OUTPUT – INPUT STABILITY and

FEEDBACK STABILIZATION

Daniel Liberzon

CDC ’03

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-ChampaignU.S.A.

MOTIVATION

? ? ?

ISS:stability(no outputs) linear: stable

eigenvalues

detectability(no inputs) linear: stable

unobserv. modes

minimumphase linear: stable zeros

stable inverse

DEFINITION

(L, Sontag, Morse, 2002) Call the system

output-input stable if integer N and functions

s.t.

where

Example:

UNDERSTANDING OUTPUT-INPUT STABILITY

1 Output-input stability:

Uniform detectability w.r.t. extended output:2

1 2 3<=> +

3 Input-bounding property:

SISO SYSTEMS

For systems analytic in controls, canreplace the input-bounding property by

where is the first derivative containing u

doesn’t have this property

For affine systems:

this reduces to relative degree ( )

For affine systems in global normal form,

output-input stability ISS internal dynamics

MIMO SYSTEMS

Existence of vector relative degree not necessary

For linear systems reduces to usual minimum phase notion

Input-bounding property – via nonlinear structure algorithm

Equation for is ISS w.r.t.

Example:Input-bounding property:

Detectability:

APPLICATION: FEEDBACK DESIGN

Output stabilization state stabilization

Global asymptotic stabilization by static state feedback

Output-input stability closed-loop GAS

No global normal form is needed

See also Astolfi, Ortega, Rodriguez (2002)

Example:

APPLICATIONS of OUTPUT-INPUT STABILITY

• Analysis of cascade systems

• Adaptive control

• Input / output operators

Potential other applications:

• Stabilization & small gain (Jiang, Praly, et. al.)

• Output regulation, disturbance decoupling (Isidori)

• Bode integrals and entropy (Iglesias)

• Bode integrals and cheap control (Seron et. al.)

• More ?

See L, Sontag, Morse (TAC 2002), L (MTNS 2002 )