1. Quasi-Sliding Mode Control of Systems with Unmatched Uncertainty
S. Janardhanan

2. Overview
We have already seen ‘matched’ uncertainty. In this lecture we see
Unmatched Uncertainty
uncertainty
System
Net  0

3. State based DSMC for Unmatched Uncertainty
Misawa et al had proposed a single sliding surface based SFB-DSMC control algorithm for unmatched uncertainty.
See: C. Y. Tang, E. Misawa, “Discrete variable structure control for linear multivariable systems : a state feedback approach”, Proc. American Control Conference, pp , June 1998

4. Misawa’s Control The state feedback control by Misawa is of form
Which is based on a single sliding surface.

5. Where …

6. Why one sliding surface only ?
Argument : s1  0 means a close to asymptotically stable dynamics. s2  0 would also mean the same
Counter – Simultaneously s1  0, s2  0 is much tighter criterion, and there is control (available) to achieve this.
AND – Output is available …

7. The Logic
X(0)
X(k*)

8. MROF-DSMC in Unmatched Uncertainty
Basic idea :
System states and sliding function are computed assuming average disturbance
And control is computed as
where the gains satisfy certain conditions

9. MROF-DSMC …
The basic method to prove this point is yo show that the value of the sliding function s(k) decreases monotonically whenever the system state is outside a a priori known quasi-sliding mode band.
For a detailed discussion see : S. Janardhanan and B. Bandyopadhyay, “Discrete-time Sliding Mode Control”, Springer-Verlag, 2005

10. The Control Law

11. Illustrative Example – Cart System
Model is 6th order, 2 input , 3 output system (possibility of 2 sliding surfaces)

12. Results – Sliding Function - SFB

13. Results- Sliding Function - MROF-DSMC

14. Results – Control Input - SFB

15. Results – Control Input – MROF-DSMC

16. Advantage - Disadvantage
As in matched – expected uncertainty band is higher than SFB . (But, may not be so in all cases during implementation)
Uses all available input freedom for controller design.
Uses bias in disturbance to advantage (d0 in control)