1. Some Aspects of Continuous-time Sliding Mode Control
S. Janardhanan

2. Overview
Robustness
Multivariable Sliding Mode
‘Almost’ Sliding Mode

3. Hence, sliding mode is robust.
Robustness of CSMC
In Lecture 2,
When in sliding mode, entire system dynamics is governed by sliding surface parameters and not original system parameters.
Hence, sliding mode is robust.

4. Disturbance Consider the system with disturbance
Disturbance comes through input channel
How does sliding mode behave in such a situation.

5. Disturbance Rejection
The control law is designed so as to bring the system to the sliding surface.
Let us see dynamics confined to the sliding surface
Thus,
Therefore,
And
Again, dynamics independent of disturbance.
Hence disturbance rejection.

6. What if more than one input ?
If system has more than one input, then the system can be transformed to the form
With having more than one elements.
Thus, will also have multiple rows. Hence, the system can have more than one sliding surface

7. Approach to sliding surface
Sliding mode will start when all sliding functions are zero. I.e, intersection of all sliding surfaces.
Approach to the intersection
Direct to intersection (Eventual)
Surface by surface
In particular order (Fixed Order)
First approach (Free order)

8. Eventual Sliding Mode
In this type of sliding mode, the state trajectory moves to the intersection of the sliding surfaces through a connected subset in the state space.
It does not necessary stay on any one of the sliding surfaces on approaching it.

9. Eventual Sliding Mode

10. Fixed order Sliding Mode
In fixed order sliding mode, the state trajectory moves to one pre-specified sliding surface and staying on it moves to the intersection of the first surface with the next pre-specified sliding surface

11. Free order sliding mode
In free order sliding mode, the state trajectory remains on a sliding surface once the state approaches it. However, there is no particular order in which the surfaces are reached

12. Ordered Sliding Mode

13. Chattering Refreshed
A conventional sliding mode behaviour would have a sliding surface dynamics of the form
However, due to finite bandwidth of the actuator, the input cannot switch fast enough near the sliding surface
Chattering – Finite frequency, finite amplitude oscillations about the sliding surface

14. Almost Sliding Mode
To remedy chattering, the strict requirement of “movement on sliding surface” is relaxed.
We try to get ‘Almost’ – sliding mode (Quasi sliding mode)

15. Saturation function based Sliding Mode Control
Instead of
Inside the band |s|<, the reaching law is linear as
This is also called ‘boundary layer technique

16. The motion
S=
S=-
S=0

17. Disadvantage
‘Almost’ is NOT exact

18. Discrete Time Sliding Mode and Control
NEXT
Discrete Time Sliding Mode and Control