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Unformatted text preview: Chapter 2 Stability Concepts In this chapter, we discuss two notions of stability, namely input/output (I/O) stability, and internal (or Lyapunov) stability. They are related, but not exactly the same. The latter is more of a state space concept, whereas the first is of an input/output (i.e. transfer function in linear systems) notion. 2.1 I/O Stability Preliminaries: Vector and function norms 1. Sup norms are used for vectors for simplicity: bardbl x bardbl = max i  x i  . Other norms are also okay 2. Induced matrix norms: let A n n , ( i stands for induced) bardbl A bardbl i, := sup x bardbl Ax bardbl bardbl x bardbl = max j summationdisplay i A ji 3. vector function norms: for x : [0 , ) n , the L norm is used, i.e. bardbl x ( ) bardbl = sup t bardbl x ( t ) bardbl = sup t  x i ( t )  . Definition A system (not necessarily linear) F : u ( ) y ( ) is I/O stable iff there exists k < , s.t. for all bounded u ( ), bardbl y ( ) bardbl k bardbl u ( ) bardbl For a linear possibly time varying system, (ignoring initial conditions) the output is given by: y ( t ) = integraldisplay t H ( t, ) u ( ) d (2.1) note that this is a linear map between the input function space and the output function space. Theorem The linear system (2.1) is I/O stable iff sup t braceleftbiggintegraldisplay t bardbl H ( t, ) bardbl i d bracerightbigg := k < . (2.2) Because norms are equivalent on finite dimensional spaces, the norms on vectors ( x ( t ) and y ( t )) do not have to be infinity norms....
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 Fall '08
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