Module10 root locus

The introduction

Root locus:

Objective: To find the closed loop poles

Ideas: geometry —- solving closed-loop characteristic equations: algebra

Method: Draw a picture

Essence: Search algorithm

Presentation: Rule of thumb

Advantages: Find all possible positions and trends of closed loop poles at once

Disadvantages: incompleteness

Rules for drawing root traces

Analysis of the version

K monotonically increases to form the root locus

N: number of open-loop poles

M: number of open loop zeros

The root locus is continuous and symmetric with respect to the real axis

Root trajectories cannot coincide, they must be separated if they meet

Rule1: The starting point of the root locus (K=0) is the open-loop pole, and the end point (K=∞) is the open-loop zero or infinity; There are n root trajectories (generally n≥m, Max (n,m)); We have n minus m asymptotes

Rule3 Rule4:The numerator just takes the real part, because the imaginary parts of the complex conjugate roots cancel each other out

Rule5: exit Angle of pole =180- “pole” + “zero”; Incident Angle =180- “zero” + “pole”; “Zero” and “pole” refer to the included Angle between the line of two points and the real axis; Only for the complex root poles and the complex root zeros, the poles and zeros on the real axis are not needed

Rule6: Solve the intersection of root locus and imaginary axis :s=jω substitute 1+GH=0 to make the real part and imaginary part 0 to solve ω; The K at the intersection with the imaginary axis is the critical stable K value

Rule7:K value is not necessarily monotonically increasing, there will be many local extreme points, so there will be confluence points and separation points. Method: Differentiate with respect to K

Easy wrong! : The common separation point of the root locus is the separation point of the two root locus branches on the real axis == the position between two poles/between two zeros/multiple poles; It doesn’t have to be either!! See below

Rule8: separation point and separation Angle of root locus; The separation point is the position between two poles on the real axis or between two zeros or multiple poles with an Angle of 360 over the number of relevant poles

Rule9: Given the closed-loop pole S, calculate K

| h | = 1 K = s attachment to all pole length of the product/s to all the products of the zero line lengthCopy the code

Note: a K value corresponds to n poles (n is the number of open loop poles/the number of root trajectories/the number of closed loop poles); Because 1+GH=0 given a K, since s is of n highest degree, there are n solutions

proof:

Version: to solve the problem.

Write the open loop transfer function G(s)H(s)

Rule1: The starting point is the open-loop pole and the ending point is the open-loop zero or infinity; There are n root trajectories; We have n minus m asymptotes

Rule2: Determines the root locus on the real axis

If there is an asymptote: Rule3, Rule4: determine the Angle and intersection of the asymptote and the real axis (technique: about the real axis symmetry, only one side) if there is no asymptote test write: Rule3 Rule4: none

The numerator just takes the real part

If there are complex roots: Rule5: used only for complex roots: (pole) exit Angle =180- pole + zero; Angle of incidence =180- zero + pole

If there is an intersection point between the root locus and the imaginary axis: Rule6: Solve the intersection point between the root locus and the imaginary axis :s=jω substitute 1+GH=0 and make the real part 0 to solve ω

☆ To solve the separation point: Rule7 Rule8: separation point and separation Angle of the root trajectory:

Separation point:

Separation Angle of the Angle included with the real axis:.Represents the number of overlapping points at the point leaving the real axis(Or combined with symmetry, the Angle between l rays is 360/ L)

Rule9: Given the closed-loop pole S, calculate K

K is equal to the product of the lengths from s to the poles divided by the product of the lengths from s to the zerosCopy the code

Rule10: n-m≥2, then the sum of the real parts of the closed loop poles = the sum of the real parts of the open loop poles

Handout by Mr. Wang

Analyze system performance with root locus

After the open-loop poles are added in ↑:②, K* must be limited from the original infinity to (0,6) if the system is to be stable.

DR