VSC Design via Sliding ModesHarry G. Kwatny

Department of Mechanical Engineering & MechanicsDrexel University

Outline VS systems, sliding modes, reaching Example: undersea vehicle Design based on normal form

VSC Design via Sliding Modes: Setup( , )

, , smoothn m

x f x ux R u R f

=

∈ ∈

discontinuous across switching surface ( ) 0

( ), ( ) 0( ) , 1, ,

( ), ( ) 0

( ), ( ), smooth

i i

i ii

i i

i i

u s x

u x s xu x i m

u x s x

u x u x

+

−

+ −

=

>= =

<

System:

Control:

VSC Design via Sliding Modes: Strategy( )

( )

( )

1. Choose switching surfaces, , so that sliding mode has desired dynamics.2. Choose control functions, , so that sliding mode is reached in finite time.Approach:1. How does choice of

i

i

i

s x

u x

s x

±

( )

affect sliding behavior? equivalent control method2. How can we choose to insure finite time reaching? Lyapunov method

iu x±

⇒

⇒

Equivalent Control ~ 1( ) ( ) ( )

( ) ( )( ){ }

, 0, , ,

, 0

Suppose 0 contains a sliding domain.

We can obtain the dynamics in by identifying the .equivalent control

i ii

i i

ns

s

u t x s xx f t x u u

u t x s x

M x R s x

M

+

−

>= = <

= ∈ =

Equivalent Control ~ 2( )

( ) ( )

( ) ( )n

I

t

f

r

the

l

re exists a control , such that

, , and 0, then is referredto as the and denoted

, , 0

if a solution exists for , then

equivalent

0

c

on 0,and

, ,

o o eq

eq

eq

eq

u t x

x f t x u s x uu

ss x f t x ux

u s s

x f t x u

= =

∂= ≡

∂= ≡

=

( )( ) dynamics in sliding,t x ⇐

Equivalent Control – Special Cases

( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )( )

( )

1

1

1

1

0

det 0

,

(assuming det 0)

eq

eq

eq

x f x G x u

s x S x f x S x G x u

SG u S x G x S x f x

x I G x S x G x S x f x

x Ax Bu s x Cx

u CB CAx CB

x I B CB C Ax

−

−

−

−

= +

= + ≡

≠ ⇒ = − = −

= + =

= ≠

= −

Systems 'linear in control'

Linear Systems

Reaching

( ) ( ) ( ) ( )

( ) ( ) ( )

( )( ) ( )

2

Choose

2 ,

Suppose 2 , ( ) on where is an open set containing

is a sliding domain and it is reached in finite time from any initial point in .

T

T

Ts

s s

s

V x s x s x s x

sV x s x f x ux

s x s x f x u s x D MD D M

DD

ρ

= = ⇒

∂=

∂∂ ∂ < − −

⊂ ⇒

Linear Example ~ Reaching( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )( ) ( )

( )( ) ( ) ( )

( )

2

, ,

2 ( ) 2

We can only affect the second term. Choose,sgn , 0

2 ( ) sgn

p

This insures b

rovid

that is a g

,

/

lo

ed

x f x u Ax bu s x cx

V s x V s x cAx s x cbu

u k x s x cb k x x

V s x cAx cbk x s x cb s x

cbk x c

b

s x

Ax x

k x cAx c

ρ

ρ

ρ

= = + =

= ⇒ = +

= − > ∀

= − ≤ − − ≥ ∀

= +

.al sliding domain

Linear SISO Example ~ 1( )

( )

( )

[ ]

( )

1

1

1 1

, 0 0

: 0

Define a matrix whose columns span ker , i.e.,, 0

Notice that ker and Im ker . Define a state transformation

, ,

eq eq

n i

s x cx s s

cx cAx cbu u cb cAx

x I b cb c Ax

V cV v v cv

b c X b c

x w z x Vw bz

−

−

−

= ≡ ⇒ ≡

= + = ⇒ = −

= −

= =

∉ = ⊕

= +

( ) ( )

1

1

,nw R z R

Vw bz I b cb c A Vw bz

−

−

∈ ∈

+ = − +

Sliding dynamics

Linear SISO Example ~ 2

[ ] ( ) [ ]

[ ] ( )

( ) [ ] [ ]

1

1 11

1

00 1

0

0 0

n

w wV b I b cb c A V b

z z

U UV Ub IV b d cb c

d dV db

w U w U wI b cb c A V b A V b

z d z z

w UAV UAb wz z

−

− −−

−

= −

= ⇒ = ⇒ =

= − =

=

[ ]sliding: 0z

w UAV w→

=

( ) 1 0

note:

I b cbd c− − =

Designing the Sliding Surface

( ) ( )( )

( )( ) ( )

0

1 1 1 2

2 1 1 2 2 1 2

1 2 2 0

Consider the system

rank around

satisfies controllability rank conditionTransform to :

,

, ,

, ,det 0 around n m m

x f x G x u

G x mx

x f x x

x f x x G x x u

x R x R G x−

= +

=

=

= +

∈ ∈ ≠

regular form

( )( )( )

( ) ( )

2 0 1

1 1 1 0 1

1 2 0 1 2

Strategy:1) choose so that

,

has desired behavior,2) choose to enforce sliding on

,

x s x

x f x s x

us x x s x x

= −

= −

= +

Design dlidingdynamics

Reaching

Example: Linear SISO Design ~ 1( )

12 2

2

11 111 1 2

122

11 12 1

21 22

, rank 1, , controllablereorder states to obtain

, with , det 0

transform to :

, ,00

0t

1

regular form

nn

n

x Ax bu b A b

bb b R b

b

I bI b bz Tx T T

bb

A Az z u

A A

−−−−

−

−

= + =

= ∈ ≠

− − = = =

= +

( )11 12he pair , is controllableA A

Example: Linear SISO Design ~ 2

( )

( )

( ) ( ) ( ) ( )

( )

12 1 1 11 12 1 1 2

1 2

2

To shape , choose, ,

Choose by any means, pole placement, LQG

i

, etc.Now, To d

sliding dynamics

reach ng contresign the take1 ,

ol1 0

2 2

n

T TQ

T

z Kz z A A K z z R z RK

s z Kz zu

V z s z s z Q s z Q Q

V z s

−= ⇒ = + ∈ ∈

= − +

= = = >

=

[ ] [ ]( ) ( ) ( )

( ) [ ] [ ]

* *

- 1 - 1

sgn , ,

- -

TT T

i i i

TTm m

Q s z K Q K Az s Qu

u z s s z Qs z

z z K I Q K I Az

κ

κ

= +

= − =

>

Example: Linear MIMO Design ~ 1( )

12 2

2

1111 2

122

11 12

21 22

, rank , , controllablereorder states to obtain

, with , det 0

transform to :

, ,00

0t

regular form

m m

n mn m

x Ax Bu B m A B

BB B R B

B

I BI B Bz Tx T T

BB

A Az z u

A A I

×

−−−−

−

= + =

= ∈ ≠

− − = = =

= +

( )11 12he pair , is controllableA A

Example: Linear MIMO Design ~ 2

( )

( )

( ) ( ) ( ) ( )

( ) [ ] [ ]

2 1 1 11 12 1

1 2

2

To shape ,

Choose by any means, pole placement, LQG, etc.Now,

sliding dynamic

d

s

reaching contTo esign the take1 1 , 02 2

- -

rol

T TQ

TT Tm m

z Kz z A A K zK

s z Kz zu

V z s z s z Q s z Q Q

V z s Q s z K I Q K I A

= ⇒ = +

= − +

= = = >

= =

( ) ( ) ( )( ) [ ] [ ]

* *sgn , ,

- -

T

i i i

TTm m

z s Qu

u z s s z Qs z

z z K I Q K I Az

κ

κ

+

= − =

>

Example: Underwater Vehicle[ ]

( )

[ ]

2

, , uncertain, ,

1

1. Choose a sliding surface: .Why? Because 0 ( stabilizes first eq)

2. Choose reaching control based on , ,

2 1

T

mx cx x u m c u U Ux v

cv v v um m

s v xs x x v x

V x v s s s

vV s c v v

m

λλ λ

λ

+ = ∈ −

=

= − +

= +≡ ⇒ = − = −

= =

=−

0120

U ss u u

U sm− > + ⇒ = <

Control Based on Normal Form

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

F z uz Az E z z uy Cz

ξ ξα ξ ρ ξ

== + +=

( ) ( ) ( )Choose such that 0 0

( ) ( ) 0( )

( ) ( ) 0i i

ii i

s x s x Kz x

u x s xu x

u x s x

+

−

= ⇔ =

>=

<

Recall Brunovsky structure of A,E

( ) ( )( )

x f x G x uy h x

= +=ρ−1( )x

α ( )x

Kz x( )

-x

yuv

Sliding Dynamics

( )

( ) ( ) ( ) ( )

[ ] ( )

1 1

0

( ) 0 ( ) 0

( ) 0

, 0

eq

eq

s x Kz x

Kz KAz KE x x u

u x KAz x x x

z I EK Az Kz t

α ρ

ρ ρ α− −

= ⇔ =

⇓

= + + = ⇓

= − −

⇓

= − =

KE I=Note : same as feedback linearizing

control

Sliding Dynamics

Choosing K1

1

,1 , 1

0 0

, 1

0 0

r

i

rm

i i i r

m

k

K k a a

k

←→

−

←→

= =

,1 ,2 , 1

0 1 0, 1, ,

are 0 and are0 0 1

ii i i r

i mm r m

a a a

λ

−

= − − − −

One choice:

Eigenvalues of (A+EK) are:Sliding eigenvalues

ReachingConsider the positive definite quadratic form in s

QssxV T=)(

Upon differentiation we obtain

[ ] QKzuQKzKAzQssVdtd TTTT ρα 222 ++==

If the controls are bounded, 0>≤ iiUu ( 0 0> ≤ ≤ >U u Ui i imin, max, ) then choose

uU s xU s xi

i i

i i

=><

RSTmin,

*

max,*

a fa f

00

, mi ,,1= , )()()(* xQKzxxs Tρ=

VSC Summary