Solutions2HW9

# Solutions2HW9 - 9.1 9.2 Find one minimal and one...

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Unformatted text preview: 9.1 9.2 Find one minimal, and one non-minimal realization for the transfer ﬁmction 3 2 _ g“): s 3+6.9 +3.3 10 s +4s2+s—6 Solution: 21) This gives the differential equation ji+4j5+jzrﬁy=ﬁ+6i£+3ﬁh10u Integrator realization: This gives 0 l 0 0 i: 0 0 1 + 0 u 6 —I —4 l y=[—4 2 2]x+u b) Cancel poles and zeros at —2 and I : 5+5 g(s)=—, or y+3y:n+5u. 5+3 y=2x+u x':ﬂ3x+u Find a minimal reaiization for the transfer function 2+3 (2 =— g ) zz+7z+10 Solution: 9.4 Given below are two different realizations. a) Show that they are each minimal realizations of the same system. b) Find the similarity transformation matrix Tthat relates the two. 0 1 0 o 1 A- o o 1 A 1 — 1 0 0 1- 517 91* _1 0 0 —12 —19 —s o —1 —9o 29 —55 —1 ‘2 0 1 0 A2: —3 1 —2 152: 0 1 c2=[o I 0] 133 —42 31 1 3 — Solution: 3) Two realizations have the same impulse response, i.e., 121(51 — A,)'ib. : 132(51' n A2)"b2 1 3+3 sz+9s+26 (s+l)(s+3)(s+4) —(s+8) —(32+9s+26) b) P2 =(Q§92)"Q§QIPI = TP. 0 l 0 l 0 0 o w] 0 —l o o —3 1 —2 0 l 0 92— 3 —1 2 ‘ Q" 0 —1 0 l 72 l 0 0 l -1 2 —1 0 0 —1 —3 —l —2 T ‘1 T T=[QzQz) 9291: 0 O ...
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Solutions2HW9 - 9.1 9.2 Find one minimal and one...

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