-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathbodeplot.dtx
2273 lines (2244 loc) · 104 KB
/
bodeplot.dtx
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
% \iffalse meta-comment
%
% Copyright (C) 2021-2024 by Rushikesh Kamalapurkar <[email protected]>
% -----------------------------------------------------------
%
% This file may be distributed and/or modified under the conditions of
% the LaTeX Project Public License, either version 1.3c of this license
% or (at your option) any later version. The latest version of this
% license is in:
%
% http://www.latex-project.org/lppl.txt
%
% and version 1.3c or later is part of all distributions of LaTeX
% version 2006/05/20 or later.
%
% \fi
%
% \iffalse
%<package>\NeedsTeXFormat{LaTeX2e}[2018/04/01]
%<package>\DeclareRelease{}{2024-02-06}{bodeplot-2024-02-06.sty}
%<package>\DeclareCurrentRelease{}{2024/04/18}
%<package>\ProvidesPackage{bodeplot}[2024/04/18 v1.2 Generate Bode, Nichols, and Nyquist plots]
%<package>\RequirePackage{pdftexcmds}
%<package>\RequirePackage{ifplatform}
%<package>\RequirePackage{xparse}
%<package>\RequirePackage{pgfplots}
%<package> \pgfplotsset{compat=1.18}
%<package> \usepgfplotslibrary{groupplots}
%
%<*driver>
\documentclass{ltxdoc}
\usepackage{cprotect}
\usepackage[declutter]{bodeplot}
\usepackage[colorlinks]{hyperref}
\usepackage{fancyvrb}
\usepackage{iftex}
\iftutex % LuaTeX, XeTeX
\usepackage{fontspec}
\setmonofont{DejaVuSansMono}[Scale=MatchUppercase]
\else % old engines
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[scaled]{DejaVuSansMono}
\fi
\usepackage{showexpl}
\lstset{
explpreset={numbers=none},
language=[LaTeX]Tex,
basicstyle=\ttfamily\tiny,
commentstyle=\itshape\ttfamily\tiny,
showspaces=false,
showstringspaces=false,
breaklines=true,
backgroundcolor=\color{white!90!black},
breakautoindent=true,
captionpos=t
}
\usepackage{geometry}
\geometry{lmargin=2in,rmargin=1in,tmargin=1in,bmargin=1in}
\usetikzlibrary{decorations.markings,arrows.meta,spy,backgrounds}
\usepackage[nottoc]{tocbibind}
\sloppy
\EnableCrossrefs
\CodelineIndex
\RecordChanges
\begin{document}
\DocInput{bodeplot.dtx}
\PrintChanges
\PrintIndex
\end{document}
%</driver>
% \fi
%
% \CheckSum{1857}
%
% \changes{v1.0}{2021/10/25}{Initial release}
% \changes{v1.0.4}{2021/11/05}{Fixed unintended optional argument macro expansion}
% \changes{v1.0.6}{2021/11/18}{Fixed issue \#3}
% \changes{v1.0.7}{2021/12/02}{Removed unnecessary semicolons}
% \changes{v1.0.7}{2022/01/18}{Updated documentation}
% \changes{v1.0.8}{2022/07/06}{Added a new class option `declutter'}
% \changes{v1.1.0}{2022/07/06}{Fixed phase wrapping in gnuplot mode}
% \changes{v1.1.1}{2022/07/31}{Enable Hz and rad units}
% \changes{v1.1.7}{2024/02/06}{Detect and turn off shorthands to improve `babel' compatibility}
% \changes{v1.2}{2024/04/18}{Removed global option to process pgf commands in radians}
%
% \GetFileInfo{bodeplot.sty}
% \DoNotIndex{\newcommand,\xdef,\gdef,\def,\edef,\addplot,\approx,\arabic,\opt,\typ,\obj,\else,\if@pgfarg,\if@Hzarg,\if@radarg,\if@declutterarg,\if@babel,\fi,\begin,\end,\feature,\footnotesize,\draw,\detokenize,\DeclareOption,\foreach,\ifdim,\ifodd,\Im,\Re,\let,\newif,\nextgroupplot,\noexpand,\expandafter,\unexpanded,\PackageError,\PackageWarning,\relax,\RequirePackage,\tikzset,\pgfmathsetmacro,\pgfmathtruncatemacro,\ProcessOptions}
%
% \title{The \textsf{bodeplot} package\\version 1.2}
% \author{Rushikesh Kamalapurkar \\ \texttt{[email protected]}}
%
% \maketitle
% \tableofcontents
% \clearpage
% \section{Introduction}
%
% Generate Bode, Nyquist, and Nichols plots for transfer functions in the canonical (TF) form \begin{equation}G(s) = e^{-Ts}\frac{b_ms^m+\cdots+b_1s+b_0}{a_ns^n+\cdots+a_1s+a_0}\label{eq:TF}\end{equation} and the zero-pole-gain (ZPK) form \begin{equation}G(s) = Ke^{-Ts}\frac{(s-z_1)(s-z_2)\cdots(s-z_m)}{(s-p_1)(s-p_2)\cdots(s-p_n)}.\label{eq:ZPK}\end{equation} In the equations above, $b_m,\cdots,b_0$ and $a_n,\cdots,a_0$ are real coefficients, $T\geq 0$ is the loop delay, $z_1,\cdots,z_m$ and $p_1,\cdots,p_n$ are complex zeros and poles of the transfer function, respectively, and $K\in \Re$ is the loop gain.
%
% For transfer functions in the ZPK format in (\ref{eq:ZPK}) \emph{with zero delay}, this package also supports linear and asymptotic approximation of Bode plots.
%
% By default, all phase plots use degrees as units. Use the |rad| package option or the optional argument |tikz/{phase unit=rad}| to generate plots in radians. The |phase unit| key accepts either |rad| or |deg| as inputs and needs to be added to the |tikzpicture| environment that contains the plots.
%
% By default, frequency inputs and outputs are in radians per second. Use the |Hz| package option or the optional argument |tikz/{frequency unit=Hz}| to generate plots in hertz. The |frequency unit| key accepts either |rad| or |Hz| as inputs and needs to be added to the |tikzpicture| environment that contains the plots.
% \subsection{External Dependencies}
% By default, the package uses |gnuplot| to do all the computations. If |gnuplot| is not available, the |pgf| package option can be used to do the calculations using the native |pgf| math engine. Compilation using the |pgf| math engine is typically slower, but the end result should be the identical (other than phase wrapping in the TF form, see limitations below).
%\subsection{Directory Structure}
% Since version 1.0.8, the |bodeplot| package places all |gnuplot| temporary files in the working directory. The package option |declutter| restores the original behavior where the temporary files are placed in a folder called |gnuplot|.
% \subsection{Limitations}
% \begin{itemize}
% \item Before version 1.2, in |pgf| mode, the package set |trig format plots| to |rad| globally. Version 1.2 onwards, this option is passed to each |addplot| command individually so that it does not affect other plots in the document. To roll back to the pre-1.2 behavior, load the package with |\usepackage[pgf]{bodeplot}[=2024-02-06]|.
% \item In |pgf| mode, Bode phase plots and Nichols charts in TF form wrap angles so that they are always between -180 and 180$^\circ$ or $-\pi$ and $-\pi$ radian. As such, these plots will show phase wrapping discontinuities. Since v1.1.1, in |gnuplot| mode, the package uses the |smooth unwrap| filter to correct wrapping discontinuities. As of now, I have not found a way to do this in |pgf| mode, any merge requests or ideas you may have are welcome! Since v1.1.4, you can redefine the |n@mod| macro using the commands |\makeatletter\renewcommand{\n@mod}{\n@mod@p}\makeatother| to wrap the phase between 0 and 360$^\circ$ or $0$ and $2\pi$ radian. The commands |\makeatletter\renewcommand{\n@mod}{\n@mod@n}\makeatother| will wrap the phase between -360 and 0$^\circ$ or $-2\pi$ and $0$ radian.
% \item Use of the |declutter| option with other directory management tools such as a |tikzexternalize| prefix is not recommended.
% \end{itemize}
% \clearpage
% \section{TL;DR}
% All Bode plots in this section are for the transfer function (with and without a transport delay)
% \begin{equation}
% G(s) = 10\frac{s(s+0.1+0.5\mathrm{i})(s+0.1-0.5\mathrm{i})}{(s+0.5+10\mathrm{i})(s+0.5-10\mathrm{i})} = \frac{s(10s^2+2s+2.6)}{(s^2+s+100.25)}.
% \end{equation}
% \iffalse
%<*ignore>
% \fi
\hrulefill
{\centering Bode plot in ZPK format
\begin{LTXexample}[pos=l,hsep=20pt,width=0.5\textwidth]
\BodeZPK{%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10%
}
{0.01}
{100}
\end{LTXexample}
\hrulefill
Same Bode plot over the same frequency range but supplied in Hz, in TF format with arrow decoration, transport delay, unit, and color customization (the phase plot may show wrapping if the |pgf| package option is used)
\begin{LTXexample}[pos=r,width=0.5\textwidth]
\BodeTF[%
samples=1000,
plot/mag/{blue,thick},
plot/ph/{green,thick},
tikz/{%
>=latex,
phase unit=rad,
frequency unit=Hz%
},
commands/mag/{
\draw[->](axis cs:0.1,40) -- (axis cs:{10/(2*pi)},60);
\node at (axis cs: 0.08,30) {\tiny Resonant Peak};
}%
]
{%
num/{10,2,2.6,0},
den/{1,1,100.25}%
}
{0.01/(2*pi)}
{100/(2*pi)}
\end{LTXexample}
\hrulefill
\clearpage
\hrulefill
Linear approximation with customization
\begin{LTXexample}[pos=l,hsep=20pt,width=0.5\textwidth]
\BodeZPK[%
plot/mag/{red,thick},
plot/ph/{blue,thick},
axes/mag/{ytick distance=40},
axes/ph/{ytick distance=90},
approx/linear%
]{%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10%
}
{0.01}
{100}
\end{LTXexample}
\hrulefill
Plot with delay and customization
\begin{LTXexample}[pos=r,hsep=20pt,width=0.5\textwidth]
\BodeZPK[%
plot/mag/{blue,thick},
plot/ph/{green,thick},
axes/mag/ytick distance=40,
axes/ph/ytick distance=90%
]{%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10,
d/0.01%
}
{0.01}
{100}
\end{LTXexample}
\hrulefill
\clearpage
\hrulefill
Individual gain and phase plots with more customization
\begin{minipage}[t]{0.45\textwidth}
\begin{LTXexample}[pos=t,width=\columnwidth]
\begin{BodeMagPlot}[%
axes/{height=2cm,
width=4cm}%
]
{0.01}
{100}
\addBodeZPKPlots[%
true/{black,thick},
linear/{red,dashed,thick},
asymptotic/{blue,dotted,thick}%
]
{magnitude}
{%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10%
}
\end{BodeMagPlot}
\end{LTXexample}
\end{minipage}\hfill
\begin{minipage}[t]{0.45\textwidth}
\begin{LTXexample}[pos=t,width=\columnwidth]
\begin{BodePhPlot}[%
height=2cm,
width=4cm,
ytick distance=90
]
{0.01}
{100}
\addBodeZPKPlots[%
true/{black,thick},
linear/{red,dashed,thick},
asymptotic/{blue,dotted,thick}%
]
{phase}
{%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10%
}
\end{BodePhPlot}
\end{LTXexample}
\end{minipage}
\hrulefill
Nichols chart
\begin{LTXexample}[pos=l,hsep=20pt,width=0.5\textwidth]
\NicholsZPK[samples=1000]
{%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10,
d/0.01%
}
{0.001}
{500}
\end{LTXexample}
\hrulefill
Same Nichols chart in TF format (may show wrapping in |pgf| mode)
\begin{LTXexample}[pos=r,hsep=20pt,width=0.5\textwidth]
\NicholsTF[samples=1000]
{%
num/{10,2,2.6,0},
den/{1,1,100.25},
d/0.01%
}
{0.001}
{500}
\end{LTXexample}
\hrulefill
\clearpage
\hrulefill
Multiple Nichols charts with customization
\begin{LTXexample}[pos=l,hsep=20pt,width=0.5\textwidth]
\begin{NicholsChart}[%
ytick distance=20,
xtick distance=30
]
{0.001}
{100}
\addNicholsZPKChart [red,samples=1000] {%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10%
}
\addNicholsZPKChart [blue,samples=1000] {%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/5%
}
\end{NicholsChart}
\end{LTXexample}
\hrulefill
Nyquist plot
\begin{LTXexample}[pos=r,hsep=20pt,width=0.5\textwidth]
\NyquistZPK[plot/{red,thick,samples=1000}]
{%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10%
}
{-30}
{30}
\end{LTXexample}
\hrulefill
Nyquist plot in TF format with arrows
\begin{LTXexample}[pos=l,width=0.5\textwidth]
\NyquistTF[%
plot/{%
samples=1000,
postaction=decorate,
decoration={%
markings,
mark=between positions 0.1 and 0.9 step 5em with {%
\arrow{Stealth [length=2mm, blue]}
}
}
}%
]
{%
num/{10,2,2.6,0},
den/{1,1,100.25}%
}
{-30}
{30}
\end{LTXexample}
\hrulefill
\clearpage
\hrulefill
Multiple Nyquist plots with customization
\begin{LTXexample}[pos=r,hsep=20pt,width=0.5\textwidth]
\begin{NyquistPlot}{-30}{30}
\addNyquistZPKPlot [red,samples=1000] {%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/10%
}
\addNyquistZPKPlot [blue,samples=1000] {%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/5%
}
\end{NyquistPlot}
\end{LTXexample}
\hrulefill
Nyquist plots with additional commands, using two different macros
\begin{minipage}[t]{0.48\textwidth}
\begin{LTXexample}[pos=t,width=\columnwidth]
\begin{NyquistPlot}[%
tikz/{
spy using outlines={%
circle,
magnification=3,
connect spies,
size=2cm
}
}%
]
{-30}{30}
\addNyquistZPKPlot [blue,samples=1000] {%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/0.5%
}
\coordinate (spyon) at (axis cs:0,0);
\coordinate (spyat) at (axis cs:-22,5);
\spy [green] on (spyon) in
node [fill=white] at (spyat);
\end{NyquistPlot}
\end{LTXexample}
\end{minipage}\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{LTXexample}[pos=t,width=\columnwidth]
\NyquistZPK[%
plot/{blue,samples=1000},
tikz/{
spy using outlines={%
circle,
magnification=3,
connect spies,
size=2cm
}
},
commands/{
\coordinate (spyon) at (axis cs:0,0);
\coordinate (spyat) at (axis cs:-22,5);
\spy [green] on (spyon) in
node [fill=white] at (spyat);
}%
]
{%
z/{0,{-0.1,-0.5},{-0.1,0.5}},
p/{{-0.5,-10},{-0.5,10}},
k/0.5%
}
{-30}
{30}
\end{LTXexample}
\end{minipage}}
\hrulefill
\clearpage
% \iffalse
%</ignore>
% \fi
%
% \section{Usage}
% \noindent In all the macros described here, the frequency limits supplied by the user are assumed to be in |rad/s| unless either the |Hz| package option is used or the optional argument |tikz/{frequency unit=Hz}| is supplied to the |tikzpicture| environment. All phase plots are getenrated in degrees unless either the |rad| package option is used or the optional argument |tikz/{frequency unit=rad}| is supplied to the |tikzpicture| environment.
%
% \subsection{Bode plots}
% \DescribeMacro{\BodeZPK}
% |\BodeZPK| \oarg{obj1/typ1/\marg{opt1},obj2/typ2/\marg{opt2},...}\\
% \hspace*{2em}\marg{z/\marg{zeros},p/\marg{poles},k/\marg{gain},d/\marg{delay}}\\
% \hspace*{2em}\marg{min-freq}\marg{max-freq}
%
% \noindent Plots the Bode plot of a transfer function given in ZPK format using the |groupplot| environment. The three mandatory arguments include: (1) a list of tuples, comprised of the zeros, the poles, the gain, and the transport delay of the transfer function, (2) the lower end of the frequency range for the $x-$axis, and (3) the higher end of the frequency range for the $x-$axis. The zeros and the poles are complex numbers, entered as a comma-separated list of comma-separated lists, of the form |{{real part 1,imaginary part 1},| |{real part 2,imaginary part 2},...}|. If the imaginary part is not provided, it is assumed to be zero.
%
% The optional argument is comprised of a comma separated list of tuples, either |obj/typ/{opt}|, or |obj/{opt}|, or just |{opt}|. Each tuple passes options to different |pgfplots| macros that generate the group, the axes, and the plots according to:
% \begin{itemize}
% \item Tuples of the form |obj/typ/{opt}|:
% \begin{itemize}
% \item |plot/typ/{opt}|: modify plot properties by adding options |{opt}| to the |\addplot| macro for the magnitude plot if |typ| is |mag| and the phase plot if |typ| is |ph|.
% \item |axes/typ/{opt}|: modify axis properties by adding options |{opt}| to the |\nextgroupplot| macro for the magnitude plot if |typ| is |mag| and the phase plot if |typ| is |ph|.
% \item |commands/typ/{opt}|: add any valid TikZ commands (including the the parametric function generator macros in this package, such as |\addBodeZPKPlots|, |\addBodeTFPlot|, and |\addBodeComponentPlot|) to the magnitude plot if |typ| is |mag| and the phase plot if |typ| is |ph|. The commands passed to |opt| need to be valid TikZ commands, separated by semicolons as usual. For example, a TikZ command is used in the description of the |\BodeTF| macro below to mark the gain crossover frequency on the Bode Magnitude plot.
% \end{itemize}
% \item Tuples of the form |obj/{opt}|:
% \begin{itemize}
% \item |plot/{opt}|: adds options |{opt}| to |\addplot| macros for both the magnitude and the phase plots.
% \item |axes/{opt}|: adds options |{opt}| to |\nextgroupplot| macros for both the magnitude and the phase plots.
% \item |group/{opt}|: adds options |{opt}| to the |groupplot| environment.
% \item |tikz/{opt}|: adds options |{opt}| to the |tikzpicture| environment.
% \item |approx/linear|: plots linear approximation.
% \item |approx/asymptotic|: plots asymptotic approximation.
% \end{itemize}
% \item Tuples of the form |{opt}| add all of the supplied options to |\addplot| macros for both the magnitude and the phase plots.
% \end{itemize}
% The options |{opt}| can be any |key=value| options that are supported by the |pgfplots| macros they are added to.
% For example, given a transfer function \begin{equation}G(s) = 10\frac{s(s+0.1+0.5\mathrm{i})(s+0.1-0.5\mathrm{i})}{(s+0.5+10\mathrm{i})(s+0.5-10\mathrm{i})},\label{eq:ZPKExample}\end{equation} its Bode plot over the frequency range $[0.01,100]$ can be generated using
%\begin{verbatim}
%\BodeZPK [blue,thick]
% {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10}
% {0.01}{100}
%\end{verbatim}
% which generates the plot in Figure \ref{simpleBode}. In this example, a delay is not specified, so it is assumed to be zero. A gain is not specified, so it is assumed to be 1. A single comma-separated list of options |[blue,thick]| is passed, so it is passed on to the |\addplot| commands in both the magnitude and the phase plots. The default plots are thick black lines and each of the axes, excluding ticks and labels, are 5cm wide and 2.5cm high.
%
% \begin{figure}
% \begin{center}
% \BodeZPK[blue,thick]{z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10}{0.01}{100}
% \cprotect\caption{\label{simpleBode}Output of the |\BodeZPK| macro.}
% \end{center}
% \end{figure}
%
% The width and the height, along with other properties of the plots, the axes, and the group can be customized using native |pgf| keys. For example, a linear approximation of the Bode plot with customization of the plots, the axes, and the group can be generated using
%\begin{verbatim}
%\BodeZPK[%
% plot/mag/{red,thick},
% plot/ph/{blue,thick},
% axes/mag/{ytick distance=40,xmajorticks=true,xlabel={Frequency (rad/s)}},
% axes/ph/{ytick distance=90},
% group/{group style={group size=2 by 1,horizontal sep=2cm,width=4cm,height=2cm}},
% approx/linear]
% {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10}
% {0.01}{100}
%\end{verbatim}
% which generates the plot in Figure \ref{customBode}.
%
% \begin{figure}
% \begin{center}
% \BodeZPK[plot/mag/{red,thick},plot/ph/{blue,thick},axes/mag/{ytick distance=40,xmajorticks=true,xlabel={Frequency (rad/s)}},axes/ph/{ytick distance=90},approx/linear,group/{group style={group size = 2 by 1,horizontal sep=2cm},width=4cm,height=2cm}] {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},g/10} {0.01} {100}
% \cprotect\caption{\label{customBode}Customization of the default |\BodeZPK| macro.}
% \end{center}
% \end{figure}
%
% \DescribeMacro{\BodeTF}
% |\BodeTF| \oarg{obj1/typ1/\marg{opt1},obj2/typ2/\marg{opt2},...}\\
% \hspace*{2em}\marg{num/\marg{coeffs},den/\marg{coeffs},d/\marg{delay}}\\
% \hspace*{2em}\marg{min-freq}\marg{max-freq} \rmfamily
%
% \noindent Plots the Bode plot of a transfer function given in TF format. The three mandatory arguments include: (1) a list of tuples comprised of the coefficients in the numerator and the denominator of the transfer function and the transport delay, (2) the lower end of the frequency range for the $x-$ axis, and (3) the higher end of the frequency range for the $x-$axis. The coefficients are entered as a comma-separated list, in order from the highest degree of $s$ to the lowest, with zeros for missing degrees. The optional arguments are the same as |\BodeZPK|, except that linear/asymptotic approximation is not supported, so |approx/...| is ignored.
%
% For example, given the same transfer function as (\ref{eq:ZPKExample}) in TF form and with a small transport delay, \begin{equation}G(s) = e^{-0.01s}\frac{s(10s^2+2s+2.6)}{(s^2+s+100.25)},\label{eq:TFExample}\end{equation} its Bode plot over the frequency range $[0.01,100]$ can be generated using
%\begin{verbatim}
%\BodeTF[%
% commands/mag/{\node at (axis cs: 2.1,0) [circle,fill,inner sep=0.05cm,
% label=below:{$\omega_{gc}$}]{};}]
% {num/{10,2,2.6,0},den/{1,1,100.25},d/0.01}
% {0.01}{100}
%\end{verbatim}
% which generates the plot in Figure \ref{simpleBodeTF}. Note the $0$ added to the numerator coefficients to account for the fact that the numerator does not have a constant term in it. Note the semicolon after the TikZ command passed to the |\commands| option.
%
% \begin{figure}
% \begin{center}
% \BodeTF[commands/mag/{\node at (axis cs: 2.1,0) [circle,fill,inner sep=0.05cm,label=below:{$\omega_{gc}$}] {};}]{num/{10,2,2.6,0},den/{1,1,100.25},d/0.01}{0.01}{100}
% \cprotect\caption{\label{simpleBodeTF}Output of the |\BodeTF| macro with an optional TikZ command used to mark the gain crossover frequency.}
% \end{center}
% \end{figure}
%
% \DescribeEnv{BodeMagPlot}
% |\begin{BodeMagPlot}|\oarg{obj1/\marg{opt1},obj2/\marg{opt2},...}\\
% \hspace*{3em}\marg{min-frequency}\marg{max-frequency}\\
% \hspace*{2em}|\addBode...|\\
% \hspace*{1.5em}|\end{BodeMagPlot}|\\
% The |BodeMagPlot| environment works in conjunction with the parametric function generator macros |\addBodeZPKPlots|, |\addBodeTFPlot|, and |\addBodeComponentPlot|, intended to be used for magnitude plots. The optional argument is comprised of a comma separated list of tuples, either |obj/{opt}| or just |{opt}|. Each tuple passes options to different |pgfplots| macros that generate the axes and the plots according to:
% \begin{itemize}
% \item Tuples of the form |obj/{opt}|:
% \begin{itemize}
% \item |tikz/{opt}|: modify picture properties by adding options |{opt}| to the |tikzpicture| environment.
% \item |axes/{opt}|: modify axis properties by adding options |{opt}| to the |semilogaxis| environment.
% \item |commands/{opt}|: add any valid TikZ commands inside |semilogaxis| environment. The commands passed to |opt| need to be valid TikZ commands, separated by semicolons as usual.
% \end{itemize}
% \item Tuples of the form |{opt}| are passed directly to the |semilogaxis| environment.
% \end{itemize}
% The frequency limits are translated to the x-axis limits and the domain of the |semilogaxis| environment. Example usage in the description of |\addBodeZPKPlots|, |\addBodeTFPlot|, and |\addBodeComponentPlot|.
%
%\DescribeEnv{BodePhPlot}
% |\begin{BodePhPlot}|\oarg{obj1/\marg{opt1},obj2/\marg{opt2},...}\\
% \hspace*{3em}\marg{min-frequency}\marg{max-frequency}\\
% \hspace*{2em}|\addBode...|\\
% \hspace*{1.5em}|\end{BodePhPlot}|\\
% Intended to be used for phase plots, otherwise same as the |BodeMagPlot| environment
%
% \DescribeMacro{\addBodeZPKPlots}
% |\addBodeZPKPlots| \oarg{approx1/\marg{opt1},approx2/\marg{opt2},...}\\
% \hspace*{2em}\marg{plot-type}\\
% \hspace*{2em}\marg{z/\marg{zeros},p/\marg{poles},k/\marg{gain},d/\marg{delay}}
%
% \noindent Generates the appropriate parametric functions and supplies them to multiple |\addplot| macros, one for each |approx/{opt}| pair in the optional argument. If no optional argument is supplied, then a single |\addplot| command corresponding to a thick true Bode plot is generated. If an optional argument is supplied, it needs to be one of |true/{opt}|, |linear/{opt}|, or |asymptotic/{opt}|. This macro can be used inside any |semilogaxis| environment as long as a domain for the x-axis is supplied through either the |approx/{opt}| interface or directly in the optional argument of the |semilogaxis| environment. Use with the |BodePlot| environment supplied with this package is recommended. The second mandatory argument, |plot-type| is either |magnitude| or |phase|. If it is not equal to |phase|, it is assumed to be |magnitude|. The last mandatory argument is the same as |\BodeZPK|.
%
% For example, given the transfer function in (\ref{eq:ZPKExample}), its linear, asymptotic, and true Bode plots can be superimposed using
%\begin{verbatim}
%\begin{BodeMagPlot}[height=2cm,width=4cm] {0.01} {100}
% \addBodeZPKPlots[%
% true/{black,thick},
% linear/{red,dashed,thick},
% asymptotic/{blue,dotted,thick}]
% {magnitude}
% {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10}
%\end{BodeMagPlot}
%
%\begin{BodePhPlot}[height=2cm, width=4cm, ytick distance=90] {0.01} {100}
% \addBodeZPKPlots[%
% true/{black,thick},
% linear/{red,dashed,thick},
% asymptotic/{blue,dotted,thick}]
% {phase}
% {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10}
%\end{BodePhPlot}
%\end{verbatim}
% \begin{figure}
% \begin{center}
% \begin{BodeMagPlot}[height=2cm,width=4cm]{0.01}{100}
% \addBodeZPKPlots[%
% true/{black,thick},
% linear/{red,dashed,thick},
% asymptotic/{blue,dotted,thick}]
% {magnitude}
% {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10}
% \end{BodeMagPlot}
% \begin{BodePhPlot}[height=2cm,width=4cm,ytick distance=90]{0.01}{100}
% \addBodeZPKPlots[%
% true/{black,thick},
% linear/{red,dashed,thick},
% asymptotic/{blue,dotted,thick}]
% {phase}
% {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10}
% \end{BodePhPlot}
% \end{center}
% \caption{\label{multiBodeZPK}Superimposed approximate and true Bode plots using the \texttt{BodeMagPlot} and \texttt{BodePhPlot} environments and the \texttt{\textbackslash addBodeZPKPlots} macro.}
% \end{figure}
% which generates the plot in Figure \ref{multiBodeZPK}.
%
% \DescribeMacro{\addBodeTFPlot}
% |\addBodeTFPlot|\oarg{plot-options}\\
% \hspace*{2em}\marg{plot-type}\\
% \hspace*{2em}\marg{num/\marg{coeffs},den/\marg{coeffs},d/\marg{delay}}
%
% \noindent Generates a single parametric function for either Bode magnitude or phase plot of a transfer function in TF form. The generated parametric function is passed to the |\addplot| macro. This macro can be used inside any |semilogaxis| environment as long as a domain for the x-axis is supplied through either the |plot-options| interface or directly in the optional argument of the container |semilogaxis| environment. Use with the |BodePlot| environment supplied with this package is recommended. The second mandatory argument, |plot-type| is either magnitude or |phase|. If it is not equal to |phase|, it is assumed to be |magnitude|. The last mandatory argument is the same as |\BodeTF|.
%
% \DescribeMacro{\addBodeComponentPlot}
% |\addBodeComponentPlot|\oarg{plot-options}\marg{plot-command}
%
% \noindent Generates a single parametric function corresponding to the mandatory argument |plot-command| and passes it to the |\addplot| macro. The plot command can be any parametric function that uses |t| as the independent variable. The parametric function must be |gnuplot| compatible (or |pgfplots| compatible if the package is loaded using the |pgf| option, \textbf{with angles passed to trigonometric functions in radian}). The intended use of this macro is to plot the parametric functions generated using the basic component macros described in Section \ref{sec:BasicComponents} below.
%
% \subsubsection{Basic components up to first order\label{sec:BasicComponents}}
%
% \DescribeMacro{\TypeFeatureApprox}
% |\TypeFeatureApprox|\marg{real-part}\marg{imaginary-part}
%
% \noindent This entry describes 20 different macros of the form |\TypeFeatureApprox| that take the real part and the imaginary part of a complex number as arguments. The |Type| in the macro name should be replaced by either |Mag| or |Ph| to generate a parametric function corresponding to the magnitude or the phase plot, respectively. The |Feature| in the macro name should be replaced by one of |K|, |Pole|, |Zero|, or |Del|, to generate the Bode plot of a gain, a complex pole, a complex zero, or a transport delay, respectively. If the |Feature| is set to either |K| or |Del|, the |imaginary-part| mandatory argument is ignored. The |Approx| in the macro name should either be removed, or it should be replaced by |Lin| or |Asymp| to generate the true Bode plot, the linear approximation, or the asymptotic approximation, respectively. If the |Feature| is set to |Del|, then |Approx| has to be removed. For example,
% \begin{itemize}
% \item |\MagK{k}{0}| or |\MagK{k}{400}| generates a parametric function for the true Bode magnitude of $ G(s) = k $
% \item |\PhPoleLin{a}{b}| generates a parametric function for the linear approximation of the Bode phase of $ G(s) = \frac{1}{s-a-\mathrm{i}b} $.
% \item |\PhDel{T}{200}| or |\PhDel{T}{0}| generates a parametric function for the Bode phase of $ G(s) = e^{-Ts} $.
% \end{itemize}
% All 20 of the macros defined by combinations of |Type|, |Feature|, and |Approx|, and any |gnuplot| (or |pgfplot| if the |pgf| class option is loaded) compatible function of the 20 macros can be used as |plot-command| in the |addBodeComponentPlot| macro. This is sufficient to generate the Bode plot of any rational transfer function with delay. For example, the Bode phase plot in Figure \ref{multiBodeZPK} can also be generated using:
%\begin{verbatim}
%\begin{BodePhPlot}[ytick distance=90]{0.01}{100}
% \addBodeComponentPlot[black,thick]{%
% \PhZero{0}{0} + \PhZero{-0.1}{-0.5} + \PhZero{-0.1}{0.5} +
% \PhPole{-0.5}{-10} + \PhPole{-0.5}{10} + \PhK{10}{0}}
% \addBodeComponentPlot[red,dashed,thick] {%
% \PhZeroLin{0}{0} + \PhZeroLin{-0.1}{-0.5} + \PhZeroLin{-0.1}{0.5} +
% \PhPoleLin{-0.5}{-10} + \PhPoleLin{-0.5}{10} + \PhKLin{10}{20}}
% \addBodeComponentPlot[blue,dotted,thick] {%
% \PhZeroAsymp{0}{0} + \PhZeroAsymp{-0.1}{-0.5} + \PhZeroAsymp{-0.1}{0.5} +
% \PhPoleAsymp{-0.5}{-10} + \PhPoleAsymp{-0.5}{10} + \PhKAsymp{10}{40}}
%\end{BodePhPlot}
%\end{verbatim}
%\begin{figure}
% \begin{center}
% \begin{BodePhPlot}[ytick distance=90]{0.01}{100}
% \addBodeComponentPlot[black,thick] {\PhZero{0}{0} + \PhZero{-0.1}{-0.5} + \PhZero{-0.1}{0.5} + \PhPole{-0.5}{-10} + \PhPole{-0.5}{10} + \PhK{10}{0}}
% \addBodeComponentPlot[red,dashed,thick] {\PhZeroLin{0}{0} + \PhZeroLin{-0.1}{-0.5} + \PhZeroLin{-0.1}{0.5} + \PhPoleLin{-0.5}{-10} + \PhPoleLin{-0.5}{10} + \PhKLin{10}{20}}
% \addBodeComponentPlot[blue,dotted,thick] {\PhZeroAsymp{0}{0} + \PhZeroAsymp{-0.1}{-0.5} + \PhZeroAsymp{-0.1}{0.5} + \PhPoleAsymp{-0.5}{-10} + \PhPoleAsymp{-0.5}{10} + \PhKAsymp{10}{40}}
% \end{BodePhPlot}
% \end{center}
% \caption{\label{multiBodeComponents}Superimposed approximate and true Bode Phase plot using the \texttt{BodePhPlot} environment, the \texttt{\textbackslash addBodeComponentPlot} macro, and several macros of the \texttt{\textbackslash TypeFeatureApprox} form.}
%\end{figure}
% which gives us the plot in Figure \ref{multiBodeComponents}.
%
% \subsubsection{Basic components of the second order}
%
% \DescribeMacro{\TypeSOFeatureApprox}
% |\TypeSOFeatureApprox|\marg{a1}\marg{a0}
%
% \noindent This entry describes 12 different macros of the form |\TypeSOFeatureApprox| that take the coefficients $ a_1 $ and $ a_0 $ of a general second order system as inputs. The |Feature| in the macro name should be replaced by either |Poles| or |Zeros| to generate the Bode plot of $G(s)=\frac{1}{s^2+a_1 s+a_0}$ or $G(s)=s^2+a_1 s+a_0$, respectively. The |Type| in the macro name should be replaced by either |Mag| or |Ph| to generate a parametric function corresponding to the magnitude or the phase plot, respectively. The |Approx| in the macro name should either be removed, or it should be replaced by |Lin| or |Asymp| to generate the true Bode plot, the linear approximation, or the asymptotic approximation, respectively.
%
% \DescribeMacro{\MagSOFeaturePeak}
% |\MagSOFeaturePeak|\oarg{draw-options}\marg{a1}\marg{a0}
%
% \noindent This entry describes 2 different macros of the form |\MagSOFeaturePeak| that take the the coefficients $ a_1 $ and $ a_0 $ of a general second order system as inputs, and draw a resonant peak using the |\draw| TikZ macro. The |Feature| in the macro name should be replaced by either |Poles| or |Zeros| to generate a peak for poles and a valley for zeros, respectively. For example, the command
%\begin{verbatim}
%\begin{BodeMagPlot}[xlabel={}]{0.1}{10}
% \addBodeComponentPlot[red,dashed,thick]{\MagSOPoles{0.2}{1}}
% \addBodeComponentPlot[black,thick]{\MagSOPolesLin{0.2}{1}}
% \MagSOPolesPeak[thick]{0.2}{1}
%\end{BodeMagPlot}
%\end{verbatim}
% generates the plot in Figure \ref{BodePeak}.
%
% \begin{figure}
% \begin{center}
% \begin{BodeMagPlot}[xlabel={}]{0.1}{10}
% \addBodeComponentPlot[red,dashed,thick]{\MagSOPoles{0.2}{1}}
% \addBodeComponentPlot[black,thick]{\MagSOPolesLin{0.2}{1}}
% \MagSOPolesPeak[very thick]{0.2}{1}
% \end{BodeMagPlot}
% \end{center}
% \cprotect\caption{\label{BodePeak} Resonant peak in asymptotic Bode plot using |\MagSOPolesPeak|.}
% \end{figure}
%
% \DescribeMacro{\TypeCSFeatureApprox}
% |\TypeCSFeatureApprox|\marg{zeta}\marg{omega-n}
%
% \noindent This entry describes 12 different macros of the form |\TypeCSFeatureApprox| that take the damping ratio, $ \zeta $, and the natural frequency, $ \omega_n $ of a canonical second order system as inputs. The |Type| in the macro name should be replaced by either |Mag| or |Ph| to generate a parametric function corresponding to the magnitude or the phase plot, respectively. The |Feature| in the macro name should be replaced by either |Poles| or |Zeros| to generate the Bode plot of $G(s)=\frac{1}{s^2+2\zeta\omega_n s+\omega_n^2}$ or $G(s)=s^2+2\zeta\omega_n s+\omega_n^2$, respectively. The |Approx| in the macro name should either be removed, or it should be replaced by |Lin| or |Asymp| to generate the true Bode plot, the linear approximation, or the asymptotic approximation, respectively.
%
% \DescribeMacro{\MagCSFeaturePeak}
% |\MagCSFeaturePeak|\oarg{draw-options}\marg{zeta}\marg{omega-n}
%
% \noindent This entry describes 2 different macros of the form |\MagCSFeaturePeak| that take the damping ratio, $ \zeta $, and the natural frequency, $ \omega_n $ of a canonical second order system as inputs, and draw a resonant peak using the |\draw| TikZ macro. The |Feature| in the macro name should be replaced by either |Poles| or |Zeros| to generate a peak for poles and a valley for zeros, respectively.
%
% \DescribeMacro{\MagCCFeaturePeak}
% |\MagCCFeaturePeak|\oarg{draw-options}\marg{real-part}\marg{imaginary-part}
%
% \noindent This entry describes 2 different macros of the form |\MagCCFeaturePeak| that take the real and imaginary parts of a pair of complex conjugate poles or zeros as inputs, and draw a resonant peak using the |\draw| TikZ macro. The |Feature| in the macro name should be replaced by either |Poles| or |Zeros| to generate a peak for poles and a valley for zeros, respectively.
%
% \subsection{Nyquist plots}
% \DescribeMacro{\NyquistZPK}
% |\NyquistZPK| \oarg{plot/\marg{opt},axes/\marg{opt}}\\
% \hspace*{2em}\marg{z/\marg{zeros},p/\marg{poles},k/\marg{gain},d/\marg{delay}}\\
% \hspace*{2em}\marg{min-freq}\marg{max-freq}
%
% \noindent Plots the Nyquist plot of a transfer function given in ZPK format with a thick red $+$ marking the critical point (-1,0). The mandatory arguments are the same as |\BodeZPK|. Since there is only one plot in a Nyquist diagram, the |\typ| specifier in the optional argument tuples is not needed. As such, the supported optional argument tuples are |plot/{opt}|, which passes |{opt}| to |\addplot|, |axes/{opt}|, which passes |{\opt}| to the |axis| environment, and |tikz/{opt}|, which passes |{\opt}| to the |tikzpicture| environment. Asymptotic/linear approximations are not supported in Nyquist plots. If just |{opt}| is provided as the optional argument, it is interpreted as |plot/{opt}|. Arrows to indicate the direction of increasing $\omega$ can be added by adding |\usetikzlibrary{decorations.markings}| and |\usetikzlibrary{arrows.meta}| to the preamble and then passing a tuple of the form
%\begin{verbatim}
%plot/{postaction=decorate,decoration={markings,
% mark=between positions 0.1 and 0.9 step 5em with {%
% \arrow{Stealth| |[length=2mm, blue]}}}}
%\end{verbatim}
%\textbf{Caution:} with a high number of samples, adding arrows in this way may cause the error message |! Dimension too big|.
%
% For example, the command
%\begin{verbatim}
%\NyquistZPK[plot/{red,thick,samples=2000},axes/{blue,thick}]
% {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10}
% {-30}{30}
%\end{verbatim}
% generates the Nyquist plot in Figure \ref{simpleNyquistZPK}.
%
% \begin{figure}
% \begin{center}
% \NyquistZPK[plot/{red,thick,samples=2000},axes/{blue,thick}] {z/{0,{-0.1,-0.5},{-0.1,0.5}},p/{{-0.5,-10},{-0.5,10}},k/10} {-30} {30}
% \cprotect\caption{\label{simpleNyquistZPK}Output of the |\NyquistZPK| macro.}
% \end{center}
% \end{figure}
%
% % \DescribeMacro{\NyquistTF}
% |\NyquistTF| \oarg{plot/\marg{opt},axes/\marg{opt}}\\
% \hspace*{2em}\marg{num/\marg{coeffs},den/\marg{coeffs},d/\marg{delay}}\\
% \hspace*{2em}\marg{min-freq}\marg{max-freq}
%
% \noindent Nyquist plot of a transfer function given in TF format. Same mandatory arguments as |\BodeTF| and same optional arguments as |\NyquistZPK|. For example, the command
%\begin{verbatim}
%\NyquistTF[plot/{green,thick,samples=500,postaction=decorate,
% decoration={markings,
% mark=between positions 0.1 and 0.9 step 5em
% with{\arrow{Stealth[length=2mm, blue]}}}}]
% {num/{10,2,2.6,0},den/{1,1,100.25}}
% {-30}{30}
%\end{verbatim}
% generates the Nyquist plot in Figure \ref{simpleNyquistTF}.
%
% \begin{figure}
% \begin{center}
% \NyquistTF[plot/{green,thick,samples=500,postaction=decorate,decoration={markings,mark=between positions 0.1 and 0.9 step 5em with {\arrow{Stealth[length=2mm, blue]}}}}] {num/{10,2,2.6,0},den/{1,1,100.25},d/0.01} {-30} {30}
% \cprotect\caption{\label{simpleNyquistTF}Output of the |\NyquistTF| macro with direction arrows. Increasing the number of samples can cause |decorations.markings| to throw errors.}
% \end{center}
% \end{figure}
%
% \DescribeEnv{NyquistPlot}
% |\begin{NyquistPlot}|\oarg{obj1/\marg{opt1},obj2/\marg{opt2},...}\\
% \hspace*{3em}\marg{min-frequency}\marg{max-frequency}\\
% \hspace*{2em}|\addNyquist...|\\
% \hspace*{1.5em}|\end{NyquistPlot}|\\
% The |NyquistPlot| environment works in conjunction with the parametric function generator macros |\addNyquistZPKPlot| and |\addNyquistTFPlot|. The optional argument is comprised of a comma separated list of tuples, either |obj/{opt}| or just |{opt}|. Each tuple passes options to different |pgfplots| macros that generate the axes and the plots according to:
% \begin{itemize}
% \item Tuples of the form |obj/{opt}|:
% \begin{itemize}
% \item |tikz/{opt}|: modify picture properties by adding options |{opt}| to the |tikzpicture| environment.
% \item |axes/{opt}|: modify axis properties by adding options |{opt}| to the |axis| environment.
% \item |commands/{opt}|: add any valid TikZ commands inside |axis| environment. The commands passed to |opt| need to be valid TikZ commands, separated by semicolons as usual.
% \end{itemize}
% \item Tuples of the form |{opt}| are passed directly to the |axis| environment.
% \end{itemize}
% The frequency limits are translated to the x-axis limits and the domain of the |axis| environment.
%
% \DescribeMacro{\addNyquistZPKPlot}
% |\addNyquistZPKPlot|\oarg{plot-options}\\
% \hspace*{2em}\marg{z/\marg{zeros},p/\marg{poles},k/\marg{gain},d/\marg{delay}}
%
% \noindent Generates a twp parametric functions for the magnitude and the phase a transfer function in ZPK form. The generated magnitude and phase parametric functions are converted to real and imaginary part parametric functions and passed to the |\addplot| macro. This macro can be used inside any |axis| environment as long as a domain for the x-axis is supplied through either the |plot-options| interface or directly in the optional argument of the container |axis| environment. Use with the |NyquistPlot| environment supplied with this package is recommended. The mandatory argument is the same as |\BodeZPK|.
%
% \DescribeMacro{\addNyquistTFPlot}
% |\addNyquistTFPlot|\oarg{plot-options}\\
% \hspace*{2em}\marg{num/\marg{coeffs},den/\marg{coeffs},d/\marg{delay}}
%
% \noindent Similar to |\addNyquistZPKPlot|, with a transfer function input in the TF form.
%
% \subsection{Nichols charts}
% \DescribeMacro{\NicholsZPK}
% |\NicholsZPK| \oarg{plot/\marg{opt},axes/\marg{opt}}\\
% \hspace*{2em}\marg{z/\marg{zeros},p/\marg{poles},k/\marg{gain},d/\marg{delay}}\\
% \hspace*{2em}\marg{min-freq}\marg{max-freq}
%
% \noindent Nichols chart of a transfer function given in ZPK format. Same arguments as |\NyquistZPK|.
%
% \DescribeMacro{\NicholsTF}
% |\NicholsTF| \oarg{plot/\marg{opt},axes/\marg{opt}}\\
% \hspace*{2em}\marg{num/\marg{coeffs},den/\marg{coeffs},d/\marg{delay}}\\
% \hspace*{2em}\marg{min-freq}\marg{max-freq}
%
% \noindent Nichols chart of a transfer function given in TF format. Same arguments as |\NyquistTF|. For example, the command
%\begin{verbatim}
%\NicholsTF[plot/{green,thick,samples=2000}]
% {num/{10,2,2.6,0},den/{1,1,100.25},d/0.01}
% {0.001}{100}
%\end{verbatim}
% generates the Nichols chart in Figure \ref{simpleNicholsTF}.
%
% \begin{figure}
% \begin{center}
% \NicholsTF[plot/{green,thick,samples=2000}] {num/{10,2,2.6,0},den/{1,1,100.25},d/0.01} {0.001} {100}
% \cprotect\caption{\label{simpleNicholsTF}Output of the |\NyquistZPK| macro.}
% \end{center}
% \end{figure}
%
%
% \DescribeEnv{NicholsChart}
% |\begin{NicholsChart}|\oarg{obj1/\marg{opt1},obj2/\marg{opt2},...}\\
% \hspace*{3em}\marg{min-frequency}\marg{max-frequency}\\
% \hspace*{2em}|\addNichols...|\\
% \hspace*{1.5em}|\end{NicholsChart}|\\
% The |NicholsChart| environment works in conjunction with the parametric function generator macros |\addNicholsZPKChart| and |\addNicholsTFChart|. The optional argument is comprised of a comma separated list of tuples, either |obj/{opt}| or just |{opt}|. Each tuple passes options to different |pgfplots| macros that generate the axes and the plots according to:
% \begin{itemize}
% \item Tuples of the form |obj/{opt}|:
% \begin{itemize}
% \item |tikz/{opt}|: modify picture properties by adding options |{opt}| to the |tikzpicture| environment.
% \item |axes/{opt}|: modify axis properties by adding options |{opt}| to the |axis| environment.
% \item |commands/{opt}|: add any valid TikZ commands inside |axis| environment. The commands passed to |opt| need to be valid TikZ commands, separated by semicolons as usual.
% \end{itemize}
% \item Tuples of the form |{opt}| are passed directly to the |axis| environment.
% \end{itemize}
% The frequency limits are translated to the x-axis limits and the domain of the |axis| environment.
%
% \DescribeMacro{\addNicholsZPKChart}
% |\addNicholsZPKChart|\oarg{plot-options}\\
% \hspace*{2em}\marg{z/\marg{zeros},p/\marg{poles},k/\marg{gain},d/\marg{delay}}
%
% \noindent Generates a twp parametric functions for the magnitude and the phase a transfer function in ZPK form. The generated magnitude and phase parametric functions are passed to the |\addplot| macro. This macro can be used inside any |axis| environment as long as a domain for the x-axis is supplied through either the |plot-options| interface or directly in the optional argument of the container |axis| environment. Use with the |NicholsChart| environment supplied with this package is recommended. The mandatory argument is the same as |\BodeZPK|.
%
% \DescribeMacro{\addNicholsTFChart}
% |\addNicholsTFChart|\oarg{plot-options}\\
% \hspace*{2em}\marg{num/\marg{coeffs},den/\marg{coeffs},d/\marg{delay}}
%
% \noindent Similar to |\addNicholsZPKChart|, with a transfer function input in the TF form.
%
% \StopEventually{\PrintIndex}
% \clearpage
% \section{Implementation}
% \subsection{Initialization}
% \begin{macro}{\n@mod}
% \begin{macro}{\n@mod@p}
% \begin{macro}{\n@mod@n}
% \begin{macro}{\n@pow}
% \begin{macro}{gnuplot@id}
% \begin{macro}{gnuplot@prefix}
% \changes{v1.0.3}{2021/11/03}{Added jobname to gnuplot prefix}
% \changes{v1.0.8}{2022/07/06}{Fixed issue \#6}
% \changes{v1.1.4}{2023/10/12}{Changed phase wrapping in pgf mode}
% \changes{v1.2}{2024/04/18}{Removed global option to process pgf commands in radians}
% We start by processing the class options.
% \begin{macrocode}
\newif\if@pgfarg\@pgfargfalse
\DeclareOption{pgf}{
\@pgfargtrue
}
\newif\if@declutterarg\@declutterargfalse
\DeclareOption{declutter}{
\@declutterargtrue
}
\newif\if@radarg\@radargfalse
\DeclareOption{rad}{
\@radargtrue
}
\newif\if@hzarg\@hzargfalse
\DeclareOption{Hz}{
\@hzargtrue
}
\ProcessOptions\relax
% \end{macrocode}
% Then, we define new macros to unify |pgfplots| and |gnuplot|. New macros are defined for the |pow| and |mod| functions to address differences between the two math engines.
% \begin{macrocode}
\newcommand{\n@mod}[2]{(#1)-((round((#1)/(#2)))*(#2))}
\newcommand{\n@mod@p}[2]{(#1)-((floor((#1)/(#2)))*(#2))}
\newcommand{\n@mod@n}[2]{(#1)-((floor((#1)/(#2))+1)*(#2))}
\if@pgfarg
\newcommand{\n@pow}[2]{(#1)^(#2)}
\else
\newcommand{\n@pow}[2]{(#1)**(#2)}
% \end{macrocode}
% Then, we create a counter so that a new data table is generated and for each new plot. If the plot macros have not changed, the tables, once generated, can be reused by |gnuplot|, which reduces compilation time. The |declutter| option is used to enable the |gnuplot| directory to declutter the working directory.
% \begin{macrocode}
\newcounter{gnuplot@id}
\setcounter{gnuplot@id}{0}
\if@declutterarg
\edef\bodeplot@prefix{gnuplot/\jobname}
\else
\edef\bodeplot@prefix{\jobname}
\fi
\tikzset{
gnuplot@prefix/.style={
id=\arabic{gnuplot@id},
prefix=\bodeplot@prefix
}
}
% \end{macrocode}
% If the operating system is not Windows, and if the |declutter| option is not passed, we create the |gnuplot| folder if it does not already exist. \changes{v1.0.2}{2021/11/01}{Fixed issue \#1}
% \begin{macrocode}
\ifwindows\else
\if@declutterarg
\immediate\write18{mkdir -p gnuplot}
\fi
\fi
\fi
% \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \end{macro}
% \begin{macro}{\if@babel}
% \begin{macro}{\shorthand@list}
% Check if the |babel| package is loaded and generate a list of shorthands if it is. The code is based on \href{https://tex.stackexchange.com/a/708797/110602}{this stackexchange answer}.\changes{v1.1.6}{2024/01/14}{Detect `babel-french' using `frenchbsetup'}\changes{v1.1.7}{2024/02/06}{Directly detect shorthands instead of detecting the language.}
% \begin{macrocode}
\newif\if@babel\@babelfalse
\AtBeginDocument{%
\@ifpackageloaded{babel}{%
\@babeltrue
\let\shorthand@list\@empty
\def\do#1{%
\begingroup
\lccode`\~=`#1\relax
\lowercase{\ifbabelshorthand~{\g@addto@macro\shorthand@list{~}}{}}
\endgroup
}
\dospecials
}{}
}
% \end{macrocode}
% \end{macro}
% \end{macro}
% \begin{macro}{bode@style}
% Default axis properties for all plot macros are collected in this |pgf| style.
% \begin{macrocode}
\pgfplotsset{
bode@style/.style = {
label style={font=\footnotesize},
tick label style={font=\footnotesize},
grid=both,
major grid style={color=gray!80},
minor grid style={color=gray!20},
x label style={at={(ticklabel cs:0.5)},anchor=near ticklabel},
y label style={at={(ticklabel cs:0.5)},anchor=near ticklabel},
scale only axis,
samples=200,
width=5cm,
log basis x=10
}
}
% \end{macrocode}
% \end{macro}
% \begin{macro}{freq@filter}
% \begin{macro}{freq@label}
% \begin{macro}{freq@scale}
% \begin{macro}{ph@scale}
% \begin{macro}{ph@x@label}
% \begin{macro}{ph@y@label}
% These macros handle the |Hz| and |rad| class options and two new |pgf| keys named |frequency unit| and |phase unit| for conversion of frequency and phase units, respectively. \changes{v1.1.1}{2022/07/31}{New macros to enable `Hz' and `rad' units for frequency and phase, respectively}
% \begin{macrocode}
\pgfplotsset{freq@filter/.style = {}}
\def\freq@scale{1}
\pgfplotsset{freq@label/.style = {xlabel = {Frequency (rad/s)}}}
\pgfplotsset{ph@x@label/.style = {xlabel={Phase (deg)}}}
\pgfplotsset{ph@y@label/.style = {ylabel={Phase (deg)}}}
\def\ph@scale{180/pi}
\if@radarg
\pgfplotsset{ph@y@label/.style = {ylabel={Phase (rad)}}}
\pgfplotsset{ph@x@label/.style = {xlabel={Phase (rad)}}}
\def\ph@scale{1}
\fi
\if@hzarg
\def\freq@scale{2*pi}
\pgfplotsset{freq@label/.style = {xlabel = {Frequency (Hz)}}}
\if@pgfarg
\pgfplotsset{freq@filter/.style = {x filter/.expression={x-log10(2*pi)}}}
\fi
\fi
\tikzset{
phase unit/.initial={deg},
phase unit/.default={deg},
phase unit/.is choice,
phase unit/deg/.code={
\renewcommand{\ph@scale}{180/pi}
\pgfplotsset{ph@x@label/.style = {xlabel={Phase (deg)}}}
\pgfplotsset{ph@y@label/.style = {ylabel={Phase (deg)}}}
},
phase unit/rad/.code={
\renewcommand{\ph@scale}{1}
\pgfplotsset{ph@y@label/.style = {ylabel={Phase (rad)}}}
\pgfplotsset{ph@x@label/.style = {xlabel={Phase (rad)}}}
},
frequency unit/.initial={rad},
frequency unit/.default={rad},
frequency unit/.is choice,
frequency unit/Hz/.code={
\renewcommand{\freq@scale}{2*pi}
\pgfplotsset{freq@label/.style = {xlabel = {Frequency (Hz)}}}
\if@pgfarg
\pgfplotsset{freq@filter/.style = {x filter/.expression={x-log10(2*pi)}}}
\fi
},
frequency unit/rad/.code={
\renewcommand{\freq@scale}{1}