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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 ∈ ℜ 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 , ∞ ) → ℜ 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,τ ) u ( τ ) dτ (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,τ ) bardbl i dτ 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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This note was uploaded on 02/07/2012 for the course ME 8281 taught by Professor Staff during the Fall '08 term at Minnesota.
- Fall '08