[Automatic Control] Plot a Root-locus Approximately

2017. 6. 23. 22:51 글(https://publisher2016.tistory.com/62)을 이동

 

[자동제어] 근궤적 근사적으로 그리는 법

 

Given a G(s)H(s) : Loop Transfer Function. 

Then, you can derive the information of a characteristic equation from a given loop transfer function. 

Follow next steps for plotting a Root-Locus.

Characteristics.

1. Number of branches = the order of the polynomial(the number of closed-loop poles)

2. Root Loci is symmetrical about the real axis. 

 

 

1. Recognize points of start and end
   If a loop transfer function is given by KG(s)H(s) where K is a constant, then the characteristic equation is 1 + KG(s)H(s) = 0.
   This form can be changed like G(s)H(s) = -1/K. 
   At first, when K = 0, that is G(s)H(s) -> inf. as K->0, s is the pole of G(s)H(s).
   Next, when K = inf, that is G(s)H(s) -> 0 as K->inf, s is the zero of G(s)H(s).
   Plot poles and zeros on a Re-Im plane. 
2. Interval of the existence of roots on the real axis. 
   At here G(s)H(s) = -1/K, if 0<K<inf, then G(s)H(s) < 0. You just solve and find the interval of this inequality G(s)H(s) < 0. However, there is a more simple way. If odd number of poles or zeros are on the right side of some interval, then that interval is a root locus. But even, it isn't. You can find the reason of this from an angle condition.
3. Asymptotes(Behavior at infinity)
   θ = (2*l + 1)π / |n - m| where l = 1, 2, 3, ..., |n-m| -1, n≠m, n is the number of finite poles and m is the number of finite zeros.
   Intersection of the asymptotes : σ = (Σ Poles of G(s)H(s) - Σ Zeros of G(s)H(s) ) / (n - m)
4. Breakaway point
   From -K = 1 / (G(s)H(s)), solve this equation, -dK/ds = d( 1 / (G(s)H(s)) )/ds
5. Angles of departure and arrival
   From G(s)H(s) = -1/k, ∠(G(s)H(s)) = ∠(-1/K)
   => ∠(G(s)H(s)) = (2*l + 1)π ( or you just write π). 
   Next, expand and arrange the equation in the LHS(Left Hand Side). 
   And a term that you want to know an angle of departure or arrival is set by θ. 
   A variable s in the each term in the LHS is set by bar of s which is so close to original truly s.  
6. Intersection of the root locus with the imaginary axis
   Use the Rout-Hurwitz Test.

 

간단히 말해서

1. K -> 0, K -> ∞ 출발점과 도착점
2. 실수축에 존재하는 근의 범위
3. s가 무한대일때 출발점과 도착점(점근선 각과 실수축 교차점)
4. 분기점
5. 출발각, 도착각
6. 허수축 교차점