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Root Locus. Closed-loop control system with a variable parameter K. Unity feedback control system. The gain K is a variable parameter. Root locus for a second-order system when K e <K 1 <K 2 . The locus is shown in color. Evaluation of the angle and gain at s 1 ,for gain K=K 1.

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Root locus for a second-order system when K parameter.e<K1<K2. The locus is shown in color.


Evaluation of the angle and gain at s parameter.1,for gain K=K1.



(a) The zero and poles of a second-order system,(b) the root locus segments,and (c) the magnitude of each vector at s1.


A fourth-order system with (a) a zero and (b) root locus. locus segments,and (c) the magnitude of each vector at s1.


Illustration of the breakaway point (a) for a simple second-order system and (b) for a fourth-order system.


A graphical evaluation of the breakaway point. second-order system and (b) for a fourth-order system.


Closed-loop system. second-order system and (b) for a fourth-order system.


Evaluation of the (a) asymptotes and (b) breakaway point. second-order system and (b) for a fourth-order system.


Illustration of the angle of departure:(a) Test point infinitesimal distance from p1;(b) actual departure vector at p1


Evaluation of the angle of departure. infinitesimal distance from p



1.Write the characteristic equation in pole-zero form so that the parameter of interest k appears as 1+kF(s)=O.

2.Locate the open-loop poles and zeros of F(s) in the s-plane.

3.Locate the segments of the real axis that are root loci.

4.Determinethenumberofseparateloci.

5.Locate the angles of the asymptotes and the center of the asymptotes.

6.Determine the break away point on the real axis(if any).

7.By utilizing the Routh-Hurwitz criterion, determine the point at which the locus crosses the imaginary axis(if it does so).

8.Estimate the angle of locus departure from complex poles and the angle locus arrival at complex zeros.


Root loci as a function of that the parameter of interest k appears as 1+kF(s)=O.α and β:(a) loci as α varies; (b) loci as β varies for one value of α = α1


A region in the s-plane for desired root location. that the parameter of interest k appears as 1+kF(s)=O.


A feedback control system. that the parameter of interest k appears as 1+kF(s)=O.


Closed-loop system with a controller. that the parameter of interest k appears as 1+kF(s)=O.







Root locus of the system if K arm.2=0,and K1 is variedfrom K1=0 to K1=∞,and Gc(s)=K1.





















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