From 1rst order to Higher order Sliding modes

Transcription

1 From 1rst order to Higher order Sliding modes W. Perruquetti Ecole Centrale de Lille, Cité Scientifique, BP 48, F Villeneuve d Ascq Cedex - FRANCE. tel : fax : wilfrid.perruquetti@ec-lille.fr November 2010 / Douz

2 Table of Contents 1 Differential Inclusion: Notion of solution 2 3

3 Introduction A simple stabilization problem: double integrator How to stabilize? ẍ = u (1) ẋ 1 = x 2 ẋ 2 = u (2) F1 car.

4 Introduction A simple stabilization problem: double integrator u Classical solution: State feedback State feedback which is to frequency approach or polynomial approach x x Assume that (x 1, x 2 ) is available Time (secs) 1 0 x Time (secs) Stabilization even with a bounded control!! Time (secs) x1 Sate feedback: stabilization of (2) with u = x x 2

5 Introduction A simple stabilization problem: double integrator A variable structure controller If the speed is not available : Observer (dynamic extension) An alternative solution with output feedback? The simplest function being: u = f(x 1 ) (variable α) u = αx 1 ẋ 1 = x 2 ẋ 2 = α x 1 (3)

6 Introduction A simple stabilization problem: double integrator Strategy 1: position x available and the signum of ẋ (in fact of xẋ ) How to play with α? ẍ + αx = 0 x 1 (t) = x 0 cos( αt) + ẋ0 α sin( αt) (4) x 2 (t) = x 0 α sin( αt) + ẋ0 cos( αt) (5) (x 0 αx1 + ẋ0 α x 2 ) 2 + (ẋ 0 x 1 x 0 x 2 ) 2 = (x 2 0 α + ẋ 2 0 α ) 2 (6) Solutions are ellipsoids

7 Introduction A simple stabilization problem: double integrator Area I : x 1 x 2 < 0, α I = 1 Area II : x 1 x 2 > 0, α II = 2 After 2k + 1 switching: l 2k+1 = α I α II l 0

8 Introduction A simple stabilization problem: double integrator Strategy 2: position x and velocity ẋ are available How to play with α? α = 1 ẍ + x = 0 Solutions are ellipsoids Phase portrait

9 Introduction A simple stabilization problem: double integrator α = 1 Phase portrait ẍ x = 0 Solutions are hyperbolas x(t) = x 0 cosh(t) + ẋ 0 sinh(t) One stable and one unstable manifold

10 Introduction A simple stabilization problem: double integrator Area I : α I = 1 4 Commande par mode glissant yzone I : k Z 1.5 x 1 < 0 x 1 + x 2 X 0 X X n WI y 1 or X x 1 > 0 x 1 + x 2 X X m Zone II : k Area II : α II = 1 y X m X X x 1 0 x 1 + x y 2 < X 0n X or X -1.5 WW IW 1 WW 1.5 x 1 0 x 1 + x 2 > 0 W. PERRUQUETTI 14

11 Introduction Some first questions Problems: Notion of solution, Discontinuous Control (damaging the actuators), How to find the switching logic? Panis (Canada 97)

12 Introduction Variable Structure System Differential Inclusion: Notion of solution 4 Commande par mode glissant General Problem formualtion for VSS: Problématiques générales ẋ = f i (t, x, u i ) SS : Find the switching logic and the DX control? F DT I T X U I X2 I ouver les commandes et les iques de commutations!! s 2 2 s s s e e

13 Introduction Sliding Mode Control CMG : SMC (1rst order and Higher): 4 Commande par mode glissant n n i Ewc{cx Slap principle Le principe des «baffes» ve{'f i Ewc{cx W. PERRUQUETTI

14 Introduction Sliding Mode Control Objective To constrain the trajectories of system ẋ = f(x) + g(x)u to reach, in a finite time, and then, to stay onto the sliding surface chosen according to the control objectives Sliding mode control { u u = + (s) if sign(s(x)) > 0 u s) if sign(s(x)) < 0 A simple sliding mode control design with u + u u = u eq + u disc given by s = ṡ = 0, W. (invariance Perruquetti of1rst the to HOSM sliding surface)

15 Introduction Sliding Mode Control Objective To constrain the trajectories of system ẋ = f(x) + g(x)u to reach, in a finite time, and then, to stay onto the sliding surface chosen according to the control objectives Sliding mode control { u u = + (s) if sign(s(x)) > 0 u s) if sign(s(x)) < 0 A simple sliding mode control design with u + u u = u eq + u disc given by s = ṡ = 0, W. (invariance Perruquetti of1rst the to HOSM sliding surface)

16 Introduction Sliding Mode Control Objective To constrain the trajectories of system ẋ = f(x) + g(x)u to reach, in a finite time, and then, to stay onto the sliding surface chosen according to the control objectives Sliding mode control { u u = + (s) if sign(s(x)) > 0 u s) if sign(s(x)) < 0 A simple sliding mode control design with u + u u = u eq + u disc given by s = ṡ = 0, (invariance of the sliding surface) u disc = ksign(s), (convergence in finite time onto the surface)

17 Introduction Sliding Mode Control Objective To constrain the trajectories of system ẋ = f(x) + g(x)u to reach, in a finite time, and then, to stay onto the sliding surface chosen according to the control objectives Sliding mode control { u u = + (s) if sign(s(x)) > 0 u s) if sign(s(x)) < 0 A simple sliding mode control design with u + u u = u eq + u disc given by s = ṡ = 0, (invariance of the sliding surface) u disc = ksign(s), (convergence in finite time onto the surface)

18 Introduction Sliding Mode Control: Advantages vs disadvantages Advantages: System order reduction Finite time convergence (adjust time response) Robustness w.r.t. parametric uncertainties and disturbances Disadvantages: Chattering phenomena (actuator damage) Noise sensitivity (??) Output feedback (??)

19 Introduction Sliding Mode Control: One more time... Sliding mode control design: hitting phase (or reaching phase), and the sliding phase. stability/attractivity concepts: existence of sliding motions is a contraction property (locally), shaping procedure: stabilization problem ( tune the shape of the sliding : in sliding minimum phase system).

20 Introduction Sliding Mode Control: One more time... { ẋ1 = x 2 (1+x 2 2 ) 2 x 1x 2 u 1+x 2 2 ẋ 2 = u (7) This system seems complex, however, if we set z 1 = x 1 (1 + x 2 2) z 2 = x 2 (note that it defines a global diffeomorphism), then one obtains

21 Introduction Sliding Mode Control: One more time... { ż1 = z 2 ż 2 = u (8) and it becomes obvious that if in sliding mode z 2 = z 1, then z 1 converges asymptotically to zero (ż 1 = z 2 = z 1 ) and thus z 2 also converges. In this step of design (the sliding phase ), the shape of the sliding manifold arises naturally.

22 Introduction Sliding Mode Control: One more time... Now, we need to force the system to evolve on the constraint z 2 = z 1. For this, let us define the sliding surface as S = {z R 2 : s(z) = 0} (9) s(z) = z 2 + z 1 (10) Then, according to the equivalent control method, we need the control to satisfy { u u(z) = + (z) if s(z) > 0 u (z) if s(z) < 0 min(u + (z), u (z)) < u eq = z 2 < max[u + (z), u (z)] in order to ensure that a sliding mode exists on S.

23 Introduction Sliding Mode Control: One more time... This leads to various design controls, for example, { 1 if s(z) > 0 u(z) = 1 if s(z) < 0 which ensures a finite time convergence to S as soon as the initial conditions are close enough to the surface and satisfy z 2 < 1. But, can we provide a better characterization of the initial conditions leading to a sliding mode?

24 Introduction Sliding Mode Control: One more time... An alternative to this control is u(z) = { z2 1 if s(z) > 0 z if s(z) < 0 (11) which ensures a finite time convergence to S, whatever the initial conditions. But since the chattering problem remains, can we stabilize the system while reducing the chattering?

25 Differential Inclusion: Notion of solution ẋ = f(x, x), x X \ S (12) where X is the state manifold (locally diffeomrophic to R n ). Problem : f is not defined on a manifold of codimension one (if S = {x R n : s(x) = 0} and s is a scalar function) thus Cauchy-Lipschitz and Peano Theorem does not apply (existence (and uniqueness) of solutions). Notion of solutions on the manifold : extend the vector field f on the manifold S. (Aizerman, Filipov, Utkin,....)

26 Differential Inclusion: Notion of solution Main points of view : Real world (system is not discontinuous), just take into account (delays, hysterisis, saturation) in a small vicinity of the sliding manifold S ε = {x R n : s(x) ε} ( ε radius), then use the usual results, then ε 0: Sliding mode are then limit of classical solution. That is Aizerman s point of view embed the discontinuous system into a Differential Inclusion (Filipov), Equivalent control theory (Utkin).

27 Differential Inclusion: Notion of solution Filipov s points of view : replace the ODE with discontinuous right-hand side ẋ = f(t, x), x X \ S R n with the following differential inclusion ẋ F (t, x) which capture the behaviors of the original system, where F (t, x) = conv(f(t, B ε (x) M)) ε>0 µ(m)=0

28 Differential Inclusion: Notion of solution DT ẋ = sign(x), x R 4 Commande 1 par mode if glissant x < 0 F (x) = 1 if x > 0 [ 1, 1] if x = 0 DX Z SIGNX X 2 SIXSIX; Z = SI X AX 2 8(%) % W. PERRUQUETTI 30 Signum

29 Differential Inclusion: Notion of solution 4 Commande par mode glissant y x S ẋ = f 0 (t, x) X - Xw F T X f 0 should be in T x S. F T X F T X F Z T X A6 4 X - f 0 (t, x) conv{f + (t, x), f (t, x)} T x S W. PERRUQUETTI 32

30 Differential Inclusion: Notion of solution f 0 = αf + + (1 α)f, α [ 1, 1] f 0 (t, x) T x S ds, f 0 = 0 α ds, f + + (1 α) ds, f = 0 α = ds, f ds, f f + ẋ = ds, f f 0 = ds, f f + f + ds, f + ds, f f + f

31 Differential Inclusion: Notion of solution Utkin s points of view : On the sliding manifold, replace the dynamics by the following ODE (equivalent dynamics) ẋ = f eq (t, x, u e q) where f eq (t, x, u e q) ensure invariance of the sliding manifold that is f eq (t, x, u e q) : s(x(t)) = 0, t > 0 thus s is identically zero which implies that ṡ is also zero Remark Filipov and Utkin thechnics are equivalent only for system linear in the control that is ẋ = f(x) + g(x)u.

32 Differential Inclusion: Notion of solution Counter example : ẋ 1 = 0.3x 2 + x 1 u (13) ẋ 2 = 0.7x 1 + 4x 1 u 3 (14) Sliding manifold defined by: s(x) = x 2 + x1 Control: u = sign(s(x)x 1 ).

33 Differential Inclusion: Notion of solution Sliding mode occurs if sṡ < 0 (close to ) : ṡ = 0.3x 2 + x 1 u 0.7x 1 + 4x 1 u 3 (15) ṡ = 0.3x 2 0.7x 1 + x 1 u(1 + 4u 2 ) (16) If s(x) 0, x 2 x 1 ṡ = x 1 + x 1 u(1 + 4u 2 ) (17) sṡ = sx 1 5 sx 1 < 0 (18) Yes sliding will occur

34 Differential Inclusion: Notion of solution Equivalent dynamics (Filipov): ( ) ( f + 0.3x2 + x (x) = 1, f 0.3x2 x (x) = 1 3.3x 1 4.7x 1 ( ) 1 1 f (x) α = ( ) ( ) = 0.3x 2 5.7x 1 2x x 1 8x 1 When x close to the sliding manifold (x 2 x 1 ) we have α = 6x 1 10x 1 = 6 10 thus the equivalent dynamics is ), (19) ẋ 1 = α(0.3x 2 + x 1 ) + (1 α)(0.3x 2 x 1 ) (20) AS = 6 10 (0.3x 2 + x 1 ) (0.3x 2 x 1 ) (21) = 0.3x x 1 = 0.1x 1 (22)

35 Differential Inclusion: Notion of solution Equivalent dynamics (Utkin): ṡ = 0.3x 2 + x 1 u 0.7x 1 + 4x 1 u 3 When x close to the sliding manifold (x 2 x 1 ) we have ṡ = x 1 + x 1 u(1 + 4u 2 ) = 0 (23) u(1 + 4u 2 ) = 1 x 1 = 0 (24) u(1 + 4u 2 ) = 1 u = 0.5, u R ẋ 1 = 0.3x x 1 When x close to the sliding manifold (x 2 x 1 ) ẋ 1 = 0.2x 1 Unstable

36 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Non Linear affine systems ẋ = f(x) + g(x)u (25) Sliding manifold being defined by a C 1 function (same dimension as u) S = {x R n : s(x) = 0} (26)

37 Table of Contents Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance 1 Differential Inclusion: Notion of solution 2 Attractivity condition and invariance condition of the sliding manifold Sliding mode equivalent dynamics Robustness with respect to matched disturbance 3

38 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Attractivity and invariance condition u = { u + if s(x) > 0 u if s(x) < 0 (27) Attractivity condition: s T ṡ < 0 min(u +, u ) < u eq < max(u +, u ) (28) Invariance condition: ṡ = 0 u = u eq (29)

39 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Attractivity and invariance condition Be careful, this condition does not imply that the sliding manifold is reached in finite time. Thus, this condition (for the existence of a sliding mode) should be replaced by a more restrictive condition for example (mu-reachability condition) s T ṡ < µs (30) Show that V (s) = s T s goes to zero in finite time

40 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Attractivity and invariance condition Let us consider a linear system with a linear sliding surface Equivalent control if CB invertible ẋ = Ax + Bu (31) S = {x R n : s(x) = Cx} (32) s T ṡ = s T C(Ax + Bu) < 0 (33) ṡ = 0 u eq = (CB) 1 CAx (34) If u = (CB) 1 Ksign(s) + u eq, s T ṡ = m i=1 k i s i < 0

41 Table of Contents Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance 1 Differential Inclusion: Notion of solution 2 Attractivity condition and invariance condition of the sliding manifold Sliding mode equivalent dynamics Robustness with respect to matched disturbance 3

42 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Sliding mode equivalent dynamics ṡ = s x (f(x) + g(x)u) = L f s + L g su Equivalent control if L g s invertible Thus the equivalent dynamics are ṡ = 0 u eq = (L g s) 1 L f s (35) ẋ = f(x) + g(x) ( (L g s) 1 L f s ) (36) ( ( )) 1 s = Id g(x) (L g s) f(x) (37) x ( ( Id g(x) (Lg s) 1 s x)) is a projection operator

43 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Sliding mode equivalent dynamics Let us consider a linear system ẋ = Ax + Bu (38) Equivalent control if CB invertible u eq = (CB) 1 CAx (39) ẋ = (Id B(CB) 1 C)Ax (40) = A eq x (41) A eq = (Id B(CB) 1 C)A (42)

44 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Sliding mode equivalent dynamics Using a change of coordinates one can obtain (B 2 M m (R)) ( ) ( ) 0 A11 A B =, A = 12, C = ( ) C B 2 A 21 A 1 C ( ) A A eq = 11 A 12 C2 1 C 1A 11 C2 1 C (43) 1A 12 ( = P 1 A11 A 12 C2 1 C ) 1 A 12 P (44) 0 0 A eq has at least m zero eigenvalues and at most n m non zero ones.

45 Table of Contents Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance 1 Differential Inclusion: Notion of solution 2 Attractivity condition and invariance condition of the sliding manifold Sliding mode equivalent dynamics Robustness with respect to matched disturbance 3

46 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Robustness with respect to matched disturbance ẋ = Ax + Bu + p (45) p span(b) (46) (46) is called the matching condition, thus we have p = Bp. Put (45) into a controllable canonical form (hereafter m = 1) ẋ c = A c x + B c (u + p ) (47) A c = , B c = (48) 1 a c a cn

47 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Robustness with respect to matched disturbance Select a linear sliding manifold S = {x R n : s(x) = 0} where n 1 s(x) = x cn + a i x ci i=1 ṡ = = n i=1 n 1 a ci x ci + u + p + a i x ci+1 (49) i=1 n a cix ci + u + p, a ci = a ci + a ci 1 (50) i=1

48 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Robustness with respect to matched disturbance Control n u = ksign(s) a cix ci (51) i=1 sṡ = k s + p s, (52) If the disturbance is bounded sup p <, then take k = µ + sup p sṡ < µ s

49 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Robustness with respect to matched disturbance Equivalent dynamics ẋ c1 = x c2 (53). =. (54) ẋ cn 2 = x cn 1 (55) n 1 ẋ cn 1 = x cn = a i x ci (56) a i (Hurwitz) No influence of the perturbation once the sliding manifold is reached (only the hitting phase is influenced) i=1

50 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Robustness with respect to matched disturbance Example: double integrator ẋ 1 = x 2 (57) ẋ 2 = u + p, sup p < (58) Sliding manifold: S = {x R n : s(x) = 0}, s(x) = x 2 + a 1 x 1 Compute Equivalent control (without disturbance p = 0) ṡ = 0 = u eq + a 1 x 2 u eq = a 1 x 2

51 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control : First order sliding mode Robustness with respect to matched disturbance Example: double integrator Control driving the solutions to S in finite time u = u eq + u disc, u disc = ksign(s) sṡ = s(u disc + p) < µ s, k = µ + sup p.

52 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control: First order sliding mode Advantages Insensibility against perturbations (matching perturbations) The choice of surface s(x, t) = 0 allow to choose a priori the closed-loop dynamics Disadvantages Trajectory s = 0 Chattering Chattering phenomenon s(x, t) must have a relative degree equal to 1 wrt. u The trajectories are not robust against perturbations during the reaching phase

53 Attractivity condition and invariance condition of the sliding manifo Sliding mode equivalent dynamics Robustness with respect to matched disturbance Sliding Mode Control: First order sliding mode Advantages Insensibility against perturbations (matching perturbations) The choice of surface s(x, t) = 0 allow to choose a priori the closed-loop dynamics Disadvantages Trajectory s = 0 Chattering Chattering phenomenon s(x, t) must have a relative degree equal to 1 wrt. u The trajectories are not robust against perturbations during the reaching phase

54 Higher Order Sliding Mode Control Objective Introduction Commande par modes glissants d ordre supérieur optimale Expérimentation sur deux benchmarks industriels Conclusion et perspectives To constrain the system trajectories to evolve onto the sliding surface: { } Modes S r glissants d ordre r = ρ = x R n : s = ṡ =... = s (r 1) 0 trajectoire chattering s =0 s =0 trajectoire Commande par modes Commande par modes Stabilis ż 1 ż 2 mode glissant d ordre r =1 mode glissant d ordre r > 1 ż ρ s

55 Higher Order Sliding Mode Control Introduced by A. Levant (Ph. D. supervisor Emel yanov) in 87 Ideal : { } S r = x R n : s = ṡ =... = s (r 1) = 0 Real : s = O(Ts r ) (59) ṡ = O(Ts r 1 ) (60)... =... (61) s (r 1) = O(Ts ) (62) T s = sampling period (63) With respect to a bounded deterministic Lebesgue-measurable noise (bounded by ε): s = O(ε 1/2r 1 )

56 Higher Order Sliding Mode Control Advantages: Robustness w.r.t. bounded matching perturbation, Reduce the of the sliding dynamics up to at most (n r) (in fact if counting the added integrators exactly (n r)). Finite Time convergence to S r, Chattering reduction (sometimes see relative degree of s), Higher convergence accuracy.

57 Higher Order Sliding Mode Control S r = { } x R n : s = ṡ =... = s (r 1) = 0 (64) Let the set S r be non-empty and assume that it consists of Filippov s trajectories of the discontinuous dynamic system. Definition Any motion (Filipov sense) in the set S r is called an r-sliding mode with respect to the constraint function s.

58 Higher Order Sliding Mode Control Sliding mode and relative degree ẋ = f(t, x, u), s = s(t, x) Theorem (H. Sira-Ramirez 89) A first order sliding mode exists iff the relative degree of s w.r.t the above defined system is one. Equivalent dynamics is stable system is minimum phase w.r.t s. Relative degree r strictly greater than one : Only an r-sliding mode algorithm leads to a finite time convergence on the sliding manifold.

59 Higher Order Sliding Mode Control Problem : find algorithms ensuring higher order sliding modes. There exist for r = 1, 2 and 3 for any r > 3 there no satisfactory constructive algorithm (only the structure is proposed and existence is proved for large enough parameters) Ideal Algorithms: Real Algorithms: The necessary information increase with the order Twisting and Super-twisting [Levant] Sub-optimal [Bartolini] Nested HOSM [Levant] Quasi-continuous HOSM [Levant] Good approximation for 2nd order Drift algorithm [Emel yanov] Discretized version of ideal ones

60 Higher Order Sliding Mode Control 2-order sliding mode algorithms Hypothesis: s = a(t, x) + b(t, x, u)u 1 For any continuous u(t) s.t. u U M, U M > 1 the solution of the system is well defined for all t. 2 u 1 (0, 1) s.t. for any continuous function u(t) with u(t) > u 1, t 1, s.t s(t)u(t) > 0 for each t > t 1. (u(t) = sign[s(t 0 )], enforces s = 0 in finite time) 3 s 0 > 0, u 0 < 1, Γ m > 0, Γ M > 0 such that if s(t, x) < s 0 then 0 < Γ m b(t, x, u) Γ M, u U M, x X (65) and the inequality u > u 0 entails ṡu > 0. 4 A > 0 s.t within s(t, x) < s 0 the following inequality holds t, x X, u U M a(t, x) A (66)

61 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] r(s) = 1 y 1 = s, y 2 = ṡ, after some transient a(t, x) A, 0 < Γ m b(t, x, u) Γ M, A > 0. { ẏ1 = y 2 = a(t, x) + b(t, x, u) u ẏ 2 (67) with y 2 (t) unmeasured but with a possibly known sign.

62 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] u if u > 1, u(t) = λ m sign(y 1 ) if y 1 y 2 0; u 1, λ M sign(y 1 ) if y 1 y 2 > 0; u 1. Sufficient conditions: λ M > λ m λ m > 4Γ M s0 λ m > A Γ m Γ m λ M A > Γ M λ m + A. (68) (69)

63 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant]

64 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] r(s) = 2 u(t) = { λm sign(y 1 ) if y 1 y 2 0 λ M sign(y 1 ) if y 1 y 2 > 0

65 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Example ẋ 1 = x 2 (70) ẋ 2 = x 3 (71) ẋ 3 = x 1 x 2 + u + p(t) (72) sup p(t) = π (73) t R

66 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Example Using (70) : for s 1 (x) = x 2 + ax 1 (74) we have ṡ 1 = x 3 + ax 2, (75) s 1 = x 1 x 2 + u + p(t) + ax 3 (76) r(s 1 ) = 2, (77) thus if s 1 (x) = ṡ 1 (x) = 0 in finite time then the equiv. dynamics is ẋ 1 = ax 1 thus x 1 (t) 0, x 2 (t) 0, x 3 (t) 0.

67 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Example Using (70) + (71) (ẋ 1 = x 2, ẋ 2 = x 3 ) : for we have s 2 (x) = x 3 + (ω 2 nx 1 + 2ζω n x 2 ) (78) ṡ 2 = x 1 x 2 + u + p(t) + (ω 2 nx 2 + 2ζω n x 3 ), (79) r(s 2 ) = 1, (80) thus if s 2 (x) = 0 in finite time then the equiv. dynamics is ẋ 1 = x 2, ẋ 2 = (ω 2 nx 1 + 2ζω n x 2 ) thus x 1 (t) 0, x 2 (t) 0, x 3 (t) 0.

68 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Example Case 1: 1rst order SM using s 2 s 2 (x) = x 3 + (ω 2 nx 1 + 2ζω n x 2 ), ṡ 2 = x 1 x 2 + u + p(t) + (ω 2 nx 2 + 2ζω n x 3 ), Compute equiv. control (without p): u eq = x 1 x 2 (ω 2 nx 2 + 2ζω n x 3 ) u = u eq + u disc, u disc = ksign(s), k > π + µ sṡ = k s + sp < µ s

69 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Example Case 2: 2nd order SM using s 2 Since r(s 2 ) = 1 we add : Chattering removal ẋ 1 = x 2 (81) ẋ 2 = x 3 (82) ẋ 3 = x 1 x 2 + u + p(t) (83) u = v (84)

70 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Example Thus we have s 2 (x) = x 3 + (ωnx ζω n x 2 ), ṡ 2 = x 1 x 2 + u + p(t) + (ωnx ζω n x 3 ), s 2 = x 1 x 3 + x v + ṗ + (ωnx ζω n (x 1 x 2 + u + p(t))) = a(x) + v + (ṗ + 2ζω n p)

71 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Example Compute equiv. control (without p): v eq = a(x) v = v eq + v disc, { λm sign(s v disc = T A(s 2 ) = 2 ) if s 2 ṡ 2 0 λ M sign(s 2 ) if s 2 ṡ 2 > 0 u = v C 0

72 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Example Case 3: 2nd order SM using s 1 Since r(s 1 ) = 2 we can directly use TA (Chattering!!) ṡ 1 = x 3 + ax 2, (85) s 1 = x 1 x 2 + u + p(t) + ax 3 (86) Compute equiv. control (without p): v eq = x 1 x 2 ax 3 u = u eq + u disc, { λm sign(s u disc = T A(s 1 ) = 1 ) if s 1 ṡ 1 0 λ M sign(s 1 ) if s 1 ṡ 1 > 0

73 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Twisting Algorithm (TA) [Levant] Convergence acceleration TA + pole placement (at the same location!!) { u = α 2 λm sign(s) if sṡ 0 s 2αṡ + λ M sign(s) if sṡ > 0

74 Higher Order Sliding Mode Control 2-order sliding mode algorithms: sub-optimal [Bartolini et al.] y 1 = s, y 2 = ṡ, after some transient a(t, x) A, 0 < Γ m b(t, x, u) Γ M, A > 0. { ẏ1 = y 2 = a(t, x) + b(t, x, u)u. ẏ 2 (87) (y 1, y 2 ) Trajectories are confined within limit parabolic arcs. Control: v(t) = α(t)λ { M sign(y 1 (t) 1 2 y 1 M ), α if [y α(t) = 1 (t) 1 2 y 1 M ][y 1M y 1 (t)] > 0 1 if [y 1 (t) 1 2 y 1 M ][y 1M y 1 (t)] 0, (88) where y 1M is the last maximum of y 1 (t), i.e. the last value of y 1 for t s.t. y 2 = ẏ 1 = 0

75 Higher Order Sliding Mode Control 2-order sliding mode algorithms: sub-optimal [Bartolini et al.] Sufficient conditions: α (0, 1] (0, 3Γm Γ M ), λ M > max ( Φ α Γ m, ) 4Φ 3Γ m α Γ M. (89)

76 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Super twisting Algorithm (STA) [Levant] The control is given by: u(t) = u { 1 (t) + u 2 (t) u if u > 1 u 1 (t) = { W sign(y 1 ) if u 1 λ s0 u 2 (t) = ρ sign(y 1 ) if y 1 > s 0 λ y 1 ρ sign(y 1 ) if y 1 s 0 (90)

77 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Super twisting Algorithm (STA) [Levant] Sufficient conditions : W > Φ Γ m λ 2 4A Γ 2 m 0 < ρ 1 2 Γ M (W +A) Γ m(w A) (91)

78 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Super twisting Algorithm (STA) [Levant] Simplified version if b does not depend on control, u does not need to be bounded and s 0 = : u = λ s ρ sign(y 1 ) + u 1, u 1 = W sign(y 1 ).

79 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Drift Algorithm (DA) [Emelyanov] Control (s relative degree is 1): u if u > 1 u = λ m sign( y 1i ) λ M sign( y 1i ) if y 1 y 1i 0; u 1 if y 1 y 1i > 0; u 1 (92) where λ m > 0, λ M > 0 are proper positive constants such that λ m < λ M and λ M λm is sufficiently large, and y 1i = y 1 (t i ) y 1 (t i τ), t [t i, t i+1 ).

80 Higher Order Sliding Mode Control 2-order sliding mode algorithms: Drift Algorithm (DA) [Emelyanov] Similar controller (when s is relative degree 2) : { λm sign( y u = 1i ) if y 1 y 1i 0 λ M sign( y 1i ) if y 1 y 1i > 0

81 Higher Order Sliding Mode Control r-order sliding mode algorithms: Hommogeneous SM [Levant] Let p least common multiple of 1, 2,..., r s (r) [ C, C] + [K m, K M ]u (93) ϕ 0,r = s (94) N 1,r = s r 1 r (95) ϕ i,r = s (i) + β i N i,r sign(ϕ i 1,r ) (96) N i,r = ( s p r s (i 1) p r i r i+1 ) p (97) u = λsign(ϕ r 1,r (s, ṡ,..., s (r) )) (98) β i hard to find but can be set in advance and λ should be large enough!

82 Higher Order Sliding Mode Control r-order sliding mode algorithms: Quasi continuous Hommogeneous SM [Levant] s (r) [ C, C] + [K m, K M ]u ϕ 0,r = s N 0,r = s Ψ 0,r = ϕ 0,r = sign(s) N 0,r ϕ i,r = s (i) r i r i r i+1 + β i N i 1,r Ψ i 1,r N i,r = s (i) r i+1 + β i N i 1,r Ψ i 1,r Ψ i,r = ϕ i,r N i,r u = λsign(ψ r 1,r (s, ṡ,..., s (r) )) β i hard to find but can be set in advance and λ should be large enough!

83 Table of Contents 1 Differential Inclusion: Notion of solution 2 3

84 Higher Order Sliding Mode Control ẋ = f(x) (99) where f is a continuous vector field or differential inclusion ẋ F (x) (100) where F is set valued map.

85 Higher Order Sliding Mode Control Sufficient condition for ODE (or DI) to be finite time stable: Lemma Suppose there exists a Lyapunov function V (x) defined on a neighborhood U R n of the origin of system (99) and some constants τ, γ > 0 and 0 < β < 1 such that d dt V (x) (99) τv (x) β + γv (x), x U\{0}. Then{ the origin of system (99) } is FTS. The set Ω = x U : V (x) 1 β < τ γ is contained in the domain of attraction of the origin. The settling time satisfies T (x) ln(1 γ τ V (x)1 β ) γ(β 1), x Ω.

86 Higher Order Sliding Mode Control Let λ > 0, r i > 0, i {1,..., n} called weights one can define: the vector of weights r = (r 1,..., r n ) T, the dilation matrix Λ r = diag{λ r i } n i=1, (101) note that Λ r x = (λ r 1 x 1,..., λ r i x i,..., λ rn x n ) T. let r denotes the finite product r 1 r 2... r n then the r homogeneous norm of x R n is defined by: n r (x) = ( x 1 r r x i r r i x n r rn ) 1 r. (102)

87 Higher Order Sliding Mode Control Definition A function h : R n R is r homogeneous with degree d r,h R if for all x R n we have (Hermes 90) : λ d r,h h(λ r x) = h(x). (103) When such a property holds, we write deg r (h) = d r,h.

88 Higher Order Sliding Mode Control Let us note that for any positive real number λ: λ 1 n r (Λ r x) = n r (x), (104) this is deg r (n r ) = 1. Let us introduce the following compact set S r = {x R n : n r (x) = 1}, (105)

89 Higher Order Sliding Mode Control Remark In fact in stead of dealing with S r one can take any closed curve properly chosen diffeomorphic to S n 1.

90 Higher Order Sliding Mode Control Such homogeneity notion can be also defined for vector fields, ordinary differential system (99) Definition A vector field f : R n R n is r homogeneous with degree d r,f R, with d r,f > min i {1,...,n} (r i ) if for all x R n we have (see Hermes 90) : λ d r,f Λ 1 r f(λ r x) = f(x), (106) which is equivalent to all i-th component f i being r homogeneous function of degree r i + d r,f. When such a property holds, we write deg r (f) = d r,f. The system (99) is r homogeneous of degree d r,f if the vector field f is homogeneous of degree d r,f.

91 Higher Order Sliding Mode Control Theorem If the system (99) is locally AS and r homogeneous with negative degree then it is FTS.

92 Higher Order Sliding Mode Control ẋ = f(x) + g(x)u, s(t, x) with r(s) = cte = ρ N +. HOSM FTS for the following system : ż 1 = z 2 ż 2 = z 3. (107) ż ρ 1 = z ρ ż ρ = a(x, t) + b(x, t)u z = [z 1, z 2,..., z ρ 1, z ρ ] T = [ s, ṡ,..., s (ρ 2), s (ρ 1)] T a(x, t) = L ρ f s(x, t) b(x, t) = L g L ρ 1 f s(x, t)

93 Higher Order Sliding Mode Control a, b have known nominal part denoted by a, b their unknown part being described by δ a, δ b, this is: { a = a + δa b = b + δ b

94 Higher Order Sliding Mode Control Assumptions: Assume that the nominal part b is invertible. Using: u = b 1 (w a) (108) where w R is the knew input (107) leads to: ż 1 = z 2 ż 2 = z 3. ż ρ 1 = z ρ ż ρ = ϑ(x, t) + (1 + ζ(x, t)) w where ϑ, ζ are given by: { ϑ = δ a δ b b 1 a ζ = δ b b 1 (109)

95 Higher Order Sliding Mode Control Assumption: ϑ(x, t), ζ(x, t) bounded: a(x) > 0 and 0 < b 1 s.t.: { ϑ(x, t) a(x) (110) ζ(x, t) 1 b Idea: FTS the unperturbed chain of integrator using homogeneity

96 Higher Order Sliding Mode Control ż 1 = z 2. ż ρ = w (111)

97 Higher Order Sliding Mode Control Theorem (Bhat 2005) Let k 1,..., k ρ positives ctes s.t. p ρ + k ρ p ρ k 2 p + k 1 is Hurwitz. Then ɛ (0, 1) s.t ν (1 ɛ, 1), (111) is FTS by: w(z) = k 1 sign (z 1 ) z 1 ν 1... k ρ sign (z ρ ) z ρ νρ (112) where ν 1,..., ν ρ are given by: with ν ρ = ν et ν ρ+1 = 1. ν i 1 = ν iν i+1 2ν i+1 ν i, i = 2,..., ρ, (113)

98 Table of Contents 1 Differential Inclusion: Notion of solution 2 3

99 Integral Sliding Mode Control Objective To remove the reaching phase To guarantee the robustness properties against perturbations in the model from the initial time instance Philosophy To choose the sliding variable such that the system trajectories are already on the sliding surface at the initial time instance

100 Integral Sliding Mode Control Objective To remove the reaching phase To guarantee the robustness properties against perturbations in the model from the initial time instance Philosophy To choose the sliding variable such that the system trajectories are already on the sliding surface at the initial time instance

101 Integral Sliding Mode Control w nom (z) FTS the unperturbed system(111). w disc (z) is built to cope with ϑ(x, t) and ζ(x, t) for (109). Leading to a ρ order sliding mode for s(x, t).

102 Table of Contents 1 Differential Inclusion: Notion of solution 2 3

103 Problem setup Differential Inclusion: Notion of solution Reference trajectory ẋ ref ẏ ref θ ref = Objective cos θ ref 0 sin θ ref [ vref w ref ] j yre f (t0) y(t0) Robot réel t1 θ t2 t4 t3 t3 t2 t1 θre f Robot de référence t4 Individual tracking of the optimal planned trajectory for each robot i To stabilize the tracking errors: e x e y e θ = x x ref y y ref θ θ ref O x(t0) xre f (t0) i Figure 1:

104 Problem setup Differential Inclusion: Notion of solution Reference trajectory ẋ ref ẏ ref θ ref = Objective cos θ ref 0 sin θ ref [ vref w ref ] j yre f (t0) y(t0) Robot réel t1 θ t2 t4 t3 t3 t2 t1 θre f Robot de référence t4 Individual tracking of the optimal planned trajectory for each robot i To stabilize the tracking errors: e x e y e θ = x x ref y y ref θ θ ref O x(t0) xre f (t0) i Figure 1:

105 Problem setup Differential Inclusion: Notion of solution Difficulties Presence of perturbations and parametric uncertainties in the model: ẋ ẏ θ = cos θ 0 sin θ [ v w ] + p(q, t)

106 Algo. 1 Differential Inclusion: Notion of solution Assumptions Perturbations satisfy the matching condition Perturbations are bounded by known positive functions Reference velocities are continuous and bounded No stop point

107 Algo. 1 Differential Inclusion: Notion of solution The tracking errors asymptotically converge toward zero under: u = u nom + u disc Continuous term u nom [Jiang et al., 2001] u nom stabilize the tracking errors without perturbation [ vref cos e 3 + µ 3 tanh e 1 u nom = w ref + µ1v ref e 2 sin e 3 1+e 2 1 +e2 2 e 3 + µ 2 tanh e 3 ] with e 1 e 2 e 3 = cos θ sin θ 0 sin θ cos θ e x e y e θ

108 Algo. 1 Differential Inclusion: Notion of solution The tracking errors asymptotically converge toward zero under: u = u nom + u disc Discontinuous term u disc u disc reject the effect of the perturbation from the initial time instance [ ] G u disc = 1 (e)sign(σ 1 ) G 2 (e)sign( e 2 σ 1 + σ 2 ) with σ = [σ 1, σ 2 ] T given by: σ = σ0 (e) + e aux σ 0 (e) = [ e 1, e 3 ] T : linear [ combinaison] of [ state vref cos e ė integral part aux = 3 1 e2 w ref 0 1 e aux = σ 0 (e(0)) ] u nom (e)

109 Algo. 1 Differential Inclusion: Notion of solution The tracking errors asymptotically converge toward zero under: u = u nom + u disc Discontinuous term u disc u disc reject the effect of the perturbation from the initial time instance [ ] G u disc = 1 (e)sign(σ 1 ) G 2 (e)sign( e 2 σ 1 + σ 2 ) with σ = [σ 1, σ 2 ] T given by: σ = σ0 (e) + e aux σ 0 (e) = [ e 1, e 3 ] T : linear [ combinaison] of [ state vref cos e ė integral part aux = 3 1 e2 w ref 0 1 e aux = σ 0 (e(0)) ] u nom (e)

110 Algo. 1 Differential Inclusion: Notion of solution The tracking errors asymptotically converge toward zero under: u = u nom + u disc Discontinuous term u disc u disc reject the effect of the perturbation from the initial time instance [ ] G u disc = 1 (e)sign(σ 1 ) G 2 (e)sign( e 2 σ 1 + σ 2 ) with σ = [σ 1, σ 2 ] T given by: σ = σ0 (e) + e aux σ 0 (e) = [ e 1, e 3 ] T : linear [ combinaison] of [ state vref cos e ė integral part aux = 3 1 e2 w ref 0 1 e aux = σ 0 (e(0)) ] u nom (e)

111 Experimental results: Algo. 1 Single nominal control 6 traj. référence traj. réelle 5 obstacle obstacle augmenté 4 ISMC 5 4 traj. référence traj. réelle obstacle obstacle augmenté y (m) x (m) y (m) erreur (m) erreur (m) t(s) t(s)

112 Algo. 1 Differential Inclusion: Notion of solution Limitations conservative assumptions discontinuities on velocities perturbations must satisfy the matching condition Solution Practical stabilization using second order ISMC x δ x(0) 2ε t δ T

113 Algo. 1 Differential Inclusion: Notion of solution Limitations conservative assumptions discontinuities on velocities perturbations must satisfy the matching condition Solution Practical stabilization using second order ISMC x δ x(0) 2ε t δ T

114 Experimental results ISM of Order 1 ISM of Order traj robot 3 traj robot 1 traj robot traj robot 2 traj robot 1 traj robot y [m] 0.2 y [m] x [m] x [m]

115 Experimental results ISM of Order 1 ISM of Order erreur l 12 l 12,des 0.14 erreur l 12 l 12,des erreur l 13 l 13,des 0.14 erreur l 13 l 13,des [m] [m] t [s] t [s] 0.8 erreur ψ 12 ψ 12,des erreur ψ 12 ψ 12,des 0.6 erreur ψ 13 ψ 13,des 0.6 erreur ψ 13 ψ 13,des [rad] 0 [rad] t [s] t [s]

116 Video Video

117 Video with 3 miabot Video

118 Video with 7 miabots Video