Four Generations of Higher Order Sliding Mode Controllers. L. Fridman Universidad Nacional Autonoma de Mexico Aussois, June, 10th, 2015

Transcription

1 Four Generaions of Higher Order Sliding Mode Conrollers L. Fridman Universidad Nacional Auonoma de Mexico Aussois, June, 1h, 215

2 Ouline 1 Generaion 1:Two Main Conceps of Sliding Mode Conrol 2 Generaion 2: Second Order Sliding Modes 3 Generaion 3: Super-Twising Algorihm 4 Generaion 4:Arbirary Order Sliding Mode Conrollers 5 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers 6 Conclusions 2 / 71

3 Generaion 1:Two Main Conceps of Sliding Mode Conrol The equivalen conrol ẋ = f (x, ) + B(x, )u wih u disconinuous as previously defined. To find he value of conrol sliding on he he surface σ(x) =, i.e.: σ = Gf + GBu =, G = grad σ If GB is no singular (x, ) han an equivalen conrol The sliding mode dynamics u eq (x, ) := [G(x)B(x, )] 1 G(x)f (x, ) ẋ = f B(GB) 1 Gf 3 / 71

4 Generaion 1:Two Main Conceps of Sliding Mode Conrol Sliding surface Desired error dynamics hen: σ := x 2 + cx 1 = = x 1 () = x 1 e c, x 2 () = cx 2 e c The manifold σ = is known as he sliding surface The surface characerizes he desired dynamics The conrol objecive of sliding mode conrol is o reach σ = in finie ime Once on he surface, he conrol mus keep he rajecories sliding on he surface: sliding mode 4 / 71

5 Generaion 1:Two Main Conceps of Sliding Mode Conrol Example: Firs Order Sliding Mode ẋ 1 = x 2 ẋ 2 = u + f x 1, x 2 are he saes u is he conrol f = 2 + 4sin(/2) +.6sin(1). σ = x 1 + x 2 u, f x Convergence of saes x2 5 Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion / 71

6 Generaion 1:Two Main Conceps of Sliding Mode Conrol Firs Order Sliding Mode: Precision τ = 1e 3 6 x Firs Generaion x x / 71

7 Generaion 1:Two Main Conceps of Sliding Mode Conrol SUMMARY(Early 8h) Advanages Compensaes mached uncerainies heoreically exacly if i supposed ha he saes are available Reduces SMC design o conrol selecion for wo reduced order sysems Finie-ime convergence o he sliding surface CRISIS (Early 8h) Chaering High order derivaives are needed o design sliding surfaces The heory was no complee:heoreically exac compensaion needs heoreically exac differeniaion The dimension of sliding dynamics is reduced jus for 1. 7 / 71

8 Generaion 1:Two Main Conceps of Sliding Mode Conrol SUMMARY(Early 8h) Open problems Subsiue disconinuous conrol wih coninuous one Theoreically exac differeniaors are needed o realize heoreically exac compensaion of he mached uncerainies Reduce he dimension of he sliding dynamics for more han 1 uni 8 / 71

9 Secion 2 Generaion 2: Second Order Sliding Modes 9 / 71

10 Generaion 2: Second Order Sliding Modes Chaering as he relaive degree problem Figure: Prof. Levan and Prof. Fridman 1 / 71

11 Generaion 2: Second Order Sliding Modes Second Order Sliding Modes ẋ 1 = x 2 ẋ 2 = u + f (x, ) σ = x 1 f (x, ) unknown uncerainies/perurbaions. All he parial derivaives of f (x, ) are bounded on compacs Main Objecive Figure: Prof. Emelyanov,Prof. Korovin and Prof. Levan. To design a conrol u such ha he origin of sysem is finie-ime sable, in spie of he uncerainies/perurbaions f (x, ), wih f (x, ) < f + for all, x 11 / 71

12 Generaion 2: Second Order Sliding Modes Twising algorihm u = a sign(x 2 ) b sign(x 1 ), b > a + f +, a > f +. Known bounds f + a and b chosen appropriaely (Emelyanov e al. 86), Ensures finie-ime exac convergence for boh x 1 and x 2 12 / 71

13 Generaion 2: Second Order Sliding Modes Twising Algorihm ẋ 1 = x 2 ẋ 2 = u + f x 1, x 2 are he saes u is he conrol f = 2 + 4sin(/2) +.6sin(1). 1 Convergence of saes Phase porai of x1 and x2 1 x x x1 Conrol Inpu and Negaive of he Perurbaion u, f / 71

14 Generaion 2: Second Order Sliding Modes Comparison Firs Generaion vs Second Generaion:Precision 14 / 71

15 Generaion 2: Second Order Sliding Modes Ani-chaering Sraegy Ẋ = F (, X ) + G(, X )u, X R n, u R, F < F +, The swiching variable σ(x ) : σ = f (σ, ) + g(σ, )u. Add an Inegraor in conrol inpu: If u = v = a sign( σ()) b sign(σ()), so u is a Lipschiz coninuous conrol signal ensuring finie-ime convergence o σ = Criicism(1987) If i is possible o measure σ = f (, σ) + g(, σ)u, hen he uncerainy f (, σ) = σ g(, σ)u is also known and can be compensaed wihou any disconinuous conrol! Couner-argumen If g is uncerain so σ depends on u hrough uncerainy! The ani-chaering sraegy is reasonable for he case of uncerain conrol gains. 15 / 71

16 Generaion 2: Second Order Sliding Modes Discussion abou SOSM Advanages of SOSM 1 Allows o compensae bounded mached uncerainies for he sysems wih relaive degree wo wih disconinuous conrol signal 2 Allows o compensae Lipschiz mached uncerainies wih coninuous conrol signal using he firs derivaive of sliding inpus 3 Ensures quadraic precision of convergence wih respec o he sliding oupu 4 For one degree of freedom mechanical sysems: he sliding surface design is no longer needed. 5 For sysems wih relaive degree r: he order of he sliding dynamics is reduced up o (r 2). The design of he sliding surface of order (r 2) is sill necessary! 16 / 71

17 OPEN PROBLEMS:EARLY 9h To reduce he chaering subsiuing disconinuous conrol signal wih coninuous one he derivaive of he sliding inpu sill needed! The problem of exac finie-ime sabilizaion and exac disurbance compensaion for SISO sysems wih arbirary relaive degree remains open. More deep decomposiion is sill needed Theoreically exac differeniaors are needed o realize heoreically exac compensaion of he Lipschiz mached uncerainies 17 / 71

18 Generaion 2: Second Order Sliding Modes Firs Generaion vs Second Generaion u, f x Convergence of saes x2 5 Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion Figure: Firs Generaion x u, f 1 Convergence of saes x2 Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion Figure: Second Generaion 18 / 71

19 Generaion 2: Second Order Sliding Modes Terminal Algorihm ẋ 1 = x 2, ẋ 2 = u(x), u(x) = α sign(s(x)), s(x) = x 2 + β x 1 sign(x 1 ). Figure: Prof. Z. Man 19 / 71

20 Generaion 2: Second Order Sliding Modes Relaive Degree of Terminal Sliding Variable Time derivaive of he swiching surface ṡ(x) = ẋ 2 + β x 2 2 x 1 = α sign(s(x)) + β x 2 2 x 1. s(x) is singular for x 1 =, and he relaive degree of he swiching surface does no exis On x 2 = β x 1 sign(x 1) ṡ = α sign(s(x)) β2 2 sign(x1). Two ypes of behavior for he soluion of he sysem are possible 2 / 71

21 Generaion 2: Second Order Sliding Modes Terminal mode: β 2 < 2α, Trajecories of he sysem reach he surface s(x) = and remain here. Twising mode β 2 > 2α Trajecories do no slide on he surface s(x) = 21 / 71

22 Generaion 2: Second Order Sliding Modes How o overcame he Singulariy? Singulariy can be overcome by rewriing he funcion s s(x) = β 2 x 1 + x 2 2 sign(x 2 ). 22 / 71

23 Secion 3 Generaion 3: Super-Twising Algorihm 23 / 71

24 Generaion 3: Super-Twising Algorihm The Super Twising Algorihm (STA) Emalyanov, Korovin, Levanovsky, 199, Levan 1993 Paricular Case of STA ẋ = f (x, ) + g(x, )u, u = k 1 x 1 2 sign(x) + v, v = k 2 sign(x), f (x(), ) is Lipschiz coninuous uncerainy/disurbance Coninuous conrol signal Exac finie ime convergence o x() = ẋ() =, T Thee derivaive of x is no used!!! If x is measured, he STA is an oupu-feedback conroller 24 / 71

25 Generaion 3: Super-Twising Algorihm Example Super-Twising Algorihm ẋ 1 = x 2 ẋ 2 = u + f x 1, x 2 are he saes u is he conrol f = 2 + 4sin(/2) +.6sin(1). σ = x 1 + x 2 x Convergence of saes x Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion u, f / 71

26 x x x Generaion 1:Two Main Conceps of Sliding Mode Conrol Generaion 2 Generaion 3 Generaion 4 Generaion 5 Conclusions Generaion 3: Super-Twising Algorihm Second Generaion vs Third Generaion Convergence of saes x2 Phase porai of x1 and x Convergence of saes x2 Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion x1 Conrol Inpu and Negaive of he Perurbaion 1 u, f u, f Figure: Firs Generaion Figure: Second Generaion Convergence of saes x Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion u, f Figure: Third Generaion 26 / 71

27 Generaion 3: Super-Twising Algorihm Comparison Second Generaion o Third Generaion:Precision 27 / 71

28 Generaion 3: Super-Twising Algorihm Robus Exac Differeniaor, Levan(1998) Signal o differeniae: f () Assume f () L Find an observer for f () bounded perurbaion. STA observer ẋ 1 = x 2, ẋ 2 = f, y = x 1, ˆx 1 = k 1 ˆx 1 y 1 2 sign(ˆx1 y) + ˆx 2, ˆx 2 = k 2 sign(ˆx 1 y) Convergence of STA assures:(f ˆx 1) = (ḟ ˆx2) = afer finie ime. 28 / 71

29 Generaion 3: Super-Twising Algorihm SUMMARY Advanages 1 Coninuous conrol signal compensaing Lipschiz uncerainies 2 Chaering aenuaion. bu no is complee removal! (Boiko, Fridman 25). 3 Differeniaor obained using he STA: Finie-ime exac esimaion of derivaives in he absence of boh noise and sampling, Bes possible asympoic approximaion in he sense of Kolmogorov / 71

30 Generaion 3: Super-Twising Algorihm Open problems: End of 2h cenury Open problems: End of 2h cenury 1 Relaive degree r 2: Need sliding surface. Consequenly he saes converge o he origin asympoically. Deeper decomposiion is needed! 2 STA based differeniaor for he sliding surface design is no enough: he can no provide he bes possible precision for highes derivaives 3 / 71

31 Secion 4 Generaion 4:Arbirary Order Sliding Mode Conrollers 31 / 71

32 Generaion 4:Arbirary Order Sliding Mode Conrollers Arbirary Order Sliding Mode Conrollers Ẋ = F (, X ) + G(, X )u, X R n, u R σ = σ(x, ), R. σ has a fixed and known relaive degree r. Conrol problem is ransformed ino he finie-ime sabilizaion of an uncerain differenial equaion σ (r) = f (, X ) + g(, X )u, (1) and corresponding differenial inclusion where C, K m and K M are known consans. σ (r) [ C, C] + [K m, K M ]u, (2) 32 / 71

33 Generaion 4:Arbirary Order Sliding Mode Conrollers Nesed arbirary order sliding-mode conrollers 21: Nesed arbirary order SM conroller Solve he finie-ime exac sabilizaion problem for an oupu wih an arbirary relaive degree. Bounded Lebesgue measurable uncerainies. Nesed higher order sliding-mode(hosm) conrollers are consruced using a recursion Figure: Prof. Levan 33 / 71

34 Generaion 4:Arbirary Order Sliding Mode Conrollers Nesed Third Order Singular Terminal Algorihm Third Order ( u = α sign σ + 2( σ 3 + σ 2 ) 1 6 ) sign( σ + σ 2 3 sign(σ)) Figure: 3rd Order Nesed SM Fourh Order (... u = α sign σ + 3( σ 6 + σ 4 + σ 3 ) 12 1 )) sign ( σ + ( σ 4 + σ 3 ) sign( σ +.5 σ 4 sign(σ)) Finie-ime sabilizaion of σ = and is successive derivaives up o r / 71

35 Generaion 4:Arbirary Order Sliding Mode Conrollers HOSM Differeniaor The Nesed Conroller needs he oupu and is successive derivaives Insrumen: HOSM arbirary order differeniaor Le σ() signal o be differeniaed k 1 imes Assume ha σ (k) L. 3-h order HOSM differeniaor ż = v = 3L 1 4 z σ 3 4 sign(z σ) + z 1, ż 1 = v 1 = 2L 1 3 z1 v 2 3 sign(z1 v ) + z 2, ż 2 = v 2 = 1.5L 1 2 z2 v sign(z2 v 1) + z 3 ż 3 = 1.1L sign(z 3 v 2) (3) z i rue derivaive σ (i) (). 35 / 71

36 Generaion 4:Arbirary Order Sliding Mode Conrollers Concep of HOSM Figure: HOSM concep 36 / 71

37 Generaion 4:Arbirary Order Sliding Mode Conrollers Advanages of nesed HOSM for SISO sysems wih relaive degree r Theoreically exac disurbance compensaion basing on oupu informaion only Full dynamical collapse: ensures σ = σ = σ = = σ (r 1) = in finie-ime Ensures he r-h order precision for he sliding oupu wih respec o he discreizaion sep and fas parasiic dynamics The sliding surface design is no longer needed 37 / 71

38 Open problems: Afer 25 For SISO sysems wih relaive degree r sill produces a disconinuous conrol signal Ani-chaering sraegy: he informaion abou σ (r) conaining perurbaions is needed 38 / 71

39 Generaion 4:Arbirary Order Sliding Mode Conrollers Example Nesed 3rd Order Singular Terminal Conroller wih ani-chaering sraegy ẋ 1 = x 2 ẋ 2 = u + f u = u 2 x 1, x 2 are he saes u 2 is he conrol f = 2 + 4sin(/2) +.6sin(1). σ = x 1 5 Convergence of saes x x1 Conrol Inpu and Negaive of he Perurbaion 2 ẋ2 1 Phase porai of x1, x2 and ẋ2 1 u, f / 71

40 x x x Generaion 1:Two Main Conceps of Sliding Mode Conrol Generaion 2 Generaion 3 Generaion 4 Generaion 5 Conclusions Generaion 4:Arbirary Order Sliding Mode Conrollers Firs Generaion o Fourh Generaion Convergence of saes x2 Phase porai of x1 and x Convergence of saes x2 Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion x1 Conrol Inpu and Negaive of he Perurbaion 1 u, f u, f Figure: Firs Generaion Figure: Second Generaion Convergence of saes x Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion 5 Convergence of saes x x1 Conrol Inpu and Negaive of he Perurbaion 2 ẋ2 1 Phase porai of x1, x2 and ẋ2 1 u, f u, f Figure: Third Generaion Figure: Fourh Generaion 4 / 71

41 Generaion 4:Arbirary Order Sliding Mode Conrollers Comparison Third Generaion o Fourh Generaion:Precision 41 / 71

42 Secion 5 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers 42 / 71

43 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Arbirary Order Sliding-Mode Conrollers Figure: J. Moreno, B. Bandyopadhyay, S. Kamal, A. Chalanga 43 / 71

44 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous arbirary order sliding-mode conrollers Disconinuous Inegral Algorihm(D-I), (Zamora, Moreno,213), Seeber, Horn (215) Two versions of he Coninuous Terminal Sliding Mode Algorihm(CTSMA) (Mexico- India) (a) Coninuous Singular Terminal Sliding Mode Algorihm (CSTSMA); (b) Coninuous Nonsingular Terminal Sliding Mode Algorihm (CNTSMA). Coninuous Twising Algorihm(CTA)(Torres, Fridman, Moreno,215) 44 / 71

45 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Properies of CHOSM For he sysems wih relaive degree r Coninuous conrol signal Finie-ime convergence o he (r + 1)-h order sliding-mode se Derivaives of he oupu up o he (r 1) order 45 / 71

46 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Twising Algorihm (CTA) ẋ 1 = x 2 ẋ 2 = u + f (x, ) σ = x 1 u = k 1 x 1 1/3 k 1 x 2 1/2 where k 1, k 2, k 3, k 4 are appropriae posiive gains. New Noaion: z p = z p sgn(z) (k 3 x 1 (τ) + k 4 x 2 (τ) )dτ, (4) 46 / 71

47 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Twising Algorihm (CTA) Closed Loop Sysem ẋ 1 = x 2 ẋ 2 = k 1 x 1 1/3 k 2 x 2 1/2 + x 3 ẋ 3 = k 3 x 1 k 3 x 2 + ρ, (5) ρ = f ẋ + f, and ρ. x Twising srucure o rejec perurbaions. 47 / 71

48 u x Generaion 1:Two Main Conceps of Sliding Mode Conrol Generaion 2 Generaion 3 Generaion 4 Generaion 5 Conclusions Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Simulaion: CTA 6 Convergence of saes 5 Phase porrai of x 1 and x x 1 x 2 x 3 x ime [sec] x 1 Conrol Inpu 2 Perurbaion and esimaion 1 1 esimaion perurbaion ime [sec] ime [sec] Figure: Numerical resuls for a double inegraor wih perurbaion 48 / 71

49 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Discussion abou he CTA Advanages Algorihms homogeneous of degree δ f = 1, wih weighs ρ = 3, 2, 1. The only informaion needed o manain finie ime convergence of all hree variables x 1, x 2 and x 3 is he oupu (x 1) and is derivaive (x 2) Precision corresponds o a 3rd order sliding mode 49 / 71

50 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Singular Terminal Sliding Mode Algorihm (CSTSMA) ẋ 1 = x 2 ẋ 2 = u + f (x, ) σ = x 1 u = k 1 φ 1/2 k 3 φ dτ, (6) where φ = ( x 2 + k 2 x 1 2/3), and k 1, k 2, k 3 are appropriae posiive gains. 5 / 71

51 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Singular Terminal Sliding Mode Algorihm (CSTSMA) Closed Loop Sysem ẋ 1 = x 2 ẋ 2 = k 1 φ 1/2 + x 3 ẋ 3 = k 3 φ + ρ, (7) ρ = f ẋ + f, and ρ. x Combinaion of he Super-Twising algorihm wih he Singular Terminal Sliding mode. 51 / 71

52 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Nonsingular Terminal Sliding Mode Algorihm (CNTSMA) ẋ 1 = x 2 ẋ 2 = u + f (x, ) σ = x 1 u = k 1 φ N 1/3 k 3 φ N dτ, (8) where φ N = ( x 1 + k 2 x 2 3/2), and k 1, k 2, k 3 are appropriae posiive gains. 52 / 71

53 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Nonsingular Terminal Sliding Mode Algorihm (CNTSMA) Closed Loop Sysem ẋ 1 = x 2 ẋ 2 = k 1 φ N 1/3 + x 3 ẋ 3 = k 3 φ N + ρ, (9) ρ = f x ẋ + f and ρ. Combinaion of he Super-Twising algorihm wih he Nonsingular Terminal Sliding Mode algorihm. 53 / 71

54 x Generaion 1:Two Main Conceps of Sliding Mode Conrol Generaion 2 Generaion 3 Generaion 4 Generaion 5 Conclusions Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Simulaion:CSTSMA 6 Convergence of saes 4 2 x 1 x 2 x (a) 4 x x x 1 x 2 x (b) 54 / 71

55 u f x x 2 Generaion 1:Two Main Conceps of Sliding Mode Conrol Generaion 2 Generaion 3 Generaion 4 Generaion 5 Conclusions Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Singular Terminal Sliding Mode Conrol(CSTSMC) Twising conroller-like Behavior. 4 2 Convergence of saes x 1 x Phase porrai of x 1 and x 2 φ = x vs x (a) x 1 (b) 15 1 Conrol Inpu 8 6 Perurbaion and Esimaion Perurbaion Esimaion (c) (d) Figure: Numerical example uncerain double inegraor 55 / 71

56 u x x 1 x 3 f x1 Generaion 1:Two Main Conceps of Sliding Mode Conrol Generaion 2 Generaion 3 Generaion 4 Generaion 5 Conclusions Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Nonsingular Terminal Sliding Mode Conrol(CNTSMC) 1 Convergence of saes x 2 x x (a) (b) 2 15 Conrol inpu 8 6 Perurbaion and Esimaion Perurbaion Esimaion (c) (d) Figure: Numerical example uncerain riple inegraor 56 / 71

57 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Sliding-Like Behavior of CNTSMC.5 Phase porrai of x 1 and x 2 x 1 vs. x 2 φ =.5 x x 1 Figure: Phase porrai of Plan s saes x 1 and x 2, and locus of he swiching curve φ = φ N =, showing a Sliding-Like behavior of he CNTSMC 57 / 71

58 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Twising-Like Behavior of CNTSMC Phase porrai of x 1 and x 2 x 1 vs. x 2 φ = x x 1 Figure: Phase porrai of Plan s saes x 1 and x 2, and locus of he swiching curve φ = φ N =, showing a Twising-Like behavior of he CNTSM conroller. 58 / 71

59 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Disconinuous-Inegral Algorihm(D-I) ẋ 1 = x 2 ẋ 2 = u + f (x, ) σ = x 1 u = k 1 x 1 1/3 k 1 x 2 1/2 where k 1, k 2, k 3 are appropriae posiive gains. (k 3 x 1 (τ) )dτ, (1) 59 / 71

60 x x2 ẋ2 x1 x x2 x x2 Generaion 1:Two Main Conceps of Sliding Mode Conrol Generaion 2 Generaion 3 Generaion 4 Generaion 5 Conclusions Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers All five Generaions Convergence of saes Phase porai of x1 and x Conrol Inpu and Negaive of he Perurbaion Convergence of saes 1 Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion Convergence of saes Phase porai of x1 and x x1 Conrol Inpu and Negaive of he Perurbaion u, f 1 u, f u, f Figure: Firs Generaion Figure: Second Generaion Figure: Third Generaion 5 Convergence of saes x x1 Conrol Inpu and Negaive of he Perurbaion 2 1 Phase porai of x1, x2 and ẋ2 1 u, f Figure: Fourh Generaion Figure: Fifh Generaion:CSTSMC Figure: Fifh Generaion:CNTSMC 6 / 71

61 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Fourh Generaion vs Fifh Generaion:Precision Same precision and smoohness of conrol wihou using σ 61 / 71

62 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Third Generaion vs Fifh Generaion:Precision 62 / 71

63 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Comparison 5 Generaions Algorihm Conrol Signal Informaion Sabiliy Order of sliding w.r.. σ Firs SMC Disconinuous σ, σ Asympoic 1 2SMC Disconinuous σ, σ Finie ime 2 Super-wising Coninuous σ, σ Asympoic 2 3SMC + ani-chaering Coninuous σ, σ, σ Finie ime 3 Coninuous 2SMC Coninuous σ, σ Finie ime 3 63 / 71

64 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Arbirary Order CTA 3-CTA ẋ 1 = x 2 ẋ 2 = x 3 (11) (12) ẋ 3 = k 1 x k2 x k3 x x4 (13) ẋ 4 = k 4 x 1 k 5 x 2 (14) Same properies ha Coninuous Twising Algorihm. 64 / 71

65 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers 3-CTA 6 x x x x 5 x 1 x 2 x 3 x ime [sec] x 1 x 2 x 3 x ime [sec] ime [sec] ime [sec] ime [sec] Figure: 3-CTA and saes precision wih τ =.1 65 / 71

66 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Coninuous Nesed Sliding Mode Algorihm 3-CSNSMA is proposed as follows ẋ 1 = x 2 ẋ 2 = x 3 ẋ 3 = k 1 φ 2 1/2 sign (φ 2) + x 4 ẋ 4 = k 4sign (φ 2) + ρ (15) φ 2 = x 3 + k 3 ( x x2 ) 4 1 ( ) 6 sign x 2 + k 2 x sign(x1) (16) Same properies ha Coninuous Terminal Sliding Mode. 66 / 71

67 x Generaion 1:Two Main Conceps of Sliding Mode Conrol Generaion 2 Generaion 3 Generaion 4 Generaion 5 Conclusions Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers 3-CSNSMA 6 Convergence of saes x 1 4 x 2 x 3 2 x (a) 8 x x x x 1 x 2 x 3 x (b) 67 / 71

68 Generaion 5: Coninuous Arbirary Order Sliding-Mode Conrollers Discussion abou C2SM C2SM Homogeneous of degree δ f = 1, wih weighs ρ = 3, 2, 1. The only informaion ha needed o manain finie ime convergence of all hree variables x 1, x 2 and x 3 is he oupu (x 1) and is derivaive (x 2) Can work for an uncerain sysem wih relaive degree 2 wih respec o is oupu. Can compensae bounded Lebesgue measurable perurbaions. r-chosm Homogeneous of degree δ f = 1, wih weighs ρ = r, r 1,, 2, 1. Can be used for uncerain sysem wih relaive degree r 1 wih respec o oupu. Insensible o perurbaions whose ime derivaive is bounded! (can no grow faser han a linear funcion of ime!) 68 / 71

69 Secion 6 Conclusions 69 / 71

70 Conclusions Conclusions Las hree decades new generaions of conrollers: second order slding mode conrollers(1985); super-wising conrollers(1993); arbirary order sliding-mode conrollers(21,25). We have presened he nex generaion: wo families of coninuous nesed sliding-mode conrollers, ha can be used on Lipschiz sysems wih relaive degree r, providing a coninuous conrol signal. New conrollers ensure a finie-ime convergence of he sliding oupu o he (r + 1) h-order sliding se using informaion on he sliding oupu and is derivaives up o he order r 1 7 / 71

71 Conclusions References Y. Shessel, C. Edwards, L. Fridman, A. Levan. Sliding Mode Conrol and Observaion, Series: Conrol Engineering, Birkhauser: Basel, 213. L. Fridman, J. Moreno, B. Bandyopadhyay, S. Kamal, A. Chalanga Coninuous Nesed Algorihms : The Fifh Generaion of Sliding Mode Conrollers. In: Recen Advances in Sliding Modes: From Conrol o Inelligen Mecharonics. X. Yu, O. Efe (eds), Sudies in Sysems, Decision and Conrol 24, Springer, Swizerland 215. pp / 71