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  University of West FloridaDepartment of Electrical & Computer EngineeringEEL4657L — Linear Control Systems LaboratorySpring 2018 Lab #6Root Locus: Graphical & Analytical Analysis Using MATLAB Objective The objective of this laboratory exercise is to utilize MATLAB to obtain the root locus for varioussystems. In addition, analytical techniques implemented in MATLAB are used to verify graphical  ω  axis crossing points with associated gain as well as break away/in points with associated gain. R ( s )  ✲ ✒✑✓✏  + − ✲  K   ✲  G ( s )  ✲ Y   ( s )      ✻ Figure 1 . Unity negative feedback system with variable gain  K  . Background Theory When the root locus technique is employed, one typically assumes a unity negative feedback systemwith open loop gain  G ( s ) cascaded with a variable gain parameter  K   as shown in Figure 1. Then,the objective is to investigate how varying  K   affects the stability as well as transient behavior of the system. In the following, the analytical techniques that are employed to obtain an estimate to aroot locus plot are presented. In addition, the necessary procedure for using the MATLAB ControlSystem Toolbox as well as ancillary  m  files to obtain required information is also discussed. Analytical Techniques  While MATLAB, as described later, can plot the root locus and numerically determine for whichrange of   K   the system is stable or which  K   leads to a certain type of transient behavior, it isnecessary to have an understanding of how to do this analytically based on the abbreviated stepsdescribed below.1. Determine the closed loop transfer function for the system shown in Figure 1 and write it inthe standard form T  ( s ) =  Y  ( s ) R ( s ) =  KG ( s )1 + KG ( s ) .  (1) EEL4657L Lab #6 1 of  6 Last revised: 17 December, 2017  In the event that the system is not a unity negative feedback system, or, in the case of a unitynegative feedback system where the parameter  K   is not a numerator factor of   G ( s ), someadditional work is required to obtain the form shown in (1). However, these situations are notconsidered in this laboratory exercise.2. Once the form for the transfer function shown in (1) is obtained, identify and plot in thecomplex frequency ( s ) plane the poles of   G ( s ), using an  × , as well as the zeros of   G ( s ), usingan  ◦ . In addition, determine the number of infinite zeros of   G ( s ). Recall that this is equal tothe order of the denominator polynomial of   G ( s ) minus the order of the numerator polynomialof   G ( s ).3. Sketch the required root locus segment(s) on the real axis in the complex  s  plane. Recall thatthe root locus segment(s) on the real axis are to the left of an odd number of poles and zeroslocated on the real axis.4. Determine whether or not the root locus for the system has break away points or break inpoints. Break away points occur if poles on the real axis come together and go to finite orinfinite zeros in the complex  s  plane. Break in points occur if poles in the complex  s  planego to finite zeros on the real axis or to infinite zeros on the real axis. Note that at breakaway or break in points, the second order (sub)system has a critically damped response in thetime domain. For break away points, values of   K   before the break away  K   value lead to anover damped (sub)system response while values of   K   after the break away  K   lead to an underdamped (sub)system. For break in points, values of   K   before the break in  K   value lead to anunder damped (sub)system response while values of   K   after the break in  K   lead to an overdamped (sub)system.If break away or break in points are determined to exist, then they respectively occur atmaxima or minima for  K   on the real axis. As such, these points are determined by finding theroots of 0 =  ddsK   =  dds  − 1 G ( s )  .  (2)That is, (2) is determined by equating the denominator of (1) to zero, solving for  K   and thendifferentiating the resulting equation with respect to  s .Note that (2) may yield more values of   s  than there are break away or break in points. Theextraneous points, which are determined by noting that break away or break in points must bereal and must occur between poles or zeros on the real axis, are eliminated leaving the actualbreak away or break in points as  s  =  s break . Furthermore, the value of   K   which leads to thebreak away or break in point at the location  s  =  s break  is determined as K  break  =  −   j ( s +  p  j )  i ( s + z i )  s = s break =  −  p ( s ) z ( s )  s = s break .  (3)5. Determine whether or not the root locus for the system crosses the imaginary (  ω ) axis. Thiscan occur under numerous circumstances. For instance, it can occur when poles in the left half  EEL4657L Lab #6 2 of  6 Last revised: 17 December, 2017  s  plane follow a trajectory to finite or infinite zeros in the right half   s  plane. It can also occurwhen poles in the right half   s  plane follow trajectories to finite or infinite zeros in the left half  s  plane. If the gain value leading to a crossing on the  ω  axis is determined to be  K  cross , then,in the first situation, the system will be stable if the gain satisfies 0  < K < K  cross  while in thesecond situation, the system will be stable if   K > K  cross .To determine whether or not a  ω  axis crossing exists, form the Routh–Hurwitz table for theclosed loop system  T  ( s ) shown in (1). That is, if   G ( s ) =  z ( s ) /p ( s ), then the polynomial for theRouth–Hurwitz table is  p ( s )+ Kz ( s ). Since  ω  axis crossings occur when an entire row in theRouth–Hurwitz table is zero, determine if there are any  K   values that lead to a row of zerosin the Routh–Hurwitz table by starting with the first row and proceeding to the last row inthe Routh–Hurwitz table. If   K   does exist, then  K  cross  =  K  . To determine the exact locationon the  ω  axis where the crossing occurs, one utilizes the auxiliary equation. Recall that theauxiliary equation is obtained from the row immediately preceeding the just determined zerorow. Then, for the auxiliary equation, each value of   K   is replaced by  K  cross , and the equationis solved for  s , which yields the  ω  axis crossings points.6. To ultimately plot the complete root locus trajectory, note that all root locus segments muststart at the poles of   G ( s ) and ultimately end at the zeros of   G ( s ) with numerical values of anybreak away, break in or  ω  axis crossing points determined as noted previously. Furthermore,the root locus trajectory must by symmetric with respect to the horizontal (  ω  = 0) axis. MATLAB The above procedure can be automated by using the MATLAB  rlocus  command, which requiresthe Control System Toolbox. The  rlocus  command, at a minimum, requires one argument, whichdescribes the open loop system shown in Figure 1 ( doc rlocus ). Typically, this is done by providinga vector  num  and a vector  den  where the vector components respectively are the coefficients of the numerator and denominator polynomial of   G ( s ) and then forming the system using  sys =tf(num,den) . Then, to obtain various pieces of information concerning the system at a specificpoint of the root locus trajectory (e.g., gain value  K  , pole location, etc.), one clicks on the desiredlocation of the resulting root locus plot.Finally, as seen in Lab #5,  m  files supplied by other users can enhance the functionality of MATLAB.For this lab,  routh.m , which is in the file at ,can be used to analytically determine the  ω  axis crossing points, which are the boundary betweena stable and unstable system. The use of this  m  file is described through various examples inthe  routh_handout.pdf  supplemental handout at , which is also con-tained in the zip file . EEL4657L Lab #6 3 of  6 Last revised: 17 December, 2017  Procedure 1. Consider the system in Figure 1 with open loop gain G ( s ) = 1( s + 1)( s  + 3) 2  = 1 s 3 + 7 s 2 + 15 s + 9 . Use the MATLAB  rlocus  command to plot the system root locus. Present this plot in thefinal report. Next, click on the end of the left root locus and bottom root locus trajectoresto show the gain value (as well as other information) associated with the end of each of thethree root locus trajectories. Now, click on the top  ω  crossing point to label this point. Theplot with the four noted points is presented in the final report in addition to the unlabeledone. (Ensure that the plot window is sufficiently large so that the boxes associated with thelabels do not overlap, but that it is not so large that when it is copied to the word processedlab document that the labels become too small to read.)Next, based on the information presented in the handout  routh handout.pdf  (focus on thefifth example in the handout) as well as in the Background Theory section of this document,use  routh.m  with additional MATLAB code to have MATLAB “analytically” determine the  ω  axis crossing value as well as associated gain value  K  . In particular, utilize the  routh command as  routh(den+k*num,epsilon)  where  k  and  epsilon  are defined to be symbolic(i.e.,  syms ) and  den  and  num  are equal length vectors (i.e., if leading zeros are required tomake the vector lengths equal, then they must be used) containing respectively the coefficientsof the denominator and numerator of   G ( s ). Then, based on the results of   routh , identify thefirst row which can be entirely zero (0) and add the additional necessary MATLAB code tofind the associated  k . Once  k  is determined via MATLAB, append additional MATLAB codeto utilize the immediately preceding row in the Routh table and solve for  s ; this will yield the  ω  crossing points as seen in lecture. In implementing the MATLAB code, the  eval ,  solve and  subs  commands will be needed. (Note that above approach requires appending MATLABcode as intermediate results are obtained in order to ultimately obtain the final result; thatis, the final result is not obtained until the code is run several times where after each runadditional code is added based on previous results. While one can write code that will obtainthe final result without any intermediate user modifications, it would be too time consumingfor a time restricted lab exercise that has additional topics that are covered.)For the final report, present the two root locus plots in figures with descriptive figure captions.Also, indicate what the gain value  K   should be at the end of the root locus trajectory. Doesthis correspond to what was found in the second root locus plot. Explain any discrepancy. Inaddition, show the MATLAB code as well as Command Window input/output that utilizes the routh.m  file to determine the  ω  crossing points and associated gain value  K   in two additionalfigures. Do these “analytical” results for the crossing points and gain match those that werefound from the root locus plot? Again, explain any discrepancy.2. Consider the system in Figure 1 with open loop gain G ( s ) = ( s + 4)( s  + 5) s ( s + 2) =  s 2 + 9 s + 20 s 2 + 2 s . EEL4657L Lab #6 4 of  6 Last revised: 17 December, 2017
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