Question

QS. (a) A system has the transfer function 5+1 G(s) s'+33-10s - 24 Use the Routh-Hurwitz stability check to determine whether this system is stable or not stable, and state why. [10 marks] (b) Consider the system shown in Figure 5.1, where R(s) is the system input, Y(s) is the system output, K, represents a proportional controller, G(s)=- s? +45 +8 1 and - 5 R(5) Y(s) K G(s) H(s) ) Figure 5.1 Determine the range of values of the gain K, for which the closed-loop system will be stable. Show all your working. [10 marks) (ii) State the critical gain obtained in (i) above, and determine the corresponding cross-over frequency. [5 marks] [Total marks for Q5:25] 06 41 M

Answer 1

$+37-05-14 using Rover stability Confteria, 833-105-24 etaAbm Characterstic S3 2010 A system ů 82 stable only 3 24 when all the the S 3x610) - 1x624) 144 3 so - 24 elements of column of Routh table ů (tue), given system & stable. then the 53 1 -10 82 ď w -24 s' so -24 o As, the values the elements of 1 column has Eve) in the Routh Stability table, hence system Unstable. the given

$+37-05-14 using Rover stability Confteria, 833-105-24 etaAbm Characterstic S3 2010 A system ů 82 stable only 3 24 when all the the S 3x610) - 1x624) 144 3 so - 24 elements of column of Routh table ů (tue), given system & stable. then the 53 1 -10 82 ď w -24 s' so -24 o As, the values the elements of 1 column has Eve) in the Routh Stability table, hence system Unstable. the given

Ang ay R61 بر کا ko stuste /st3 f در گاو kp g²+uste 1 /st3 --> TSV GCS) IF GBS)HCS) Ke +५+8 1 1 + kp Х getusti (5+3) D Ko (5+3) (5+3)($+45+8)+K ke(5+3) TS) Transfer function s+as+208 +(34 +Kp)

characters the equation, $+75 +205 +(24+ kp) sº osing Routh stability Criteria Now, 7- an darron as (sutike) 140-124+K) 7 SC zutke By Routh stability table, fes a system to be stable, the Istr column elements have to be positive (sojo 0 oo using this, 24+kp > 0 2) kp>-24 1 >0 7 140-(24+kp) 116-kp so ke L116 - is for the given system to be stable -24 kp <116

۱- Here, Transfer function, Kp (5+3) (5+3)(8+45+8)+ kp Putting go kp (jw+B) (ju+3) (j=W²+ u jw +B) + kp 32 =(-1) 1) ke ljw+3) (ju+3) (-w²tuju +8) + xp He low+3) -Jw3-4W²-3w²th o rojo +24 tke kp (jw +3) -jw3_7w²trojw trutke ke (jw+3) -762 +Dutke +jW ( 20-02) Gain cross over frequency Me (jiwa), 14 Gola ) = 1.