UNAM Dr Leonid Fridman NEW TRENDS IN SLIDING

UNAM Dr. Leonid Fridman NEW TRENDS IN SLIDING CONTROL MODE L. Fridman Universidad Nacional Autónoma de México División de Posgrado, Facultad de en Ingeniería Edificio ‘A’, Ciudad Universitaria C. P. 70 -256, México D. F. lfridman@verona. fi-p. unam. mx 14 MAYO DE 2004 1

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control Given a system f(x, t) u x 0 2

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control 3

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control Motivations Given a system f(x, t) u x 0 Problem formulation: Design control function u to provide asymptotic stability in presence of bounded uncertain term uncertainties and external disturbances. , that contains model 4

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control Basics of Sliding Mode Control x(0) Desired compensated error dynamics (sliding surface): • The purpose of the Sliding Mode Controller (SMC) is to drive a system's trajectory to a user-chosen surface, named • sliding surface, and to maintain the plant's state trajectory on this surface thereafter. The motion of the system on the sliding surface is named • sliding mode. The equation of the sliding surface must be selected such that the system will exhibit the desired (given) behavior in the sliding mode that will not depend on unwanted parameters (plant uncertainties and external disturbances). 5

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control 1. Sliding surface design x 2 x(0) reaching phase x 1 sliding phase 2. SMC design Sliding mode existence condition Equivalent control 6

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control WHY Sliding mode control? WHEN Sliding mode control? More than Robustness(insensitivity!!!!) to disturbances and uncertainties Control plants that operate in presence of unmodeled dynamics, parametric uncertainties and severe external disturbances and noise: aerospace vehicles, robots, etc. 7

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control Numerical example: Features: 1. Invariance to disturbance 2. High frequency switching 8

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control Continuous and smooth sliding mode control 1. Continuous approximation via saturation function sign s sat(s/e) 1 s e s -1 Numerical example: 9

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control Simulations Features: 1. Invariance to disturbance is lost to some extend 2. Continuous asymptotic control 10

UNAM Dr. Leonid Fridman Second order Sliding mode control 1. Twisting Algorithm Features: 1. Convergence in finite time for and 2. Robustness INSENSITIVITY!!!! 3. Convergence 11

UNAM Dr. Leonid Fridman New trends in sliding mode control Chattering avoidance whit Twisting Algorithm (continuous control) Features: 1. Convergence in finite time for 2. Robustness 3. Convergence and 12

UNAM Dr. Leonid Fridman Continuous Second order Sliding mode control 2. Super Twisting Algorithm Features: 1. Invariance to disturbance 2. Continuous control 13

UNAM Dr. Leonid Fridman Sliding mode observers/differentiators 3. Second Order ROBUST TO NOISE Sliding Mode Observer 14

UNAM Dr. Leonid Fridman Higher order Sliding mode control 4. High order slides modes controllers of arbitrary order Features: 1. Convergence in finite time for 2. Robustness 3. Convergence 4. r-Smooth control 15

UNAM Dr. Leonid Fridman Higher order Sliding mode control High order slides modes controllers of arbitrary order 16

UNAM Dr. Leonid Fridman CHATTERING ANALISYS Frequency analysis 1. Frecuency Methods modifications. Boiko, Castellanos LF IEEE TAC 2004 2. Universal Chattering Test. Boiko, Iriarte, Pisano, Usai, LF 3. Chattering Shaping. Boiko, Iriarte, Pisano, Usac, LF 17

UNAM Dr. Leonid Fridman CHATTERING ANALISYS Singularly Perturbed Approach (s, x) S PLANT ACTUA TOR S Integral Manifold LF IEEE TAC 2001 Averaging LF IEEE TAC 2002 Second Order Sliding Mode Controllers 18

UNAM Dr. Leonid Fridman UNDERACTUATED SYSTEMS SMC + H_{∞} Fernando Castaños & LF SMC + Optimal multimodel Poznyak, Bejarano & LF 19

UNAM Dr. Leonid Fridman OBSERVATION & IDENTIFICATION VIA 2 -SMC §Uncertainty identification §Parameter identification §Identification of the time variant parameters J. Dávila & LF 20

UNAM Dr. Leonid Fridman RELAY DELAYED CONTROL Countable set of periodic solutions=sliding modes Shustin, E. Fridman LF 93 Set of Steady modes 21

UNAM Dr. Leonid Fridman CONTROL OF OSCILLATIONS AMPLITUDE Only Is accessible FFS 93 -----s(t-1) is accessible Strygin, Polyakov, LF IJC 03, IJRNC 04 22

UNAM Dr. Leonid Fridman APPLICATIONS §Investigation and implementation of 2 -SMC §Shaping of Chattering parameters 23