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 )
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