me677c11p1 slidingmodecontrol t

Sliding Mode Control Lecture Notes by B.Yao

Sliding Mode Control

First-Order Nonlinear Systems with Uncertainties

( , ) , ( , ) ( , ) ( , )Tx f x t u f x t x t x t (S1) where

( , ) : known basis functions Px t R : unknown weights PR : uncertain nonlinearity

Assumption: min max: (S2)

: ( , ) ( , )x t x t (S3)

where min max, ,and ( , ) are knownx t .

Objective: Want ( ) ( ) or ( ) ( ) 0 asd dx t x t z x t x t t .

Error Dynamics:

d d

Td

z x x f u x

u x

(S4)


Control Structure: ˆ, T

a s a du u u u x (S5)

ˆ( )T Ts sz u u (S6)

where ˆ : parameter estimation error

Choose a bounding function such that ( , ) Th x t

For example,

max min( , ) ( , ) ( , )Th x t x t x t

Ideal Sliding Mode Control (SMC) Law:

( , ) sgn( )s ou h x t h z (S7) where and

1 0

sgn( ) undetermined for 01 0

zz z

z

. (S8)

( , )h x t

0oh


Theoretical Performance of Ideal SMC Law: When z > 0,

T Ts o

To o

z u h h

h h h

(S9)

which indicates that z decreases with a rate no less than oh , and thus,

will reach 0z after a finite time (0)o

zh

When z < 0, T T

o oz h h h h oh (S10) which indicates that z increases and will reach to 0z after a finite time.

Note that the above conditions can be concisely described by: sgn( )o oz z h z z h z (S11)

The above analysis shows that z reaches to zero in a finite time and stays at zero thereafter, which is also called sliding mode. During the sliding mode,

0, 0z z the control input for z staying at zero should be: T

sequ (S12)

which is the so-called equivalent control for the discontinuous ideal SMC control law (S7). Note that sequ is not known in advance and is theoretically obtained by the discontinuous control law (S7) switching at infinite fast.


SISO Systems in Normal Form System:

1 2

2 3

( )Tn

x xx x

x x u

, 1

1

, , Tnx x x

y x

(S12)

Objective: Want ( ) asdy y t t . Let

1 1

2 2

( 1)

( )( )

( )

d

d

nn n d

x x y tx x y t

x x y t

(S13)

Error Dynamics:

(S14)

1 2

2 3

( ) ( )

n T nn n d d

x xx x

x x y u y


Step 1: Target Surface Design (or Sliding Surface)

1 1 1 1

( 1) ( 2)1 1 1 1 1 n n n

n nn

s x c x c x

x c x c x

(S15)

If we choose 1 2 1, , , nc c c such that 1 21 1

n nnc c is Hurwitz, then,

when the target surface is reached (i.e., 0s ), 1( ) 0x t exponentially as . For example, let

1

11 1or

nk n k

o n k n ods x c Cdt

(S16)

Step 2: Reaching Phase

Synthesize a control law such that s reaches to zero in a finite time and stay on 0s thereafter. For this, note

1 1 1 1

( )1 1 1 1

( )1 1 2

n n n

nn d n n

T nd n n

s x c x c x

x y c x c x

u y c x c x

(S17)

t


Control Structure:

( )

1 1 2

,ˆ

a s

T na d n n

u u u

u y c x c x

(S18)

T

ss u (S19) Ideal SMC Law:

( , ) sgn( )s ou h x t h s (S20) where

( , ) Th x t (S21)

os s h s (S22)

which will force s to zero in a finite time

2 1os x x

1x

2x0s

00,s s h

00,s s h

2 1os x x

1x

2x0s

00,s s h

00,s s h

Equivalent Control on Sliding Surface:

0

Tseq

s

u


Chattering Problem of Ideal SMC In reality, we cannot have the perfect switching; then, the control will switch between positive value and negative value and s behaves in a “zig-zag” way.

1x

2x0s

00,s s h

00,s s h

1x

2x0s

00,s s h

00,s s h


Smoothed SMC Instead of using sgn( )z which causes control input chattering, let us try to use some continuous functions to approximate it. For example:

( , ) sat

: boundary layer thickness

s osu h x t h

When which indicates that the system will reach the boundary layer s in a finite time and stay inside the boundary layer thereafter as in the idle SMC law. Inside the boundary layer:

( )To

To

ss h h

h hs s

(S23)

which can be thought as having an equivalent proportional feedback gain of

, ,os ss h s

0

sgn( )s1

1


os

h hk

for s dynamics. Thus, at the steady-state, the tracking error is

bounded by:

min

( ) ( )( )T

s

sk

which can be made arbitrarily small by using a small enough boundary layer thickness – the smaller is, the larger the resulting sk and the smaller the above upper bound is. However, too small may lead to a sk so large that the resulting control law is so sensitive to measurement noise (thus control input chattering) or excite the neglected high-frequency dynamics leading local instability as in the idle SMC law. Therefore, in reality, there will be a trade-off in selecting a suitable value for .


Unknown Input Gain

(S24)

Use the same sliding surface as in (S15):

( )1 1 2

T nd n ns bu y c x c x

Let us say:

( )1 1 2

1 ˆ, ˆT n

a s a d n nu u u u y c x c xb

(S25)

( )

1 1 2

ˆ ( )

T na s d n n

Ta s

Ta s

s b u b u y c x c x

b b u b u

b u b u

Let h(x,t) be a bounding function that

( , ) Tah x t b u (S26)

For example, ( , )TM M ah x t b u , max minM , max minMb b b

1 2

min max

, 0Tn

x x

x bu b b b


Then

min

( ( , ) ) sgn( )os

h x t hu sb

(S27)

guarantees that

min

( ) sgn( )

( )

Ta o

o o

bss s b u h h sb

sh h h s h s


Systems with Matched Uncertainties Consider SISO only:

( ) ( )[ ( , , )]( ) n

x f x g x u t x uy h x x R

(S28)

where h, f, g are known functions and is uncertain nonlinearity, bounded by:

( , , ) ( , ) , 1o u ut x u x t u (S29) Assume that the output y=h(x) has a relative degree of for the nominal system x f g u . Then, using the coordinate transformation introduced in I/O feedback:

( )( )xx

where

1

( )( )

( )

( )

f

f

h xL h x

x

L h x

and

( ) ( ) 0x g xx


1 2 2 3

1 1

1

( ) ( )( ) ( )( )

( ) ( )

f f

f g f

L h x L h xx f x g x u

x xL h x L L h x u

[ ( )] ( , )x f g u f fx x x

(S30)

1y Objective:

( ) asdy y t t Let:

1 1

2 2

( 1)

( )( )

( )

d d

d

d

e y y y te y t

e y t

1 2

2 3

1 ( )( ) [ ] ( )f g f d

e ee e

e L h x L L u y t

(S31)


Step 1: Synthesize the switching function

1 1 1 1 1 1 11 2 ( )d ds e k e k e k k e tdt dt

(S32)

which will be stable if 1 21 1s k s k

is Hurwitz. Step 2:

1 1 1 1

1 ( )1 1 2 = ( ) ( )[ ] ( )f g f d

s e k e k e

L h x L L h x u k e k e y t

(S33)

( )1 1 21

1 ( ) ( )( )

a s

a f dg f

u u u

u L h x k e k e y tL L h x

(S34)

Then 1 ( )[ ]g f ss L L h x u

1 1

1 1 g f s g f

g f s g f o u a s

ss sL L h u sL L h

L L h su s L L h u u

(S35)


Let

11 ( , )sgn( )

( )s sg f

u k x t sL L h x (S36)

1

1 (1 )

s g f o u a u s

s u g f o u a

ss k s s L L h u s k

k s s L L h u

Thus, by choosing

111s s g f o u a

u

k L L h u

(S37)

the existence of the sliding mode condition will be satisfied:

sss s

Again, to be able to implement the above sliding mode controller, it is necessary to make sure all internal states are bounded as well. Thus, the same as in the I/O feedback linearization, the proof of the bounded-input-bounded-state stability of the internal dynamics is essential to complete the overall design and analysis.

me677c11p1 slidingmodecontrol t

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