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 L , [0 , ) } u is a truncation of u : u ( t ) = braceleftBigg u ( t ) , t , t > L e is a linear space and L L e Example: u ( t ) = t, u ( t ) = braceleftBigg t, t , t > u / L but u 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 ) = ( Hu ) p. 4/1 5 Definition: A mapping H : L m e L q e is L stable if K such that bardbl ( Hu ) bardbl L ( bardbl u bardbl L ) + ,...
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Lect_20 - Nonlinear Systems and Control Lecture # 20...

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