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Unformatted text preview: for t hese e quations is shown i n Fig. 13.8b. I t c an b e s een from Fig. 13.8a
t hat t he s tates Z , a nd Z2 a re d ecoupled, whereas t he s tates X , a nd X 2 (Fig. 13.8b) a re
c oupled. I t s hould b e r emembered t hat Figs. 13.8a a nd 1 3.8b a re s imulations o f t he s ame
s ystem·t. o C omputer E xample C 13.5
Solve E xample 13.10 using MATLAB.
C aution: T he a nswer for 13 is n ot u nique.
A =[O 1 ;-2 - 3]; B =[I; 2];
[ V, L ]=eig(A);
L ambda=P* A * inv(P);
B hat=P*B 0 I f C im = 0, t hen t he s tate Z m will not appear in t he expression for Y i. Since all t he
s tates a re decoupled because o f t he diagonalized n ature of t he e quations, t he s tate
Z m c annot be observed directly o r i ndirectly (through other s tates) a t t he o utput
Y i. Hence t he m th m ode e Arnt will not be observed a t t he o utput Y i. I f C lm, C 2m, . ..
, C km ( the m th c olumn in m atrix C) a re all zero, t he s tate Z m will n ot b e observable
a t a ny o f t he k o utputs, a nd t he s tate Z m is unobservable. I n c ontrast, if a t least
one element in t he m t h column o f C is nonzero, Z m is observable a t least a t one
o utput. Thus, a s ystem w ith diagonalized equations o f t he form in Eqs. (13.78) is
c ompletely observable i f a nd o nly i f t he m atrix C h as no c olumn o f zero elements.
I n t he above discussion, we a ssumed distinct eigenvalues; for repeated eigenvalues,
t he modified criteria c an be found in t he l iterature.!,2
I f t he s tate-space description is n ot in diagonalized form, it m ay b e converted
into diagonalized form using t he procedure in Example 13.10. I t is also possible
t o t est for controllability a nd observability even if t he s tate-space description is in
undiagonalized f orm'!' 2t
• 1 3.5 E xample 1 3.11
I nvestigate t he c ontrollability a nd o bservability o f t he s ystems i n F igs. 13.9a a nd
13.9b. Controllability and Observability Consider a diagonalized state-space description of a system
(13.78b) Y = C z+Df (a) W e shall assume t hat all n eigenvalues AI, A2, . .. , An a re distinct. T he s tate
e quations (13.78a) are of t he form
m =I,2, . .. n If b ,
ml bm2 , . .. , bmj ( the m th row in matrix B) a re all zero, t hen a nd t he variable Z m is uncontrollable because Z m is n ot c onnected t o any of the
i nputs. Moreover, Z m is decoupled from all t he remaining ( n - 1) s tate variables
because of t he diagonalized n ature of the variables. Hence, there is no direct or
indirect coupling o f Z m w ith any of the inputs, and the system is uncontrollable. In
contrast, if a t l east one element in the m th row of B is nonzero, Z m is coupled t o
a t least one i nput a nd is therefore controllable. Thus, a s ystem w ith a diagonalized t Here we only have a simulated state equation; the outputs are not shown. The outputs are
linear combinations of s tate variables (and inputs). Hence, the o utput equation can be easily
incorporated into these diagrams (see Fig. 13.7). f~
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