ECE-320 Lab 2: Root Locus For Controller Design

Transcription

1 ECE-320 Lab 2: Root Locu For Controller Deign In thi Lab you will exlore the ue of the root locu technique in deigning controller. The root locu indicate the oible location of the cloed loo ole of a ytem a a arameter (uually the gain ) varie from mall to large value. Thi aignment i to be done with Matlab' iotool.(io come from Single Inut Single Outut) Some thing to ee in mind about the root locu: The only oible cloed loo ole are on the root locu. The root locu tart ( = 0) on ole and end ( ) on zero. All of the ole and zero you add to the ytem hould be in the left half lane. Memo Your (brief) memo for thi lab hould jut be a tatement that you did all of the required wor. Controller Configuration In thi lab we will be auming a unity feedbac controller of the following form, where C() i the controller and G() i the lant (thi i the notation iotool ue). + - Common Controller Tye Proortional (P) Controller: C() i Integral (I) Controller: C () i ( z) Proortional+Integral (PI) Controller: C() Proortional+Derivative (PD) Controller: C( ) ( z) Proortional+Integral+Derivative (PID) Controller: * i ( z)( z ) ( z1)( z2) C() d In iotool, you will ee the tranfer function form of the controller. To determine the arameter, i, and d you need to equate ower of of the tranfer function with the form above. Note that if the cloed loo ytem i table, the I, PI, and PID controller will roduce a teady tate error of zero for a te inut unle the lant being controlled ha a zero at the origin. A iotool ummary i at the end of thi lab. d 1

2 Part A Let' aume the lant we are trying to control ha the tranfer function G () ( 5)( 6) 2 Thi i econd order ytem with two real ole, located at -5 and -6. Our general goal will be to eed u the reone of the ytem and roduce a ytem with a teady tate error for a unit te inut of 0.1 or le, a ercent overhoot of 10% or le, and a ettling time of le than 1 econd. Getting Started Enter the tranfer function for the lant, G(), in your worace. Tye iotool in the Matlab command window Clic cloe when the hel window come u Clic on Vew, the Deign Plot Configuration, and turn off all lot excet the Root Locu lot (et the Plot Tye to Root Locu for Plot 1 and et the Plot Tye to None for all other lot) It i eaiet if you ue the zero/ole/gain format for the comenator. To do thi, in the SISO Deign window clic on Edit SISO Tool Preference Otion and clic on zero/ole/gain. Loading the Tranfer Function In the SISO Deign window, Clic on File Imort. We will uually be aigning G() to bloc G (the lant). Under the Sytem heading, clic on the line that indicate G, then clic on Browe. Chooe the Available Model that you want aigned to G (clic on the aroriate line) and then clic on Imort and then on Cloe. Clic OK on the Sytem Data (Imort Model) window Once the tranfer function ha been entered, the root locu (for a roortional controller) i dilayed. Mae ure the ole and zero of your lant are where you thin they hould be. Generating the Ste Reone In the SISO Deign window, clic on Analyi Reone to Ste Command (the ytem may tae a while to reond) You will robably have two curve on your te reone lot. To jut get the outut, in the SISO Deign window elect Analyi Other Loo Reone. If you only want the outut, then only r to y hould checed. However, ometime you will alo want the r to u outut, ince it how the control effort for P, I, and PI controller. In the SISO Deign window you can move the location of the ole in the root locu lot by utting the curor over the in button and holding the left button down a you move the ole location. You hould note that the te reone change a the ole location change. The bottom of the root locu window will how you the cloed loo ole correonding to the curor location (you need to clic the left button). However, if you need all of the cloed loo ole you have to loo at all of the branche. Note that the aumed inut for iotool i alway 1, o the teady tate error i one minu the teady tate value of the outut. 2

3 Adding Contraint in the -lane In the SISO Deign window, right clic on the root locu lot, and chooe Deign Requirement then either New to add new contraint, or Edit to edit exiting contraint. At thi oint you have a choice of variou tye of contraint. Enter the ettling time contraint (le than 1 econd), and then enter the ercent overhoot contraint (le than 10%). You will not be able to enter the teady tate error contraint, becaue that i not really controlled by ole location in the -lane. To meet the contraint, you want the ole to tay within the white area and avoid the haded (yellow) area. Remember thee contraint are only exact for ideal econd order ytem!!! Adding Contraint in the Ste Reone Grah Right clic in the LTI Viewer window There are a number of way to add contraint to the outut. Uually the bet i to elect Characteritic, and then chooe that you want. Each of the characteritic will caue a blue dot to aear on the outut creen. If you hover over the blue dot you can find the correonding value. Note that the aumed inut for iotool i alway 1, o the teady tate error i one minu the teady tate value of the outut. Proortional (P) Control (Thi i the default for iotool) We will now loo at the root locu lot for thi ytem with a roortional controller. Thi i the default for iotool and i what we are currently looing at. If you want to now the value of the roortional gain ( ) you can do thi in one of two way: In the SISO Deign window, clic on Deign Edit Comenator In the Control and Etimation Tool Manager, clic on Comenator Editor In thi window, the controller i written a C(), and you hould be able to read off the value of the roortional gain. Next, loo at the te reone a the gain varie (liding the in box on the root locu lot i the eaiet way to do thi). You hould notice a few thing: a increae, the imaginary art of the cloed loo ole increae, thu the PO (Percent Overhoot) increae a increae, the teady tate error decreae there i no value of for which the ytem i untable 3

4 Integral (I) Control In the Control and Etimation window, right clic in the Dynamic window to enter and remove ole and zero. You will be able to change thee value very eaily later. At thi oint jut add an integrator. 1 In thi window, the controller i written a C() i and you hould be able to read off the value of the integral gain. Loo at the form of C to be ure it' what you intended, and then loo at the root locu with the comenator. You can again ee how the te reone change with the comenator by moving the location of the ole (grab the in dot and lide it). Next loo at the te reone a the value of the integral gain varie. You hould notice a few thing There are ome value of i ( i 11 or o) for which the ytem i untable. To find thi value of the gain, you may have to enter the value directly into the gain box (thi i how you can enter recie value) The ytem i much lower now than with the roortional control (the ettling time i generally larger) The teady tate error for a te inut i now zero. Proortional+Integral (PI) Control ( z) The form of a PI controller i C ().Since we already have the 1 art we need to add a real zero. For a PI controller, one ole i alway fixed at the origin. The zero of the comenator can move, but the ole cannot. Siotool will, in fact, let you move the ole, but you will no longer have a PI controller. Right clic in the Dynamic window and add a real zero. Note that the location of the zero will robably default to -1 (or omewhere ele). We can move it later o do not worry about it. In the Dynamic window, left clic on the Real Zero box and a new Location box will o u. You can enter the value you want for the real zero in thi box and then hit Enter. The new location hould how u in the Dynamic box. In the SISO Deign window, you can now grab the real zero and move it a well a grabbing the ic quare and move them (correonding to changing the gain). You hould lay around with thi a bit while looing at how the te reone i changing. See if you can get a ettling time of le than one econd and a ercent overhoot of le than 10% while eeing z 5. Note that you hould be looing at the te reone to determine thi, not the root locu lot! Since we have a tye 1 ytem, the teady tate error i zero. Proortional+Derivative (PD) Control The form of a PD controller i C( ) ( z). To get thi form from our integral controller, we need to remove the ole at the origin. In the Dynamic window, right clic on the Integrator line, and elect Delete Pole/Zero. Loo at the form of the controller and it hould loo lie a PD controller. We no longer have a tye 1 ytem, o the teady tate error i no longer zero. 4

5 Set the zero at -1 and the gain to 4. The SISO Deign window how the root locu lot that i retty hard to ee much, ince it i alo howing a ole near To zoom in near zero, in the white ace in the root locu lot right clic the moue and elect Proertie. Then elect Limit, unchec the Auto-Scale boxe, and et the Real Axi limit from -10 to 0, and the Imaginary Axi limit from -1.0 to 1.0, and then elect Cloe. Try to adjut the gain and the zero o the teady tate error i le than 0.1 and the ercent overhoot i le than 10% while eeing the zero between 0 and -5. Next, wet the zero to omething le than -6 and you will ee a very different root locu lot. (You may have to right clic in the white area, elect Proertie, then Limit, and then chooe Auto-Scale.) It hould be much eaier to meet all of the deign requirement now (teady tate error le than 0.1, PO le than 10%, and a ettling time le than 1 econd.) However, thi tye of controller generally require a large control effort. Proortional+Integral+Derivative (PID) Control There are two general form for a PID controller, deending on whether the two zero are real or comlex * ( z1)( z2) ( z)( z ) conjugate air, C() Note that for a PID controller you are free to move the zero, but the ole mut remain at the origin. Siotool will allow you to move the ole, but you will no longer have a PID controller and you will not have a tye 1 ytem. PID control with real zero In the Dynamic box, add an integrator and another zero. Modify your controller o you meet all of the contraint a well a eeing 5, 5, 0.2 PID control with comlex conjugate zero In the Dynamic box, remove the real zero and add comlex conjugate zero. i d Modify your controller o you meet all of the contraint a well a eeing 5, 5, 0.2 Printing/Saving the Figure: To ave a figure iotool ha created, clic File Print to Figure i d 5

6 Part B Let' aume the lant ha the tranfer function G () You need to ue the zero/ole/gain format for the comenator. To do thi clic on Edit SISO Tool Preference Otion and clic on zero/ole/gain. You hould try and meet the following contraint e T ec PO.. 25% You hould do your bet to meet each one of thee, but in any event you mut have i d In addition, for the I and PI controller you mut have ut ( ) Your te reone grah hould alo how the control effort for thee two tye of controller! You need to try to meet thee deign contraint for a I controller (hard to meet ettling time, robably need T 4ec ) PD controller PI controller (hard to meet ettling time, robably needt 3.5ec ) PID controller with real zero PID controller with comlex conjugate zero 6

7 Part C Let' aume the lant ha the tranfer function G () You need to ue the zero/ole/gain format for the comenator. To do thi clic on Edit SISO Tool Preference Otion and clic on zero/ole/gain. You hould try and meet the following contraint e T ec PO.. 25% You hould do your bet to meet each one of thee, but in any event you mut have i d In addition, for the I and PI controller you mut have ut ( ) Your te reone grah hould alo how the control effort for thee two tye of controller! You need to try to meet thee deign contraint for a I controller (hard to meet ettling time, robably need T 4ec ) PD controller (robably won't wor, do the bet you can) PI controller (really hard to meet ettling time, robably needt 10ec ) PID controller with real zero PID controller with comlex conjugate zero 7

8 iotool ummary Getting Started Enter the tranfer function for the lant, G(), in your worace. Tye iotool in the command window Clic cloe when the hel window come u Clic on Vew, the Deign Plot Configuration, and turn off all lot excet the Root Locu lot (et the Plot Tye to Root Locu for Plot 1 and et the Plot Tye to None for all other lot) Loading the Tranfer Function In the iotool Deign Window, Clic on File Imort. We will uually be aigning G() to bloc G (the lant). Under the Sytem heading, clic on the line that indicate G, then clic on Browe. Chooe the available Model that you want aigned to G (clic on the aroriate line) and then clic on Imort and then on Cloe. Clic OK on the Sytem Data (Imort Model) window Once the tranfer function ha been entered, the root locu i dilayed, mae ure the ole and zero of your lant are where you thin they hould be. Generating the Ste Reone Clic on Analyi Reone to Ste Command You will robably have two curve on your te reone lot. To jut get the outut, tye Analyi Other Loo Reone. If you only want the outut, then only r to y i checed. However, ometime you will alo want the r to u outut, ince it how the control effort for P, I, and PI controller. You can move the location of the ole in the root locu lot by utting the curor over the in button and holding the left button down a you move the ole location. You hould note that the te reone change a the ole location change. The bottom of the root locu window will how you the cloed loo ole correonding to the curor location (you need to clic the left button). However, if you need all of the cloed loo ole you have to loo at all of the branche. Entering a Comenator (Controller) Clic on Deign Edit Comenator Right clic in the Dynamic window to enter real ole and zero. You will be able to change thee value very eaily later. You can either edit the ole/zero location in thi window, or by grabbing the ole and zero in the root locu lot and moving them. However, ometime you jut have to go bac to thi window. Loo at the form of C to be ure it' what you intended, and then loo at the root locu with the comenator. You can again ee how the te reone change with the comenator by moving the location of the ole (grab the in dot and lide it). You can alo change the location of the ole and zero of the comenator by grabbing them and liding them. Be careful not to change the ole and zero of the lant though! 8

9 Adding Contraint Right Clic on the root locu lot, and chooe Deign Requirement then either New to add new contraint, or Edit to edit exiting contraint. At thi oint you have a choice of variou tye of contraint. Remember thee contraint are only exact for ideal econd order ytem!!! Printing/Saving the Figure: To ave a figure iotool ha created, clic File Print to Figure 9