ECE 421

Root Locus Example #2
Plotting and Reshaping the Root Locus



This example looks at the root locus plot for a particular open-loop transfer function, Gp(s). This transfer function would represent some system which is to be controlled. We will also look at how the root locus of Gp(s) can be reshaped so that the final root locus goes through a particular point in the s-plane. The reshaping is done by placing another transfer function, Gc(s), in series with Gp(s). The poles and zeros of Gc(s) are chosen to make the total phase shift of Gc(s)Gp(s) at the chosen point equal to 180 degrees (K > 0) or 0 degrees (K < 0). The gain of Gc(s) would be chosen to make the total magnitude of Gc(s)Gp(s) equal to 1 at the chosen point. The transfer function Gp(s) is

A particular point s = s1 will be specified as a desired closed-loop pole. If the root locus of Gp(s) passes through s1, then s = s1 can be made a closed-loop pole merely by placing a gain Kc in series with Gp(s) and adjusting the value of Kc to satisfy the magnitude criterion. The phase criterion will already be satisfied if the root locus of Gp(s) passes through s1. If the root locus of Gp(s) does not pass through s1, then additional positive or negative phase shift will be needed to satisfy the phase criterion at s1.

In general, the point s = s1 would be chosen as a desired closed-loop pole under the assumption that time-domain performance requirements, such as overshoot and settling time, will be satisfied if s1 is a closed-loop pole. Often, 2nd-order approximations are used to decide an appropriate s1. For this example, the value for s1 is chosen rather arbitrarily for illustration, without worrying about time-domain performance. The point s1 is

First, we will compute the magnitude and phase of the transfer function Gp(s), evaluated at the point s = s1. Graphically, we can draw vectors from each of the open-loop poles and zeros to the point s1. The magnitude of Gp(s) at s1 will be the gain (8) times the lengths of each of the vectors drawn from zeros to s1 divided by the lengths of each of the vectors drawn from poles to s1. In MATLAB, the "polyval" function can be used to evaluate the numerator and denominator polynomials at s1, and the "abs" and "angle" functions can be used to determine the magnitude and phase.
Evaluating Gp(s) at s = s1

At the point s = s1, Gp(s) has a magnitude of 1.0346 and a phase of -150.6201 degrees. Since the phase shift is not an integer multiple of 180 degrees (either even or odd), then the root locus of Gp(s) does not pass through s1 for either K > 0 or K < 0.

Since Gp(s) has 5 poles and 3 zeros, there will be 2 branches of the root locus which go to infinity as the magnitude of K goes to infinity. The directions that these branches take depends on the sign of K. The root locus plots assume that there is a variable gain K in series with Gp(s). The root locus plots for K > 0 and K < 0 are provided. On each plot, the point s1 is also shown. The plots obviously show that neither of the plots pass through s1.
Root Locus for Gp(s), K > 0
Root Locus for Gp(s), K < 0

Since the phase shift of Gp(s) at s = s1 is approximately -150 degrees, the easiest way to satisfy the phase angle criterion is to add a -30 degrees of phase shift, measured at s1. Since poles provide negative phase shift (since they are in the denominator), the minimum amount of adjustment to Gp(s) would be one pole placed so as to give approximately -30 degrees of phase shift at s1, and a gain which could be adjusted to satisfy the magnitude criterion. A simple bit of trig shows that one pole located at s = -7.3285 provides the -29.3799 degrees necessary to make the root locus of Gc1(s)Gp(s) pass through s1. The magnitude of Gc1(s)Gp(s) without any gain is 0.1692. Therefore, the compensator gain must be 5.9103 so that the magnitude criterion is satisfied at s1. The transfer function for the compensator Gc1(s) is shown below. The root locus of Gc1(s)Gp(s) is plotted, and it is seen that the root locus of the compensated system does pass through s1. Since the gain has been chosen to satisfy the magnitude criterion at s1, then s1 is an actual closed-loop pole for the compensated system.
Root Locus for Gc1(s)Gp(s), K > 0

The transfer function Gc1(s) is the minimum configuration required to work with Gp(s) to make the specified s1 a closed-loop pole. Notice that the total number of poles in Gc1(s)Gp(s) is now 6, while the number of zeros is still 3. Since there are now 3 more poles than zeros, there will be 2 branches of the root locus which move into the right-half of the s-plane with increasing gain. This might not be desired. An alternate way of achieving the necessary phase shift is to use a compensator with 1 zero and 1 pole, with their locations chosen so as to satisfy the phase angle criterion at s1. With this approach, the excess number of poles over zeros is not increased. These locations are not unique, although there are constraints on the locations. The gain of the compensator would again be chosen to satisfy the magnitude criterion at s1. Details of the design procedure will be covered later. For a more detailed root locus design example, see Root Locus Design Example #1.
Root Locus for Gc2(s)Gp(s), K > 0

MATLAB Code

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Lastest revision on Wednesday, June 7, 2006 12:16 PM