Higher Order Super-Twisting Algorithm Shyam Kamal 1 Asif Chalanga 2 - - PowerPoint PPT Presentation

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 Ingenier´

ıa Universidad Nacional Aut´

• noma de M´

exico (UNAM)

4Facultad de Ingenier´

ıa Universidad Nacional Aut´

• noma de M´

exico (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 Chattering First SMC Discontinuous σ and ˙ σ Asymptotic Yes STC Continuous σ and ˙ σ Asymptotic No Twisting 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 output σ and its derivative ˙ σ

It is clear from the table that finite time control under the absolutely continuous control signal without explicit knowledge of disturbance is still unexplored. Similar kind of situation is also true for the system with higher relative degree.

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 the

• utput and r is the relative degree of the system with respect to output

using information of σ, ˙ σ, ..., σ(r−1) which generates the absolutely continuous control signal for the arbitrary relative degree; compensates theoretically exactly Lipschitz in time on the system trajectories 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 real power p ∈ R, ⌊z⌉p = |z|psgn(z), therefor ⌊z⌉2 = |z|2sgn(z) = z2. If p is an

• dd 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

• f degree δ ∈ R with the dilatation

dκ : (x1, x2, · · · , xn) → (κ̺1x1, κ̺2x2, · · · , κ̺nxn), where ̺ = (̺1, ̺2, · · · , ̺n) are some positive numbers (called the weights), if for any κ > 0 the following identity f(x) = κ−δd−1

κ f(dκx) holds.

Definition A scalar function V : Rn → R is called homogeneous of degree δ ∈ R with the dilatation 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 of x1, x2, · · · , xn where σ = x1 is the output. 2-STA is given as follows ˙ x1 = −k1|x1|

1 2 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

6 sign

• x2 + k2|x1|

3 4 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

30 sign (l1)

• and

l1 = x3 + k3

• |x1|12 + |x2|15 1

20 sign

• x2 + k2|x1|

4 5 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 the states 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 parameter based 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 gains 3-STA

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

4-STA

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

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

1 2 3 4 5 6 7 8 9 10 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 Time (sec) States x1 x2 x3 3 3.05 3.1 −4 −2 2 x 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

1 2 3 4 5 6 7 8 9 10 −4 −3 −2 −1 1 2 3 4 Time (sec) States x1 x2 x3 x4 6 6.05 6.1 −5 5 x 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 its derivative (x2) information are only needed for the finite time convergence of all three variables x1, x2 and x3. Proposed algorithm can be work as controller for the uncertain system with 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 relative degree n − 1 with respect to output. The main idea behind construction of this algorithm is that add one extra discontinuous integral term which is able to reconstruct the perturbation and also nullify. But it is necessary that perturbations must be Lipschitz continuous, meaning is that the first derivative is exit almost everywhere and its also be bounded, but perturbations might not be bounded. It is necessary to specify here that large number of second order uncertain 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 the stability analysis of (3.2) V(x) = p1|x1|

4 3 − p12⌊x1⌉ 2 3

• x2 + k2⌊x1⌉2/3

+ p2

• x2 + k2⌊x1⌉2/3
• 2

+ p13⌊x1⌉

2 3 ⌊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 order super-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 ∈ R3, x = 0). Lyapunov function (4.1) can also be expressed as in quadratic form in the vector ΞT =

• ⌊x1⌉

2 3 φ ⌊x3⌉2

, where φ =

• x2 + k2⌊x1⌉2/3

, i.e. V(x) = ΞT PΞ, where P =   p1 − 1

2p12 1 2p13

− 1

2p12

p2 − 1

2p23 1 2p13

− 1

2p23

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 > 1 4p2

12,

p1

• p2p3 − p2

23

• + 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), whose trajectories converges in finite time to the origin x = 0 for every value of the perturbation |ρ| < ∆. The convergence time of a trajectory starting at the initial condition x0 can be estimated as T(x0) ≤ 4 κV

1 4 (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 and calculating first time derivative along (3.2), one can always find κ for the set

• f gains k1, k2, k3 for which states of 3-STA (3.2) converges to the equilibrium

point 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 − 1 with 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 the Lipschitz (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 and taking 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) as n-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 third

• rder 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 the Lipschitz (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 the Lipschitz (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 k3sign (φ1) dτ (5.6) and u3 = −k1 |φ2|1/2 sign (φ1) − t k4sign (φ2) dτ (5.7) where φ1 and φ2 are defined as (3.2) and (3.3) respectively. Following parameters are used for the simulation uncertain double order integrator (5.4)

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

uncertain third order integrator (5.5)

initial conditions x1(0) = −1, x2(0) = 3 and x3(0) = 1 gains 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

2 4 6 8 10 12 14 16 18 20 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 Time (sec) States x1 x2 4 4.02 4.04 4.06 4.08 4.1 −2 −1 1 2 x 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

2 4 6 8 10 12 14 16 18 20 −12 −10 −8 −6 −4 −2 2 4 6 Time (sec) Control

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

2 4 6 8 10 12 14 16 18 20 −3 −2 −1 1 2 3 4 Time (sec) States x1 x2 x3 5 5.05 5.1 5.15 5.2 −10 −5 5 x 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

2 4 6 8 10 12 14 16 18 20 −12 −10 −8 −6 −4 −2 2 4 6 Time (sec) Control

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 using the absolutely continuous control signal in the presence of matched Lipschitz (in time) uncertainties. For the above mentioned goal, generalization of the Super-Twisting algorithm (STA) for r relative degree system ensuring finite time convergence for the set σ, ˙ σ, ..., σ(r) where σ represents the output using information of σ, ˙ σ, ..., σ(r−1) has been proposed. The convergence conditions for the 3-STA algorithm have been proposed. The formula for algorithm of arbitrary order has been also suggested. The Lyapunov function based convergence conditions for the 4 and higher STA are still open problem, which will look in the future. The simulations results are confirmed the efficiency of the proposed algorithm.

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