Question

4) Using the Routh-Hurwitz Criterion, determine whether the following Polynomials are Stable or Unstable. Please Show Supporting Work: 1) H(s) = s? + 10s + 5 = 0 Stable Unstable 11) H(s) = s4 +53 + 5s2 + 20s + 10 = 0 Stable Unstable 111) H(s) = 83 + 4Ks2 + (5 + K)s + 10 = 0 The Range of K for a Stable System is: a. b. K > 0.46 K< 0.46 0<K <0.46 Unstable for all K ن ن IV) H(s) = 33 + 10s? + 50s + K = 0 Determine the Range of K for System Stability:

Answer 1

1) HS) = S²+ 105 +5. RH Table 2 1 St 10 s 5 5 o r column in 6 ooooo Since there the given are no sign changes system tis) is STABIE a ) H () = + 5+ 6s+ 20 + 0 = 0 RH Table: 84 1 s3,1 s² 20 5 10 200 10 o so 10 There are two sign first column changes in So, the given system is UNISTABIE 11 H) +4 ks2+15+15+10 20 RH Table! s3 1 5+k s² 4K 10 S (4)5+)-100 41

2 For stable system, in first column should all elements that, be of some size, for (4*) 5+k)>10 1675 Bur 20x + 4 K ²10 4k?+20k -10 > 0 Qk? 10K -5 > KC-5.45 We get the and solution [K30,46 / Nor possible Since ko So, Range of kis k>OME for stable system -IV) Hol= S²+105² +50stk 3 1 50 st 10 K s 500- o (9 for Stable system - 500K > 0 (k < 500 and ko To <k (500) for stability,