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Unformatted text preview: e e xternal t erminals, b ut i ts t ransfer
f unction H(8) =
is s ilent a bout i t. T he s ystem i n F ig. 13.9a is controllable b ut
n ot o bservable, whereas t he s ystem in Fig. 13.9b is observable b ut n ot c ontrollable.
T he t ransfer f unction d escription o f a s ystem l ooks a t a s ystem o nly f rom t he
i nput a nd o utput t erminals. C onsequently, t he t ransfer d escription c an s pecify o nly
t he p art o f t he s ystem w hich is coupled t o t he i nput a nd t he o utput t erminals.
F igures 1 3.lOa a nd 1 3.lOb show t hat i n b oth c ases only a p art o f t he s ystem t hat
h as a t ransfer f unction H (8) =
is c oupled t o t he i nput a nd t he o utput t erminals.
T his is t he r eason w hy b oth s ystems h ave t he s ame t ransfer f unction H (8) =
T he s tate v ariable d escription ( Eqs. 13.79 a nd 13.82), o n t he o ther h and, c ontains a ll t he i nformation a bout t hese s ystems t o d escribe t hem c ompletely. T he
r eason is t hat t he s tate v ariable d escription is a n i nternal d escription, n ot t he e xternal d escription o btained f rom t he s ystem b ehavior a t e xternal t erminals.
M athematically, t he r eason t he t ransfer f unction fails t o d escribe t hese s ystems
c ompletely is t he f act t hat t heir t ransfer f unction h as a c ommon f actor 8 - 1 i n
t he n umerator a nd d enominator; t his c ommon f actor is canceled o ut w ith a c onsequent loss o f t he i nformation a bout t hese s ystems. S uch a s ituation o ccurs w hen
a s ystem is uncontrollable a nd/or u nobservable. I f a s ystem is b oth c ontrollable
a nd o bservable (which is t he c ase w ith m ost o f t he p ractical s ystems) t he t ransfer
f unction d escribes t he s ystem c ompletely. I n s uch a c ase t he i nternal a nd e xternal
d escriptions a re e quivalent. S!l S!l a nd
(13.84a) (13.84b) S!l T he first row of 13 is zero. Hence t he second mode (corresponding to A l l ) is not
controllable. However, since none of t he columns of C vanish, both modes are observable
a t the output. Hence the system is observable but not controllable.
We reach to the same conclusion by realizing the system with the s tate variables Z l
a nd Z 2. T he two s tate equations are z =Az+Bf y =Cz
F rom Eqs. (13.83) and (13.84), we have :h = Zl a nd t hus
y = Zl + Z2 (13.85) Figure 13.lOb shows a realization of these equations. Clearly, each of t he two modes is
observable a t t he o utput, but t he mode corresponding t o Al = 1 is not controllable. • o C omputer E xample C 13.6
Solve Example 13.11 using MATLAB.
A =[l 0 ;1 - 1]; B =[l; 0]; C =[l -2];
[V, L ]=eig(A);
P =inv(V); 1 3.6 s! S tate-Space Analysis o f D iscrete-Time Systems W e have shown t hat a n n th-order d ifferential e quation c an b e e xpressed i n
t erms o f n f irst-order differential equations. I n t he following analogous p rocedure,
we s how t hat a n n th-order d ifference e quation c an b e e xpressed in t erms o f n firstorder d ifference e quations.
C onsider t he z -transfer f unction
8 24 13 S tate-Space Analysis 13.6 S...
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