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

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

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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

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

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

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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

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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

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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

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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:

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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)

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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”)

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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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TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,

TOWARDS ROBUST LIE-ALGEBRAIC STABILITY CONDITIONS for SWITCHED LINEAR SYSTEMS 49 th CDC, Atlanta, GA, Dec 2010 Daniel Liberzon Univ. of Illinois, Urbana-Champaign,

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