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Higher Order Sliding Mode Control

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1 Higher Order Sliding Mode Control
Department of Engineering Higher Order Sliding Mode Control 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 2. Bartolini et al.: ‘Output tracking control of uncertain nonlinear second order systems’, Automatica, 1997, 33(12) pp H. Sira-ranirez, ‘On the sliding mode control of nonlinear systems’, Syst.Contr.Lett.1992(19) pp 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 sliding surface Making sliding surface attractive

4 High frequency switching of control Robustness Chattering

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

6 Sliding variable must have relative degree one w.r.t. control.
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 What about reachability condition?
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 1-sliding vs 2-sliding s ds 1-sliding τ s ds 2-sliding τ τ2
Sliding error = O(τ) Sliding error = O(τ2)

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

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

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

13 Simulation

14 Examples continue … Finite time 2-sliding super-twisting algorithm
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 HOSM can be used to avoid chattering
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

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