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(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
ction 6.1 The Concept of Stability 跨越华盛顿州 Puget Sound 的塔科马峡谷的首座大 桥。开始晃动时 测速
跨越华盛顿州Puget Sound 的 塔 科 马 峡 谷 的 首 座 大 桥——开始晃动时
≈? 灾准发生时
灾难发生时
9.2 Algebra Stability Criterion (代数稳定判据) +G(s The closed loop transfer function is C(S) G(s) H(S) R(S) 1+G(S)H(S) To insure stability we require that the roots of the following characteristic equation are on the left half plane 1+G(s)H(S)=0 Let's assume that the characteristic equation is an n-th order polynomial 1+G(s)H(s)=aoS"+aS"++amS+a And its characteristic roots are 12
9.2 Algebra Stability Criterion (代数稳定判据） And its characteristic roots are n s , s , s 1 2
On the basis of the relationships between roots and coefficients of the equation we know that ∑ If we request that these roots have all negative real-part, those values ∑ must all be ∑SsSk k=1 positive, and must be no zero ≠k Otherwise there is one positive real-part root at least (D'Is The necessary condition of stable system
• On the basis of the relationships between roots and coefficients of the equation, we know that: = − = − = = − = = = = i n i n n n i j k i j k i j k n i j i j i j n i i s a a s s s a a s s a a s a a 1 0 , , 1 0 3 , 1 0 2 1 0 1 ( 1) must all be If we request that these roots s1 -sn have all negative real-part, those values 0 0 2 0 1 , , a a a a a a n positive, and must be no zero. Otherwise there is one positive real-part root at least. The necessary condition of stable system
The suflicient and Necessary condition on Stable system The routh's method 系统稳定的充要条件)(P146) To determine the stability we construct an array of coefficients called routh array 1+G(SH(s)=aos"+a,"+.+an-5+an n-1 Original data 5 a, b2 b 6.as-a 0 h, h, h3, Calculated data
The Sufficient and Necessary Condition on Stable System —— The Routh’s Method (系统稳定的充要条件）( P146 ) Original data Calculated data