Question

3. For the feedback control system shown in Figure Q3 below, the forward-path transfer function given by G(s) and the sensor transfer function is given by H(s). R(s) C(s) G(s) H(s) Figure Q3 It is known that G(s) -- K(+20) S(+5) H(s) = and K is the proportional gain. (S+10) i. Determine the closed-loop transfer function and hence the characteristic equation of the system. [6 marks] ii. Using the Routh-Hurwitz criterion, determine the stability of the closed-loop system. Determine the range of K, in which the system is stable. [8 marks] Find the value of K for marginal-stable and the frequency of oscillation for marginal stability [6 marks]

Answer 1

Solution- CCS) ROS) GCS) HOS) Jis the above quation dat of control System Forward - path transfer fuckion gloven by = G CS) The Sensor transfer function is given by HCS) Here KC 5420) SC 5+5) GCS) = & K is the proportional again H(S) (S410) K-10 RCS) HCS) = 005) H(S) = K lo lo (1tk) at (14101141 Operation feedback Closed toop Control System formula CCS) GCS) RCS) I+GCS) HCS) 5 ( st2b)/s (545) [ Stro] Ho CXC SERO)) S(545) 5 SCS +5) (5710) s(5 + 25 + 105) (5+10) s (5+5+105) 5 RIS)

SC S+5+55) SV +55 +55" 65 +5.5 S (6+5) ... CCS) RCS) = Ils 10 charactersilic Equation of the System | + GS) HCS) = 0 It loc $220) $(545) 1 + 200 10 [ co 1 trov) х 575 It 1000 lt I S15 - 'S+6 5+5 ng (b] The character the c equation of oth order combills system can be Waile as ansh+anis + aist do = P Stability caiterion is applied using Routh table +-.- The which is sh any an an-2 Ign- an- an-3 an-I- b3.. - bi b2 - G C2 (3- -

ao re coefficient of the charactreshic where an ans equation bi- an an-2.. an an-3 -1/2 last an-1 bz an- an-y-anan-5 4 an-l bi an-3-an- 62 3.5 bi (2 b. an-5 - au-l -1 bi s² tlosv + 20 S +100 GICS) HiCS) = 4(5+6) S+5 S . - 30 s: 0 sº : 50 IG CSI = 53HOST+5 °+205 +100 cc) The value of “x4 for Stability and frequency of oscillahon for marginal Slablely The characterishe equahon has roots on imaginary axis (jo ars) with the steady-stale output be a sustained Oscillation. the

K cstz) TCS) = (S+PI) ( stwmv) Frequency : w fwn K GCS) SCH+0345) (+0.255) 10 2 (1+0.4 (21) (+0.25 (2) 16 21 1-8)(1.5

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