OUTPUT INPUT STABILITY and FEEDBACK STABILIZATION Daniel Liberzon

OUTPUT – INPUT STABILITY and FEEDBACK STABILIZATION Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng. , Univ. of Illinois at Urbana-Champaign U. S. A. CDC ’ 03

MOTIVATION ISS: stability (no outputs) detectability (no inputs) minimum phase linear: stable eigenvalues linear: stable unobserv. modes linear: stable zeros stable inverse ? ? ?

DEFINITION (L, Sontag, Morse, 2002) Call the system output-input stable if s. t. where Example: integer N and functions

UNDERSTANDING OUTPUT-INPUT STABILITY 1 Output-input stability: 2 Uniform detectability w. r. t. extended output: 3 Input-bounding property: 1 <=> 2 + 3

SISO SYSTEMS For systems analytic in controls, can replace the input-bounding property by where is the first derivative containing u For affine systems: this reduces to relative degree ( ) doesn’t have this property 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 Example: Input-bounding property: ü ü Detectability: ü Equation for is ISS w. r. t. ü

APPLICATION: FEEDBACK DESIGN Global asymptotic stabilization by static state feedback Output stabilization state stabilization ü Example: ü Output-input stability closed-loop GAS No global normal form is needed See also Astolfi, Ortega, Rodriguez (2002)

APPLICATIONS of OUTPUT-INPUT STABILITY • Analysis of cascade systems • Adaptive control • Input / output operators See L, Sontag, Morse (TAC 2002), L (MTNS 2002 ) 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 ?