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

Higher Order Super-Twisting Algorithm Shyam Kamal 1 Asif Chalanga 2 Prof.J.A.Moreno 3 Prof.L.Fridman 4 and Prof.B.Bandyopadhyay 5 125 Indian Institute of Technology Bombay, Mumbai-India 3 Instituto de Ingenier´ ıa Universidad Nacional Aut´ onoma de M´ exico (UNAM) 4 Facultad de Ingenier´ ıa Universidad Nacional Aut´ onoma 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 Motivation 1 Higher Order STA 2 Convergence Condition for the 3-STA 3 Controller Design based on Generalized STA 4 Simulation Results 5 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 + ρ 1 ¨ (2.1) where ρ 1 is a non vanishing Lipschitz disturbance and | ˙ ρ 1 | < ρ 0 . Algorithm Control Signal Information Stability Chattering First SMC Discontinuous σ and ˙ σ Asymptotic Yes STC Continuous σ and ˙ Asymptotic No σ 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: σ, ..., σ ( r ) where σ represents the finite time convergence for the set σ, ˙ output and r is the relative degree of the system with respect to output σ, ..., σ ( r − 1 ) which generates the absolutely using information of σ, ˙ 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 | p sgn ( z ) , therefor ⌊ z ⌉ 2 = | z | 2 sgn ( z ) � = z 2 . If p is an odd number, this notation does not change the meaning of the equation, i.e. ⌊ z ⌉ p = z p . Therefore ⌊ z ⌉ 0 = sgn ( z ) , ⌊ z ⌉ 0 z p = | z | p , ⌊ z ⌉ 0 | z | p = ⌊ z ⌉ p ⌊ z ⌉ p ⌊ z ⌉ q = | z | p sgn ( z ) | z | q sgn ( z ) = | z | p + q (2.2) Also, σ = x 1 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 : R n → R n (or a differential inclusion) is called homogeneous of degree δ ∈ R with the dilatation d κ : ( x 1 , x 2 , · · · , x n ) �→ ( κ ̺ 1 x 1 , κ ̺ 2 x 2 , · · · , κ ̺ n x n ) , where ̺ = ( ̺ 1 , ̺ 2 , · · · , ̺ n ) are some positive numbers (called the weights), if for any κ > 0 the following identity f ( x ) = κ − δ d − 1 κ f ( d κ x ) holds. Definition A scalar function V : R n → R is called homogeneous of degree δ ∈ R with the dilatation d κ if for any κ > 0 the following identity V ( x ) = κ − δ V ( d κ x ) 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 x 1 , x 2 , · · · , x n where σ = x 1 is the output. 2-STA is given as follows 1 x 1 = − k 1 | x 1 | 2 sign ( x 1 ) + x 2 ˙ x 2 = − k 2 sign ( x 1 ) + ρ ˙ (3.1) where x 1 , x 2 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 x 1 = x 2 ˙ x 2 = − k 1 | φ 1 | 1 / 2 sign ( φ 1 ) + x 3 ˙ x 3 = − k 3 sign ( φ 1 ) + ρ ˙ (3.2) where φ 1 = x 2 + k 2 | x 1 | 2 / 3 sign ( x 1 ) , x 1 , x 2 , x 3 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 x 1 = x 2 ˙ x 2 = x 3 ˙ x 3 = − k 1 | φ 2 | 1 / 2 sign ( φ 2 ) + x 4 ˙ x 4 = − k 4 sign ( φ 2 ) + ρ ˙ (3.3) where | x 1 | 3 + | x 2 | 4 � 1 6 sign � � 3 � φ 2 = x 3 + k 3 x 2 + k 2 | x 1 | 4 sign ( x 1 ) (3.4) and x 1 , x 2 , x 3 , x 4 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 x 1 = x 2 ˙ x 2 = x 3 ˙ x 3 = x 4 ˙ x 4 = − k 1 | φ 3 | 1 / 2 sign ( φ 3 ) + x 5 ˙ x 5 = − k 5 sign ( φ 3 ) + ρ ˙ (3.5) where | x 1 | 12 + | x 2 | 15 + | x 3 | 20 � 1 �� � 30 sign ( l 1 ) φ 3 = x 4 + k 4 and | x 1 | 12 + | x 2 | 15 � 1 20 sign � � 4 � l 1 = x 3 + k 3 x 2 + k 2 | x 1 | 5 sign ( x 1 ) and x 1 , x 2 , x 3 , x 4 , x 5 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 x 1 = x 2 ˙ x 2 = x 3 ˙ . . . x n − 1 = − k 1 | φ n − 2 | 1 / 2 sign ( φ n − 2 ) + x n ˙ x n = − k n sign ( φ n − 2 ) + ρ ˙ (3.6) where φ n − 2 we define later part of the paper, x 1 , x 2 , · · · , x n 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 φ n − 2 is given as follows:- r R 1 , r − 1 = | x 1 | r + 1 where r represents the relative degree of algorithm with respect to x 1 . � q i � | x 1 | r 1 + | x 2 | r 2 + · · · + | x i − 2 | r i − 2 � R i , r − 1 = � where i = 2 , 3 , · · · , ( r − 1 ) , r 1 , r 2 , · · · , r i − 2 and q i is designed parameter based on the homogeneity weight of the x i + 1 . S 0 , r − 1 = x 1 S 1 , r − 1 = x 2 + k 2 R 1 , r − 1 sign ( x 1 ) S i , r − 1 = x i + 1 + k i + 1 R i , r − 1 sign ( S i − 1 , r − 1 ) where i = 2 , 3 , · · · , ( r − 1 ) Finally φ n − 2 = s r − 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 x 1 ( 0 ) = − 1, x 2 ( 0 ) = − 3 and x 3 ( 0 ) = 1 gains k 1 = 6 , k 2 = 4 and k 3 = 4 4-STA initial conditions x 1 ( 0 ) = − 1 , x 2 ( 0 ) = 3, x 3 ( 0 ) = 1 and x 4 ( 0 ) = 1 gains k 1 = 4 , k 2 = 2 , k 3 = 1 and k 4 = 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 3.5 x 1 3 x 2 −3 x 10 2.5 2 x 3 2 0 1.5 States −2 1 0.5 −4 3 3.05 3.1 0 −0.5 −1 −1.5 0 1 2 3 4 5 6 7 8 9 10 Time (sec) 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 4 −3 x 10 x 1 5 3 x 2 x 3 0 2 x 4 1 −5 6 6.05 6.1 States 0 −1 −2 −3 −4 0 1 2 3 4 5 6 7 8 9 10 Time (sec) Figure : Evolution of States of 4-STA w.r.t. time Prof.L.Fridman — Higher Order Super-Twisting Algorithm 15