1. Discrete-time Sliding Mode Control
S. Janardhanan

2. Rethink: Quasi –Sliding mode
Advantage : Is in discrete-time, more practical
Disadvantage :
Is Quasi. Not Exactly Sliding Mode
There is always chattering. Even in theory.

3. Why the chattering ?
QSMC is derived from ‘discretizing’ continuous SMC logic.
The signum term is required in CSMC

4. Chattering .. Why ? In discrete-time system Result : Always chattering
The sign(s) changes abruptly near s=0
Control cannot be changed at any time
Result : Always chattering

5. Fresh Approach
Let us again see the aim of sliding mode control. But, now in discrete-time systems
Aim : To get the system to the sliding surface and maintain the state on the surface
To get to the surface : s(k+1)=0 (with s(k)  0)
To maintain : Again s(k+1)=0

6. Discrete-time sliding mode
So, apparently the reaching law is simply
s(k+1)=0
For this the control would be

7. Problem The control action Problem Solution
Brings the system to sliding mode in one step
Keeps RP on sliding surface thereafter.
Problem
Probability of too much control in the ‘one step’
Solution
GO SLOW
But, be sure you are going forward

8. Mathematically speaking …
Let the maximum applicable control be u0
Use the control as

9. To go in right direction
The control u1 tries to make s(k+1)=0
But, it may require too much ‘effort’.
So, with limited resources we can apply a maximum of only u0.
Now, to be sure this is ‘enough’. Ie., we are moving forward, we get the condition

10. The condition Thus, the condition for this logic to work is
See : G. Bartolini, A. Ferrara and V. Utkin, “Adaptive sliding mode control in discrete-time systems”, Automatica, Vol. 31, No. 5, pp , May 1995.

11. Extension to Uncertain Systems
The same algorithm can be extended to systems with matched uncertainty.
In this case, the condition would become

12. Another Algorithm
Another approach to the DSMC problem is to design a control algorithm that would inherently ‘go slow’.
Instead of trying to reach the surface in one step. Try to reach it in say k* steps.

13. The Problem Statement Consider the design of DSMC for system
Bartoszewicz proposed the reaching law
cTx=s(k) is the desired sliding surface
See : A. Bartoszewicz, “Discrete-time quasi-sliding mode control strategies”, IEEE Trans. Ind. Electron., Vo. 45, No. 1, pp , 1998

14. Bartoszewicz QSMC … sd is an a priori known function such that
 sd(k) is always of the same sign

15. Feasible Sd(k)
A feasible function for sd(k) that satisfies the above said conditions can be given as

16. Bartoszewicz Control Law
The control input is of the form

17. Proof of Convergence
The Bartoszewicz control law satisfies the reaching condition
For hence
and to s(k)=0 in deterministic systems atleast in theory.

18. Proposed Algorithm
Combines Bartoszewicz’s approach and FOS to get multirate output feedback based QSMC control strategy that does not have switching function and hence no chatter.
Reaching Law Modified

19. Proposed Control Law
The Control is of the form

20. Proof of Convergence
The proposed control law satisfies the reaching condition
For Hence
and to s(k)=0 in deterministic systems atleast in theory.