eac saplin instant so te alorit was coputationally proibitive in practical situations. Te reforulation of nonlinear systes into linear paraeter varyin PV systes is a proisin tecnique to reduce te coplexity of nonlinear systes. PV systes are linear systes wose dynaics depend on tie-varyin paraeters tat can be easured on-line. Terefore, nonlinear systes can be reforulated as PV systes witout any loss of enerality Park and Jeon, 004; Tot, 00; Juners, et al., 0. Off-line MPC for PV systes was previously developed by Buroonsri and Keawo 0b. ltou te on-line coputational tie was sinificantly reduced, te stabilizable reion of te alorit was sall because te ellipsoidal invariant sets were used in proble forulation. In order to enlare te size of stabilizable reion, an off-line forulation of robust MPC usin polyedral invariant sets was proposed by Buroonsri and Keawo 0c. Te polyedral invariant sets were used in te proble forulation so a sinificantly larer stabilizable reion was obtained. ltou te stabilizable reion was enlared, te conservativeness was obtained because te scedulin paraeter was not incorporated into te controller desin. In tis researc, an efficient forulation of off-line MPC usin polyedral invariant sets is presented. Te sequences of explicit control laws correspondin to te sequences of polyedral invariant sets are pre-coputed off-line. t eac saplin instant, te current state and te 6 Porncai Buroonsr Pornpun rundecaca and Sooratep Keawo

3 scedulin paraeter are easured. Te real-tie control law is ten calculated by linear interpolation between te pre-coputed control laws. Te proposed alorit can ive a relatively lare stabilizable reion because te polyedral invariant sets are coputed in te off-line proble forulation. Moreover, te scedulin paraeter is used in real-tie interpolation between te pre-coputed control laws so better control perforance copared to an off-line robust MPC alorit of Buroonsri and Keawo 0c can be obtained. Tis article is oranized as follows. In section, te proble description is presented. Te proposed alorit is presented in section 3. In section 4, te proposed alorit is applied to a case study and te results are discussed. Finally, te paper is concluded in section 5.. Proble Description Te odel considered ere is te followin linear paraeter varyin PV systes Te tecniques to transfor nonlinear systes into PV systes can be found in Tot 00. x k + p k x k + Bu k, y k Cx k, were xk is a vector of states, u k is a vector of control inputs and y k is a vector of outputs. In tis researc, we assue tat te scedulin paraeters p k [ p k, p k,..., p ] are easurable on-line at eac saplin instant. Moreover, we k assue tat p k Ω, Ω Co{,,..., } 3, were Ω is te polytope, Co is te convex ull, are te vertices of te convex ull and is te nuber of vertices of Ω. ny p k witin Ω is a linear cobination of te vertices suc tat p k p k, p k, 0 p k 4, *Correspondin autor P.Buroonsri. Tel: Ext.60. E-ail address: porncai.bu@aidol.ac.t. 04. International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies. Volue 5 No. ISSN eissn Online available at ttp://tuenr.co/v05/005.pdf. 7

4 Te obective is to find a state feedback control law tat stabilizes PV systes and subect to te followin input and output constraints u u k + i / k u,,,..., nu, 0,,,..., 5,, in,ax i yr yr k + i / k yr,ax, r,,..., ny, 0,,,..., 6,, in i were n u is te nuber of control inputs, n y is te nuber of outputs, u in and u ax are te vectors of input constraints, y in and y ax are te vectors of output constraints. 3. Te Proposed Off-line MPC lorit In tis section, an off-line MPC forulation for nonlinear systes is developed. Te nonlinear systes are reforulated as PV systes so teir coplexity is sinificantly reduced. Most of te solutions of te control proble are calculated off-line so te on-line coputational tie is sinificantly reduced. 3. Off-line Procedures 3.. Off-line Step : Copute te Sequences of Off-line State Feedback Gains Coose a sequence of states x i i {,,...,N}, and solve te optiization proble presented by Buroonsri and Keawo 0b off-line to obtain te sequences of state feedback ains K i,,, i,,..., N,,,... were N is te nuber of te cosen states and is te nuber of vertices of Ω. 3.. Off-line Step : Copute te Sequences of Polyedral Invariant Sets Given te state feedback ains polyedral invariant set { x M x d } S i, / i, i, K i, fro 3... For eac K i,, te correspondin is coputed by followin tese steps Set M ], T T T T T [ C, C, K, K T T T T T [ yax, yin, uax, uin and d ]. Select row fro M, d and ceck l, l,..., weter + is redundant wit respect to M, d M, l Bl K x d, by solvin te followin proble 8 Porncai Buroonsr Pornpun rundecaca and Sooratep Keawo

5 ax 7, W i,,, l s.t. W,, l M, l + Bl K x d, 8, M i x d, 9, If W 0, te constraint M, l + Bl K x d, is non-redundant wit respect to i,,, l > M, d. Ten, add non-redundant constraints to M, d by assinin M ] T T T [ M, M, l + Bl K and T T T d [ d, d, ]. 3 et + and return to step. If is strictly larer tan te nuber of rows in M, d ten terinate. 3. On-line Procedures t eac saplin tie, easure te current state x k and te scedulin paraeter p k. Wen x k S i,, x k S i +,,,,...,, i N, te real-tie state feedback ain + k p k Ki+, K α k α k p k K α can be calculated fro α k obtained by solvin te followin optiization proble in α k 0, s.t. u in K k x k α u, ax p k x k + BK α k x k Si,,,,...,, 0 < α k 3, It is seen tat te on-line optiization proble is only linear prorain so it can be efficiently solved Boyd and Vandenbere, 004. is for uaranteein input constraint satisfaction and is for uaranteein tat te next predicted state still lies in te polyedral *Correspondin autor P.Buroonsri. Tel: Ext.60. E-ail address: porncai.bu@aidol.ac.t. 04. International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies. Volue 5 No. ISSN eissn Online available at ttp://tuenr.co/v05/005.pdf. 9

6 invariant sets coputed off-line. 4. Results and Discussion Consider te nonlinear two-tank syste nel et al., 000 wic is described by te followin equation. S + u 4, S. 5, were is te water level in tank, is te water level in tank and u is te inlet water flow. Te sceatic diara of te nonlinear two-tank syste is sown in Fiure. Fiure : Te sceatic diara of te nonlinear two-tank syste. Te operatin paraeters of te nonlinear two-tank syste are sown in Table. Table : Te operatin paraeters of te nonlinear two-tank syste. Paraeter Value Unit S,500 c S,600 c 9 c 4 c 980 c/s 0.00 k/c 3,eq 4 c,eq 70 c 30 Porncai Buroonsr Pornpun rundecaca and Sooratep Keawo

7 et eq,, eq, and eq u u u were subscript eq is used to denote te correspondin variable at equilibriu condition, te obective is to reulate to te oriin by anipulatin u. Te input and output constraints are iven as follows.5 u k/s, 3 c, 50 c 6, By evaluatin te Jacobian atrix of 4 and 5 alon te vertices of te constraints set 6, we ave tat all te solutions of 4 and 5 are also te solutions of te followin differential inclusion u p S S + 7, were 4,,..., are iven by, 0,in,in,in,in,ax,ax 0, 0,ax,in,in 3,ax,ax,ax 4 0 8, and 4,,..., p are iven by / / / /. / / / /,in,ax,ax,in,ax,ax p *Correspondin autor P.Buroonsri. Tel: Ext.60. E-ail address: porncai.bu@aidol.ac.t. 04. International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies. Volue 5 No. ISSN eissn Online available at ttp://tuenr.co/v05/005.pdf. 3

8 p p 3 / / /,ax / / /,ax,ax,in,in /. / / /. /,in /,ax,ax,ax / / / /,in,in,in / /,in / /,in p. 9, 4 /,ax /,in /,ax /,in Te discrete-tie odel is obtained by discretization of 7 usin Euler first-order approxiation Sebor, et al., 004 wit a saplin period of 0.5 s and it is oitted ere for brevity. Fiure sows te polyedral invariant sets coputed off-line by te proposed alorit. Fiure 3 sows te polyedral invariant sets coputed off-line by an off-line robust MPC alorit of Buroonsri and Keawo 0c. For bot alorits, te polyedral invariant sets are coputed by coosin te sae sequence of states, i {,,...,5} x i. Note tat wit te sae nuber of cosen states, te proposed alorit requires larer nuber of polyedral invariant sets tan an off-line robust MPC alorit of Buroonsri and Keawo 0c. Tis is due to te fact tat for te proposed alorit, te nuber of sequences of polyedral invariant sets coputed is equal to te nuber of te vertices of te polytope Ω. In coparison, only a sequence of polyedral invariant sets needs to be coputed off-line for an off-line robust MPC alorit of Buroonsri and Keawo 0c.. S, i, i {,,..., 5} 3 Porncai Buroonsr Pornpun rundecaca and Sooratep Keawo

9 . S i,, i {,,..., 5}.3 S, i, 3 i {,,..., 5}.4 S, i, 4 i {,,..., 5} Fiure : Te polyedral invariant sets coputed off-line by te proposed alorit. *Correspondin autor P.Buroonsri. Tel: Ext.60. E-ail address: porncai.bu@aidol.ac.t. 04. International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies. Volue 5 No. ISSN eissn Online available at ttp://tuenr.co/v05/005.pdf. 33

10 Fiure 3: Te polyedral invariant sets coputed off-line by an off-line robust MPC alorit of Buroonsri and Keawo 0c. Fiure 4 sows te reulated output. For te proposed alorit, te scedulin paraeter is easured on-line at eac saplin tie so less conservativeness copared to an off-line robust MPC alorit of Buroonsri and Keawo 0c can be obtained. It can be observed tat te proposed alorit requires less tie to enter and reain witin te settlin band ± 0.% of,equilibriu and Keawo 0c. copared to an off-line robust MPC alorit of Buroonsri Fiure 4: Te reulated output. Te control input is sown in Fiure 5. For te proposed alorit, te pre-coputed state 34 Porncai Buroonsr Pornpun rundecaca and Sooratep Keawo

11 feedback ains are interpolated on-line so a sooter input response is obtained. Fiure 5: Te control input. Te overall coputational burdens are sown in Table. ltou te proposed alorit requires larer off-line coputational tie tan an off-line robust MPC alorit of Buroonsri and Keawo 0c, te on-line coputation is tractable because only linear prorain needs to be solved on-line. ll of te siulations ave been perfored in Intel Core i-5.4ghz, GB RM, usin SeDuMi Stur, 999 and Yalip öfber, 0 witin Matlab 008a environent. Table : Te overall coputational burdens. lorit Off-line CPU tie s On-line CPU tie s n off-line robust MPC alorit Te proposed alorit Conclusion In tis researc, an efficient forulation of off-line MPC for nonlinear systes usin polyedral invariant sets as been developed. Te results sow tat te proposed alorit can ive better control perforance tan te previously developed off-line robust MPC alorit. Tis is due to te fact te scedulin paraeter is incorporated into te controller desin. Te controller desin is illustrated wit an exaple of nonlinear two-tank syste. *Correspondin autor P.Buroonsri. Tel: Ext.60. E-ail address: porncai.bu@aidol.ac.t. 04. International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies. Volue 5 No. ISSN eissn Online available at ttp://tuenr.co/v05/005.pdf. 35

12 6. cknowledeents Tis researc proect is supported by Maidol University. 7. References nel D., Casavola,., and Mosca, E Constrained predictive control of nonlinear plants via polytopic linear syste ebeddin, Int. J. Robust Nonlin., 03, Boyd, S., and Vandenbere, Convex Optiization, Cabride University Press, Cabride. Buroonsr P., and Keawo, S. 0a. MPC for PV systes based on paraeter-dependent lyapunov function wit perturbation on control input stratey. Enineerin Journal, 6, 6-7. Buroonsr P., and Keawo, S. 0b. n ellipsoidal off-line odel predictive control stratey for linear paraeter varyin systes wit applications in ceical processes. Syst. Control ett., 63, Buroonsr P., and Keawo, S. 0c. n off-line robust MPC alorit for uncertain polytopic discrete-tie systes usin polyedral invariant sets. J. Process Contr., 6, Juners, M., Oliveira, R.C..F., and Peres, P..D. 0. MPC for PV systes wit bounded paraeter variations. Int. J. Control, 84, ee, J.H. 0. Model Predictive Control: review of te tree decades of developent. Int. J. Control uto., 93, öfber, J. 0. utoatic robust convex prorain. Opti. Metod Softw., 7, 5-9. Man., Nicolao, G.D., Manan., and Scattolin R. 00. stabilizin odel-based predictive control alorit for nonlinear systes. utoatica, 379, Manent F. 0. Considerations on nonlinear odel predictive control tecniques. Coput. Ce. En., 35, Mayne, D.Q., Rawlins, J.B., Rao, C.V., and Scokaert, P.O.M Constrained odel predictive control: stability and optiality. utoatica, 366, Morar M., and ee, J.H Model predictive control: past, present and future. Coput. Ce. En., 34, Park, P.G., and Jeon, S.C Constrained RHC for PV systes wit bounded rates of paraeter variations, utoatica, 405, Qin, S.J., and Badwell, T survey of industrial odel predictive control tecnoloy. utoatica, 7, Raes, K., Sukor, S.R.., and ziz, N Nonlinear odel predictive control of a 36 Porncai Buroonsr Pornpun rundecaca and Sooratep Keawo

13 distillation colun usin NRX odel. Cop. id C., 7, Sebor, D.E., Edar, T.F., and Mellicap, D Process Dynaics and Control, Jon Wiley & Sons, New York. Stur, J.F Usin SeDuMi.0, a Matlab toolbox for optiization over syetric cones. Opti. Metod Softw.,, Suzuk H., and Suie, T Model predictive control for linear paraeter varyin constrained systes usin ellipsoidal set prediction. Int. J. Control, 80, Tot, R. 00. Modelin and Identification of inear Paraeter-Varyin Systes, Spriner, ondon. Yu, S., Bö, C., Cen, H., and llöwer, F. 0. Model predictive control of constrained PV systes. Int. J. Control, 856, Dr. P. Buroonsri is a lecturer in te Departent of Ceical Enineerin, Maidol University. He received is B.En. fro Culalonkorn University in 008. He obtained is M.En and D.En fro Culalonkorn University in 009 and 0, respectively. His current interests involve robust MPC syntesis, odelin and optiization in ceical processes. P. rundecacai is a raduate student in te Departent of Ceical Enineerin, Culalonkorn University. Se received er B.En fro te Departent of Ceical Enineerin, Kon Kaen University in 0. Her researc interests are in optiization and printed electronics. Dr. S. Keawo is an ssistant Professor in te Departent of Ceical Enineerin, Culalonkorn University. He earned is B. En fro Culalonkorn University in 997. He continued is study at te University of Tokyo were e received is M.En and P.D. in 00 and 004, respectively. He as been workin on te use of statistics and optiization, life cycle and printed electronics. Peer Review: Tis article as been internationally peer-reviewed and accepted for publication accordin to te uidelines in te ournal s website. Note: Oriinal version of tis article was accepted and presented at te Tird International-Tai Ceical Enineerin and pplied Ceistry TICE Conference, ointly oranized by Departent of Ceical Enineerin, Faculty of Enineerin, Kon Kaen University and Tai Institute of Ceical Enineerin and pplied Ceistry, at Pullan Kon Kaen Raa Orcid Hotel, Kon Kaen, THIND, October 7-8, 03. *Correspondin autor P.Buroonsri. Tel: Ext.60. E-ail address: porncai.bu@aidol.ac.t. 04. International Transaction Journal of Enineerin, Manaeent, & pplied Sciences & Tecnoloies. Volue 5 No. ISSN eissn Online available at ttp://tuenr.co/v05/005.pdf. 37