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Unformatted text preview: orced responses appear in Eq. 2.51b. Here we shall
solve this problem by the classical method, which requires the initial conditions a t t = 0+.
These conditions, already found in Eq. (2.16), are • and
The loop equation for this system [see Example 2.2 or Eq. (1.55)J is
(D2 + 3D + 2) y(t) = D f(t) T he characteristic polynomial is ) ,2 + 3), + 2 = (), + 1)(), + 2). Therefore, the natural
Yn(t) = K 1e- t + K 2e-2t
T he forced response, already found in Example 2.10 ( a), is T he total response is
y (t) = K le- t + K 2e- 2t _ 1 5e- 3t Differentiation o f this equation yields Setting t = 0+ a nd s ubstituting y(O+) = 0, y(O+) = 5 in these equations yields f ='S·t+3'j
y t=dsolve('D2y+3.Dy+2.y=f' , 'y(O) = 2', ' Dy(O) = 3', ' t')
y t = - 9/4+5/2.t-19/4*exp(-2*t)+9*exp(-t) 0 Assessment o f t he Classical Method
T he d evelopment in t his s ection shows t hat t he c lassical m ethod is r elatively
simple c ompared t o t he m ethod o f finding t he r esponse a s a s um of t he z ero-input
a nd z ero-state c omponents. Unfortunately, t he classical m ethod h as a s erious drawback b ecause i t yields t he t otal r esponse, which c annot b e s eparated i nto components a rising from t he i nternal c onditions a nd t he e xternal i nput. I n t he s tudy o f
s ystems i t is i mportant t o b e a ble t o e xpress t he s ystem r esponse t o a n i nput f (t) a s
a n e xplicit function of f (t). T his is n ot p ossible in t he classical m ethod. Moreover,
t he classical m ethod is r estricted t o a c ertain class o f i nputs; i t c annot b e a pplied
t o a ny i nput. A nother m inor p roblem is t hat b ecause t he classical m ethod yields
t otal r esponse, t he a uxiliary conditions m ust b e o n t he t otal r esponse which exists
only for t ~ 0 +. I n p ractice we a re m ost likely t o know t he c onditions a t t = 0 (before t he i nput is applied). Therefore, we n eed t o d erive a new s et o f a uxiliary
conditions a t t = 0 + from t he k nown conditions a t t = 0 -.
I f we m ust solve a p articular l inear differential e quation o r find a response o f
a p articular L TI s ystem, t he classical m ethod m ay b e t he b est. I n t he t heoretical
s tudy o f l inear systems, however, t he classical m ethod is p ractically useless. 2 .6 System Stability B ecause o f t he g reat v ariety of possible s ystem b ehaviors, t here a re several
definitions o f s tability i n t he l iterature. H ere we s hall consider a definition t hat is
s uitable for causal, linear, time-invariant (LTI) systems.
I n o rder t o u nderstand s ystem s tability i ntuitively, let us examine t he s tability
c oncept as a pplied t o a r ight c ircular cone. S uch a cone c an b e m ade t o s tand
forever on its circular base, o n i ts apex, o r o n i ts side. For t his r eason t hese t hree
s tates o f t he c one a re s aid t o b e e qUilibrium s tates. Q ualitatively, however, t he
t hree s tates show very different behavior. I f t his cone, s tanding o n i ts circular bas...
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