Unformatted text preview: t he
c omplex p lane a nd d etermine w hether i t is asymptotically stable, marginally stable, or unstable.
( a) D (D + 2 )y(t) = 3 f(t)
( b) D 2(D + 3 )y(t) = ( D + 5 )f(t)
( c) ( D + l )(D + 2 )y(t) = ( 2D + 3 )f(t) + 1)(D2 + 9)y(t) ( d) (D2 ( e) ( D + 1) (D2 - 4D + 9)yet) = (D + 7 )f(t) = (D2 + 2D+4) f (t) Answer: ( a) m arginally s table ( b) u nstable ( c) s table ( d) m arginally s table ( e) u nstable. 'V , . .... . ' (e) ( I) ..... , t- 2.6-1 System Response t o Bounded Inputs
F rom t he e xample of t he r ight circular cone, i t a ppears t hat w hen a system is
in s table e quilibrium, application of a small force (input) produces a small response.
In c ontrast, w hen t he s ystem is in u nstable e quilibrium, a small force (input) produces a n u nbounded response. Intuitively we feel t hat every b ounded i nput s hould
produce a b ounded r esponse in a stable system, whereas in a n u nstable system this
would n ot b e t he case. We shall now verify this hunch a nd show t hat i t is indeed
Recall t hat for a n L TIC system i:
i: yet) = h(t)
(g) = (h) F ig. 2 .16 Location of characteristic roots and the corresponding characteristic modes. Therefore ly(t)1 :::;
The characteristic polynomials of these systems are
( b) ( c)
( d) + 1) (A2 + 4A + 8) = (A + 1)(A + 2 - j2)(A + 2 + j 2)
(A - 1) (A2 + 4A +8) = (A - 1)(A + 2 - j2)(A + 2 + j 2)
(A + 2)(A2 + 4) = (A + 2)(A - j2)(A + j 2)
(A + 1)(A2 + 4)2 = (A + 2)(A _ j 2)2(A + j 2)2
(A * j (t)
(2.64) h (r)j(t - r)dr I h(r)llj(t - r)1 dr Moreover, if j (t) is b ounded, t hen I j(t - r)1 < K l < 0 0, a nd 152 2 T ime-Domain Analysis of Continuous-Time Systems 2.7 Intuitive Insights into System Behavior 153 Because h (t) c ontains t erms o f the form eA;t o r tkeA;t, h(t) decays exponentially
w ith t ime if Re A j < O. Consequently, for an asymptotically stable s ystemt a nd i: Ih(T)ldT < K2 < (2.65) 00 T hus, for a n a symptotically stable system, a bounded i nput always produces a
bounded o utput. Moreover, we c an show t hat for an unstable or a marginally
stable system, t he o utput y(t) is u nbounded for some bounded i nput (see Problem
2.6-4). These r esults lead t o t he formulation of a n a lternative definition of stability
known as b ounded-input, b ounded-output ( BIBO) s tability: a s ystem is
B IBO s table if, a nd only if, a b ounded i nput p roduces a bounded o utput. Observe
t hat a n a symptotically s table system is always B IBO stable.:I: However, a marginally
s table s ystem is B IBO u nstable.
Intuition can cut the math jungle instantly!
Implications o f Stability
All p ractical signal processing systems must be stable. Unstable systems a re
useless from t he viewpoint of signal processing because any set of intended or unintended initial conditions leads t o a n u nbounded response t hat e ither destroys t he
s ystem or ( more likely) leads it t o some s aturation conditions t hat change t he nature of t he s ystem. Even if t he discernible initial conditions are zero, s tray voltages
or t hermal n oise si...
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