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**Higher Order Sliding Mode Control**

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

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**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.

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**Review: Sliding Mode Control**

Consider a NL system Design consists of two steps Selection of sliding surface Making sliding surface attractive

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High frequency switching of control Robustness Chattering

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**Pros and cons Order reduction Full state availability**

Robust to matched uncertainties Chattering at actuator Sliding error = O(τ) Simple to implement

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**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.

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**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

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**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

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**1-sliding vs 2-sliding s ds 1-sliding τ s ds 2-sliding τ τ2**

Sliding error = O(τ) Sliding error = O(τ2)

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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. r-sliding (r p) is the suitable choice. 2-sliding design is done to avoid chattering.

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**2-sliding algorithms: examples**

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

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**Pendulum The model: Sliding variable: Sliding variable dynamics:**

Twisting Controller coefficients: α = 0.1, VM = 7

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Simulation

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**Examples continue … Finite time 2-sliding super-twisting algorithm**

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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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.

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**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

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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 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

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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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Thank You

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