Module g Routh's method. Root locus Magnitude and Phase Equations (4 hours)
Module 9 Routh’s Method, Root Locus : Magnitude and Phase Equations (4 hours)
9.1 Routh's stability Criterion About Stability: (P145. Section 1) 9.1.1 Define on the Stability of Closed-loop System When the transfer function of system is q(s), the output C(s)=中(s)R(S)= G(s) R(S) KM(S R(s) 1+G(s)H(s) S-S Suppose C(s) KM(S-3 r()=(),R(S)=1 S-S S-S
9.1 Routh’s Stability Criterion About Stability: ( P145. Section 1) 9.1.1 Define on the Stability of Closed-loop System: When the transfer function of system is Ф(s), the output is: ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 R s s s K M s R s G s H s G s C s Φ s R s i n i − = + = = = Suppose r(t) = (t), R(s) =1 = = − = − = n i i i i n i s s c s s KM s C s 1 1 ( ) ( ) ( )
CO)=LC()=∑ce if V re s;<0,Ve→>0(t→>∞) The system Is stable, f any Res; >0, the ene→>o(t→) Then total response C()=∑ce→>o The system is unstable
s t n i i i C t L C s c e = − = = 1 1 ( ) [ ( )] if Re s 0, e →0 (t →) st i i The system is stable; if any Re s 0, then e → (t → ) s t j j = → = s t n i i i C t c e 1 Then ( ) total response The system is unstable
Conclusion The system is stable only when all the closed loop poles are located in the left-hand half of the s complex plane ap) The system become unstable as soon as one closed-loop pole is located in the right-hand half of the s complex plane (rhp) Stable Unstable
Conclusion • The system is stable only when all the closedloop poles are located in the left-hand half of the s complex plane (LHP); • The system become unstable as soon as one closed-loop pole is located in the right-hand half of the s complex plane (RHP)
(a) stable (b) Neutral (c) Unstable