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

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References 1.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. 3.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 ECC2001.

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

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RobustnessChattering High frequency switching of control

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

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Isnt it restrictive? Sliding variable must have relative degree one w.r.t. control.

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

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

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1-sliding vs 2-sliding s ds 2-sliding τ τ2τ2 s ds 1-sliding τ Sliding error = O(τ)Sliding error = O(τ 2 )

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

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

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Pendulum The model: Sliding variable: Sliding variable dynamics: Twisting Controller coefficients: α = 0.1, V M = 7

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Simulation

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

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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.

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Is it possible to stabilise sliding surface with relative degree 2 in to 2-sliding set using only s, not its derivative? Answer: yes! 1.by designing observer 2. using modified super-twisting algorithm. Question:

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Modified super-twisting algorithm System type: Where λ, u 0, k and W are positive design constants 1. Sinusoidal oscillations for = u 0 2. Unstable for < u 0 3. Stable for > u 0

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Phase plot Sufficient conditions for stability

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

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Simulation Results Controller coefficients:

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Results continued …

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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

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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.

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