Higher Order Sliding Mode Control

1
Higher Order Sliding Mode Control
Department of Engineering

• M. Khalid Khan
• Control Instrumentation group

2
References

• Levant, A. Sliding order and sliding accuracy
  in
• sliding mode control, Int. J. Control,
  1993,58(6)
• pp.1247-1263.
• 2. Bartolini et al. Output tracking control of
  uncertain
• nonlinear second order systems,
  Automatica, 1997,
• 33(12) pp.2203-2212.
• H. Sira-ranirez, On the sliding mode control of
• nonlinear systems, Syst.Contr.Lett.1992(19)
  pp.303-312
• 4. M.K. Khan et al. Robust speed control of an
• automotive engine using second order
  sliding modes,
• In proc. of ECC2001.

3
Review Sliding Mode Control
Consider a NL system
Design consists of two steps

• Selection of sliding surface

• Making sliding surface attractive

4
Robustness
Chattering
5
Pros and cons

• Order reduction

• Full state availability

• Robust to matched uncertainties

• Chattering at actuator

• Sliding error O(t)

• Simple to implement

6
Isnt it restrictive?
Sliding variable must have relative degree one
w.r.t. control.
7
Higher Order Sliding Modes
Consider a NL system
Sliding surface

• rth-order sliding set -

• rth-order sliding mode- motion in rth-order
  sliding set. Sliding variable (s) has relative
  degree r

8
So traditional sliding mode control is now 1st
order sliding mode control!
But What about reachability condition?
There is no generalised higher order
reachability condition available
9
1-sliding vs 2-sliding
Sliding error O(t)
Sliding error O(t2)
10
Sliding variable dynamics
Selected sliding variable, s, will have

• relative degree, p ? 2

• relative degree, p 1

• 1-sliding design is
• possible.

• r-sliding (r ? p) is the
• suitable choice.

• 2-sliding design is done
• to avoid chattering.

11
2-sliding algorithms examples

• Consider system represented in sliding variable
  as

Finite time converging 2-sliding twisting
algorithm
Sliding set
12
Pendulum
The model
Sliding variable
Sliding variable dynamics
Twisting Controller coefficients a 0.1, VM 7
13
Simulation
14
Examples continue

• Consider a system of the type

Finite time 2-sliding super-twisting algorithm
Sliding set
15
Review 2-sliding algorithms

• Twisting algorithm forces sliding variable (s)
  of relative degree 2 in to the 2-sliding set but
  uses

• Super Twisting algorithm do not uses but
  sliding variable (s) has relative degree only one.

16
Is it possible to stabilise sliding surface with
relative degree 2 in to 2-sliding set using only
s, not its derivative?
Question
Answer yes!

1. by designing observer

2. using modified super-twisting algorithm.
17
Modified super-twisting algorithm
System type
Where ?, u0 , k and W are positive design
constants

1. Sinusoidal oscillations for ? u0
2. Unstable for ?lt u0
3. Stable for ?gt u0

18
Phase plot
Sufficient conditions for stability
19
Application Anti-lock Brake System (ABS)
ABS model
Can be written as
20
Simulation Results
Controller coefficients
21
Results continued
22
Conclusions

• The restriction over choice of sliding variable
  can be relaxed by HOSM.

• HOSM can be used to avoid chattering

• A new 2-sliding algorithm which uses only
  sliding variable s (not its derivative) has been
  presented together with sufficient conditions
  for stability.

• The algorithm has been applied to ABS system and
  simulation results presented

23
Future Work

• The algo can be extended for MIMO systems.

• Possibility of selecting control dependent
• sliding surfaces is to be investigated.

• Stability results are local, need to find global
  
• results.

24
Thank You