1. Terminal Sliding Mode
S. Janardhanan

2. Terminal Sliding Mode
Sliding mode control concept which tries to make x=0 in finite time (not just s=0)
In continuous time it is accomplished by using a nonlinear sliding surface of form (given here for 2nd order)

3. Discretization
Let us assume the system is discretized at a sampling interval , and the system state is moving along the sliding surface (somehow)
If there is a k* such that x=0 after k*, then

4. The possibilities

5. The Improbability
There are only a finite number of points from which the states can go to origin.
It is highly unlikely that the system would cross these points.
Hence, it can be assumed that due to discretization, the finite-time part of terminal sliding mode is no longer true.

6. Stability … Atleast Analyzing the discrete system
around origin, it is found that
It is required that the |derivative|<1, so system is unstable around origin. Diverges from origin

7. Periodicity
Analysis shows that if f(y*)=-y*, then {y*,-y*} form a limit set. (Not much can be said in this case)
Further, this is the only possible 2 period.
If no such y* exists then there are no periodic orbits (Sarkovskii Theorem)
Since System is not stable around origin (the only stationary point), the system would diverge. (while still on the sliding surface)

8. Analysis Discretization of Continuous Terminal SM
Almost never leads to finite time convergence
Certainly leads to an instability around origin
May lead to periodic/ chaotic behaviour
Failing which system is unstable
Discrete-terminal sliding mode should be handled differently from continuous time terminal sliding mode

9. DTSM Aim : Given a discrete-time system
The terminal sliding surface is such that the system dynamics confined to the surface (brought about by control) has the property
Nilpotent function

10. Algorithm for DTSM
Using appropriate transformation , transform the system into brunowsky canonical form
Sliding surface is
Design appropriate control to achieve DSM
Note : Control Should not be based on CSMC idea ( Bartoszewicz, Bartolini-Utkin)
Can be converted to MROF also.

11. Example
Consider the system
In a transformed co-ordinate frame

12. Results