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

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

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

4. UNDERSTANDING OUTPUT-INPUT STABILITY 1Output-input stability: Uniform detectability w.r.t. extended output: 2 123 + 3 Input-bounding property:

5. SISO SYSTEMS For systems analytic in controls, can replace 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

6. 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:

7. 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:

8. 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 )