Control package

The control package is part of the Octave Forge project. Its mercurial repository lies in control at the Octave Forge

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
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
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
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
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 ${\textstyle -3\,dB}$ magnitude at ${\textstyle f=1/T_{1}=2.5\,rad/s}$ .

Inverted Pendulum

Model

A nonlinear model of the inverted pendulum can be derived by

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

where the variables are defined as

Name	Definition
${\textstyle M}$	Mass of the cart
${\textstyle m}$	Mass of the pendulum
${\textstyle c}$	Translational damping coefficient
${\textstyle l}$	Length of the pendulum
${\textstyle I}$	Inertia of the pendulum
${\textstyle b}$	Rotational damping coefficient

Schematic drawing of the inverted pendulum.

Linearization around the point ${\textstyle \theta =\pi }$ and substitution of ${\textstyle \theta '=\theta -\pi }$ (from here on in the analysis, ${\textstyle \theta '}$ will be written as ${\textstyle \theta }$ , but it should be noted that ${\textstyle \theta }$ now measures from a new reference) leads to

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

This can be expressed in the frequency domain like

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

Transfer functions

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

${\begin{aligned}G_{1}(s)&={\frac {X(s)}{F(s)}}={\frac {\alpha _{2}s^{2}+\alpha _{1}s+\alpha _{0}}{\beta _{4}s^{4}+\beta _{3}s^{3}+\beta _{2}s^{2}+\beta _{1}s}}\\G_{2}(s)&={\frac {\Theta (s)}{F(s)}}={\frac {\gamma _{1}s}{\beta _{4}s^{3}+\beta _{3}s^{2}+\beta _{2}s+\beta _{1}}}\end{aligned}}$

with

{\textstyle \alpha _{2}=ml^{2}+I}

{\textstyle \alpha _{1}=b}

{\textstyle \alpha _{0}=-mgl}

{\textstyle \beta _{4}=(M+m)(ml^{2}+I)-m^{2}l^{2}}

{\textstyle \beta _{3}=(M+m)b+(ml^{2}+I)c}

{\textstyle \beta _{2}=-(M+m)mgl+bc}

{\textstyle \beta _{1}=-mglc}

{\textstyle \gamma _{1}=-ml}

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.

Bode diagram of the cart movement transfer function.
Bode diagram of the pendulums rotation transfer function.