Control package: Difference between revisions
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! Chapter !! Function !! Implemented !! File | ! Chapter !! Function !! Implemented !! File | ||
|- | |- | ||
| rowspan="11" | Basic Models || tf || style="background: green;" | yes || @tf/tf.m | | rowspan="11" | Basic Models || tf || style="background: green;" | yes || @tf/tf.m | ||
|- | |- | ||
| zpk || style="background: green;" | yes || zpk.m | | zpk || style="background: green;" | yes || zpk.m | ||
|- | |- | ||
| ss || style="background: green;" | yes || @ss/ss.m | | ss || style="background: green;" | yes || @ss/ss.m | ||
|- | |- | ||
| frd || style="background: green;" | yes || @frd/frd.m | | frd || style="background: green;" | yes || @frd/frd.m | ||
|- | |- | ||
| pid || style="background: red;" | no | | pid || style="background: red;" | no || | ||
|- | |- | ||
| pidstd || style="background: red;" | no | | pidstd || style="background: red;" | no || | ||
|- | |- | ||
| pid2 || style="background: red;" | no | | pid2 || style="background: red;" | no || | ||
|- | |- | ||
| dss || style="background: green;" | yes || @lti/dss.m | | dss || style="background: green;" | yes || @lti/dss.m | ||
|- | |- | ||
| drss || style="background: red;" | no | | drss || style="background: red;" | no || | ||
|- | |- | ||
| filt || style="background: green;" | yes || filt.m | | filt || style="background: green;" | yes || filt.m | ||
|- | |- | ||
| rss || style="background: red;" | no | | rss || style="background: red;" | no || | ||
|- | |- | ||
| rowspan="24" | Tunable Models || ltiblock.gain || style="background: red;" | no | | rowspan="24" | Tunable Models || ltiblock.gain || style="background: red;" | no || | ||
|- | |- | ||
| ltiblock.pid || style="background: red;" | no | | ltiblock.pid || style="background: red;" | no || | ||
|- | |- | ||
| ltiblock.pid2 || style="background: red;" | no | | ltiblock.pid2 || style="background: red;" | no || | ||
|- | |- | ||
| ltiblock.ss || style="background: red;" | no | | ltiblock.ss || style="background: red;" | no || | ||
|- | |- | ||
| ltiblock.tf || style="background: red;" | no | | ltiblock.tf || style="background: red;" | no || | ||
|- | |- | ||
| realp || style="background: red;" | no | | realp || style="background: red;" | no || | ||
|- | |- | ||
| AnalysisPoint || style="background: red;" | no | | AnalysisPoint || style="background: red;" | no || | ||
|- | |- | ||
| genss || style="background: red;" | no | | genss || style="background: red;" | no || | ||
|- | |- | ||
| genfrd || style="background: red;" | no | | genfrd || style="background: red;" | no || | ||
|- | |- | ||
| genmat || style="background: red;" | no | | genmat || style="background: red;" | no || | ||
|- | |- | ||
| getLoopTransfer || style="background: red;" | no | | getLoopTransfer || style="background: red;" | no || | ||
|- | |- | ||
| getIOTransfer || style="background: red;" | no | | getIOTransfer || style="background: red;" | no || | ||
|- | |- | ||
| getSensitivity || style="background: red;" | no | | getSensitivity || style="background: red;" | no || | ||
|- | |- | ||
| getCompSensitivity || style="background: red;" | no | | getCompSensitivity || style="background: red;" | no || | ||
|- | |- | ||
| getPoints || style="background: red;" | no | | getPoints || style="background: red;" | no || | ||
|- | |- | ||
| replaceBlock || style="background: red;" | no | | replaceBlock || style="background: red;" | no || | ||
|- | |- | ||
| getValue || style="background: red;" | no | | getValue || style="background: red;" | no || | ||
|- | |- | ||
| setValue || style="background: red;" | no | | setValue || style="background: red;" | no || | ||
|- | |- | ||
| getBlockValue || style="background: red;" | no | | getBlockValue || style="background: red;" | no || | ||
|- | |- | ||
| setBlockValue || style="background: red;" | no | | setBlockValue || style="background: red;" | no || | ||
|- | |- | ||
| showBlockValue || style="background: red;" | no | | showBlockValue || style="background: red;" | no || | ||
|- | |- | ||
| showTunable || style="background: red;" | no | | showTunable || style="background: red;" | no || | ||
|- | |- | ||
| nblocks || style="background: red;" | no | | nblocks || style="background: red;" | no || | ||
|- | |- | ||
| getLFTModel || style="background: red;" | no | | getLFTModel || style="background: red;" | no || | ||
|- | |- | ||
| rowspan="9" | Model with Time Delays || pade || style="background: red;" | no | | rowspan="9" | Model with Time Delays || pade || style="background: red;" | no || | ||
|- | |- | ||
| absorbDelay || style="background: red;" | no | | absorbDelay || style="background: red;" | no || | ||
|- | |- | ||
| thiran || style="background: green;" | yes || thiran.m | | thiran || style="background: green;" | yes || thiran.m | ||
|- | |- | ||
| hasdelay || style="background: red;" | no | | hasdelay || style="background: red;" | no || | ||
|- | |- | ||
| hasInternalDelay || style="background: red;" | no | | hasInternalDelay || style="background: red;" | no || | ||
|- | |- | ||
| totaldelay || style="background: red;" | no | | totaldelay || style="background: red;" | no || | ||
|- | |- | ||
| delayss || style="background: red;" | no | | delayss || style="background: red;" | no || | ||
|- | |- | ||
| setDelayModel || style="background: red;" | no | | setDelayModel || style="background: red;" | no || | ||
|- | |- | ||
| getDelayModel || style="background: red;" | no | | getDelayModel || style="background: red;" | no || | ||
|- | |- | ||
| rowspan="26" | Model Attributes || get || style="background: green;" | yes || @iddata/get.m | | rowspan="26" | Model Attributes || get || style="background: green;" | yes || @iddata/get.m | ||
|- | |- | ||
| set || style="background: green;" | yes || @iddata/set.m | | set || style="background: green;" | yes || @iddata/set.m | ||
|- | |- | ||
| tfdata || style="background: green;" | yes || @lti/tfdata.m | | tfdata || style="background: green;" | yes || @lti/tfdata.m | ||
|- | |- | ||
| zpkdata || style="background: green;" | yes || @lti/zpkdata.m | | zpkdata || style="background: green;" | yes || @lti/zpkdata.m | ||
|- | |- | ||
| ssdata || style="background: green;" | yes || @lti/ssdata.m | | ssdata || style="background: green;" | yes || @lti/ssdata.m | ||
|- | |- | ||
| frdata || style="background: green;" | yes || @lti/frdata.m | | frdata || style="background: green;" | yes || @lti/frdata.m | ||
|- | |- | ||
| piddata || style="background: red;" | no | | piddata || style="background: red;" | no || | ||
|- | |- | ||
| pidstddata || style="background: red;" | no | | pidstddata || style="background: red;" | no || | ||
|- | |- | ||
| piddata2 || style="background: red;" | no | | piddata2 || style="background: red;" | no || | ||
|- | |- | ||
| pidstddata2 || style="background: red;" | no | | pidstddata2 || style="background: red;" | no || | ||
|- | |- | ||
| dssdata || style="background: green;" | yes || @lti/dssdata.m | | dssdata || style="background: green;" | yes || @lti/dssdata.m | ||
|- | |- | ||
| chgFreqUnit || style="background: red;" | no | | chgFreqUnit || style="background: red;" | no || | ||
|- | |- | ||
| chgTimeUnit || style="background: red;" | no | | chgTimeUnit || style="background: red;" | no || | ||
|- | |- | ||
| isct || style="background: green;" | yes || @lti/isct.m | | isct || style="background: green;" | yes || @lti/isct.m | ||
|- | |- | ||
| isdt || style="background: green;" | yes || @lti/isdt.m | | isdt || style="background: green;" | yes || @lti/isdt.m | ||
|- | |- | ||
| isempty || style="background: green;" | yes || built-in function | | isempty || style="background: green;" | yes || built-in function | ||
|- | |- | ||
| isfinite || style="background: green;" | yes || built-in function | | isfinite || style="background: green;" | yes || built-in function | ||
|- | |- | ||
| isParametric || style="background: red;" | no | | isParametric || style="background: red;" | no || | ||
|- | |- | ||
| isproper || style="background: red;" | no | | isproper || style="background: red;" | no || | ||
|- | |- | ||
| isreal || style="background: green;" | yes || built-in function | | isreal || style="background: green;" | yes || built-in function | ||
|- | |- | ||
| isiso || style="background: red;" | no | | isiso || style="background: red;" | no || | ||
|- | |- | ||
| isstable || style="background: green;" | yes || @lti/isstable.m | | isstable || style="background: green;" | yes || @lti/isstable.m | ||
|- | |- | ||
| isstatic || style="background: red;" | no | | isstatic || style="background: red;" | no || | ||
|- | |- | ||
| order || style="background: red;" | no | | order || style="background: red;" | no || | ||
|- | |- | ||
| ndims || style="background: green;" | yes || built-in function | | ndims || style="background: green;" | yes || built-in function | ||
|- | |- | ||
| size || style="background: green;" | yes || built-in function | | size || style="background: green;" | yes || built-in function | ||
|- | |- | ||
| rowspan="5" | Model Arrays || stack || style="background: red;" | no | | rowspan="5" | Model Arrays || stack || style="background: red;" | no || | ||
|- | |- | ||
| nmodels || style="background: red;" | no | | nmodels || style="background: red;" | no || | ||
|- | |- | ||
| permute || style="background: green;" | yes || built-in function | | permute || style="background: green;" | yes || built-in function | ||
|- | |- | ||
| reshape || style="background: green;" | yes || built-in function | | reshape || style="background: green;" | yes || built-in function | ||
|- | |- | ||
| repsys || style="background: green;" | yes || repsys.m | | repsys || style="background: green;" | yes || repsys.m | ||
|- | |- | ||
|} | |} | ||
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! Function !! Implemented !! File | ! Function !! Implemented !! File | ||
|- | |- | ||
| feedback || style="background: green;" | yes || @lti/feedback.m | | feedback || style="background: green;" | yes || @lti/feedback.m | ||
|- | |- | ||
| connect || style="background: green;" | yes || @lti/connect.m | | connect || style="background: green;" | yes || @lti/connect.m | ||
|- | |- | ||
| sumblk || style="background: green;" | yes || sumblk.m | | sumblk || style="background: green;" | yes || sumblk.m | ||
|- | |- | ||
| series || style="background: | | series || style="background: green;" | yes || @lti/series.m | ||
|- | |- | ||
| parallel || style="background: | | parallel || style="background: green;" | yes || @lti/parallel.m | ||
|- | |- | ||
| append || style="background: green;" | yes || append.m | | append || style="background: green;" | yes || append.m | ||
|- | |- | ||
| blkdiag || style="background: green;" | yes || @lti/blkdiag.m | | blkdiag || style="background: green;" | yes || @lti/blkdiag.m | ||
|- | |- | ||
| imp2exp || style="background: red;" | no | | imp2exp || style="background: red;" | no || | ||
|- | |- | ||
| inv || style="background: green;" | yes || built-in function | | inv || style="background: green;" | yes || built-in function | ||
|- | |- | ||
| lft || style="background: red;" | no | | lft || style="background: red;" | no || | ||
|- | |- | ||
| connectOptions || style="background: red;" | no | | connectOptions || style="background: red;" | no || | ||
|- | |- | ||
|} | |} | ||
Line 317: | Line 317: | ||
| getPeakGain || style="background: red;" | no || || || || || | | getPeakGain || style="background: red;" | no || || || || || | ||
|- | |- | ||
| rowspan=" | | rowspan="11" | Stability Analysis || pole || style="background: green;" | yes || @lti/pole.m || || || || | ||
|- | |- | ||
| zero || style="background: green;" | yes || @lti/zero.m || || || || | | zero || style="background: green;" | yes || @lti/zero.m || || || || | ||
|- | |- | ||
| damp || style="background: | | damp || style="background: green;" | yes || damp.m || || || || | ||
|- | |- | ||
| dsort || style="background: | | dsort || style="background: green;" | yes || dsort.m || || || || | ||
|- | |- | ||
| esort || style="background: | | esort || style="background: green;" | yes || esort.m || || || || | ||
|- | |- | ||
| tzero || style="background: red;" | no || || || || || | | tzero || style="background: red;" | no || || || || || | ||
|- | |||
| pzmap || style="background: green;" | yes || pzmap.m || || || || | |||
|- | |- | ||
| pzplot || style="background: red;" | no || || || || || | | pzplot || style="background: red;" | no || || || || || | ||
Line 373: | Line 375: | ||
| pidtuneOptions || style="background: red;" | no || || || || || | | pidtuneOptions || style="background: red;" | no || || || || || | ||
|- | |- | ||
| rowspan=" | | rowspan="5" | SISO Feedback Loops || rlocus || style="background: green;" | yes || rlocus.m || || || || | ||
|- | |||
| sgrid || style="background: red;" | no || || || || || | |||
|- | |- | ||
| rlocusplot || style="background: red;" | no || || || || || | | rlocusplot || style="background: red;" | no || || || || || | ||
Line 452: | Line 456: | ||
=== PT1 / Low-pass filter step response === | === PT1 / Low-pass filter step response === | ||
<!-- {{SyntaxHighlight| --> | <!-- {{SyntaxHighlight| --> | ||
{{Code|Creating a transfer function and plotting its response|<syntaxhighlight lang="octave" style="font-size:14px"> | {{Code|Creating a transfer function and plotting its response|<syntaxhighlight lang="octave" style="font-size:14px"> | ||
T1 = 0.4; # time constant | T1 = 0.4; # time constant | ||
P = tf([1], [T1 1]) | P = tf([1], [T1 1]); # create transfer function model | ||
step(P, 2) # plot step response | step(P, 2) # plot step response | ||
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[[File:Pt1.png|600px]] | [[File:Pt1.png|600px]] | ||
==== Bode Diagram with TikZ/PGFplots ==== | |||
We use the same system as before but we draw now with <code style="font-size:14px; border:solid lightgray 1px; padding: 2px">bode(P)</code> a bode diagram. | |||
The output is written to a .csv file. | |||
<!-- {{SyntaxHighlight| --> | |||
{{Code|Creating a transfer function and calculating its bode diagram|<syntaxhighlight lang="octave" style="font-size:14px"> | |||
T1 = 0.4; # time constant | |||
P = tf([1], [T1 1]); | |||
[mag, pha, w] = bode(P); | |||
csvwrite("dat/bode_p.csv", [w', 20*log10(mag), pha]); | |||
</syntaxhighlight>}} | |||
We can then invoke LaTeX with the following .tex file | |||
<!-- {{SyntaxHighlight| --> | |||
{{Code|Creating a bode diagram with pgfplots|<syntaxhighlight lang="latex" style="font-size:14px"> | |||
\documentclass[tikz]{standalone} | |||
\usepackage{pgfplotstable} | |||
\usepackage{siunitx} | |||
\usetikzlibrary{pgfplots.groupplots} | |||
\begin{document} | |||
\pgfplotstableread[col sep=comma]{\detokenize{/path/to/csv/file/dat/bode_p.csv}}\datatable | |||
\begin{tikzpicture} | |||
\begin{groupplot}[ | |||
group style={rows=2}, | |||
width=0.8\textwidth, | |||
height=0.4\textwidth, | |||
xmajorgrids, | |||
ymajorgrids, | |||
enlarge x limits=false, | |||
xmode=log, | |||
] | |||
\nextgroupplot[ | |||
title={Bode Diagram of $P$}, | |||
ylabel={Magnitude / \si{\decibel}}, | |||
] | |||
\addplot[blue,line width=1pt] table[x index=0,y index=1]{\datatable}; | |||
\nextgroupplot[ | |||
xlabel={Frequency / \si{\radian\per\second}}, | |||
ylabel={Phase / \si{\degree}}, | |||
] | |||
\addplot[red,line width=1pt] table[x index=0,y index=2]{\datatable}; | |||
\end{groupplot} | |||
\end{tikzpicture} | |||
\end{document} | |||
</syntaxhighlight>}} | |||
to generate a beautiful bode diagram | |||
[[File:bode_p.png]] | |||
It can be seen that a first order low-pass filter has <math display="inline">-3\,dB</math> magnitude at <math display="inline">f=1/T_1=2.5\,rad/s</math>. | |||
=== Inverted Pendulum === | === Inverted Pendulum === | ||
==== Model ==== | |||
A nonlinear model of the inverted pendulum can be derived by | A nonlinear model of the inverted pendulum can be derived by | ||
Line 506: | Line 568: | ||
|} | |} | ||
[[File:inverted_pendulum_new.png]] | [[File:inverted_pendulum_new.png|frame|Schematic drawing of the inverted pendulum.]] | ||
Linearization around the point <math display="inline">\theta=\pi</math> and substitution of <math display="inline">\theta'=\theta-\pi</math> (from here on in the analysis, <math display="inline">\theta'</math> will be written as <math display="inline">\theta</math>, but it should be noted that <math display="inline">\theta</math> now | Linearization around the point <math display="inline">\theta=\pi</math> and substitution of <math display="inline">\theta'=\theta-\pi</math> (from here on in the analysis, <math display="inline">\theta'</math> will be written as <math display="inline">\theta</math>, but it should be noted that <math display="inline">\theta</math> now | ||
Line 527: | Line 589: | ||
</math> | </math> | ||
==== Transfer functions ==== | |||
By dividing the state variables by the input one obtains the transfer functions | |||
<math> | <math> | ||
Line 560: | Line 624: | ||
<!-- {{SyntaxHighlight| --> | <!-- {{SyntaxHighlight| --> | ||
{{Code|Creating the transfer functions for the inverted pendulum|<syntaxhighlight lang="octave" style="font-size:14px"> | {{Code|Creating the transfer functions for the inverted pendulum|<syntaxhighlight lang="octave" style="font-size:14px"> | ||
m = 0.15; | |||
l = 0.314; | |||
M = 1.3; | |||
I = 0; | |||
g = 9.80665; | |||
b = 20; | |||
c = 20; | |||
G1 = tf([m*l^2+I b -m*g*l], | |||
[(M+m)*(m*l^2+I)-m^2*l^2 (M+m)*b+(m*l^2+I)*c -(M+m)*m*g*l+b*c -m*g*l*c 0]); | |||
G2 = tf([-m*l], | |||
[(M+m)*(m*l^2+I)-m^2*l^2 (M+m)*b+(m*l^2+I)*c -(M+m)*m*g*l+b*c -m*g*l*c]); | |||
</syntaxhighlight>}} | </syntaxhighlight>}} | ||
Line 577: | Line 642: | ||
<!-- {{SyntaxHighlight| --> | <!-- {{SyntaxHighlight| --> | ||
{{Code|Creating Bode plots for the inverted pendulum|<syntaxhighlight lang="octave" style="font-size:14px"> | {{Code|Creating Bode plots for the inverted pendulum|<syntaxhighlight lang="octave" style="font-size:14px"> | ||
bode(G1); | |||
bode(G2); | |||
</syntaxhighlight>}} | </syntaxhighlight>}} | ||
one obtains the | one obtains the bode diagrams of the two transfer functions. | ||
<div><ul> | |||
<li style="display: inline-block;"> [[File:bode_g1.png|frame|Bode diagram of the cart movement transfer function.]] </li> | |||
<li style="display: inline-block;"> [[File:bode_g2.png|frame|Bode diagram of the pendulums rotation transfer function.]] </li> | |||
</ul></div> | |||
[[Category:Octave | [[Category:Octave Forge]] |
Revision as of 10:05, 14 June 2019
The control package is part of the Octave Forge project.
Function list
Linear System Representation
Chapter | Function | Implemented | File |
---|---|---|---|
Basic Models | tf | yes | @tf/tf.m |
zpk | yes | zpk.m | |
ss | yes | @ss/ss.m | |
frd | yes | @frd/frd.m | |
pid | no | ||
pidstd | no | ||
pid2 | no | ||
dss | yes | @lti/dss.m | |
drss | no | ||
filt | yes | filt.m | |
rss | no | ||
Tunable Models | ltiblock.gain | no | |
ltiblock.pid | no | ||
ltiblock.pid2 | no | ||
ltiblock.ss | no | ||
ltiblock.tf | no | ||
realp | no | ||
AnalysisPoint | no | ||
genss | no | ||
genfrd | no | ||
genmat | no | ||
getLoopTransfer | no | ||
getIOTransfer | no | ||
getSensitivity | no | ||
getCompSensitivity | no | ||
getPoints | no | ||
replaceBlock | no | ||
getValue | no | ||
setValue | no | ||
getBlockValue | no | ||
setBlockValue | no | ||
showBlockValue | no | ||
showTunable | no | ||
nblocks | no | ||
getLFTModel | no | ||
Model with Time Delays | pade | no | |
absorbDelay | no | ||
thiran | yes | thiran.m | |
hasdelay | no | ||
hasInternalDelay | no | ||
totaldelay | no | ||
delayss | no | ||
setDelayModel | no | ||
getDelayModel | no | ||
Model Attributes | get | yes | @iddata/get.m |
set | yes | @iddata/set.m | |
tfdata | yes | @lti/tfdata.m | |
zpkdata | yes | @lti/zpkdata.m | |
ssdata | yes | @lti/ssdata.m | |
frdata | yes | @lti/frdata.m | |
piddata | no | ||
pidstddata | no | ||
piddata2 | no | ||
pidstddata2 | no | ||
dssdata | yes | @lti/dssdata.m | |
chgFreqUnit | no | ||
chgTimeUnit | no | ||
isct | yes | @lti/isct.m | |
isdt | yes | @lti/isdt.m | |
isempty | yes | built-in function | |
isfinite | yes | built-in function | |
isParametric | no | ||
isproper | no | ||
isreal | yes | built-in function | |
isiso | no | ||
isstable | yes | @lti/isstable.m | |
isstatic | no | ||
order | no | ||
ndims | yes | built-in function | |
size | yes | built-in function | |
Model Arrays | stack | no | |
nmodels | no | ||
permute | yes | built-in function | |
reshape | yes | built-in function | |
repsys | yes | repsys.m |
Model Interconnection
Function | Implemented | File |
---|---|---|
feedback | yes | @lti/feedback.m |
connect | yes | @lti/connect.m |
sumblk | yes | sumblk.m |
series | yes | @lti/series.m |
parallel | yes | @lti/parallel.m |
append | yes | append.m |
blkdiag | yes | @lti/blkdiag.m |
imp2exp | no | |
inv | yes | built-in function |
lft | no | |
connectOptions | no |
Model Transformation
Chapter | Function | Implemented | File | Number of Tests | Status | SLICOT functions | Priority (0-2) |
---|---|---|---|---|---|---|---|
Model Type Conversion | pidstd2 | no | |||||
make1DOF | no | ||||||
make2DOF | no | ||||||
getComponents | no | ||||||
Continuous-Discrete Conversion | c2d | yes | @lti/c2d.m | ||||
d2c | yes | @lti/d2c.m | |||||
d2d | yes | @lti/d2d.m | |||||
upsample | no | ||||||
c2dOptions | no | ||||||
d2cOptions | no | ||||||
d2dOptions | no | ||||||
Model Simplification | hsvd | yes | hsvd.m | ||||
hsvplot | no | ||||||
sminreal | yes | @lti/sminreal.m | |||||
balred | no | ||||||
minreal | yes | @lti/minreal.m | |||||
balreal | no | ||||||
modred | no | ||||||
balredOptions | no | ||||||
hsvdOptions | no | ||||||
State-Coordinate Transformation | canon | no | |||||
prescale | yes | @lti/prescale.m | |||||
ss2ss | no | ||||||
xperm | yes | @lti/xperm.m | |||||
Modal Decomposition | modsep | no | |||||
stabsep | no | ||||||
freqsep | no | ||||||
stabsepOptions | no | ||||||
freqsepOptions | no |
Linear Analysis
Chapter | Function | Implemented | File | Number of Tests | Status | SLICOT functions | Priority (0-2) |
---|---|---|---|---|---|---|---|
Time-Domain Analysis | linearSystemAnalyzer | no | |||||
impulseplot | no | ||||||
initialplot | no | ||||||
lsimplot | no | ||||||
stepplot | no | ||||||
covar | yes | covar.m | |||||
impulse | yes | impulse.m | |||||
initial | yes | initial.m | |||||
lsim | yes | lsim.m | |||||
step | yes | step.m | |||||
lsiminfo | no | ||||||
stepinfo | no | ||||||
stepDataOptions | no | ||||||
Frequency-Domain Analysis | bodeplot | no | |||||
nicholsplot | no | ||||||
nyquistplot | no | ||||||
sigmaplot | no | ||||||
bode | yes | bode.m | |||||
nichols | yes | nichols.m | |||||
nyquist | yes | nyquist.m | |||||
sigma | yes | sigma.m | |||||
evalfr | no | ||||||
freqresp | yes | @lti/freqresp.m | |||||
bandwidth | yes | built-in function | |||||
dcgain | yes | @lti/dcgain.m | |||||
getGainCrossover | no | ||||||
getPeakGain | no | ||||||
Stability Analysis | pole | yes | @lti/pole.m | ||||
zero | yes | @lti/zero.m | |||||
damp | yes | damp.m | |||||
dsort | yes | dsort.m | |||||
esort | yes | esort.m | |||||
tzero | no | ||||||
pzmap | yes | pzmap.m | |||||
pzplot | no | ||||||
iopzplot | no | ||||||
allmargin | no | ||||||
margin | yes | margin.m | |||||
Plot Customization | bodeoptions | no | |||||
hsvoptions | no | ||||||
nicholsoptions | no | ||||||
nyquistoptions | no | ||||||
pzoptions | no | ||||||
sigmaoptions | no | ||||||
timeoptions | no | ||||||
setoptions | no | ||||||
getoptions | no | ||||||
ctrlpref | no | ||||||
updateSystem | no |
Control Design
Chapter | Function | Implemented | File | Number of Tests | Status | SLICOT functions | Priority (0-2) |
---|---|---|---|---|---|---|---|
PID Controller Tuning | pidTuner | no | |||||
pidtune | no | ||||||
pidtuneOptions | no | ||||||
SISO Feedback Loops | rlocus | yes | rlocus.m | ||||
sgrid | no | ||||||
rlocusplot | no | ||||||
controlSystemDesigner | no | ||||||
sisoinit | no | ||||||
Linear-Quadratic-Gaussian Control | lqr | yes | lqr.m | ||||
lqry | no | ||||||
lqi | no | ||||||
dlqr | yes | dlqr.m | |||||
lqrd | no | ||||||
kalman | yes | kalman.m | |||||
kalmd | no | ||||||
lqg | no | ||||||
lqgreg | no | ||||||
lqgtrack | no | ||||||
augstate | no | ||||||
norm | yes | @lti/norm.m | |||||
Pole Placement | estim | yes | estim.m | ||||
place | yes | place.m | |||||
reg | no |
Matrix Computations
Function | Implemented | File | Number of Tests | Status | SLICOT functions | Priority (0-2) |
---|---|---|---|---|---|---|
lyap | yes | lyap.m | ||||
lyapchol | yes | lyapchol.m | ||||
dlyap | yes | dlyap.m | ||||
dlyapchol | yes | dlyapchol.m | ||||
care | yes | care.m | ||||
dare | yes | dare.m | ||||
gcare | no | |||||
gdare | no | |||||
ctrb | yes | ctrb.m | ||||
obsv | yes | obsv.m | ||||
ctrbf | yes | ctrbf.m | ||||
obsvf | yes | obsvf.m | ||||
gram | yes | gram.m | ||||
bdschur | no |
Examples
PT1 / Low-pass filter step response
Code: Creating a transfer function and plotting its response |
T1 = 0.4; # time constant
P = tf([1], [T1 1]); # create transfer function model
step(P, 2) # plot step response
#add some common markers like the tangent line at the origin, which crosses lim(n->inf) f(t) at t=T1
hold on
plot ([0 T1], [0 1], "g")
plot ([T1 T1], [0 1], "k")
plot ([0 T1], [1-1/e 1-1/e], "m")
hold off
|
Bode Diagram with TikZ/PGFplots
We use the same system as before but we draw now with bode(P)
a bode diagram.
The output is written to a .csv file.
Code: Creating a transfer function and calculating its bode diagram |
T1 = 0.4; # time constant
P = tf([1], [T1 1]);
[mag, pha, w] = bode(P);
csvwrite("dat/bode_p.csv", [w', 20*log10(mag), pha]);
|
We can then invoke LaTeX with the following .tex file
Code: Creating a bode diagram with pgfplots |
\documentclass[tikz]{standalone}
\usepackage{pgfplotstable}
\usepackage{siunitx}
\usetikzlibrary{pgfplots.groupplots}
\begin{document}
\pgfplotstableread[col sep=comma]{\detokenize{/path/to/csv/file/dat/bode_p.csv}}\datatable
\begin{tikzpicture}
\begin{groupplot}[
group style={rows=2},
width=0.8\textwidth,
height=0.4\textwidth,
xmajorgrids,
ymajorgrids,
enlarge x limits=false,
xmode=log,
]
\nextgroupplot[
title={Bode Diagram of $P$},
ylabel={Magnitude / \si{\decibel}},
]
\addplot[blue,line width=1pt] table[x index=0,y index=1]{\datatable};
\nextgroupplot[
xlabel={Frequency / \si{\radian\per\second}},
ylabel={Phase / \si{\degree}},
]
\addplot[red,line width=1pt] table[x index=0,y index=2]{\datatable};
\end{groupplot}
\end{tikzpicture}
\end{document}
|
to generate a beautiful bode diagram
It can be seen that a first order low-pass filter has magnitude at .
Inverted Pendulum
Model
A nonlinear model of the inverted pendulum can be derived by
where the variables are defined as
Name | Definition |
---|---|
Mass of the cart | |
Mass of the pendulum | |
Translational damping coefficient | |
Length of the pendulum | |
Inertia of the pendulum | |
Rotational damping coefficient |
Linearization around the point and substitution of (from here on in the analysis, will be written as , but it should be noted that now measures from a new reference) leads to
This can be expressed in the frequency domain like
Transfer functions
By dividing the state variables by the input one obtains the transfer functions
with
This can be expressed in Octave
Code: Creating the transfer functions for the inverted pendulum |
m = 0.15;
l = 0.314;
M = 1.3;
I = 0;
g = 9.80665;
b = 20;
c = 20;
G1 = tf([m*l^2+I b -m*g*l],
[(M+m)*(m*l^2+I)-m^2*l^2 (M+m)*b+(m*l^2+I)*c -(M+m)*m*g*l+b*c -m*g*l*c 0]);
G2 = tf([-m*l],
[(M+m)*(m*l^2+I)-m^2*l^2 (M+m)*b+(m*l^2+I)*c -(M+m)*m*g*l+b*c -m*g*l*c]);
|
and by invoking the bode
command
Code: Creating Bode plots for the inverted pendulum |
bode(G1);
bode(G2);
|
one obtains the bode diagrams of the two transfer functions.