Difference between revisions of "HW3.A Brian Thomas ECE301Fall2008mboutin" - Rhea

Revision as of 15:42, 18 September 2008

Problem: 5 - Give a formal definition of a “stable system”. Give a formal definition of an unstable system.

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 '''Problem:''' 5 - Give a formal definition of a “stable system”. Give a formal definition of an unstable system.
-'''Solution:''' Consider system f, where input x(t) yields output y(t) = f(x(t)).  Consider bounded x(t), i.e. <math>\exists M_1 \in \mathbb{R} \text{ s.t. } \forall t \in \mathbb{R}, |x(t)|\leq M_1</math>.  The system f can be considered '''stable''' iff <math>\forall x(t), \exists M_2 \in \mathbb{R} \text{ s.t. } \forall t \in \mathbb{R}, |f(x(t))| \leq M_2</math>
-(ie, The system f is stable iff for all x(t), if x(t) is bounded, then f(x(t)) is bounded as well.)
-Consider system f, where input x(t) yields output y(t) = f(x(t)).  Consider bounded x(t), i.e. <math>\exists M_1 \in \mathbb{R} \text{ s.t. } \forall t \in \mathbb{R}, |x(t)|\leq M_1</math>.  The system f can be considered '''unstable''' iff <math>\exists x(t) \text{ s.t. } \nexists M_2 \in \mathbb{R} \text{ s.t. } \forall t \in \mathbb{R}, |f(x(t))| \leq M_2</math>
-(ie, The system f is stable iff there exists a bounded x(t) for which f(x(t)) is not bounded.)