UNAM

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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_at_verona.fi-p.unam.mx 14 MAYO DE 2004
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UNAM
Dr. Leonid Fridman
Intuitive theory of Sliding mode control

• Given a system

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UNAM
Dr. Leonid Fridman
Intuitive theory of Sliding mode control
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UNAM
Dr. Leonid Fridman
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
0
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UNAM
Dr. Leonid Fridman
Intuitive theory of Sliding mode control
Basics of Sliding Mode Control

• 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).

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UNAM
Dr. Leonid Fridman
Intuitive theory of Sliding mode control
x
1. Sliding surface design
2
reaching phase
x(0)
x
1
sliding phase
2. SMC design
Sliding mode existence condition
Equivalent control
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UNAM
Dr. Leonid Fridman
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.
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UNAM
Dr. Leonid Fridman
Intuitive theory of Sliding mode control
Numerical example
Features 1. Invariance to disturbance 2. High
frequency switching
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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
s
e
-1
Numerical example
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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
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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
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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
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UNAM
Dr. Leonid Fridman
Continuous Second order Sliding mode control
2. Super Twisting Algorithm
Features 1. Invariance to disturbance 2.
Continuous control
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UNAM
Dr. Leonid Fridman
Sliding mode observers/differentiators
3. Second Order ROBUST TO NOISE Sliding Mode
Observer
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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
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UNAM
Dr. Leonid Fridman
Higher order Sliding mode control
High order slides modes controllers of arbitrary
order
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UNAM
Dr. Leonid Fridman
CHATTERING ANALISYS

1. Frecuency Methods modifications. Boiko,
   Castellanos LF IEEE TAC2004
2. Universal Chattering Test. Boiko, Iriarte,
   Pisano, Usai, LF
3. Chattering Shaping. Boiko, Iriarte, Pisano, Usac,
   LF

Frequency analysis
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UNAM
Dr. Leonid Fridman
CHATTERING ANALISYS
Singularly Perturbed Approach
Integral Manifold Averaging
LF IEEE TAC 2001 LF IEEE TAC 2002
Second Order Sliding Mode Controllers
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UNAM
Dr. Leonid Fridman
UNDERACTUATED SYSTEMS
SMC H_8 Fernando Castaños LF SMC Optimal
multimodel Poznyak, Bejarano LF
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UNAM
Dr. Leonid Fridman
OBSERVATION IDENTIFICATION VIA 2 -SMC

• Uncertainty identification
• Parameter identification
• Identification of the time variant parameters
• J. Dávila LF

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UNAM
Dr. Leonid Fridman
RELAY DELAYED CONTROL
Countable set of periodic solutionssliding modes
Shustin, E. Fridman LF 93
Set of Steady modes
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UNAM
Dr. Leonid Fridman
CONTROL OF OSCILLATIONS AMPLITUDE
Only
Is accessible FFS 93------ s(t-1) is
accessible Strygin, Polyakov, LF IJC 03, IJRNC
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UNAM
Dr. Leonid Fridman
APPLICATIONS

• Investigation and implementation of 2-SMC
• Shaping of Chattering parameters