Lect_20 - Nonlinear Systems and Control Lecture 20...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 20 Input-Output Stability – p. 1/1 5 Input-Output Models y = Hu u ( t ) is a piecewise continuous function of t and belongs to a linear space of signals The space of bounded functions: sup t ≥ bardbl u ( t ) bardbl < ∞ The space of square-integrable functions: integraltext ∞ u T ( t ) u ( t ) dt < ∞ Norm of a signal bardbl u bardbl : bardbl u bardbl ≥ and bardbl u bardbl = 0 ⇔ u = 0 bardbl au bardbl = a bardbl u bardbl for any a > Triangle Inequality: bardbl u 1 + u 2 bardbl ≤ bardbl u 1 bardbl + bardbl u 2 bardbl – p. 2/1 5 L p spaces: L ∞ : bardbl u bardbl L ∞ = sup t ≥ bardbl u ( t ) bardbl < ∞ L 2 ; bardbl u bardbl L 2 = radicalBigg integraldisplay ∞ u T ( t ) u ( t ) dt < ∞ L p ; bardbl u bardbl L p = parenleftbigg integraldisplay ∞ bardbl u ( t ) bardbl p dt parenrightbigg 1 /p < ∞ , 1 ≤ p < ∞ Notation L m p : p is the type of p-norm used to define the space and m is the dimension of u – p. 3/1 5 Extended Space: L e = { u | u τ ∈ L , ∀ τ ∈ [0 , ∞ ) } u τ is a truncation of u : u τ ( t ) = braceleftBigg u ( t ) , ≤ t ≤ τ , t > τ L e is a linear space and L ⊂ L e Example: u ( t ) = t, u τ ( t ) = braceleftBigg t, ≤ t ≤ τ , t > τ u / ∈ L ∞ but u τ ∈ L ∞ e Causality: A mapping H : L m e → L q e is causal if the value of the output ( Hu )( t ) at any time t depends only on the values of the input up to time t ( Hu ) τ = ( Hu τ ) τ – p. 4/1 5 Definition: A mapping H : L m e → L q e is L stable if ∃ α ∈ K β ≥ such that bardbl ( Hu ) τ bardbl L ≤ α ( bardbl u τ bardbl L ) + β,...
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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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Lect_20 - Nonlinear Systems and Control Lecture 20...

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