Analysis of Steady State Behavior of Second Order Sliding Mode Algorithm I. Boiko, L. Fridman, R....
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Transcript of Analysis of Steady State Behavior of Second Order Sliding Mode Algorithm I. Boiko, L. Fridman, R....
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
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
Universidad Nacional Autónoma de México
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
Universidad Nacional Autónoma de México
Caractheristics of TA and STA
Universidad Nacional Autónoma de México
Super Twisting Algorithm StructureSuper Twisting Algorithm Structure
)(
)(
)(
)()()(
0
2
1
21
ysigny
ysignu
ysignu
tututu
0
0
yif
yif
ρ = 0.5 (square root)
Universidad Nacional Autónoma de México
Methods of analysis
Poincaré maps Describing functions analysis . . .
Universidad Nacional Autónoma de México
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
Universidad Nacional Autónoma de México
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
Universidad Nacional Autónoma de México
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
Universidad Nacional Autónoma de México
22
2
2
13092.11
11329.18986.0
),(
1
y
y
y
A
jA
AN
1;8.0;6.0 22
4321
Universidad Nacional Autónoma de México
ExampleExample
21
212
21
xxy
uxxx
xx
a
uuu aa
01.0
1
1
101.0
1)(
2
ss
s
ssW
Universidad Nacional Autónoma de México
Negative reciprocal of DF –N-1(Ay) and the Nyquist plot W(j)
Universidad Nacional Autónoma de México
Negative reciprocal of DF –N-1(Ay) and the Nyquist plot W(j) (zoomed)
Universidad Nacional Autónoma de México
)(Im
141
jWAy
16.66 41033.2 yA
0)(Re
1128.1
)(Im
14)(
2
11
jWjWF
0
Universidad Nacional Autónoma de México
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.
Universidad Nacional Autónoma de México
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.