1. Higher Order Sliding Mode Control
Department of Engineering
Higher Order Sliding Mode Control
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
2. Bartolini et al.: ‘Output tracking control of uncertain
nonlinear second order systems’, Automatica, 1997,
33(12) pp
H. Sira-ranirez, ‘On the sliding mode control of
nonlinear systems’, Syst.Contr.Lett.1992(19) pp
4. M.K. Khan et al.: ‘Robust speed control of an
automotive engine using second order sliding modes’,
In proc. of ECC’2001.

3. Review: Sliding Mode Control
Consider a NL system
Design consists of two steps
Selection of sliding surface
Making sliding surface attractive

4. High frequency
switching of control
Robustness
Chattering

5. Pros and cons Order reduction Full state availability
Robust to matched uncertainties
Chattering at actuator
Sliding error = O(τ)
Simple to implement

6. Sliding variable must have relative degree one w.r.t. control.
Isn’t 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. What about reachability condition?
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 s ds 1-sliding τ s ds 2-sliding τ τ2
Sliding error = O(τ)
Sliding error = O(τ2)

10. Sliding variable dynamics
Selected sliding variable, s, will have
relative degree, p= 1
relative degree, p  2
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
 < 1
Sliding set:

12. Pendulum The model: Sliding variable: Sliding variable dynamics:
Twisting Controller coefficients: α = 0.1, VM = 7

13. Simulation

14. Examples continue … Finite time 2-sliding super-twisting algorithm
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. Question:
Is it possible to stabilise sliding surface with relative degree 2 in to 2-sliding set using only s, not its derivative?
Answer: yes!
by designing observer
2. using modified super-twisting algorithm.

17. Modified super-twisting algorithm
System type:
Where λ, u0 , k and W are
positive design constants
Sinusoidal oscillations for = u0
Unstable for < u0
Stable for > 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 HOSM can be used to avoid chattering
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