INTRODUCING FRACTIONAL SLIDING MODE CONTROL

Transcription

1 II Enconro de Jovens Invesigadores do LAETA FEUP, Poro, April 2012 INTRODUCING FRACTIONAL SLIDING MODE CONTROL Duare Valério IDMEC / IST, TULisbon Av Rovisco Pais 1, Lisboa, Porugal duarevalerio@isulp Key words: Fracional derivaives, Fracional Calculus, Fracional conrol, Sliding mode conrol, Nonlinear conrol Absrac Tis survey paper presens fracional sliding mode conrol Sliding mode conrol is a nonlinear conrol sraegy a consiss in providing a conrol acion a resrics e plan o conrol o funcion according o some pre-defined simple dynamic Tis desired dynamic is called sliding surface Wen e plan follows e sliding surface, i is in sliding mode Fracional sliding mode conrol is e use of sliding mode conrol wi fracional plans, or e use of sliding mode conrol wi a sliding surface corresponding o a fracional order dynamic, or bo Here, fracional refers o e inclusion of fracional derivaives in e differenial equaion governing e dynamic a correspond o fracional powers of s wen a Laplace ransform is applied, or fracional powers of jω wen a Fourier ransform is applied Forcing a plan o beave according o some pre-defined simple fracional dynamic may be desirable since suc a dynamical beaviour as significan robusness properies 1 Inroducion Sliding mode conrol is a nonlinear conrol sraegy a consiss in providing a conrol acion a resrics e plan o conrol o funcion according o some pre-defined simple dynamic Tis desired dynamic is called sliding surface Wen e plan follows e sliding surface, i is in sliding mode For an inroducion o is sraegy, see for insance [5] Fracional sliding mode conrol is e use of sliding mode conrol wi fracional plans, or e use of sliding mode conrol wi a sliding surface corresponding o a fracional order dynamic, or bo Here, fracional refers o e inclusion of fracional derivaives in e differenial equaion governing e dynamic a correspond o fracional powers of s wen a Laplace ransform is applied, or fracional powers of jω wen a Fourier ransform is applied Forcing a plan o beave according o some pre-defined simple fracional dynamic may be desirable since suc a dynamical beaviour as significan robusness properies Tis survey paper is organised as follows: secion 2 presens basic noions of fracional derivaives; secion 3 presens fracional sliding mode conrol; secion 4 concludes e paper wi a simulaion example 2 Fracional derivaives For more deails on fracional derivaives, see, for insance, [7, 6] and e bibliograpy quoed erein

2 Duare Valério 21 Ineger order derivaives By definiion, and so f lim f f f f f lim lim f f f f 2 f 2f +f 2 lim 2 2 f 3 f f lim lim f 2f +f 2 2 f 2f +f 2 2 lim f 3f +3f 2 f n 1 k n k k0 f k f n lim n 4 A rigourous proof of e las equaliy can be found by maemaic inducion Recall a combinaions of a ings, b a a ime are defined as a b 22 Te Grünwald-Lenikoff definiion a! b!a b! Combinaions of a ings, b a a ime can be defined also for a, b / N 0 using funcion Γ: a Γa +1 6 b Γb +1Γa b +1 I can be easily sown again, using maemaical inducion a is expression is equivalen o a 1b Γb a 7 b Γb +1Γ a As e applicabiliy range of 6 and 7 is no exacly e same, combinaions are beer defined joining bo expressions ogeer and adding some poins oerwise undefined bu were 6 or 7convergeo0oge Γa +1, if a, b, a b/ Z Γb +1Γa b +1 a 1 b Γb a b Γb +1Γ a, if a Z b Z , if b Z b a N a/ Z Because of is, we can apply 4 also wen e order is no naural:? 1 k z f k k f z k0 lim z 9 1 5

3 Duare Valério Te upper limi of e summaion is cosen so a 9 will, for z Z, resul in a Riemann inegral Making z 1 in 9, and using 7, f 1 lim lim? 1 1 k k k0 1 f k? 1 k 1k Γk +1 f k 10 Γk + 1Γ1 k0 If is is o be a Riemann inegral c fd, en e upper limi of e summaion mus be c, and mus be resriced o posiive values If e good beaviour of f includes differeniabiliy, is will bring no problems of differen rig derivaives and lef derivaives a any poin c is called e erminal We are us led o a definiion of fracional derivaive due o e works of Grünwald and of Lenikoff: c 1 k z k k0 f k f z lim + z 11 I can be sown a z 2, z 3 and so on also lead o e corresponding Riemann inegrals Noice a c is diverging o + Wen z is naural, e summaion can be runcaed afer z erms, because 8 sows a from en on all erms are zero as k z N Tis is e only case e summaion can be runcaed; in all oers ere will be an infiniy of erms Wa ismeansisawenz is naural e derivaive will ave a value independen of c; oerwise e derivaive will depend on c Or, on oer words, naural order derivaives are local operaors; all oers are no local, since ey depend on c, wicforz 1 is e lower limi of inegraion In oer words sill, if is idenified wi ime, e resul as a memory of wa appens o funcion f for some ime before, since ime c Fracional derivaives always look like inegrals in a respec weer z is a posiive real or a negaive real, even weer z is complex and no like derivaives 23 Te Riemann-Liouville definiion Wile 11 urns ou o be convenien for numerical purposes, i is very ard o use o find any derivaives analyically Forunaely, an equivalen expression can be found: τ z 1 fτdτ, if Rz R Γ z c c Dz f f, if z 0 d d cdz 1 f, if Rz 0 I[z] 0 12 d Rz d Rz cd z Rz f, if R[a] R + Tis is proved in [3] Tis alernaive definiion is due o e works of Riemann and Liouville, and for ineger negaive values of z reduces o c D n x x f x fd d c c }{{} n inegraions x c x n 1 fd 13 n 1!

4 Duare Valério wic can be proven once more by maemaical inducion For ineger posiive values of z we ge a paricular case of e law of exponens, wic says a c D m c Dn f c Dm+n f oldsif m, n Z + 0,orifm, n Z 0,orifm Z+ n Z Definiion 12 clearly sows a D is a linear operaor of funcion f 3 Fracional sliding mode conrol for a SISO plan Wile fracional sliding mode conrol can be applied o MIMO plans, in is paper only e SISO case will be addressed Le e plan o conrol be 0 Dα1 x x 2 0 Dα2 x 2 x 3 14 were: y is e oupu; α i > 0, i 1,,n; 0D αn 1 x n 1 x n all iniial condiions are assumed o be zero; 0 Dαn x n f A x + f B xu y x vecor x called pseudo-sae vecor is given by x x 2 x x 3 0 Dα1 x x 0D α2 0D α1 x x n 1 x n 0 Dαn 1 0 D α2 0 Dα1 x x x x 0D α1 0D α1+α2 0D n 1 k1 α k x n elemens 15 Noice a is las equaliy is a consequence of all iniial condiions being zero, oerwise i would no be necessarily rue Also noice a, wile plan 14 is in general fracional, if α i N, i 1,,n i will be of ineger order We wan e oupu y x o follow a reference r, and for is we will make x follow a reference vecor r, given by r r r 0D αn 1 0 Dα2 0 Dα1 r r 0 Dα1 0 D α2 0D α1 r 0 Dα1 0D α1+α2 0D r r n 1 k1 α k r 16

5 Duare Valério Error ɛ x r is ɛ ɛ ɛ ɛ 0 Dα1 0D α1+α2 0D n 1 k1 α k ɛ [x r] [x r] [x r] 0 Dα1 0D α1+α2 0D n 1 k1 α k [x r] Sliding surface Le us define a sliding surface sx as a linear combinaion of fracional derivaives of e error, including all orders found in x and in r, and in ɛ, and evenually some oers: k 1,,n 1, i : β i Applying e Laplace ransformaion o 19, we ge sx 0 18 m sx μ i 0 Dβi ɛ 19 i1 k α p 20 p1 Es SXs 1 m μ i0 D βi i1 21 I is of course reasonable o coose coefficiens μ i so a ransfer funcion 21 is sable Forcing sx 0, we will obain ɛ 0, and ence x r Wile i is likely a sx canno be always made exacly equal o 0, if sx is bounded, an upper bound for e ensuing error can be found 32 Example: commensurable plan, uncerainy in f A x Le us consider a plan wi α 1 α 2 α n a plan in is case is said commensurable, wi f B x 1: 0 Dnα x f A x + u 22 Suppose a f A x is no known exacly, bu a an approximae value ˆf A x is known We wan sx o go o zero and say ere, wic means a dsx d 0 Tis can be done devising a conrol law a ensures 1 ds 2 x 2 d 2 η sx sx dsx d σsx dsx d η sx η 23

6 Duare Valério were η R + is a value cosen by e conrol designer We will define e sliding surface as n 1 sx 0D β α β + λ ɛ 24 0 D n 1α ɛ+sɛ 0 D n 1α [x r] + Sɛ 0 D α [f A x + u] 0 D n 1α r+sɛ Here, λ and β are design parameers, wi β R + cosen so a α β N in oer words, α mus be an ineger muliple of β All erms save 0 D n 1α ɛ are colleced in Sɛ, and from now on we will omi mos dependencies on, x and ɛ, aswellaseerminalsofoperaord, o alleviae e noaion Hence and so Ds D 1 α f A + D 1 α u D 1+nα α r + DS 25 Ds 0 u f A + D nα r D α S 26 bu since f A is unknown and only ˆf A is known, we will raer implemen e conrol acion as u ˆf A + D nα r D α S kd α 1 σs 27 were e reason for e coice of e nonlineariy kd α 1 σs will be clear in a momen, wen desirable values for consan k are found Replacing 27 in 25 we obain and replacing is in 23 we obain Ds D 1 α f A D 1 α ˆfA + D 1+nα α r DS kσs D 1+nα α r + DS D 1 α f A ˆf A kσs 28 D 1 α f A ˆf A σs k η k η + D 1 α f A ˆf A σs 29 Te erm D 1 α f A ˆf A σs may be eier posiive or negaive If we can find an upper bound Ϝ for e error commied wi approximaion D 1 α f A D 1 α ˆf A,aisosay,if D 1 α f A ˆf A < Ϝ, enwemakek η + Ϝ in conrol law 27 and ereby guaranee a 23 is verified, even in e presence of uncerainy in f A 33 Example: ineger plan, uncerainy in f B x Consider plan { D 2 x bu y x were we only know a variaion range for b We define sx as sx 0 D 1+α ɛ+λɛ, α > 0 31 wic means a, even oug e plan is of ineger order, wen e sliding surface sx 0 is being followed, i will beave wi fracional order dynamics We will ave alleviaing again from now on e noaion as done in e example above 30 Ds D 2+α ɛ + λdɛ λdɛ + bd α u D 2+β r 32

7 Duare Valério and so Ds 0 bu D 2 r λd 1 α ɛ 33 bu since b is unknown and only ˆb is known, we implemen e conrol acion as and from ere we ge wic, replaced in 23, gives 1 bˆb ũ D 2 r λd 1 α ɛ 34 u ũ 1ˆb D ˆb r λd 1 α ɛ kd α σs 35 Ds λdɛ + bˆbd 2+α r bˆb λdɛ bˆb kσs D 2+α r 1 bˆb λdɛ D 2+α r bˆbkσs 36 k ˆb b η + λdɛ D 2+α rσs bˆb k η 37 ˆb b 1 D 2+α r + λdɛ σs } {{ } D α ũ and consequenly, aking ino accoun a 1 β ˆb b β, and making we guaranee a 23 is verified k βη +β 1 D α ũ Avoiding caering Because e sign funcion σξ, in e definiion of e conrol law, is a ard non-lineariy, e sysem will likely oscillae around sx 0, causing conrol caering, wic is undesirable Tis can be improved replacing σξ in e definiion of conrol acion u waever i may be wi some funcion ςξ a smoos e non-lineariy Two possible coices for ςξ are, for insance, { ξ, if ξ 1 ς 1 ξ 39 σξ, if ξ > 1 or ς 2 ξ 2 arcan ζξ 40 π were ζ R + is cosen o regulae ow far ς 2 ξ isfromσξ noice a, wen ζ +, ς 2 ξ approaces σξ Te price o pay for avoiding caering wi suc replacemens is a performance degradaion 4 Example Consider e plan wi uncerainy in f A x and in f B x f Ax {}}{ 0D 3 4 x π b {}}{ x 0 D 1 4 x+ 2 u 41

8 Duare Valério wi ˆf A x 3 x 0 D 1 4 x andˆb 3 2,wereα 1 4, x [x 0 D 1 4 x 0 D 1 2 x] T and n 3 Ifwemakeβ α, k 18 andλ 10, we ge sx 0 D 1 2 ɛ+20 0 D 1 4 ɛ + 100ɛ 42 Sɛ 20 0 D 1 4 ɛ + 100ɛ 43 u 2 [ 3 x 3 0 D 1 4 x 20 0 D 1 2 x D 1 4 x + 0 D 3 4 r+20 0 D 1 2 r D 1 4 r 18 0 D 3 4 σ 0D 1 2 x+20 0 D 1 4 x + 100x ] 0 D 1 2 r 20 0 D 1 4 r 100r 44 Figure 1 sows simulaions resuls obained wi e Grünwald-Lenikoff definiion, T s 01 and making σξ 2 π arcan 20ξ o avoid caering 1 05 x, r r x u Figure 1: Simulaion resuls from secion 4 for a sinusoidal reference For oer applicaion examples, see for insance [2, 8, 4, 1]

9 Duare Valério Acknowledgmens Tis work was parially suppored by Fundação para a Ciência e a Tecnologia, roug IDMEC under LAETA References [1] H Delavari, R Gaderi, A Ranjbar, and S Momani Fuzzy fracional order sliding mode conroller for nonlinear sysems Communicaions in Nonlinear Science and Numerical Simulaion, 154: , 2010 [2] Concepción A Monje, YangQuan Cen, Blas M Vinagre, Dingyü Xue, and Vicene Feliu Fracional-order Sysems and Conrols Springer, London, 2010 [3] I Podlubny Fracional differenial equaions: an inroducion o fracional derivaives, fracional differenial equaions, o meods of eir soluion and some of eir applicaions Academic Press, San Diego, 1999 [4] Amar Si-Ammour, Said Djennoune, and Maamar Beayeb A sliding mode conrol for linear fracional sysems wi inpu and sae delays Communicaions in Nonlinear Science and Numerical Simulaion, 145: , 2009 [5] J Sloine and W Li Applied Nonlinear Conrol Prenice Hall, Englewood Cliffs, 1991 [6] Duare Valério Fracional and variable order conrollers In I Enconro de Jovens Invesigadores do LAETA, Lisboa, 2010 [7] Duare Valério and José Sá da Cosa An inroducion o single-inpu, single-oupu Fracional Conrol IET Conrol Teory & Applicaions, 58: , 2011 [8] Cun Yin, Sou-ming Zong, and Wu-fan Cen Design of sliding mode conroller for a class of fracional-order caoic sysems Communicaions in Nonlinear Science and Numerical Simulaion, 17: , 2012