Question

NASC

NESSARY & sufficient criterior

Use the Routh - Hurwitz technique for the Derive the N.A.S.C. for a 5th order system with arbitrary coefficients {an.... a3 (n=5). Clearly state the conditions on the a.). - Assess the stability and state the N.A.S.C. for each system in #p6ol, page 405 (also shown on Canvas in Files Homewor. of the text. Note: This problem has parte "a)" through "g). [Routh- Hurwitz is optional for 3.] - For a system given as: y = 0 (n=3) estem stable? Why or why not: :) Is the system Explain (show) ;;) What form of control algorithm should ult) have to ensure stability of the system for any initial condition set? {ý (0) = a, , ; (o) = La, y(0) = d;} 21,2,3 = any constant value set

Answer 1

) a) 5455+2 20 ast as tao zo compare equation De 2 Š I as ao šli a so acao-az-o o 1000 o stability, there should not be any sign clonges in the first column of the above tabulation Here no sign changes in the 18t column 7 so, System is stable. No. of Lelf half of poles = 2. No. of Right half of poles = 0 6) 83 45 +85+4=0 1 53 11 8 sv 4 4 s 32-42 o ³ no sign changes in so | Ax4-0 o 1st column. so system is stable no. of Left half of poles = 3 no. of right half of poles 20 7 19

e) 3 + 25 or 6S + 20 =0 $311 -6 12 20 s' 12-20 o here a sign changes 2 ss 2014 so system is unstable. No. of Right half of poles=2 No. of left half of poles = 1 d) 54 + s3+ 25 + 125+ 10 =0 5+1 53 2 12 10 0 -10x12-10 -10 13x10-0 13 s' -10x12-10 o y here a sign changes So, System is unstable No. of Right half of poles 22 neo. of left half of poly = 2 e) st+5+ 35 f 25 tk 30 5411 3 K $ 3-2 K-o o s' (2x1 - xxl o so (Q-kk-0 o 1 (26)

for system to be stable KO & (2-x) to 27K KE2 in the range of stabity is ocker 54 253+ st6=0 gr 0-(-5) Toys (+5) (5/2)-6x2 2 o here, ano. of Sign changes 161-49/5)-0 o 415 so, system is stable No. of Right half of poles = 2 No. of Left half of polss = 3 g) 56+54 +252 + s'estk 0. salik SV 1- (1-4) ko o s' k(1-K) - k o

for system to be stable KEO & KCl-K)-k to k KCl-K) - K to K-k_k to - krzo κ Σο the range of t for stability is kto 1. consider gth order system Agst aq s t aq s²t Azs + As+ AG=0. Routh array - s5 l ao ar at st a az as 33 br= (aan, dome) b;= (2.10.0.) sva (1930 - otok) = (2,6,56,62) s' die (16221-612) dz-(0) so le= di C2-40)

Nesessary condition for Stability o coefficients of the characteristic polynomial should be positive @ There should not be any missing term is thie Characteristic equation sufficient condition for stability ☺ All the elements (ao, ai, bi, ci, di, e) of the 1st column of the routh array should have the same sign.