introduction in smc

8/3/2019 Introduction in SMC

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P R E S E N T E D B Y

ARINDAM KI SKU

Sliding Mode Control-An Introduction


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Outline

What is this Sliding Mode and how did its studystart ?

How to design a controller using this concept ?


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First Formal steps

The first steps of sliding mode control Theoryoriginated in the early 1950s initiated by S.V.Emelyanov.

Started as VSC Variable Structure Control

i.e. Varying system structure for stabilization.


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Variable Structure Control-Constituent System

Model 2Model 1


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Piecing together

Model 3


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Properties of VSC

Both constituent system were oscillatory ( i.e. model-1 & model-2 ) and were not asymptotically stable.

Combined system is asymptotically stable( i.e. model- 3 ).

Property not present any of the constituent system isobtain by VSC.


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Another Example Unstable Constituent System

Model 1 Model 2


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Analyses-

Both system were unstable.

Only stable mode is one mode of system

where

If the following VSC is employed


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Combine:- model-1 & model- 2


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In these case.

Again property is no present in the constituentsystems is found in the combined system.

A stable system can be obtain by varying between

two unstable structures. However, a more interesting behavior can be

observed if we use a different switching logic


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The Regions


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Sliding Mode

New trajectory that was not present in any of the two originalsystems


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Sliding Mode ?

Defined : Motion of the system trajectory along achosen line/plane/surface of the state space.

Sliding Mode Control : Control designed with theaim to achieve sliding mode.

. Is usually of VSC type

. Eg: Previous problem can be perceived as

0

sgn( )

x x u

u xs x


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What is the advantage ?

Consider a n-th order system represented in thephase variable form

Also consider the sliding surface defined as

1

1 1

, 1, 2, , 1i i

n n n

x x i n

x a x a x bu

1 1 2 2 1 1

1 1 2 2 1 1

1

1 2 2 3 2 1 1

1

0T

n n n

n n n

n

n n n n i i

i

s c x c x c x c x x

x c x c x c x

x c x c x c x c c x


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Advantage.

Thus entire dynamics of the system is govern by thesliding line/surface parameters only

In sliding mode, dynamics independent of systemparameter (a1,a2).

1

1

1 2 2 3 2 1 1

1

, 1, 2, , 1i i

n

n n n n i i

i

x x i n

x c x c x c x c c x


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Required Properties

Sliding mode should have the following properties-

1) System stability confined to sliding surface.

(unstable sliding mode is NOT sliding mode at all)

2) Sliding mode should not take forever to start.


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Stable Surface

Consider the system

If the sliding surface is designed asthen confined to the surface ( ) ,the dynamics of can be written as

1 11 1 12 2

2 21 1 22 2

x A x A x

x A x A x Bu

1

2

1T

x

s K c xx

1 20s Kx x

1x

1 11 1 12 2 11 12 1x A x A x A A K x


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The Surface

If K is so designed that has eigenvalues onLHP only , then the dynamics of is stable.

Since , the dynamics of is also stable.

Hence sliding surface is designed as , thesystem dynamics confined to s=0 is stable.

(Requirement 1)

Note :It is not necessary for s to be a linear function of x.

11 12A A K

1x

1 20Kx x

2x

1 2

Ts Kx x c x


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References

V. Utkin, Variable Structure Systems with SlidingMode,IEEE Trans. Automat. Contr., AC-12, No.2, pp. 212-222, 1977

Discrete-time Sliding Mode Control by BijnanBandyopadhyay & Janardhanan Sivaramakrishnan.

Introduction to Control System by D.K. Anand &R.B. Zmood.

introduction in smc

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