OUTPUT-INPUT STABILITY:

A NEW VARIANT OF THE

MINIMUM-PHASE PROPERTY FOR

NONLINEAR SYSTEMS

D. LiberzonUniv. of Illinois at Urbana-Champaign, USA

A. S. MorseYale University, USA

E. D. Sontag Rutgers University, USA

MOTIVATION

stability(no outputs)

detectability(no inputs)

minimumphase

,),( uxfx )(xhy

ISS:

)||()||( uxV )|)(|sup()|,)0((||)(|

0sutxtx

ts

linear: stable unobserv. modes )||()||( yxV

)|)(|sup()|,)0((||)(|0

sytxtxts

stable inverse:? ? ?

00 xu

linear: stableeigenvalues

00 xy

0,0 xuy

linear: stable zeros,)(ry u

MOTIVATION: Adaptive Control

If: • the system in the box is output-stabilized• the plant is minimum-phase

Then the closed-loop system is detectablethrough e (“tunable” – Morse ’92)

Controller

Plant

Design e

y

u ymodel

DEFINITION

),( uxfx )(xhy

)|)(|sup(),|)0(|(0)(

)(sytx N

tstu

tx

Call the system

output-input stable if integer N and functions

s.t.

KKL , 1,)0( NCux

where ),...,,(: )( Nyyyy N

Example: uy

Reduces to min phase for linear systems (MIMO too)

DISCUSSION

Split into two:

This is detectability (w.r.t. extended output)

Have a Lyapunov sufficient condition

This is related to relative degree

)|)(sup(|),|)0(|()(0

sytxtx N

ts

)|)(sup(|),|)0(|()(0

sytxtu N

ts

RELATIVE DEGREE

)||(|)(||| )(21

ryxu

Call r a uniform relative degree if )1(,...,, ryyy don’t depend on u and

for some K21,

However, the system

doesn’t have a relative degree

uy arctan

For affine systems:

this reduces to the usual definition ( )

uxbxay r )()()( 0)( xb

SOME RESULTS

• Detectability w.r.t. extended output: )|)(sup(|),|)0(|()(

0sytxtx N

ts

plus relative degree: )||()||(|| )(21

ryxu

imply output-input stability: )|)(|sup(),|)0(|(

0)()( sytx N

tstutx

• Output-input stability implies detectability

• Main result: for SISO systems analytic in controls, output-input stability implies relative degree

Example: affine systems uxbxay r )()()(

Not true for MIMO systems !

FEEDBACK DESIGN

21 ...

ubar ),(),(

),( q

Apply u to have with A stable A

If the system is output-input stable then

implies 0x

No global normal form is needed

With global normal form:

output-input stability corresponds to ISS internal dynamics

1y , relative degree r

)|)(sup(|),|)0(|()(0

sytxtx N

ts

CASCADE SYSTEMS

u y1 2

If: • is detectable (IOSS)• is output-input stable (N=r)

12

Then the cascade system is detectable (IOSS) w.r.t. u and extended output ry

For linear systems recover usual detectability (observability decomposition)

ADAPTIVE CONTROL

Planty

u

y

Controller Designmodel

ey

If: • the plant is output-input stable (N=r)

• the system in the box is input-to-output

stable (IOS) from to

Then the closed-loop system is detectablethrough (“weakly tunable”)

1re ry

re