1. OUTPUT – INPUT STABILITY Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign MTNS ’02

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

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:

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

6. 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

7. MIMO SYSTEMS Existence of relative degree no longer necessary For linear systems reduces to usual minimum phase notion Input-bounding property – via Hirschorn’s algorithm Example: Extensions: Singh’s algorithm, non-affine systems

8. INPUT / OUTPUT OPERATORS Operator is output-input stable if A system is output-input stable if and only if its I/O mapping (for zero i.c.) is output-input stable under suitable minimality assumptions

9. APPLICATION: FEEDBACK DESIGN Output-input stability guarantees closed-loop GAS No global normal form is needed ( r – relative degree) Output stabilization state stabilization Apply u to have with A stable Example:

10. 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)

11. 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”)

12. SUMMARY New notion of output-input stability  applies to general smooth nonlinear control systems  reduces to minimum phase for linear (MIMO) systems  robust variant of Byrnes-Isidori minimum phase notion  relates to ISS, detectability, left-invertibility  extends to input/output operators Applications:  Feedback stabilization  Cascade connections  Adaptive control  More ?