Higher Order Sliding Mode Control

1. Higher Order Sliding Mode Control Department of Engineering M. Khalid Khan Control & Instrumentation group

2. References • Levant, A.: ‘Sliding order and sliding accuracy in sliding mode control’, Int. J. Control, 1993,58(6) pp.1247-1263. 2. Bartolini et al.: ‘Output tracking control of uncertain nonlinear second order systems’, Automatica, 1997, 33(12) pp.2203-2212. • H. Sira-ranirez, ‘On the sliding mode control of nonlinear systems’, Syst.Contr.Lett.1992(19) pp.303-312 4. M.K. Khan et al.: ‘Robust speed control of an automotive engine using second order sliding modes’, In proc. of ECC’2001.

3. Review: Sliding Mode Control Consider a NL system Design consists of two steps • Selection of slidingsurface • Making sliding surface attractive

4. High frequency switching of control Robustness Chattering

5. Order reduction Full state availability Pros and cons • Robust to matched uncertainties • Chattering at actuator • Sliding error = O(τ) • Simple to implement

6. Isn’t it restrictive? Sliding variable must have relative degree one w.r.t. control.

7. Higher Order Sliding Modes Consider a NL system Sliding surface • rth-order sliding set: - • rth-order sliding mode:- motion in rth-order sliding set. Sliding variable (s) has relative degree r

8. So traditional sliding mode control is now 1st order sliding mode control! But What about reachability condition? There is no generalised higher order reachability condition available

9. ds ds τ τ τ2 s s 1-sliding 2-sliding 1-sliding vs 2-sliding Sliding error = O(τ) Sliding error = O(τ2)

10. Sliding variable dynamics Selected sliding variable, s, will have • relative degree, p 2 • relative degree, p= 1 • 1-sliding design is possible. • r-sliding (r  p) is the suitable choice. • 2-sliding design is done to avoid chattering.

11.  < 1 2-sliding algorithms: examples • Consider system represented in sliding variable as Finite time converging 2-sliding twisting algorithm Sliding set:

12. Pendulum The model: Sliding variable: Sliding variable dynamics: Twisting Controller coefficients: α = 0.1, VM = 7

13. Simulation

14. Examples continue … • Consider a system of the type Finite time 2-sliding super-twisting algorithm Sliding set:

15. Review: 2-sliding algorithms • Twisting algorithm forces sliding variable (s) of relative degree 2 in to the 2-sliding set but uses • Super Twisting algorithm do not uses but sliding variable (s) has relative degree only one.

16. Question: Is it possible to stabilise sliding surface with relative degree 2 in to 2-sliding set using only s, not its derivative? Answer: yes! • by designing observer 2.using modified super-twisting algorithm.

17. Modified super-twisting algorithm System type: Where λ, u0 , k and W are positive design constants • Sinusoidal oscillations for = u0 • Unstable for< u0 • Stable for > u0

18. Phase plot Sufficient conditions for stability

19. Application: Anti-lock Brake System (ABS) ABS model: Can be written as:

20. Simulation Results Controller coefficients:

21. Results continued …

22. Conclusions • The restriction over choice of sliding variable can be relaxed by HOSM. • HOSM can be used to avoid chattering • A new 2-sliding algorithm which uses only sliding variable s (not its derivative) has been presented together with sufficient conditions for stability. • The algorithm has been applied to ABS system and simulation results presented

23. Future Work • The algo can be extended for MIMO systems. • Possibility of selecting control dependent sliding surfaces is to be investigated. • Stability results are local, need to find global results.

24. Thank You