Frtool – The User's Guide
Contents
1. 1 3 1 11 Status frame This frame will be used by FRtool to communicate with the user For example if a gain equal to zero will be set FRtool will display a message to inform the user that he should change the gain to another value because that one is not allowed 1 3 2 Responses window Simulations of closed loop responses and characteristics can be seen in the responses window see Figure 1 9 This window is shown hidden automatically by FRtool as follows if no responses or characteristics are needed the window will be hidden once at least one response 15 requested it will be automatically shown All requested responses and characteristics will share the same window and will be updated every time a change 1s made to the system or controller parameters Right clicking on a characteristic plot users can select features like grid on off peak response settling time rising time or zoom The Print option in the File menu of the responses window will display the Matlab print preview window to print all the responses currently displayed Frequency Response Design Tool Characteristics Eile step Response Impulse Response Fr 7 1 From 1 a qi a ts 09 m pi E TEES y E Time sec Time sec Bode Diagrams Nyquist Diagrams From mm tn RA qi i a 12 metas T e i E CL Frequency rad sec Real Axis Figure 1 9 Responses2. The Export option of the File menu allows user to save current loop configuration by saving F P and H transfer functions in Matlab workspace in case that system export is requested or by saving K transfer function in case of controller export request These transfer functions can be found in the Matlab workspace with the following names sysF sysP sysH and sysK Print Nichols and Print options of the File menu will open the Matlab print preview window First option will print only the Nichols curve while the second one will print the main window as it is displayed Exit ends the application and closes all sub windows of FRTool 1 3 1 2 Tools menu The Tools menu has two submenus the Constraints submenu that allows user to define the design specs robustness overshoot settling time phase margin and gain margin and the Simulation submenu that allows user to define the final time for closed loop simulations FRTool will automatically determine a final time for simulation but in case this value is not satisfactory the user can define his own final time For defining these restrictions FRTool will display a constraint window as shown in Figure 1 11 1 3 1 3 Help menu At this time only general help and release information are available via the options of this menu but home page navigation and other features will be added in a future version 1 3 1 4 Nichols plot axes Once the system and or the controller are defin
3. 1 5 Im Ro sin u 1 6 M 20 108104 1 7 Im 180 zr 180 arctan 1 8 Re ee AB 1 1 TT 77 1 ele 1 1 um um mm m om 70 200 180 160 140 120 100 BO 40 2 0 Open Loop Phase deg Figure 1 4 Robustness curves on Nichols plot Once the real and imaginary parts of a complex number are known its modulus and phase can be computed using relations 1 7 and 1 8 As a result of this conversion on the Nichols plot we will obtain the curves shown in Figure 1 4 corresponding to the circles from Nyquist Robustness spec is fulfilled only when the Nichols curve of the open loop system remains outside the restriction curve 40 dd B Open Loop Gain dB c I I 1 um um mm mm 40 220 200 180 160 140 120 100 80 60 40 20 0 Open Loop Phase deg Figure 1 5 Gain and phase margins measured on the Nichols plot Besides these 3 restrictions we also considered gain and phase margins because these two design specs are easy to read on the Nichols characteristics The gain margin 1s the vertical line that starts in the point having 0 dB gain and 180 degrees phase and goes downward while the phase margin restriction 1s the horizontal line starting in the same point and going to the right side of the graphic Figure 1 5 illustrates these exp
4. 40 Open Loop Gain dB 100 Open Loop Phase deg Kp own v Edit compensalor Figure 1 16 Nichols curve of the K P open loop We don t know yet the effect of changing the controller s zero pole and gain note the static gain of the phase lead compensator has not been shown explicitly 1n formula 1 11 Now we shall test the effect of each one First we will move the zero from 2 to 5 and we shall try to see the effect on the Nichols curve Figure 1 17 shows the new position of the Nichols curve Figure 1 18 shows the effect of placing the zero in 1 We can see that the Nichols shape becomes more convex when he zero becomes more negative and that the Nichols curve starts having a peak when the zero 1s less negative Next step is to see the effect of the pole s position on the Nichols curve Therefore we will first move the pole to 60 and will obtain the Figure 1 19 The effect of placing the pole at 10 1s shown in Figure 1 20 At this time we know the effect of changing the zero and pole s position and we also know from theory the effect of the loop gain on the Nichols plot illustrated 1n Figure 1 21 and 1 22 when increasing the gain higher values than 1 the curve will be shifted up when decreasing the gain values between 0 and 1 the Nichols curve will be shifted down Frequency Hesponze Design Tool _ i a a i eee i i 4 i zero change
5. we will call it system import window one for importing a predefined designed controller in ZPK format named controller import window and one for setting up a design spec called constraint window 1 3 1 Main window The main window is used to set up the design tool and it is shown in Figure 1 6 As you see the window contains three menus File Tools and Help zooming tool axis for plotting the open loop Nichols curve button for loop configuration system definition frame controller gain frame activation deactivation checkbox compensator editor activation button responses frame constraints frame and status frame In the following subchapters we will discuss the features available in this window 1 3 1 1 FILE menu This menu is used for file operations and it has the following options import export of processes and controllers Nichols plot main window prints and of course application exit The Import option of this menu can be called to get transfer functions from the Matlab workspace It has two submenus one for importing a process and another one for importing a controller in both cases placed in the blocks from the control loop shown in Figure 1 7 or Figure 1 8 In case that system import 1s requested the system import window will be activated and in case of controller import request the controller import window will be shown see Figures 1 12 and 1 13 Figure 1 8 Control loop configuration
6. window of FR Tool 1 3 3 Design window This 1s the second most important window of the frequency design tool see Figure 1 10 because it allows users to change the position of the controller poles and zeros and also to add remove new existing poles and zeros The limits of the axes will be adjusted in correlation with the pole zero placements and current limits will be changed using the zooming tool available in this window which works in the same way as the one available in the main window Using drag amp drop the user can change the position of the poles and zeros and while dragging the roots he can see the changes of the Nichols curve in real time FRtool will take care of complex conjugated poles moving the conjugated pole in the appropriate position By releasing the mouse button the new position of the pole or zero will be set and the responses window will be updated too Compensators poles and zeros Zoom Zoom X Zoom Y Zoom Dut EH um M HF E FE FF BF E Position i Add zero Erase current Add pole Figure 1 10 Controller design window 1 3 4 0ther windows of FRTool Whenever a design specification has to be defined FRTool displays a window similar to the one shown in Figure 1 11 The window will contain the restriction s name type lt gt lt gt value and measurement units Figure 1 11 shows the overshoot case and it can be read as Overshoot has to be lower t
7. Frtool The User s Guide Frequency Response Controller Design Tool Cristian VLASIN Robin DE KEYSER Cristian Vlasin aut utcluj ro autoctrl rug ac be 1 1 Introduction Control engineers students can design a controller in one of following ways applying a mathematical design method or using a controller design tool In case they choose to apply a mathematical method they usually have to simplify the mathematics Using a controller design tool they can tune the controller s parameters without simplifying the process transfer function and can obtain in most of the cases better best closed loop performance At this time Matlab offers a controller design tool based on the root locus method rltool This tool cannot handle systems with time delay without approximating the dead time by a rational transfer function To solve this problem we choose to work with frequency diagrams Nichols diagrams where we know that dead time Tg will have no influence on the gain but it will just shift the phase with Tg o 1 2 Design specs An important feature of a controller design tool is the possibility to define design constraints which will guide the designer in the tuning process These design specs have to be converted into graphical restrictions to make designer s job easier Graphical restrictions are obtained from numerical specs using relations defined for the second order system relations which can be extended to hi
8. d to 5 ith New Nichols curve w 17 Figure 1 T 0 C zd E l1 cc d laced in 1 is p 18 Nichols curve when zero 1 Figure Frequency Hesponse Design Tool p5 aE EO Pole i 1 i i i Frequency A Figure 1 20 Nichols curve with controller pole at 10 Frequency Response Design Tool Open Loop Gain 98 200 150 100 50 D Open Loop Phase io e iar n ih 2 150 Phase ded Figure 1 22 Nichols curve shifted with a gain of 0 3 Playing a few minutes with the pole zero placement and gain we can shape the Nichols curve so that all design specs are fulfilled The picture of the resulting Nichols curve 15 shown in Figure 1 23 You can see that the curve does not cross the OS restriction and that the bandwidth frequency 1s above the 3dB green line This means that all design specs are fulfilled and we can prove this by taking a look at the closed loop response of the system see Figure 1 24 The controller parameters resulting from this design exercise are listed in Figure 1 23 and the values of overshoot and settling time obtained are listed in Figure 1 24 1 5 FAQ 1 5 1 How can tune controller parameters Controller parameters are gain poles and zeros The gain can be tuned using the controller gain frame p
9. ecreasing of the gain and a textbox for the step size how much will be added or subtracted to from the current value of the gain to obtain the new value Another way to change the controller s gain 1s dragging the bandwidth frequency with the mouse only vertical movement is possible because this is the effect of changing the gain of the open loop 1 3 1 8 Ngrid and controller editor activation frame Clicking on the Ngrid checkbox the user can show or hide the constant gain and phase curves for the Nichols plot the so called Ngrid The Edit compensator button can be used to open or reopen the design window 1 3 1 9 Responses frame Using the checkboxes available in this frame the user can select which responses or characteristics have to be displayed For instance if step response 1s requested the user will click on the Step checkbox to activate this type of response Multiple responses and or characteristics can be active at the same time and they will be displayed in the same window the responses window 1 3 1 10 Constraints frame This frame is similar with the responses frame but it is used to notify FRtool that a certain design spec should be taken into account when drawing the graphical constraints If the value of that design spec 1s not already set the constraint window will be displayed to input the value of that design spec Even if defined a design spec can be omitted if the corresponding checkbox is unchecked
10. ed the Nichols curve of the open loop system 1s displayed in the main window to guide the user in the design process The design specs robustness settling time overshoot gain margin and phase margin will be also displayed as graphical restrictions on the same plot see Figure 1 6 Watching this plot the designer can see the effect of changing controller parameters and can adjust these parameters until he fulfils all design specs 1 3 1 5 Loop configuration button The button displays continuously the current loop configuration see Figures 1 7 and 1 8 and allows user to change the position of the controller from direct loop to feedback Placing the process in different blocks on the control loop and changing the loop configuration the user can obtain closed loop responses for setpoint and perturbances in different points of the control loop 1 3 1 6 System definition frame The system contained in the P block of the control loop can be defined as well using the system definition frame In this case only the numerator and denominator of the P system will be affected and no dead time systems can be defined Once a system 1s imported using the menu option see import feature these two textboxes will be disabled and they will be re enabled only when all the systems placed in F P or H blocks are erased 1 3 1 7 Controller gain frame This frame contains a textbox with the value of the gain two buttons which allow increasing and d
11. gher order systems as a rule of thumb We considered as important design specs the following ones robustness settling time maximum overshoot gain and phase margins Given the maximum overshoot as design specification we can compute the damping factor using relation 1 1 ni OS e 11 Straightforward we can obtain the peak value M of the closed loop frequency domain magnitude Bode as defined in relation 1 2 which will be a curve on the Nichols plot as shown in Figure 1 1 You can see that increasing the value of OS from 5 to 35 the curves become less restrictive they become smaller 1 2 2 26l t 20743 t 13 1 3 To fulfil this specification the Nichols curve of the open loop system has to be outside or tangent to the correspondent M curve QOpen Loop Gain dB m m m 0 220 200 180 160 140 120 100 5 0 5 0 4 0 2 0 Open Loop Phase deu Figure 1 1 Constant M curves corresponding to overshoot values The second important design specification 1s the settling time Given the settling time and the maximum overshoot specs we can use relations 1 1 and 1 3 to obtain the natural frequency p of the equivalent second order system and the damping factor Once we know the damping factor and the natural frequency the value of the 3dB bandwidth frequency for the closed loop is obtained with relation 1 4 It will be
12. gure 1 23 FRTool s main window with all specs fulfilled Frequency be Design Tool Charac LESTIE IEEE 2 lt 5 t 0 5745 lt 0 85 Figure 1 24 FRTool s responses window Step response 1 5 6 How will find my exported systems and controller On File Export S ystem request FRtool will send to Matlab workspace the systems from all blocks except the K block and the given names will be sysF sysP and sysH On File Export Controller request FRTool will send to the same Matlab workspace the controller in ZPK format with the sysK name 1 5 7 Why after define the settling time it is not displayed on the Nichols curve Settling time restriction s graphical equivalent is considered to be the bandwidth frequency which is computed using damping factor and natural frequency To compute the damping factor you have to define a maximum overshoot Therefore you cannot use only the settling time restriction without the overshoot restriction However overshoot restriction can be used without defining settling time spec
13. han ____ percents Set restriction Figure 1 11 Constraints window of FRTool As we mentioned before the process can be placed in three positions P F H on the control loop see Figure 1 7 or 1 8 Therefore the process import window shown in Figure 1 12 has at its left side a list of processes defined as TF or ZPK in the Matlab workspace and a set of three buttons one for each block from the control loop To insert a process from the list into block P it has to be selected with the mouse and the P button has to be pressed When everything 1s set the Use button will validate the loop configuration Similar with the process import window is the controller import window The difference is that in this case the list will contain only ZPK systems defined in the workspace To use a predefined controller it has to be selected form the list and inserted into block with t Import systems from workspace Selected system s E Figure 1 13 Controller import window 1 4 Design example application In most of the cases the easiest way to learn how to use software is to follow a short demonstration Therefore we picked an application and we will try to give a demo of the design procedure Task considering that the plant has the transfer function given in relation 1 9 design a phase lead controller such that the unity feedback closed loop system will satisfy the following specifica
14. laced in the main window or by dragging the red circle bandwidth frequency displayed on the Nichols curve The controller poles and zeros can be easily tuned using drag amp drop features available in the controller design window 1 5 2 How I define restrictions design specs When selecting a restriction FRtool checks first if there is a value defined for this restriction If no value is found the constraint window is displayed to force the user to give a value to the restriction Another way is to select the Tools Specs submenu options and insert the value in the window that is displayed In this case the checkbox that activates the restriction has to be clicked otherwise the spec is not displayed 1 5 3 How I erase all systems defined in the F P and blocks By simply opening the system import window from File Import System menu option and clicking on Use without selecting any system the blocks from the control loop will be set to empty 1 5 4 How do I have to define the system in workspace In order to be able to import a system the user has to define it as TF or ZPK Only these types of systems are displayed in the system import window s list 1 5 5 How does the controller have to be defined In order to be able to import a controller the user has to define it as ZPK Only this type of system is displayed in the controller import window s list Fie quency Response Design Tool Fi
15. lanations To fulfill these two specs the Nichols curve has to go on the right side of the phase margin line and below the gain margin line without crossing any of them With all design specs active and fulfilled the Nichols plot from the main window of the tool should look like the one shown in Figure 1 6 A Frequency Response Design Tool File Toole Help Gain dB 200 150 100 50 0 Cioen Loop Phase deg zoom Zoom zoom Y Zoom us Figure 1 6 Main window of FRTool Specs fulfilled on Nichols 1 3 GUI The FRtool s graphical user interface GUI is powerful offering a set of useful features that make this tool easy to use It has the following windows one in which the Nichols curve of current open loop and restrictions are shown and which 1s used to set design specs do file operations and design a part of the controller its gain from now on we will call it main window one in which the user can see step and impulse responses as well as Bode or Nyquist characteristics used for checking whether design specs are fulfilled or not this one will be called responses window one for placing controller poles and zeros this will be called controller design window because this is the window where the user will play with controller s parameters in which the user can select the transfer functions for the P and blocks see Figures 1 7 1 8 in the control loop
16. o relation 1 12 1 Is Kis G l 1 11 s 1 11 ole gt Zero 1 12 To have the structure defined in relation 1 11 the controller s pole and zero have to be both pure real roots Furthermore to fulfill relation 1 12 we will choose the pole s location 25 and the zero s location 2 To define them we have to go to the controller design window and do the following actions write 25 in the real part textbox and then press pole button to define the pole and write 2 in the same textbox and press Add zero button to define the zero At this time the controller design window will look like in Figure 1 15 and the main window will be similar to Figure 1 16 Frequency Response Design Tool ii EL LENT 5 T AC it alta bg PN ek x Fe FTP See TEn ee i i i L3 P L 1 TRp Li 4 a oo ow i i i i mr mr mc mv me mm mmm mm I L 1 i i i 1 am Gu gs s sm m 7 77 0770 Bode T Mun Re F 0S Lompensalor s poles and zeros T m um oum oam oam mou oam uoa uo Lx o o og css 25i pe 0 4h Figure 1 15 Controller pole zero placement i Frequency Response Design Fit Tool Help
17. tions e maximum overshoot lt 5 e settling time lt 0 8 seconds 2500 025 a s s 25 1 9 Supposing that FRtool is already installed and working properly on our system we define the process in the Matlab workspace using relation 1 10 Sys 2500 125 0 inputDelay 0 2 1 10 If FRtool is not started yet it can be started with frtool command Once the tool s main window is visible we can start using the tool by importing the transfer function of the process Clicking on File Import System menu the system import window will appear with the Sys transfer function in the list After clicking on the Sys name on the P button and finally on the Use button the system is placed into the P block of the control loop and in the main window is displayed the Nichols plot of the open loop At this time the open loop contains only the P transfer function our system Next step is to define the design specs Clicking on the Ts checkbox constraint window will be displayed to enter the value of the settling time spec because it was not defined yet After entering this value we proceed with the overshoot design spec in the same way Afterwards the main window of the tool will be the one shown in Figure 1 14 From now on we have to pick the structure of the controller and design its parameters We have to design a phase lead controller defined by relation 1 11 which leads us t
18. used as graphical restriction because frequency is the parameter along the Nichols curve 0 41 25 J407 407 2 1 4 Figure 1 2 shows how the bandwidth frequency is displayed on the Nichols plot 40 20 0 067234 1 Open Loop Gain dB c Wb 7 8805 40 220 200 180 160 1440 120 100 80 60 40 20 Open Loop Phase ideg Figure 1 2 Nichols plot with highlighted frequencies You can see that along the Nichols curve there are multiple red circles corresponding to the variation of p To fulfill the settling time design specification the circle corresponding to the related value of has to be above the green 3dB line and because the value of p was obtained using also the overshoot spec the overshoot spec has to be respected too Robustness design spec can be taken into account according to the following definition in the Nyquist plot see Figure 1 3 Robustness is the radius of the circle and it can have values that respect the relation 0 lt Ro lt 1 Because we are using the Nichols plot for the design process we need to convert this robustness spec from Nyquist to Nichols Given the value of Ro we can compute the values of the real and imaginary part of the Ro vector with relation 1 5 and 1 6 where wis the angle of Ro vector measured as shown in Figure 1 3 Qpen Loop Gain 48 Figure 1 3 Definition of robustness on Nyquist Re 1 Ro cos w
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