1. 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 , México D. F.
14 MAYO DE 2004

2. 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

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

4. 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

5. 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).

6. 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

7. 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.

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

9. 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:

10. 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

11. 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

12. 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

13. 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

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

15. 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

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

17. 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

18. 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

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

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

21. 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

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

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