NEW TRENDS IN SLIDING CONTROL MODE 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

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

UNAM Dr. Leonid Fridman Intuitive theory of Sliding mode control

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

UNAM Dr. Leonid Fridman Basics of Sliding Mode Control 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).

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

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

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

Continuous and smooth sliding mode control 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:

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

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

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

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

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

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

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

UNAM Dr. Leonid Fridman CHATTERING ANALISYS Frecuency Methods modifications. Boiko, Castellanos LF IEEE TAC2004 Universal Chattering Test. Boiko, Iriarte, Pisano, Usai, LF Chattering Shaping. Boiko, Iriarte, Pisano, Usac, LF Frequency analysis

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

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

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

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

CONTROL OF OSCILLATIONS AMPLITUDE 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

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