1. OUTPUT-INPUT STABILITY: A NEW VARIANT OF THE MINIMUM-PHASE PROPERTY FOR NONLINEAR SYSTEMS D. Liberzon Univ. of Illinois at Urbana-Champaign, USA A. S. Morse Yale University, USA E. D. Sontag Rutgers University, USA

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

3. MOTIVATION: Adaptive Control If: the system in the box is output-stabilized the plant is minimum-phase Then the closed-loop system is detectable through e (“tunable” – Morse ’92) Controller Plant Design model

4. DEFINITION Call the system output-input stable if integer N and functions s.t. where Example: Reduces to min phase for linear systems (MIMO too)

5. DISCUSSION Split into two: This is detectability (w.r.t. extended output) Have a Lyapunov sufficient condition This is related to relative degree

6. RELATIVE DEGREE Call r a uniform relative degree if don’t depend on u and for some However, the system doesn’t have a relative degree For affine systems: this reduces to the usual definition ( )

7. SOME RESULTS Detectability w.r.t. extended output: plus relative degree: imply output-input stability: Output-input stability implies detectability Main result: for SISO systems analytic in controls, output-input stability implies relative degree Example: affine systems Not true for MIMO systems !

8. FEEDBACK DESIGN...... Apply u to have with A stable If the system is output-input stable then implies No global normal form is needed With global normal form: output-input stability corresponds to ISS internal dynamics, relative degree

9. CASCADE SYSTEMS If: is detectable (IOSS) is output-input stable ( N=r ) Then the cascade system is detectable (IOSS) w.r.t. u and extended output For linear systems recover usual detectability (observability decomposition)

10. ADAPTIVE CONTROL Plant Controller Design model 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 detectable through (“weakly tunable”)