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Related Manuals for National Instruments Xmath Interactive Control Design Module ICDM
Summary of Contents for National Instruments Xmath Interactive Control Design Module ICDM
- Page 1 NI MATRIXx Xmath Interactive Control Design Module Xmath Interactive Control Design Module April 2007 370754C-01...
- Page 2 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 Instruments Corporation. All rights reserved.
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Page 3: Important Information
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. - Page 4 Conventions 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.
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Page 5: Table Of Contents
What the ICDM Main Window Plots Show ...2-7 Controller/Synthesis Window Compatibilities...2-7 Using ICDM ...2-9 General Plotting Features ...2-11 Ranges of Plots and Sliders...2-11 Zooming ...2-12 Data-Viewing Plots ...2-12 Interactive Plot Re-ranging ...2-13 © National Instruments Corporation Xmath Interactive Control Design Module... - Page 6 Contents Graphically Manipulating Poles and Zeros... 2-13 Editing Poles and Zeros ... 2-13 Editing Poles and Zeros Graphically ... 2-14 Complex Poles and Zeros ... 2-14 Isolated Real Poles and Zeros... 2-14 Nonisolated Real Poles and Zeros and Almost Real Pairs ... 2-14 Adding/Deleting Poles and Zeros...
- Page 7 Synthesis Modes ...7-3 Opening the LQG Synthesis Window ...7-3 Setup and Terminology ...7-4 Standard LQG (All Toggle Buttons Off)...7-4 Integral Action...7-5 Exponential Time Weighting ...7-5 Output Weight Editing ...7-6 State-Space Interpretation ...7-7 © National Instruments Corporation Xmath Interactive Control Design Module Contents...
- Page 8 Contents Manipulating the Design Parameters... 7-7 Manipulating the Design Parameters Graphically ... 7-7 Ranges ... 7-8 Chapter 8 H-Infinity Synthesis H-Infinity Synthesis Window Anatomy... 8-1 Opening the Synthesis Window ... 8-3 Setup and Synthesis Method ... 8-3 Central H-Infinity Controller ... 8-4 Output Weight Editing ...
- Page 9 Standard LQG (All Toggle Buttons “Off”) ...12-11 Integral Action...12-11 Exponential Time Weighting ...12-12 Weight Editing...12-12 How to Select w, u, y, and z...12-13 H-Infinity Solution ...12-14 Manipulating the Design Parameters ...12-16 Main Window...12-16 Ranges ...12-17 © National Instruments Corporation Xmath Interactive Control Design Module Contents...
- Page 10 Contents Chapter 13 Multi-Loop Synthesis Multi-Loop Window Anatomy... 13-1 Setup and Synthesis Method ... 13-3 Multi-Loop Versus Multivariable Design... 13-3 Opening the Multi-Loop Synthesis Window ... 13-7 Designing a Multi-Loop Controller... 13-7 Graphical Editor ... 13-7 Selecting and Deselecting Loops ... 13-7 Editing and Deleting Loops ...
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Page 11: Using This Manual
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,... - Page 12 Chapter 1 Introduction • • • • • • • • • • Xmath Interactive Control Design Module Chapter 5, Root Locus Synthesis, 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 closed-loop poles.
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Page 13: Commonly-Used Nomenclature
• • • • © National Instruments Corporation 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. -
Page 14: Matrixx Help
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. - Page 15 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 Have a user’s understanding of Xmath (enough to create a plant transfer function).
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Page 16: Introduction To Siso Design
Basic SISO Terminology 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. © National Instruments Corporation Design. C(s) –... - Page 17 Chapter 2 Introduction to SISO Design The equations describing this system are as follows: where In ICDM, the plant and controller transfer function are required to be rational, that is, the ratio of two polynomials: where n plant denominator, controller numerator, and controller denominator, respectively.
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Page 18: Overview Of Icdm
• • • • © National Instruments Corporation 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 = n . -
Page 19: Icdm Main Window
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, MIMO forms. -
Page 20: Lqg Synthesis Window
Key Transfer Functions and Data Flow in ICDM ICDM has three key transfer functions: • • • © National Instruments Corporation Synthesis. The plant transfer function P The alternate plant transfer function P The current controller transfer function C Chapter 2... -
Page 21: Summary
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 H . -
Page 22: What The Icdm Main Window Plots Show
There are some restrictions on the controllers that each synthesis window can accept (read): • • © National Instruments Corporation list. The current controller is the active or selected entry on the list of saved controllers. section. 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. - Page 23 Chapter 2 Introduction to SISO Design • • • • 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.
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Page 24: Using Icdm
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 Interactively design a controller. Switch synthesis methods and continue designing. - Page 25 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 P which is automatically exported to the ICDM Main window for analysis and display.
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Page 26: General Plotting Features
The Ranges window can be opened by selecting Ranges on the View or Plot menu, or by pressing <Ctrl-R> in the window in question. In addition, every ICDM © National Instruments Corporation Chapter 2 Figure 2-4. Simple ICDM Session... -
Page 27: Zooming
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. -
Page 28: Interactive Plot Re-Ranging
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 Chapter 2 2-13... -
Page 29: Editing Poles And Zeros Graphically
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. -
Page 30: Adding/Deleting Poles And Zeros
(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... - Page 31 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 <Ctrl-P> 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.
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Page 32: Icdm Main Window
The ICDM Main window, shown in Figure 3-1, consists of the following elements, from top to bottom: • • • © National Instruments Corporation Communicate with Xmath—for example, transfer plants/controllers from/to Xmath Display warning and log messages Display a variety of standard plots... -
Page 33: Communicating With Xmath
Chapter 3 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 A line that identifies the type and source of the current controller. -
Page 34: Most Common Usage
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 Xmath Interactive Control Design Module... -
Page 35: Reading A Controller From Xmath Into Icdm
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. -
Page 36: Icdm Plots
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. © National Instruments Corporation Loop transfer function magnitude Loop transfer function phase... -
Page 37: Ranges Of Plots
Chapter 3 ICDM Main Window 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. -
Page 38: Plot Magnify Windows
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 Xmath Interactive Control Design Module... - Page 39 Chapter 3 ICDM Main Window of Plots 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.
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Page 40: Selecting A Synthesis Or History Window
The Edit menu has two important entries: • • © National Instruments Corporation Selecting EditAdd to History, or typing 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. -
Page 41: Pid Synthesis
Gain Integral (I) Integral time constant Derivative (D) Derivative time constant © National Instruments Corporation Multi-Loop Synthesis. A menu bar with entries Special, Edit, View, and Help. A text area that displays the transfer function of the current PID controller. -
Page 42: Toggling Controller Terms On And Off
Chapter 4 PID Synthesis Table 4-1. PID Controller Terms and Parameters (Continued) Term HF rolloff 1 HF rolloff time 1 HF rolloff 2 HF rolloff time 2 Toggling Controller Terms On and Off For each parameter, the toggle button at the left of the row is used to toggle the terms on and off. - Page 43 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: 1 sT © National Instruments Corporation Xmath Interactive Control Design Module...
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Page 44: Opening The Pid Synthesis Window
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: ICDM enforces a proper controller transfer function, that is, a finite high frequency gain. -
Page 45: Time Versus Frequency Parameters
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 Chapter 4 section of Chapter 2, Introduction to SISO . -
Page 46: Derivative Term Normalization
Chapter 4 PID Synthesis Derivative Term Normalization The derivative term is low-frequency normalized, which means that at low frequencies (below 1/T 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, so static tracking, static actuator effort, and so on are not affected by the derivative term. -
Page 47: Root Locus Synthesis
The closed-loop poles, which are on the locus, also are shown. © National Instruments Corporation Displays selected gain and phase contours in the complex plane of the loop transfer function. - Page 48 Chapter 5 Root Locus Synthesis The Root Locus Synthesis window consists of, from top to bottom: • • • • Xmath Interactive Control Design Module Figure 5-1. Root Locus Synthesis Window A menu bar with entries Special, Edit, View, and Help. A slider and variable edit box for the gain.
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Page 49: Opening The Root Locus Synthesis Window
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 Edit menu or by typing the accelerators in the Root Locus window. A more detailed description appears following. -
Page 50: Plotting Styles
Chapter 5 Root Locus Synthesis 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 one of many possible plotting styles. In all cases, the (open-loop) controller and plant poles and zeros are shown on the plot. -
Page 51: Phase Contours
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 The plot shows the locus of points where the phase angle of the loop transfer function is 180 . -
Page 52: Slider And Plot Ranges
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 one of the plots. This allows you to find the gain associated with a particular point on a phase contour, for example. -
Page 53: Design
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 Chapter 5 Xmath Interactive Control Design Module... -
Page 54: Interpreting The Nonstandard Contour Plots
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. - Page 55 Chapter 5 Root Locus Synthesis Figure 5-3. Root Locus Synthesis Window with the 0 dB Magnitude Contour © National Instruments Corporation Xmath Interactive Control Design Module...
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Page 56: Pole Place Synthesis
• • • • • © National Instruments Corporation Normal mode (integral action not enforced) Integral action mode A menu bar with entries Special, Edit, View, and Help. A toggle button used to set normal or integral action mode. A slider and variable-edit box used to time or frequency-scale the closed-loop poles. -
Page 57: Pole Place Modes
Chapter 6 Pole Place Synthesis Pole Place Modes In Pole Place, the user selects either closed-loop poles (in normal mode) or 2n + 1 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. -
Page 58: Normal Mode
In normal mode, the controller transfer function has order n and is strictly proper: where Therefore, the closed-loop characteristic polynomial has degree 2n: where © National Instruments Corporation n–1 (s) = s (s) = b C(s) = n (s)/d n–1... -
Page 59: Integral Action Mode
Chapter 6 Pole Place Synthesis We can write this polynomial equation as follows: These 2n linear equations are solved to find the 2n controller parameters , ..., x 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-loop poles. -
Page 60: State-Space Interpretation
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 F dynamics. © National Instruments Corporation Graphically Manipulating Poles and Zeros Introduction to SISO Design, for a general discussion of , ...,... -
Page 61: Butterworth Configuration
Chapter 6 Pole Place Synthesis A circle of radius F circle to change F Butterworth Configuration Click the Butterworth button to move the poles to a Butterworth configuration, preserving F set to Butterworth. Editing the Closed-Loop Poles You can change the closed-loop poles two ways: by editing or by grabbing and dragging them. -
Page 62: Lqg Synthesis
• • These parameters are described in greater detail later in this chapter. © National Instruments Corporation A menu bar with entries Special, Edit, View, and Help. A message area that describes the synthesis mode (type of controller); for example,... - Page 63 Chapter 7 LQG Synthesis • • Xmath Interactive Control Design Module Figure 7-1. LQG Synthesis Window A control panel used to graphically edit the output weight transfer function. A plotting area that contains the following plots: – The symmetric root locus plots of the control and estimator closed-loop poles.
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Page 64: Synthesis Modes
LQG window has not been opened in this ICDM session. © National Instruments Corporation If the decay rate is enabled, it is shown as a vertical line that can be dragged. -
Page 65: Setup And Terminology
Chapter 7 LQG Synthesis Setup and Terminology The different modes are described using the following basic terminology: Figure 7-1 shows a block diagram with the basic setup for LQG synthesis, where The noises w spectral densities (PSDs). The parameter to the PSD of w C(s) Standard LQG (All Toggle Buttons Off) In LQG synthesis mode, the controller minimizes a weighted sum of the... -
Page 66: Integral Action
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 ∫ Xmath Interactive Control Design Module... -
Page 67: Output Weight Editing
Chapter 7 LQG Synthesis Output Weight Editing When Weight Zero Edit is enabled, the LQG controller is based on y ˜ integral action, the controller minimizes the quantity: and with integral action, the quantity: where The transfer function W is the output weighting transfer function. When W = 1, this reduces to the standard LQG controller described previously. -
Page 68: State-Space Interpretation
• • • © National Instruments Corporation and a can be manipulated using the sliders or variable 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 or appropriately. -
Page 69: Ranges
Chapter 7 LQG Synthesis Ranges To change the ranges of the sliders or plots, select View»Ranges or enter 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. -
Page 70: H-Infinity Synthesis
The H 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 ( ) –... - Page 71 Chapter 8 H-Infinity Synthesis Figure 8-1. H-Infinity Synthesis Window Xmath Interactive Control Design Module ni.com...
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Page 72: Opening The Synthesis Window
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: where © National Instruments Corporation y is the plant output signal is a normalized (input-referred process) noise is a normalized (sensor) noise... -
Page 73: Central H-Infinity Controller
Chapter 8 H-Infinity Synthesis C(s) 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 and w The design is based on H, the closed-loop transfer matrix relating the noises 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. -
Page 74: Output Weight Editing
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 (unweighted) design. © National Instruments Corporation Chapter 8 becomes close to , where the H -norm is... -
Page 75: Manipulating The Design Parameters
Chapter 8 H-Infinity Synthesis Manipulating the Design Parameters The parameters , , and 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. -
Page 76: Ranges
Introduction to SISO Selecting View»Auto-scale or pressing <Ctrl-A> in the H window will cause new ranges to be assigned to the sliders and plots, based on the current controller. © National Instruments Corporation Chapter 8 Interactive Plot Re-ranging Design. Xmath Interactive Control Design Module... -
Page 77: History Window
The History window is shown in Figure 9-1. From top to bottom, it consists of: • • © National Instruments Corporation Introduction to MIMO A menu bar with Special, Edit, and Help menus. A scrolled list that shows the designs saved on the history list. Columns... -
Page 78: Selecting The Active Controller
Chapter 9 History Window • • 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. -
Page 79: Deleting History List Entries
ICDM, and then save to Xmath by selecting File»Write Controller in the ICDM Main window. © National Instruments Corporation Using an Xmath GUI Synthesis button at the bottom of the History window. This does two... -
Page 80: Using The History List
Chapter 9 History Window Using the History List 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. -
Page 81: Alternate Plant Window
You always can use data-viewing to determine which plot corresponds to the plant and which corresponds to the alternate plant. © National Instruments Corporation Introduction to MIMO 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. -
Page 82: Alternate Plant Window Anatomy
Chapter 10 Alternate Plant Window Alternate Plant Window Anatomy The Alternate Plant window is shown in Figure 10-1. From top to bottom, it consists of: • • • • • • • Xmath Interactive Control Design Module 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. -
Page 83: Opening The Alternate Plant Window
Using the Special menu, you can read the plant from ICDM or any transfer function from Xmath into the alternate plant. © National Instruments Corporation Chapter 10 Figure 10-1. Alternate Plant Window... -
Page 84: Normalization
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: where K is the gain (shown in the slider and Variable Edit box), are the zeros, and plant is required to be proper, that is, have at least as many poles as zeros For high-frequency normalization there is no restriction on the poles or... -
Page 85: Using The Alternate Plant Window
• • • © National Instruments Corporation = P and then adding a pole-zero pair is a good way to see 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. -
Page 86: Ranges Of Sliders And Plot
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 <Ctrl-R> 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. -
Page 87: Introduction To Mimo Design
The equations describing this system are: 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 Synthesis. , that subtracts from the actuator signal. This signal P u d –... -
Page 88: Transfer Functions
Chapter 11 Introduction to MIMO Design u denotes the plant input or actuator signal, 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 n P denotes the plant transfer function, which is a matrix of size n C denotes the controller transfer function, which is a matrix of size n Figure 11-1 shows standard feedback connections and signals used in... - Page 89 • • • • • © National Instruments Corporation 2 block matrix that relates the three output vector signals to the two 1 – PC I to e, u, and y, have standard names and interpretations (which agree The sensitivity transfer function is denoted S and given by –1...
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Page 90: Integral Action
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 In addition to these transfer functions you encounter two (complementary) open-loop transfer functions:... -
Page 91: Overview Of Icdm For Mimo Design
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 ICDM Main window LQG/H window... -
Page 92: Mimo Plot Window
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 selected. MIMO Plot Window For a more detailed MIMO transfer function plot, an option labeled MIMO plot is available under the Main window menu bar. -
Page 93: History Window
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 Chapter 11 Figure 11-3. Complete Set of Plot Choices... - Page 94 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.
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Page 95: Lqg/H-Infinity Synthesis
• • • © National Instruments Corporation A menu bar with entries Special, Edit, View, and Help. A message area that describes the synthesis mode (type of controller), for example, LQG with integral action. A control panel for changing the five design parameters: –... -
Page 96: Lqg/H-Infinity Weights Window
Chapter 12 LQG/H-Infinity Synthesis • • These parameters are described in greater detail later in this chapter. LQG/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: •... - Page 97 • • © National Instruments Corporation descriptions are for the control cost parameter display. The noise level display is similar in appearance. A table with n rows, having in each row: – A toggle button to include the input in the set of control inputs –...
- Page 98 Chapter 12 LQG/H-Infinity Synthesis • • • The control weight parameter in the main LQG/H window is related to the weights in this window by 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.
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Page 99: Decay Rate Window
• H-Infinity Performance Window The H Performance window, shown in Figure 12-4, consists of, from top to bottom: • • © National Instruments Corporation , and are then replaced with noise variances u, i y, j , and . The noise level parameter in the main LQG/H y, j A menu bar with entries Special, Edit, View, and Help. -
Page 100: Frequency Weights Window
Chapter 12 LQG/H-Infinity Synthesis Frequency Weights Window The Frequency Weights window is shown in Figure bottom, it consists of: • • Xmath Interactive Control Design Module value. If a lower bound on the minimal value of is known, it also is displayed. -
Page 101: Synthesis Modes And Window Usage
The Weight Edit toggle button enables and disables output weight editing. The Hinf Bound toggle button enables and disables design mode. © National Instruments Corporation Figure 12-5. LQG/H-Infinity Frequency Weights Window Integral action Exponential time weighting (guaranteed decay rate). This feature is only enabled in the case of LQG design. -
Page 102: Opening The Lqg/H-Infinity Synthesis Window
Chapter 12 LQG/H-Infinity Synthesis Opening the LQG/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/H window is opened, the current controller is read into the LQG/H window;... - Page 103 • • • The measured output consists of the following: • • • © National Instruments Corporation u ˜ Filtered inputs ( ) Plant states (x y ˜ Filtered plant outputs ( ) Integrated, filtered plant outputs (y General LQG state disturbances (w...
- Page 104 Chapter 12 LQG/H-Infinity Synthesis In the block diagram, set of plant outputs as measurements. Similarly, 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.
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Page 105: Standard Lqg (All Toggle Buttons "Off")
E denotes expectation. Integral Action When Integral action is enabled, the controller minimizes a variation on the LQG cost: ⎛ ⎜ ⎜ ⎝ where © National Instruments Corporation ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ ------- -... -
Page 106: Exponential Time Weighting
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. -
Page 107: How To Select W, U, Y, And Z
(w), actuators (u weighted outputs (z). • • • © National Instruments Corporation and W u, i y, j 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. -
Page 108: H-Infinity Solution
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. - Page 109 Therefore, a better control performance is often obtained by trying to lower the H norm not to its absolute minimum, but rather to a slightly larger value. © National Instruments Corporation -- - is a square matrix such that -- -...
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Page 110: Manipulating The Design Parameters
Chapter 12 LQG/H-Infinity Synthesis Manipulating the Design Parameters Main Window The design parameters and 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 the and sliders are moved. -
Page 111: Ranges
Introduction to SISO Selecting View»Auto-Scale or pressing <Ctrl-A> in the LQG window causes new ranges to be assigned to the sliders and plots, based on the current controller. © National Instruments Corporation Chapter 12 Interactive Plot Re-ranging Design. 12-17... -
Page 112: Multi-Loop Synthesis
• • • • © National Instruments Corporation A menu bar with entries Special, Edit, and Help. A plot area where SISO control loops can be created graphically. We will call this area the graphical editor. A scrolled list of loop names, with actuator and output labels. - Page 113 Chapter 13 Multi-Loop Synthesis 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).
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Page 114: Setup And Synthesis Method
In general, all components of the resulting controller transfer function will be nonzero. The multi-loop synthesis method allows the user to close © National Instruments Corporation Chapter 13 13-3... - Page 115 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.
- Page 116 – – – © National Instruments Corporation Figure 13-4. Multi-Loop Configuration with 3-Sensor and 2-Actuator Plant 13-5 Chapter 13 Multi-Loop Synthesis Xmath Interactive Control Design Module...
- Page 117 Chapter 13 Multi-Loop Synthesis Figure 13-4 shows an example multiloop configuration for the 3-sensor, 2-actuator plant. There are two loops: one from sensor 1 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.
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Page 118: Opening The Multi-Loop Synthesis Window
When a loop is selected, it is indicated in the graphical editor by a thick line. © National Instruments Corporation By clicking a box in the column to the left (controller inputs), then clicking a box in the column to the right. -
Page 119: Editing And Deleting Loops
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. - Page 120 You can run several demos simultaneously. Note © National Instruments Corporation Select a demo (for example, Variable Binding). Click OK. from the...
- Page 121 Appendix A Using an Xmath GUI Tool Each demo has a Help menu in its menu bar, near 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.
- Page 122 • • • © National Instruments Corporation Figure A-2. Programmable GUI Examples Do It Dialog 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.
- Page 123 Appendix A Using an Xmath GUI Tool • • • Xmath Interactive Control Design Module 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: –...
- Page 124 • • • © National Instruments Corporation 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 <Return>.
- Page 125 Appendix A Using an Xmath GUI Tool • Xmath Interactive Control Design Module 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.
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Page 126: Technical Support And Professional Services
Technical Support and Professional Services Visit the following sections of the National Instruments Web site at ni.com • • • If you searched your local office or NI corporate headquarters. Phone numbers for our worldwide offices are listed at the front of this manual. You also can visit... - Page 127 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...
- Page 128 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, NI support and services, B-1 noise power, 2-5 nomenclature, 1-3 PID synthesis window, 2-4, 2-7 plant degree, 2-2...
- Page 129 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 software (NI resources), B-1 step response, 2-3 plot, 2-7 support, technical, B-1 system object, 2-1 technical support, B-1...
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