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National Instruments Xmath Interactive Control Design Module ICDM User's Manual

1. gt gt W UF gt Xp gt Z o o o y w r y BE at gt gt P e gt Wy 9 gt u Uact y Yi sens l sens e gt Ysens z gt sens gt py y Yp sens gt sens gt Yp lt 4 C I4 lt Figure 12 6 LQG H Infinity Control Design Configuration In the block diagram 6 represents a matrix that selects a subset of the set of plant outputs as measurements Similarly 6 selects a subset of the plant inputs as control inputs These subsets are determined by the toggle buttons in the weights window These allow the user to quickly investigate the effect of including excluding sensors and actuators without having to redefine the plant model Xmath Interactive Control Design Module 12 10 ni com Chapter 12 LQG H Infinity Synthesis The system equations of plant filters and integrators are as follows Plant P Xp Apxy Bpaci Bow Wy fee CX p w w Output filter F 45x5 Boy Wwe y Cx w Input filter F y Agxy Bact u C Yaar gt Daci Integrator y Wyw Yr Xt w Standard LQG All Toggle Buttons Off In LQG synthesis mode the controller C minimizes a weighted sum of the steady state actuator and output variance n n J lim E T T Reg Ryu x 2 2 Eo p ai R R T FS Purti D Pr pd ux uul u i l j
2. cece eseesceseeseeeseceeeeeeaeeeseceeeeeeeeeaees 3 7 Selecting a Synthesis or History Window 0 eee eee eeeessceeeeseeeeeseeseseeeesaeeseenseeneees 3 9 Edit Ments e AEE EE sats sen adi paste A E A a 3 9 Chapter 4 PID Synthesis Window Anatomy e n a a E E E aE O RN ia 4 1 PID Controller Terms arinaa hed Hees A a E Ra aR 4 1 Toggling Controller Terms On and Off sseeesssessesesseesesresrsresrsresresrsresreerssese 4 2 Opening the PID Synthesis Window ssssssssssssseessessseesreersresrsesresreresresesreses 4 4 Manipulating the Controller Parameters 000 cece es esceseeseceseeseeeseeseeesecesesesseeeaeesees 4 4 Time Versus Frequency Parameters ccceceeeeseeeeceseeseeesecseeeseeseenseessenaes 4 5 Ranges of Sliders and PlOtss c 5 cccstesssccscosacsaevesvessevassnsenssaeveesessavseosvontetsvceseve 4 5 Controller Term Normalization 000 eceeeesceseeeseeseeeseeeeeseceeeseeseeeaetsees 4 5 Integral Term Normalization eee eseesecereeseeeseseeeeseseeeeaeenees 4 5 Derivative Term Normalization cece ceeeseeeeeeeeeeeeeeeseeenees 4 6 Rolloff Term Normalization cc cceeceeseeeeseeseceseeseeeseeseeseeeeeseees 4 6 Xmath Interactive Control Design Module vi ni com Contents Chapter 5 Root Locus Synthesis OVERVIEW nnana a a a nedeetin K e A O ated O toa deegs tenet yeaa ee 5 1 Window Anatomy e eonia E RER EO EA A ER eee 5 1 Opening the Root Locus Synthesis Window ssssssssssesssssessssesesesesnssese
3. where K is the gain shown in the slider and Variable Edit box Z4 Zm are the zeros and p p are the poles For DC normalization the poles and zeros are restricted to be nonzero If you want the alternate plant to have either poles or zeros at s 0 you must use high frequency normalization Notice that with DC normalization the gain is exactly the DC gain of the alternate plant that is K P 0 Manipulating the Parameters The gain K can be changed using the slider or the Variable Edit box The poles and zeros of Pan can be manipulated graphically using the buttons to the right of the plot Refer to the Graphically Manipulating Poles and Zeros section of Chapter 2 Introduction to SISO Design for a general discussion of how to graphically edit poles and zeros You cannot add a zero if the addition would result in an improper alternate plant transfer function Similarly you cannot delete a pole if the deletion would result in an improper alternate plant transfer function With DC normalization you cannot create any poles or zeros at s 0 and you cannot move existing poles or zeros to s 0 Xmath Interactive Control Design Module 10 4 ni com Chapter 10 Alternate Plant Window You can switch between high frequency and DC normalization by clicking the appropriate buttons If the alternate plant has a pole or zero at s 0 then you cannot switch to DC normalization Using the Alternate Plant Window The
4. P u Wproc u C y Wsens y Wy Figure 7 1 shows a block diagram with the basic setup for LQG synthesis where uis the actuator signal output of the controller Wproc 18 an input referred process noise y is the plant output signal y is the weighted output signal Wsens 18 a Sensor noise P is the plant transfer function C is the controller transfer function W is the output weight transfer function The noises W oc and Wsens are white that is they have constant power spectral densities PSDs The parameter vv is the ratio of the PSD of Wens to the PSD of Wy o r 0 u y y C s gt e gt P s gt W s gt Wproc Wsens Figure 7 2 Block Diagram Showing the Basic Setup for LQG Synthesis Standard LQG All Toggle Buttons Off In LQG synthesis mode the controller minimizes a weighted sum of the steady state actuator and output variance J lim E pu t y t where E denotes expectation p is the Control cost parameter and the parameter v gives the ratio of the intensity of the sensor noise w to the intensity of the process Wroc noise which are assumed to be white Xmath Interactive Control Design Module 7 4 ni com Integral Action Chapter 7 LQG Synthesis When integral action is enabled the controller minimizes a variation on the LQG cost J lim E pu t y t 2 t where 2 1 1 Tig f yada 0 As in the st
5. PID Synthesis Window sesesesnseesesessesntuseserreressesessensesersensreeesesssse 2 4 Root Locus Synthesis Window eee sseeseeseceseesecseeeseeneeeseesees 2 4 Pole Place Synthesis Window eceeeesesseeseeseceseeeeeeeeeeeeeeeseeaees 2 4 LQG Synthesis Window 0 scescesesccessesecceseseccoeenesonsssessesnsenees 2 5 H Infinity Synthesis Window 00 ieee sees eeeeseceeeeseeneeeseeneeeneenees 2 5 History Window wi scecscssseseaesiagiesescevesesveas ceases nii a a a 2 5 Alternate Plant WINdoWissiiinnnno ninam nn nunana e ee 2 5 Key Transfer Functions and Data Flow in ICDM ssesessesseessseesersrerrsrsersreer 2 5 SUMMA a a E E E einen AT E eo 2 6 Origin of the Controller srir a A aa 2 6 What the ICDM Main Window Plots Show ssseseeesseeeseesrereeereeeer 2 7 Controller Synthesis Window Compatibilities tee eee eeeeeeeeeereeeeeees 2 7 Using CDM siaaa evita eesti habs Seeley 2 9 General Plotting Feature asss E E E ERTE 2 11 Ranges of Plots and Sliders eee eeceeseeseeseeeeeeseseeeesesseenseesees 2 11 ZOOMING fxs cesesceedevsies saved AI E ETEA EIEEE havens 2 12 Data Viewing Plot sicrie inet aa a E 2 12 Interactive Plot Re ranging sseesssesesesrsresrereresrrsrsresreresresrsresreerse 2 13 National Instruments Corporation v Xmath Interactive Control Design Module Contents Graphically Manipulating Poles and Zeros eee eeeeeeseeeseeeeeeeeeseeeseenees 2 13 Editing Poles and Zeros
6. PID controllers can be synthesized using the Multiloop Synthesis window Refer to Chapter 13 Multi Loop Synthesis Window Anatomy The PID Synthesis window is shown in Figure 4 1 It consists of the following from top to bottom e A menu bar with entries Special Edit View and Help e A text area that displays the transfer function of the current PID controller e A control panel with five rows each of which corresponds to one design parameter A Bode plot of the controller transfer function with handles for graphically manipulating the design PID Controller Terms The overall controller transfer function is given by the product of up to five terms each of which depends on one parameter The five parameters and corresponding terms in the controller are shown in Table 4 1 from top to bottom Table 4 1 PID Controller Terms and Parameters Term Parameter Symbol Controller Proportional P Gain K K Integral I Integral time constant Tint 1 1 sT Derivative D Derivative time constant Taig Lt slug National Instruments Corporation 4 1 Xmath Interactive Control Design Module Chapter 4 PID Synthesis Table 4 1 PID Controller Terms and Parameters Continued Term Parameter Symbol Controller HF rolloff 1 HF rolloff time 1 Tap 1 I sTy py HF rolloff 2 HF rolloff time 2 Trr2 1 1 sThy2 Toggling Controller Terms On and Off For each paramete
7. To make the plot range smaller grab and drag the appropriate axis to the desired location A dashed line shows what the new plot range will be To make the plot range larger click the left mouse button on the appropriate axis and while holding the button down move the cursor away from the plot axis In this case you will not see a dashed line showing the new plot range Instead a small box will appear that tells you what the new range will be The new range is given by extrapolation of the cursor position You can move the cursor over other plots and even out of the plotting window while increasing the range of a plot If a plot range is symmetric then the new range also will be symmetric That is for a symmetric plot range the minimum and maximum values for X or Y are the same except for sign Changing the maximum will also change the minimum These changes will be exported to the Ranges window Graphically Manipulating Poles and Zeros In many of the ICDM windows the user can grab and drag poles and zeros graphically The paradigm of grabbing and dragging poles and zeros is uniform across windows Remember that you cannot always grab and drag every pole or zero you see in an ICDM plot for example in the Root Locus window you can grab and drag any controller pole or zero but you cannot grab or drag a plant pole or zero Editing Poles and Zeros If there is a push button labeled Edit near the plotting area you can use it to
8. eee ee eseesecseceeeseeeseeseesseeseenseeaeenes Multi Loop Versus Multivariable Design e cee eeseeeeeeeee Opening the Multi Loop Synthesis Window eee Designing a Multi Loop Controller eee eee eeeeeeseeeeseeeeeeseeeees Graphical Editors si nenwiico cnt rennin deicuein ieee ss Selecting and Deselecting Loops ce eceeeeeeesseeeeeseeeseeees Editing and Deleting Loops 200 0 eeseesceeeneeeeeeeeceeeeeeeeeeeaeee Loop Gain Magnitude and Phase eee ee eeeeeeeeeseeseeeeeens Appendix A Using an Xmath GUI Tool Appendix B Technical Support and Professional Services Index Xmath Interactive Control Design Module X ni com Introduction The Xmath Interactive Control Design Module ICDM is a complete library of classical and modern interactive control design functions that takes full advantage of Xmath s powerful object oriented graphical environment It provides a flexible intuitive interactive control design framework This manual provides an overview of different aspects of linear systems analysis describes the Xmath Interactive Control Design function library and gives examples of how you can use Xmath to solve problems rapidly Using This Manual This manual is meant to complement the Xmath Help system The Xmath Help system can be used to find answers to specific questions such as In the Root Locus window how can I add a new pair of complex poles to the controller In contrast
9. Alternate Plant window is used to analyze the robustness of a given controller to changes or unmodeled dynamics in the plant Robustness to Plant Variations The simplest test is to start with the plant which is the default value of the alternate plant and then vary the gain and the poles and zeros of the alternate plant A robust system will not show excessive differences between the responses with the plant and the alternate plant With this method you can easily see the effects of plant gain pole and zero variations With DC normalization varying the poles and zeros affects at high frequencies but does not change the DC gain This is appropriate when the plant variations and modeling errors are more pronounced at high frequencies Adding Unmodeled Dynamics Starting with Pa P and then adding a pole zero pair is a good way to see the effects of a little unmodeled plant dynamics on the system Notice that when you add a pole zero pair you have not yet changed the alternate plant transfer function The change occurs smoothly as you drag the zero away from the pole e To add a little excess phase and rolloff in the loop create a real pole zero pair and separate them a bit with the pole to the right of the zero This simulates the effect of unmodeled diffusion dynamics in the system e To add a little lightly damped dynamics create a lightly damped pole zero pair that is with small negative real part and then drag
10. Control Design Module Chapter 2 Introduction to SISO Design window has an autoscale feature which can be invoked by selecting Autoscale on the View or Plot menu of the window When you invoke Autoscale ICDM tries to assign some reasonable values to the slider and plot scales Zooming You can enlarge any portion of an ICDM plot using plot zooming Clicking the middle mouse button with the cursor anywhere in the plot creates a small box containing a magnified version of the plot near the cursor The middle mouse button can be held down and dragged which creates an effect similar to dragging a magnifying glass across the plot Pressing lt Ctrl gt along with the middle mouse button on UNIX increases the size of the magnified box Clicking with the middle mouse button increases the zoom factor Pressing lt Shift Ctrl gt along with middle mouse button yields a large zoom box with a large magnification factor Zooming is a good way to read text in ICDM plots for example titles axis labels and so on These were intentionally made small because zooming is easy Data Viewing Plots Pointing at or near plotted information within the ICDM windows and clicking the right mouse button causes a small window to appear that identifies the plot and gives the coordinates of the nearest data point for example Loop Gain L 10 1Hz l 11 2dB along with its index This feature is called data viewing If the right mouse button is clicked an
11. MIMO modes depending on the plant that is read in To try out the MIMO features described here and in the next two chapters you must first either read in a MIMO plant or select the MIMO plant from the Default Plants submenu in the Read Plant menu Basic Terminology for MIMO Systems The following sections define the basic terminology and notation used to refer to MIMO systems in ICDM for analysis and plotting The LQG Heo Synthesis window uses additional terminology described in Chapter 12 LQG H Infinity Synthesis Feedback System Configuration ICDM uses the feedback configuration shown in Figure 11 1 The configuration and signal names agree with the standard SISO configuration shown in Figure 2 1 with two differences Here all signal paths represent vector signals whereas in the SISO setup the signals are scalar There is also a new signal dacn that subtracts from the actuator signal This signal can be thought of as an actuator referred disturbance signal an actuator noise or just a fictitious input that allows you to see the transfer functions from the actuator to various other signals The equations describing this system are y P u d c u Ce e r y where as shown in Figure 11 1 y denotes the plant output or sensor signal which is a vector of size n National Instruments Corporation 11 1 Xmath Interactive Control Design Module Chapter 11 Introduction to MIMO Design u denotes the plant input or actuator sign
12. SYSTEM IS CUSTOMIZED AND DIFFERS FROM NATIONAL INSTRUMENTS TESTING PLATFORMS AND BECAUSE A USER OR APPLICATION DESIGNER MAY USE NATIONAL INSTRUMENTS PRODUCTS IN COMBINATION WITH OTHER PRODUCTS IN A MANNER NOT EVALUATED OR CONTEMPLATED BY NATIONAL INSTRUMENTS THE USER OR APPLICATION DESIGNER IS ULTIMATELY RESPONSIBLE FOR VERIFYING AND VALIDATING THE SUITABILITY OF NATIONAL INSTRUMENTS PRODUCTS WHENEVER NATIONAL INSTRUMENTS PRODUCTS ARE INCORPORATED IN A SYSTEM OR APPLICATION INCLUDING WITHOUT LIMITATION THE APPROPRIATE DESIGN PROCESS AND SAFETY LEVEL OF SUCH SYSTEM OR APPLICATION Conventions 3 bold italic monospace monospace bold monospace italic The following conventions are used in this manual The symbol leads you through nested menu items and dialog box options to a final action The sequence File Page Setup Options directs you to pull down the File menu select the Page Setup item and select Options from the last dialog box This icon denotes a note which alerts you to important information Bold text denotes items that you must select or click in the software such as menu items and dialog box options Bold text also denotes parameter names Italic text denotes variables emphasis a cross reference or an introduction to a key concept Italic text also denotes text that is a placeholder for a word or value that you must supply Text in this font denotes text or characters that you should enter fro
13. Whenever a loop is disabled or enabled the corresponding loop gain magnitude and phase plots in the main window also will change line type from dashed to solid or vice versa Xmath Interactive Control Design Module 13 8 ni com Using an Xmath GUI Tool This appendix describes the basics of using an Xmath GUI tool Overview ICDM was developed using the programmable Xmath GUI Graphical User Interface Using a graphical tool such as ICDM is quite different from using a toolbox that has a traditional command line user interface To see a menu of Programmable GUI examples enter guidemo from the Xmath command area This displays the menu of GUI demos shown in Figure A 1 1 Select a demo for example Variable Binding 2 Click OK In a few seconds the demo will appear Your window manager may require you to position the window s generated by the demo this is done by dragging the window to the desired location 3 Note You can run several demos simultaneously National Instruments Corporation A 1 Xmath Interactive Control Design Module Appendix A Using an Xmath GUI Tool Programmable GUI Examples Variable Binding wv More Variable Binding wv Fourier Analysis wv Puzzle Game vy Quadratic Optimization wv Lead Lag Control Desiqner wv Hinf LQG Control Desiqner wv lt Retwm to Previous Menu gt w lt Exit Demo gt Figure A 1 Programmable GUI Examples Each demo has a Help menu in its menu bar nea
14. appropriate entries in the File menu Reading a plant into ICDM is often the first thing you do in a design session Before a plant is read in the plots will be empty and you will be unable to open any synthesis windows Similarly writing the ICDM controller back to Xmath is often the last thing you do in an ICDM design session before quitting or exiting If the current controller has not been written to Xmath and you attempt to exit ICDM a dialog box will open and ask for confirmation before exiting Default Plants By selecting FileRead Default Plant a dialog box will open which you can use to read one of three default plants into ICDM This default plant dialog box is only meant to be used when you are learning how to use ICDM and need a quick way to enter a plant It saves you the trouble of creating a plant in Xmath and then reading it into ICDM It has no real use except in the unlikely event that your plant happens to be one of the default plants Saving and Restoring an ICDM Session The FileSave Tool button saves the entire state of the ICDM tool into an Xmath save file You can continue the design session at another time or on another computer using the FileRestore Tool button Reading Another Plant into ICDM When you first start ICDM you can read a plant from Xmath using the FileRead plant from Xmath button After there is a plant defined in ICDM you can only read a new plant into ICDM from Xmath when there is no synth
15. common tasks are interactively designing a controller and interactively studying the robustness of a given controller Figure 2 2 shows a simplified schematic representation of the interactive design loop ICDM Synthesis Cis ICDM Main Designer __ Window __ ____y Window i Figure 2 2 Simple Representation of the Interactive Design Loop The solid lines indicate graphical or alpha numeric communication The dashed line shows the automatic export of the controller from the synthesis window to the ICDM Main window Notice that only one synthesis window can be open at any given time Also notice that for the purposes of design the user interacts only with the synthesis window and not with the ICDM Main window National Instruments Corporation 2 9 Xmath Interactive Control Design Module Chapter 2 Introduction to SISO Design Figure 2 3 shows a simplified schematic representation of the interactive robustness analysis loop Here the user interacts with the Alternate Plant window interactively changing the alternate plant transfer function Par which is automatically exported to the ICDM Main window for analysis and display The user receives graphical information from the Alternate Plant window displays and also the ICDM Main window Figure 2 3 shows a simplified schematic representation of the interactive design loop Alternate Pat 8 ICDM Main Designer ___p Pl
16. delete operation click the left mouse button with the skull and crossbones cursor away from any pole or zero Xmath Interactive Control Design Module 5 6 ni com Chapter 5 Root Locus Synthesis Design This section gives short descriptions of how the Root Locus window can be used to design or analyze controllers This section also provides some interpretations and describes some uses of the nonstandard contour plots Adding a Pole Zero Pair Adding a pole zero pair is a good way to add a little lead or lag action to an existing controller When you first add the pole zero pair you will not have changed the controller transfer function As you grab the zero and drag it away from the pole you will induce a smooth change in the controller transfer function By dragging the zero a little closer to the origin you will add a small amount of lead action to the controller that is increase the controller phase for frequencies between the pole and zero and increase the magnitude at frequencies larger than the pole Similarly by dragging the zero away from the origin you will create some lag action that is decrease the loop phase between the zero and pole and increase the gain below the pole frequency Deleting Pole Zero Pairs Deleting a controller pole zero pair is a good way to do interactive controller model reduction Suppose that you have synthesized a suitable controller and need to find a lower order controller that has nearl
17. disturbance rejection only on a subspace of dimension r so do not be surprised if some or many diagonal entries of T are not one or off diagonal entries are not zero Finally unlike a SISO plant a MIMO plant can have both poles and zeros at s 0 and such situations will constraint what types of integral action are possible In all cases however ICDM will warn you if the integral action you have selected results in an unstable closed loop system Overview of ICDM for MIMO Design The following sections provide an overview of ICDM for MIMO Design ICDM MIMO Windows The most important windows for MIMO design are e ICDM Main window e LQG Hee window e Multi Loop Synthesis window e History window e MIMO Alternate Plant window e MIMO Plot window Some of windows used for SISO design are not available in MIMO design mode for example PID Root Locus and Pole Place Synthesis Some others are very similar or even the same for example the Main window and the History window Others have different forms that depend on the mode for example LQG Synthesis Alternate Plant and History Some windows only work in MIMO mode for example Multiloop Synthesis and MIMO plot Windows that are not available or applicable in the current mode are dimmed in the menus and cannot be selected Main Window The ICDM Main window is almost the same as in SISO mode The greatest difference is that a different set of plots is available in MIMO mo
18. loop is done by clicking the button at the bottom of the Multi Loop window A loop that has been disabled is represented by a dashed line in the graphical editor There are two choices for editing a loop PID and Root Locus After an option has been selected the regular SISO Design window opens The only difference with the regular SISO Synthesis window is the top part of the window where loop sensor and actuator name are listed Refer to Figure 13 5 where Root Locus synthesis was selected While the SISO Synthesis window is open it is impossible to do any kind of manipulation in the Multi Loop window Only after the SISO window is closed the Multi Loop window will become active again It is therefore not possible for instance to select some other loop and disable it during the SISO design phase Loop Gain Magnitude and Phase In the plot area of the main window two plots are displayed where the loop gain magnitude and phase of each loop that was closed The loop gain is the SISO transfer function LOC COP als where C is the transfer function of the controller of the selected loop and where P jy 18 the SISO equivalent plant of the ith loop Here i refers to index of the loop in the scrolled list The loop displayed in a thick line type is the one that was selected that is the one that is currently being edited The loops are displayed in the same line type in the loop gain magnitude and phase plots in the main window
19. loop poles In this case the 2n 1 degrees of freedom in the closed loop poles along with the constraint that the controller must have at least one pole at s 0 exactly determine the controller transfer function In fact the closed loop poles give a complete parameterization of all controllers with at least one pole at s 0 and n or fewer other poles Equations similar to those shown in the Normal Mode section are used to determine the controller parameters given the closed loop pole locations Xmath Interactive Control Design Module 6 4 ni com Chapter 6 Pole Place Synthesis State Space Interpretation In a state space framework it is common to classify the closed loop poles as n control eigenvalues and n estimator eigenvalues But in fact it makes no difference in the final controller transfer function how you classify the closed loop poles In other words in a state space framework swapping a control eigenvalue and an estimator eigenvalue will result in different feedback and estimator gains but the same final controller Opening the Pole Place Window The Pole Place window can accept any controller with n poles or n 1 poles provided the controller has at least one pole at s 0 The Integral Action toggle button will be properly set In particular it accepts all LQG and Hee controllers This allows the user to manually tune the closed loop poles in a design that was originally LQG or H In
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22. sec reee ro E L r AAE A O E Eaa E E 10 4 Manipulating the Parametersig iiini a aN a e E E RS 10 4 Using the Alternate Plant Window 0 ccc cceeesesecsssseesessessessesseseceseseeseseceesesaesaneas 10 5 Robustness to Plant Variations eee ceeeescseeeeceeeeseeseeesecseeesecseeeaeeaeenees 10 5 Adding Unmodeled Dynamics eee eeeeceeseeeseeseeseeeseeeeeaeeneeeaesneeeaeesees 10 5 Ranges of Sliders and Plot oo eee eseeseceeeeeeeeeeseeeseeseeesesaecneeesesneeeaeenees 10 6 Xmath Interactive Control Design Module viji ni com Contents Chapter 11 Introduction to MIMO Design Basic Terminology for MIMO Systems 0000 cee eeeeseeseceeeseeecceseceeecaeensecseeesetseenseesees 11 1 Feedback System Configuration ec ceeeseeseeesecseceseeseeeseceenseeeeeaeseeeeaees 11 1 Transfer PUnCUONS eannan dere eed eh ese ate ee te ha aed eves 11 2 Inte eral AGON petig rece AERE ch cewcarvnes se E E E A 11 4 Overview of ICDM for MIMO Desig cece cece eeeeseceseceeeseeeeesecseeeaecseenseeneenaes 11 5 TC DM MIMO Wid Ows krenienn Ae sevtes sities EEE 11 5 Main Window vis 8 22s eect Retr n a SR aves 11 5 MIMO PIOt Wn OW cuss ee eee a EE ER TT REE 11 6 History WindOw easan a E E R E E NA 11 7 Alternate Plant Window MIMO Version ccc ccccssscesssecesseeeeeseeesssesenseees 11 7 Chapter 12 LOG H Infinity Synthesis Window Anatomy ss scisit a heeds ited aod tienes E ereesean te ese asin dade ides 12 1 LQG H Infinity Main Window 000 eee ce
23. so static tracking static actuator effort and so on are not affected by the derivative term You can start by making the Tj term small and then gradually increasing it until you get a good balance between better stability margins and excessive actuator effort Rolloff Term Normalization The two HF rolloff terms are low frequency normalized so they have little effect at frequencies below Tj You can start with small values for these parameters and then gradually increase them until you start to notice a degradation in stability Xmath Interactive Control Design Module 4 6 ni com Root Locus Synthesis This chapter describes the user interface terminology and parameters used for root locus synthesis Overview The Root Locus window performs two main functions e Displays selected gain and phase contours in the complex plane of the loop transfer function e Allows the user to manipulate the controller transfer function graphically by dragging controller poles and zeros or dragging the closed loop poles along the root locus plot The Root Locus window only works in SISO mode Window Anatomy The Root Locus Synthesis window is shown with the standard default contour in Figure 5 1 The branches of the locus connect the zeros and poles of the loop transfer function which are shown in the plot The closed loop poles which are on the locus also are shown National Instruments Corporation 5 1 Xmath Interactive Con
24. synthesis window is compatible with the current controller that you have just read in the parameters in the synthesis window will be set appropriately If the synthesis window is not compatible with the controller you will be warned that opening the synthesis window will overwrite the controller When you read a controller from Xmath it is represented as a transfer function This means that you cannot get a controller from Xmath into the LQG or synthesis windows Writing the Plant Back to Xmath Because you cannot change the plant transfer function from inside ICDM the only reason to write the plant back to Xmath is if you have forgotten what the plant transfer function is or if you fear that you may have changed it in Xmath Writing the Alternate Plant back to Xmath If you want to write the alternate plant transfer function to Xmath use the Special menu in the Alternate Plant window Writing a Controller on the History List to Xmath If you want to write a controller that has been saved on the history list to Xmath you first must make it the current controller by opening the History window and selecting it Then use the FileWrite Controller button to write the current controller to Xmath Xmath Interactive Control Design Module 3 4 ni com Chapter 3 ICDM Main Window ICDM Plots Various plots can be shown at the bottom of the main ICDM window The Plot menu is used to select which plots are shown and also to magnify a plot o
25. that have been saved on the history Xmath Interactive Control Design Module 2 6 ni com Chapter 2 Introduction to SISO Design list The current controller is the active or selected entry on the list of saved controllers Only one synthesis window or the History window is allowed to be open at any given time which eliminates any possible confusion over the source of the current controller Remember the simple rule If any synthesis window or the History window is open it is the source of the current controller What the ICDM Main Window Plots Show The plots in the ICDM Main window always use the plant and the current controller For example the step response plot shows the step response of the closed loop system formed by the plant transfer function and the current controller transfer function Optionally the plots also can show the response of the alternate plant connected with the current controller In this case the responses with the plant and the alternate plant are shown in different line types or colors and can always be distinguished by data viewing Refer to the Data Viewing Plots section Controller Synthesis Window Compatibilities As much as possible ICDM allows you to switch from one synthesis method to another while keeping the current controller the same As an example suppose the LQG Synthesis window is open so the current controller is an LQG controller You then can open the Root Locus Synthesis window w
26. the pole and zero away from each other This will create a resonance typical of a neglected mode in a mechanical system e To simulate the effect of an unmodeled time delay in the loop create a real pole zero pair at s 2 T and then drag the zero to s 2 T4 National Instruments Corporation 10 5 Xmath Interactive Control Design Module Chapter 10 Alternate Plant Window Ranges of Sliders and Plot To change the ranges of the Gain slider or the pole zero plot select View Ranges or press lt Ctrl R gt in the Alternate Plant window The slider range also will be changed automatically if you type a new value which is outside the current range into the variable edit box The plot can also be re ranged interactively by grabbing and dragging the plot axes refer to the Interactive Plot Re ranging section of Chapter 2 Introduction to SISO Design Selecting View Auto scale or pressing lt Ctrl A gt in the Alternate Plant window will cause new ranges to be assigned to the slider and plot based on the current alternate plant Xmath Interactive Control Design Module 10 6 ni com Introduction to MIMO Design The following chapters describe the use of ICDM for MIMO design NI assumes the reader is familiar with the use of ICDM for SISO design In many cases the texts describe the differences between SISO and MIMO design This chapter provides an introduction to MIMO design ICDM automatically switches between SISO and
27. the SISO version For a MIMO plant there are many parameters that the user might want to vary in a robustness analysis for example the gain in each actuator the gain in each sensor various poles zeros and their residues for each entry of P and so on There are so many parameters that the user might want vary in MIMO robustness analysis that no simple user interface could suffice Instead the user manipulates the plant in Xmath in the ways appropriate for the problem at hand and simply reads in the set of alternate National Instruments Corporation 11 7 Xmath Interactive Control Design Module Chapter 11 Introduction to MIMO Design plants Therefore the MIMO Alternate Plant window looks very much like the History window the user can read various alternate plants into a list and select one as the alternate plant The semantics of the Alternate Plant window are identical in SISO and MIMO versions ICDM Matrix Plot Plot choices View Step Response Pressure reference Steam flow reference Temperature reference Pressure Steam flow Temperature 10 20 30 f Hz T PC inv PC Closed loop transfer function P Plant L PC ___Her S inv I PC Hyr T PC inv IsPe Hur C inv Iepc Compensator Lact CP Hed P inv I CP Hyd P inv 1 CP Hud cP inv I cP 1 Figure 11 4 MIMO Plot Window with a Step Response Plot of
28. the Transfer Function Xmath Interactive Control Design Module 11 8 ni com LOG H Infinity Synthesis This chapter describes the MIMO LQG H Synthesis window The LQG Hee window is used to synthesize both LQG and He controllers The two design methods have been combined in a single window because of the similarity regarding the use of weights constant weights frequency dependent weights and integrators Window Anatomy The MIMO LQG He Synthesis window consists of a main window and four auxiliary windows for editing constant weights frequency dependent weight functions decay rate and performance level LOG H Infinity Main Window The LQG He Main window is shown in Figure 12 1 From top to bottom it consists of e A menu bar with entries Special Edit View and Help e A message area that describes the synthesis mode type of controller for example LQG with integral action e A control panel for changing the five design parameters Control cost parameter p Sensor noise parameter v Integral action time constant Tin Decay rate or exponential time weighting parameter a Hee performance level y National Instruments Corporation 12 1 Xmath Interactive Control Design Module Chapter 12 LQG H Infinity Synthesis T ICDM MIMO LQG Hinf Synthesis Special Edit View Type 0 001 Control cost Sensor noise P 220 gpi 3888 i e 0 001 1000 m 0 01 x sec 8 4333 3880 L
29. third sensor to the second actuator as shown in Figure 13 4 In Figure 13 4 the two SISO controllers have been labeled C and C respectively When at some moment you are editing modifying deleting disabling or enabling controller C the transfer function of the corresponding SISO plant is that of P as shown in Figure 13 5 so that P equiv 18 the SISO equivalent plant resulting from cutting the loop between the first sensor and the first actuator P thus depends on which loop you are designing Xmath Interactive Control Design Module 13 4 ni com Chapter 13 Multi Loop Synthesis ry Cj oo Wena nneneaees u4 y p A i R y ct ead l r2 C2 P y2 gt gt i Us o gt _ i o y Y3 rg e3 c 2 oad 1 gt E Figure 13 4 Multi Loop Configuration with 3 Sensor and 2 Actuator Plant National Instruments Corporation 13 5 Xmath Interactive Control Design Module Chapter 13 Multi Loop Synthesis ICDM Multi loop RL Synthesis Figure 13 5 Root Locus Window During the Multi Loop Design Figure 13 4 shows an example multiloop configuration for the 3 sensor 2 actuator plant There are two loops one from sensor to actuator 1 and one from sensor 3 to actuator 2 In multiloop design you can alternate between designing each of the SISO controller transfer functions with the other fixed Figure 13 5 shows an example multiloop config
30. this manual is intended for describing the general concepts and operation of the ICDM Document Organization This manual includes the following chapters National Instruments Corporation Chapter 1 Introduction starts with an outline of the manual and some use notes It also contains an overview of the Interactive Control Design Module Chapter 2 Introduction to SISO Design outlines the types of linear systems the system object represents and then discusses the implementation of a system within Xmath Chapter 3 CDM Main Window describes the use of the ICDM Main Window which includes communication with Xmath displaying warning and log messages displaying a variety of standard plots selecting a synthesis method for controller design and controlling auxiliary windows Chapter 4 PID Synthesis discusses the PID synthesis window This window is used to synthesize various types of standard classical SISO controllers such as P PI PD PID lead lag and lag lead 1 1 Xmath Interactive Control Design Module Chapter 1 Introduction e Chapter 5 Root Locus Synthesis describes the user interface terminology and parameters used for root locus synthesis e Chapter 6 Pole Place Synthesis discusses the Pole Place synthesis window which is used to design a SISO controller by assigning the closed loop poles e Chapter 7 LOG Synthesis discusses the LQG synthesis window which is used to synthesize a linear quadratic Gaus
31. 1 where E denotes expectation Integral Action When Integral action is enabled the controller minimizes a variation on the LQG cost Ra Ryu lx tina life Sl Ee Zo se jel j l where t yO yar int National Instruments Corporation 12 11 Xmath Interactive Control Design Module Chapter 12 LQG H Infinity Synthesis Penalizing the running integral of the plant output forces the power spectral density of the plant output to vanish at zero frequency In classical control terms this forces a pole at s 0 in the loop transfer function that is integral control As with PID design the parameter T gives the time scale over which the effects of the integral action will take place Exponential Time Weighting When this feature is enabled the plant is first changed to P s a where a is the Decay Rate parameter In other words the plant is made less stable its poles and zeros are shifted to the right by the value a Then the LQG controller for this destabilized plant is computed Finally the poles and zeros of this controller are shifted left by the Decay Rate parameter a One effect of this shifting is that the closed loop poles are guaranteed to have real part less than the Decay Rate parameter a or in other words the closed loop time domain responses are guaranteed to decay at least as fast as exp at This is why the parameter is called Decay Rate Weight Editing When Weight Edit
32. 2 c cs ccsscessessseasescszessecedeapeviescnsessssepssiseedceve 2 13 Editing Poles and Zeros Graphically cc eceeeceessceesseeseeeseceeeesecseeeseeeenaes 2 14 Complex Poles and Zeros sisisi iana a 2 14 Isolated Real Poles and Zeros nesesseeessereseersseerrseeresrererresesresreeese 2 14 Nonisolated Real Poles and Zeros and Almost Real Pairs 2 14 Adding Deleting Poles and Zeros cceceeeeeeceesseesetseeeeeeseeseeenees 2 15 Adding Deleting Pole Zero Pairs eeeeseeseeeseeseeeseeseeeseeseees 2 15 Chapter 3 ICDM Main Window Wandow Anatomy cc s c vccssecuscvdee siveteesssnacus vue end ondua EE Mask spevscende fides A eE 3 1 Communicating with Xmath oe eee eeeeseceeeeseeeeesseeeseeeesaeeeesseeaeees 3 2 Most Common Usage si ederse ouine e ehair 3 3 Default Plants inon A Rese cect E 3 3 Saving and Restoring an ICDM Session s sssssssssssssesessssesiseserereses 3 3 Reading Another Plant into ICDM sseesseeseseeeeeseesrrereereresrsresreeese 3 3 Reading a Controller from Xmath into ICDM eee eee 3 4 Writing the Plant Back to Xmath s s sssessssssesssssssesssesssssessrsesesesesesse 3 4 Writing the Alternate Plant back to Xmath cece eeeeeeee ees 3 4 Writing a Controller on the History List to Xmath oo 3 4 TEDW PIOUS arse edere se eia fob cos Mend cob E E E E EER 3 5 Selecting Plots sii icnaos 3 5 Ran ees Of Plots ensanse pois set ets ottevied REET 3 6 Plot Magnify Windows
33. DUCTS ARE NOT DESIGNED WITH COMPONENTS AND TESTING FOR A LEVEL OF RELIABILITY SUITABLE FOR USE IN OR IN CONNECTION WITH SURGICAL IMPLANTS OR AS CRITICAL COMPONENTS IN ANY LIFE SUPPORT SYSTEMS WHOSE FAILURE TO PERFORM CAN REASONABLY BE EXPECTED TO CAUSE SIGNIFICANT INJURY TO A HUMAN 2 IN ANY APPLICATION INCLUDING THE ABOVE RELIABILITY OF OPERATION OF THE SOFTWARE PRODUCTS CAN BE IMPAIRED BY ADVERSE FACTORS INCLUDING BUT NOT LIMITED TO FLUCTUATIONS IN ELECTRICAL POWER SUPPLY COMPUTER HARDWARE MALFUNCTIONS COMPUTER OPERATING SYSTEM SOFTWARE FITNESS FITNESS OF COMPILERS AND DEVELOPMENT SOFTWARE USED TO DEVELOP AN APPLICATION INSTALLATION ERRORS SOFTWARE AND HARDWARE COMPATIBILITY PROBLEMS MALFUNCTIONS OR FAILURES OF ELECTRONIC MONITORING OR CONTROL DEVICES TRANSIENT FAILURES OF ELECTRONIC SYSTEMS HARDWARE AND OR SOFTWARE UNANTICIPATED USES OR MISUSES OR ERRORS ON THE PART OF THE USER OR APPLICATIONS DESIGNER ADVERSE FACTORS SUCH AS THESE ARE HEREAFTER COLLECTIVELY TERMED SYSTEM FAILURES ANY APPLICATION WHERE A SYSTEM FAILURE WOULD CREATE A RISK OF HARM TO PROPERTY OR PERSONS INCLUDING THE RISK OF BODILY INJURY AND DEATH SHOULD NOT BE RELIANT SOLELY UPON ONE FORM OF ELECTRONIC SYSTEM DUE TO THE RISK OF SYSTEM FAILURE TO AVOID DAMAGE INJURY OR DEATH THE USER OR APPLICATION DESIGNER MUST TAKE REASONABLY PRUDENT STEPS TO PROTECT AGAINST SYSTEM FAILURES INCLUDING BUT NOT LIMITED TO BACK UP OR SHUT DOWN MECHANISMS BECAUSE EACH END USER
34. In the ICDM Main window the Plot Choices dialog box is used to select any combination of the eight plots This dialog box is modal so you cannot interact with any other Xmath window until you dismiss it Ranges of Plots The ranges for the plots can be set in the Ranges window shown in Figure 3 3 The Ranges window can be made to appear by selecting Plot Ranges or pressing lt Ctrl R gt in the ICDM Main window The Ranges window also is used to determine the number of points used in the plots ICDM provides two convenient ways to select ranges for the plots The first is to use the Autoscale feature which can be invoked from the Plot menu or from the Ranges window When Autoscale is invoked ICDM tries to assign sensible ranges to the plots but does not always succeed The second convenient method for changing the ranges of plots is to use interactive re ranging which is described in the General Plotting Features section of Chapter 2 Introduction to SISO Design The ICDM Ranges window shown in Figure 3 3 is used to set the analysis ranges and plot ranges for the ICDM Main window Xmath Interactive Control Design Module 3 6 ni com Chapter 3 ICDM Main Window Figure 3 3 ICDM Ranges Window Plot Magnify Windows In addition to the standard plotting features zooming data viewing and interactive re ranging described in the General Plotting Features section of Chapter 2 Introduction to SISO Design the plots in the ICDM Main w
35. MIMO design refer to Chapter 11 Introduction to MIMO Design Saving the Current Controller on the History List You can save the current controller on the History list at any time by selecting the Add to History button in the Edit Menu of the ICDM Main window or pressing lt Ctrl A gt in the ICDM Main Window You will be prompted for a comment that is saved along with the design Opening the History Window To make the History window appear select Synthesis History Window in the ICDM Main window This will close any open synthesis window automatically save the current controller on the history list if it has not already been saved and make the current controller the active or selected design on the history list History Window Anatomy The History window is shown in Figure 9 1 From top to bottom it consists of e A menu bar with Special Edit and Help menus e A scrolled list that shows the designs saved on the history list Columns show the history number the date and time the controller was saved the type of controller and a comment that was saved with the controller If the comments run off the visible part of the list you can scroll left and right or resize the History window wider National Instruments Corporation 9 1 Xmath Interactive Control Design Module Chapter 9 History Window e A Variable Edit box which shows which history list entry is active or currently selected The selected entry is the c
36. MIMO design The mode is determined automatically by the plant you read into ICDM The two different modes feature somewhat different plot options different synthesis options and so on NI has made the notation conventions and windows used for MIMO design as similar as possible to those used for SISO design Therefore a user familiar with version 1 0 of ICDM which handled only SISO design should have little trouble using the new MIMO synthesis tools NI also recommends that the user who wishes to use ICDM for MIMO design start by becoming familiar with its features for SISO design Chapters 2 through 10 discuss SISO design Chapters 11 through 13 discuss MIMO design The MIMO descriptions have been written for the user who is familiar with SISO design features To use ICDM you should e Have auser s understanding of Microsoft Windows or X Windows and the window manager that you use For example you should be able to move resize and iconify windows use a pull down menu and use a scrollbar Xmath Interactive Control Design Module 1 4 ni com Chapter 1 Introduction e Have a user s understanding of Xmath enough to create a plant transfer function e Know the basics of how to interact with an Xmath GUI application for example using a slider to set a parameter value a variable edit box for typing in values data viewing and plot zooming e Know the basics of classical control system design for SISO design and state
37. NI MATRIXx Xmath Interactive Control Design Module April 2007 lt 7 NATIONAL 370754C 01 INSTRUMENTS Worldwide Technical Support and Product Information ni com National Instruments Corporate Headquarters 11500 North Mopac Expressway Austin Texas 78759 3504 USA Tel 512 683 0100 Worldwide Offices Australia 1800 300 800 Austria 43 662 457990 0 Belgium 32 0 2 757 0020 Brazil 55 11 3262 3599 Canada 800 433 3488 China 86 21 5050 9800 Czech Republic 420 224 235 774 Denmark 45 45 76 26 00 Finland 385 0 9 725 72511 France 33 0 1 48 14 24 24 Germany 49 89 7413130 India 91 80 41190000 Israel 972 3 6393737 Italy 39 02 413091 Japan 81 3 5472 2970 Korea 82 02 3451 3400 Lebanon 961 0 1 33 28 28 Malaysia 1800 887710 Mexico 01 800 010 0793 Netherlands 31 0 348 433 466 New Zealand 0800 553 322 Norway 47 0 66 90 76 60 Poland 48 22 3390150 Portugal 351 210 311 210 Russia 7 495 783 6851 Singapore 1800 226 5886 Slovenia 386 3 425 42 00 South Africa 27 0 11 805 8197 Spain 34 91 640 0085 Sweden 46 0 8 587 895 00 Switzerland 41 56 2005151 Taiwan 886 02 2377 2222 Thailand 662 278 6777 Turkey 90 212 279 3031 United Kingdom 44 0 1635 523545 For further support information refer to the Technical Support and Professional Services appendix o comment on National Instruments documentation refer to the National Instruments Web site at ni com info and enter the info code feedback 2007 National Instrumen
38. The history list can be used in several ways You can save controllers as benchmarks whose performance you want to match with a simpler controller You also can save any promising designs that you find so you can later use them as the initial conditions for designing Xmath Interactive Control Design Module 9 4 ni com Alternate Plant Window This chapter describes the form of the Alternate Plant window used for SISO design refer to Chapter 11 Introduction to MIMO Design for the form used for MIMO design Role and Use of Plant and Alternate Plant In addition to the plant P ICDM can optionally maintain an alternate plant Py These two transfer functions have different uses and purposes e The plant is always used for the synthesis windows that need it For example when synthesizing an LQG controller the LQG synthesis is based on the plant P In other words the plant is used for design In contrast the alternate plant Pay is never used by any synthesis method e The alternate plant P is used only for analysis Specifically by turning on the Alternate Plant display refer to the Displaying the Alternate Plant Responses section the plots in the ICDM Main Window will show the alternate plant connected with the current controller and the plant connected with the current controller e Itis not possible to change the plant in ICDM except by reading it from Xmath In contrast the alternate plant can be manipulated usi
39. age area that describes the synthesis mode type of controller for example LQG with integral action e A control panel for changing the four design parameters Control cost parameter p Sensor noise parameter V Integral action time constant 7 Decay rate or exponential time weighting parameter a These parameters are described in greater detail later in this chapter National Instruments Corporation 7 1 Xmath Interactive Control Design Module Chapter7 LQG Synthesis Control Sym Root Locus Weight poles amp zeros Weight transfer function freq Hz Figure 7 1 LQG Synthesis Window e A control panel used to graphically edit the output weight transfer function e A plotting area that contains the following plots The symmetric root locus plots of the control and estimator closed loop poles The control cost and sensor noise parameters can be changed by dragging the closed loop poles along the plot Xmath Interactive Control Design Module 7 2 ni com Chapter 7 LQG Synthesis If the decay rate is enabled it is shown as a vertical line that can be dragged A plot showing the poles and zeros of the output weight transfer function If weight zero editing is enabled the zeros can be edited graphically A plot showing the magnitude of the output weight transfer function Synthesis Modes In addition to standard LQG synthesis the LQG Synthesis window supports any combina
40. al which is a vector of size n r denotes the reference or command input signal which is a vector of size n e denotes the error signal which is a vector of size ny dact denotes the actuator disturbance signal which is a vector of size n P denotes the plant transfer function which is a matrix of size n X n C denotes the controller transfer function which is a matrix of size n x n Figure 11 1 shows standard feedback connections and signals used in ICDM for MIMO design All of the signals are vectors dact Figure 11 1 Standard Feedback Connections and Signals for MIMO Design Transfer Functions In ICDM the plant and controller transfer function are required to be strictly proper that is P s C sI A B C s Coont SI Acon Beont where A B and C are matrices stored in the system object corresponding to the plant and similarly for the controller The plant order or plant McMillan degree is the size of A that is the number of plant states Similarly the size of Acon 1s the controller order or controller degree In ICDM the multivariable transfer function is required to be strictly proper that is have zero feedthrough term D 0 The controller is required to be proper Xmath Interactive Control Design Module 11 2 ni com Chapter 11 Introduction to MIMO Design The standard feedback system has two vector input signals r and dacn and three vector output signals
41. alues For multivariable systems the optimal control and the optimal estimator play different roles in the control system But in the single actuator single sensor case the roles are completely symmetric In particular swapping the parameters p and v yields the same final LQG controller This symmetry is broken if you use either output weighting or integral action however Manipulating the Design Parameters The design parameters p and v can be changed using the associated sliders or the variable edit boxes If you type in a value that is outside the current slider range the slider range will automatically adjust You can change the ranges for the sliders using the Ranges window The parameters T7 and a can be manipulated using the sliders or variable edit boxes provided the associated toggle button is on If the toggle button is off then the slider and variable edit box are insensitive you cannot drag the slider handle and you cannot type into the variable edit box When the toggle buttons are turned on again the parameters are restored to their previous or default values Manipulating the Design Parameters Graphically The design parameters also can be manipulated graphically as follows e The closed loop poles are shown on the two symmetric root locus plots They can be dragged along the root locus plot which results in setting the parameters p or v appropriately e When the Decay Rate toggle button is on a dashe
42. andard mode the sensor noise parameter v is the ratio of the sensor noise intensity to the input referred process noise intensity Penalizing the running integral of the plant output forces the power spectral density of the plant output to vanish at zero frequency In classical control terms this forces a pole at S 0 in the loop transfer function that is integral control As with PID design the parameter T gives the time scale over which the effects of the integral action will take place Exponential Time Weighting When this feature is enabled the plant is first changed to P s a where a is the Decay Rate parameter In other words the plant is made less stable its poles and zeros are shifted to the right by the value a Then the LQG controller for this destabilized plant is computed Finally the poles and zeros of this controller are shifted left by the Decay Rate parameter a One effect of this shifting is that the closed loop poles are guaranteed to have real part less than the Decay Rate parameter a In other words the closed loop time domain responses are guaranteed to decay at least as fast as exp at This is why the parameter is called Decay Rate National Instruments Corporation 7 5 Xmath Interactive Control Design Module Chapter 7 LQG Synthesis Output Weight Editing When Weight Zero Edit is enabled the LQG controller is based on y Wy which is a filtered version of the plant outp
43. ange of the magnified plot By selecting Plot New Plot Magnify or pressing lt Ctrl N gt in the ICDM Main window you can select a plot for magnification In this case the new plot will appear in a new Plot Magnify window When multiple Plot Magnify windows are open the Plot Magnify command will send the selected plot to the most recently created Plot Magnify window New Plot Magnify will create a new Plot Magnify window Xmath Interactive Control Design Module 3 8 ni com Chapter 3 ICDM Main Window Selecting a Synthesis or History Window The Synthesis menu in the ICDM Main window is used to select which synthesis window will be active If the current controller is compatible with the requested synthesis window then the synthesis window opens and is initialized with the current controller If the current controller is not compatible with the Synthesis menu selected then a dialog box appears that gives the user several options If the user proceeds in this case the current controller will be replaced with the previous design in the synthesis window selected For example if the Root Locus Synthesis window is open so that the current controller is a general transfer function and the user requests the LQG Synthesis window a dialog box will issue a warning that the current design will be overwritten and give the user the option of cancelling the request proceeding or writing the current Root Locus controller to the history list b
44. ant pe npr Window Window E Figure 2 3 Simple Representation of the Interactive Robustness Analysis Figure 2 4 shows a simple ICDM session The ICDM Main window is shown at upper left and the Pole Place Synthesis window is at lower right The user can drag the closed loop poles in the Pole Place window The controller that is synthesized is automatically exported to the ICDM Main window for analysis and plotting Notice that the user s graphical input is mostly through the Pole Place window The ICDM Main window is used mostly for graphical output Xmath Interactive Control Design Module 2 10 ni com Chapter 2 Introduction to SISO Design ICDM Main Window 0 1 2 44 6 8 Loop phase Act step response Figure 2 4 Simple ICDM Session General Plotting Features All of the plots in the ICDM Main and other windows support several useful features arbitrary re ranging zooming data viewing and interactive graphical re ranging Ranges of Plots and Sliders Every ICDM window has an associated Ranges window that can be used to set the ranges of the sliders and plots appearing in the window as well as other parameters such as numbers of points plotted The Ranges window can be opened by selecting Ranges on the View or Plot menu or by pressing lt Ctrl R gt in the window in question In addition every ICDM National Instruments Corporation 2 11 Xmath Interactive
45. ay Rate toggle button controls exponential time weighting The Weight Edit toggle button enables and disables output weight editing The Hinf Bound toggle button enables and disables design mode National Instruments Corporation 12 7 Xmath Interactive Control Design Module Chapter 12 LQG H Infinity Synthesis Opening the LOG H Infinity Synthesis Window The LQG H window can only accept LQG H controllers If the current controller is of type LQG H perhaps from the History window and the LQG Hee window is opened the current controller is read into the LQG Hee window that is the push buttons and parameters are set to the appropriate values If the current controller is not of type LQG H and the user attempts to open the Synthesis window a dialog box opens and warns the user that proceeding with opening the Synthesis window will overwrite the current controller with the LQG He controller The LQG He window remembers its parameter settings that is when it is opened the parameters will be exactly as they were when the LQG Hee window was last closed or set to default values if the LQG He window has not been opened in this ICDM session Setup and Terminology The input and output signals are distinguished in the following categories e Disturbances e Actuators Uac e Measurements y e Costed outputs z Whether an input signal is a disturbance or an actuator and whether an output is a measurement or a costed o
46. ctronic or mechanical including photocopying recording storing in an information retrieval system or translating in whole or in part without the prior written consent of National Instruments Corporation National Instruments respects the intellectual property of others and we ask our users to do the same NI software is protected by copyright and other intellectual property laws Where NI software may be used to reproduce software or other materials belonging to others you may use NI software only to reproduce materials that you may reproduce in accordance with the terms of any applicable license or other legal restriction Trademarks MATRIXx National Instruments NI ni ccom and Xmath are trademarks of National Instruments Corporation Refer to the Terms of Use section on ni com legal for more information about National Instruments trademarks Other product and company names mentioned herein are trademarks or trade names of their respective companies Members of the National Instruments Alliance Partner Program are business entities independent from National Instruments and have no agency partnership or joint venture relationship with National Instruments Patents For patents covering National Instruments products refer to the appropriate location Help Patents in your software the patents txt file on your CD or ni com patents WARNING REGARDING USE OF NATIONAL INSTRUMENTS PRODUCTS 1 NATIONAL INSTRUMENTS PRO
47. d Zeros If the pole or zero that you grab is real and not very close to another real pole or zero then the pole or zero motion will be constrained to the real axis You cannot drag the pole or zero off the real axis Nonisolated Real Poles and Zeros and Almost Real Pairs If a pair of nearby poles or zeros is very near the real axis that is two nearby real poles or zeros or a pair of complex poles and zeros with very small imaginary part then the dragging motion will depend on how you originally drag the poles or zeros If you drag it up or down then the pair acts as a complex pair and there is no constraint on how you can drag it For example two real poles that are very close to each other can be split into a complex pair by grabbing either one and dragging it away from the real axis On the other hand if you drag the selected pole or zero left or right then the pair act as a real pair the selected pole or zero then can be dragged only along the real axis and the other pole or zero becomes real Xmath Interactive Control Design Module 2 14 ni com Chapter 2 Introduction to SISO Design if it was not already but otherwise does not move Thus to make a pair of complex poles real you first drag one of them near the real axis and release Then you select one of these poles again and this time drag it left or right This will cause the pair to become real Adding Deleting Poles and Zeros This section describes how you are all
48. d all inputs are actuators Input referred disturbances and measurement noise can be selected using the toggle buttons in the second column of the noise level display of the Weights window Setting a toggle button to the off position corresponds to setting the noise variance of the corresponding signal to zero Weighted outputs and inputs can be selected using the toggle buttons in the second column of the Weights window Setting a toggle button to the off position corresponds to setting the weight of the corresponding signal to zero 12 13 Xmath Interactive Control Design Module Chapter 12 LQG H Infinity Synthesis By clicking the button at the bottom of the Weights window arbitrary weight matrices can be loaded from Xmath The noise variances and weights selected in this way are simply added to the diagonal weight and noise matrices determined by the push buttons and sliders of the Weights window There are certain limitations and restrictions e IfR is zero none of the weight sliders on the actuators can be disabled This is because of the nonsingularity requirement of the input weight matrix for the regulator problem e If Q is zero none of the noise variance sliders of the sensors can be disabled This is because of the nonsingularity requirement of the output weight matrix for the estimator problem e Ifa smaller number of actuators have been selected than there are sensors setpoint tracking cannot be expected i
49. d dragged the selected plot is tracked even if another plot comes close Pressing lt Shift gt along with the right mouse button allows the user to get values on the piecewise linear plot that interpolates the data values In this case index 45 7 means that the selected plot point is between the 45th and 46th X coordinate entries Because all ICDM plots have extensive data viewing features the number of labels used to identify plots are minimal For example the Root Locus plot has red and black poles and zeros shown but no indication or label in the plot saying what these colors mean On a black and white display you cannot distinguish between the red and black poles zeros You can read in the Help file message that the red ones correspond to the plant and the black ones to the controller However the easiest way to find out what the Xmath Interactive Control Design Module 2 12 ni com Chapter 2 Introduction to SISO Design poles and zeros are and indeed the only way on a black and white display is to use data viewing As a general rule To find out the meaning purpose or value of an object pole zero curve and so on in an ICDM plot use data viewing Most objects in the ICDM Plot windows support data viewing Interactive Plot Re ranging The range for any plot can be set in the appropriate Ranges window Alternatively the ranges for plots can be interactively changed by grabbing and dragging the axes of the plots
50. d line appears in the symmetric root locus plots showing Ns a You can drag this line left and right to set the Decay Rate parameter e When Weight Zero Editing is enabled the user can graphically manipulate the zeros of the weight transfer function W on the plot labeled Weight Poles amp Zeros Refer to the Graphically Manipulating Poles and Zeros section of Chapter 2 Introduction to SISO Design for a general discussion of how to move add delete or edit these zeros graphically National Instruments Corporation 7 7 Xmath Interactive Control Design Module Chapter 7 LQG Synthesis Ranges To change the ranges of the sliders or plots select View Ranges or enter R in the LQG window The slider ranges also will be changed automatically if you type a new value which is outside the current range into the corresponding variable edit box The plot also can be re ranged interactively by grabbing and dragging the plot axes refer to the Interactive Plot Re ranging section of Chapter 2 Introduction to SISO Design Selecting View Auto scale or pressing lt Ctrl A gt in the LQG window causes new ranges to be assigned to the sliders and plots based on the current controller Xmath Interactive Control Design Module 7 8 ni com H Infinity Synthesis This chapter describes the Hoo Synthesis window used for SISO plants The H Synthesis window is used to synthesize a central controller Such controllers are sometimes called l
51. de and the default plot selections are different When the user selects Plot Options from the ICDM Main window menu bar the window shown in Figure 11 2 appears It shows the Plot Choices window for the MIMO case This window contains a subset of the complete set of plot options which are the ones most likely to be used To get access to the complete set of plot National Instruments Corporation 11 5 Xmath Interactive Control Design Module Chapter 11 Introduction to MIMO Design options the user clicks the Show all options button after which the plot options window shown in Figure 11 3 opens From this window all transfer functions mentioned in the Transfer Functions section can be selected y MIMO Plot Options Dialog Frequency response singular value plots F LC J S inv I Pc JP F T PC inv I Pc m e Miscellaneous plots 1 Closed loop poles F Step response 1 Actuator effort Show all options OK Apply Cancel rr _ Figure 11 2 Plot Choices Window for the MIMO Case MIMO Plot Window For a more detailed MIMO transfer function plot an option labeled MIMO plot is available under the Main window menu bar This plot will display MIMO responses in a matrix format where each element of the transfer function is displayed individually in one element of the plot matrix Figure 11 4 Figure 11 4 shows the MIMO Plot window with a step response plot of the transfer function arranged as a matrix The MIMO Plot wind
52. e gt RECEP TE 1 Int tine _ Decay rate 1 Hinf boud Figure 12 1 LQG H Infinity Main Window e A pull down menu for frequency dependent weight selection on inputs W nol 1 4 7 u and outputs W ppd 1 5M e A button for recomputing the controller These parameters are described in greater detail later in this chapter LOG H Infinity Weights Window The Weights window is for defining control cost and noise level parameters and is shown in Figure 12 2 From top to bottom the Weights window consists of e A menu bar with entries Special Edit View and Help e Arow with two radio buttons which toggle the contents of the window between control cost and noise level parameters The following Xmath Interactive Control Design Module 12 2 ni com Chapter 12 LQG H Infinity Synthesis descriptions are for the control cost parameter display The noise level display is similar in appearance e A table with n rows having in each row A toggle button to include the input in the set of control inputs A toggle button to include the input in the set of costed inputs labeled with the signal name A slider defining the constant weight factor of the input u i eee p pi l at A variable edit box for the same constant weight factor m ICDM LQG Weights Special Edit View A Control cost wy Noise level F Actuators on off fF Water flow F Actuators on off F Sensors
53. e u and y It can therefore be described by the 3 x 2 block matrix that relates the three output vector signals to the two input vector signals 5 u CHU PC CP I CPY d 2 P P I CP 1 1 J PC I PC P 1 CP The entries of this block matrix that is the transfer functions from r and dact tO e u and y have standard names and interpretations which agree with the standard SISO notation e The sensitivity transfer function is denoted S and given by S I PCY The sensitivity transfer function is the transfer function from reference input r to the error signal e e The closed loop transfer function T is given by T PC I PCY Tis the transfer function from r to y T can be expressed in several other ways for example T PC UI CP 1 PC PC 1 S e The actuator effort transfer function C I PCY is the transfer function from r to u and so is related to the actuator effort required For example its step response matrix shows the closed loop step responses from each reference input signal to each actuator signal e The transfer function from d to e P T CP is denoted Sac and called the actuator referred sensitivity transfer function The actuator referred sensitivity transfer function determines the errors generated by actuator referred disturbances It also can be expressed as I PC P Notice that it is complementary to the transfer function described just above that is C PC in the sense tha
54. e contour plotting style The plot shows the set of all possible closed loop pole locations as the gain is swept from 0 to 160 200 10 The plot shows the loci of points where the phase angle of the loop transfer function is 160 170 180 190 or 200 These plots show the set of all possible closed loop pole locations as the gain is swept from 0 to and there is an additional phase shift of 20 10 in the loop transfer function None No phase contours are plotted 0 dB The plot shows the locus of points where the magnitude of the loop transfer function including the delay if applicable is 0 dB 2 2 dB The plot shows the loci of points where the magnitude of the loop transfer function is 2 1 0 1 and 2 dB respectively None No magnitude contours are plotted This is the default magnitude contour plotting style Notice that by selecting None for both phase and magnitude contours the plot shows only the controller and plant poles and zeros This is useful for graphically editing the controller poles and zeros If any phase contours are plotted the closed loop poles are shown in blue on a color display They can be dragged along the 180 contour plot National Instruments Corporation 5 5 Xmath Interactive Control Design Module Chapter 5 Root Locus Synthesis All of the plots support data viewing click the right mouse button with the cursor positioned near a pole zero or on
55. e of the plots This allows you to find the gain associated with a particular point on a phase contour for example Slider and Plot Ranges To change the ranges of the Gain slider or the root locus plot select View Ranges or press lt Ctrl R gt in the Root Locus window The slider range also will be changed automatically if you type a new value which is outside the current range in the corresponding variable edit box The plot also can be re ranged interactively by grabbing and dragging the plot axes Refer to the discussion of plot re ranging in section s plots features Selecting View Auto scale or pressing lt Ctrl A gt in the Root Locus window will cause new ranges to be assigned to the slider and plot based on the current controller Manipulating the Parameters The Graphically Manipulating Poles and Zeros section of Chapter 2 Introduction to SISO Design describes how to graphically manipulate the controller poles and zeros The Root Locus window enforces a proper controller that is the controller must have at least as many poles as zeros If you attempt to add one zero or a pair of zeros that would result in more controller zeros than poles a warning is issued Similarly you cannot delete one or more controller poles if the deletion would result in an improper controller You can select Edit Undo to restore the deleted pole s and or zero s provided you have not made any other changes since deleting To abort a
56. e output weight transfer function National Instruments Corporation 8 3 Xmath Interactive Control Design Module Chapter 8 H Infinity Synthesis u 24 1S gt r 0 y 22 C s P s gt W s K gt gus Figure 8 2 Block Diagram Showing the Basic Setup for H Infinity Synthesis Figure 8 2 shows a block diagram with the basic setup for Heo synthesis where closed loop transfer matrix H relates the two exogenous inputs w and w to the two outputs z and z2 The design is based on H the closed loop transfer matrix relating the noises w and w to the signals z and z2 H is given by the following equation H L JpPC JpvCc t PW JvPCW The entries of the closed loop transfer matrix can be interpreted as the normalized transfer functions from the process and sensor noises to the actuator and output respectively The singular values of H are shown in the top left plot of the Hoo Synthesis window Central H Infinity Controller The controller C is chosen to minimize the y entropy of the closed loop transfer matrix H given by E Y 1 1 L H foe og 1_law poe 1 6 y 1 6 0 y where and O are the singular values of H jo Xmath Interactive Control Design Module 8 4 ni com Chapter 8 H Infinity Synthesis If either of these singular values is equal to or exceeds y the y entropy is defined to be o In other words the y entro
57. e p and v sliders are moved The user can change the ranges for the sliders using the Ranges window The parameters 7 and a can be manipulated using the sliders or variable edit boxes provided the associated toggle button is On If the toggle button is Off then the slider and variable edit box are insensitive you cannot drag the slider handle and you cannot type into the variable edit box When the toggle buttons are turned On again the parameters are restored to their previous or default values The design parameters also can be manipulated graphically e When the Decay Rate toggle button is on a dashed line appears in the Decay Rate window showing Rs a The user can drag this line left and right to set the Decay Rate parameter e When Weight Edit is enabled the user can graphically manipulate the poles and zeros of the weight transfer functions or on the plot labeled Weight Poles amp Zeros Refer to the Editing Poles and Zeros Graphically section of Chapter 2 Introduction to SISO Design for a general discussion of how to move add delete or edit these zeros graphically e When the H performance level toggle button is enabled the user can graphically manipulate y by moving the dashed line in the Hee Performance window vertically Xmath Interactive Control Design Module 12 16 ni com Chapter 12 LQG H Infinity Synthesis Ranges To change the ranges of the sliders or plots select View Ranges or press lt Ctr
58. e required phase shift 10 and gain increase 3 dB at for example by adding an appropriate pole zero pair On the other hand if the complex number is a poor place for a closed loop pole for example very lightly damped or unstable then the current compensator is not robust since only a 10 phase shift along with 3 dB of gain change in loop gain most likely the plant would result in a closed loop pole at s In this case you turn to the problem of synthesizing new compensation which decreases the phase and magnitude of the loop transfer function at the frequency s This has the effect of making the closed loop system less likely to have a pole at s when the plant transfer function is changed that is it results in a more robust design Figure 5 3 shows the Root Locus window with the phase contours turned off and the 0 dB magnitude contour turned on The locus shows the set of all possible closed loop poles for the modified loop transfer function L s e L s as O varies from zero to 27 By data viewing the contour you can find the phase shift value of O that corresponds to any point on the locus Xmath Interactive Control Design Module 5 8 ni com Chapter5 Root Locus Synthesis Figure 5 3 Root Locus Synthesis Window with the 0 dB Magnitude Contour National Instruments Corporation 5 9 Xmath Interactive Control Design Module Pole Place Synthesis This chapter discusses the Pole Place Synthesis window which is u
59. e transfer function C 1 PC which is the transfer function from r to u Integral action means that the controller C has a pole at s 0 Roughly speaking this means that the loop gain is very large at low frequencies Integral action implies that S O 0 so if r is constant the error e converges to zero that is the output y t approaches r as t gt Overview of ICDM This section provides a broad overview of the architecture concepts and major functions of ICDM restricting our discussion to the case of SISO plants and controllers This section also provides a summary of how ICDM works and what it does ICDM Windows ICDM supports many windows that serve a variety of functions The most important windows are National Instruments Corporation ICDM Main window PID Synthesis window Root Locus Synthesis window Pole Place Synthesis window LQG Synthesis window Hoo Synthesis window History window Alternate Plant window 2 3 Xmath Interactive Control Design Module Chapter 2 Introduction to SISO Design These are briefly described in the following sections and in more detail in later chapters Several of these windows have different forms for SISO and MIMO design This chapter restricts the discussion to the SISO forms Refer to Chapter 11 Introduction to MIMO Design for a discussion of the MIMO forms ICDM Main Window The most important window is the ICDM Main window which is used to e Communicate w
60. ed inputs and outputs The sliders represent the noise intensities instead of the control cost The button is meant to load the noise variance matrix of states and outputs including cross terms Qx Q y Q yx Q y Q Xmath Interactive Control Design Module 12 4 ni com Chapter 12 LQG H Infinity Synthesis The weights P i Py j Pu and p are then replaced with noise variances Vu i Vy js Vu and V The noise level parameter in the main LQG Hee window is related to the noise levels in this window by v v V Decay Rate Window The Decay Rate window is shown in Figure 12 3 From top to bottom it consists of e A menu bar with entries Special Edit View and Help e A plotting area that contains two plots A plot of controller poles and a vertical line indicating the decay rate A plot of estimator poles and a vertical line indicating the decay rate ICDM LQG Poles Estimator poles Figure 12 3 LQG H Infinity Decay Rate Window H Infinity Performance Window The H Performance window shown in Figure 12 4 consists of from top to bottom e A menu bar with entries Special Edit View and Help e A plot area that contains a plot of the closed loop transfer matrix along with the parameter y which can be grabbed and dragged to a new National Instruments Corporation 12 5 Xmath Interactive Control Design Module Chapter 12 LQG H Infinity Synthesis value If a lower b
61. edit poles and zeros If you click the Edit button the cursor will become a pencil symbol Select a pole or zero by clicking the left mouse button with the cursor positioned at the desired pole or zero A dialog box will open that National Instruments Corporation 2 13 Xmath Interactive Control Design Module Chapter 2 Introduction to SISO Design contains variable edit boxes for the value of the pole or zero the real and imaginary part when the pole or zero is complex and if appropriate its multiplicity After you enter new values you can select OK which will make the changes and dismiss the dialog box or Cancel which will dismiss the dialog box without making the changes The values you type in will not be accepted if they are invalid for example a negative multiplicity for a pole or zero Editing Poles and Zeros Graphically The easiest way to change a pole or zero is to grab it by clicking the left mouse button and dragging it to the desired location You can only drag poles and zeros in sensible ways For example you cannot drag a single real pole or zero off the real axis to a complex location More precisely the dragging of poles and zeros works as described in the following sections Complex Poles and Zeros If the pole or zero that you grab is complex then the complex conjugate pole or zero will automatically move as required In this case you can drag the pole or zero in any direction Isolated Real Poles an
62. efore proceeding If the user proceeds then the Root Locus window will close the LQG window will open and the current controller will be overwritten with the controller from the LQG Synthesis window To open the History window select Synthesis History Because all controllers are compatible with the History window the History window will open with the current controller active In other words the current controller will be saved on the history list if it has not already been saved and made the active or selected controller on the history list Edit Menu The Edit menu has two important entries e Selecting EditAdd to History or typing h in the ICDM Main window will cause the current controller to be saved on the history list You will be prompted for a comment that will be saved along with the current controller Refer to Chapter 9 History Window e Selecting Edit Alternate Plant window causes the Alternate Plant window to appear Refer to Chapter 10 Alternate Plant Window National Instruments Corporation 3 9 Xmath Interactive Control Design Module PID Synthesis This chapter discusses the PID Synthesis window This window is used to synthesize various types of standard classical SISO controllers such as P PI PD PID lead lag and lag lead However the controller that is designed by the PID Synthesis window will be referred to as a PID controller even if it has some other form such as PI Multivariable MIMO
63. equency or time by graphical input or assigned to a Butterworth configuration The pole place window supports integral action as an option Xmath Interactive Control Design Module 2 4 ni com Chapter 2 Introduction to SISO Design LOG Synthesis Window The LQG Synthesis window synthesizes LQG controllers and therefore can be used only with strictly proper plants The user can vary weights for the ratio of control input to regulation output cost and the ratio of sensor output noise power to process input noise power Optionally the user can specify a guaranteed decay rate and integral time constant By dragging zeros on a symmetric root locus plot the user can vary the state weighting or perform LTR design There also is a MIMO LQG window described in Chapter 12 LQG H Infinity Synthesis H Infinity Synthesis Window The Heo Synthesis window synthesizes central He controllers also called minimum entropy risk sensitive or LEQG controllers The user can vary weights for the ratio of control input to regulation output cost the ratio of sensor output noise power to process input noise power and the risk sensitivity or Heo bound parameter y The user can vary the state weighting or equivalently the output weight transfer function by dragging zeros History Window The History window is used to display and manipulate the design history list which is a list of controllers that have been explicitly saved during the d
64. erator 2 2 order 2 2 transfer function 2 2 2 5 11 2 plant McMillan degree 11 2 Plot Choices window 11 5 plots ICDM 3 5 Nichols 3 5 Nyquist 3 5 zooming 2 12 ni com Pole Place Modes 6 2 Synthesis window 2 4 6 1 window 2 8 poles 1 1 closed loop 2 3 2 4 polynomial 2 3 process noise 12 9 programming examples NI resources B 1 proper polynomials 2 2 R Ranges window 2 11 risk sensitivity 2 5 robustness analysis 2 10 Root Locus plot 2 4 window 2 4 2 7 S sensitivity transfer function 2 2 11 3 sensor noise 12 9 noise parameter 7 1 7 5 8 1 12 1 signal 2 2 11 1 signal 2 2 error 2 2 Simple ICDM Session 2 11 SISO 1 4 National Instruments Corporation l 3 Index software NI resources B 1 step response 2 3 plot 2 7 support technical B 1 system object 2 1 T technical support B 1 training and certification NI resources B 1 transfer function 2 2 2 5 alternate plant 2 5 closed loop 2 3 controller 2 2 current controller 2 5 loop 2 2 plant 2 2 troubleshooting NI resources B 1 W Web resources B 1 Weight Edit 12 16 weight transfer function 7 7 Weight Zero Edit 7 6 8 5 Z Zooming 2 12 Xmath Interactive Control Design Module
65. eseesceseeeeeseeneeeseeeseceeeeeseensereeeeeees 12 1 LQG H Infinity Weights Window 00 eee ceeeseceeceeeeseeeseceeeeeetseeeseseeeaee 12 2 Decay Rate W Indo Wini orara Seba coat gobs ons co yenbte fue E aa 12 5 H Infinity Performance Window ce eceeeeseseeeeseeeeeeseeseeeeeessenseeaesneeeaeenees 12 5 Frequency Weights Window nieee e ateue aE 12 6 Synthesis Modes and Window Usage e seeseeessessesesseerssrsresrsreereresresrsresresesrestsrrsrestnt 12 7 Opening the LQG H Infinity Synthesis Window eee eee eects reeeeeees 12 8 Setup and Terminology is cessed i n o e ta aes 12 8 Standard LQG All Toggle Buttons OfP esseeseeseeeseeseseseeseseersseeseresreseesses 12 11 Integral Acto na vider date bia laa ean 12 11 Exponential Time Weighting insiren ceceseeseeseeseeeseesseeaeeseeeaeeneeeseseeeaees 12 12 Weight Editing nene aie aie es en ea ea 12 12 HOW to Selectw iw yand Ze vce aa e e E i A AEE E Weavers cutee EKR 12 13 HeInfinity Solution sieni n a nee wide aia tee 12 14 Manipulating the Design Parameters 000 00 eee eeeeeeseceeeeseceeetseeeeessecseeeseceseeaseneenaes 12 16 Main Window 32 fendi dala athe ea Hie ese Oe tes 12 16 IRAN QOS se EE PER OREA EE INER EERO E P A 12 17 National Instruments Corporation ix Xmath Interactive Control Design Module Contents Chapter 13 Multi Loop Synthesis Multi Loop Window Anatomy cceeceesccesceseseceseesececeeeeseeseatessneeneeeeee Setup and Synthesis Method oo
66. esign process The History window can be used to rapidly cycle through and compare a subset of the saved designs Any controller on the history list can be recalled and the design process continued Alternate Plant Window The Alternate Plant window is used to study the robustness of a controller to variations or changes in the plant The user can interactively vary the plant gain or dynamics or add extra parasitic dynamics to the plant see the effect on the closed loop system and compare it to the nominal system Key Transfer Functions and Data Flow in ICDM ICDM has three key transfer functions e The plant transfer function P e The alternate plant transfer function Pa e The current controller transfer function C National Instruments Corporation 2 5 Xmath Interactive Control Design Module Chapter 2 Introduction to SISO Design The plant and the alternate plant have very different uses in ICDM and therefore different data flow characteristics The plant transfer function is read from Xmath into the ICDM Main window and is then exported to the synthesis windows that need it Pole Place LQG and Hee In other words the controllers designed using the Pole Place LQG or Heo Synthesis windows are based on the plant transfer function You cannot change the plant transfer function in ICDM except by reading in a new plant from Xmath The alternate plant transfer function can be read into ICDM from Xmath or set equal to the plant t
67. esis window and the history window is not open If you try to read a plant when a synthesis window or the History window is open a dialog box will notify you that the open synthesis or History window first must be closed Reading a new plant into ICDM when there already was a defined plant has several important consequences First all controllers on the history list that were designed by the Pole Place LQG or synthesis windows are converted National Instruments Corporation 3 3 Xmath Interactive Control Design Module Chapter 3 ICDM Main Window to a simple transfer function representation which means that you cannot read them back into the Pole Place LQG or synthesis windows because these types depend on the plant Also all synthesis windows will be reset to their initial default settings Because these side effects may be undesired the user is warned before these actions are taken Reading a Controller from Xmath into ICDM You can read a controller from Xmath into ICDM using the FileRead Controller entry This requires closing any open synthesis window or the History window Reading in a new controller will overwrite the current controller in ICDM so unless there is no current controller or you have saved the current controller to the history list or Xmath you will be warned and asked for confirmation After you have read in the new controller you can proceed with opening a synthesis window and the usual rules apply If the
68. he possibility that the plant transfer function is not strictly proper that is the plant can have as many zeros as poles Normal Mode In normal mode the order number of poles of the controller is fixed and equal to n the order of the plant so there are a total of 2n closed loop poles In this case the 2n degrees of freedom in the closed loop poles exactly determine the controller transfer function which also has 2n degrees of freedom In normal mode the controller transfer function has order n and is strictly proper C s n s d s where d s S x 18 x95 X ns ys ys 2y Therefore the closed loop characteristic polynomial has degree 2n x s n s n s d s d s s A s A2 5 Non 2n 2n 1 s 05 n where Aj Ay are the closed loop poles chosen by the user National Instruments Corporation 6 3 Xmath Interactive Control Design Module Chapter 6 Pole Place Synthesis We can write this polynomial equation as follows by 0 1 0 o bi bo ay 1 Ba bi x oe Yi Gi bn 4 bo an 1 An 2 1 rf b bai bi an an a 0 b a by 0 ay a 0 0 b 0 0 a P 0 0 b 0o 0 a a Oy ay An 0 These 2n linear equations are solved to find the 2n controller parameters Xis lt Xn and yy Yne Integral Action Mode The degree number of poles of the controller is fixed and equal to n 1 so there are a total of 2n 1 closed
69. he resulting controller transfer function will be nonzero The multi loop synthesis method allows the user to close National Instruments Corporation 13 3 Xmath Interactive Control Design Module Chapter 13 Multi Loop Synthesis one loop at a time The loops that are not closed are considered to have a transfer function equal to zero During the design phase the user can modify delete disable or enable controller components of loops that were designed earlier When the user is designing a controller for one specific sensor and one specific actuator of a multivariable plant this SISO plant has no obvious direct relationship with the original multivariable open loop transfer function The reason for this is that each of the earlier designed controller components results in a modified transfer function for all other pairs of sensors and actuators For example consider the case of a plant with two actuators and three sensors in Figure 13 3 This figure shows standard feedback connection with scalar signals shown for a plant with three sensors outputs and two actuators inputs r e1 u y e p Oooo tyl er A r e y2 Z e y C uz P o gt gt Ig C3 Y3 a gt e gt Figure 13 3 Standard Feedback Connection Suppose that at some moment in the design phase two loops have been closed one from the first sensor to the first actuator and one from the
70. hich will be initialized with the current LQG controller Moreover opening the Root Locus window will cause the LQG synthesis window to close You now can continue the design using the Root Locus window For example you might delete some controller poles and zeros that is do some interactive controller model reduction When you have deleted some controller poles and zeros the controller will no longer be an LQG controller so you cannot expect to be able to open the LQG window and retain the current controller There are some restrictions on the controllers that each synthesis window can accept read e The PID Synthesis window can accept any PID controller The PID Synthesis window is intuitive enough to figure out if a given controller has PID form and if so set its parameters appropriately e The Root Locus window accepts all controllers so it can be opened at any time The current controller will be read into the Root Locus National Instruments Corporation 2 7 Xmath Interactive Control Design Module Chapter 2 Introduction to SISO Design window Thus the Root Locus Synthesis window can be used to interactively tweak or model reduce a controller designed by another method such as LQG e The Pole Place window accepts any controller with the same number of poles as the plant or one more pole than the plant if it has integral action In particular the Pole Place window can accept any LQG or Heo controller with or without i
71. hted design National Instruments Corporation 8 5 Xmath Interactive Control Design Module Chapter 8 H Infinity Synthesis Manipulating the Design Parameters The parameters y p and v can be changed using the associated slider or variable edit box If the user types in a value that is outside the current slider range the slider range will automatically adjust The user can change the ranges for the sliders using the Ranges window Refer to the Infeasible Parameter Values section for what happens when the requested value of y is infeasible The parameter y also can be changed graphically by grabbing and dragging the dashed horizontal line in the singular value plot Manipulating the Weight Transfer Function When Weight Zero Editing is enabled the user can graphically manipulate the zeros of the weight transfer function W on the plot labeled Weight Poles amp Zeros Refer to the Graphically Manipulating Poles and Zeros section of Chapter 2 Introduction to SISO Design for a general discussion of how to move add delete or edit these zeros graphically infeasible Parameter Values If the user requests an infeasible value for y then it will be reset to an approximation of the optimal that is smallest possible feasible value y In this case four logarithmic bisection iterations are used to determine y a feasible value of y such that log Yprev 108 Yeg rev re log Ynew log Yopi lt 16 where Ynew iS
72. ial applications this is the way control systems are designed for complex multivariable plants e Appendix A Using an Xmath GUI Tool describes the basics of using an Xmath GUI tool Throughout this manual extended examples following each function discussion help pinpoint the flexibility and applicability of the Interactive Control Design function library This appendix describes the basics of using an Xmath GUI tool Xmath Interactive Control Design Module 1 2 ni com Chapter 1 Introduction Commonly Used Nomenclature This manual uses the following general nomenclature Related Publications Matrix variables are generally denoted with capital letters vectors are represented in lowercase G s is used to denote a transfer function of a system where s is the Laplace variable G q is used when both continuous and discrete systems are allowed H s is used to denote the frequency response over some range of frequencies of a system where s is the Laplace variable H q is used to indicate that the system can be continuous or discrete A single apostrophe following a matrix variable for example x denotes the transpose of that variable An asterisk following a matrix variable for example A indicates the complex conjugate or Hermitian transpose of that variable For a complete list of MATRIXx publications refer to Chapter 2 MATRIXx Publications Help and Customer Support of the MATRIXx Getting Started Guide The foll
73. ials that is they have at least as many poles as zeros In other words the degree of n is less than or equal to the degree of d which is N and similarly for n and d In some situations the plant and controller are required to be strictly proper which means that there are more poles than zeros Other important terms include e The loop transfer function L is defined as L PC The loop gain is the magnitude of the loop transfer function e The sensitivity transfer function is denoted as S and given by S 1 1 PC The sensitivity transfer function is the transfer function from the reference input r to the error signal e Xmath Interactive Control Design Module 2 2 ni com Chapter 2 Introduction to SISO Design The closed loop transfer function T is given by T PC 1 PC T is the transfer function from r to y The characteristic polynomial of the system is defined as X nen ded Its degree is equal to the order of the plant plus the order of the controller The closed loop poles are the zeros of the characteristic polynomial This definition avoids any problem with unstable pole zero cancellations between the plant and controller The closed loop zeros are the zeros of n n The output response to a unit step input or just the step response is the step response of the transfer function T that is the response of y when the command input r is a unit step The actuator step response is the step response of th
74. ich in this case means debt P O 0 Suppose the controller has the form C s 1 s Ry C s where C has no poles at s 0 and the constant matrix is nonsingular for example Rp J Then you have 7 0 7 so that you have perfect asymptotic decoupling and tracking of constant reference inputs You also have perfect asymptotic rejection of constant actuator disturbances The condition on C means the controller has an integrator in each of its channels In this case the integrators can be thought of as either acting on the sensor signals or acting on the actuator signals You often have integrators associated with some of the sensors or some of the actuators Then the matrix can be less than full rank and you generally get the benefits of integral action in only some of the I O channels that is only some of the entries of T 0 are 1 or 0 When n n things get more complicated The rank of Ro is at most r max n n If you ask ICDM to insert more than r integrators say using an integrator in each sensor channel for a plant with three actuators and five sensors you will have a closed loop system that cannot be stabilized it will have an uncontrollable or unobservable mode at s 0 If this happens ICDM will warn you Even if you do not have excess integrators you should realize that you will get perfect asymptotic tracking Xmath Interactive Control Design Module 11 4 ni com Chapter 11 Introduction to MIMO Design or
75. ign Module Chapter 5 Root Locus Synthesis Plotting Styles Selecting View Locus Select or pressing lt Ctrl L gt in the Root Locus window produces a dialog box in which the user can choose one of many possible plotting styles In all cases the open loop controller and plant poles and zeros are shown on the plot On color displays e Controller poles and zeros are black e Plant poles and zeros are red This serves as a mnemonic you can manipulate black but not red poles or zeros On monochrome displays the plant poles and zeros are lighter than the controller poles and zeros You can always use data viewing to obtain more information about a pole or zero Refer to the Data Viewing Plots section of Chapter 2 Introduction to SISO Design Figure 5 2 shows a dialog box for choosing plotting style with standard default contours selected Root Locus Select Dialog Phase contours A off y 180 deg vy 160 200 10 deg Magnitude contours v Off v 2 2 dB Close Figure 5 2 Root Locus Select Dialog for Choosing Plotting Style Xmath Interactive Control Design Module 5 4 ni com Phase Contours Chapter 5 Root Locus Synthesis For each of magnitude and phase contours you can choose one of three possible plotting styles Magnitude Contours 180 The plot shows the locus of points where the phase angle of the loop transfer function is 180 This yields a conventional root locus display This is the default phas
76. indow support another feature plot magnify windows Selecting Plot Plot Magnify or pressing lt Ctrl M gt or in the ICDM Main window will cause the cursor to change into a crosshairs symbol Positioning the cursor over an ICDM plot and clicking the left mouse button causes the plot to appear in a new window called a Plot Magnify window as shown in Figure 3 4 This window can be resized using the window manager and can be independently re ranged Refer to the Ranges National Instruments Corporation 3 7 Xmath Interactive Control Design Module Chapter 3 ICDM Main Window of Plots section If another plot is subsequently selected for magnifying it will replace the current plot in the plot magnify window The Plot Magnify window is a separate window that shows one of the ICDM main plots The Plot Magnify window shown in Figure 3 4 can be independently resized by the window manager The ranges of the Plot Magnify window can also be independently set 7 ICDM Plot Magnify Special View Help Act step response Figure 3 4 Plot Magnify Window It also is possible to select a portion of a plot for magnification Click and drag the left mouse button with the cursor in an ICDM plot While holding the left mouse button down you can drag out a box shown in dashed lines when you release the dashed box becomes the range for the magnified plot You also can drag out a box in the magnified plot itself This effectively changes the r
77. inear exponential quadratic Gaussian LEQG or minimum entropy controllers For a description of the MIMO Synthesis window refer to Chapter 12 LQG H Infinity Synthesis H Infinity Synthesis Window Anatomy The Heo Synthesis window is shown in Figure 8 1 From top to bottom it consists of National Instruments Corporation A menu bar with entries Special Edit View and Help A control panel for changing the three design parameters H performance level y Control cost parameter p Sensor noise parameter v These parameters are described in greater detail later in this chapter A control panel used to edit the output weight transfer function described in the Output Weight Editing section A plotting area that contains four plots A singular value plot of the normalized closed loop transfer matrix along with the parameter y which can be grabbed and dragged to a new value If a lower bound on the minimal value of y is known it also is displayed A plot that shows the closed loop poles A plot that shows the poles and zeros of the output weight transfer function When Weight Zero Edit is enabled the user can grab and drag the zeros to new locations A plot that shows the magnitude of the output weight transfer function 8 1 Xmath Interactive Control Design Module Chapter 8 H Infinity Synthesis Singular Values 01 q 2 0 freq Hz real Weight Poles amp Zeros Output Weigh
78. ing the new synthesis window from the current design Roughly speaking ICDM tries to keep the current controller when you select a new synthesis window As an example suppose the LQG window is used to design an LQG controller The user then can open the Pole Place window which will be initialized with the LQG controller and continue the design by dragging the closed loop poles to new locations At this point the user cannot expect to import the current controller back into the LQG Synthesis window because the controller is no longer an LQG controller The user can however open the Root Locus window which will be initialized with the current Xmath Interactive Control Design Module 2 8 ni com Chapter 2 Introduction to SISO Design controller Using the Root Locus window the user could reduce the controller to a PI controller by deleting poles and zeros at which point the PID window can be opened initialized at the current controller Using ICDM ICDM can be used in many ways For example you might e Interactively design a controller e Switch synthesis methods and continue designing e Review and compare your best designs and perhaps start designing again from a previous design e Analyze the robustness of one or more controllers with respect to variations in the plant transfer function export one or more controllers to Xmath such as for a nonlinear simulation or downloading to an AC 100 for real time testing The most
79. is enabled the LQG controller is based on u Wi Mj and y W jY which are filtered versions of the plant inputs and outputs u and y i 1 np j 1 n Without integral action the controller minimizes the quantity z n Rox Rou x 1 tnd Leaf PF edie Do i 1 j l and with integral action the quantity u leae ii i l j 1 where t 50 E AY 0 Xmath Interactive Control Design Module 12 12 ni com Chapter 12 LQG H Infinity Synthesis The transfer functions W and W are the input and output weighting transfer functions respectively When W 1 and W 1 this reduces to the previously described standard LQG controller Notice that integral action also can be accomplished by defining filters that have poles on s 0 This is useful if integral action is required for a subset of the outputs The standard toggle button for integral action applies to all outputs How to Select w u y and z The user has complete freedom in designating components of the input and output vector as external disturbances w actuators tact sensors y and weighted outputs z National Instruments Corporation Sensors and actuators are disabled enabled using the leftmost column of toggle buttons in the Weights window This is useful for situations where you want to know what the value of individual sensors and actuators is for the achievable control performance By default all outputs are sensors an
80. ith Xmath for example transfer plants controllers from to Xmath e Display warning and log messages e Display a variety of standard plots e Select a synthesis method for controller design e Control several auxiliary windows PID Synthesis Window The PID Synthesis window is used to synthesize a PID controller with up to two additional poles usually used for high frequency rolloff Each term can be separately toggled on and off so the PID window can be used to synthesize P PD PI PID lead lag and lag lead controllers The design parameters can be typed in manipulated graphically by slider controls or manipulated graphically on a Bode plot of the controller transfer function Root Locus Synthesis Window The Root Locus window can be used in many ways for synthesis and analysis of controllers It can display a conventional root locus in near real time while the user drags controller poles and zeros The user can graphically create or destroy controller poles and zeros The closed loop poles can be dragged along the root locus plot which causes the gain parameter to be set automatically Nonconventional phase and gain contours can be plotted as an aid to controller synthesis or robustness analysis Pole Place Synthesis Window The Pole Place Synthesis window is used to design a controller by assigning the closed loop poles The closed loop poles can be typed in or dragged on a plot The closed loop poles can be scaled in fr
81. ittle effect on the transfer function If you want to delete a pole or zero that is very near a zero or pole respectively then you may have to first separate them a little bit Otherwise the delete command may be interpreted as a delete pair command Xmath Interactive Control Design Module 2 16 ni com ICDM Main Window This chapter describes the use of the ICDM Main window which is used to perform the following functions e Communicate with Xmath for example transfer plants controllers from to Xmath e Display warning and log messages e Display a variety of standard plots e Select a synthesis method for controller design e Control several auxiliary windows for example Ranges Alternate Plant Notice that the ICDM Main window is not directly used to design the controller It is used to make high level decisions such as which synthesis method to use and to view or analyze the response with the current controller This chapter is limited to the discussion of SISO design For MIMO design information refer to Chapter 11 Introduction to MIMO Design Window Anatomy The ICDM Main window shown in Figure 3 1 consists of the following elements from top to bottom e A menu bar with File Edit Plot Synthesis and Help menus e A scrolled text area for warnings and messages You can resize this area independently of the rest of the ICDM Main window The log messages that appear here are meant to give a rough trace
82. king the middle mouse button anywhere in the plot creates a box containing a magnification of a small area of the plot centered at the cursor The middle mouse button can be held down and dragged which creates an effect similar to dragging a magnifying glass across the plot The center of the zoomed window corresponds to the tip of the cursor Pressing lt Ctrl gt along with the middle mouse button increases the size of the magnified box Pressing lt Shift gt along with the middle mouse button increases the zoom factor Pressing lt Shift Ctrl gt along with middle mouse button yields a large zoom box with a large magnification factor By pointing at or near a curve or object in a plot and pressing the right mouse button a small window will appear that identifies the curve or object and gives the coordinates and index of the nearest data value If you press and drag the right mouse button the selected curve will be tracked even if another curve comes close Pressing lt Shift gt along with the right mouse button allows the user to get values on the piecewise line curve that interpolates the data values In this case index 45 7 means that the selected plot point is between the 45th and 46th curve index entries A 6 ni com Technical Support and Professional Services Visit the following sections of the National Instruments Web site at ni com for technical support and professional services Support Online technical support
83. l R gt in the LQG window The slider ranges also will be changed automatically if the user types a new value which is outside the current range into the corresponding variable edit box The plot also can be re ranged interactively by grabbing and dragging the plot axes refer to the Interactive Plot Re ranging section of Chapter 2 Introduction to SISO Design Selecting View Auto Scale or pressing lt Ctrl A gt in the LQG window causes new ranges to be assigned to the sliders and plots based on the current controller National Instruments Corporation 12 17 Xmath Interactive Control Design Module Multi Loop Synthesis This chapter describes multi loop synthesis The Multi Loop window is used to synthesize a MIMO controller using PID and Root Locus methods applying them one loop at a time In many practical industrial applications this is the way control systems are designed for complex multivariable plants Multi Loop Window Anatomy The Multi Loop Synthesis window is shown in Figure 13 1 From top to bottom it consists of e A menu bar with entries Special Edit and Help e A plot area where SISO control loops can be created graphically We will call this area the graphical editor e A scrolled list of loop names with actuator and output labels e Alabel Active loop name and a variable edit box for editing the name of the highlighted loop e A label Status and a button which can be clicked to change the stat
84. lementation of a guaranteed decay rate Another difference is that certain parts of the combined closed loop system are not allowed to have zeros on the imaginary axis If that is the case an error message is reported in a window When the Heo performance level y is large the Heo controller is approximately the same as the LQG controller By reducing with the slider or by dragging the horizontal dashed line in the singular value plot the algorithm will decrease the maximal singular value to its lower bound For practical Heo design the following should be considered In order to achieve the lower bound the Heo algorithm will sometimes place the controller poles very far to the left in the complex plane Also the gain at high frequencies is often increased significantly which increases the noise sensitivity Therefore a better control performance is often obtained by trying to lower the Heo norm not to its absolute minimum but rather to a slightly larger value National Instruments Corporation 12 15 Xmath Interactive Control Design Module Chapter 12 LQG H Infinity Synthesis Manipulating the Design Parameters Main Window The design parameters p and v can be changed using the associated sliders or the variable edit boxes If the user types in a value that is outside the current slider range the slider range will automatically adjust Notice that the slider positions in the Weights window are simultaneously updated when th
85. lly the poles and zeros of plant can be shown Bottom right Buttons to Add Delete Edit poles and or zeros of the alternate plant Xmath Interactive Control Design Module 10 2 ni com Chapter 10 Alternate Plant Window Figure 10 1 Alternate Plant Window Opening the Alternate Plant Window When the Alternate Plant window is first opened the alternate plant is initialized to the plant transfer function This is convenient because in most cases the alternate plant is some sort of variation on the plant Using the Special menu you can read the plant from ICDM or any transfer function from Xmath into the alternate plant National Instruments Corporation 10 3 Xmath Interactive Control Design Module Chapter 10 Alternate Plant Window Normalization The form of the transfer function of the alternate plant depends on the normalization selected With high frequency normalization the alternate plant transfer function is s Z 8 Z Pals K it s p s p where K is the gain shown in the slider and Variable Edit box z Zm are the zeros and p p are the poles shown in the plot The alternate plant is required to be proper that is have at least as many poles as zeros n m For high frequency normalization there is no restriction on the poles or Zeros With DC normalization the alternate plant transfer function is 1 s z d s z Pauls K g 1 s5s p 1 8 p
86. m the keyboard sections of code programming examples and syntax examples This font is also used for the proper names of disk drives paths directories programs subprograms subroutines device names functions operations variables filenames and extensions Bold text in this font denotes the messages and responses that the computer automatically prints to the screen This font also emphasizes lines of code that are different from the other examples Italic text in this font denotes text that is a placeholder for a word or value that you must supply Contents Chapter 1 Introduction Usine This Manual seerne nia a EE ees de sceewte ecesevevbdp thee A 1 1 Document Organization a A N A a E Eea aa 1 1 Commonly Used Nomenclature oo eee eceeeeceesecesesseeseeesecseeeseesecesesneseaeesees 1 3 Related Publications 23 s s nnia wi ea tala G a es 1 3 MATRIX Helpen ar A a E el ieee eects eaten 1 4 ICOM Overview ra E aE eee A ee eet en o aie a 1 4 SISO Versus MIMO Desig socn aes enen e eie r ER OE E EEE 1 4 Starting ICDM ienna a ath eee RU O E a 1 4 Chapter 2 Introduction to SISO Design SISO Desipir Overvie wens eie i seca E EET EE E E a E 2 1 Basic SISO Terminology eneninda aereas eaa 2 1 Overview of ICDM i s s cic cctsttcaseaisethl ecaas e O A EE E states ek SEEE 2 3 IC DM Windows 32s 2esieidesistees n aea A a a a e a 2 3 ICDM Main Window sieer ietsiire striss es tirtapisoigiso atisa enades sea iiai neare e oai 2 4
87. n case the integrator toggle button has been enabled H Infinity Solution The H controller design is done in entirely the same setting as the LQG controller Selection of sensors and actuators and extension with frequency weighting and integrators is identical to LQG The interpretation of weights and noise levels is slightly different The objective here is to minimize the maximal singular value of the transfer function from a normalized version w of w to a normalized version z of z The normalization is based on the weights and noise levels as determined by the Weights window More precisely assume that in the LQG formulation Eww Qww and that zis weighted in the quadratic criterion by a positive semi definite symmetric matrix n Q Then w is of the form T A w QpW and z is of the form 1 2 ZS R22 where w and z are normalized quantities Xmath Interactive Control Design Module 12 14 ni com Chapter 12 LQG H Infinity Synthesis 1 I 3 Here Q is a square matrix such that cia Oww OF 1 and Re is a square matrix such that a RR RR The H solution is defined as the one that minimizes the maximum singular value of the transfer function from w to Z The only difference in the user interface with the LQG design is that the decay rate option cannot be selected The reason for this is that the solution does not have a separation property like the LQG solution which is required for the imp
88. ndle Some GUI tools might do something for example change a plot as the handle is dragged In other cases nothing will happen until the handle is released at the new value The handle can be set to a new value by clicking with the middle button at the new value The value can be increased or decreased a small amount by clicking the left button away from the handle Holding the button down makes the handle steadily move towards the cursor A 5 Xmath Interactive Control Design Module Appendix A Using an Xmath GUI Tool A slider might also appear like a bar graph Its tip represents the value but it will be read only that is the user cannot change its value by dragging the handle Often a value is displayed with a slider and a variable edit box refer to for example the leadlag demo This allows the value to be changed either by dragging the slider or entering a new value from the keyboard e GUI windows might contain plots which can accept graphical input from the user The left mouse button is used for graphical input the middle for plot zooming and the right for plot data value viewing Xmath Interactive Control Design Module The function of the left mouse button depends upon the particular tool and plot Often a tool will allow a curve to be grabbed and dragged by depressing the left mouse button with the cursor near the curve dragging the mouse with the button down and then releasing at a new position Clic
89. ng the Alternate Plant window You can think of the plant P as the plant model used for control system design and the alternate plant Pax as the plant model used for control system validation Displaying the Alternate Plant Responses The ICDM Main window plots always show the response of the controller connected with the plant By turning on the Alternate Plant display the responses of the controller with the alternate plant also are shown in these plots You always can use data viewing to determine which plot corresponds to the plant and which corresponds to the alternate plant National Instruments Corporation 10 1 Xmath Interactive Control Design Module Chapter 10 Alternate Plant Window Alternate Plant Window Anatomy The it co Alternate Plant window is shown in Figure 10 1 From top to bottom nsists of A menu bar with Special Edit and View menus A toggle button for controlling whether the plots in ICDM main will include the response with the alternate plant A toggle button that is used to display the plant poles and zeros in the plot refer to Figure 10 1 Two toggle buttons that select DC or high frequency normalization refer to Figure 10 1 A slider and variable edit box for the gain of the alternate plant These controls are used to show and also to change the gain of the alternate plant Bottom left A plot for displaying and manipulating the poles and zeros of the alternate plant Optiona
90. ng the Weight Transfer Function cece eeeeeeseeeeeeneeeeeeeeeees 8 6 Infeasible Parameter Valles i raccorder oaen iA RESE 8 6 Rapes e a eet RO ed eee Re isdn lata ahs 8 7 Chapter 9 History Window Saving the Current Controller on the History List tees eeecesereeeteeeseeeeeeeees 9 1 Opening the History Window eee cacane aaea e es 9 1 History Window Amatomy s icescesesssscudis eseep decuves sobiveus AE ESE ESEE EAE E EAEE yS 9 1 Selecting the Active Controller oo eee ececseeseceseeseeesecseeeseeaeeesecaecnsessesnaessesseseaeesgees 9 2 Editing the Comments asioita 9 2 Deleting History Eist Entries eens eere er a e E E a S 9 3 To Continue Designing from a Saved Controller sesesseseseseesesreseeresrsresresreresrrereersees 9 3 Cycling Through Designs ienne a E sit te ERS 9 3 Writing a Saved Design to Xmath ew eee eececeeseceeeseceeeeseseeeeseceeeesesseeeeeseeeasenseeas 9 3 Using the History Listi iene te cteagt lectins e a EA A Ea 9 4 Chapter 10 Alternate Plant Window Role and Use of Plant and Alternate Plant sssseeseseeeeesesressseessesreresresrsresresrsresrsrese 10 1 Displaying the Alternate Plant Responses essessseesseeeseeesrsrsrrsrssrsrestsresresrseesrrereresrs 10 1 Alternate Plant Window Anatomy seesssssssesssseessesrsresrertsrestesesresreesresrnresrenenenreereese 10 2 Opening the Alternate Plant WindowW cccecececesessesecsecessessesscssesseseseeseeseesecseees 10 3 Normalization
91. nge of items can be selected by pressing the left mouse button dragging the mouse and releasing Pressing lt Shift gt and the left mouse button selects all the items from the current item to the previous item that was selected with the left mouse button Pressing lt Ctrl gt and the left mouse button augments rather than replaces the existing selections This allows discontiguous ranges of items to be selected This type of list is used in the history sorting and history column dialogs in the leadlag demo When you select one or more items from a list you then choose some action such as Delete or Display GUI tools can display a dialog A dialog is a small window that could contain a message and one or more buttons For example a dialog may have a single button and a message giving a warning or indicating an error Usually a dialog is modal you cannot interact with any other GUI or Xmath windows until the dialog has been removed If you find you cannot interact with Xmath or other GUI windows then look for a modal dialog that might have been accidently covered by another window GUI tools allow detailed Help messages to be displayed These are often listed under a Help pull down menu at the top right of the GUI window The Help message appears in a new window that provides scrollbars as needed The scrollbars are operated with the left and middle mouse buttons The window is dismissed by clicking the Close button Xmath Interacti
92. ntegral action This allows the user to manually tune the closed loop poles in a design that was originally LQG or H e The LQG window only accepts controllers that were generated by the LQG synthesis window e The Heo Synthesis window only accepts controllers that were generated by the Heo Synthesis window e The History window which can be considered as a synthesis window since it exports a controller to the ICDM Main window is compatible with all controllers If the current controller has been saved on the history list then the History window opens with the current controller the active controller on the history list If the current controller has not been saved on the history list it is first automatically saved on the history list then the History window opens with the current controller active These restrictions are important when you select a new synthesis window or read a controller from Xmath into ICDM If the controller is not compatible with the synthesis window the user is warned and given several options about how to proceed In general these restrictions on controllers and synthesis windows should be transparent to the user ICDM is designed to do something sensible whenever a conflict can arise and to warn the user before any damaging actions are taken When the new controller and synthesis window will be compatible the new synthesis window is initialized with the controller The user can simply start designing us
93. nu in the ICDM Main Window If the synthesis window you select is compatible with the controller it will appear initialized with the current controller and the History window will disappear You now can continue designing Alternatively you can select a design on the history list and then click the Synthesis button at the bottom of the History window This does two things first it makes that entry active that is the current controller and second it opens the appropriate synthesis window which closes the History window Cycling Through Designs The Cycle button is used to quickly compare some of the designs on the history list First select several designs from the list and then click the Cycle button Refer to Appendix A Using an Xmath GUI Tool for a discussion of how to select multiple non contiguous entries in a list Clicking Cycle causes the current controller to cycle among the selected entries Therefore the Cycle button is used both to select a subset of designs for cycling and to cycle the current controller among the selected designs Writing a Saved Design to Xmath To write a design that has been saved on the history list to Xmath select it so it becomes the current controller for ICDM and then save to Xmath by selecting File Write Controller in the ICDM Main window National Instruments Corporation 9 3 Xmath Interactive Control Design Module Chapter 9 History Window Using the History List
94. nventions used in the manual iv NI resources B 1 drivers NI resources B 1 E error signal 2 2 11 2 estimator eigenvalues 6 5 examples NI resources B 1 Exponential Time Weighting 12 12 F feedback configuration 2 1 FileRead Controller 3 4 FileWrite Controller button 3 4 filter noise 12 9 Xmath Interactive Control Design Module Index G gain loop 2 2 graphical editor 13 1 H Help 1 5 help technical support B 1 high frequency normalization 10 4 H Infinity performance level 12 1 Synthesis Window 2 5 History window 2 5 2 6 2 8 9 1 ICDM Help 1 5 ICDM Main Window 2 4 3 2 elements 3 1 input referred disturbances 12 9 instrument drivers NI resources B 1 integral action 2 3 12 11 mode 6 4 time constant 7 1 12 1 interactive design loop 2 9 K Kalman filter 7 7 KnowledgeBase B 1 L LEQG controllers 2 5 linear exponential quadratic Gaussian LEQG controllers 8 1 loop gain 2 2 transfer function 2 2 11 4 Xmath Interactive Control Design Module LQG synthesis window 2 5 window 2 8 LTR design 2 5 MATRIXx Help 1 4 1 5 MIMO LQG H Infinity synthesis window 12 1 Plot window 11 6 transfer function plot 11 6 model reduction 2 7 multi loop synthesis 13 1 National Instruments support and services B 1 NI support and services B 1 noise power 2 5 nomenclature 1 3 P PID synthesis window 2 4 2 7 plant degree 2 2 denominator 2 2 num
95. of your ICDM design session It records major actions such as reading a new controller or plant in opening a new synthesis window saving controllers to the history list and so on e A line that gives the plant name National Instruments Corporation 3 1 Xmath Interactive Control Design Module Chapter 3 ICDM Main Window A line that identifies the type and source of the current controller The source is either the currently active synthesis window or the history list A plotting area for the various plots 7 ICDM Main Window File Edit Plot Synthesis 9 31 45 PID synthesis window selected 9 31 32 Using default plant 3 9 31 21 First read a plant using the File menu 9 31 21 Welcome to ICDM Plant Default plant 3 SISO order 3 Compensator PID Step response Loop gain pg 15 1 0 5 D D1 o1 ge 2 4 Loop phase Act step response 2 5 2 1 5 1 0 5 a Figure 3 1 ICDM Main Window Communicating with Xmath The File menu is used to communicate with Xmath that is to read controllers and or plants from Xmath into ICDM and to write controllers and or plants from ICDM back to Xmath Xmath Interactive Control Design Module 3 2 ni com Chapter 3 ICDM Main Window Most Common Usage In most cases you will read a plant from Xmath at the beginning of an ICDM design session and write one or more controllers back to Xmath during or at the end of an ICDM design session This is done by selecting the
96. omy ssesesseessseeseeesestssesresrsresreestestsrestenenresreresresrseestsees 7 1 Synthesis ModE S ne ereere tees cae vost E E E O A RE E 7 3 Opening the LQG Synthesis Window cei eeeeseeeceseeseeeseceeeeeeteeeaereeeeaees 7 3 Setup and Terminology sess ioeie vei a Sarees 7 4 Standard LQG All Toggle Buttons Off cee eeceeseeseeceeeeeseeeneersaeeeneeeneees 7 4 Inte etal Actionin crocuses int a E T A E E tenis apt iene 7 5 Exponential Time Weighting 00 00 00 eee ceeeseesceeeeseeseeseeeaecnseesessseseenseeaees 7 5 Output Weight Baiting err Se becets cde ae ee a E dad aiea 7 6 State Space Interpretations sssrini ioi i i i 7 7 National Instruments Corporation vii Xmath Interactive Control Design Module Contents Manipulating the Design Parameters cceesceessceseesceceseeeseeeeseeeaceseneseaeeeeaeeeneeeneeeeas 7 7 Manipulating the Design Parameters Graphically 0 eee eee eeeeteeeeeeees 7 7 RANGES seein ivsccnseesdesiey sea edees Bint vacnd bead R aad E AEE EA aA 7 8 Chapter 8 H Infinity Synthesis H Infinity Synthesis Window Anatomy eee eseesceseesseeeceeeeseeneeeaeeneeeaeeeseeaeenaes 8 1 Opening the Synthesis WindOW iseseisana e o Eaa nS ei 8 3 Setup and Synthesis Method sssini i a a EE SS 8 3 Central H Infinity Controller oo eee reee uai h t E RE 8 4 Output Weight Editing 0 eee aaia E 8 5 Manipulating the Design Parameters eis eeeeceeseeeeeseeceseeseeseeesecseeeaeeneeeaseneeeaes 8 6 Manipulati
97. on off rF Sensors on off r Sensors on off E Temperature Log Av al 100 ey 2 Weight matrices from lt None gt Figure 12 2 LQG H Infinity Weights Window e A slider representing the logarithmic average of the input weights p that is 1 logp 7 BP ui a National Instruments Corporation 12 3 Xmath Interactive Control Design Module Chapter 12 LQG H Infinity Synthesis e A table with Ny TOWS with in each row A toggle button to include the output in the set of measured outputs A toggle button to include the output in the set of costed outputs labeled with the signal name A slider defining the constant weight factor of the output Py pi es n A variable edit box for the same constant weight factor e A slider representing the logarithmic average of the weights for each output P that is 1 logp 7 logp j y A e A button for entering an Xmath variable name of type matrix of the form Rex Ryu Rp R nuu R containing the weights on states and inputs including cross terms The control weight parameter p in the main LQG H window is related to the weights in this window by p p Py If the radio buttons in the first row are set to select the noise level display the contents of the window looks almost exactly the same The leftmost column of toggle buttons has the same meaning but the second column of toggle buttons is used to enable disable the noise on select
98. ontroller exported to ICDM for plotting e Buttons for manipulating the history list Selecting the Active Controller You can type a number in the Variable Edit box that shows the selected controller or you can select a controller in the list which will become highlighted and then click Select at the bottom of the History window Notice that you can consider the History window as a type of synthesis window with one simple design parameter the integer that gives the selected design kad ICDM Controller History Special Edit mom date time type comment 17 5 14 51 Hinf first design 175 14 52 PP butterworth 2 Current Design 3 Select Edit gt Synthesis Delete Cycle Undo Figure 9 1 History Window Editing the Comments To change the comment stored with a design select an entry from the list and click Edit Xmath Interactive Control Design Module 9 2 ni com Chapter 9 History Window Deleting History List Entries Any number of designs on the history list can be deleted by selecting them and then clicking Delete To renumber the remaining designs you can select Edit Renumber Refer to Appendix A Using an Xmath GUI Tool for a discussion of how to select multiple non contiguous entries in a list To Continue Designing from a Saved Controller First select the desired design to make it the ICDM current controller Then select an appropriate synthesis window from the Synthesis me
99. ound on the minimal value of y is known it also is displayed Fi ICDM LQG Hinf Gamma entropy level Special Edit View Help Singular Values freq Hz Figure 12 4 LQG H Infinity Performance Level Window Frequency Weights Window The Frequency Weights window is shown in Figure 12 5 From top to bottom it consists of e A menu bar with entries Special Edit View and Help e A plot area with two plots A plot that shows the poles and zeros of the selected input or output weight transfer function The user can grab and drag the poles and zeros to new locations A plot with the frequency response magnitude of the weight function Xmath Interactive Control Design Module 12 6 ni com Chapter 12 LQG H Infinity Synthesis ICDM LQG Frequency weights Weight poles amp zeros Weight transfer function T T To am freq Hz a ia ea Pressure y Figure 12 5 LQG H Infinity Frequency Weights Window Synthesis Modes and Window Usage In addition to the standard LQG H synthesis any combination of three additional features is supported e Integral action e Exponential time weighting guaranteed decay rate This feature is only enabled in the case of LQG design e Input and output weight editing In the Main window the synthesis mode is reported in the text in the titlebar at the top The Int Time integral action time toggle button enables and disables integral action The Dec
100. ow offers a choice between either e Step response e Frequency response magnitude e Frequency response phase The type of transfer function displayed can be selected by clicking one of the buttons at the bottom of the window When an alternate plant is selected the MIMO plot will contain two responses in each element of the plot matrix one for the plant and one for the alternate plant Xmath Interactive Control Design Module 11 6 ni com Chapter 11 Introduction to MIMO Design Notice that having the MIMO Plot window on the screen may increase the required computational response time of ICDM Closing the window using the Special option of the MIMO Plot menu bar will then result in a speed up i EA MIMO Plot Options Dialog 2 Frequency response singular value plots Compensator H Plant P Loop gain L PC Sensitivity Her S inv I PC Actuator effort Hur C inv I PC Closed loop transfer fumction Hyr T PC inv I PC Actuator referred loop gain Lact CP Actuator referred sensitivity Hed Sact inv I CP Actuator referred act effort Hud P inv I CP l r z w w l Actuator referred clp tfn Hyd Tact CP inv I CP Miscellaneous plots Closed loop poles F Step response 4 Actuator effort r caer Figure 11 3 Complete Set of Plot Choices History Window The History window is exactly the same in SISO and MIMO modes Alternate Plant Window MIMO Version The MIMO version of the alternate plant window differs from
101. owed to add or delete poles or zeros in some windows Bear in mind that ICDM may not allow you to add or delete a zero or pole in certain cases for example if the action would result in a nonproper controller In this situation you will be warned with a dialog box which opens To add a zero click the Add Zero button that is near the plotting area or select the Add Zero entry from the Edit menu In some cases there is an accelerator for this such as typing z in the window These actions will cause the cursor to become a crosshairs symbol If you click the left mouse button with the cursor very near the real axis then you will create one real zero If you click the left mouse button with the cursor farther from the real axis then you will create a pair of complex conjugate zeros Creating a pole is similar typing p in the window is the accelerator for creating a pole or complex conjugate pole pair To abort a pole or zero add operation click the left mouse button with the cursor outside the plot area To delete a pole or zero press the lt Ctrl gt key near the plotting area select the Delete entry from the Edit menu or enter d in the window These actions will cause the cursor to become a skull and crossbones symbol Then click the left mouse button with the cursor near the pole or zero that you want to delete If the pole or zero is complex then its complex conjugate also will be destroyed You always can select Undo in the Edit menu
102. owing documents are particularly useful for topics covered in this manual National Instruments Corporation MATRIXx Getting Started Guide Xmath User Guide Xmath Control Design Module Xmath Interactive Control Design Module Xmath Interactive System Identification Module Part 1 Xmath Interactive System Identification Module Part 2 Xmath Module Reduction Module Xmath Optimization Module Xmath Robust Control Module Xmath Xu Module 1 3 Xmath Interactive Control Design Module Chapter 1 Introduction MATRIXx Help Interactive Control Design Module function reference information is available in the MATRIXx Help The MATRIXx Help includes all Interactive Control Design functions Each topic explains a function s inputs outputs and keywords in detail Refer to Chapter 2 MATRIXx Publications Help and Customer Support of the MATRIXx Getting Started Guide for complete instructions on using the MATRIXx Help feature ICDM Overview This section provides an overview of the Interactive Control Design Module a tool for interactive design of continuous time linear time invariant controllers ICDM runs under Xmath using the Xmath Graphical User Interface GUI SISO Versus MIMO Design Starting ICDM Version 2 0 of ICDM handles full multivariable design that is design of multi input multi output MIMO controllers for MIMO plants Thus ICDM 2 0 operates in two basic modes SISO design single input single output and
103. py is finite only for H lt y and rapidly increases to as H becomes close to y where the Heo norm is defined as IHl max o H j 0 lt lt Refer to Chapters 5 and 12 of Linear Controller Design Boyd and Barratt Prentice Hall 1991 for some interpretations of the y entropy Therefore the controller designed will always satisfy H lt y For this reason yis sometimes called the H performance level For y which is too small there may be no controller that can achieve the required performance level For large y the y entropy of H is very nearly the same as the LQG cost with the same parameters p and v so the Heo controller will be nearly the same as the LQG controller with the same values of p and v Output Weight Editing When Weight Zero Edit is enabled the user can graphically edit the output weight transfer function W The weighting transfer function is given by ns W s TO Its denominator is fixed and equal to the numerator of the plant transfer function its numerator can be manipulated by the user The lower left plot shows the poles and zeros of the weight transfer function W When Weight Zero Edit is enabled the user can grab and drag the zeros shown or Add Delete Edit zeros using the push buttons The lower right plot shows the magnitude of the weight transfer function When it is flat and equal to 0 dB for all frequencies you have W 1 that is standard unweig
104. r The graphical editor consists of two columns of square boxes where in the leftmost column each box represents a sensor and where in the rightmost column each box represents an actuator A line between a sensor box and a actuator box represents a control loop These connections can be made as follows e By clicking a box in the column to the left controller inputs then clicking a box in the column to the right e By drawing a lasso around one or more boxes in the column to the left with the left mouse button then drawing a lasso around the same number of boxes in the column to the right For each connection that is made an entry is added to the scrolled list with a default name for the loop The default name is of the form Loop_ lt i gt _ lt j gt Selecting and Deselecting Loops A loop can be selected by clicking it in the graphical editor or by clicking the corresponding entry in the scrolled list A loop can be deselected by clicking it again When a loop is selected it is indicated in the graphical editor by a thick line National Instruments Corporation 13 7 Xmath Interactive Control Design Module Chapter 13 Multi Loop Synthesis Editing and Deleting Loops When a loop is highlighted it can be edited deleted disabled or enabled Here editing means designing a SISO controller for the selected loop The editing and deleting options are accessible under the Edit pull down menu Disabling or enabling a
105. r the toggle button at the left of the row is used to toggle the terms on and off On means that the corresponding controller term appears in the overall controller transfer function and the slider and variable edit box can be used to change the parameter Off means that the controller term does not appear in the overall controller transfer function In this case the slider and the variable edit box are read only you cannot drag the slider and you cannot type in the variable edit box When the button is turned on again the parameter value is restored to its previous value You can use the buttons to do a quick A B comparison of a PID controller with or without a given term for example to see the effect of a high frequency rolloff term or integral action Xmath Interactive Control Design Module 4 2 ni com Chapter 4 PID Synthesis Magnitude freq H2 phase Figure 4 1 PID Synthesis Window As an example suppose that the P and I toggle buttons are on and the D and HF rolloff buttons are off The controller transfer function will then have the following form C s K 1 sT ing National Instruments Corporation 4 3 Xmath Interactive Control Design Module Chapter 4 PID Synthesis Notice that there are at least two other commonly used forms for a PID control law that differ from the one used in ICDM C s K 1 1 T 5 Taff and C s K 1 T ins Dgiges ICDM enforces a proper controller tran
106. r set the plotting ranges The user can choose any combination of the following e Loop transfer function magnitude e Loop transfer function phase e Sensitivity and complementary sensitivity magnitude e Closed loop poles and zeros e Output response to a unit step input e Actuator response to a unit step input e Nyquist plot of loop transfer function e Nichols plot of loop transfer function Refer to the Basic SISO Terminology section of Chapter 2 Introduction to SISO Design for definitions of these terms The default plots are loop transfer function magnitude and phase and output and actuator response to a unit step input For more information about other plots available for MIMO design refer to Chapter 11 Introduction to MIMO Design Selecting Plots Selecting Plot Plot Choices or pressing lt Ctrl P gt in the main ICDM window will cause a plot selection dialog box to appear as shown in Figure 3 2 The plot selection dialog box that appears is modal which means that you cannot interact with any other Xmath window until you have dismissed this dialog by clicking Cancel or OK Plot Options Dialog Loop gain magnitude F Loop gain phase S T magnitude 4 Poles amp Zeros Step response F Actuator step response Nyquist Nichols Apply 7 Cancel Lew Figure 3 2 ICDM Main Window Plot Choices Dialog National Instruments Corporation 3 5 Xmath Interactive Control Design Module Chapter 3 ICDM Main Window
107. r the upper right side of the window The Help messages explain how to interact with the demo and what it does It may be helpful to read the rest of this appendix before or while you try the demos You can exit a demo by selecting the Special Exit option Interacting with a GUI Application This section describes the mechanics of interacting with GUI windows Tools that use the GUI will create windows that contain control elements such as buttons sliders pull down menus plots and lists For example Figure A 2 shows the Programmable GUI PGUI Example Do It dialog Xmath Interactive Control Design Module A 2 ni com GUI Functions Appendix A Using an Xmath GUI Tool kal PGUI Example Do It 4 Toggle button v value Figure A 2 Programmable GUI Examples Do It Dialog Many functions are controlled by the left mouse button For example a button is activated or selected by pointing at the button and clicking the left mouse button The PGUI Example dialog has two buttons Do It and 12 GUI Objects Other objects behave as follows National Instruments Corporation A button square shaped is either on or off Its indicator is filled in when it is on It can be toggled by pointing and clicking the left mouse button The button shown in Figure A 2 is off Activating a button causes some action to be performed Radio buttons diamond shaped are a group of buttons that have radio behavior which means tha
108. ransfer function Its properties are very different from the plant transfer function however e Using the Alternate Plant window the user can graphically manipulate the alternate plant transfer function e The alternate plant transfer function is never exported to that is used by the synthesis windows that need to know the plant Pole Place LQG Ho The alternate plant transfer function is used to verify a controller design that was based on the plant transfer function The alternate plant transfer function is used only to show the alternate plant plots in the ICDM Main window Refer to the What the ICDM Main Window Plots Show section Summary The plant transfer function is used for design the alternate plant transfer function is used for robustness analysis or validation The distinction is not so important for PID and root locus design because the controller does not depend on the plant Origin of the Controller The controller can originate from that is be designed by several possible sources e An Open Synthesis window For example if the Pole Place Synthesis window is open then the current controller is determined by the Pole Place Synthesis window When you interact with the Pole Place window by dragging a closed loop pole to a new location you will be changing the current controller transfer function C e The History window If the History window is open the controller comes from the list of controllers
109. res section of Chapter 2 Introduction to SISO Design Controller Term Normalizations Each of the controller terms is normalized in a way that is convenient for most PID design tasks as described in the following sections Integral Term Normalization The integral term is high frequency normalized which means that it is approximately one for frequencies above 1 T Therefore you can adjust the integral time constant 1 7 without significantly affecting the controller transfer function at high frequencies For example you can add integral action to a controller without significantly affecting the stability margins or closed loop dynamics by adding the integral term with 1 7 well below the crossover frequency that is 1 T large In this case your controller will enforce steady state tracking but over a time period longer than the closed loop system dynamics You then can slowly decrease 1 T until you get a good balance between fast integral action and the degradation of stability margins National Instruments Corporation 4 5 Xmath Interactive Control Design Module Chapter 4 PID Synthesis Derivative Term Normalization The derivative term is low frequency normalized which means that at low frequencies below 1 7 i it is nearly one and so has little effect on the overall controller transfer function at low frequencies In particular the loop transfer function at s 0 is not affected by the derivative term at all
110. s windows ICDM function reference material is available in the MATRIXx Help Refer to Chapter 2 MATRIXx Publications Help and Customer Support of the MATRIXx Getting Started Guide for additional instructions on using the MATRIXx Help National Instruments Corporation 1 5 Xmath Interactive Control Design Module Introduction to SISO Design Xmath provides a structure for system representation called a system object This object includes system parameters in a data structure designed to reflect the way these systems are analyzed mathematically Operations on these systems are likewise defined using operators that mirror as closely as possible the notation control engineers use This chapter outlines the types of linear systems the system object represents and then discusses the implementation of a system within Xmath The functions used to create a system object and to extract data from this object are an intrinsic part of the object class and are also described Finally this chapter discusses the functions check discretize and makecontinuous which use information stored in the system object to convert systems from one particular representation to another SISO Design Overview This section provides an overview of what ICDM does and how it works restricting the discussion to SISO design If your interest is MIMO design you first should read this chapter and then Chapter 11 Introduction to MIMO Design Basic SISO Terminolog
111. sed to design a SISO controller by assigning the closed loop poles Pole Place operates in two modes e Normal mode integral action not enforced e Integral action mode The Pole Place Synthesis window cannot be used to design MIMO controllers Window Anatomy The Pole Place window is shown in Figure 6 1 From top to bottom it consists of e A menu bar with entries Special Edit View and Help e A toggle button used to set normal or integral action mode e A slider and variable edit box used to time or frequency scale the closed loop poles e A plot used to display and manipulate the closed loop poles e Buttons used to manipulate the closed loop poles National Instruments Corporation 6 1 Xmath Interactive Control Design Module Chapter6 Pole Place Synthesis Figure 6 1 Pole Place Synthesis Window Pole Place Modes In Pole Place the user selects either closed loop poles in normal mode or 2n closed loop poles in integral action mode These poles uniquely determine the controller transfer function This process can be described in terms of the coefficients of the plant and controller numerators and denominators The plant transfer function is given by P s n s d s Xmath Interactive Control Design Module 6 2 ni com Chapter 6 Pole Place Synthesis where d s s ays ans dp ny S bos bys ab Notice that the order of the plant is n and allow t
112. sesensnsrnesenesesesns 5 3 Terminology iea r E a E E E E a n 5 3 Plotting Styles onnie a sa E E A a RE are Ga 5 4 Phase ContOUrS anenee ii E E EK ERE EEA 5 5 Magnitude Contours snien aM a ea A aa EENAA 5 5 Slider and Plot Ranges ae r R R E AE E E Ss 5 6 Manipulating the Parameters isinen erer eii ees Eea A SEREEN ATEREA 5 6 IDIIN Ea a EEE E set casi A AE E hate ee cen eA E E 5 7 Adding a Pole Zero Paf asenn ionan EGA eines We Aa iad ees 5 7 Deleting Pole Zero Palts i e ses ore ere a E edness E etsy 5 7 Interpreting the Nonstandard Contour Plots 0 cccecsesssesseseseeseesesseneenees 5 8 Chapter 6 Pole Place Synthesis Window Anatomy eonenna ie EEEE aces N EEE OA E tonsa ERA A 6 1 Pole Place Modes eienenn n A N a deeded 6 2 Norma Mode e aren cock gee EE EE E E E e E EES 6 3 Integral Action Mode sninen na a E A a e essen aed 6 4 State Space Interpretation 22 0 0 teris a EE E RRE 6 5 Opening the Pole Place WindoW s sssesssessssssesesssrsssesessssesesesessnnrsesenensresse 6 5 Manipulating the Closed Loop Poles ss ssesssessssesssesssrssesrssrsresrsresresesresreernresrsresreresreet 6 5 Time and Frequency Scaling 0 eee ee nnes a n e a E 6 5 Butterworth Configuration seisein prais a EEE i E 6 6 Editing the Closed Loop Poles sssessseeessesesresseeessrsrsesrretsrestssrsresreresresesreses 6 6 sliderand Plot Range Sai ee ue aeae a Erea AE E eager Abate 6 6 Chapter 7 LQG Synthesis LQG Synthesis Window Anat
113. sfer function that is a finite high frequency gain Therefore if the D term is on ICDM will require at least one HF rolloff term also to be on Opening the PID Synthesis Window When you select the PID window from the Synthesis menu in the ICDM Main window the PID window first decides whether the current controller transfer function has the form of a PID controller If it does then the PID window sets its parameters including the push buttons to the values that would yield the current controller and then opens In this case the current controller remains unchanged If the current controller does not have the form of a PID controller then a dialog box appears and warns the user and offers several alternatives Manipulating the Controller Parameters Each parameter can be changed using the slider variable edit box or graphically To change the sign of the parameter or to change the parameter to a value outside the current slider or plot range you must use the variable edit box Notice that negative values are allowed but often are not what you want The parameters also can be changed graphically by grabbing and dragging the controller Bode plot in the following ways e To change the gain with the gain parameter turned on grab the magnitude Bode plot anywhere except near the handles dark circles on the plot You now can drag the Bode plot up and down which changes the gain e To change the other parameters listed in col
114. sian LQG controller for a SISO plant e Chapter 8 H Infinity Synthesis describes the Hoo synthesis window used for SISO plants The Heo synthesis window is used to synthesize a central controller Such controllers are sometimes called linear exponential quadratic Gaussian LEQG or minimum entropy controllers e Chapter 9 History Window describes the History window used for SISO plants The History window is used to display and manipulate the design history list which is a list of controllers that have been explicitly saved during the design process e Chapter 10 Alternate Plant Window describes the form of the Alternate Plant window used for SISO design e Chapter 11 Introduction to MIMO Design provides an introduction to MIMO design building on the earlier discussions of SISO design ICDM automatically switches between SISO and MIMO modes depending on the plant that is read in e Chapter 12 LOG H Infinity Synthesis describes the MIMO LQG Hce synthesis window The LQG H window is used to synthesize both LQG and H controllers The two design methods have been combined in a single window because of the similarity regarding the use of weights constant weights frequency dependent weights and integrators e Chapter 13 Multi Loop Synthesis describes multi loop synthesis The multi loop window is used to synthesize a MIMO controller using PID and Root Locus methods applying them one loop at a time In many practical industr
115. space design for MIMO design An introduction to Xmath and a basic introduction to X Windows can be found in the Xmath User Guide There are several ways you can find out about the basics of interacting with an Xmath GUI application e Refer to Appendix A Using an Xmath GUI Tool e Enter guidemo in the Xmath Command window to start up the GUI demo applications this allows you to try out sliders push buttons scrollbars data viewing and so on After you have mastered the basic mechanics of using an Xmath GUI application you should be ready to get started To start up ICDM enter icdm in the Xmath Command window Your window manager may require you to position a window that is created using the left or middle mouse button After the ICDM Main Window appears the Xmath command prompt will return You now can use Xmath and ICDM simultaneously The user interface for ICDM is designed to be intuitive that is things mostly work the way you would assume that they should work so you should be able to start using ICDM immediately NI recommends that you read Chapter 2 Introduction to SISO Design before using the module ICDM includes a complete Help system In the menu bar of every CDM window there is a Help menu The Help messages contain detailed descriptions of every feature and function of ICDM You can get a good overview of the features of ICDM by scanning the entries in the menu bars and reading the Help messages in the variou
116. t freq Hz Figure 8 1 H Infinity Synthesis Window Xmath Interactive Control Design Module 8 2 ni com Chapter 8 H Infinity Synthesis Opening the Synthesis Window The Heo window can only accept H controllers If the current controller is of type H perhaps from the History window and the Heo window is opened the current controller is read into the Hee window that is the parameters are set to the appropriate values If the current controller is not of type H and the user attempts to open the Hoo Synthesis window a dialog box appears and warns the user that proceeding with opening the synthesis window will overwrite the current controller with the controller The Heo window remembers its parameter settings When it is opened the parameters will be exactly as they were when the window was last closed or they will be set to default values if the Heo window has not been opened in this ICDM session Setup and Synthesis Method This section describes the closed loop transfer matrix refer to Figure 8 2 The H synthesis procedure can be described using the following standard setup y Put wy u C v Ayw Zz pu Zz W y where yis the plant output signal w is a normalized input referred process noise w is a normalized sensor noise z is the normalized actuator signal z2 is the weighted plant output signal P is the plant transfer function C is the controller transfer function W is th
117. t at most one can be on at any time Like the station selection buttons on a radio selecting one button automatically turns off any other button that is on A pull down menu is displayed by depressing and holding the left mouse button As the mouse is dragged the various menu selections usually buttons are highlighted Releasing the mouse activates the selected button A cascaded menu is indicated by a small arrow to the right of the text in the button The cascaded menu is displayed by moving the mouse to the right A text entry area behaves like the command input area in Xmath Input is terminated by a new line Before you can type in it you must focus the keyboard at it by clicking the left mouse button Focus is indicated by a border highlight A 3 Xmath Interactive Control Design Module Appendix A Using an Xmath GUI Tool A list is a vertical list of items strings that can be selected highlighted Depending on the application a list can be configured to allow various types of selection A ssingle selection list allows only a single line to be selected Clicking the left mouse button selects a line This is the type of list that appears in the window shown in Figure A 1 A multiple selection list allows multiple lines to be selected The selection of a single line is toggled by clicking with the left mouse button Anextended selection list also allows multiple lines to be selected A contiguous ra
118. t of this manual You also can visit the Worldwide Offices section of ni com niglobal to access the branch office Web sites which provide up to date contact information support phone numbers email addresses and current events National Instruments Corporation B 1 Xmath Interactive Control Design Module Index A actuator disturbance signal 11 2 effort transfer function 11 3 loop transfer function 11 4 signal 2 2 11 2 step response 2 3 actuator referred actuator effort transfer function 11 3 closed loop transfer function 11 3 sensitivity transfer function 11 3 Alternate Plant display 10 1 transfer function 2 5 3 4 window 2 5 2 10 10 2 Autoscale 2 12 B Bode plot 2 4 Butterworth 6 6 C characteristic polynomial 2 3 closed loop poles 2 3 2 4 transfer function 2 3 11 3 zeros 2 3 command input signal 2 2 11 2 complementary loop transfer function 11 4 Control cost parameter 7 4 8 1 12 1 control eigenvalues 6 5 controller current 3 2 degree 2 2 National Instruments Corporation denominator 2 2 numerator 2 2 order 2 2 transfer function 2 2 2 5 11 2 conventions used in the manual iv Cycle button 9 3 D data viewing 2 12 DC normalization 10 4 Decay Rate 7 1 12 1 parameter 7 7 12 12 toggle button 7 7 12 16 Default Plants 11 1 default plot values 3 5 diagnostic tools NI resources B 1 disturbance input vector 12 9 documentation co
119. t the two transfer functions can be obtained from each other by swapping P and C The transfer function from dacs to u CP I CP is called the actuator referred actuator effort transfer function Notice that it is related to the closed loop transfer function by swapping P and C It can also be expressed as C I PC P e The transfer function from d e to y P I CP is denoted T and called the actuator referred closed loop transfer function National Instruments Corporation 11 3 Xmath Interactive Control Design Module Chapter 11 Introduction to MIMO Design Notice that in the SISO case these complementary pairs of transfer functions obtained by swapping P and C are the same It is important to remember that in the MIMO case they can be different they even have different dimensions if n n In addition to these transfer functions you encounter two complementary open loop transfer functions e The loop transfer function L is defined as L PC This is the transfer function of the loop cut at the sensor or the error e e The actuator loop transfer function or complementary loop transfer function L is defined as Lact CP This is the transfer function of the loop cut at the actuator Integral Action Integral action can be quite complicated in the MIMO setting The simplest case occurs when the plant is square that is n n the plant has no poles at s 0 and no zeros at s 0 wh
120. th vg Editing the Closed Loop Poles You can change the closed loop poles two ways by editing or by grabbing and dragging them Both of these methods are described in the General Plotting Features section of Chapter 2 Introduction to SISO Design Slider and Plot Ranges To change the ranges of the Frequency Scaling slider or the plot of closed loop poles select View Ranges or press lt Ctrl R gt in the Pole Place window The slider range also will be changed automatically if you type a new value which is outside the current range in the variable edit box The plot also can be re ranged interactively by grabbing and dragging the plot axes refer to the General Plotting Features section of Chapter 2 Introduction to SISO Design Selecting View Auto scale or pressing lt Ctrl A gt in the Root Locus window will cause new ranges to be assigned to the slider and plot based on the current controller Xmath Interactive Control Design Module 6 6 ni com LQG Synthesis This chapter discusses the LQG Synthesis window which is used to synthesize a linear quadratic Gaussian LQG controller for a SISO plant If you select LQG synthesis with a MIMO plant you will get the MIMO LQG Synthesis window described in Chapter 12 LOG H Infinity Synthesis LOG Synthesis Window Anatomy The LQG Synthesis window is shown in Figure 7 1 From top to bottom it contains e A menu bar with entries Special Edit View and Help e A mess
121. the value that y is reset to Yopt 18 the optimal smallest possible feasible value of y Yprey S the previous value of y Yreq is the value of y requested by the user After the Heo Synthesis window has determined a lower bound on Yop it is displayed in the singular value plot It will disappear if the user changes the control cost sensor noise or output weight parameters Xmath Interactive Control Design Module 8 6 ni com Chapter 8 H Infinity Synthesis Ranges To change the ranges of the sliders or plots select View Ranges or press lt Ctrl R gt in the Heo window The slider ranges also will be changed automatically if you type a new value which is outside the current range in the corresponding variable edit box The plot also can be re ranged interactively by grabbing and dragging the plot axes refer to the Interactive Plot Re ranging section of Chapter 2 Introduction to SISO Design Selecting View Auto scale or pressing lt Ctrl A gt in the Hee window will cause new ranges to be assigned to the sliders and plots based on the current controller National Instruments Corporation 8 7 Xmath Interactive Control Design Module History Window This chapter describes the History window used for SISO plants The History window is used to display and manipulate the design history list which is a list of controllers that have been explicitly saved during the design process For a description of the History window used for
122. this case you cannot read the resulting controller back into the LQG or window since the controller no longer has this special form Manipulating the Closed Loop Poles The closed loop poles and zeros can be dragged and edited interactively Refer to the Graphically Manipulating Poles and Zeros section of Chapter 2 Introduction to SISO Design for a general discussion of manipulating poles graphically Time and Frequency Scaling The slider and variable edit box show the average value of the closed loop pole magnitudes and therefore can be interpreted as roughly the bandwidth of the closed loop system The average frequency is given by Fave Aid ii hon van where Aj Az are the closed loop poles Notice that the average is geometric You can change F by dragging the slider or typing into the variable edit box The effect is that the closed loop poles are all multiplied by a scale factor in such a way that becomes the requested value Therefore by changing Favg you are time or frequency scaling the closed loop dynamics National Instruments Corporation 6 5 Xmath Interactive Control Design Module Chapter 6 Pole Place Synthesis A circle of radius F also is displayed in the plot You also can drag the circle to change Favg Butterworth Configuration Click the Butterworth button to move the poles to a Butterworth configuration preserving F ye The initial pole configuration also is set to Butterwor
123. tion of three optional features e Integral action e Exponential time weighting guaranteed decay rate e Output weight editing The synthesis mode is reported in the text at the top of the window The toggle Int Time integral action time button enables and disables integral action The Decay Rate toggle button controls exponential time weighting The Weight Zero Edit toggle button enables and disables output weight editing Opening the LOG Synthesis Window The LQG window can only accept LQG controllers If the current controller is of type LQG perhaps from the History window and the LQG window is opened the current controller is read into the LQG window That is the push buttons and parameters are set to the appropriate values If the current controller is not of type LQG and the user attempts to open the LQG Synthesis window a dialog box appears and warns the user that proceeding with opening the LQG Synthesis window will overwrite the current controller with the LQG controller The LQG window remembers its parameter settings When it is opened the parameters will be exactly as they were when the LQG window was last closed or it will be set to default values if the LQG window has not been opened in this ICDM session National Instruments Corporation 7 3 Xmath Interactive Control Design Module Chapter 7 LQG Synthesis Setup and Terminology The different modes are described using the following basic terminology y
124. to restore deleted poles or zeros back provided you have not made any other changes since deleting To abort a delete operation click the left mouse button with the skull and crossbones cursor in a free area of the plot Adding Deleting Pole Zero Pairs When you add or delete a pole or zero you can drastically change the transfer function that you are editing In some cases it may be better to add or delete a pole zero pair that is a pole and zero in exactly the same location Adding a pole zero pair does not change the transfer function at all until the pole and zero are moved apart National Instruments Corporation 2 15 Xmath Interactive Control Design Module Chapter 2 Introduction to SISO Design To add a pole zero pair click the Add Pair button select the Add Pair entry on the Edit menu or press lt Ctrl P gt in the window As with poles and zeros the pole zero pair you create will be either real or a complex conjugate pair depending on how close the cursor is to the real axis when you click the left mouse button After the pair is created you can drag the pole and zero away from each other which results in a smooth change to the transfer function By convention the cursor first grabs the zero in a pole zero pair To delete a pole and zero that are very near each other click the Delete button position the cursor near the pole and zero and click the left mouse button This will remove the pole and zero but have l
125. trol Design Module Chapter5 Root Locus Synthesis Figure 5 1 Root Locus Synthesis Window The Root Locus Synthesis window consists of from top to bottom e A menu bar with entries Special Edit View and Help e A slider and variable edit box for the gain These controls are used to show and also to change the controller gain The gain also can be changed graphically by dragging the closed loop poles along the root locus e Bottom left The root locus plot The plot shows selected phase or gain contours of the loop transfer function along with the plant and controller poles and zeros A more detailed description appears following e Bottom right Buttons to add delete edit poles and or zeros Poles zeros and pole zero pairs also can be created and destroyed using the Xmath Interactive Control Design Module 5 2 ni com Chapter 5 Root Locus Synthesis Edit menu or by typing the accelerators in the Root Locus window A more detailed description appears following The Root Locus Synthesis window is shown in Figure 5 1 with the standard default 180 contour The branches of the locus connect the zeros and poles of the loop transfer function which are shown in the plot The closed loop poles which are on the locus are also shown Opening the Root Locus Synthesis Window Terminology The Root Locus window can accept any type of controller so it can always be opened It simply reads the current controller from ICDM Yo
126. ts Corporation All rights reserved Important Information Warranty The media on which you receive National Instruments software are warranted not to fail to execute programming instructions due to defects in materials and workmanship for a period of 90 days from date of shipment as evidenced by receipts or other documentation National Instruments will at its option repair or replace software media that do not execute programming instructions if National Instruments receives notice of such defects during the warranty period National Instruments does not warrant that the operation of the software shall be uninterrupted or error free A Return Material Authorization RMA number must be obtained from the factory and clearly marked on the outside of the package before any equipment will be accepted for warranty work National Instruments will pay the shipping costs of returning to the owner parts which are covered by warranty National Instruments believes that the information in this document is accurate The document has been carefully reviewed for technical accuracy In the event that technical or typographical errors exist National Instruments reserves the right to make changes to subsequent editions of this document without prior notice to holders of this edition The reader should consult National Instruments if errors are suspected In no event shall National Instruments be liable for any damages arising out of or related to this document
127. u then can use the Root Locus window to manipulate the controller poles zeros and gain After you have changed the controller using the Root Locus window the controller loses any special form it may have had for example LQG It is represented by its transfer function Thus you can use the Root Locus window to change the zeros poles and gain of a controller originally designed using the LQG window but you then cannot read the controller back into the LQG Synthesis window since it is no longer an LQG controller The loop transfer function is expressed in the following product form S Z S Z L KETA G s p S P where K is called the gain notice that the gain is high frequency normalized the z values are the zeros of the loop transfer function and the p values are the poles of the loop transfer function Each of these poles and zeros is associated with either the plant or the controller The Root Locus window allows you to change the gain change or delete any controller pole or zero or create new controller poles and zeros as long as the controller transfer function remains proper that is has finite gain at high frequencies The Root Locus window will not allow you to change or delete any plant pole or zero The Alternate Plant window can be used to modify the plant interactively and see the effect on the closed loop system performance National Instruments Corporation 5 3 Xmath Interactive Control Des
128. umn four of Table 4 1 grab the appropriate handle dark circle on either plot and drag it left and right to the desired frequency which is the inverse of the time parameter The associated slider and variable edit box also will be updated as you drag the handle Xmath Interactive Control Design Module 4 4 ni com Chapter 4 PID Synthesis Time Versus Frequency Parameters Notice that the sliders and variable edit boxes use time parameters whereas the Bode plot handles use frequencies that is the inverses of the time parameters If you think of integral action as being parameterized by a characteristic time then you may prefer to use the slider If you think of integral action as being parameterized by a characteristic frequency reset rate then you may prefer to manipulate the Bode plot handle Ranges of Sliders and Plots The ranges for the sliders and plots can be changed in several ways If you enter a value that lies outside the slider range in the corresponding variable edit box the range of the slider will automatically adjust to accommodate the new value You also can change the range of a slider using the Ranges window which appears when you select View Ranges or press lt Ctrl R gt in the PID window Selecting View Auto Scale will cause ICDM to select sensible values for the slider and plot ranges based on the current controller The ranges for the plots also can be changed interactively Refer to the General Plotting Featu
129. uration for the 3 sensor 2 actuator plant considered redrawn showing C connected to Peguiv which is the plant with all other loops closed in this case just one other loop While designing C it is useful to think of it as the SISO controller for the SISO equivalent plant P Again notice that P depends on which loop you are designing Xmath Interactive Control Design Module 13 6 ni com Chapter 13 Multi Loop Synthesis Opening the Multi Loop Synthesis Window The multi loop window can accept any type of MIMO controller and will decompose it into its SISO components one for each loop Control loops are categorized as being of type PID or type Root Locus If a loop is not of type PID then it will be categorized as a Root Locus controller Remember that the Root Locus Synthesis window accepts any type of SISO controller A warning is issued in a dialog box that appears when the order of the multi loop imported controller is very high For instance acceptance of an LQG H controller will generally lead to n n SISO control loops each of which has the full state order This usually leads to an unacceptably high order and the user gets the chance to start from scratch by pressing the Reset button in the dialog box that appears Designing a Multi Loop Controller This section describes the multi loop controller including the graphical editor how to manage loops and loop gain magnitude and phase Graphical Edito
130. us from disabled to enabled and vice versa National Instruments Corporation 13 1 Xmath Interactive Control Design Module Chapter 13 Multi Loop Synthesis Pressure 1 Water flow Steam flow CK Temperature C gt Fuel flow Figure 13 1 Multi Loop Main Window After the Multi Loop window is opened two plots are added at the bottom of the ICDM Main window for display of the loop gain magnitude and phase of the control loops that will be synthesized with the Multi Loop method refer to Figure 13 2 Xmath Interactive Control Design Module 13 2 ni com Chapter 13 Multi Loop Synthesis ICDM Main Window File Edit Plot Synthesis 8 26 13 Closed loop system is unstable 8 8 25 54 Closed loop system is wnstable 8 25 52 Root locus synthesis window selected Fi Plant Default plant 4 MM0 size 3 2 8 Compensator Singular values of PC ogaingular values of inv l PC PID oT 100 xingular values of PC inv l PC a 40 g 60 80 109 7 gt i0 To 100 SISO loop gain magnitudes Figure 13 2 Multi Loop Gain and Phase Plots Added to the ICDM Main Window Setup and Synthesis Method This section describes the setup and synthesis method for multi loop synthesis Multi Loop Versus Multivariable Design In most multivariable control design methods such as synthesis no specific assumptions are made about which loops should be closed and which ones not In general all components of t
131. ut signal y Without integral action the controller minimizes the quantity J lim E pu t y t too and with integral action the quantity f 2 2 2 J lim E pu t y t z t to where t 20 1 Tind POA 0 The transfer function W is the output weighting transfer function When W 1 this reduces to the standard LQG controller described previously The weighting transfer function is given by W s n s n s Its denominator is fixed and equal to the numerator of the plant transfer function Its numerator can be manipulated by the user The lower left plot shows the poles and zeros of the weight transfer function W When Weight Zero Edit is enabled the user can grab and drag the zeros shown or Add Delete Edit zeros using the push buttons The lower right plot shows the magnitude of the weight transfer function When it is flat and equal to 0 dB for all frequencies you have W 1 that is standard LQG design based on the plant output y When for example W is larger than 0 dB at low frequencies this means that the LQG controller is based on a filtered version of y that emphasizes low frequencies which presumably results in a controller with larger loop gain at low frequencies Xmath Interactive Control Design Module 7 6 ni com Chapter 7 LQG Synthesis State Space Interpretation In LQG theory the closed loop poles consist of n optimal control eigenvalues and n estimator Kalman filter eigenv
132. utput is determined by the toggle buttons in the weights window The objective of the control design is to minimize the expectation of the power of LQG or the maximal singular value of the transfer function from w to z He The LQG control design problem is discussed first followed by a discusion of how the control design is interpreted in the same setting An essential part of the LQG H gt formulation is the selection of frequency dependent weights control inputs and measurements The control design is based on the diagram shown in Figure 12 6 Xmath Interactive Control Design Module 12 8 ni com Chapter 12 LQG H Infinity Synthesis The weighted output vector z consists of the following e Filtered inputs e Plant states x e Filtered plant outputs y e Integrated filtered plant outputs y The disturbance input vector w consists of the following e General LQG state disturbances w e General LQG output disturbances w Input referred disturbances or process noise w e Measurement or sensor noise w e Filter noise wp e Reference noise for the purpose of setpoint tracking w The measured output consists of the following e Integrated filtered measured plant outputs y sens e Filtered measured plant outputs Yens e Measured plant outputs Vp sens National Instruments Corporation 12 9 Xmath Interactive Control Design Module Chapter 12 LQG H Infinity Synthesis
133. ve Control Design Module A 4 ni com National Instruments Corporation Appendix A Using an Xmath GUI Tool GUI windows might contain buttons that display some value The value can be changed by clicking the button whereupon a text entry area will appear in place of the button You can enter a new value followed by pressing lt Return gt If the GUI tool does not like your new value it reserves the right to change it to an acceptable value that is displayed again in the button These buttons are called variable edit boxes The button labeled 12 shown in Figure A 2 is a variable edit box displaying the value of the variable w If you click this button it is replaced by the w value text entry area as shown in Figure A 3 After a value is entered from the keyboard the text entry area is replaced by the button for example the 12 button sf PGUI Example Do It I Toggle button v value Figure A 3 PGUI Example Dialog after Pressing the 12 Button GUI windows might contain sliders which resemble linear potentiometers and whose values are changed by a linear motion of the handle The position of the slider s handle represents its value Usually the limits of the slider are shown at its ends Figure A 3 shows a slider with minimum value 0 and maximum value 10 Its value is about 6 The value of a slider can be changed in several ways The handle can be grabbed and dragged by clicking the left mouse button on the ha
134. y This section describes the basic terminology and notation for SISO plants and controllers used in ICDM and this manual ICDM uses the standard classical feedback configuration shown in Figure 2 1 C s P s gt Figure 2 1 Standard Classical Feedback Configuration Used in ICDM National Instruments Corporation 2 1 Xmath Interactive Control Design Module Chapter 2 Introduction to SISO Design The equations describing this system are as follows Pu u Ce e r y where y denotes the plant output or sensor signal u denotes the plant input or actuator signal r denotes the reference or command input signal e denotes the error signal P denotes the plant transfer function C denotes the controller transfer function In ICDM the plant and controller transfer function are required to be rational that is the ratio of two polynomials Py eget d s d s where n dp ne and d are polynomials called the plant numerator plant denominator controller numerator and controller denominator respectively The symbols n and d are mnemonics for numerator and denominator The degree of d is the plant order or plant degree Similarly the degree of d is the controller order or controller degree The poles and zeros of these transfer functions are the zeros roots of the denominator and numerator polynomials respectively In ICDM P and C are required to be proper polynom
135. y the same performance Using the Root Locus window you want to move stable controller poles or zeros near each other without sacrificing controller performance Good candidates are poles and zeros substantially outside the control system bandwidth or pairs of nearby poles and zeros After you have moved a controller pole or zero or both so that they are near each other and hopefully control system performance has not changed too much then you can delete the pair without severely affecting the controller transfer function You have just reduced the controller order by one or two if you deleted a complex conjugate pair of poles and zeros National Instruments Corporation 5 7 Xmath Interactive Control Design Module Chapter 5 Root Locus Synthesis Interpreting the Nonstandard Contour Plots The Root Locus window can display phase contours other than the standard 180 as well as various magnitude contour plots The meaning of these curves is simple if L s a then s would be a closed loop pole if the loop transfer function were multiplied by 1 a at the frequency s For example a point s labeled LCs 3 dB on one of the 170 curves would be a closed loop pole if the loop transfer function at the frequency were to increase in magnitude by 3 dB and increase in phase by 10 This simple observation works two ways Continuing the previous example to have a pole at s try to change the current controller to achieve th

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