Higher Order Super-Twisting Algorithm Shyam Kamal 1 Asif Chalanga 2 Prof.J.A.Moreno 3 Prof.L.Fridman 4 and Prof.B.Bandyopadhyay 5 125 Indian Institute of Technology Bombay, Mumbai-India 3 Instituto de Ingenier´ ıa Universidad Nacional Aut ´ onoma de M ´ exico (UNAM) 4 Facultad de Ingenier´ ıa Universidad Nacional Aut ´ onoma de M ´ exico (UNAM) VSS14, Nantes, June 29 July 2 2014
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Higher Order Super-Twisting Algorithm
Shyam Kamal1
Asif Chalanga2 Prof.J.A.Moreno3 Prof.L.Fridman4 and Prof.B.Bandyopadhyay5
125Indian Institute of Technology Bombay, Mumbai-India
3Instituto de Ingenierıa Universidad Nacional Autonoma de Mexico (UNAM)
4Facultad de Ingenierıa Universidad Nacional Autonoma de Mexico (UNAM)
VSS14, Nantes, June 29 July 2 2014
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Outline
1 Motivation
2 Higher Order STA
3 Convergence Condition for the 3-STA
4 Controller Design based on Generalized STA
5 Simulation Results
6 Conclusion
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 2
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Motivation
Consider the second order system
σ = u + ρ1 (2.1)
where ρ1 is a non vanishing Lipschitz disturbance and |ρ1| < ρ0 .
Algorithm Control Signal Information Stability ChatteringFirst SMC Discontinuous σ and σ Asymptotic Yes
STC Continuous σ and σ Asymptotic NoTwisting Discontinuous σ and σ Finite time Yes
Third SMC Continuous σ and σ and disturbance Finite time No
Table : Different control strategies for the second order uncertain integrator with outputσ and its derivative σ
It is clear from the table that finite time control under the absolutelycontinuous control signal without explicit knowledge of disturbance is stillunexplored.Similar kind of situation is also true for the system with higher relativedegree.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 3
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Generalized Order Super-Twisting
Generalized order Super-twisting which has following properties:
finite time convergence for the set σ, σ, ..., σ(r ) where σ represents theoutput and r is the relative degree of the system with respect to outputusing information of σ, σ, ..., σ(r−1) which generates the absolutelycontinuous control signal for the arbitrary relative degree;
compensates theoretically exactly Lipschitz in time on the systemtrajectories uncertainties/perturbations;
precision of the output σ corresponding to (r + 1)th order sliding mode;
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 4
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Notation
In this paper the following notation is used, for a real variable z ∈ R to a realpower p ∈ R, ⌊z⌉p = |z|psgn(z), therefor ⌊z⌉2 = |z|2sgn(z) 6= z2. If p is anodd number, this notation does not change the meaning of the equation, i.e.⌊z⌉p = zp . Therefore
⌊z⌉0 = sgn(z), ⌊z⌉0zp = |z|p, ⌊z⌉0|z|p = ⌊z⌉p
⌊z⌉p⌊z⌉q = |z|psgn(z)|z|qsgn(z) = |z|p+q (2.2)
Also, σ = x1 represents the output for the generalized n-STA.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 5
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Definition
Following standard definition existing in literature [?]:
Definition
A vector field f : Rn → Rn (or a differential inclusion) is called homogeneous
of degree δ ∈ R with the dilatationdκ : (x1, x2, · · · , xn) 7→ (κ1 x1, κ
2x2, · · · , κn xn), where = (1, 2, · · · , n)
are some positive numbers (called the weights), if for any κ > 0 the followingidentity f (x) = κ−δd−1
κ f (dκx) holds.
Definition
A scalar function V : Rn → R is called homogeneous of degree δ ∈ R with thedilatation dκ if for any κ > 0 the following identity V (x) = κ−δV (dκx) holds.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 6
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Higher Order STA
In this section generalization of STA is presented.
For the simplicity of notation algorithm is expressed in the term ofx1, x2, · · · , xn
where σ = x1 is the output.
2-STA is given as follows
x1 = −k1|x1|12 sign(x1) + x2
x2 = −k2sign (x1) + ρ (3.1)
where x1, x2 represent the states and the perturbation ρ satisfied |ρ| ≤ ∆.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 7
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Higher Order STA
3-STA is proposed as follows
x1 = x2
x2 = −k1 |φ1|1/2 sign (φ1) + x3
x3 = −k3sign (φ1) + ρ (3.2)
where φ1 = x2 + k2|x1|2/3sign(x1),
x1, x2, x3 represent the states and the perturbation ρ satisfied |ρ| ≤ ∆.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 8
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Higher Order STA
4-STA is proposed as follows
x1 = x2
x2 = x3
x3 = −k1 |φ2|1/2 sign (φ2) + x4
x4 = −k4sign (φ2) + ρ (3.3)
where
φ2 = x3 + k3
(
|x1|3 + |x2|
4) 1
6sign
(
x2 + k2|x1|34 sign(x1)
)
(3.4)
and x1, x2, x3, x4 represent the states and the perturbation ρ satisfied |ρ| ≤ ∆.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 9
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Higher Order STA
5-STA is proposed as follows
x1 = x2
x2 = x3
x3 = x4
x4 = −k1 |φ3|1/2 sign (φ3) + x5
x5 = −k5sign (φ3) + ρ (3.5)
where
φ3 = x4 + k4
[
(
|x1|12 + |x2|
15 + |x3|20) 1
30sign (l1)
]
and
l1 = x3 + k3
(
|x1|12 + |x2|
15) 1
20sign
(
x2 + k2|x1|45 sign(x1)
)
and x1, x2, x3, x4, x5 represent the states and the perturbation ρ satisfied|ρ| ≤ ∆.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 10
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Higher Order STA
n-STA is proposed as follows
x1 = x2
x2 = x3
...
xn−1 = −k1 |φn−2|1/2 sign (φn−2) + xn
xn = −knsign (φn−2) + ρ (3.6)
where φn−2 we define later part of the paper, x1, x2, · · · , xn represent thestates and the perturbation ρ satisfied |ρ| ≤ ∆.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 11
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Higher Order STA
Definition of φn−2 is given as follows:-
R1,r−1 = |x1|r
r+1
where r represents the relative degree of algorithm with respect to x1.
Ri,r−1 =∣
∣|x1|r1 + |x2|
r2 + · · ·+ |xi−2|ri−2
∣
∣
qi
where i = 2, 3, · · · , (r − 1), r1, r2, · · · , ri−2 and qi is designed parameterbased on the homogeneity weight of the xi+1.
S0,r−1 = x1
S1,r−1 = x2 + k2R1,r−1sign(x1)
Si,r−1 = xi+1 + ki+1Ri,r−1sign(Si−1,r−1)
where i = 2, 3, · · · , (r − 1)
Finally φn−2 = sr−1,r−1.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 12
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation of 3-STA and 4-STA
Under the following value of initial conditions and gains3-STA
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 13
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation of 3-STA and 4-STA
0 1 2 3 4 5 6 7 8 9 10−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time (sec)
Sta
tes
x
1
x2
x3
3 3.05 3.1−4
−2
0
2x 10
−3
Figure : Evolution of States of 3-STA w.r.t. time
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 14
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation of 3-STA and 4-STA
0 1 2 3 4 5 6 7 8 9 10−4
−3
−2
−1
0
1
2
3
4
Time (sec)
Sta
tes
x
1
x2
x3
x4
6 6.05 6.1−5
0
5x 10
−3
Figure : Evolution of States of 4-STA w.r.t. time
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 15
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Discussion about 3-STA and other Generalized STA
3-STA (3.2) is homogeneous of degree δf = −1 with weights = [3, 2,1], and its solution can be provide in the sense of Flippov.
The main advantage of this algorithm is that, output (x1) and itsderivative (x2) information are only needed for the finite timeconvergence of all three variables x1, x2 and x3.
Proposed algorithm can be work as controller for the uncertain systemwith relative degree 2 with respect to, output in the case of 3-STA.
Similarly, n-STA homogeneous of degree δf = −1 with weights = [n, n − 1, · · · , 2,1] and used for the uncertain system with relativedegree n − 1 with respect to output.
The main idea behind construction of this algorithm is that add one extradiscontinuous integral term which is able to reconstruct the perturbationand also nullify.
But it is necessary that perturbations must be Lipschitz continuous,meaning is that the first derivative is exit almost everywhere and its alsobe bounded, but perturbations might not be bounded.
It is necessary to specify here that large number of second orderuncertain systems contain this class of perturbations.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 16
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Convergence condition of 3-STA
Consider the following continuous candidate Lyapunov function for thestability analysis of (3.2)
V (x) = p1|x1|43 − p12⌊x1⌉
23
(
x2 + k2⌊x1⌉2/3
)
+ p2
∣
∣
∣x2 + k2⌊x1⌉
2/3∣
∣
∣
2+ p13⌊x1⌉
23 ⌊x3⌉
2
− p23
(
x2 + k2⌊x1⌉2/3
)
⌊x3⌉2 + p3|x3|
4 (4.1)
V (x) is homogeneous of degree δV = 4, with weights = [3, 2,1].
It is differentiable everywhere but it is not locally Lipschitz at x1 = 0.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 17
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Convergence condition of 3-STA
Our main aim to derive the conditions for the coefficient(p1, p12, p2, p13, p23, p3) and for the gains (k1, k2, k3) of the third ordersuper-twisting algorithm (3.2).
Such that V (x) > 0 and time derivative of Lyapunov function (4.1) along(3.2) is negative definite (V < 0 for all x ∈ R
3, x 6= 0).
Lyapunov function (4.1) can also be expressed as in quadratic form in the
vector ΞT =[
⌊x1⌉23 φ ⌊x3⌉
2]
, where φ =(
x2 + k2⌊x1⌉2/3
)
, i.e.
V (x) = ΞT PΞ, where P =
p1 − 12 p12
12 p13
− 12 p12 p2 − 1
2 p2312 p13 − 1
2 p23 p3
(4.2)
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 18
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Proposition-1
Consider the continuous and homogeneous function V (x) given by (4.2).V (x) is positive definite and radially unbounded if and only if (P > 0)
p1 > 0, p1p2 >14
p212,
p1
(
p2p3 − p223
)
+p12
2
(
−p12p3
2+
p13p23
4
)
+p13
2
(p12p23
4−
p2p13
2
)
> 0. (4.3)
In this case V (x) satisfies the differential inequalities
V ≤ −κV 3/4 (4.4)
for some positive κ and it is a Lyapunov function for the system (3.2), whosetrajectories converges in finite time to the origin x = 0 for every value of theperturbation |ρ| < ∆. The convergence time of a trajectory starting at theinitial condition x0 can be estimated as
T (x0) ≤4κ
V14 (x0) (4.5)
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 19
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Proof
Proposition Proof
It is obvious that by taking (4.1) as Lyapunov candidate function andcalculating first time derivative along (3.2), one can always find κ for the setof gains k1, k2, k3 for which states of 3-STA (3.2) converges to the equilibriumpoint in the finite time.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 20
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Controller Design based on Generalized STA
Consider the following perturbed integrator system with relative degree n − 1with respect to output x1
x1 = x2
x2 = x3
...
xn−1 = u + d (5.1)
where x1, · · · , xn−1 are the states of the perturbed integrator and d is theLipschitz (in time) disturbance, which satisfied |d | < ∆.
Then nth order Super-Twisting Control (n-STC) for the (5.1) is given as
u = −k1 |φn−2|1/2 sign (φn−2) + xn
xn = −knsign (φn−2) (5.2)
where φn−2 is the same as (3.6).
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 21
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Controller Design based on Generalized STA
After applying the control u and defining the new variable zn = xn + d andtaking the first time derivative of zn, the system can be further written as
x1 = x2
x2 = x3
...
xn−1 = −k1 |φn−2|1/2 sign (φn−2) + zn
zn = −knsign (φn−2) + d (5.3)
The closed loop system (5.3) is the same as n-STA (3.6). Therefore,convergence condition remains the same for the proposed controller (5.2) asn-STA (3.6).
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 22
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation Results
For verifying the proposed technique of the n-STC following second and thirdorder system are considered
x1 = x2
x2 = u2 + d(5.4)
where x1, x2 are the states, u2 is the control and d = 2 + 3sin(t) is theLipschitz (in time) disturbance. Similarly
x1 = x2
x2 = x3
x3 = u3 + d
(5.5)
where x1, x2, x3 are the states, u3 is the control and d = 2 + 3sin(t) is theLipschitz (in time) disturbance.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 23
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation Results
The controller for the systems (5.4) and (5.5) are designed as
u2 = −k1 |φ1|1/2 sign (φ1)−
∫ t
0k3sign (φ1)dτ (5.6)
and
u3 = −k1 |φ2|1/2 sign (φ1)−
∫ t
0k4sign (φ2)dτ (5.7)
where φ1 and φ2 are defined as (3.2) and (3.3) respectively.
Following parameters are used for the simulationuncertain double order integrator (5.4)
initial conditions x1(0) = −1 and x2(0) = 3gains k1 = 6, k2 = 4 and k3 = 4
uncertain third order integrator (5.5)initial conditions x1(0) = −1, x2(0) = 3 and x3(0) = 1gains k1 = 5, k2 = 2, k3 = 1 and k4 = 4
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 24
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation Results
0 2 4 6 8 10 12 14 16 18 20−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time (sec)
Sta
tes
x
1
x2
4 4.02 4.04 4.06 4.08 4.1−2
−1
0
1
2x 10
−6
Figure : Evolution of states w.r.t., time (uncertain double integrator)
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 25
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation Results
0 2 4 6 8 10 12 14 16 18 20−12
−10
−8
−6
−4
−2
0
2
4
6
Time (sec)
Con
trol
Figure : Evolution of control w.r.t., time (uncertain double integrator)
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 26
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation Results
0 2 4 6 8 10 12 14 16 18 20−3
−2
−1
0
1
2
3
4
Time (sec)
Sta
tes
x
1
x2
x3
5 5.05 5.1 5.15 5.2−10
−5
0
5x 10
−3
Figure : Evolution of states w.r.t., time (uncertain triple integrator)
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 27
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Simulation Results
0 2 4 6 8 10 12 14 16 18 20−12
−10
−8
−6
−4
−2
0
2
4
6
Time (sec)
Con
trol
Figure : Evolution of control w.r.t., time (uncertain triple integrator)
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 28
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Conclusion
The paper discussed the realization of higher order sliding mode usingthe absolutely continuous control signal in the presence of matchedLipschitz (in time) uncertainties.
For the above mentioned goal, generalization of the Super-Twistingalgorithm (STA) for r relative degree system ensuring finite timeconvergence for the set σ, σ, ..., σ(r ) where σ represents the output usinginformation of σ, σ, ..., σ(r−1) has been proposed.
The convergence conditions for the 3-STA algorithm have beenproposed.
The formula for algorithm of arbitrary order has been also suggested.
The Lyapunov function based convergence conditions for the 4 andhigher STA are still open problem, which will look in the future.
The simulations results are confirmed the efficiency of the proposedalgorithm.
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 29
Motivation Higher Order STA Convergence Condition for the 3-STA Controller Design based on Generalized STA Simulation Results Conclusion
Thank You!
Prof.L.Fridman — Higher Order Super-Twisting Algorithm 30