ECE 320 Lab 2 Root Locus For Controller Design In this Lab you will explore the use of the root locus technique in designing controllers The root locus indicates the possible location of the closed loop poles of a system as a parameter usually the gain k varies from small to large values Often we implement controllers compensators to put the dominant poles of a system in a location that will produce a more desirable response This assignment is to be done with Matlab s sisotool Some things to keep in mind about the root locus The only possible closed loop poles are on the root locus The root locus starts k 0 on poles and ends k on zeros If you click on the root locus plot it gives you the value of gain k and the location of the closed loop poles on that branch for that value of k It does not tell you all of the closed loop pole locations if there is more than one branch All of the poles and zeros you add to the system should be in the left half plane Memo Your brief memo for this lab should indicate that you did all of the required work and that you understand that you are worthless scum and will be royally screwed if you did not It will be very obvious to me on the next two labs if you did not do this lab and I will change your grade accordingly Be creative in your memo Controller Configuration In this lab we will be assuming a unity feedback controller of the following form where C s is the controller and G s is the plant this is the notation sisotool uses Common Controller Types Proportional P Controller C s k p k ki k s s Derivative D Controller C s kd s ks Integral I Controller C s ki k s z s s Proportional Derivative PD Controller C s k p kd s k s z Proportional Integral PI Controller C s k p Proportional Integral Derivative PID Controller C s k p ki k s z s z k s z1 s z2 kd s s s s 1 For the PID controller we can have either two complex conjugate zeros or two real zeros In sisotool you will see the transfer functions form of the controllers To determine the parameters k p ki and kd you need to equate powers of s of the transfer functions with the forms above It is easiest if you use the zero pole gain format for the compensators To do this click on Edit SISO Tool Preferences Options and click on zero pole gain Note that if the system is stable the I PI and PID controllers will produce a steady state error of zero unless the system being controlled has a zero at the origin which would cancel with the controllers pole at the origin Introduction to sisotool Getting Started Enter the transfer function for the plant G s in your workspace Type sisotool in the command window Click close when the help window comes up Click on Vew then Design Plots Configuration and turn off all plots except the Root Locus plot set the Plot Type to Root Locus for Plot 1 and set the Plot Type to None for all other Plots Loading the Transfer Function In the SISO Design window Click on file import We will usually be assigning G s to block G the plant Under the System heading click on the line that indicates G then click on Browse Choose the available Model that you want assigned to G Click on the appropriate line and then click on Import and then on Close Click OK on the System Data Import Model window Once the transfer function has been entered the root locus is displayed Make sure the poles and zeros of your plant are where you think they should be Generating the Step Response Click on Analysis Response to Step Command You will probably have two curves on your step response plot To just get the output type Analysis Other Loop Responses If you only want the output then only r to y is checked and then click OK However sometimes you will also want the r to u output since it shows the control effort for P I and PI controllers You can move the location of the pole in the root locus plot by putting the cursor over the pink button and holding the left mouse button down as you move the pole locations You should note that the step response changes as the pole locations change The bottom of the root locus window will show you the closed loop poles corresponding to the cursor location if you hold down the left mouse button However if you need all of the closed loop poles you have to look at all of the branches 2 Entering a Compensator Controller Type Designs Edit Compensator Right click in the Dynamics window to enter real poles and zeros You will be able to changes these values very easily later You can either edit the pole zero locations in this window or by grabbing the poles and zeros in the root locus plot and moving them However sometimes you just have to go back to this window Look at the form of C to be sure it s what you intended and then look at the root locus with the compensator You can again see how the step response changes with the compensator by moving the locations of the poles grab the pink dot and slide it You can also change the location of the pole and zeros of the compensator by grabbing them and sliding them Be careful not to change the poles and zeros of the plant though Adding Constraints Right Click on the Root Locus plot and choose Design Requirements then either New to add new constraints or Edit to edit existing constraints At this point you have a choice of various types of constraints Remember these constraints are only exact for ideal second order systems Printing Saving the Figures To save a figure sisotool has created click File Print to Figure Odds and Ends You may want to fix the axes To do this Right click on the Root Locus Plot Choose Properties Choose Limits Set the limits and turn the Auto Scale off You may also want to put on a grid as another method of checking your answers To do this right click on the Root Locus plot then choose Grid It is easiest if you use the zero pole gain format for the compensators To do this click on Edit SISO Tool Preferences Options and click on zero pole gain 3 Part A Lab partners work together on this part Let s assume the plant we are trying to control has the transfer function G s 30 30 s 11s 30 s 5 s 6 2 This is second order system with two real poles located at 5 and 6 Our general goal will be to speed up the response of …