higher order super-twisting algorithmvss2014.irccyn.ec-nantes.fr/slides/fridman1_2.pdf ·...

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

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.



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;



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.



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.



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| ≤ ∆.



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| ≤ ∆.



Higher Order STA


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| ≤ ∆.



Higher Order STA


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| ≤ ∆.



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| ≤ ∆.



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.



Simulation of 3-STA and 4-STA

Under the following value of initial conditions and gains3-STA

initial conditions x1(0) = −1, x2(0) = −3 and x3(0) = 1gains k1 = 6, k2 = 4 and k3 = 4

4-STAinitial conditions x1(0) = −1, x2(0) = 3, x3(0) = 1 and x4(0) = 1gains k1 = 4, k2 = 2, k3 = 1 and k4 = 2




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




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



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.



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.



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)



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)



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.



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).



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).



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.



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



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)



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)



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)



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)



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.



Thank You!


higher order super-twisting algorithmvss2014.irccyn.ec-nantes.fr/slides/fridman1_2.pdf ·...

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