# Introduction to Root Locus. What is root locus?

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1 Introduction to Root Locus What is root locus? A graphical representation of the closed loop poles as a system parameter (Gain K) is varied Method of analysis and design for stability and transient response Can be used to solve first and second order systems but its real power lies in its ability to provide solutions for system of higher order than 2 Can be used to qualitatively describe the performance of the system as various parameters are changed. o Transient response: The effect of varying gain upon % overshoot, settling time and peak time. o Stability: can see the range of gain for stability, range of gain for instability and the conditions that cause the system to break into oscillation The control system problem The poles of the open loop transfer function are easily found (typically known by inspection and do not change with changes in system gain) The poles of close loop transfer function are more difficult to find (typically cannot be found without factoring the close-loop system s characteristic polynomial which is the denominator of the closed-loop transfer function). The closed-loop poles change with changes in system gain. Types of transfer functions: o Feedforward transfer function : o Open-loop transfer function o Closed-loop transfer function

2 Let and, then where N and D are factored polynomials and signify numerator and denominator terms respectively. Therefore a zero of T(s) consist of zero of G(s) and poles of H(s) but the poles of T(s) are not immediately known and in fact can change with K. Since the system s transient response and stability are dependent upon the poles of T(s), we have no knowledge of the system s performance unless we factor the denominator for specific values of K Vector representation of complex number Any complex number σ + jω, described in Cartesian coordinate can be graphically represented by a vector. The complex number can also be described in polar form with magnitude M and angle θ as M θ. If the complex number is substituted into a complex function F(s), another complex number will result. If F(s)=s + a and s= σ + jω, therefore F(s) = σ + jω + a= (σ + a) + jω. With F(s)=s + a, we know that F(s) has a zero at a and if we translate the vector a unit to the left, we have an alternative representation of complex number that originates at the zero of F(s) and terminates on the point s= σ + jω. We conclude that (s + a) is a complex number and can be represented by a vector drawn from the zero of the function to the point s. For example (s + 7) with s = 5 + j2, is a complex number drawn from the zero of the function (-7) to the point s which is 5 + j2. In an actual case, there will be more than one zero and one pole, therefore the calculation of final M and θ can be done as follows:

3 Evaluation of complex function via vectors Given Find G(s) at the point s = -3 + j4 Solution 4 (s) 3 2 (s+2) (s+1) Figure 1.1: Vector representation of G(s) Let the required point Label all poles and zeros: Find the magnitude and angle of all poles and zeros:

4 The angle given by tan -1 is an acute angle and since the location of the required pole p d is in the second quadrant of p 1, the actual angle is = Therefore G(s) at the point s = -3 + j4 is Defining the Root Locus R(s) + E(s) C(s) - Figure 1.2

5 The root locus technique can be used to analyze and design the effect of loop gain upon the system transient response and stability. Figure 1.2 is a block diagram representation of a tracking system where the closedloop poles of the system change location as the gain K is varied. Table 1 was formed by applying the quadratic formula to the denominator of the transfer function in Figure 1.2. It shows the variation of pole location for different values of gain K. Table 1 Pole location as a function of gain for the system in Figure 1.2 K Pole 1 Pole j j j j j j j j j5-5 - j5 Root Locus Properties Closed-loop poles are used in analyzing the transient response of a system. Since the closed-loop poles is obtained by setting the denominator of the closed-loop transfer function to zero, therefore we can conclude the denominator of the close loop transfer function is contributing to the close loop poles.

6 Since the denominator of the close loop transfer function is always equals to 1 + open-loop transfer function (G(s)H(s)), therefore open loop transfer function can be used in determining the transient response and stability of the system. Open loop poles are obtained from the denominator of an open-loop transfer function G(s)H(s). We then plot the open loop poles on the s-plane to calculate whether the given point lies on the root locus of the system or if the given point is a close loop pole of the system. Example: R(s) + E(s) C(s) - Figure 1.3 Given a system in Figure 1.3. Determine if the following points are on the root locus: a) b) If the point is on the root locus, find the gain K Solution: a) Let the required point Label all poles and zeros:

7 If this point is a closed-loop pole for some value of gain, then the angle of the zeros minus the angle of the poles must equal an odd multiple of Therefore -2 + j3 is not a point on the root locus or alternatively we can say -2 + j3 is not a closed-loop pole for any gain. No need to find K b) Let the required point Label all poles and zeros: If this point is a closed-loop pole for some value of gain, then the angle of the zeros minus the angle of the poles must equal an odd multiple of

8 Therefore is a point on the root locus or alternatively we can say is a closed-loop pole for some value of gain K. Find gain K,

9 Reference: 1. Norman S. Nise, Control System Engineering, 4 th Edition, Wiley International Edition (2004)

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