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Program CC USER GUIDE

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1. S 5 1 4142136 2 1 4142136 2 CC gt ilt b b t 1 1 6282743 sin 1 4142136t 0 6435083 exp 1 4142136t exp 20t for t gt 0 0 0230261 The Feedback Operator Definition The feedback operator is defined as gih g 1 h g The feedback operator returns the closed loop system for the following block diagram The variables g and h can be of any compatible type and the dimensions of g and h must be the same Unity Feedback For unity feedback systems y compute the closed loop system by gli If g is multivariable then it must be square and the unity gain in the feedback path expands to the correct size identity matrix Precedence The feedback operator has higher precedence than multiplication so the closed loop response of must be computed using parentheses g k h Note that for multivariable systems g k and k g are quite a bit different Transfer Functions For transfer functions g ng dg and h nh dh the feedback operator returns n d g h n d gdp Due to cancellations that take place during transfer function algebra the same result is returned whether g h or g 1 h g is used The feedback operator is convenient but not necessary In the following example a closed loop system is computed and displayed CC gt g 10 5 1 5 5 2 CC gt h 3 5 4 CC cl 2g h CC pzf cl 10 s 1 5 4 s 0 898 s 2 551 2 5 188 2 Quadruples Th
2. foo a b returns a b c at hb Enter this function using Program CC s text window or any other text editor and store in an ASCII file named foo mac somewhere on Program CC s path The first set of lines starting with are echoed in response to help foo Call this function at the command line by CC gt foo 2 3 Variables defined inside the function are local to that function unless explicitly declared inside the function as global Functions and script files can be nested to any level User defined unctions when called are compiled and the compiled version is executed and then saved for possible later use Script files A script file does not have a function declaration and hence does not have any input or output arguments All of the variables in the script file are at the same level as the calling expression This means all of the currently defined variables at the command level are available inside a script file if called from the command level and any variables defined in the script file are available after returning from the script file Programming Program CC uses a high level language both on the command line and in user defined functions and scripts The language contains expressions of the form lt lhs gt lt rhs gt lt rhs gt Declaration and typing of all the variables is done at run time It is this late typing that makes the high level language so powerful and convenient The price to b
3. ylim 20 10 sets y axis limits The labels title and text were added using the plot dialog box The pieces of the text are leftarrow Top to bottom n zeta 2 4 6 8 1 g s 1Nover Nzeta 1 To enter the last one for example using the text function look at the location in the plot change dialog box and then recreate the text by text g s lNover Nzeta 1 16 10 7 fontsize 12 Partial Fraction Expansions Two Real Poles Partial fraction expansions pfe s are used so that high order transfer functions can be decomposed into terms with just one pole each Usually this is done so that the inverse Laplace transform can be applied to each term It is also a first step for hold equivalence discretization methods Enter a second order transfer function CC gt g 4 s 1 s 2 CC gt display g And then compute the pfe and the ilt inverse Laplace transform CC gt pfe g 4 4 GNS S SSH s 1 s 2 CC ilt 9 g t 4 exp t 4 exp 2t for t gt 0 By the way when copying pfe s to a text editor use the Courier New font so that the spacing and lines are all correct The general case for pfe s is rather tedious but for a strictly proper transfer function more poles than zeros with no multiple poles the pfe is n 1L where p g s __ 5 i Pi 8 9 _ 5 You could if you wanted to compute the residues the ai s CC gt substitute s
4. 3 3 CC gt gl bessel 3 3 CC gt g2 chebyshev 3 3 3 A Bode plot of the Butterworth filter is created by CC gt bode g which should look something like Bode plot Magnitude dB Magnitude Phase degrees 10 Freq rad sec Plot Dialog Box The plot dialog box was used to add the title and the text next to each line and to increase the size of the font Popup this dialog box by double clicking on the plot or using the construction toolbar icon or using the menu or using the right mouse click popup menu The plot dialog box is shown below with the y1 axis the magnitude axis change options Plot options X aaa a a Magnitude dB s s s s s Try the following change the dB range from 60 to 20 dB push Apply to make the change and leave the dialog box open click the Title radiobutton and add the title Bode plot click the Font radiobutton and change the font size to 14pt click the Text radiobutton and add the text strings Magnitude and Phase close the dialog box and move the text by clicking and dragging copy the plot to the clipboard using the copy icon on the toolbar Frequency Response Comparison Now compare the frequency response of all three filters CC bode g gl 92 When comparing more than one Bode plot it is best to compare just the magnitude Use the Bode radiobutton on the plot dialog box to make this change Also try to
5. 5 See also sim time timevec Printing and Copying Hardcopy output can be created by printing and copying Copying and pasting uses the standard Windows hotkeys When copying a selection of text the Courier New font is recommended This is the font used in the command window and it is recommended because it is non proportional needed so that matrix and transfer function displays line up and because it correctly displays the line used as part of the transfer function displays Character 227 is used for this line which is not always a line in other font selections When copying a plot a bitmap copy is used The resolution defaults to the same as used for the screen 72 dpi appropriate for documents such as this help file that are designed to be displayed on a computer screen Double and quadruple resolution can also be used for copying respectively 150 and 300 dpi The resolution can be selected using the Copy Option menu item or by the copy and copyoption functions The print menu item and the print function can be used to print a figure The default for the print function is a full page portrait print of the current figure preserving the screen aspect ratio A figure can be created with a mixture of plots and or text and then printed as a single unit on the printer The location of the figure on the printer page can be set using options included with either the plot plotoption or print functions For example the figure can b
6. The dimensions of x and y must be equal x1 x 2 x 4 6 x plot xl y xlim 1 7 ylim 1 1 0 5 0 5 1 1 0 1 2 3 4 5 6 7 For plot x y if x and or y is complex then the real part is used plot y s Changes one or more of the line style line color symbol style line width and symbol height according to the code y yellow solid line point m magenta dashed line circle i dotted line x x mark r red dash dot line plus g green long dashed star b blue dash dot dot box w white S stairsteps t triangle k black spikes lt gt diamond la light gray a gray In addition dy dm dc dr dg db and da are dark colors lwn where n 1 specifies the line width default 1 shn where n 1 specifies the symbol height default 6 Here s an example seed 1 y rand 1 20 plot y r 1w2 sh12 1 0 8 0 6 0 4 0 2 The example specified x red solid line 1w2 line width 2 boxes sh12 symbol height 12 The plot change dialog box line option can be used to made the same changes Log10 Scales The x and or y axes can be plotted using log10 scales To plot the y axis with a log10 scale semilogy y po 10 5 0 5 10 15 20 To plot the x axis with a log10 scale semilogx y And to plot both axes with log10 scales loglog y Cursors Axis Cursor Create a plot of random numbers CC gt rand seed 1 CC gt p
7. ilt g roc 1 5 g t 4 exp 2t for t gt 0 4 exp t fort lt 0 CC gt t 3 05 3 CC y ilt g t roc 1 5 Non causal Impulse Response US ead s 1 s 2 git 4 exp C20 fort gt 0 g t A expct fort 0 The Autocorrelation and Variance Now take the same transfer function with input u s and output y s g s If the input is zero mean unit intensity white noise the power spectral density of the output is 9 5 0 computed as follows CC yd g g s The stable ilt is the autocorrelation The autocorrelation evaluated at t 0 is variance of the output stochastic process CC gt ilt yd 0 stable ans 1 3333333 This is done more simply by CC gt var g ans 1 3333333 And for the standard deviation CC std g ans 1 1547005 A plot of the autocorrelation is created as follows CC gt t 3 05 3 CC gt y ilt stable CC gt plot t y Autocorrelation variance 1 33 Classical Control Example When Delay is a Problem The Controlled Element The controlled element is a 3rd order Butterworth filter with a 0 5 second delay Enter this system as follows CC gt g butter 3 3 pade 5 3 CC gt format mini CC gt sho g 27 9 289 0 724 10 17 0 5 3 3 9 289 0 724 10 17 A 3rd order Pade approximation of the delay is used The format mini command is used to display just 4 signficant digits The system g s is displayed using the
8. 1 ans 0 9241708 0 3801822j CC gt gk gk abs gk 3 1 CC gt k gk g CC gt k Compute the closed loop response by using the feedback operator gt 1 1 Look at the Bode plot of the loop transfer function gk s and the closed loop step response CC gt subplot 121 bode gk subplot 122 time cl Loop transfer function Closed loop unit step response 0 2 a 2 E 20 90 45 a o 0 180 1 20 270 0 5 40 S 2200 0 10 10 10 310 10 0 5 10 15 20 Freq rad sec Time The subplot function is used to place the plots side by side in the same window a 1 by 2 set of tiles with the plots at index 1 and then 2 Robustness Margins The classical robustness margins are CC margin gk At w 1 r s Phase margin 22 36 deg Delay margin 0 39 sec At w 1 2 r s Mp 3 89 11 79 dB At w 1 32 r s Gain margin 1 32 2 42 dB The cursor on the Bode plot can also be used to determine the robustness margins or more conveniently using the Robustness item on the right mouse click popup menu The robustness margins are not part of the specification but the closed loop step response obviously does not come close to the 3 second settling time specification Second Try The Internal Model Control Problem The combined phase drop of the Butterworth filter and the delay make this a tricky closed loop design Another approach is to select a desired closed loop response and then compute a compensator that comes clos
9. 1 g 1 ans 4 CC gt substitute s 2 g 2 ans 4 Complex Poles The above formula is valid for complex poles in which case the poles and residues are complex and occur in complex conjugate pairs Try the following CC gt g 4 5 1 s 2 2 s 5 CC gt pte g al s 1 AAA s 1 5 1 2 2 2 CC gt ilt 9 g t exp t cos 2t exp t for t gt 0 The complex pole is at 1 2j displayed as s Re 2 Im 2 The pfe command combines the complex conjugate terms so that the display only has real numbers Multiple Poles Now try a case with multiple poles CC gt g 4 s 1 5 2 2 CC gt pfe 9 CC ilt 9 g t 4 exp t 4t 4 exp 2t for t gt 0 The residue calculations use derivatives with respect to s and the ilt has a polynomial in t and it s best just to let Program CC swet the details The general case is described in the reference manual Extracting Parts of the PFE You can use the partial function to extract parts of the pfe The stable flag keeps poles with real parts less than the boundary and the unstable flag keeps poles to the right CC gt partial g stable 1 5 5 2 2 CC gt partial g unstable 1 5 The partial function defaults to stable with a boundary of zero which is the stable partial factor The pfe2vec function converts the coefficients of the pfe to a vector but the description is complicated and we will l
10. number of states is specified gt 0 1 0 0 b 0 1 c 0 1 d 0 CC gt p a b c d CC gt p nstates a nrows Or more simply CC gt p pack a b c d State space quadruples can also represent a vector difference equation k 1 Axk Bug CX Duk digital quadruple is automatically created by functions such as convert or is explicitly specified by CC p mode digital CC p delta 1 40 String variables are defined by characters enclosed in single quotes CC str string variable Two single quotes inside a string is interpreted as a single quote Vector string variables can have ragged lengths and are created like rows in a matrix using semicolons CC gt vstr first row second row Operators Program CC has a rich variety of operators arranged below in groups Arithmetic Addition subtraction and multiplication operations both left and right division the feedback operator and exponentiation pos E M oup ow The top part of the left or right slash points to the variable to be inverted The feedback operator is defined by a b a eye b a A matrix plus or minus a number adds or subtracts the number from each matrix element This is different from Program CC Version 4 A matrix plus or minus a row or column vector adds or subtracts the vector from each row or column Dot arithmetic The dotted versions of the arithmetic operations work element by element when applied to matrices
11. shorthand form whereby a equals sa and zeta omega equals s 2 2 zeta omega s omega 2 Requirements The objective is to design a control system that rejects constant output disturbances and the step response settles to 2 of the final value within 3 seconds with an overshoot less than 10 Use the following control architecture Open Loop Analysis First look at the open loop step response CC time g 12 Open Loop Unit Step Reponse AN W 0 8 0 6 0 4 0 2 0 0 2 0 1 2 3 4 5 Time 2 12 Real 1 081 The trace cursor is used to determine the overshoot of 8 1 and the settling time of 2 7 seconds The open loop response meets the specfication except for the output disturbance rejection and indeed the output disturbance specification is the only reason to use feedback at all The slight bobble in the first half second is due to the use of a Pade approximation for the delay The bobble is more obvious if you use the zoom cursor to blow up this region Ctrl Z click and drag First Try Integral Control Classical control design uses guided trial and error An integral control is needed to reject a constant output disturbance and the closed loop bandwidth should be about 1 3 the required rise time or about 1 rad sec Set the controller equal an integrator and then adjust the gain so that the crossover frequency of gk s is 1 rad sec The following commands do so algebraically CC gt gk g s CC gt gk 3
12. the response window Save Save the file Run Execute the function or script Run What Enter the command used for testing The syntax check results are displayed in a response window that is attached to the bottom of the text editor window The menu items from Again to Run What are also available as buttons on the response window Source lines in the response window with syntax errors can be double clicked to jump to the corresponding lines for changes The command string selected by Run what is echoed to the command window and the text results if any are displayed in the command window The preliminary release for Program CC also contains theSyntax menu items Tokens and Parse The Tokens menu item converts the ascii text to the token stream used by the parser Parse shows the pseudo commands output by the parser that actually get executed These features are interesting but really only useful for debugging the parser Directories The Program CC path starts with the current working directory and then proceeds in the order listed by the path command The current working directory is set by the cd command CC cd c ccv5 functions The cd function with no parameter and the pwd function display the current working directory The following versions of the path command displays sets and expands the path CC gt path CC gt path c project1 CC path path c project2 The currently defined variables are saved and
13. with the commands CC gt g 50 5 2 s s 10 CC gt g 50 s 2 s s 10 The frequency response is g s evaluated for s on the j omega axis The Bode plot is the magnitude and phase of the frequency response plotted using log10 and dB axes Type the following to create a Bode plot with automatic scaling CC gt bode g 40 tn tn a Z 5 S 20 a 0 20 40 10 Freq radisec Making Changes A right mouse click on the plot pops up the following menu Select Asymptotes to put magnitude asymptotes on the plot Popup the plot change dialog box in any of the following ways Select Change using the right mouse click popup menu Select Change using the menu bar Plot menu Double click anywhere on the plot Use the Construction toolbar icon Change to the title screen shown below by clicking the Title radio button Enter the title in the edit box change the font if desired and then press the Apply button E Plot options X Bode Plot s g e s e s Change to the text screen shown below by pushing the Text radio button Put each text entry the plot using the Apply push button the text is placed by default in the middle of the plot Close or Cancel out of the plot change dialog box and then move the text by clicking and dragging with the mouse or using the arrow keys Plot options 9 86664 Midd
14. 39 sec At w 1 2 r s Mp 3 89 11 79 dB At w 1 32 r s Gain margin 1 32 2 42 dB The cursor on the Bode plot can also be used to determine the robustness margins or more conveniently using the Robustness item on the right mouse click popup menu The robustness margins are not part of the specification but the closed loop step response obviously does not come close to the 3 second settling time specification Second Try The Internal Model Control Problem The combined phase drop of the Butterworth filter and the delay make this a tricky closed loop design Another approach is to select a desired closed loop response and then compute a compensator that comes close to this desired response while maintaining closed loop stability The internal model control problem does this as long as the controlled element is stable CC gt desired bessel 1 3 CC gt k imc g desired CC gt sho k 0 5561 3 0 5 3 9 289 0 0 724 10 17 0 0 437 3 996 0 693 10 86 11 46 The compensator is 6th order rather high but this level of complexity is what it takes to meet the specifications Compute the loop transfer function LTF and the closed loop response CC gt gk g k CC gt cl gk 1 CC gt sho cl 15 59 289y 0 07245 10 17 2 322 0 724 2 542 9 289 0 724 10 17 The closed loop system is stable The Bode plot of gk s and the unit step response of cl s are shown below CC gt subplot 121 bode gk red
15. Block matrices can be combined using the same operators The matrix is complex if any of the elements is a complex number Matrices defined using parenthese and braces will zero fill as needed Matrices defined using square brackets require the number of rows or columns of adjacent blocks to be the same which is sometimes safer Square brackets also allow spaces to be used in place or commas and carriage returns in place a semicolons CC d 1 2 0 More 3 4 0 More gt 0 0 5 The number of rows and columns in a matrix can be determined using the size function or by using structure members a nrows and a ncols Transfer functions are entered using the transform variable CC gt g 5 s 5 The variable s is by default equal to the Laplace transform variable and if overwritten can be reset using s Laplace Z transforms are entered using the variable z which is reset if needed using z ztransform The sample period of a z transform is undefined unless specified as part of a function such as convert or unless explicity specified CC gd 5 2 1 z 8 CC gd delta 1 40 Transfer function matrices are created whenever matrix elements are transfer functions CC h g 1 0 1 s State space quadruples contain the four matrices used to define the the vector differential equation Ax Bu DU The matrices have compatible dimensions and can be combined into a block matrix but this block matrix is not a quadruple until the
16. Now try a slightly more involved example to enter 10s 1 s s 2 000 s 107 g1 s 7 type the following CC gt g1 10 5 1 s s 2 2 1 10 s 10 2 CC gl S s 2 25 100 Use the Courier New Font By the way when copying transfer functions from the command window into documents use the Courier New font so that the dividing line is an actual line and not the strange symbol used by the Courier and Fixedsys fonts Polynomials Polynomials are treated as transfer functions with a unit denominator gt 5 3 3 5 2 10 5 100 p s s 3 35 2 105 100 Filters Several functions are available to generate different types of transfer functions Try few of the following CC gt butter 2 2 CC chebyshev 2 2 2 CC gt bessel 2 2 CC gt itae 2 2 CC gt pade 1 2 CC gt leadlag 2 30 CC integrator eCConotch 2 17 5 CC gt onepole 2 CC gt twopoles 7 2 CC gt onezero 2 CC gt twozeros 7 2 For help on any of these CC gt help butter or CC gt winhelp butter or if just the name is entered then a dialog can be used to enter the parameters CC gt butter Transfer Function Displays Transfer functions can be displayed in several different ways as suggested by the names of the following function Try the following without the comments CC gt display displays coefficients as they are stored CC gt shorthand or sho the shorthand form CC gt single b multiplies polyno
17. Program CC USER GUIDE Version 5 0 01 01 2002 Copyright c Systems Technology Inc and Peter M Thompson Material seleccionado para el curso Control de Procesos de la Facultad de Ciencias Exactas y Tecnolog a de la U N T Who Who should use Program CC Program CC is designed for students engineers and consultants who occasionally or frequently perform linear system analysis control system analysis and control system design We do not want to make the entrance requirements too high but users should at least know what a matrix is and not be surpised that functions of time can be converted to and from the frequency domain The required background material is taught in courses about differential equations linear algebra complex numbers control theory and linear systems theory Program CC is an excellent complement to any of these courses The building blocks of the program are real and complex numbers and matrices Laplace transforms z transforms Fourier transforms and vector differential equations The Student Version is designed primarily for education Limits are placed on the system order 20th and on the total amount of data storage 64 Kbtyes A matrix of size 90x90 will use up the available space The Professional Version removes these limitation and provides user support The available commands are the same What What does Program CC do Program CC is used for matrix analysis transfer function analysis and s
18. Ro qu a X x al i Assignment Assignment is treated as an operation so that multiple assignments can be made for example a b c 1 As in C assignment operations can be used whereby a b is the same as a op b Unitary The unitary operators are plus and negation not factorial only for positive integers hermitian and transpose te Sx See Relational and logic operators These are most often used in conditional statements The C notation is used for or and and amp amp S gt lt lt amp amp Matrix building operators Row column and diagonal block augmentation are denoted respectively by and space lt ret gt Matrices can be surrounded by or but they work differently Matrices surrounded by square backets can use space and lt ret gt respectively for row and column augmentation and the number of rows or columns for augmented blocks must be equal Matrices surrounded by parentheses or braces cannot use space and lt ret gt and use zero fill as needed when combining blocks Colon The colon operator i j expands to a row vector i i 1 i 2 j and I j k expands to i i j i 2 j Parentheses Parentheses square brackets and braces must be used in pairs and have the same definitions except that matrices built with square brackets have different rules tJ Ld Punctuation Statements are ended
19. STATE and DATA Now there is only one command level In Version 4 the programs are called macros Now the programs are called functions and scripts A macro with no arguments is equivalent to a script There is no equivalent to a macro with arguments Despite this difference the same file type mac is used in Version 5 In a Version 4 macro with or without arguments all of the variables used in the macro are global In a Version 5 function the arguments and variables are local and in a Version 5 script file the variables are global A Version 4 macro with arguments has the syntax foo a b c d This is now replaced by the syntax a b foo c d The at sign is no longer used and a distinction is made between input and output arguments In Version 4 the macro arguments are embedded in the programming text with the symbols amp 1 82 amp n and are replaced at run time using ASCII substitution This syntax is no longer used If ASCII substitution is needed then the eval and feval functions can be used In Version 4 command arguments that expect text replies the text does not need to be surrounded by single quotes for example bode redo Now all text arguments are treated as string variables that must be surrounded by single quotes for example bode g redo In Version 4 the statements are terminated by amp or ret In Version 5 the semicolon can also be used In Versio
20. The result is 0 5 10 15 20 The buttons on the Lines page can be used to delete an existing line select an existing line or create a new line Using the hold function Hold the current plot so that the next plot command will add a new line to the existing axis CC gt hold on CC gt plot 2 2 rand 1 20 Clear the hold by CC gt hold off Using the line function The line function adds a line to the current plot CC gt line 6 2 rand 1 20 Using the plotoption function Any of the plot options and parameters can be changed using the plotoption function A new line is entered by CC gt plotoption addline 2 rand 1 20 lt gt Multiple Plots Multiple plots are created in the following ways 1 The figure function is used to put the next plot in an existing or new window 2 The subplot function is used to divide a figure into tiles and to put the next plot in one of the tiles 3 The stripchart function plots vector time responses on a set of axes Figure Two plots in separate windows are created by Program CC 15 XI File Edit Functions Window Help ran Plot View 1 liol Plot View 2 gt Command Window CC rand seed l CC plot rand 1 20 CC gt figure CC plot rand 1 50 CC The function figure is echoed to the screen by the toolbar icon The above tiling of the windows is created using the toolbar icon B For an a
21. ack at all The slight bobble in the first half second is due to the use of a Pade approximation for the delay The bobble is more obvious if you use the zoom cursor to blow up this region Ctrl Z click and drag First Try Integral Control Classical control design uses guided trial and error An integral control is needed to reject a constant output disturbance and the closed loop bandwidth should be about 1 3 the required rise time or about 1 rad sec Set the controller equal an integrator and then adjust the gain so that the crossover frequency of gk s is 1 rad sec The following commands do so algebraically CC gt gk g s CC gk j 1 ans 0 9241708 0 3801822j CC gk gk abs gk 3 1 CC gt k gk g CC gt k Compute the closed loop response by using the feedback operator gt 1 1 Look at the Bode plot of the loop transfer function gk s and the closed loop step response CC gt subplot 121 bode gk subplot 122 time cl Loop transfer function gt Closed loop unit step response 40 a aj 2 20 90 45 c 0 180 1 20 270 0 5 40 1 5 2900 0 10 10 10 10 10 0 5 10 15 20 Freq Time The subplot function is used to place the plots side by side in the same window a 1 by 2 set of tiles with the plots at index 1 and then 2 Robustness Margins The classical robustness margins are CC gt margin gk At w 1 r s Phase margin 22 36 deg Delay margin 0
22. als in the denominator the polynomials Each polynomial is entered by the polynmoal order followed by the coefficients ordered high to low To go back and forth with the vector of coefficients CC gt vec tovec gl CC gt vec CC gt g2 enter vec CC gt g2 Entering Using the Shorthand Form The shorthand form can be used to enter transfer functions For the g1 s example type CC gt gl senter 2 0 10 1 1 2 1 0 2 1 10 CC gt sho 91 0 0 1 10 The coefficients are the number of numerator factors the factors the number of denominator factors the factors Constants b are entered as 0 0 first order factors a are entered as 1 a and second order factors zeta omega are entered as 2 zeta omega Display Example If you didn t type of the different display as suggested above why not this segment is finished with all of them CC gt b butter 2 2 pade 1 1 s CC gt display b 4 s 20 b s S s 2 2 8284271s 4 s 20 CC gt sho b 4 20 b s 0 0 7071068 2 20 CC single b S 4 22 8284275 3 60 5685425 2 805 CC gt unitary b 4 s 20 B s A c S s 2 2 82842715 4 s 20 CC gt pzf 4 s 20 b S ir A eerie ie eee 5 5 1 4142136 2 1 4142136 2 s 20 CC tcf 0 05 1 2 PIN PSSS SS AA E s 0 255 2 0 70710685 1 0 055 1 CC pfe b Ji 0 9769739843 2238219 DS O SS A
23. an be embedded in the text Start this example by creating a plot with random numbers CC gt rand seed 1 CC gt plot 1 20 Adding Text Using a Right Click Right click where you want the text to be located popup menu will appear AutoScale Ctrl 4 Change Text Trace Cursor Ctrl T Zoom On Ctrl Z Standard Select Text and then the plot options dialog box will pop up with the text page showing Enter the text and then press the Apply or Ok button The text will then be copied to the plot with the anchor point at the original right click location 1 0 8 0 6 Sample text 0 4 0 2 0 0 5 10 15 20 Moving the Text Move the text by clicking and dragging Double click the text to popup the plot options dialog box with the Text page Plot options X Sample T exl v Li y MiddleLeft s s e s s e 2 s HETTOVESIERE KENO ERE 27027077 Use the location and y editlines to change the location of the anchor point in the current axis scales Changing the Text Change the text using the plot options dialog box The Use defaults checkbox resets everything except the text and location The Font button is used to change the font for just this piece of text The Background Color changes the background color The Anchor Point is point on the frame for which the text location is defined The Rotation sets th
24. argin bandwidth airplanebw point point g w Returns text containing g s at s j w or g z at z exp w delta as shown in the following example CC gt b butter 2 2 CC gt point b 3 At 5 0 33 b s 0 2061856 0 3499085j Magnitude 0 4061385 7 8265175 dB Phase 120 50896 deg See also bode freq phasemargin x phasemargin g Returns a two element row vector containing the unit magnitude crossover frequency in rad sec and the phase margin in degrees The input g is a tf or siso quadruple Searches for unit magnitude crossovers from 0 001 to 1000 rad sec Puts multiple occurances if found in a 2 column matrix Returns null if no crossovers are found The phase margin is 180 plus the phase at unit magnitude The phase margin is the phase lag that destabilizes the closed loop system x phasemargin g w Returns the closest phase margin to w rad sec or null if none found x phasemargin g wlow whigh npts Searches from wlow to whigh rad sec with npts number of points Example CC b butter 2 2 s CC phasemargin b ans 0 9731041 47 965209 See also margin delaymargin gainmargin mpmargin bandwidth airplanebw gainmargin x gainmargin g Returns a two element row vector containing the 180 degree phase crossover frequency in rad sec and the gain margin as a linear gain and nat in dB The input g is a tf or siso quadruple Searches for 180 phase crossovers from 0 001 to 1000 rad sec Puts mult
25. change the axes to linear scales and then add the title and text to create Three Low Pass Filters bh Magnitude 0 8 0 6 0 4 Butterworth 0 2 0 0 5 10 15 Freq rad sec The following command uses plot options to change to linear scales omega versus magnitude and to change the axis limits to those shown above CC bode g gl1 g2 type wm bodeoption 1 xlim 0 15 ylim 0 1 2 The cursor is also shown on the above plot Click anywhere on the plot to add a cursor The crossbar cursor reads its values off the axes The trace cursor is always attached to the data traces the data and reads off data values Switch between the two types of cursors using Ctrl T or the cursor change toolbar icon The trace cursor is shown above Step Response Comparison Compare the unit step responses with the command CC gt time 90 91 92 12 Unit Step Responses Bessel Butterworth Make these changes set the time axis range to 8 add the title and text use the line radio button to change the line widths to 2 What has been learned How to use the bode and time commands and the plot dialgog box Also the Butterworth frequency response is the flattest in the passband at the expense of overshoot in the step response and the Bessel step resonse has the smallest amount of distortion Frequency Response Analysis Bode Plot Enter and display the following transfer function 50 s42 868 s s 10
26. ction plots vectors and matrices Examples are given of the different combinations that can be plotted Duplicate the plots by copying the commands pasting them on the command line and then hitting return plot y where y real vector The index of y is plotted versus the vector elements rand seed 1 y rand 1 20 plot y what y 1 by 20 Real Matrix 1 0 8 0 6 0 4 0 2 0 5 10 15 20 plot y where y real matrix The row index is plotted versus each column randn seed 1 y 1 2 3 randn 20 3 25 plot y what y 20 by 3 Real Matrix 4 0 5 10 15 20 plot y where y complex vector The real part is plotted versus the imaginary part x 0 1 500 1 y exp 3 10 pi 3 x plot y what y 1 by 501 Complex Matrix 0 5 0 5 plot y where y complex matrix The real part versus the imaginary part of each column is plotted yl y y 1 plot yl what y1 501 by 2 Complex Matrix 1 0 5 LG 0 5 1 1 0 5 0 0 5 1 1 5 2 plot x y where x and y are real vectors Plots x versus y They must be the same length 2 01 2 2 plot plot x y where x is a real vector and y is a real matrix Plots x versus each column of y The length of x and the number of rows in y must be the same 2 3 4 plot axis 1 1 1 1 1 0 5 0 5 plot x y where x and y are real matrices Plots columns of x versus respective columns of y
27. e angle of the text measured clockwise from the positive real axis Use the Previous Text and New Text buttons to select existing or new text The font size and color of the Sample Text is changed below 1 0 8 0 6 04 Sample text 0 2 0 0 5 10 15 20 Frames and Callouts The Frame checkbox puts a box around the text The Callout checkbox draws a line connected from the frame to the callout x and y location This location also uses the axis scales The callout line can be adjusted using the cursor Examples are shown below 1 Framed text 0 8 0 6 04 Sample text 0 2 Deleting the Text Select the text by clicking on it and then use the Del key to delete it Use the Remove Text button on the Text page Tex Commands Tex is an equation editing language for laying out equations Tex commands use regular text and can be imbedded in plot text plot titles and axis labels A subset of Tex is implemented in Program CC and can be used for subscripts superscripts fractions Greek letters and selected mathematical symbols See the reference manual for a complete list For this example CC gt xlabel a 2 b 3 c 4 5 alpha beta gamma CC gt text g s approx s 1l over 5 2 25 10 15 20 0 br Sample text tJ 5 1 gis 52 25 10 0 5 10 15 20 2 5 a Fancier Text Callouts Copy the plot to Word add callouts
28. e feedback operator is more important for state soace quadruples because it preserves minimal order Convert the above transfer functions to state space CC pg ccf g CC ph ccf h The total number of states is 3 Check the number of states of the closed loop system when it is computed two different ways CC pg ph nstates ans 3 CC pg 1 ph pg nstates ans 5 The feedback operator preserves minimal order You will also find that it is often easier to determine the correct closed loop response using the feedback operator Some Algebra Define the state space realizations o B u Xh Bry y C Xg u r C Xp Combine them to create the following realization of the closed loop system o _ BCh Xg 4 B Xh BnCg An Xh 0 x g The augmented state is defined with the xg state on top This is the special case realization computed by the feedback operator The general case when there are nonzero D terms in the quadruples is considerably more complicated Linear Fractional Transformation Another way to define a closed loop system is the linear fractional transformation It is used to define the H2 and Hinfinity optimal control problems The closed loop system is the response from d to e The feedback operator with some additional algebra can be used to compute the closed loop system but it is easier to use the Ift function mcl lft The Plot Function The plot fun
29. e nargin inside the function equals the number of input arguments used in the calling expression The output arguments are defined in the function and assigned using the assignment operator to the variables on the left hand side Too many output arguments results in a runtime error Fewer output arguments can be used and the remaining are not assigned The variable nargout inside the function equals the number of output argument used in the calling expression The ouptut and input arguments can be surrounded by or The arguments can be separated by spaces if square brackets are used If a function is used in an expression then only the first output argument is returned and used in the expression for example gt 1 sin pi 4 Many functions do not have any output arguments These functions perform actions as plotting and should not be used in expressions They actually do return something a non printing null variable Program CC Version 5 is function oriented yet the prompt is called a command prompt A command is an expression built up using functions variables and operators User Defined Functions and Scripts User defined functions Any number of user defined functions and script files can be placed on the path and therefore made available to the user A user defined function has three parts a function definition help and a set of commands A simple user defined function is function c foo a b
30. e paid is slower execution times Operations should be defined using vectors matrices and functions when possible as opposed to looping over matrix elements so that the overall execution time is acceptable T The language contains the following conditional looping and control statements a elseif else end for while break continue goto function return Each if for and while statement must be terminated and is always paired with the nearest end A function is compiled and the compiled version is executed The compiled version is stored for possible and faster later use Each command line is also compiled and then executed but the compiled version is not stored Text Editor Window The text editor window is used to create edit and debug function and script files A text editor window is opened using the file menu items new and open using tool bar buttons or from the command line CC edit CC edit function name Standard text editing features are available Only one font can be defined at a time The file size of the file is limited to about 30 Kbytes Any text file can be edited but the main purpose of the text window is to edit Program CC mac files The text editor window can be used for syntax checking and test runs of functions and scripts Use the syntax menu items to do this Check Perform a syntax check XRef Display a cross reference table of variables Again Perform the previous operation again Close Close
31. e set to fill the printer page The full resolution of the printer is used Another approach to printing is to place more than one figure on the printer page and also to mix the plots with text Buttons on the print dialog box and options in the print command can be used to easily print full page landscape or 2 or 3 to a page portrait Print options can also be used to divide the printer page into subplots or place a plot to any rectangle on the printer page The Ipdisplay function can be used to print variables and text The Iptext function is also used to display text with more flexibility Other Ip functions also can be used mainly to display transfer functions in different ways such as Ipsho A cursor is used to automatically march down the printer page and to page eject when the page is full After a plot the cursor automatically moves to just after the plot or if a page eject is used as part of the plot to the top of the next page A figure that is copied using screen resolution will be an exact copy of what is on the screen Copying at higher resolution and printing should result in an image that is close to what is on the screen but it may not be exactly the same particularly if automatic scaling is used Another consideration is that larger size fonts relative to the dimensions of a plot usually look better on printed images Use the fontsize plotoption to increase the size of the fonts before copying or printing the fig
32. e the poles and residues are complex and occur in complex conjugate pairs Try the following CC gt g 4 5 1 s 2 2 s 5 Cc gt pfe g al s 1 QuS e s 1 5 1 2 2 2 CC gt ilt 9 g t exp t cos 2t exp t for t gt 0 The complex pole is at 1 2j displayed as s Re 2 Im 2 The pfe command combines the complex conjugate terms so that the display only has real numbers Multiple Poles Now try a case with multiple poles CC gt g 4 s 1 5 2 2 CC gt pfe g 4 4 4 GAS CS cd lcu Ay NN s 1 5 2 2 5 2 CC gt ilt 9 g t 4 exp t 4t 4 exp 2t for t gt 0 The residue calculations use derivatives with respect to s and the ilt has a polynomial in t and it s best just to let Program CC swet the details The general case is described in the reference manual Extracting Parts of the PFE You can use the partial function to extract parts of the pfe The stable flag keeps poles with real parts less than the boundary and the unstable flag keeps poles to the right CC gt partial g stable 1 5 842 2 CC gt partial g unstable 1 5 The partial function defaults to stable with a boundary of zero which is the stable partial factor The pfe2vec function converts the coefficients of the pfe to a vector but the description is complicated and we will leave it to the reference manual to sort out the details
33. e to this desired response while maintaining closed loop stability The internal model control problem does this as long as the controlled element is stable CC gt desired bessel 1 3 CC gt k imc g desired CC gt sho k 0 556 3 0 5 3 9 289 0 724 10 17 0 0 437 3 996 0 693 10 86 11 46 The compensator is 6th order rather high but this level of complexity is what it takes to meet the specifications Compute the loop transfer function LTF and the closed loop response CC gt gk g k CC gt cl gk 1 CC gt sho cl 15 9 289 0 724 10 17 2 322 0 724 2 542 9 289 0 724 10 17 The closed loop system is stable The Bode plot of gk s and the unit step response of cl s are shown below CC gt subplot 121 bode gk redo subplot 122 time cl T Loop transfer function Closed loop unit step response 0 a a a 1 20 an a gt gt 00 0 5 20 270 40 f 2200 0 10 10 10 10 10 0 1 2 3 4 5 Freq rad sec Time The redo option is used on the bode function to preserve the axis limits Using the trace cursor the overshoot is less than 1 the 2 settling time is 2 6 seconds the phase margin is 63 degrees and the gain margin is 8 6 dB The specifications for the design have been satisfied The imc function can be used to reduce the rise time further Use the up arrow to recall and change the following commands or better yet put the followi
34. eave it to the reference manual to sort out the details Inverse Laplace Transforms The Causal ILT The inverse laplace transform ilt function computes and displays the ilt of a transfer function by 1 computing the poles 2 computing the partial fraction expansion using the residue formula 3 using a table look up to compute the ilt of each term and 4 displaying the result Try the following CC gt g 4 s 1 5 2 CC ilt 9 g t 4 exp t 4 exp 2t for t gt 0 Plotting the Impulse Response If the 2nd parameter of the ilt function is a vector of times then the function returns with the ilt computed at these times Use this feature to plot the impulse response of g s CC t20 05 5 CC y ilt g t CC plot t y 1 Impulse Response 4 0 8 gis DE 0 6 git 4 expct 4 exp 20 0 4 0 2 The transfer function was displayed above after using the plot dialog box to enter the following Tex commands g s 4 5 1 5 2 The same plot can be created using CC gt time g s Noncausal ILT The ilt functon defaults to the causal ilt nonzero only for positive time The anti causal ilt is returned by GCO ilt g anti g t 0 for t gt 0 4 exp t 4 exp 2t fort lt 0 A boundary parallel to the jw axis can be defined and poles to the left are causal and poles to the right are anti causal Do so below with a boundary at 1 5 and then plot the non causal ilt cO
35. es a m by n set of tiles and puts the next plot in the p th tile counting row wise In the above example a 2x1 set of tiles are created and each tile is filled Any existing tiles that are overwritten in whole or in part are deleted The subplot position left bottom width height defines a tile using fractional locations 0 to 1 defined left to right and bottom to top so that left bottom defines the left lower corner of the axes and left width bottom height defines the right upper location of the axes Room should be left for the axis and title If subplot mnp is used then the axes are automatically indented to make room for axis labels and the title unless the plots are very very small Stripchart Use the stripchart function to put a vector time series on separate vertical axes with a common horizontal axis What is actually done is that each column of a matrix is plotted on a separate vertical axis In the following example random numbers are plotted versus the row numbers Program CC CC randn seed l CC y randn 50 4 CC gt stripchart ly CC Use the plot options dialog box and the stripchart radio button to plot the stripchart using overlapping y axes as shown below Time Graphics Example Bode Plot The claim is that graphics are easy and interactive so let us now demonstrate this In this multi part example 3 different low pass filters are compared First create them CC gt g butter
36. es the command and displays the text results if any in the command window The Close button closes the dialog box echos the command line to the command window but does not execute the command The Cancel button closes the dialog box and nulls out the current command Program CC Version 4 Compatibility The key words are not very Program CC Version 5 differs in many ways from Version 4 The 16 17 19 20 21 22 differences for user s familiar with Version 4 are listed below All of the Version 4 commands are replaced with function calls For example the Version 4 command eig a d is replaced by the function call d eig a In Version 4 the command arguments are prompted if not included on the command line This is no longer supported all of the arguments must now be included as part of a function call In Version 4 only the non ambiguous part of a command name needs to be entered For example display can be shortened to dis Now the full name needs to be entered In Version 4 the variable and commands names are case insensitive In Version 5 the variable and function names are case sensitive In Version 4 the variable names are limited to 6 characters followed by an optional dot and up to a 3 character suffix In Version 5 there is no limit to the length of a command name Dots cannot be used Underlines can now be part of a name In Version 4 there are different command levels the main ones being CC
37. g s CC t20 05 5 CC y ilt g t CC plot t y 1 Impulse Response 0 8 g s 8 1 6 2 0 6 g t 4 4 expC20 0 4 0 2 The transfer function was displayed above after using the plot dialog box to enter the following Tex commands g s 4 5 1 5 2 The same plot can be created using CC gt time g s Noncausal ILT The ilt functon defaults to the causal ilt nonzero only for positive time The anti causal ilt is returned by Co 2ilt g anti g t 0 for t gt 0 4 exp t 4 exp 2t fort lt 0 A boundary parallel to the jw axis can be defined and poles to the left are causal and poles to the right are anti causal Do so below with a boundary at 1 5 and then plot the non causal ilt Gc2 ilt g x098 1 5 g t 4 exp 2t for t gt 0 4 exp t fort lt 0 CC t 3 05 3 CC y ilt g t roc 1 5 Non causal Impulse Response g s s 1 s 2 g t 4 exp 20 fort gt 0 gif A expcb fort 0 The Autocorrelation and Variance Now take the same transfer function with input u s and output y s gls If the input is zero mean unit intensity white noise the power spectral density of the output is g s g s computed as follows CC yd g g s The stable ilt is the autocorrelation The autocorrelation evaluated at t 0 is variance of the output stochastic process CC ilt yd 0 stable ans 1 3333333 This is do
38. g ut u Returns the response to a time series input yt y sim g input_option input_parameters delta imax Returns the response to different types of inputs and with the specified delta and tmax yt y sim g input_option input_parameters sim_option parameter General form Parameters yt vector of output times y vector or matrix of output values g tf tf matrix or quadruple ut vector of input times u vector or matrix of input values if a matrix then each column is the input signal for the corresponding input channel delta sample period tmax maximum of output times Input options and parameters timeseries ut u step height impulse area pulse height width whitenoise sigma seed zero Simulation options and parameters delta delta sample period tmax tmax time ic xO i c vector quadruple only channel n selected input channel Step reponse example delta and tmax are automatically determined b butter 2 2 yt y sim b Pulse response example with delta 1 and tmax 10 yt 5 b pulse 1 1 1 10 Initial condition example with zero input and with delta imax automatically determined p ocf x0 1 0 yt 51 zero ic x0 Time series example with delta 01 and tmax 5 ut 0 1 2 5 u 0 1 0 0 yt y sim b ut u 01 5 To plot the simulation output plot yt y To plot both the input and outpu
39. gin g Returns text containing phase delay gain and mp margins Does so by searching from 0 001 to 1000 rad sec for 180 phase crossovers unit gain crossovers or local maxima of g 1 g Returns multiple occurances if they occur The input g can be a tf or siso quadruple margin g w Returns text containing the closest occurance to w rad sec of each type of robustness margin margin g wlow whigh npts Searches from wlow to whigh rad sec with npts number of points text mat margin g w text mat margin g wlow whigh npts Returns the text the same as displayed in the text variable text and returns the margins in the matrix mat packed as follows 1st column type 1 phase 2 delay 3 gain 3 mp 2nd column omega r s 3rd column the margin The phase is returned in degrees delay in seconds and gain and mp as unitless gains not dBs Definitions phase margin 180 phase where magnitude is unity delay margin delay with phase lag equal to phase margin gain margin 1 gain where phase is 180 degrees mp margin local maximum of g 1 g Example CC gt b butter 2 2 s CC gt margin b At w 0 973 r s Phase margin 47 97 deg Delay margin 0 86 sec At w 1 22 r s Mp 1 3 2 29 dB At w 2 r s Gain margin 2 83 9 03 dB CC text mat margin b CC gt mat mat 1 0 9731041 47 965209 2 0 9731041 0 8602891 4 1 2216030 1 3023584 3 2 0000003 2 8284279 See also phasemargin delaymargin gainmargin mpm
40. ifornia copyrighted by Peter M Thompson and STI Hawthorne is located in the greater Los Angeles area Our Web site is www systemstech com STI primarily does consulting and government contract work STI has been in existence for more than forty years founded by Duane T McRuer and Irving L Ashekenas and now headed by R Wade Allen and Thomas T Myers The company has been involved throughout its history in a wide variety of systems dynamics human operator and controls activities We have worked on just about the entire inventory or U S fighter and transport aircraft the NASA Space Shuttle landing system several Army and Navy helicopters the National Aerospace Plane and several rocket boosters We have worked on exotics such as blimps parachutes the Gossamer the Smithsonian s Quintocotalus Northropi and old and new versions of Northrop s flying wing STI has also worked extensively on ground vehicle dynamics and manual control of ground vehicles everything from motorcycles cars and trucks and the thankfully no longer popular 3 wheel off road vehicles Recently STI has branched into software development and PC based low cost simulators Our simulator products include automobile and truck simulators and a parachute simulator Program CC has been a software product since 1986 Version 5 is our most recent release and the first to emerge from the DOS world and take full advantage of Microsoft Windows Our expertise and experience in c
41. iple occurances if found in a 2 column matrix Returns null if no crossovers are found The gain margin is the inverse gain at the 180 degree phase crossover frequency The gain margin is the gain increase that destabilizes the closed loop system x gainmargin g w Returns the closest gain margin to w rad sec or null if none found x gainmargin g wlow whigh npts Searches from wlow to whigh rad sec with npts number of points Example CC b butter 2 2 s CC gainmargin b ans 2 0000003 2 8284279 See also margin phasemargin delaymargin mpmargin bandwidth airplanebw time time g Plots the unit step response time g ut u Plots the response to a time series input time g input delta tmax Plots the response to different types of inputs time simoption param plotoption param General form See sim and plotoption Plots the result of a linear simulation Use in place of sim without having to store the output times and values Provides the convenience of automatically changing tmax to match the plotted x axis The example b butter 2 2 time b is the same as yt y sim b plot yt y To plot vector outputs with axes for each output use yt y sim p ut u stripchart yt y See also input plot sim stripchart timevec sim yt y sim g Returns the unit step response yt y sim g delta tmax Returns the unit step response with the specified delta and tmax yt y sim
42. ize of the line and or symbols See line styles bode g1 s1 gn sn option params The general form of the bode function includes plot options Example b butter 2 2 s bode b Plot options particular to bode include asymptotes bodeoption siggy and phase Robustness margins can be determined by using the cursor or by using the right mouse click pop up menu See also freg margin nichols nyquist point plot siggy freq y freq g s Returns the frequency response g s where g tf tf matrix or quadruple but must be single input s complex vector of frequencies in rad sec y freq g low high npts option Generates a vector of frequencies and then computes the frequency response where low low frequency high high frequency npts number of points option 0 rad sec log spaced default 1 rad sec linear spaced 2 Hz log spaced 3 Hz linear spaced The analog or w plane frequency response is g s The digital frequency response is g z at z exp s g delta If g delta 0 then g delta 1 is used For the usual frequency response s should be complex and on the jw axis as returned by freqvec and dfreqvec To use a different delta then reset the the structrual member g delta newDelta y freq g s Examples w freqvec 01 100 100 y freq g w bode w y y freq g w freqvec 01 1000 100 bode w y bode g See also dfreqvec freqvec point substitute svfreg and point margin mar
43. leLeft E awo H ml b oo HETTOVESIERE s s e s s s s KENO ERE 27027077 Put a cursor on the plot by clicking dragging Change to the trace cursor using the Cursor toolbar icon which toggles between Copy the result to the clipboard using the Copy toolbar icon After not too much effort my changed plot looks like Bode plot 40 0 on m mm wo D E E o 20 45 E o Freq 149 9 ris Mag 0 3328 9 557 dB Phase 86 95 deg Comparison Plots In the next Bode plot 2nd order systems with 5 different damping ratios are compared Enter the transfer functions with into a tf matrix CC gt h for i 1 5 h i 1 twopoles 2 i 1 end Type the following CC gt sho h CCbode h Here just to be different the magnitude and phase are placed in separate plots CC gt subplot 211 bode h b bodeoption 1 xlim 1 10 ylim 20 10 CC gt subplot 212 bode h b bodeoption 2 xlim 1 10 ylim 180 0 Magnitude dB 10 Top to bottom t 2 4 6 8 1 107 2 34 10 2 34 10 Freq The plotoptions entered as part of the bode function or in a separate call to the plotoption function are how to recreate a plot in a function The above plot options are forces each line to be solid blue bodeoption 1 selects magnitude bodeoption 2 selects phase xlim 1 10 sets x axis limits
44. loaded by the save and load commands CC gt save datafile_name CC gt load datafile_name The datafile_name cc file is saved in the current working directory or in any other directory if included in the name The same file is searched for loading in the current working directory and then anywhere in the path or just in the directory if included in the name User defined functions can be created in the text editor window and are by default saved in the current working directory These same functions when called are searched for first in the current working directory and then along the path The first occurance is compiled executed and saved for later use The function is recompiled when used again if the text file has changed since the last occurance The best way to check if a function is located on the path is to use the help command CC gt help name Plotting The plot commands are bode freq response magnitude and or phase loglog log10 versus log10 nichols freq response phase versus magnitude nyquist freq response real versus imaginary plot x Versus y rootlocus real versus imaginary of roots semilogx log10 versus linear semilogy linear versus log10 stripchart x versus multi axis y time simulation response Recommended use Each of these plot commands are interactive Create the plot by using simple versions of the commands such as plot x y or bode g which will autoscale the axes Then pop up a plot option dialog box by do
45. lot rand 1 20 And then single click anywhere on the plot to popup an axis cursor 1 0 8 0 6 0 4 0 2 0 5 10 15 20 x 8 535 y 0 4476 It is called the axis cursor because it reads values off the axes Trace Cursor Use the Ctrl T hotkey to switch to the trace cursor 1 0 0 5 10 15 20 x 9 y 0 6711 The diamond shaped cursor jumps to the nearest data point and reads off the value It is called the trace cursor because it traces data points Toggle between the axis and trace cursors by any of the following The Ctrl T hotkey The Plot menu item that toggles between Axis cursor and Trace cursor The right click popup menu item with the same names The tool bar icon that toggles between EE Zoom cursor The axis scales can be zoomed in and out using the zoom cursor Change to the zoom cursor by The Ctrl Z hotkey The Plot menu item that toggles between Zoom on and Zoom off The right click popup menu item with the same names The toolbar icon that toggles between Ve Left click to zoom in and right click to zoom out The zoom cursor is shown below 0 8 0 7 0 6 0 5 0 4 4 8 10 12 Click and drag to select a region 0 8 0 7 06 0 5 0 4 4 6 8 10 12 The zoom cursor changes when over an axis and then changes just that axis or just that axis limit Z Ry Rig Ry Ng Change back to automatic scaling with any of the followi
46. lternative tiling use Named Figures The figure function can set the number name of the figure Program CC 15 x File Edit Functions Window Help 0 ucvBm Plot View 1 E Plot View 10 CC gt plot rand 1 100 CC gt figure String Title CC gt oplot rand 1 200 The two new figures are named Plot View 10 and String Title Making Changes The plot change dialog box can be called for any of the figures and used to make changes Functions such as title and plotoption can be used to make changes from the command line These functions always work on the current figure which is the last one plotted or the last one referred to using the figure function For example to add a title to Plot View 2 CC gt figure 2 CC gt title New title The gcf function returns the number or name or the current figure in this case CCO gcf ans 2 Subplots Close all of the above figures using the Windows Close All menu item Clear the command window using Ctrl A and pressing Del twice Create a new figure with two vertical tiles each with its own plot Program CC 10 x File Edit Functions Window Help gt Command Window ES Plot oxi CC gt rand seed 1 CC gt subplot 211 CC gt plot rand 1 20 CC gt subplot 212 CC gt plot rand 1 50 CC gt The windows were arranged using mi The subplot mnp function creat
47. mial factors CC gt unitary sets each high order coefficient to one CC gt pzf pole zero form CC tcf b time constant form CC pfe b partial fraction expansion CC ilt b inverse Laplace transform The shorthand form is popular is some locales As the name suggests the display is fairly short and is defined by a s b a b sls 2605 o O o g s Usually at least around here the function name is shortened to sho Note that the damping and natural frequency of complex modes are displayed which are usually more meaningful than the real and imaginary parts but if you want to see the real and imaginary parts then use the pzf pole zero form function defined by a s b BO ee s s Re 2 Im The tcf time constant form function is defined by 1 2 81 1 s s 0 20510 1 g s The coefficient multiplying the s term is the effective delay or advance if negative of that mode The pfe partial fraction expansion and ilt inverse Laplace transform are described in separate sections of the tutorial Entering Using Coefficients Transfer funtions can also be entered a vector of coefficients The second example can be re entered by CC gt gl enter 2 0 10 1 1 1 2 1 1 0 2 1 2 1 10 10 2 Most people prefer the symbolic form and who can blame them The order of the coefficients is the number of polynomials in the numerator the polynomials the number of polynomi
48. n 4 the statements cannot be continued to the next line In Version 5 this can be done using ellipses In Version 4 macros the comments are lines that begin with an apostrophe in the first column This type of comment is still supported and in addition any text between the percent sign 96 and the next ret is a comment The percent sign can be anywhere on the line including after executable text In Version 4 the macro execution continued after an error Now function and script execution does not begin if a compile error is found and the execution stops whenever a runtime error is found In Version 4 the for and while loops were quite slow Now they are much much faster In Version 4 the macros are interpreted Each line is tokenized and parsed immediately executed and then forgotten Each line in a for or while loop is interpreted in each pass which is why Version 4 looping is so slow In Version 5 an entire function or script is tokenized and parsed i e compiled before the first line is executed and then the executable version is saved for later use In Version 4 the macros echo text to the command line using echo text This is done in Version 5 using disp text A conversion utility from Version 4 macros to Version 5 functions is not available In Version 4 all of the systems transfer functions and quadruple are either analog or digital depending on the current mode and if digital all of the sample pe
49. ne more simply by CC gt var g ans 1 3333333 And for the standard deviation CC std g ans 1 1547005 A plot of the autocorrelation is created as follows CC gt t 3 05 3 CC gt y ilt stable CC plot t y Autocorrelation lt variancez1 33 Partial Fraction Expansions Two Real Poles Partial fraction expansions pfe s are used so that high order transfer functions can be decomposed into terms with just one pole each Usually this is done so that the inverse Laplace transform can be applied to each term It is also a first step for hold equivalence discretization methods Enter a second order transfer function CC gt g 4 s 1 s 2 CC gt display g And then compute the pfe and the ilt inverse Laplace transform CC gt pfe g 4 4 QS ecu s 1 s 2 CC gt ilt g g t 4 exp t 4 exp 2t for t gt 0 By the way when copying pfe s to a text editor use the Courier New font so that the spacing and lines are all correct The general case for pfe s is rather tedious but for a strictly proper transfer function more poles than zeros with no multiple poles the pfe is n a s gt L where a amp 5 _ 15 Pi m len You could if you wanted to compute the residues the ai s CC substitute s 1 g 1 ans 4 CC gt substitute s 2 g 2 ans 4 Complex Poles The above formula is valid for complex poles in which cas
50. ng The Ctrl A hotkey The Autoscale Plot menu item The Autoscale right click popup menu item The toolbar icon Apto Changing the Plot Plot Options Dialog Box The recommended way to create plots in Program CC is to use automatic scaling followed by the plot options dialog box First create a plot CC rand seed 1 CC plot rand 1 20 1 0 8 0 6 0 4 0 2 0 5 10 15 20 Then popup the plot options dialog box in any of the following ways Select Change using the right mouse click popup menu Select Change using the menu bar Plot menu Double click anywhere on the plot Use the Construction toolbar icon It is shown below with the page used to change the x axis scales Plot options X o s s s s s s The radio buttons on the left are used to select different pages of options The top group of controls are Auto Set all of the options to automatic or default values Apply Apply the changes and keep the dialog box open Ok Apply and close Cancel Close without any changes Help Popup Windows help All of the editlines except for new lines must contain numbers or text not variable names or expressions Use the plotoption function to make the same changes from the command line or from within functions or scripts Plot Options on the Toolbar The following toolbar buttons 3 Bb MQ iw dt are from left to right Print Copy Plot options Zoom cur
51. ng in a function CC gt desired bessel 3 3 k imc 9 desired gk g k cl gk 1 CC gt subplot 121 bode gk redo subplot 122 time cl redo Classical Control Example When Delay is a Problem The Controlled Element The controlled element is a 3rd order Butterworth filter with a 0 5 second delay Enter this system as follows CC gt g butter 3 3 pade 5 3 CC gt format mini CC gt sho g 27 9 289 0 724 10 17 0 5 3 3 9 289 0 724 10 17 A 3rd order Pade approximation of the delay is used The format mini command is used to display just 4 signficant digits The system g s is displayed using the shorthand form whereby a equals s a and zeta omega equals s 2 2 zeta omega s omega 2 Requirements The objective is to design a control system that rejects constant output disturbances and the step response settles to 2 of the final value within 3 seconds with an overshoot less than 10 Use the following control architecture Open Loop Analysis First look at the open loop step response CC gt time g 12 Open Loop Unit Step Reponse EN V 0 8 0 6 0 4 0 2 0 0 2 0 1 2 3 4 5 Time Time 2 12 Real 1 081 The trace cursor is used to determine the overshoot of 8 196 and the settling time of 2 7 seconds The open loop response meets the specfication except for the output disturbance rejection and indeed the output disturbance specification is the only reason to use feedb
52. nt not just for students but also for busy engineers who only occasionally have need of systems analysis Extensive capabilities As novice users transition to experts the full depth of capabilities in Program CC becomes apparent People who learn to use all of the parts of Program CC will at least become much better and hopefully become expert control systems designers These same people may even have some fun along the way Better education Sudents new to control theory are presented with a long list of algorithms and design methods and while it can safely be said that most algorithms are not in themselves difficult the total can be quite an imposing burden Many people have become good control system designers and all have put in the necessary time and effort Many more people have had a brief exposure to systems and control and have gone elsewhere Program CC and personal computers can considerably shorten the learning process It becomes less important how to draw a Bode plot or a root locus and more important to be able to interpret the results Books full of rules and theorems can be made transparent enough to allow the fundamental ideas to come forth People who only want or need a brief exposure to the subject can come away with a better understanding of the basics and better able to read someone else s report By whom Program CC is brought to you by whom Program CC is a product of Systems Technology Inc STI of Hawthorne Cal
53. o subplot 122 time cl Loop transfer function o Closed loop unit step response 40 m L ua 2 a 1 20 90 c 0 180 0 5 20 270 40 7 5 p 2300 0 10 10 10 10 10 0 1 2 3 4 5 Freq Time The redo option is used on the bode function to preserve the axis limits Using the trace cursor the overshoot is less than 1 the 2 settling time is 2 6 seconds the phase margin is 63 degrees and the gain margin is 8 6 dB The specifications for the design have been satisfied The imc function can be used to reduce the rise time further Use the up arrow to recall and change the following commands or better yet put the following in a function CC gt desired bessel 3 3 k imc 9 desired gk g k cl gk 1 CC gt subplot 121 bode gk redo subplot 122 time cl redo Inverse Laplace Transforms The Causal ILT The inverse laplace transform ilt function computes and displays the ilt of a transfer function by 1 computing the poles 2 computing the partial fraction expansion using the residue formula 3 using a table look up to compute the ilt of each term and 4 displaying the result Try the following CC g 4 s 1 s 2 CC ilt 9 g t 4 exp t 4 exp 2t for t gt 0 Plotting the Impulse Response If the 2nd parameter of the ilt function is a vector of times then the function returns with the ilt computed at these times Use this feature to plot the impulse response of
54. ontrol system design is reflected in our products Most of the commands and plotting in Program CC are standard treatments of systems and control methods but there are many others in the program that are included because we use them and they work Dr Peter M Thompson is the main author of Program CC He has a Ph D in Electrical Engineering from the Massachusetts Institute of Technology and was an Assistant Professor at the California Institute of Technology Since 1986 he has worked at Systems Technology Inc and has been actively involved in applying advance control theory to practical problems The Command Line The heart of Program CC is the command window The last line is the command line on which the user is presented with a prompt CC gt command is entered using the keyboard and then executed with a carriage return A response may be displayed after the prompt or a window or dialog box may open depending on the command and then the user is presented with another prompt In the following example a 2x2 matrix is entered and displayed and then its eigenvalues are computed and displayed CC a 1 2 3 4 gt T 2 3 4 gt ans 5 3722813 0 3722813 Expressions of the form Ihs rhs do not display the results no matter how terminated Expressions of the form lt rhs gt display the result Long expressions can be continued on the next line using ellipses gt 2 3 g
55. puter aided control system analysis and design program Why Why use Program CC Program CC s strengths are Many variable types High level analysis Interactive plotting Eclectic design Short learning curve Extensive capabilities Better education Many variable types Program CC does not try to fit the world into matrices Transfer functions and state space quadruples are available to the user as variable types High level analysis The user does not need to be a programmer Poles and zeros are displayed in a natural form with simple commands Simulations Bode plots and root loci are available using single commands Interactive plottings The plotting can be changed using a plot dialog box Cursors and zooming are available using the mouse Previous versions of plots can be redrawn and standard sizes can be stored for later use The plots can be quickly printed or copied into reports Eclectic design Program CC allows easy transitions between classical sample data modern and optimal control System models can be built up using a combination of state space equations and transfer functions Individual responses or loops can be converted to transfer function for classical analysis and design and the results converted back to state space for simulation Short learning curve The history with Program CC has been that novice users quickly enter data and quickly start doing significant things The learning curve is short This is importa
56. riods are the same In Version 5 each system is declared analog digital or w plane and the digital and w plane sample periods can be individually set Version 4 included the data file and functions of transfer functions variable types These variable types are no longer supported In Version 4 scalar plus matrix addition was defined as the scalar times an identity matrix plus a square matrix In Version 5 scalar plus matrix addition is defined as the scalar added to each element Structure members are supported in Version 5 for example a nrows In Version 4 quadruple parts are selected by p a p b p c ad p d In Version 5 the quadruple parts are selected by p a p b p c and p d Version 5 includes logic operators amp amp and Version 5 includes increment and decrement operators and Version 5 includes operation assignment operators such as and A Microsoft Windows interface 32 bit programming allowing large size variables help name displays help about function name New file formats All of the variables in a project are stored in a single cc file Version 4 files can be loaded and saved A limited ability to execute Matlab m files Transfer Functions Entering Transfer functions are entered as algebraic expressions To enter eo s 5 type the following CC gt g 5 5 5 Display the transfer function or any other variable by typing its name CC g
57. sor Auto scaling Axis and trace cursors Changing Axis Scales The page used to change axis scales is shown above The parameters checked Auto are so determined The label defaults to none for plot function Change the max value of the x axis to 10 and add a label 1 0 8 0 6 0 4 0 2 0 2 4 6 8 10 Random numbers More Axis Parameters The More Axis Parameters page is shown below E Plot options e eeeee ee x a A gt Bo se E _ mm The label font location and orientation can be changed The numbers on the axis can have their font changed and the number of significant digits changed The label and number fonts can be different The tick position and lengths can be changed More Changes The Title page is used to enter the plot title and to change its font The Font page is used to change and make all of the fonts the same for the title axis labels axis numbers and plot text The Grid page is used to change the background Some of the options are for specific plots The Text page is used to add text to the plot and is described in another tutorial section The Standard page is used make the current axis limits and options the standard for this type of plot The standard options can then be quickly set using the right mouse click popup menu Adding Text A plot can be annotated with text The location is specified using the current axes Tex commands c
58. t respectively blue and red plot b yt y r For further details see simulation overview See also input plot stripchart time timevec Simulation Overview Linear simulation The sim function computes linear simulations of transfer functions tf matrices and quadruples The simplest from is a unit step response from the first input yt y sim b For a time series input yt y sim ut u Input types that can be automatically generated include step impulse and pulse inputs See the sim function for further details Time Series The input time series has times in the vector ut and values in the vector or matrix u The output time series has times in the vector yt and values in the vector or matrix y Multiple inputs and outputs are stored in matrix columns The number of columns of u must be equal to the number of system inputs If ut is evenly spaced the yt y and if only one output parameter is included in the simulation then y is output For time series inputs the input values are linearly interpreted between the input times and are held constant after the last input time Computational Methods For analog tfs and tf matrices the inverse Laplace transform is used for step impulse and pulse inputs ZOH discretization of analog tfs and tf matrices is used for all other types of inputs including time series inputs ZOH discretization is always used for analog quadruple system The 6th order Pade approxima
59. t 4 5 Multiple commands can be entered the same line separated by semicolon or ampersand or amp In the following example a transfer function is entered and then displayed CC gt g 5 5 1 g Parameters separated from the function by spaces surrounded by quotes The following are equivalent CC gt help butter CC gt help butter Previous commands are scrolled using the arrow keys The text above the command line is treated as a regular text window with cutting pasting and copying operations The command window cannot be deleted The Alt C hotkey jumps to the command line and the Alt X hotkey pops up a dialog box containing the command line Variables Variable names must start with a letter or underline and then can contain any number of letters numbers and underlines Variable names are case sensitive Variables have structure members that return or set different attributes For example a nrows returns the number of rows Examples of each variable type defined in Program CC are presented below Real and complex numbers are entered by CC a 2 CC b 344 j The variable j is by default equal to the square root of minus one and if overwritten can be reset using j sqrt 1 Real and complex matrices are defined using parentheses square brackets or braces Elements are separated by commas rows by semicolons and diagonal numbers by poundsigns gt 1 2 3 4 5
60. t options Any of the plot options can be manually set by using the plotoption function Axis limits can be changed using the axis function New lines can be added using the line function The title xlabel ylabel and so on change these parts of the plot Typically changes are made in a user defined function which in one step creates a particular style of plot or recreates a figure used in a report All plots are 2 dimensional There are no 3 dimensional plots Examples plot rand 1 1e2 t 0 1 2 pi x cos t y sin t plot See also axis hold figure grid line plot plot options line styles subplot text title xlabel ylabel ylabels y2label bode bode g Plots log10 abs s versus the dB magnitude and phase of g s where s is an automatically generated set of frequencies The input g can be a tf tf matrix or quadruple If g is multivariable then plots the frequency response of each input output pair Plots the magnitude using the left axis and phase using the right axis In the digital case plots log10 abs s versus g z for z exp s g delta bode s g Uses the frequencies stored in the vector s Frequencies on the jw axis should be stored as complex numbers See freqvec and dfreqvec bode s y If both s and y are vectors then plots abs log10 s versus dB abs y If s and or y are matrices then does so for each column or pair of columns bode g s The string variable s specifies the color and s
61. tate space analysis A large number of internal functions are available in a command driven user interface and any number of user defined functions can be written and used to expand the available list of functions The variable types built into the program are real and complex numbers real and complex matrices transfer functions Laplace z domain or w plane transfer function matrices quadruples state space equations strings These variables can be used for systems and control analysis A physical system such as a spacecraft aircraft tracking antenna electronic circuit or chemical process is modeled using linear differential equations The equations are converted to transfer functions and or state space equations and entered into the program Program CC can then used for time and frequency domain analysis further development and validation the model and then to design and validate the control systems A wide variety of time and frequency domain analysis techniques and graphical methods and programming capabilities are available to help the user including Linear simulation Frequency response analysis Matrix decompositions Model building and model reduction Classical control plots General 2D plotting Fast Fourier transforms Statisical analysis Optimal control Conversion between analog and digital domains Conversion between classical and modern models High level programming In short Program CC is a comprehensive com
62. tion is used to compute the required matrix exponential For digital tfs tf matrices and quadruples the simulation is computed using difference equations Maximum time The maximum time tmax can be manually entered using the tmax sim option or is automatically determined using the following logic 1 For time series inputs the max time in ut is used 2 Otherwise use 5 times the maximum time constant 3 Except for integral systems use tmax 5 Sample period The sample period delta can be manually entered using the delta sim option or is automatically determined using the following logic 1 For digital systems use g delta 2 Otherwise for time series inputs with regular spacing use that spacing 3 Otherwise use delta tmax 500 Initial conditions Initial conditions can be specified for quadruple systems using the ic sim option The ic must be a vector of compatible dimensions and the ic response can be mixed with any of the input options For just the ic response use the zero input type Nonlinear simulation Nothing is automatic but nonlinear simulations can be generated by using for loops in user defined functions input yt y input input delta tmax simoption param Returns input times and values The outputs are the same as using sim with a system g 1 Use the input function for creating a time series with evenly spaced times Example yt y input step 1 delta 1 tmax
63. uble clicking on the plot or using the menu or toolbar and use the dialog box to add a title labels or text to change the axis limits and so on and then print the plot or copy the result to the clipboard Redo and standard plots You can redo the previous set of axis limits by using the redo plot option such as bode g redo You can use a standard set of axis limits by using the std plot option such as bode g std Use the plot option dialog box to reset the standard Cursors The x y values can be read off the plot using a cursor by single clicking anywhere on the plot The axis cursor reads the values from the axis locations and the trace cursor is always centered over data points and reads the values from the stored data Zooming Axis zooming can be toggled by clicking the Ctrl z keys or by selecting the plot zoom menu item in which case the left mouse button zooms in and the right mouse button zooms out Each axis can be zoomed individually by clicking on the axis Popup Menu A subset of options are available in a pop up menu called using a right mouse click These options depend on the type of plot Mulitple plot windows The plot functions will overwrite the current plot unless the hold function is used to add the new lines to the same plot or the subplot function is used to place the next plot in a different location on the same figure or the figure function is used to place the next plot in a new window Plo
64. ures for example plot x y fontsize 16 The default fontsize is 10 pts There is currently a problem printing to HP printers that are setup using Vector graphics which is the default If plots take a long time to output and or having missing parts then switch to Raster graphics using the printer properties dialog box See also plotting overview copy copyoption Ipdisplay Iptext plot plotoptions text textoptions Classical Control Plots Frequency response plots are created with the following functions CC gt b butter 2 2 CC gt bode CC gt nyquist b CC gt nichols b A root locus plot is created by CC gt rootlocus b A unit step response is plotted by CC gt time b See each of these functions for further details The recommended way to use them is to use automatically scaling and then manually adjust them using the plot option dialog box The previous set of scales or a standard set of scales can also be used CC gt bode b redo CC gt bode b std See also plotting overview bode nichols nyquist plot rootlocus time Command Dialog Box The command dialog box contains the command line and a list box with different types of lists Previous commands Path Variables Structure members All functions User defined functions Plot options Text options The command dialog box is popped up by using the Alt X hot key The Enter button echoes the command line is to the command window execut
65. with amp or lt ret gt Lines can be continued with ellipsis A comment is defined from to ret Lines that start with in column 1 are comments H amp lt ret gt Structure members Variables are separated by their structure members using the dot operator Keystrokes The following hotkeys are available when the command line is active up arrow recall previous commands down arrow recall previous commands esc delete command line Ctrl up arrow previous line The following are available at any time Alt C Alt X jump to command line open command line dialog box The remaining hotkeys are more or less standard replacements for menu items File menu Ctrl N Ctrl O Ctrl S Ctrl X Plot menu Ctrl A Ctrl L Ctrl T Ctrl Z Edit menu clipboard Edit menu search new Ctrl Z undo Ctrl F find open Ctrl X cut Ctrl E find next save Ctrl C copy Ctrl R replace cut Ctrl V paste Ctrl A_ select all Shift F change case Ctrl G goto line Window menu Autoscale Shift F5 cascade Plot text Shift F4 tile Cursor type Zoom cursor Functions All functions have the following form outl out_nargout function name inl int nargin The input arguments can be variables or expressions Variables used as input arguments are never changed Too many input arguments results in a runtime error Fewer input arguments can be used but may result in undefined variables inside the function The variabl
66. with arrows explosions word and so on For example Word C allout Sample text 5 1 0 5 5 28 10 0 5 10 15 20 2 5 a The text function To enter text from the command line CC gt text More text 1 7 Text options can be used to select the same options as when using the plot options dialog box For example the above command can be changed to include a frame and a callout CC gt text More text 1 7 frame CC gt text More text 1 7 callout See the text function for a complete description Typically the text function is used to recreate a figure used in a report Make adjustments to the original figure using the plot options dialog box and the mouse and then copy the text locations from the dialog box to the text function Add New Lines Using the Plot Option Dialog Box Create a plot with random numbers CC gt rand seed 1 CC gt plot 1 20 1 0 2 0 5 10 15 20 Popup the plot options dialog box and switch to the Lines E Plot options A rand 1 20 2 LM NEC NE NL NE Remove Lite WEVE DS Put the new y as above or x y entries into the editline Select the color linestyle symbol line width and symbol size options These are the same choices that can be surrounded by single quotes in the plot function Apply the above choices The entry will be echoed to the command line to check for errors

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