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NI MA TRIXx TM Xmath TM Interactive Control Design Module Xmath Interactive Control Design Module April 2007 370754C-01[...]

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Conventions The follo wing con ventions 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 yo u to pull down the File menu, select the Page Setup item, and select Options from the last dialog box. This icon denotes a n ote, which aler[...]

© National Instruments Corporation v Xm ath Interactive Contr ol Design M odule Contents Chapter 1 Introduction Using This Manual...................... .............. .............. .............. .............. .............. ............. 1-1 Document Organization........... ... ........... .............. .............. .............. ..........[...]

Contents Xmath Interactive Contro l Design Module vi ni.com Graphically Manipulating Poles and Zeros ................... .............. ................. ... 2-13 Editing Poles and Zeros ......................... .............. ............... .............. 2-13 Editing Poles and Zeros Graphically ........ .............. ................. ........[...]

Contents © National Instruments Co rporation vii Xmath Interactive Co ntrol Design M odule Chapter 5 Root Locus Synthesis Overview ........... .............. ............ .............. .............. .............. .............. .............. ............ .5 - 1 Window Anatomy ....................... ........... .............. .............. ..[...]

Contents Xmath Interactive Contro l Design Module viii ni.com Manipulating the Design Parameters....................... .............. .............. .............. ............ 7-7 Manipulating the Design Parameters Graphically ........... .............. ................. 7-7 Ranges .................. .............. ............... .............. [...]

Contents © National Instruments Corporation ix Xma th Interactive Control Desi gn Module Chapter 11 Introduction to MIMO Design Basic Terminology for MIMO Syst ems ....................................... ............................. .... 11 -1 Feedback System Configuratio n............... .............. .............. .............. .............[...]

Contents Xmath Interactive Contro l Design Module x ni.com Chapter 13 Multi-Loop Synthesis Multi-Loop Window Anatomy ......... ............... .............. ................. .............. .............. ... 13-1 Setup and Synthesis Method ............ ............... .............. ................. .............. .............. ... 13-3 Multi-L[...]

© National Instruments Corporation 1-1 Xm ath Interactive Contr ol Design M odule 1 Introduction The Xmath Interactiv e Control Desi gn Module (ICDM) is a complete library of classical and modern interactiv e control design functions that takes full adv antage of Xmath’ s powerful, object-oriented, graphical en vironment. It provides a fle xible[...]

Chapter 1 Introduction Xmath Interactive Contro l Design Module 1-2 ni.com • Chapter 5, Root Locus Synthesi s , describes the user interface, terminology , and parameters used for root locus synthesis. • Chapter 6, Pole Place Synthesis , discusses the Pole Place synthesis window , which is used to design a SISO controller by assigning the close[...]

Chapter 1 Introduction © National Instruments Corporation 1-3 Xm ath Interactive Contr ol Design Modu le Commonly-Used Nomenclature This manual uses the following general nomenclature: • Matrix variables are generally denoted with capital letters; vectors are represented in lowercas e. • G ( s ) is used to denote a transfer function of a syste[...]

Chapter 1 Introduction Xmath Interactive Contro l Design Module 1-4 ni.com MA TRIXx Help Interactive Control Design Mod ule function reference infor mation is available in the MATRIXx Help . The MATRIXx H elp includes all Interactive Control Design functions. Each to pic explains a function’s inputs, outputs, and keywords in detail. R efer to Cha[...]

Chapter 1 Introduction © National Instruments Corporation 1-5 Xm ath Interactive Contr ol Design Modu le • Hav e a user’ s understandin g of Xmath (enough to create a plan t transfer function). • Know the basics of ho w to interact with an Xmath GUI application—for example, using a s lider to set a param eter value, a variable-edit box for[...]

© National Instruments Corporation 2-1 Xm ath Interactive Contr ol Design Modu le 2 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 mathemati cally. Operations on these systems [...]

Chapter 2 Introduction to SISO Design Xmath Interactive Contro l Design Module 2-2 ni.com The equations describing this system are as fol lows: where y denotes the plant output or sensor signal u denotes the plant input or actuator signal r denotes the reference o r command input si gnal e denotes the error signal P denotes the plant transfer funct[...]

Chapter 2 Introduction t o SISO Design © National Instruments Corporation 2-3 Xm ath Interactive Contr ol Design Modu le • The closed-loop transfer function T is given b y T = PC /(1 + PC ). T is the transfer function from r to y . • The characteristic polynomial of the system is def ined as X = n c n p + d c d p . Its degree is equal to the o[...]

Chapter 2 Introduction to SISO Design Xmath Interactive Contro l Design Module 2-4 ni.com These are briefly described in the following sections, and in more detail in later chapters. Sev eral of these windows hav e dif ferent forms for SISO and MIMO design. This chapter restricts the discussion to the SISO forms. Refer to Chapter 11, Introductio n [...]

Chapter 2 Introduction t o SISO Design © National Instruments Corporation 2-5 Xm ath Interactive Contr ol Design Modu le LQG 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 (ou tput) cost and t[...]

Chapter 2 Introduction to SISO Design Xmath Interactive Contro l Design Module 2-6 ni.com The plant and the alternate plant hav e v ery dif ferent uses in ICDM, and therefore differe nt data flow characteristics. The plant transfer function is read from Xmath into the I CDM Main window , and is then exported to the synthesis windows that need it—[...]

Chapter 2 Introduction t o SISO Design © National Instruments Corporation 2-7 Xm ath Interactive Contr ol Design Modu le list. The current controller is the acti v e or selected entry on the list of sav ed controllers. Only one synthesis w indow , or the History wind ow , is allo wed to be open at any gi ven time, which eliminates an y poss ible c[...]

Chapter 2 Introduction to SISO Design Xmath Interactive Contro l Design Module 2-8 ni.com window . Thus, the Root Locus Synthesis window can be used to interactiv ely tweak or model-reduce a controller designed b y another method such as LQG. • The Pole Place windo w accepts any controller with the same number of poles as the plant, or one more p[...]

Chapter 2 Introduction t o SISO Design © National Instruments Corporation 2-9 Xm ath Interactive Contr ol Design Modu le controller . Using the R oot Locus window , the user could reduce the controller to a PI cont roller by deleting poles and zeros, at which point the PID window can be opened, init ialized at the current cont roller . Using ICDM [...]

Chapter 2 Introduction to SISO Design Xmath Interactive Control Design Module 2-10 ni.com Figure 2-3 shows a simplified schemati c representation of the interactive robustness analysis loop. Here, the user inter acts with th e Alternate Plant window , interacti vely changing the alternate plant transfer function P alt , which is automatical ly expo[...]

Chapter 2 Introduction t o SISO Design © National Instruments Corporation 2-11 X math Interactive Control Design Modu le Figure 2-4. Simple ICDM Session General Plotting Features All of the plots in the ICDM Main and other windows support seve ral useful features: arbitrary re-ran ging, zoomi ng, data-viewing, an d interactive (graphical) re-rangi[...]

Chapter 2 Introduction to SISO Design Xmath Interactive Control Design Module 2-12 ni.com window has an autoscale feature, which can be invoked by selecting Autoscale on the View or Plot menu of the win dow. When you invoke Autoscale, ICDM tries to assign some reasonable values to the slider and plot scales. Zooming You can enlarge any portion of a[...]

Chapter 2 Introduction t o SISO Design © National Instruments Corporation 2-13 X math Interactive Control Design Modu le poles and zeros are (and indeed, the only way on a black-and-white display) is to use data-viewing. As a general rule: T o fi nd out the meaning, purpose, or value of an object (pole, zero, curve, and so on.) in an ICDM plot, us[...]

Chapter 2 Introduction to SISO Design Xmath Interactive Control Design Module 2-14 ni.com contains variable edit boxes for the va lue of the pole or zero (the real and imaginary part when the po le or zero is complex) an d, if appropriate, its multiplicity. Afte r you enter new values, you can sel ect OK , which will make the changes and dismiss th[...]

Chapter 2 Introduction t o SISO Design © National Instruments Corporation 2-15 X math Interactive Control Design Modu le (if it was not already) but ot herwise 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[...]

Chapter 2 Introduction to SISO Design Xmath Interactive Control Design Module 2-16 ni.com T o add a pole-zero pair , click the Add Pair button, select the Add Pair entry on the Edit menu, or press <Ctrl-P> in t he window . As with poles and zeros, the pole-zero pair you creat e will be either real or a comple x conjugate pair , depending on h[...]

© National Instruments Corporation 3-1 Xm ath Interactive Contr ol Design Modu le 3 ICDM Main Window This chapter describes the use of the ICDM Main window , which is used to perform the following functions: • Communicate with Xmat h—for example, transf er plants /controllers from/to X math • Display warning and log messages • Display a v [...]

Chapter 3 ICDM Main Window Xmath Interactive Contro l Design Module 3-2 ni.com • A line that iden tifies the type a nd source of the current controller . The source is either the currentl y acti ve synthesis windo w or the history list. • A plotting area for the various plots. Figure 3-1. ICDM Main Window Communicating with Xmath The File menu [...]

Chapter 3 ICDM Main Window © National Instruments Corporation 3-3 Xm ath Interactive Contr ol Design Modu le Most Common Usage In most cases, you will re ad 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 app[...]

Chapter 3 ICDM Main Window Xmath Interactive Contro l Design Module 3-4 ni.com to a simple transfer functi on representation, which means that you canno t read them back into th e Pole Place, LQG, or sy nthesis windows because these types depend on the plant. Also, all synthesis wind ows will be reset to their initial (default) settings. Because th[...]

Chapter 3 ICDM Main Window © National Instruments Corporation 3-5 Xm ath Interactive Contr ol Design Modu le ICDM Plots Various plo ts can be shown a t the bottom of the main ICDM window. The Plot menu is used to select which plots are shown, and also to magnify a plot or set the plotti ng ranges. The us er can choose any combination of the follow[...]

Chapter 3 ICDM Main Window Xmath Interactive Contro l Design Module 3-6 ni.com In the ICDM Main window , the Plot Choices dialog box is used to select any combination of the eight plots. This di alog 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 Range[...]

Chapter 3 ICDM Main Window © National Instruments Corporation 3-7 Xm ath Interactive Contr ol Design Modu le Figure 3-3. ICDM Ranges Window Plot Magnify Windows In addition to the standard plot ting features (zooming, data-vi ewing, and interactive re-ranging) described in the General Plotting Featu res section of Chapter 2, Introduction to SIS O [...]

Chapter 3 ICDM Main Window Xmath Interactive Contro l Design Module 3-8 ni.com of Plots section. If another plot is subsequently selected for magnifying, it will replace the current plot in the plot magnify windo w . The Plot Magnify window is a separate windo w that shows one of the ICDM main p lots. The Plot Magnify win dow , sho wn in Figure 3-4[...]

Chapter 3 ICDM Main Window © National Instruments Corporation 3-9 Xm ath Interactive Contr ol Design Modu le Selecting a Synthesis or Histor y Window The Synthesis menu in the ICDM Main win dow is used to select which synthesis window will be active. If the cur rent controller is com patible with the requested synthesis window, then the synthesis [...]

© National Instruments Corporation 4-1 Xm ath Interactive Contr ol Design Modu le 4 PID Synthesis This chapter discusse s 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 win[...]

Chapter 4 PID Sy nthesis Xmath Interactive Contro l Design Module 4-2 ni.com T oggling Controller T erms On and Off For each parameter, the toggle button at th e 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 tran sfer function, and the slider and varia[...]

Chapter 4 PID Sy nthesis © National Instruments Corporation 4-3 Xm ath Interactive Contr ol Design Modu le Figure 4-1. PID Synthesis Window As an example, suppose that the P an d I toggle b uttons are on, and the D and HF rolloff b uttons are off. The controller transfer function will then hav e the following form: Cs () K p = 1 sT int ⁄ () +[...]

Chapter 4 PID Sy nthesis Xmath Interactive Contro l Design Module 4-4 ni.com Notice that there are at least two ot her commonly used forms for a PID control law that diff er from the one used in ICDM: and ICDM enforces a proper cont roller tran sfer function, that is, a finite high frequency g ain. Therefore, if the D te rm is on, ICDM will require[...]

Chapter 4 PID Sy nthesis © National Instruments Corporation 4-5 Xm ath Interactive Contr ol Design Modu le T ime Versus Frequency Parameters Notice that the sliders and variab le-edit boxes use time parameters, whereas the Bode plot handles use freque ncies, that is, the in verses of the time parameters. If you think of integral action as being pa[...]

Chapter 4 PID Sy nthesis Xmath Interactive Contro l Design Module 4-6 ni.com Derivative T erm Normalization The derivative term is low-frequency normalized, which means that at low frequencies (below 1/ T diff ) it is nearly one, and so has little eff ect on the overall controller transfer functio n at low frequencies. In particular, the loop trans[...]

© National Instruments Corporation 5-1 Xm ath Interactive Contr ol Design Modu le 5 Root Locus Synthesis This chapter describes the user interf ace, terminology, and parameters used for root locus synthesis. Over view The Root Locus window performs two main functio ns: • Displays selected gain and phase contours in the complex plane of the loop [...]

Chapter 5 Root Locus Synthesis Xmath Interactive Contro l Design Module 5-2 ni.com Figure 5-1. Root Locus Synthesis Window The Root Locus Synthesis window consists of, from top to bottom: • A menu bar with entries Special , Edit , View , and Help . • A slider and variable edit box fo r the gain. These controls are used to show and also to chang[...]

Chapter 5 Root Lo cus Synthesis © National Instruments Corporation 5-3 Xm ath Interactive Contr ol Design Modu le Edit menu or by typing the accel erators in the Root Locus windo w . A more detailed description appears following. The Root Locus Synthesis window is sho wn in Figure 5-1 with the standard (default) 180 ° contour . The branches of th[...]

Chapter 5 Root Locus Synthesis Xmath Interactive Contro l Design Module 5-4 ni.com Plotting Styles Selecting View»Locus Select or pressing <Ctrl-L> in the Root Locus window produces a dialog box in which the user can choose on e of many possible plotting styles. In all cases, the (open-loop) con troller and plant poles and zeros are shown on[...]

Chapter 5 Root Lo cus Synthesis © National Instruments Corporation 5-5 Xm ath Interactive Contr ol Design Modu le Phase Contours For each of magnitude and phase contours, you can choose one of three possible plotting styles. • 180 ° The plot sho ws the locus of points where the phase angle of the loop transfer function is 180 ° . This yields a[...]

Chapter 5 Root Locus Synthesis Xmath Interactive Contro l Design Module 5-6 ni.com All of the plots support data viewing: click the right mouse button with the cursor positioned near a pole, zero, or one of the plots. This allows you to find the g ain associated with a particular point on a phase contour , for example. Slider and Plot Ranges To cha[...]

Chapter 5 Root Lo cus Synthesis © National Instruments Corporation 5-7 Xm ath Interactive Contr ol Design Modu le Design This section gives short descripti ons of how the Root Lo cus window can be used to design or analyze controller s. This section also provides some interpretations and describes some us es of the nonstandard contour plots. Addin[...]

Chapter 5 Root Locus Synthesis Xmath Interactive Contro l Design Module 5-8 ni.com Interpreting the Nonstandard Contour Plots The Root Locus window can display phase cont ours other than the standard 180 ° as well as v arious magnitud e cont our plots. The meaning of these curves is simple: if L ( s )= a , then s w ould be a closed-loop po le if t[...]

Chapter 5 Root Lo cus Synthesis © National Instruments Corporation 5-9 Xm ath Interactive Contr ol Design Modu le Figure 5-3. Roo t Locus Synthesis Window with the 0 dB Magnitude Contour[...]

© National Instruments Corporation 6-1 Xm ath Interactive Contr ol Design Modu le 6 Pole Place Synthesis This chapter discusses the Pole Place Synthesis window, wh ich is used to design a SISO controller by assign ing the closed-loop poles. Pole Place operates in two modes: • Normal mode (integral action not enforced) • Integral action mode Th[...]

Chapter 6 Pole Place Synthes is Xmath Interactive Contro l Design Module 6-2 ni.com Figure 6-1. Pole Place Synthesis Window Pole Place Modes In Pole Place, the user selects either closed-loop poles (i n normal mode) or 2 n + 1 closed-loop poles (in integral action mode). These poles uniq uely determine the controller transfer function. This process[...]

Chapter 6 Pole Place Sy nthesis © National Instruments Corporation 6-3 Xm ath Interactive Contr ol Design Modu le where d p ( s )= s n + a 1 s n –1 + a 2 s n –2 +. . .+ a n n p ( s )= b 0 s n + b 1 s n –1 +. . .+ ab n Notice that the order of the plant is n , and allow the possibility that th e plant transfer function is not strictly proper;[...]

Chapter 6 Pole Place Synthes is Xmath Interactive Contro l Design Module 6-4 ni.com W e can write this poly nomial equatio n as follows: These 2 n linear equations are solved to f ind the 2 n controller parameters x 1 , ..., x n and y 1 ,. . . ,y n . Integral Action Mode The degree (number of poles) of the con troller is fixed and eq ual to n +1 , [...]

Chapter 6 Pole Place Sy nthesis © National Instruments Corporation 6-5 Xm ath Interactive Contr ol Design Modu le 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 cont rol ler tra[...]

Chapter 6 Pole Place Synthes is Xmath Interactive Contro l Design Module 6-6 ni.com A circle of radius F av g also is displayed in the plot. Y ou also can drag the circle to change F avg . Butter worth Configuration Click the Butterworth button to move the poles to a Butterwor th configuration, preserving F avg . The initial pole configuratio n als[...]

© National Instruments Corporation 7-1 Xm ath Interactive Contr ol Design Modu le 7 LQG Synthesis This chapter discusse s 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 describ ed in Chapte[...]

Chapter 7 LQG Synthesis Xmath Interactive Contro l Design Module 7-2 ni.com Figure 7-1. LQG Synthesis Window • A control panel used to graphically edit the output weight transfer function. • A plotting area that cont ains the following plots: – The symmetric root locus plot s of the control and estimator closed-loop poles. The control cost an[...]

Chapter 7 LQG Synthesis © National Instruments Corporation 7-3 Xm ath Interactive Contr ol Design Modu le If the decay rate is enabled, it is sho wn as a v ertical line that can be dragged. – A plot showing the poles and zeros of the output weight transfer function. If weight zero editin g is enabled, the zeros can be edited graphically . – A [...]

Chapter 7 LQG Synthesis Xmath Interactive Contro l Design Module 7-4 ni.com Setup and T erminology The different modes are described using the following basic term inology: Figure 7-1 shows a block diagram with the basic setup for LQG synthesis, where u is the actuator signal (output of th e controller) w proc is an (input referred) process noise y[...]

Chapter 7 LQG Synthesis © National Instruments Corporation 7-5 Xm ath Interactive Contr ol Design Modu le Integral Action When integral action is enabled, the controller minimizes a variatio n on the LQG cost: where As in the standard mode, the sensor noise parameter ν is the ratio of the sensor noise intensity to the input-re ferred process nois[...]

Chapter 7 LQG Synthesis Xmath Interactive Contro l Design Module 7-6 ni.com Output Weight Editing When Weight Zero Edit is enabled, the LQG controller is based on , which is a filtered version of the plant outpu t signal y . Without integral action, the controller minimizes the quantity: and with integral action, the quantity: where The transfer fu[...]

Chapter 7 LQG Synthesis © National Instruments Corporation 7-7 Xm ath Interactive Contr ol Design Modu le State-Space Interpretation In LQG theory, the closed-l oop poles consist of n “optimal control eigenvalues” and n “estimator (Kalman filt er) eigenvalues.” For multivariable systems, the op timal control and the optimal estimator play [...]

Chapter 7 LQG Synthesis Xmath Interactive Contro l Design Module 7-8 ni.com 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 cha nged automatically if you type a new value which is outside the current rang e into the corresponding variable edit box. The plot also c[...]

© National Instruments Corporation 8-1 Xm ath Interactive Contr ol Design Modu le 8 H-Infinity Synthesis This chapter describes the H ∞ Synthesis window used for SISO plants. The H ∞ Synthesis window is used to synt hesize a central controller. Such controllers are sometimes called linear exponential quadratic Gaussian (LEQG) or minimum entrop[...]

Chapter 8 H-Infinity Sy nthesis Xmath Interactive Contro l Design Module 8-2 ni.com Figure 8-1. H-Infinity Syn thesis Window[...]

Chapter 8 H-Infinity Sy nthesis © National Instruments Corporation 8-3 Xm ath Interactive Contr ol Design Modu le Opening the Synthesis Window The H ∞ window can only accept H ∞ controllers. If the current controller is of type H ∞ (perhaps from the History window) and the H ∞ window is opened, the current cont roll er is read into the H ?[...]

Chapter 8 H-Infinity Sy nthesis Xmath Interactive Contro l Design Module 8-4 ni.com Figure 8-2. Block Diagram S howing the Basic Setup for H-Infinity Synthesis Figure 8-2 shows a block diagram with the basic setup for H ∞ synthesis where closed-loop transfer matrix H relates the two exogenous inputs w 1 and w 2 to the two outpu ts z 1 and z 2 . T[...]

Chapter 8 H-Infinity Sy nthesis © National Instruments Corporation 8-5 Xm ath Interactive Contr ol Design Modu le If either of these singular values is equal to or exceeds γ , the γ -entropy is defined to be + ∞ . In other words, the γ -entropy is f inite only for , and rap idly increases to + ∞ as becomes close to γ , where the H ∞ -nor[...]

Chapter 8 H-Infinity Sy nthesis Xmath Interactive Contro l Design Module 8-6 ni.com Manipulating the Design Parameters The parameters γ , ρ , and ν 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 automati cally adjust. The user can chang[...]

Chapter 8 H-Infinity Sy nthesis © National Instruments Corporation 8-7 Xm ath Interactive Contr ol Design Modu le Ranges To change the ranges of the sliders or plots, select View»Ranges or press <Ctrl-R> in t he H ∞ window. The slider ranges also will be cha nged automatically if you type a new value which is outside the current range in [...]

© National Instruments Corporation 9-1 Xm ath Interactive Contr ol Design Modu le 9 Histor y Window This chapter describes the History window used for SISO plants. The History window is used to display an d manipulate the design history list, which is a list of controllers that have been expl icitly saved during the design process. For a descripti[...]

Chapter 9 His tory Window Xmath Interactive Contro l Design Module 9-2 ni.com • A V ariable-Edit box which shows which history list entry is activ e or currently selected. The selected entry is the controller e xported to ICDM for plottin g. • Buttons for manipulatin g the history list. Selecting the Active Controller You can type a number in t[...]

Chapter 9 History Window © National Instruments Corporation 9-3 Xm ath Interactive Contr ol Design Modu le Deleting Histor y List Entries Any number of designs on the history list can be dele ted 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 [...]

Chapter 9 His tory Window Xmath Interactive Contro l Design Module 9-4 ni.com Using the Histor y List The history list can be used in several ways. You can save controllers as “benchmarks” whose performance y ou want to match with a sim pler controller. You also can save any promising desi gns that you find so you can later use them as the init[...]

© National Instruments Corporation 10-1 X math Interactive Control D esign Module 10 Alternate Plant Window This chapter describes the fo rm 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, [...]

Chapter 10 Alternate Plant Window Xmath Interactive Control Design Module 10-2 ni.com Alternate Plant Window Anatomy The Alternate Plant window is shown in Figure 10-1. From top to bottom , it consists of: • A menu bar with Special , Edit , and View menus. • A toggle button for controlling whether the pl ots in ICDM main will include the respon[...]

Chapter 10 Alternate Plant Window © National Instruments Corporation 10-3 X math Interactive Control De sign Module Figure 10-1. Alternate Plant Window Opening the Alternate Plant Window When the Alternate Plant window is fi rst opened, the alternate plant is initialized to the plant transfer functi on. This is convenien t because in most cases th[...]

Chapter 10 Alternate Plant Window Xmath Interactive Control Design Module 10-4 ni.com Normalization The form of the transfer function of the alternate plant depends on the normalization selected. With high-fre quency normalization, the alternate plant transfer function is: where K is the gain (sho wn in the slider and V ariable Edit box), are the z[...]

Chapter 10 Alternate Plant Window © National Instruments Corporation 10-5 X math Interactive Control De sign Module Y ou can switch between hi gh frequency and DC normalization by clicking the appropriate buttons. If the alte rnate plant has a pole or zero at s =0 , t h e n you cannot switch to DC normalization. Using the Alternate Plant Window Th[...]

Chapter 10 Alternate Plant Window Xmath Interactive Control Design Module 10-6 ni.com Ranges of Sliders and Plot To change the ranges of the Gain slider or the pole zero plot, select View»Ranges or press <Ctrl-R> in the Alternate Plant window. The slider range also will be changed automatically if y ou type a new v alue which is outside t he[...]

© National Instruments Corporation 11-1 X math Interactive Control De sign Module 11 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 provid[...]

Chapter 11 Introduction to MIMO Design Xmath Interactive Control Design Module 11-2 ni.com u denotes the plant input or actuator signal, which is a vector of size n u r denotes the reference o r command input si gnal, which is a vector of size n y e denotes the error signal, which is a v ector of size n y d act denotes the actuator disturbance sign[...]

Chapter 11 Introduction to MIMO Design © National Instruments Corporation 11-3 X math Interactive Control De sign Module The standard feedback system has two vector input signals, r and d act , and three vector output signals, e , u , and y . It can therefore be described by the 3 × 2 block matrix that relates the three ou tput vector signals to [...]

Chapter 11 Introduction to MIMO Design Xmath Interactive Control Design Module 11-4 ni.com 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; the y e ven ha ve different dimensions if n y ≠ n u . [...]

Chapter 11 Introduction to MIMO Design © National Instruments Corporation 11-5 X math Interactive Control De sign Module or disturbance rejection only on a subspace of d imension r , so do not be surprised if some (or many) diagonal entries of T are n ot one, or off diagonal entries are not zero. Finally , unlike a SISO plant, a MIMO plant can hav[...]

Chapter 11 Introduction to MIMO Design Xmath Interactive Control Design Module 11-6 ni.com 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. Figure 11-2. Plot Choices Window for the[...]

Chapter 11 Introduction to MIMO Design © National Instruments Corporation 11-7 X math Interactive Control De sign Module Notice that having the MI MO Plot window on the screen may increase the required computational respon se tim e of ICDM. Closing the wi ndow using the Special option of the MIMO Plot menu bar will then result in a speed-up. Figur[...]

Chapter 11 Introduction to MIMO Design Xmath Interactive Control Design Module 11-8 ni.com plants. Therefore, the (M IMO) 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 t he Alternate Plant window are identical in SISO a[...]

© National Instruments Corporation 12-1 X math Interactive Control De sign Module 12 LQG/H-Infinity Synthesis This chapter describes the MIMO LQG/H ∞ Synthesis window. The LQG / H ∞ window is used t o synthesize both LQG and H ∞ controllers. The two design methods have been co mbined in a single window because of the similarity regarding the[...]

Chapter 12 LQG/H-In finity Synthesis Xmath Interactive Control Design Module 12-2 ni.com Figure 12-1. LQG/H-Infinity Main Window • A pull-down menu for frequency-dependent weight selection on inputs: and outputs: • A button for recomputing the controller . These parameters are described in greater detail later in this chapter . LQG/H-Infinity W[...]

Chapter 12 LQG/H-In finity Synthesis © National Instruments Corporation 12-3 X math Interactive Control De sign Module descriptions are for the control cost p arameter display . The noise le vel display is similar in appearance. • A table with n u rows, ha ving in each row: – A toggle button to include the input in the set of contr ol inputs ?[...]

Chapter 12 LQG/H-In finity Synthesis Xmath Interactive Control Design Module 12-4 ni.com • A table with n y rows with, in each ro w: – 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 nam e – A slider defining the constant weigh[...]

Chapter 12 LQG/H-In finity Synthesis © National Instruments Corporation 12-5 X math Interactive Control De sign Module The weights ρ u , i , ρ y , j , ρ u ,a n d ρ y are then replaced w ith noise variances ν u , i , ν y , j , ν u ,a n d ν y . The noise lev el parameter in the main LQG/H ∞ window is related to the noise le vels in this wi[...]

Chapter 12 LQG/H-In finity Synthesis Xmath Interactive Control Design Module 12-6 ni.com value. If a lo wer bound on the minimal value of γ is kno wn, it also is displayed. Figure 12-4. LQG/H-Infinity Performance Level Window Frequency Weights Window The Frequency Weights wi ndow is shown in Figure 12-5 . From top to bottom, it consists of: • A [...]

Chapter 12 LQG/H-In finity Synthesis © National Instruments Corporation 12-7 X math Interactive Control De sign Module Figure 12-5. LQG/H-Infinity Frequency Weights W indow Synthesis Modes and Window Usage In addition to the standard LQ G/H ∞ synthesis, any combinatio n of three additional features is supported: • Integral action • Exponenti[...]

Chapter 12 LQG/H-In finity Synthesis Xmath Interactive Control Design Module 12-8 ni.com Opening the LQG/H-Infi nity 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 windo w) and the LQG/H ∞ window is opened, the current controll er is read i[...]

Chapter 12 LQG/H-In finity Synthesis © National Instruments Corporation 12-9 X math Interactive Control De sign Module The weighted output vector z consists of the following: • Filtered inputs ( ) • Plant states ( x p ) • Filtered plant outputs ( ) • Inte grated, filt ered plant outp uts ( y I ) The disturbance input vector ( w ) consists [...]

Chapter 12 LQG/H-In finity Synthesis Xmath Interactive Control Design Module 12-10 ni.com Figure 12-6. LQG/H-Infinity Contro l Design Configuration In the block di agram, σ sens represents a matrix that selects a subset of the set of plant outputs as measurements. Similar ly , σ act selects a subset of the plant inputs as control inpu ts. These s[...]

Chapter 12 LQG/H-In finity Synthesis © National Instruments Corporation 12-11 Xmath Interactive Co ntrol Design M odule The system equations of plant , fi lt ers, and integrators are as follo w s: Standard LQG (All T oggle Buttons “Off”) In LQG synthesis mo de, the controller C minimizes a weighted sum of the steady-state actuator and output v[...]

Chapter 12 LQG/H-In finity Synthesis Xmath Interactive Control Design Module 12-12 ni.com Penalizing the “running int egral” of the plant output forces the po wer spectral density of the plant output t o v anish at zero frequency . In classical control terms, this forces a pole at s = 0 in th e loop transfer function, that is, integral control.[...]

Chapter 12 LQG/H-In finity Synthesis © National Instruments Corporation 12-13 Xmath Interactive Co ntrol Design M odule The transfer functions W u, i and W y, j are the input and output weighting transfer functions, respecti vely . When W u, i =1 a n d W u, j = 1, this reduces to the previously described standard LQG controller . Notice that integ[...]

Chapter 12 LQG/H-In finity Synthesis Xmath Interactive Control Design Module 12-14 ni.com By clicking the button at the bottom of the W eights window , arbitrary weight matrices can be loaded fr om Xmath. The noise v ariances and weights selected in this way are simp ly added to the diagonal weight and noise matrices determ ined by the push buttons[...]

Chapter 12 LQG/H-In finity Synthesis © National Instruments Corporation 12-15 Xmath Interactive Co ntrol Design M odule Here is a square matrix such that and is a square matrix such that The H ∞ solution is defined as the one that minim izes the maximum singul ar value of the transfer function from w n to z n . The only difference in the user in[...]

Chapter 12 LQG/H-In finity Synthesis Xmath Interactive Control Design Module 12-16 ni.com Manipulating the Design Parameters Main Window The design parameters ρ and ν 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 ra nge, the slider range will automatically a[...]

Chapter 12 LQG/H-In finity Synthesis © National Instruments Corporation 12-17 Xmath Interactive Co ntrol Design M odule Ranges To change the ranges of the sliders or plots, select View»Ranges or press <Ctrl-R> in the LQG wind ow. The slider ranges also will be changed automatically if the user ty pes a ne w value which is outside the curren[...]

© National Instruments Corporation 13-1 X math Interactive Control De sign Module 13 Multi-Loop Synthesis This chapter describes multi-loop synthesis. The Multi-Loop windo w is used to synthesize a MIMO controller using PID and Root Locus method s, applying them one-loop-at- a-time. In many practical industr ial applications, this is the way contr[...]

Chapter 13 Multi-Loop Synthesis Xmath Interactive Control Design Module 13-2 ni.com Figure 13-1. Mu lti-Loop Main Window After the Multi-Loop window is opened, two plots are added at the bottom of the ICDM Main wind o w for display of the loop gain magn itude and phase of the control loops that will be synthesized with the Multi-Loop method (refer [...]

Chapter 13 Multi-Lo op Synthesis © National Instruments Corporation 13-3 X math Interactive Control De sign Module 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 synthesi s method for mult i-loop synthesis. Multi-Loop V ersus Mu ltivariable Design In most m[...]

Chapter 13 Multi-Loop Synthesis Xmath Interactive Control Design Module 13-4 ni.com 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 control ler components of l oops that were designed earlier. When the user is designi[...]

Chapter 13 Multi-Lo op Synthesis © National Instruments Corporation 13-5 X math Interactive Control De sign Module Figure 13-4. Multi-Loop Configuration with 3-Sensor and 2-Actuator Plant y 1 C (1) – P u 1 r 1 r 2 r 3 – – u 2 y 2 y 3 e 1 e 2 e 3 C (2)[...]

Chapter 13 Multi-Loop Synthesis Xmath Interactive Control Design Module 13-6 ni.com Figure 13-5. Root Locus Window During the Multi-Loop Design Figure 13-4 sh ows an e xample multiloop co nfiguration for the 3-sensor , 2-actuator plant. There are two loops: one from sensor 1 to act uator 1, and one from senso r 3 to actuator 2. In multiloop design [...]

Chapter 13 Multi-Lo op Synthesis © National Instruments Corporation 13-7 X math Interactive Control De sign Module Opening the Multi-Loop Synthesis Window The multi-loop window can accept any type of MIMO controller and will decompose it into its SISO component s, one for each loop. Control loops are categorized as bein g of type PID or type Ro ot[...]

Chapter 13 Multi-Loop Synthesis Xmath Interactive Control Design Module 13-8 ni.com Editing and Deleting Loops When a loop is hig hlighted, it can be edited, deleted, disabled, or enabled. Here, “editing” means designing a SISO controller for the selected loop. The editing and deleting opti ons are accessible under the Edit pull-down menu. Disa[...]

© National Instruments Corporation A-1 Xm ath Interactive Control Design Modu le A Using an Xmath GUI T ool This appendix describes the basics of using an Xmath GUI tool. Over view ICDM was dev eloped using the programmable Xmath GUI (Graph ical User Interface). Using a graphical tool su ch as ICDM is quite dif ferent from using a toolbox that has[...]

Appendix A Using an Xmath GUI T ool Xmath Interactive Contro l Design Module A-2 ni.com Figure A-1. Programmable GUI Examples Each demo has a Help menu in its menu bar, near the upper right side of the window . The Help messages explain ho w to interact with the demo and what it does. It may be helpful to read the rest of this appendix before (or w[...]

Appendix A Using an Xmath GUI T ool © National Instruments Corporation A-3 Xm ath Interactive Control Design Modu le Figure A-2. Programmable GUI Exam ples Do It Di alog GUI Functions Many functions are controlled by th e left mouse b utton. Fo r example, a button is acti v ated or selected by poin ting at the button and clicking the left mouse bu[...]

Appendix A Using an Xmath GUI T ool Xmath Interactive Contro l Design Module A-4 ni.com •A list is a vertical list of items (strings) that can be selected (highlighted). Dependin g on the appli cation, a list can be conf igured to allow v arious types of selection: – A single-selection list allows only a single line to be selected. Clicking the[...]

Appendix A Using an Xmath GUI T ool © National Instruments Corporation A-5 Xm ath Interactive Control Design Modu le • GUI windows might contain b utton s that display some value. The v alue can be changed by clicking the b utton, whereupon a text entry area will appear in place of the b utton. Y ou can enter a ne w v alue followed by pressing &[...]

Appendix A Using an Xmath GUI T ool Xmath Interactive Contro l Design Module A-6 ni.com A slider might also appear lik e a bar graph. Its tip represents the v alue, but it will be read-only , that is, the user cannot change its value b y dragging the handle. Often a va lue is displayed with a slid er and a variable edit box (refer to for example, t[...]

© National Instruments Corporation B-1 Xm ath Interactive Control Design Modu le B T echnical Support and Professional Ser vices Visit the following sections of the National Instruments Web site at ni.com for technical support an d professional services: • Support —Online technical support resources at ni.com/support include the following: –[...]

© National Instruments Corporation I-1 Xm ath Interactive Control De sign 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 fu nction, 11-[...]

Index Xmath Interactive Contro l Design Module I-2 ni.com 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 Wi ndow, 2-5 History window, 2-5, 2-6 , 2-8, 9-1 I ICDM Help, 1- 5 ICDM Main Window, 2-4, 3-2 elements, 3-1 input-refer red disturb[...]

Index © National Instruments Corporation I-3 Xm ath Interactive Control De sign Module Pole Place Modes, 6-2 Synthesis wi ndow, 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 pol ynomials, 2-2 R Ranges window, 2-11 risk sensitivity, 2-5 robustness analysis, [...]