1. Continuous-time Sliding Mode Control
S. Janardhanan

2. Refresher Going Over the definition once again :
Sliding mode is “Motion of the system trajectory along a ‘chosen’ line/plane/surface of the state space”.

3. What is the advantage?
Consider a n-th order system represented in the phase variable form
Also consider the sliding surface defined as

4. Advantage …
Thus entire dynamics of the system is governed by the sliding line/surface parameters only
In sliding mode, dynamics independent of system parameters (a1,a2,…).
ROBUST

5. Required Properties
For sliding mode to be of any use, it should have the following properties
System stability confined to sliding surface (unstable sliding mode is NOT sliding mode at all)
Sliding mode should not take ‘forever’ to start

6. Stable Surface Consider the system
If the sliding function is designed as
then confined to this surface ( ), the dynamics of can be written as

7. The Surface …
If K is so designed that has eigenvalues on LHP only , then the dynamics of is stable.
Since , the dynamics of is also stable.
Hence, if the sliding surface is ‘designed’ as
, the system dynamics confined to
s=0 is stable. (Requirement 1)
Note : Strictly speaking, it is not necessary for s to be a linear function of x

8. Convergence to s=0
The second requirement is that sliding mode should start at a finite time.
Split the requirement into further bits
Sliding mode SHOULD start.
It should do so in finite time.

9. Run towards the surface
To be sure that sliding mode starts at some time t>0, irrespective of the initial state x(0), we should be sure that the state trajectory is always moving towards s=0, whenever s is not zero.
Mathematics …
This is called the ‘reachability condition’

10. A figure to help out …
s>0
s<0
s=0

11. Insufficient Consider the case, This gives the solution of
is not enough (Violates Requirement 2)

12. -reachability
With only , s slows down too much when close to zero to have finite time convergence
Stronger condition is needed for finite time convergence.
Defined as -reachability condition
s has a minimum rate of convergence

13. Discontinuity
Observe
So, at , is discontinuous.  -

14. Discontinuous Dynamics
Thus, for s>0, the system dynamics are
and for s<0
Thus, at s=0, the dynamics is not well defined.
The dynamics along the sliding surface is determined using continuation method

15. Continuation Method
Using continuation method as proposed by Filippov*, it is said that when s=0, the state trajectory moves in a direction in between and
*A. F. Filppov, “Differential Equations with discontinuous righthand sides”Kluwer Academic Publishers,The Netherlands, 1988

16. Diagrammatically Speaking …

17. The reaching law approach
In reaching law approach, the dynamics of the sliding function is directly expressed. It can have the general structure

18. Few Examples Constant rate reaching law Constant+Proportional rate
Power-rate reaching law

19. The Control Signal Now, consider the condition Thus, Or, control is
And the system dynamics is governed by

20. The Chattering Problem
When, s is very close to zero, the control signal switches between two structures.
Theoretically, the switching causes zero magnitude oscillations with infinite frequency in x.
Practically, actuators cannot switch at infinite frequency. So we have high frequency oscillations of non-zero magnitude.
This undesirable phenomenon is called chattering.

21. The picture
Ideal Sliding Mode
Practical – With Chattering

22. Why is chattering undesirable?
The ‘high frequency’ of chattering actuates unmodeled high frequency dynamics of the system. Controller performance deteriorates.
More seriously, high frequency oscillations can cause mechanical wear in the system.

23. Chattering avoidance/reduction
The chattering problem is because signum function is used in control.
Control changes very abruptly near s=0.
Actuator tries to cope up leading to ‘maximum-possible-frequency’ oscillations.
Solution :
Replace signum term in control by ‘smoother’ choices’

24. Chattering Avoidance…
Some choices of smooth functions
Saturation function
Hyperbolic tangent

25. Disadvantage of ‘smoothing’
If saturation or tanh is used, then we can observe that near s=0
Where represents the saturation or tanh function.
The limit in both cases is zero.
So, technically the sliding mode is lost