how to implement super-twisting controller based on...

How to Implement Super-Twisting Controller based onSliding Mode Observer?

Asif Chalanga1

Shyam Kamal2 Prof.L.Fridman3 Prof.B.Bandyopadhyay 4 and Prof.J.A.Moreno5

124Indian Institute of Technology Bombay, Mumbai-India

3Facultad de Ingenierıa Universidad Nacional Autonoma de Mexico (UNAM)

5Instituto de Ingenierıa Universidad Nacional Autonoma de Mexico (UNAM)

VSS14, Nantes, June 29 July 2 2014

Standard Sliding Mode STC Based On STO For Perturbed Double Integrator STC based on Super-Twisting Output Feedback (STOF) HOSMO based Continuous

Outline

1 Standard Sliding Mode STC Based On STO For Perturbed Double Integrator

2 STC based on Super-Twisting Output Feedback (STOF)

3 HOSMO based Continuous Control of Perturbed Double Integrator

4 Numerical Simulation

5 Conclusion

Prof.L.Fridman — How to Implement Super-Twisting Controller based on Sliding Mode Observer? 2


Motivation

Consider the dynamical system of the following form

Second order system

x1 = x2

x2 = u + ρ1

y = x1 (2.1)

where y is the output variable, ρ1 is a non vanishing Lipschitz disturbanceand |ρ1| < ρ0 .

Our aim is to reconstruct the states of the system and then designsuper-twisting controller based on the estimated information.

Although this is already reported in the literature,

We are going to show that existing methodology is not stand on thesound mathematical background.



Motivation

The super-twisting observer dynamics

˙x1 = x2 + k1|e1|12 sign(e1)

˙x2 = u + k2sign(e1) (2.2)

where the error e1 = x1 − x1.

The error dynamics is

e1 = e2 − k1|e1|12 sign(e1)

e2 = −k2sign(e1) + ρ1 (2.3)

So e1 and e2 will converge to zero in finite time t > T0, by selecting theappropriate gains k1 and k2.

For this, one can say that x1 = x1 and x2 = x2 after finite time t > T0.



Motivation

Consider the sliding manifold of the form

s = c1x1 + x2. (2.4)

The time derivative of (2.4) (for designing the super-twisting control)

s = c1x1 + ˙x2. (2.5)

After finite time t > T0, when observer start extracting the exact informationof the states, then one can substitute x1 = x2.

Also using (2.2) and (2.5), one can further write

s = c1x2 + u + k2sign(e1). (2.6)



Motivation

System (5.1) in the co-ordinate of x1 and s by using (2.4) and (2.5)

x1 = s − c1x1

s = c1x2 + u + k2sign(e1). (2.7)

Super-twisting control design (which is existing in the literature) as

u = −c1x2 − λ1|s|12 sign(s)−

∫ t

0λ2sign(s)dτ. (2.8)

where λ1 and λ2 are the designed parameters for the control.

The closed loop system after applying the control (2.8) to (2.7)

x1 = s − c1x1

s = −λ1|s|12 sign(s)−

∫ t

0λ2sign(s)dτ + k2sign(e1) (2.9)



Motivation

Claim

Second order sliding motion is never start in the (2.9)

Mathematical discussion

Because of s contains the non-differentiable term k2sign(e1).

Which exclude the possibility of lower two subsystem of (2.9) to act asthe super-twisting algorithm.

So the second order sliding motion (so that s = s = 0 in finite time)cannot be establish.

In the next, we are going to propose the possible methodology of thecontrol design such that non-differentiable term k2sign(e1) is cancel out.

The lower two subsystem of (2.9) act as the super-twisting and finallysecond order sliding is achieved.



Proposed method 1

The main aim here, is to design u, such that sliding motion occurs in finitetime.

Proposition 1

The control input u which is defined as

u = −c1x2 − k2sign(e1)− λ1|s|12 sign(s) −

∫ t

0λ2sign(s)dτ (2.10)

where, λ1 > 0 and λ2 > 0 are selecting according to (Levant), (Moreno),leads to the establishment second order sliding in finite time, which furtherimplies asymptotic stability of x1 and x2.



Proposed method 1

Proof

The closed loop system after substituting (2.10) into (2.7)

x1 = s − c1x1

s = −λ1|s|12 sign(s) + ν

ν = −λ2sign(s) (2.11)

Last two equation of (2.11) has same structure as super-twisting. Therefore,one can easily observe that after finite time t > T0, s = s = 0.

The closed loop system is given as

x1 = −c1x1

x2 = −c1x1 (2.12)

Therefore, both states x1 and x2 are asymptotic stability by choosing c1 > 0.Also, when observer estimating the exact state x2 = x2 after finite time, thenx2 also going to zero simultaneously as x2.



Existing result:STC based on Super-Twisting Output Feedback (STOF)

Step-1

Consider the following sliding sliding surface

s = c1x1 + x2 (3.1)

assuming that all states information are available.

Step-2

To realizing the control expression based on super-twisting , take the firsttime derivative of sliding surface s using (3.1)

s = c1x1 + x2 (3.2)

Step-3

Now substitute x1 and x2 from (5.1) into (3.2),

s = c1x2 + u + ρ1 (3.3)




Step-4

Now design control as

u = −c1x2 − λ1|s|12 sign(s) −

∫ t


After substituting the control (3.4) into (3.3),

s = −λ1|s|12 sign(s) + ν

ν = −λ2sign(s) + ρ1. (3.5)

Now select λ1 > 0 and λ2 > 0 according to (moreno2012), which leads tosecond order sliding in finite time provided ρ1 is Lipschitz and |ρ1| < ρ0.When s = 0, then x1 = x2 = 0 asymptotically same as discussed above byselecting c1 > 0.




The control (3.4) is not implementable because we do not haveinformation of x2, so replace x2 by x2.

It is argued that after finite time x1 = x1 and x2 = x2, therefore controlsignal applied to original system (5.1) is

u = −c1x2 − λ1|s|12 sign(s)−

∫ t


where s = c1x1 + x2,

Without considering the the dynamics of ˙x2 for which control derivation isexplicitly dependent and it contains the discontinuous term k2sign(e1).

One can easily see that average value of this discontinuous term isequal to negative of the disturbance.

So control (3.6) we are applying for the real system is only approximatenot the exact. However, the exact controller is always discontinuouswhich already discussed and mathematically proved in the abovesection.



Higher Order Sliding Mode Observer based Continuous Control ofPerturbed Double Integrator

This methods gives the correct way to implement continuous STC, when onlyoutput information of the perturbed double integrator (5.1) is available.

The HOSMO dynamics to estimate the states for the system (5.1) is given as

˙x1 = x2 + k1|e1|23 sign(e1)

˙x2 = x3 + u + k2|e1|13 sign(e1)

˙x3 = k3sign(e1) (4.1)

Let us define the error e1 = x1 − x1 and e2 = x2 − x2 and the error dynamics is

e1 = e2 − k1|e1|23 sign(e1)

e2 = −x3 − k2|e1|13 sign(e1) + ρ1

˙x3 = k3sign(e1) (4.2)



HOSMO based Proposed method

Now define the new variable e3 = −x3 + ρ1, if ρ1 is Lipschitz and |ρ1| < ρ0,

One can further write (4.2) as

e1 = e2 − k1|e1|23 sign(e1)

e2 = e3 − k2|e1|13 sign(e1)

e3 = −k3sign(e1) + ρ1 (4.3)

So e1, e2 and e3 will converge to zero in finite time t > T0, by selectingthe appropriate gains k1, k2 and k3 (Levant).

After the convergence of error, one can find that x1 = x1, x2 = x2 andx3 = ρ1 after finite time t > T0.




Consider the sliding surface (2.4) and its time derivative is

s = c1x1 + ˙x2. (4.4)

After finite time t > T0, when observer start extracting the exact informationof the sates, then one can substitute x1 = x2.

Also using (4.2) and (4.4), one can further write

s = c1x2 + u + k2|e1|13 sign(e1) +

∫ t

0k3sign(e1)dτ (4.5)

The system (5.1) in the co-ordinate of x1 and s by using (2.4) and (4.5)

x1 = s − c1x1

s = c1x2 + u + k2|e1|13 sign(e1) +

∫ t

0k3sign(e1)dτ (4.6)




Proposition 2

The control input u which is defined as

u = −c1x2 − k2|e1|13 sign(e1)−

∫ t

0k3sign(e1)dτ − λ1|s|

12 sign(s)

−

∫ t


or

u = −c1x2 −

∫ t

0k3sign(e1)dτ − λ1|s|

12 sign(s)−

∫ t


Because observer is much faster, which makes e1 = 0 in finite time.

If λ1 > 0 and λ2 > 0 are selecting according to (Levant), (Moreno), leadsto the establishment of s equal to zero in finite time, it further impliesasymptotic stability of x1 and x2.

Proof is the same as the Proposition 1.



Discussion of HOSMO based STC Design

It is clear from the STC control (4.7) expression based on HOSMO (4.1)is continuous.

Also, when we design STC control based on HOSMO then one has totune only observer gain according to the first derivative of disturbance,because it is necessary for the convergence of the error variables of theHOSMO.

However, during controller design there is no explicit gain condition forthe λ2 with respect to disturbances.

One can also observe that STC (3.6) design based STOF (2.2), (which ispropagating in the literature without any sound mathematicaljustification) requires two gains.

One is the STO observer gain k2 based on the explicit maximum boundof the direct disturbance and another is λ2, for the STC based on themaximum bound of the derivative of disturbance.



Discussion of HOSMO based STC Design

Some observation

From the above observation that sound mathematical analysis reducesthe two gains conditions with respect to disturbance by simply one gaincondition.

Also the precision of the sliding manifold is much improved by using theHOSMO based STC rather than STO based STC.

Due to the increase of this precision of sliding variable precision of thestates are also much effected.

In other word if we talk about stabilization problem, then states are muchcloser to origin in the case of HOSMO based STC rather than STObased STC.

We only talk about closeness of states variable with respect toequilibrium point, because only asymptotic stability is possible in theboth of design methodology.



Numerical Simulation

x1 = x2

x2 = u + ρ1

y = x1 (5.1)

For the simulation, initial conditions of perturbed double integrator , for the allthree cases, STC-STO, STC-STOF and STO-HOSMO, is taken as x1 = 10,x2 = 0 and ρ1 = 2 + 3 sin(t). Other gains for the all three cases are selectedas follows

STC-STOSTC gains k1 = 3 and k2 = 4STO gains λ1 = 3.5 and λ2 = 6

STC-STOFSTC gains k1 = 2 and k2 = 1STO gains λ1 = 3.5 and λ2 = 6

STC-HOSMOSTC gains k1 = 2 and k2 = 1STO gains λ1 = 6, λ2 = 11 and λ3 = 6



Numerical Simulation: without noise

0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

Time (sec)

Out

put x

1

10 12 14 16 18 20−5

0

5

10

15x 10

−4

STC−HOSMOSTC−STOSTC−STOF

Figure : Evolution of output w.r.t. time for the STC based on HOSMO, STOF and STO




0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

Time (sec)

Slid

ing

surf

ace

5 5.001 5.002 5.003 5.004 5.005−1

0

1

2x 10

−3

STC−HOSMOSTC−STOSTC−STOF

Figure : Evolution of sliding manifold w.r.t. time for the STC based on HOSMO, STOFand STO




0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

Time (sec)

Err

or e

1

4 4.005 4.01−1

0

1x 10

−7

3rd order observersuper twisting observer

Figure : Evolution of error w.r.t. time using STO and HOSMO




0 2 4 6 8 10 12 14 16 18 20−14

−12

−10

−8

−6

−4

−2

0

2

4

6

Time (sec)

Con

trol

14 14.002 14.004 14.006 14.008 14.01−5.1

−5

−4.9

−4.8

STC−HOSMOSTC−STOF

Figure : Evolution of control STC based on HOSMO and STOF w.r.t. time




0 2 4 6 8 10 12 14 16 18 20−15

−10

−5

0

5

10

Time (sec)

Con

trol

STC−STO

Figure : Evolution of control STC based on STO w.r.t. time



Numerical Simulation: with noise

0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

Time (sec)

Out

put x

1

5 10 15 20−0.1

0

0.1

0.2

0.3

STC−HOSMOSTC−STOFSTC−STO

Figure : Evolution of output w.r.t. time for the STC based on HOSMO, STOF and STOunder noisy measurement




0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

Time (sec)

Slid

ing

surf

ace

5 5.005 5.01 5.015 5.02−0.01

0

0.01

0.02

STC−HOSMOSTC−STOFSTC−STO

Figure : Evolution of sliding manifold w.r.t. time for the STC based on HOSMO, STOFand STO under noisy measurement




0 2 4 6 8 10 12 14 16 18 20−2

0

2

4

6

8

10

12

Time (sec)

Err

or e

1

5 5.002 5.004−0.01

−0.005

0

0.005

0.01

Third order observerSuper twisting observer

Figure : Evolution of error w.r.t. time for the STC based on HOSMO, STOF and STOunder noisy measurement




0 2 4 6 8 10 12 14 16 18 20−14

−12

−10

−8

−6

−4

−2

0

2

4

6

Time (sec)

Con

trol

5 5.5 6−2

0

2

4

STC−HOSMOSTC−STO

Figure : Evolution of control STC based on HOSMO and STOF w.r.t. time under noisymeasurement




0 2 4 6 8 10 12 14 16 18 20−15

−10

−5

0

5

10

Time (sec)

Con

trol

STC−STOF

Figure : Evolution of control STC based on STO w.r.t. time under noisy measurement



Conclusion

It is shown in the paper that, if one wants to implement absolutelycontinuous STC signal for the perturbed double integrator, the derivativeof the chosen manifold must be Lipschitz in the time.

Therefore, we have the need of second order observer/differentiators inthis case.

The same is also true for the higher order perturbed chain of integrators,when we want to synthesize absolutely continuous STC signal under theoutput information.

Numerical simulations are also presented to support the effectiveness ofthe proposed methodology.



Thank You!


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