Analysis of Steady State Behavior of Second Order Sliding Mode Algorithm I. Boiko, L. Fridman, R. Iriarte Universidad Nacional Autónoma de México

Frequency Domain Analysis of Super Twisting Algorithm (STA) To show In the presence of an actuactor the transient process may converges to a periodic motion. In the presence of an actuactor the transient process may 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 order SM controllers. To compare the periodic solution of system driven by STA and first order SM controllers. Also

Universidad Nacional Autónoma de México Higher Order Sliding Mode Algorithms Twisting IEEE TAC June 2004 Twisting IEEE TAC June 2004 Super Twisting STA Super Twisting STA

Twisting Supertwisting Finite time convergence Finite time convergence Plants with relative degree two Plants with relative degree two Relay control law Relay control law Finite time convergence Finite time convergence Plants with relative degree one Plants with relative degree one Continuous control law Continuous control law Universidad Nacional Autónoma de México Caractheristics of TA and STA

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Super Twisting Algorithm Structure ρ = 0.5 (square root)

Universidad Nacional Autónoma de México u 2 plot of Super Twisting Algoritm

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Methods of analysis Poincaré maps Describing functions analysis...

Universidad Nacional Autónoma de México Advantages/Disadvantages of methods Poincaré maps Poincaré maps Sufficient conditions satisfiedSufficient conditions satisfied Complicated (requires the knowledge of the general solutions of the equations)Complicated (requires the knowledge of the general solutions of the equations) A D

Universidad Nacional Autónoma de México Advantages/Disadvantages of methods Describing function analysis Describing function analysis Easy to useEasy to use Necessary conditions satified onlyNecessary conditions satified only Approximated method (low pass filtering hypothesis is nedded)Approximated method (low pass filtering hypothesis is nedded) Works with one nonlinearity (modification is done)Works with one nonlinearity (modification is done) A D DA RR

Universidad Nacional Autónoma de México DF of the super twisting algorithm Harmonic balance equation

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Plots of -1/N(A y,  );  1 >  2 >  3 >  4

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

Universidad Nacional Autónoma de México Negative reciprocal of DF –N -1 (A y ) and the Nyquist plot W(j  ) (zoomed)

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Conclusions 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 STA. 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 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 plants and SAME actuator with UNIT control amplitude for the systems driven by first order sliding modes and STA was done. The comparison between periodic solution parameters for the SAME plants and SAME actuator with UNIT control amplitude for the systems driven by first order sliding modes and STA was done.

Universidad Nacional Autónoma de México Future trends Universal chattering test. Universal chattering test. Frequency shapping. Frequency shapping. Robustness properties of systems with actuators driven by STA algorithms. Robustness properties of systems with actuators driven by STA algorithms.