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OUTPUT – INPUT STABILITY

OUTPUT – INPUT STABILITY. Daniel Liberzon. Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign. MTNS ’02. MOTIVATION. s. ISS:. stability (no outputs). linear : stable eigenvalues. detectability (no inputs).

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OUTPUT – INPUT STABILITY

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

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

  4. DEFINITION Call the system output-input stable if integer N and functions s.t. where Example:

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

  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. Operator is output-input stable if INPUT / OUTPUT OPERATORS 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 stabilization state stabilization Apply uto have with A stable Example: ( r– relative degree) Output-input stability guarantees closed-loop GAS No global normal form is needed

  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 Design model Plant Controller • the plant is output-input stable (N=r) • the system in the box is input-to-output • stable (IOS) from to If: 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 ?

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