Sliding Mode Controller for Unstable Systems

Transcription

1 S. SIVARAMAKRISHNAN e al., Sliding Mode Conroller for Unsable Sysems, Chem. Biochem. Eng. Q. 22 (1) (28) 41 Sliding Mode Conroller for Unsable Sysems S. Sivaramakrishnan, A. K. Tangirala, and M. Chidambaram* Deparmen of Chemical Engineering, Indian Insiue of Technology, Madras, Chennai 636 Original scienific paper Received: Ocober 2, 26 Acceped: Sepember 15, 27 The mehod proposed by Rojas e al. 1 for he design of sliding mode conrollers (SMC) for unsable firs order plus ime delay sysems, is exended for delay-ime consan raio () up o 1.8. The SMC seings obained for various are fied by simple equaions. Up o = 1.2, he mehod is found o be more robus han ha of laes PID Conroller proposed by Padmasree e al. 2 There is no mehod available in lieraure o sabilize unsable sysems using PID conroller for > 1.2. Simulaion resuls are also given for a nonlinear bioreacor conrol problem. Key words: Sliding Mode Conrol, Firs Order plus Time Delay Sysems (FOPDT), Unsable Sysems Inroducion Sliding Mode Conrol (SMC) is a robus and simple procedure 3 o synhesize conrollers for linear and nonlinear processes. Convenional conrollers, such as PID, lead-lag or Smih predicors, are someimes insufficienly versaile o compensae for he uncerainies in he process parameers. A SMC could be designed o conrol linear and nonlinear sysems wih he assumpion ha he robusness of he conroller will ake care of he variaions in process parameers due o noise and oher disurbances. The aim of his paper is o design a SMC for unsable firs order plus delay ime (FOPDT) processes where > 1. A mehod for uning SMC for open loop unsable processes has been presened by Rojas e al. 1 However he mehod has some inheren disadvanages. The mehod does no work for > 1. So he mehod is modified o develop uning formulae for SMC for values of > 1. The performance of he closed loop sysem is compared wih ha of laes PID Conroller seings proposed by Padmasree e al. 2 The paper is organized as follows. Secion 2 briefly presens some basic conceps abou Sliding Mode Conrol. Secion 3 gives he SMC proposed by Rojas e al. 1 and is limiaions. Secion 4 describes he procedure o overcome he limiaion. Tuning equaions for he conroller are also given in his secion. Secion 5 shows he simulaion sudies o compare he resuling SMC performance wih ha of he laes PID conroller proposed by Padmasree e al. 2 Secion 6 shows he applicaion of SMC o an unsable bioreacor. Finally he conclusions are presened. * Corresponding auhor: Tel.: ; Fax: address: chidam@ni.edu Basic conceps abou sliding mode conrol The idea behind SMC is o define a surface along which he process can slide o is desired final value. Thus he firs sep in SMC is o define he sliding surface. The seleced in his work 4 is an inegral-differenial equaion acing on he racking error expression. n d e () d (1) d where e() is he racking error, is a uning parameer, which helps o define ; his erm is seleced by he designer, and deermines he performance of he sysem on he sliding surface, n is he sysem order. Once he reference value is reached, eq. (1) indicaes ha a seady sae reaches a consan value. To mainain a his consan value, meaning ha e() has o be zero a all imes, i is desired o make d (2) d Once he sliding surface has been seleced, aenion mus be urned o design he conrol law ha drives he conrolled variable o is reference value and saisfies eq. (2). The SMC conrol law, U() consiss of wo addiive pars; a coninuous par, U C (), and a disconinuous par, U D (): U() =U C () +U D () (3) The coninuous par is given by U C () =f (X(), R()) (4)

2 42 S. SIVARAMAKRISHNAN e al., Sliding Mode Conroller for Unsable Sysems, Chem. Biochem. Eng. Q. 22 (1) (28) Where f (X(), R()) is a funcion of he conrolled variable, and he reference value. The disconinuous par, U D (), incorporaes a nonlinear elemen ha includes he swiching elemen of he conrol law. This par of he conroller is disconinuous across he sliding surface: U D () =K D (5) where K D is a uning parameer responsible for he reaching mode, is a uning parameer used o reduce he chaering problem. Chaering is a high-frequency oscillaion around he desired equilibrium poin. SMC proposed by Rojas e al. Rojas e al. 1 have proposed a mehod for sliding mode conrol for FOPDT open loop unsable processes, whose open loop ransfer funcion is given by K e s p. s1 The conrol law is given by dx() X() U() 1 e () K d Wih K D (6a) de () sign ( K) e () e () 1 d (6b) d By aking we can simplify U C as 1 X() U() e () K K p D (7a) To assure ha he sliding surfaces behave as a criical or overdamped sysem, should be 1 4 (7b) Using he Nelder-Mead opimizaion algorihm, Rojas e al. 1 have seleced he values of K D and by minimizing ISE values and have proposed he following uning equaions for he SMC uning parameers: 2 K P K D = = K P K D 1. ; (7c) (7d) Eq. (7a) will cause problems if = >1, since his will make 1 negaive which resuls in insabiliy on he surface. Exension for from 1 o 1.8 We make he erm 1 negaive, since his will make 1 posiive, which will guaranee sabiliy on he surface. In his work, we make 1 1 and calculae 1 accordingly for various values of. The SMC parameers, namely K D and are esimaed by minimizing ISE using malab leas squares mehod. Up o = 1, he uning formulae given in eqs. (7c) and (7d) are used as he iniial guesses. For > 1, he uning formulae for sable processes given by Camacho and Smih 5 are used as iniial guesses. The converged parameer values are given in Table 1. Fig. 1 shows he variaion of he SMC parameers K D and wih. The values obained for K D and are fied by he following simple equaions as: For <.8: 33. K P K D ; (8a) For.8 1: K P K D (8b) For 1 1.8: K P K D.3531 ; (8c) For 1: K P K D 1. For K P K D 1 For = 14. (9a) (9b) (9c) The maximum error in he fied equaions is less han 1 %.

3 S. SIVARAMAKRISHNAN e al., Sliding Mode Conroller for Unsable Sysems, Chem. Biochem. Eng. Q. 22 (1) (28) 43 Table 1 Values obained for K D and by opimizaion mehod for various values of Sl. No. K D Fig. 1 Variaion of K D and wih. Solid line: Fied value using eqs. 8a,8b,9a,9b,1a,1b. Doed: acual value Simulaion resuls Example 1: Le us consider an unsable FOPDT model wih K p =1, = 1 and =.8. The SMC parameer seings are 1 =.25 min 1, = min 2, K D = , = (Table 1). The PID seings given by Padmasree e al. 2 are K C = , I = , D =.487. The SMC seings for he mehod proposed by Rojas e al. 1 are 1 =.25 min 1, = min 2, K D =.9479, =.784. Fig. 2 shows he comparisons of he servo response of all he hree mehods. The presen mehod is found o be beer han he oher wo. Under perfec parameer condiions, overshoo is lesser for he presen mehod. The robusness of he conroller is evaluaed by perurbing he gain as 1.2 in he process whereas he conroller seings used are for K p = 1. I is found ha he PID conroller 2 and SMC 1 ake a long ime o sele and also show high oscillaions (Fig. 2b). Similar robus performances are also obained for uncerainy in ime consan and ime delay. I is found ha he PID Fig. 2 Comparison of servo responses for unsable sysems: K p =1; =1; =.8 wih PID Conroller,SMC conroller proposed by Rojas e al. 1 Solid Line: Presen Mehod. Doed: PID,Dashed: SMC proposed by Rojas e al. 1 (a) Perfec parameer,(b) Uncerainy of +2 % in K p,(c) Uncerainy of 2 % in,(d) Uncerainy of +2 % in.

4 44 S. SIVARAMAKRISHNAN e al., Sliding Mode Conroller for Unsable Sysems, Chem. Biochem. Eng. Q. 22 (1) (28) Conroller is unable o sabilize he sysem wih respec o uncerainies in ime consan, whereas in he case of ime delay, i has a huge overshoo (Figs. 2c and 2d). The performances under model parameer uncerainy are beer for he presen mehod. Fig. 3 shows he comparison of he manipulaed variables vs. ime for he conrollers. From he plo, i is obvious ha he SMC Conroller oupu of he presen mehod is much smooher han ha of he PID Conroller and he SMC proposed by Rojas e al. 1 which cause highly oscillaory movemen of he conrol valve. Fig. 4 Comparison of servo responses for unsable sysems: K p =1; =1; = 1.2 wih PID Conroller. Solid Line: Presen Mehod. Doed: PID. (a) Perfec parameer,(b) Uncerainy of +5 % in K p,(c) Uncerainy of 2 % in,(d) Uncerainy of +2 % in, Fig. 3 Comparison of manipulaed variables. Legend: Same as in Fig. 2 Example 2: Le us consider an unsable FOPDT model wih K P =1, = 1 and = 1.2. The SMC parameer seings by he presen mehod are 1 = 1/3 min 1, = 1/36 min 2, K D =.2233, = (Table 1). For performance comparison, we use he PID seings given by Padmasree e al. 2 The SMC proposed by Rojas e al. 1 will no work for 1. Hence we compare he presen mehod wih he laes PID conroller proposed by Padmasree e al. 2 The conroller seings given by Padmasree e al. 2 are K C = , I = , D =.687. Fig. 4 shows he servo response for hese seings. Regarding perurbaions, he PID conroller does no sabilize even a 5 % change in K p whereas he presen SMC sabilizes he process. The response is beer han ha of he PID mehod in all he cases as can be seen from he figure. The same holds good for he conroller oupu also as shown in Fig. 5. Example 3: The performance of he proposed mehod is evaluaed on he sysem e 1.4 s /(s 1).We obain he SMC seings for he presen mehod as 1 = 3/14, = 9/784, K D =.1967 and = (Table 1). Since > 1.2, he PID conroller fails o Fig. 5 Comparison of conroller oupus under perfec parameers condiion. Legend: Same as in Fig. 4 sabilize he sysem. Fig. 6 shows he performance of he SMC under perfec parameer condiions and variaions in he process parameers. The performance of he SMC is found o be very good excep for variaion in process gain where i akes a lo of ime o sele. The manipulaed variable vs. ime plo under condiions of perfec parameer is given in Fig. 7. Example 4: Le us consider he process exp(1.7 s)/(s 1) wih a larger ime delay ( = 1.7), which once again canno be sabilized by PID conroller. We obain he SMC seings for he presen mehod as 1 = 3/34 min 1, = 9/4624 min 2, K D =.565, = (Table 1). Fig. 8a shows he response for servo problem under perfec pa-

5 S. SIVARAMAKRISHNAN e al., Sliding Mode Conroller for Unsable Sysems, Chem. Biochem. Eng. Q. 22 (1) (28) 45 Fig. 6 Servo responses for unsable sysems: K p =1; =1; = 1.4 wih SMC Conroller. (a) Solid Line: Perfec parameer,doed: Uncerainy of + 1 % in K p (b) Solid Line: Uncerainy of 1 % in,doed: Uncerainy of +1 % in Fig. 8 Servo responses for unsable sysems: K p =1; =1; = 1.7 wih SMC Conroller. (a) Solid Line: Perfec parameer,doed: (a) Uncerainy of 6 %in,(b) Uncerainy of +5 % in Table 2 characerisics of SMC wih respec o process gain,ime consan and ime delay for various values of Sl. No. in K p /% in /% in /% % 2 % +2 % % 2 % +2 % % 2 % +2 % % 2 % +2 % % 2 % +2 % % 2 % +2 % Fig. 7 Plo of manipulaed variable of SMC vs. ime for K p =1; =1; = 1.4 under perfec parameers condiion rameers. Fig. 8b shows he response when here is an uncerainy of 1 % in he process ime consan. Table 2 gives he robusness characerisics of he SMC wih respec o process parameers separaely in process gain, ime consan and ime delay for various values of. From he Table, i is clear ha up o = 1.2, he presen SMC is robus wih respec o gain, ime consan and delay. However, as increases furher, he robusness wih respec o K P, and o reduces. A = 1.8, he SMC is found o wihsand up o 1 % decrease in and 1 % increase separaely in and K p % 2 % +2 % % 2 % +2 % % 2 % +2 % % 2 % +2 % % 2 % +2 % % 2 % +2 % % 2 % +2 % % 16 % % % 16 % % % 14 % % % % % +5 %

6 46 S. SIVARAMAKRISHNAN e al., Sliding Mode Conroller for Unsable Sysems, Chem. Biochem. Eng. Q. 22 (1) (28) By simulaion, we found ha he value of 1 for he erm 1 gives saisfacory resuls. However, if we decrease he value of 1 o, say.5, we find ha he robusness of he conroller decreases considerably for = 1.4. The reduced robusness characerisics are given in Table 3. If we increase he value of 1 o, say 1.5, we find ha he performance of he resulan conroller is no good; i.e. he conroller has larger overshoo and more seling ime. Hence, a value of 1 is recommended for he erm 1. following unsable ransfer funcion model e 2 s / (99.69 s 1). A measuremen delay of 2 s is considered. For his unsable FOPDT sysem, a SMC is designed based on he uning parameers given in Table 1. The values of he uning parameers are 1 =.4, 2 =.4, K D =.713 and = The PID seings given by Padmasree e al. 2 are K c = 1.616, I = and D = The regulaory responses for hese wo seings for a sep disurbance in Q from.3333 o.3 L s 1 are evaluaed on he nonlinear model eq. (1) and he responses are shown in Fig. 1. The presen mehod gives a beer performance. The regulaory responses for an uncerainy in ime delay are also evaluaed (24 s delay in he process whereas he conroller seings are designed for 2 s delay). The Table 3 characerisics of SMC wih respec o process gain,ime consan and ime delay for values of > 1.4,up o 1.8 when 1=.5 Sl. No. in K p /% in / % in /% % 16 % % % 14 % 1.63 % % 7.65 % % 5 % Applicaion o an unsable bioreacor A SMC is designed and simulaed for an isohermal bioreacor exhibiing muliple seady sae soluions. The resuls are compared wih he laes PID conroller proposed by Padmasree e al. 2 The mahemaical model equaion of he reacor is given by Liou and Chien 6 as dc Q f d V c c kc 1 ( ) ( k c1) 2 2 (1) where Q is he inle flow rae and c f is he feed concenraion. The values of he operaing condiions are given by Q =.3333 L s 1 ; V =1L;k 1 =1s 1 ; and k 2 =1Lmol 1. For he nominal value of c f = mol L 1, he seady sae soluion of he model equaion gives he following wo sable seady saes a c = mol L 1 and.1424 mol L 1. There is one unsable seady sae a c = mol L 1. Feed concenraion is considered as he manipulaed variable. Linearizaion of he model equaion around his operaing condiion c = mol L 1 gives he Fig. 9 Plo of conroller oupu vs. ime for K p =1; =1; = 1.7 under perfec parameers condiion Fig. 1 The regulaory response of nonlinear bioreacor for a sep change in q from.3333 o.3 s 1 for delay = 2 s. solid: SMC (Presen),dash: PID

7 S. SIVARAMAKRISHNAN e al., Sliding Mode Conroller for Unsable Sysems, Chem. Biochem. Eng. Q. 22 (1) (28) 47 Fig. 11 Regulaory response of nonlinear chemical reacor for +2 % uncerainy in ime delay. Legend: as in Fig. 1 Fig. 12 The regulaory response of nonlinear bioreacor for a sep change in q from.3333 o.3233 s 1 for delay = 15 s using SMC responses are shown in Fig. 11. The presen mehod gives robus performance han ha of Padmasree e al. 2 Le us consider he measuremen delay of 15 s so ha he delay ime consan raio () = 1.5. For > 1.2, he PID conroller canno sabilize he sysem. Hence we design a SMC for sabilizing he sysem. The values of he uning parameers are 1 =.9967, =.2484, K D =.15, = The regulaory response for a sep change in he inle flow rae from.3333 o.3233 s 1 is shown in Fig. 12. The regulaory response for a +1 % uncerainy in ime delay is shown in Fig. 13. Conclusions A Sliding Mode Conroller is synhesized for a FOPDT unsable sysem for delay o ime consan raio up o 1.8. Up o = 1.2, he performance of he proposed conroller is found o be more robus han he laes PID conroller proposed by Padmasree e al. 2 For beyond 1.2, here is no mehod available in he lieraure o sabilize unsable sysems using a PID conroller. The robusness of he proposed conroller wih respec o uncerainies in process gain, ime consan and ime delay is found o decrease wih increasing. Simulaion resul on conrol of a nonlinear bioreacor is also given for he case of = 1.5. Nomenclaure K p process gain process ime consan process ime delay delay o ime consan raio Fig. 13 Regulaory response of nonlinear chemical reacor using SMC for +1 % uncerainy in ime delay for delay = 15 s Abbreviaions SMC sliding mode conroller FOPDT firs order plus delay ime ISE inegral of square of error References 1. Rojas, R., Camacho, O., Gonzalez, O. L., ISA Transacions 43 (24) Padmasree, R., Chidambaram, M., Srinivas, M. N., Compuers and Chemical Engineering 28 (24) Sloine, J. J.,Li, W., Applied Nonlinear Conrol, Prenice Hall, New Jersey, Huang,Y. J.,Way, H. K., ISA Transacion 4 (21) Camacho, O., Smih,A., ISA Transacions 39 (2) Liou, C. T., Chien, Y. S., Chem. Eng. Sci. 46 (1991) 2113.