universidad nacional autónoma de méxico

Analysis of Steady State Behavior Analysis of Steady State Behavior of Second Order Sliding Mode of Second Order Sliding Mode

AlgorithmAlgorithm

I. Boiko, L. Fridman, R. IriarteI. Boiko, L. Fridman, R. Iriarte

Universidad Nacional Autónoma de México


Frequency Domain Analysis of Super Frequency Domain Analysis of Super Twisting Algorithm (STA)Twisting Algorithm (STA)

To showTo show In the presence of an actuactor the transient process may In the presence of an actuactor the transient process may

converges to a periodic motion.converges to a periodic motion.

To analyze parameters of the periodic solution.To analyze parameters of the periodic solution. To compare the periodic solution of system driven by STA and first To compare the periodic solution of system driven by STA and first

order SM controllers.order SM controllers.

AlsoAlso


Higher Order Sliding Mode Higher Order Sliding Mode AlgorithmsAlgorithms

Twisting Twisting IEEE TAC June 2004IEEE TAC June 2004

Super Twisting STASuper Twisting STA

Twisting SupertwistingTwisting Supertwisting

Finite time Finite time convergenceconvergence

Plants with relative Plants with relative degree degree two two

Relay control lawRelay control law

Finite time Finite time convergenceconvergence

Plants with relative Plants with relative degree degree one one

Continuous control lawContinuous control law


Caractheristics of TA and STA


-

+ u yex

STAController

Plant plusactuator


Super Twisting Algorithm StructureSuper Twisting Algorithm Structure

)(

)(

)(

)()()(

0

2

1

21

ysigny

ysignu

ysignu

tututu

0

0

yif

yif

ρ = 0.5 (square root)


u2 plot of Super Twisting Algoritm


+

-

+

+

u1

u2

u1s

x e yPlant plusactuator


Methods of analysis

Poincaré maps Describing functions analysis . . .


Advantages/DisadvantagesAdvantages/Disadvantagesof methodsof methods

Poincaré mapsPoincaré maps

• Sufficient conditions satisfiedSufficient conditions satisfied• Complicated Complicated (requires the knowledge of the general solutions of the equations)(requires the knowledge of the general solutions of the equations)

A

D


Advantages/DisadvantagesAdvantages/Disadvantagesof methodsof methods

Describing function analysisDescribing function analysis• Easy to useEasy to use• Necessary conditions satified onlyNecessary conditions satified only• Approximated method Approximated method (low pass filtering hypothesis is nedded) (low pass filtering hypothesis is nedded)

• Works with one nonlinearity Works with one nonlinearity (modification is done)(modification is done)

A

D

DA

RR


DF of the super twisting algorithm DF of the super twisting algorithm

yyy

AsANNAN

1128.1

14),( 21

)(),(

1

jW

AN y

Harmonic balance equation


22

2

2

13092.11

11329.18986.0

),(

1

y

y

y

A

jA

AN

1;8.0;6.0 22

4321


Plots of -1/N(Ay,); 1 > 2 > 3 > 4


ExampleExample

21

212

21

xxy

uxxx

xx

a

uuu aa

01.0

1

1

101.0

1)(

2

ss

s

ssW


Negative reciprocal of DF –N-1(Ay) and the Nyquist plot W(j)


Negative reciprocal of DF –N-1(Ay) and the Nyquist plot W(j) (zoomed)


)(1

3092.11

11329.18986.0

22

2

2

jW

A

jA

y

y


)(Im

141

jWAy

16.66 41033.2 yA

0)(Re

1128.1

)(Im

14)(

2

11

jWjWF

0


ConclusionsConclusions It was shown that for a plant plus actuactor with relative degree It was shown that for a plant plus actuactor with relative degree

more than one a periodic motion may occur in the systems with the more than one a periodic motion may occur in the systems with the STA.STA.

An algorithm to obtain the parameters of this motion was given.An algorithm to obtain the parameters of this motion was given. The comparison between periodic solution parameters for the SAME The comparison between periodic solution parameters for the SAME

plants and SAME actuator with UNIT control amplitude for the plants and SAME actuator with UNIT control amplitude for the systems driven by first order sliding modes and STA was done.systems driven by first order sliding modes and STA was done.


Future trendsFuture trends

Universal chattering test.Universal chattering test. Frequency shapping.Frequency shapping. Robustness properties of systems with actuators Robustness properties of systems with actuators

driven by STA algorithms.driven by STA algorithms.

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