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Text File | 1992-07-06 | 653.2 KB | 39,393 lines
(*^ ::[paletteColors = 128; showRuler; currentKernel; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, L1, e8, 28, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, L1, e6, 22, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, L1, e6, 16, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, L1, a20, 22, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, L1, a15, 16, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, L1, a12, 14, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, L1, 14, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, 14, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 14, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 14, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 14, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 14, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = Left Header, inactive, nohscroll, preserveAspect, L1, 14, "Times"; ; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 14, "Times"; ; fontset = Left Footer, inactive, nohscroll, preserveAspect, center, L1, 14, "Times"; ; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ;] :[font = title; inactive; preserveAspect; rightWrapOffset = 525; fontColorBlue = 65535; ] COSY_PAK Control Systems Analysis Package for Mathematica ;[s] 3:0,0;49,1;60,2;62,-1; 3:1,25,19,Times,1,28,0,0,0;1,26,20,Times,3,28,0,0,0;1,25,19,Times,1,28,0,0,0; :[font = subtitle; inactive; preserveAspect; rightWrapOffset = 525; ] By C.K. Chen N. Sreenath 1992 :[font = message; inactive; preserveAspect; center; rightWrapOffset = 525; fontName = "Courier"; ] Systems Engineering Department Case School of Engineering Case Western Reserve University Cleveland, OH, 44106-7070 :[font = message; inactive; dontPreserveAspect; center; rightWrapOffset = 525; ] with support from CWRU Information and Network Services - Dr. Ray Neff Case Alumni Association The Lilly Foundation :[font = input; preserveAspect; rightWrapOffset = 525; ] :[font = subtitle; inactive; preserveAspect; rightWrapOffset = 525; ] Chapter 3 Transient-Response Analysis ;[s] 3:0,0;9,1;10,2;37,-1; 3:1,20,15,Times,1,22,65535,0,0;1,20,15,Times,1,22,0,0,0;1,16,12,Times,1,18,0,0,65535; :[font = section; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Initialization :[font = input; initialization; preserveAspect; rightWrapOffset = 525; ] *) (* Initialization of Path *) (* Example For UNIX machine (Default) *) $Path=Join[$Path, {"~/Library/Mathematica/Packages"}]; (* Example For IBM PC *) (* $Path=Join[$Path, {"c:\winmath\packages"}]; *) (* Example For MAC *) (* $Path=Join[$Path, {"My_Harddisk:Mathematica:Package"}]; *) (* :[font = input; initialization; preserveAspect; rightWrapOffset = 525; startGroup; ] *) Needs["COSYPAK`chap3`"] (* :[font = print; inactive; preserveAspect; rightWrapOffset = 525; endGroup; endGroup; ] Support module has been loaded. The knowledge base of signal processing operators has been\ loaded. Region of convergence routines for the transform rule bases\ are loaded. Supporting routines for transform rule bases are loaded. Supporting routines and objects for the Laplace transform\ are loaded. Supporting routines for filter design are loaded. The forward Laplace transform rule base LaPlace has been\ loaded. The inverse Laplace transform rule base InvLaPlace has been\ loaded. :[font = section; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Acknowledgements :[font = text; inactive; preserveAspect; rightWrapOffset = 525; endGroup; ] Special thanks to Brian Evans of Georgia Tech for the LaPlace transform and signal packages which is a part of the Signal Processing Packages :Copyright: Copyright 1989-1991 by Brian L. Evans, Georgia Tech Research Corporation.. :[font = section; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Time Response Analysis :[font = text; inactive; preserveAspect; rightWrapOffset = 525; endGroup; ] This section contains the standard forms and the signal responses of first, second, and higher order systems. ;[s] 2:0,0;1,1;110,-1; 2:1,11,9,Courier,0,14,0,0,0;1,13,10,Times,0,14,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] First Order System :[font = subsection; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Standard Form :[font = text; inactive; preserveAspect; rightWrapOffset = 525; ] The following function will give the standard form of a first-order system for the specified system parameters. :[font = input; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] gain = 1; timeconst = 1; g1 = gain/(s timeconst+1) :[font = output; output; inactive; preserveAspect; rightWrapOffset = 525; endGroup; endGroup; ] (1 + s)^(-1) ;[o] 1 ----- 1 + s :[font = subsection; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Impulse Response :[font = text; inactive; preserveAspect; rightWrapOffset = 525; ] COSY_PAK Function Response[ transf, Input, s, {TimeVar, StartTime, EndTime}, gopts]: Plots the output of transfer function `transf' with Laplace input signal `Input'. The Laplace variable is `s'. The output graph uses the variable `TimeVar' and starts at `StartTime' and ends at `EndTime'. Returns the output variable as a function of time `t'. ;[s] 16:0,0;1,1;19,2;20,3;87,4;126,5;132,6;161,7;166,8;260,9;269,10;284,11;291,12;346,13;347,14;348,15;351,-1; 16:1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = text; inactive; preserveAspect; rightWrapOffset = 525; ] There is primarily one parameter in the first order system that dictates its behavior: the timeconstant.The gain does not change the behavior. Example : This is an example of a first order system. After viewing this you can change the timeconstant and/or the gain and watch the behavior. ;[s] 4:0,0;143,1;144,2;153,3;288,-1; 4:1,13,10,Times,0,14,0,0,0;1,11,9,Courier,0,14,0,0,0;1,11,9,Courier,1,14,0,0,0;1,11,9,Courier,0,14,0,0,0; :[font = input; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] results = Response[g1,1,s,{t,0,7},FrameLabel->{"Time","Y(t)", "Impulse Response of G(s) "," "}]; (* The time response function is in results *) results :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; rightWrapOffset = 525; pictureLeft = 34; pictureWidth = 415; pictureHeight = 256; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.037758 0.134062 0.014715 0.579506 [ [(0)] 0.03776 0 0 2 0 Minner Mrotsboxa [(1)] 0.17182 0 0 2 0 Minner Mrotsboxa [(2)] 0.30588 0 0 2 0 Minner Mrotsboxa [(3)] 0.43994 0 0 2 0 Minner Mrotsboxa [(4)] 0.57401 0 0 2 0 Minner Mrotsboxa 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((-1 + E^t)*CStep[t])/E^t ;[o] t (-1 + E ) CStep[t] ------------------ t E :[font = text; inactive; preserveAspect; rightWrapOffset = 525; endGroup; endGroup; ] The graph indicates the input (step signal indicated by dashed line) as well as the response. 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Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; rightWrapOffset = 525; ] ((1 - E^t + E^t*t)*CStep[t])/E^t ;[o] t t (1 - E + E t) CStep[t] ------------------------ t E :[font = text; inactive; preserveAspect; rightWrapOffset = 525; ] Note the steady state error that characterizes the response first-order system to a ramp input is equal to the timeconstant. ;[s] 2:0,0;1,1;127,-1; 2:1,11,9,Courier,0,14,0,0,0;1,13,10,Times,0,14,0,0,0; :[font = output; output; inactive; preserveAspect; rightWrapOffset = 525; endGroup; endGroup; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[font = section; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] First Order System (Step Response vs. Time Const) :[font = subsection; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] 2-D Animation :[font = text; inactive; preserveAspect; rightWrapOffset = 533; endGroup; ] To begin the animation (movie), select the following bracket with the mouse and type the Apple key and 'y' simultaneously. You can control the speed of the animation by typing a number between 1 (slow) and 9 (fast). Be aware that the bracket you choose should include all the graphs that you wish to animate. The arrows in the lower left corner control the direction of the movie i.e., forward (right arrow) or backward (left arrow). A continous playback option (arrow in the center) can also be selected. 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MathPictureEnd :[font = section; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Second Order System :[font = subsection; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Standard Form :[font = text; inactive; preserveAspect; rightWrapOffset = 525; ] There are primarily two parameters in the second order system : the damping factor and the natural frequency (wn). Example : This is an example of a underdamped second order system. After viewing this you can change the damping factor and the natural frequency and watch the behavior. ;[s] 4:0,0;116,1;126,2;284,3;285,-1; 4:1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,2,14,0,0,0;1,11,8,Times,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] wn = 1; damping = 1/2; g2 = wn^2/(s^2 + 2 damping wn s + wn^2) :[font = output; output; inactive; preserveAspect; rightWrapOffset = 525; endGroup; endGroup; ] (1 + s + s^2)^(-1) ;[o] 1 ---------- 2 1 + s + s :[font = subsection; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Impulse Response :[font = input; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] results = Response[g2,1,s,{t,0,12},FrameLabel->{"Time","Y(t)", "Impulse Response of Second-Order System G(s)"," "}]; (* The time response function is in results. *) results :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; rightWrapOffset = 525; 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lineto 0.77778 0.48069 lineto 0.81746 0.50522 lineto 0.85714 0.52974 lineto 0.89683 0.55427 lineto 0.93651 0.57879 lineto 0.97619 0.60332 lineto stroke grestore grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; rightWrapOffset = 525; ] (CStep[t]*(-3*E^(t/2) + 3*E^(t/2)*t + 3*Cos[(3^(1/2)*t)/2] - 3^(1/2)*Sin[(3^(1/2)*t)/2]))/ (3*E^(t/2)) ;[o] t/2 t/2 Sqrt[3] t (CStep[t] (-3 E + 3 E t + 3 Cos[---------] - 2 Sqrt[3] t t/2 Sqrt[3] Sin[---------])) / (3 E ) 2 :[font = text; inactive; preserveAspect; rightWrapOffset = 525; endGroup; endGroup; endGroup; ] Notice that the steady state error is not zero for this second order system. :[font = section; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] Second Order System (Step Response vs. Damping Ratio) :[font = subsection; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -SurfaceGraphics- :[font = text; inactive; preserveAspect; rightWrapOffset = 525; endGroup; endGroup; endGroup; ] There is an inherent trade-off between system damping and system response. Low damping ratios result in excessive overshoot while high damping ratios are slow to respond to input signals. The best compromise between damping and response lies in damping ratios between 0.4 and 0.8. :[font = subsection; inactive; Cclosed; preserveAspect; rightWrapOffset = 525; startGroup; ] 2-D Animation :[font = text; inactive; preserveAspect; rightWrapOffset = 525; ] To begin the animation (movie), select the following bracket with the mouse and type the Apple key and 'y' simultaneously. You can control the speed of the animation by typing a number between 1 (slow) and 9 (fast). Be aware that the bracket you choose should include all the graphs that you wish to animate. The arrows in the lower left corner control the direction of the movie i.e., forward (right arrow) or backward (left arrow). A continous playback option (arrow in the center) can also be selected. 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