Signal Processing and Linear Systems-B.P.Lathi copy

4 linear transformation o f s tate vector 815 taking

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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) = 1. 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 (13.86a) a 8 24 13 S tate-Space Analysis 13.6 S...
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