Editing Control package

The {{Forge|control}} package is part of the [[Octave Forge]] project.

== Function list ==

=== Linear System Representation ===

{| class="wikitable" 
|-
! Chapter !! Function !! Implemented !! File !! Number of Tests !! Status !! SLICOT functions !! Priority (0-2)
|-
| rowspan="11" | Basic Models || tf     || style="background: green;" | yes || @tf/tf.m   || || || ||
|-
|                                zpk    || style="background: green;" | yes || zpk.m      || || || ||
|-
|                                ss     || style="background: green;" | yes || @ss/ss.m   || || || ||
|-
|                                frd    || style="background: green;" | yes || @frd/frd.m || || || ||
|- 
|                                pid    || style="background: red;"   | no  ||            || || || ||
|-
|                                pidstd || style="background: red;"   | no  ||            || || || ||
|-
|                                pid2   || style="background: red;"   | no  ||            || || || ||
|-
|                                dss    || style="background: green;" | yes || @lti/dss.m || || || ||
|-
|                                drss   || style="background: red;"   | no  ||            || || || ||
|-
|                                filt   || style="background: green;" | yes || filt.m     || || || ||
|-
|                                rss    || style="background: red;"   | no  ||            || || || ||
|-
| rowspan="24" | Tunable Models || ltiblock.gain      || style="background: red;" | no || || || || ||
|-
|                                  ltiblock.pid       || style="background: red;" | no || || || || ||
|-
|                                  ltiblock.pid2      || style="background: red;" | no || || || || ||
|-
|                                  ltiblock.ss        || style="background: red;" | no || || || || ||
|-
|                                  ltiblock.tf        || style="background: red;" | no || || || || ||
|-
|                                  realp              || style="background: red;" | no || || || || ||
|-
|                                  AnalysisPoint      || style="background: red;" | no || || || || ||
|-
|                                  genss              || style="background: red;" | no || || || || ||
|-
|                                  genfrd             || style="background: red;" | no || || || || ||
|-
|                                  genmat             || style="background: red;" | no || || || || ||
|-
|                                  getLoopTransfer    || style="background: red;" | no || || || || ||
|-
|                                  getIOTransfer      || style="background: red;" | no || || || || ||
|-
|                                  getSensitivity     || style="background: red;" | no || || || || ||
|-
|                                  getCompSensitivity || style="background: red;" | no || || || || ||
|-
|                                  getPoints          || style="background: red;" | no || || || || ||
|-
|                                  replaceBlock       || style="background: red;" | no || || || || ||
|-
|                                  getValue           || style="background: red;" | no || || || || ||
|-
|                                  setValue           || style="background: red;" | no || || || || ||
|-
|                                  getBlockValue      || style="background: red;" | no || || || || ||
|-
|                                  setBlockValue      || style="background: red;" | no || || || || ||
|-
|                                  showBlockValue     || style="background: red;" | no || || || || ||
|-
|                                  showTunable        || style="background: red;" | no || || || || ||
|-
|                                  nblocks            || style="background: red;" | no || || || || ||
|-
|                                  getLFTModel        || style="background: red;" | no || || || || ||
|-
| rowspan="9" | Model with Time Delays || pade             || style="background: red;"   | no  ||          || || || ||
|-
|                                         absorbDelay      || style="background: red;"   | no  ||          || || || ||
|-
|                                         thiran           || style="background: green;" | yes || thiran.m || || || ||
|-
|                                         hasdelay         || style="background: red;"   | no  ||          || || || ||
|-
|                                         hasInternalDelay || style="background: red;"   | no  ||          || || || ||
|-
|                                         totaldelay       || style="background: red;"   | no  ||          || || || ||
|-
|                                         delayss          || style="background: red;"   | no  ||          || || || ||
|-
|                                         setDelayModel    || style="background: red;"   | no  ||          || || || ||
|-
|                                         getDelayModel    || style="background: red;"   | no  ||          || || || ||
|-
| rowspan="26" | Model Attributes || get          || style="background: green;" | yes || @iddata/get.m     || || || ||
|-
|                                    set          || style="background: green;" | yes || @iddata/set.m     || || || ||
|-
|                                    tfdata       || style="background: green;" | yes || @lti/tfdata.m     || || || ||
|-
|                                    zpkdata      || style="background: green;" | yes || @lti/zpkdata.m    || || || ||
|-
|                                    ssdata       || style="background: green;" | yes || @lti/ssdata.m     || || || ||
|-
|                                    frdata       || style="background: green;" | yes || @lti/frdata.m     || || || ||
|-
|                                    piddata      || style="background: red;"   | no  ||                   || || || ||
|-
|                                    pidstddata   || style="background: red;"   | no  ||                   || || || ||
|-
|                                    piddata2     || style="background: red;"   | no  ||                   || || || ||
|-
|                                    pidstddata2  || style="background: red;"   | no  ||                   || || || ||
|-
|                                    dssdata      || style="background: green;" | yes || @lti/dssdata.m    || || || ||
|-
|                                    chgFreqUnit  || style="background: red;"   | no  ||                   || || || ||
|-
|                                    chgTimeUnit  || style="background: red;"   | no  ||                   || || || ||
|-
|                                    isct         || style="background: green;" | yes || @lti/isct.m       || || || ||
|-
|                                    isdt         || style="background: green;" | yes || @lti/isdt.m       || || || ||
|-
|                                    isempty      || style="background: green;" | yes || built-in function || || || ||
|-
|                                    isfinite     || style="background: green;" | yes || built-in function || || || ||
|-
|                                    isParametric || style="background: red;"   | no  ||                   || || || ||
|-
|                                    isproper     || style="background: red;"   | no  ||                   || || || ||
|-
|                                    isreal       || style="background: green;" | yes || built-in function || || || ||
|-
|                                    isiso        || style="background: red;"   | no  ||                   || || || ||
|-
|                                    isstable     || style="background: green;" | yes || @lti/isstable.m   || || || ||
|-
|                                    isstatic     || style="background: red;"   | no  ||                   || || || ||
|-
|                                    order        || style="background: red;"   | no  ||                   || || || ||
|-
|                                    ndims        || style="background: green;" | yes || built-in function || || || ||
|-
|                                    size         || style="background: green;" | yes || built-in function || || || ||
|-
| rowspan="5" | Model Arrays || stack   || style="background: red;"   | no  ||                   || || || ||
|-
|                               nmodels || style="background: red;"   | no  ||                   || || || ||
|-
|                               permute || style="background: green;" | yes || built-in function || || || ||
|-
|                               reshape || style="background: green;" | yes || built-in function || || || ||
|-
|                               repsys  || style="background: green;" | yes || repsys.m          || || || ||
|-
|}

=== Model Interconnection ===

{| class="wikitable" 
|-
! Function !! Implemented !! File !! Number of Tests !! Status !! SLICOT functions !! Priority (0-2)
|-
| feedback || style="background: green;" | yes || @lti/feedback.m ||  || || ||
|-
| connect || style="background: green;" | yes || @lti/connect.m ||  || || ||
|-
| sumblk || style="background: green;" | yes || sumblk.m ||  || || ||
|-
| series || style="background: red;" | no ||  ||  || || ||
|-
| parallel || style="background: red;" | no ||  ||  || || ||
|-
| append || style="background: green;" | yes || append.m ||  || || ||
|-
| blkdiag || style="background: green;" | yes || @lti/blkdiag.m ||  || || ||
|-
| imp2exp || style="background: red;" | no ||  ||  || || ||
|-
| inv || style="background: green;" | yes || built-in function ||  || || ||
|-
| lft || style="background: red;" | no ||  ||  || || ||
|-
| connectOptions || style="background: red;" | no ||  ||  || || ||
|-
|}

=== Model Transformation ===

{| class="wikitable" 
|-
! Chapter !! Function !! Implemented !! File !! Number of Tests !! Status !! SLICOT functions !! Priority (0-2)
|-
| rowspan="4" | Model Type Conversion || pidstd2 || style="background: red;" | no ||  ||  || || ||
|-
| make1DOF || style="background: red;" | no ||  ||  || || ||
|-
| make2DOF || style="background: red;" | no ||  ||  || || ||
|-
| getComponents || style="background: red;" | no ||  ||  || || ||
|-
| rowspan="7" | Continuous-Discrete Conversion || c2d || style="background: green;" | yes || @lti/c2d.m ||  || || ||
|-
| d2c || style="background: green;" | yes || @lti/d2c.m ||  || || ||
|-
| d2d || style="background: green;" | yes || @lti/d2d.m ||  || || ||
|-
| upsample || style="background: red;" | no ||  ||  || || ||
|-
| c2dOptions || style="background: red;" | no ||  ||  || || ||
|-
| d2cOptions || style="background: red;" | no ||  ||  || || ||
|-
| d2dOptions || style="background: red;" | no ||  ||  || || ||
|-
| rowspan="9" | Model Simplification || hsvd || style="background: green;" | yes || hsvd.m ||  || || ||
|-
| hsvplot || style="background: red;" | no ||  ||  || || ||
|-
| sminreal || style="background: green;" | yes || @lti/sminreal.m ||  || || ||
|-
| balred || style="background: red;" | no ||  ||  || || ||
|-
| minreal || style="background: green;" | yes || @lti/minreal.m ||  || || ||
|-
| balreal || style="background: red;" | no ||  ||  || || ||
|-
| modred || style="background: red;" | no ||  ||  || || ||
|-
| balredOptions || style="background: red;" | no ||  ||  || || ||
|-
| hsvdOptions || style="background: red;" | no ||  ||  || || ||
|-
| rowspan="4" | State-Coordinate Transformation || canon || style="background: red;" | no ||  ||  || || ||
|-
| prescale || style="background: green;" | yes || @lti/prescale.m ||  || || ||
|-
| ss2ss || style="background: red;" | no ||  ||  || || ||
|-
| xperm || style="background: green;" | yes || @lti/xperm.m ||  || || ||
|-
| rowspan="5" | Modal Decomposition || modsep || style="background: red;" | no ||  ||  || || ||
|-
| stabsep || style="background: red;" | no ||  ||  || || ||
|-
| freqsep || style="background: red;" | no ||  ||  || || ||
|-
| stabsepOptions || style="background: red;" | no ||  ||  || || ||
|-
| freqsepOptions || style="background: red;" | no ||  ||  || || ||
|-
|}

=== Linear Analysis ===

{| class="wikitable" 
|-
! Chapter !! Function !! Implemented !! File !! Number of Tests !! Status !! SLICOT functions !! Priority (0-2)
|-
| rowspan="13" | Time-Domain Analysis || linearSystemAnalyzer || style="background: red;" | no ||  ||  || || ||
|-
| impulseplot || style="background: red;" | no ||  ||  || || ||
|-
| initialplot || style="background: red;" | no ||  ||  || || ||
|-
| lsimplot || style="background: red;" | no ||  ||  || || ||
|-
| stepplot || style="background: red;" | no ||  ||  || || ||
|-
| covar || style="background: green;" | yes || covar.m ||  || || ||
|-
| impulse || style="background: green;" | yes || impulse.m ||  || || ||
|-
| initial || style="background: green;" | yes || initial.m ||  || || ||
|-
| lsim || style="background: green;" | yes || lsim.m ||  || || ||
|-
| step || style="background: green;" | yes || step.m ||  || || ||
|-
| lsiminfo || style="background: red;" | no ||  ||  || || ||
|-
| stepinfo || style="background: red;" | no ||  ||  || || ||
|-
| stepDataOptions || style="background: red;" | no ||  ||  || || ||
|-
| rowspan="14" | Frequency-Domain Analysis || bodeplot || style="background: red;" | no ||  ||  || || ||
|-
| nicholsplot || style="background: red;" | no ||  ||  || || ||
|-
| nyquistplot || style="background: red;" | no ||  ||  || || ||
|-
| sigmaplot || style="background: red;" | no ||  ||  || || ||
|-
| bode || style="background: green;" | yes || bode.m ||  || || ||
|-
| nichols || style="background: green;" | yes || nichols.m ||  || || ||
|-
| nyquist || style="background: green;" | yes || nyquist.m ||  || || ||
|-
| sigma || style="background: green;" | yes || sigma.m ||  || || ||
|-
| evalfr || style="background: red;" | no ||  ||  || || ||
|-
| freqresp || style="background: green;" | yes || @lti/freqresp.m ||  || || ||
|-
| bandwidth || style="background: green;" | yes || built-in function ||  || || ||
|-
| dcgain || style="background: green;" | yes || @lti/dcgain.m ||  || || ||
|-
| getGainCrossover || style="background: red;" | no ||  ||  || || ||
|-
| getPeakGain || style="background: red;" | no ||  ||  || || ||
|-
| rowspan="10" | Stability Analysis || pole || style="background: green;" | yes || @lti/pole.m ||  || || ||
|-
| zero || style="background: green;" | yes || @lti/zero.m ||  || || ||
|-
| damp || style="background: red;" | no ||  ||  || || ||
|-
| dsort || style="background: red;" | no ||  ||  || || ||
|-
| esort || style="background: red;" | no ||  ||  || || ||
|-
| tzero || style="background: red;" | no ||  ||  || || ||
|-
| pzplot || style="background: red;" | no ||  ||  || || ||
|-
| iopzplot || style="background: red;" | no ||  ||  || || ||
|-
| allmargin || style="background: red;" | no ||  ||  || || ||
|-
| margin || style="background: green;" | yes || margin.m ||  || || ||
|-
| rowspan="11" | Plot Customization || bodeoptions || style="background: red;" | no ||  ||  || || ||
|-
| hsvoptions || style="background: red;" | no ||  ||  || || ||
|-
| nicholsoptions || style="background: red;" | no ||  ||  || || ||
|-
| nyquistoptions || style="background: red;" | no ||  ||  || || ||
|-
| pzoptions || style="background: red;" | no ||  ||  || || ||
|-
| sigmaoptions || style="background: red;" | no ||  ||  || || ||
|-
| timeoptions || style="background: red;" | no ||  ||  || || ||
|-
| setoptions || style="background: red;" | no ||  ||  || || ||
|-
| getoptions || style="background: red;" | no ||  ||  || || ||
|-
| ctrlpref || style="background: red;" | no ||  ||  || || ||
|-
| updateSystem || style="background: red;" | no ||  ||  || || ||
|-
|}

=== Control Design ===

{| class="wikitable" 
|-
! Chapter !! Function !! Implemented !! File !! Number of Tests !! Status !! SLICOT functions !! Priority (0-2)
|-
| rowspan="3" | PID Controller Tuning || pidTuner || style="background: red;" | no ||  ||  || || ||
|-
| pidtune || style="background: red;" | no ||  ||  || || ||
|-
| pidtuneOptions || style="background: red;" | no ||  ||  || || ||
|-
| rowspan="4" | SISO Feedback Loops || rlocus || style="background: green;" | yes || rlocus.m  ||  || || ||
|-
| rlocusplot || style="background: red;" | no ||  ||  || || ||
|-
| controlSystemDesigner || style="background: red;" | no ||  ||  || || ||
|-
| sisoinit || style="background: red;" | no ||  ||  || || ||
|-
| rowspan="12" | Linear-Quadratic-Gaussian Control || lqr || style="background: green;" | yes || lqr.m  ||  || || ||
|-
| lqry || style="background: red;" | no ||  ||  || || ||
|-
| lqi || style="background: red;" | no ||  ||  || || ||
|-
| dlqr || style="background: green;" | yes || dlqr.m  ||  || || ||
|-
| lqrd || style="background: red;" | no ||  ||  || || ||
|-
| kalman || style="background: green;" | yes || kalman.m  ||  || || ||
|-
| kalmd || style="background: red;" | no ||  ||  || || ||
|-
| lqg || style="background: red;" | no ||  ||  || || ||
|-
| lqgreg || style="background: red;" | no ||  ||  || || ||
|-
| lqgtrack || style="background: red;" | no ||  ||  || || ||
|-
| augstate || style="background: red;" | no ||  ||  || || ||
|-
| norm || style="background: green;" | yes || @lti/norm.m  ||  || || ||
|-
| rowspan="3" | Pole Placement || estim || style="background: green;" | yes || estim.m  ||  || || ||
|-
| place || style="background: green;" | yes || place.m  ||  || || ||
|-
| reg || style="background: red;" | no ||  ||  || || ||
|-
|}

=== Matrix Computations ===

{| class="wikitable" 
|-
! Function !! Implemented !! File !! Number of Tests !! Status !! SLICOT functions !! Priority (0-2)
|-
| lyap || style="background: green;" | yes || lyap.m ||  || || ||
|-
| lyapchol || style="background: green;" | yes || lyapchol.m ||  || || ||
|-
| dlyap || style="background: green;" | yes || dlyap.m ||  || || ||
|-
| dlyapchol || style="background: green;" | yes || dlyapchol.m ||  || || ||
|-
| care || style="background: green;" | yes || care.m ||  || || ||
|-
| dare || style="background: green;" | yes || dare.m ||  || || ||
|-
| gcare || style="background: red;" | no || ||  || || ||
|-
| gdare || style="background: red;" | no ||  ||  || || ||
|-
| ctrb || style="background: green;" | yes || ctrb.m ||  || || ||
|-
| obsv || style="background: green;" | yes || obsv.m ||  || || ||
|-
| ctrbf || style="background: green;" | yes || ctrbf.m ||  || || ||
|-
| obsvf || style="background: green;" | yes || obsvf.m ||  || || ||
|-
| gram || style="background: green;" | yes || gram.m ||  || || ||
|-
| bdschur || style="background: red;" | no ||  ||  || || ||
|-
|}

== Examples ==

=== PT1 / Low-pass filter step response ===

<!-- {{SyntaxHighlight| -->
{{Code|Creating a transfer function and plotting its response|<syntaxhighlight lang="octave" style="font-size:14px">
 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
</syntaxhighlight>}}

[[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,
] 
  \nextgroupplot[
    title={Bode Diagram of $P$},
    xmode=log,
    ylabel={Magnitude / \si{\decibel}},
  ]
  \addplot[blue,line width=1pt] table[x index=0,y index=1]{\datatable}; 
  \nextgroupplot[
    xmode=log,
    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 ===

A nonlinear model of the inverted pendulum can be derived by

<math>
\begin{cases}
  (M+m)\ddot{x}+c\dot{x}-ml\cos{\theta}\ddot{\theta}+ml\sin{\theta}\dot{\theta}^2&=f(t)\\
  -ml\ddot{x}\cos{\theta}+(ml^2+I)\ddot{\theta}+b\dot{\theta}+mgl\sin{\theta}&=0
\end{cases}\text{,}
</math>

where the variables are defined as

{|
! align="left" | Name
! align="left" | Definition
|-
|<math display="inline">M</math>
|Mass of the cart
|-
|<math display="inline">m</math>
|Mass of the pendulum
|-
|<math display="inline">c</math>
|Translational damping coefficient
|-
|<math display="inline">l</math>
|Length of the pendulum
|-
|<math display="inline">I</math>
|Inertia of the pendulum
|-
|<math display="inline">b</math>
|Rotational damping coefficient
|}

[[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
measures from a new reference) leads to

<math>
\begin{cases}
(M+m)\ddot{x}+c\dot{x}+ml\ddot{\theta}&=f(t)\\
ml\dot{x}+(ml^2+I)\ddot{\theta}+b\dot{\theta}-mgl\theta&=0
\end{cases}\text{.}
</math>

This can be expressed in the frequency domain like

<math>
\begin{cases}
(M+m)s^2X(s)+csX(s)+mls^2\Theta(s)&=F(s)\\
mlsX(s)+(ml^2+I)s^2\Theta(s)+bs\Theta(s)-mgl\Theta(s)&=0
\end{cases}
</math>

and by dividing the state variables by the input one obtains the transfer functions

<math>
\begin{align}
G_1(s)&=\frac{X(s)}{F(s)}=\frac{\alpha_2s^2+\alpha_1s+\alpha_0}{\beta_4s^4+\beta_3s^3+\beta_2s^2+\beta_1s}\\
G_2(s)&=\frac{\Theta(s)}{F(s)}=\frac{\gamma_1s}{\beta_4s^3+\beta_3s^2+\beta_2s+\beta_1}
\end{align}
</math>

with

{|
|<math display="inline">\alpha_2=ml^2+I</math>
|-
|<math display="inline">\alpha_1=b</math>
|-
|<math display="inline">\alpha_0=-mgl</math>
|-
|<math display="inline">\beta_4=(M+m)(ml^2+I)-m^2l^2</math>
|-
|<math display="inline">\beta_3=(M+m)b+(ml^2+I)c</math>
|-
|<math display="inline">\beta_2=-(M+m)mgl+bc</math>
|-
|<math display="inline">\beta_1=-mglc</math>
|-
|<math display="inline">\gamma_1=-ml</math>
|}

This can be expressed in Octave

<!-- {{SyntaxHighlight| -->
{{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>}}

and by invoking the <code style="font-size:14px; border:solid lightgray 1px; padding: 2px">bode</code> command

<!-- {{SyntaxHighlight| -->
{{Code|Creating Bode plots for the inverted pendulum|<syntaxhighlight lang="octave" style="font-size:14px">
bode(G1);
bode(G2);
</syntaxhighlight>}}

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>

==References==
<references />


[[Category:Octave-Forge]]