Lab 4: Root Locus For Controller Design Overview 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 summarize what you think are the benefits and drawbacks of each of the basic controller types. For each of the three systems analyzed you need to include as attachments your plots of the step response and the values of , , and for each controller. You will also need to include root locus plots for some of the systems (only if asked for!) Finally, is the control effort for the I and PI controllers larger at the initial time, or later, near steady state? pkikdk 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). ()Gs ()Cs+ -Common Controller Types Proportional (P) Controller: ()pCs k k== Integral (I) Controller: ()ikkCsss== Derivative (D) Controller: ( )dCs ks ks== Proportional+Integral (PI) Controller: ()()ipkks zCs kss+=+= Proportional+Derivative (PD) Controller: () ( )pdCs k ks ks z=+=+ Proportional+Integral+Derivative (PID) Controller: *12()(()( )()ipdkks z s zks z s zCs k ksss s++++=++ = =) 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 , , and you need to equate powers of s of the transfer functions with the forms above. It is easies t 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. pkikdk Note 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 vVew, the Design Plots Configuration, and turn off all plots except the Root Locus plot Loading the Transfer Function • Click on file → import. • We will usually be assigning G(s) to block G (the plant), so type your transfer function name next to G and then enter. You must hit enter or nothing will happen. • Once you hit enter, you should be able to click on the OK at the bottom of the window. The window will then vanish. • 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 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. However, if you need all of the closed loop poles you have to look at all of the branches.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 Constraints 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. 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.Part A Let's assume the plant we are trying to control has the transfer function 230 30()11 30 ( 5)( 6)Gsss ss==++++ 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 the system and produce a system with a steady state