MDSPACK is a software developed by MOR Digital Systems. It gathers routines to identify, simplify and analyse dynamical systems. The library is specifically built to deal with complex - e.g. large or infinite dimensional - dynamical models. It heavily relies on approximation theory at its core.
More specifically, the library provides routines to address the following problems:
In addition, the library contains numerous utilities related to the manipulation, signal processing, transformation and visualisation of dynamical systems. These latter simplify the overall manipulation of dynamical models and data.
MDSPACK is edited by MOR Digital Systems and is distributed according to different licenses:
You will be given a file
mds.lic corresponding to
your license. It must be installed on
your computer to activate the library.
To do so, refer to license
installation section below.
Store the license file
mds.lic in a chosen
directory which absolute path is denoted
<LIC_DIR> in the
sequel. It is made available to MDSPACK
through an environment variable
MDSHOME that needs to be
defined. Refer to the section below to
do so depending on your system.
MDSHOME, and its value,
<LIC_DIR>To check that the environment
variable is properly set up, you can
display its value through the command
echo %MDSHOME% in a
terminal. This should display
<LIC_DIR>.
With some text editor, open the
(hidden) files ~/.bashrc
and ~/.profile (or
~/.bashrc_profile if it
exists instead) for Linux and
~/.zprofile for MacOS and
add the following line to the end of
these files:
export MDSHOME="<LIC_DIR>"To check that the environment
variable is properly set up, you can
display its value through the command
echo $MDSHOME in a
terminal. This should display
<LIC_DIR>.
MDSPACK is available for Windows,
Linux and MacOS (x86_64
architecture). It is distributed as an
archive containing the library and its
interfaces.
This section details the installation of the library in the system. To be usable, the license file must be installed following the instructions in the previous license installation section.
First, extract the archive on a
chosen directory on your computer (not
necessarily the same as the license
file) which absolute path is denoted
<MDSPATH> in the
sequel. Installation of required
dependencies then vary between system.
Installation instructions are detailed
below for each system.
MDSPACK library and corresponding binaries must be added to the Windows search path. To do so:
Path
in “user variables” section<MDSPATH>\binMDSPACK library can then be tested by
calling mdspack --test from
a terminal.
The required dependencies are
available in most package management
systems. In Debian derived
distributions, the required elements are
available through the package
libgfortran-10-dev and its
dependencies and can be installed as
follows,
$ sudo apt-get update
$ sudo apt-get install libgfortran-10-devThe MDSPACK library can then be added
to the system libraries, e.g. in
/usr/local/lib via a
symbolink link,
# Create symbolic link towards library
$ sudo ln -s <MDSPATH>/bin/libmdspack.so /usr/local/lib/libmdspack.so
# Update the cache of available libraries
$ sudo ldconfigSimilarly, a link towards the binary
mdspack may be added to
/usr/local/bin,
# Create symbolic link towards binary
$ sudo ln -s <MDSPATH>/bin/mdspack /usr/local/bin/mdspackMDSPACK library can then be tested by
calling mdspack --test from
a terminal.
The system embeds most required
dependencies excepted from
libgfortran. The latter can
be installed in a number of ways
following the guide on the GCC
wiki.
For instance, through Homebrew, simply type
$ brew install gccThe MDSPACK library can then be added
to the system libraries, e.g. in
/usr/local/lib via a
symbolink link,
# Create symbolic link towards library
$ sudo ln -s <MDSPATH>/bin/libmdspack.dylib /usr/local/lib/libmdspack.dylibSimilarly, a link towards the binary
mdspack may be added to
/usr/local/bin,
# Create symbolic link towards binary
$ sudo ln -s <MDSPATH>/bin/mdspack /usr/local/bin/mdspackAfter rebooting, MDSPACK library can
be tested by calling
mdspack --test from a
terminal.
Apple Sillicon
processors. If your MacOS has
an arm architecture (type
of processor can be found in the about
windows), you may use the library
through Rosetta 2.
Once MDSPACK is set up properly, its features can then be accessed through its MATLAB interface.
You may contact us at contact@mordigitalsystems.fr.
MDSPACK has been developed by Charles Poussot-Vassal and Pierre Vuillemin. See this page for further information.
It relies on additional third-party open-source libraries which are listed in appendix.
This document is available both in HTML and PDF format from MOR Digital Systems website. It gathers the information associated with the MDSPACK software.
The list of acronyms used in this document are listed below,
| Acronym | Signification |
|---|---|
| LTI | Linear Time Invariant |
| SISO | Single Input Single Output |
| MISO | Multiple Input Single Output |
| SIMO | Single Input Multiple Output |
| MIMO | Multiple Input Multiple Output |
| SVD | Singular Value Decomposition |
In addition, mathematical notations are listed in the table below
| Notation | Signification |
|---|---|
| \(n,n_x\) | number of internal variables of a dynamical model |
| \(n_u\) | number of inputs of a dynamical model |
| \(n_y\) | number of outputs of a dynamical model |
| \(n_t\) | number of time samples |
| \(s\) | Complex Laplace variable |
| \(t\) | Real time-domain variable |
| \(p\) | Real parametric variable |
The MATLAB interface is provided as a
package +mdspack located in
the directory
<MDSPATH>/API/matlab.
Dependencies. This interface requires MATLAB (>=R2016b) and uses objects (such as state-space) available in the Control System Toolbox. In addition, some functions are meant to return objects from the Robust Control System Toolbox. This is specified in the associated function documentation.
Installation. Once
MDSPACK library is installed (as
detailed in the dedicated
section), you need to add
<MDSPATH> and its
subdirectories to the MATLAB path. This
may be done by adding the following
lines to your startup.m
file:
% Path towards MDSPACK
addpath <MDSPATH>/bin
addpath <MDSPATH>/API/matlab/Whether the package is functional can
then be evaluated by calling
mdspack.test_api() from the
MATLAB command window. This routine
launches a set of tests and informs the
user if the API is fully functional.
Content of the
package. The functions
available within +mdspack
are sorted by category:
Identification techniques are classified here depending on the nature of data they are handling:
Interface to the MIMO LTI frequency-domain Loewner interpolation method. It is used as follows,
[Hr, info] = mdspack.loewner(w, H, [opt])where the input data are:
w: a real \(n_w\)
vector containing strictly positive
frequencies.H: a complex \(n_y \times n_u
\times n_w\) matrix containing
the frequency responses to be
matched.opt (optional): a
structure describing the options of the
method.and the outputs are:
Hr: a descriptor model
interpolating tangentially the input
data.
info: a structure
giving information about the
interpolation process.target: scalar
associated with the dimension objective,
being the order along the \(s\)
variables. Default is inf.
Note that if a lower order is possible,
the algorithm will use it instead.freq_band: a 2
dimensional real vector describing the
frequency interval of interest. Values
of w outside this interval
are discarded. Default value is
[0,inf].normalise: a boolean to
activate the normalisation of the
frequency response data H.
Normalisation is useful in MIMO cases to
improve the look-alike between the data
and the interpolating model when
different input-output have very
different magnitudes. Default value is
false.ls_fit: a boolean to
activate a posteriori least-square
matching of the data. This should
improve the matching in MIMO case.
Default is true.stable: a boolean to
activate the projection of the
interpolating model onto a stable
subspace. Default value is
false.check_rank: a boolean
to activate the computation of the rank
condition associated with the Loewner
framework. This may be useful for
debugging purpose when the algorithm
fails but the condition can be very
restrictive. Default value is
false.status: integer
indicating the status of the algorithm
0: no error1: rank condition not
satisfiedierr: relative
interpolation error between the model
Hr and the input data
H, computed as\[ \sqrt{ \frac{1}{n_w} \sum_{i=1}^{n_w} \Vert H_i - H_r(j\omega_i) \Vert_F^2} / \sqrt{ \frac{1}{n_w} \sum_{i=1}^{n_w} \Vert H_i \Vert_F^2} \]
This example shows how some
(supposedly unknown) test model
G may be recovered from its
frequency response.
% Test model G
G = tf(1,[1/100, 0.2/10, 1]);
% Associated frequency domain samples
w = logspace(-2,2,10);
Gjw = freqresp(G,w);
% Interpolation with Loewner
[Hr, info] = mdspack.loewner(w, Gjw);
% Comparison with initial model
norm(G - Hr,2)Proposed in [1], the Loewner framework enables to find a descriptor realisation \(H_r(s) = C_r (sE_r - A_r)^{-1} B_r\) that interpolates tangentially some given frequency-domain data, i.e. \[ w_i = H_r( \lambda_i) r_i \quad v_j^T = l_j^T H_r(\mu_j), \] with \(\lambda_i, \mu_j \in \mathbb{C}\), \(w_i, l_j \in \mathbb{C}^{n_y}\), \(v_j, r_i \in \mathbb{C}^{n_u}\). Under some rank conditions, the realisation is minimal.
Interface to the parametric MIMO LTI frequency-domain Loewner interpolation method (routine extending mdspack.loewner by performing interpolation along both \(s\) and \(p\)). It is used as follows,
[Hr, info] = mdspack.ploewner(s, p, H, [opt])where the input data are:
s: a complex \(n_s\)
vector containing the complex
(frequency) points used when computing
parametric response H.
Generally, s contains
frequencies and is thus composed of pure
complex values, but any real/complex
number may be used. Note that
s should at least contain
one single complex value.p: a real \(n_p\)
vector containing the real (parametric)
points used when computing parametric
response H.H: a complex \(n_y \times n_u
\times n_s \times n_p\) matrix
containing the responses to be matched.
More specifically, each
{s,p} combination leads to
a \(n_y\)
outputs and \(n_u\)
inputs matrix.opt (optional): a
structure describing the options of the
method.and the outputs are:
Hr: set of matrices
describing a parametric descriptor
state-space model, interpolating the
data H along \(s\) and
\(p\).
Hr is the required argument
for mdspack.usubs
routine to generate a descriptor
dss model evaluated at a
frozen parametric value, or for mdspack.dssu2uss
routine to generate an uncertain
uss model.info: a structure
providing informations about the
interpolation process.target: \(1 \times
2\) positive integer vector
associated with the dimensions
objectives. In the two parameters case,
user may chose both orders along the
\(s\)
and \(p\)
variables. The first element is the
order along the \(s\)
parameter, and the second along the
\(p\)
one. Default value is
[inf,inf].real: a boolean to
enforce the constructed realisation to
be real-valued. Default value is
false. Note that if set to
true, then s
must be organised as follows: (at least)
the first element s_r must
be real and the rest complex conjugated,
as follows:
[s_r, s_1, conj(s_1), s_2, conj(s_2), ...].verbose: boolean used
to plot informations during the model
construction. It plots (i) the SVD drop
used to detect the degrees both in \(s\) and
\(p\),
(ii) the Loewner kernel computation
error. Default value is
false. The singular values
drop is a good indicator of a rational
approximation order to be used in the
optional field target.status: integer
indicating the status of the algorithm
0: no error1: method not available
(e.g. for more than one single
p parameter)2: possible inaccurate
matching due to kernel inacurate
computation
LL2: 2D Loewner matrix
\(\mathbb
L_2\), in complex form.c: 2D Loewner matrix
kernel, such that \(\mathbb L_2
c=0\).ker_err: kernel
solution error \(||\mathbb L_2
c||_2\).order: rational orders
along \(s\) and
\(p\).
Note that in general, a real-valued
model is sought and the optional field
real should be set to
true. In this case the
s vector must be organised
as follows: at least the first element
s_r must be real while the
rest s_i
(i=1,...,n) may be complex
conjugated, as follows:
[s_r, s_1, conj(s_1), s_2, conj(s_2), ...,conj(s_1), s_n, conj(s_n)].
The obtained model Hr is
a parametric continuous-time one. It is
described as a structure. Evaluating the
state-space at a frozen parameter value
may be done using either:
uss
model.Let sample the unknown system
G along
s,
being \(200\)
points linearly spaced between
0.1i and 100i
plus in 0.01, andp, being
\(20\)
points linearly spaced between
0.1 and
1,leading to H, a
frequency response matrix of dimension
\(2\times
1\times 201\times 20\). The
triplet {s,p,H} is referred
to as the original data. The Loewner
framework automatically finds
Hr, the rational parametric
function interpolating the original
data. The minimal rational orders along
\(s\)
and \(p\) (here
respectively 3 and 2), are automatically
recovered.
% Test model G(s,p)
G = @(s,p) [s^3*p; s]/(s^2+p*s+1);
% Complex-domain samples 's' (notice that s contains first a real value,
% then a sequence of complex conjugated ones)
w = linspace(.1,100,30);
s = [.01 sort([1i*w -1i*w])];
% Real-domain parametric samples 'p'
p = linspace(.1,1,20);
% Evaluation of H, the response of G(s,p) along 's' and 'p'
% (notice that H is a 2x1x201x20 complex-valued matrix)
H = mdspack.eval(G,s,{p});
% Evaluation of Hw, the response of G(s,p) along '1i*w' and 'p'
% (notice that H is a 2x1x100x20 complex-valued matrix, similar to a Bode response)
Hw = mdspack.eval(G,1i*w,{p});
% Interpolation with the 2-parameters Loewner
opt.target = [3 inf]; % target orders along s and p
opt.verbose = true; % show the process informations
opt.real = true; % enforce real-valued approximating model
[Hr,info] = mdspack.ploewner(s,p,H,opt);
Hr
info.orderNow the structure Hr may
be evaluated at a frozen parametric
value, by mean of mdspack.usubs
and /or mdspack.dssu2uss.
Then, one may compare the frequency
responses of the obtained model with
respect to the original data.
% Comparison with initial model on a thiner frequency grid
ww = union(w,logspace(log10(min(w)),log10(max(w)),300));
for ii = 1:numel(p)
% Construct dss model, being the evaluation of Hr at p(ii)
Hr_p = mdspack.usubs(Hr,p(ii));
% Plot the Bode gain and phase diagram
figure(10), hold on
mdspack.bodemag(Hw(:,:,:,ii),w,'.','Color','k')
mdspack.bodemag(Hr_p,ww,'-')
figure(20), hold on
mdspack.bodephase(Hw(:,:,:,ii),w,'.','Color','k')
mdspack.bodephase(Hr_p,ww,'-')
drawnow
end
% Construct a 'uss' model (in MATLAB format) if Robust Control Toolbox available
Hu = mdspack.dssu2uss(Hr,[min(p) max(p)]);Originally proposed in [2] and
updated in [3] for
the MIMO case, the parametric Loewner
framework enables to find a descriptor
realisation of the form \[
H_r(s,p) = C_r(p) (sE_r - A_r(p))^{-1}
B_r \text{ or } H_r(s,p) = C_r (sE_r -
A_r(p))^{-1} B_r(p)
\] where \[
\begin{array}{rcl}
A_r(p) &=& A+\sum_{j=1}^{n_p}
A_j p^j\\
B_r(p) &=& B+\sum_{j=1}^{n_p}
B_j p^j\\
C_r(p) &=& C+\sum_{j=1}^{n_p}
C_j p^j
\end{array}
\] that interpolates given
frequency / parametric domain data, i.e.
\[
w_{ij} = H_r( \lambda_i,\pi_j) \quad
v_{kl} = H_r(\mu_k,\nu_l),
\] where \(\lambda_i, \mu_k
\in \mathbb{C}\), \(\pi_j, \nu_l \in
\mathbb{R}\) and \(w_{ij},v_{ij} \in
\mathbb{C}^{n_y\times n_u}\).
More specifically, \(\lambda_i,
\mu_k\) are partitions of
s, \(\pi_j,
\nu_l\) are partitions of
p and \(w_{ij},v_{kl}\)
are the corresponding partitions of
H. Under some rank
conditions, the realisation is minimal,
otherwise, close to minimal.
Interface to the SISO LTI time-domain Pencil method. It is used as follows,
[Hr, info] = mdspack.pencil(Y, [U, opt])where the input data are:
Y: a real \(n_t\)
vector containing constantly sampled
time-domain output data. Y
is the measurement snapshot vector.U (optional): a real
\(n_t\)
vector containing constantly sampled
time-domain input data. U
is the input snapshot vector.opt (optional): a
structure describing the options of the
method.and the outputs are:
Hr: a sampled-time
state-space model which initial
response, or response to U
input snapshot (if provided) matches, or
approximates the measure snapshots
Y.
info: a structure
giving informations about the Pencil
method process.target: indicates the
target order / rank tolerance of the
resulting generated model. Default value
is 1e-10.
target defines a rational
approximation order.target defines
a normalised rank tolerance. In this
case, the rational approximation order
is automatically computed according to
the approximation error associated with
the normalised rank decomposition of the
Pencil matrix.dt : a sampling-time
given as a positive real. Default value
is 1.svd: employed singular
value decomposition method. Default
value is 'svd'.
'svd': full SVD.'rsvd': randomized
SVD.status: integer
indicating the status of the algorithm
0: no error1: error in the
processxr0: projected
generating model initial condition, with
the same dimension as
Hr.U is empty)
Hri: Hr
equivalent impulsive model, with same
dimension as Hr. The
Hri impulse response with
zero initial condition is equivalent to
Hr response with initial
condition xr0. Both lead to
Y. c2d routine.Let us simulate the time-domain response of a fourth order linear model excited by a random input signal, with zero initial conditions. The Pencil method allows recovering the underlying sampled-time linear model producing the output sequence in response to the very same exciting signal. Interestingly, if the original model is linear, the eigenvalues are also recovered.
% Test model and simulation data
dt = .1;
t = 0:dt:20;
H = ss(tf(1,[1/100 .2/10 1]) + tf(1,[1/4 .2/2 1]));
% Simulate with random input U and null condition x0
x0 = zeros(length(H.a),1);
U = rand(1,numel(t)).*exp(-.01.*t)+1;
[Y,T,~] = lsim(H,U,t,x0);
% Pencil method
opt = struct('dt',dt,'target',4);
[Hr,info] = mdspack.pencil(Y,U,opt);
xr0 = zeros(length(Hr.a),1);
[Yr,Tr,~] = lsim(Hr,U,t,xr0);
% Display results
figure,
plot(T,Y,'o',Tr,Yr,'-')
xlabel('\textbf{Time [s]}','Interpreter','latex');
ylabel('$\mathbf{y}$','Interpreter','latex');
legend({'Data','Model'},'Interpreter','latex')
% Compare the identified eigenvalues
eig(c2d(H,dt))
eig(Hr)Similarly, if no excitation signal is fed, but a non null initial condition, the same observation can be made.
% Test model and simulation data
dt = .1;
t = 0:dt:20;
H = ss(tf(1,[1/100 .2/10 1]) + tf(1,[1/4 .2/2 1]));
% No input on the system (autonomous)
H.b = [];
H.d = [];
% Simulate with null input U and random initial condition x0
x0 = rand(length(H.a),1);
[Y,T,X] = initial(H,x0,t);
% Pencil method
opt = struct('dt',dt,'svd','svd','target',1e-6);
[Hr,info] = mdspack.pencil(Y,opt);
% Initial response
[Yr,~,~] = initial(Hr,info.xr0,t);
% Equivalent impulse model response
[Yri,Tr,Xr] = impulse(info.Hri,t);
% Display results
figure,
plot(T,Y,'o',Tr,Yr,'-',Tr,Yri,'--')
xlabel('\textbf{Time [s]}','Interpreter','latex');
ylabel('$\mathbf{y}$','Interpreter','latex');
legend({'Data','Model (initial)','Model (impulse eq.)'},'Interpreter','latex')
% Compare the identified eigenvalues
eig(c2d(H,dt))
eig(Hr)The pencil framework has been
proposed in [4]. It
enables to construct a LTI SISO system
Hr which response from an
exciting signal U (if
provided) or from an initial value (if
no input provided) leads to a
time-domain output similar to the
Y data sequence. The
procedure requires either zero initial
condition and input signal
U (not necessarily
persistently exciting) or a non zero
initial condition with no input signal.
The resulting model Hr
embeds the modal content of the
generating system and the transfer from
input to output.
Routine for model complexity reduction.
Interface to the MIMO LTI modal truncation algorithm. It is used as follows,
[Hr, info] = mdspack.modal(H, [opt])where the inputs are:
H: a (descriptor)
state-space model to be reduced.opt (optional): a
structure containing the options of the
method.And the outputs are:
Hr: a reduced-order
(descriptor) state-space model.info: a structure
containing the informations about the
reduction process.target: indicates the
target and/or order of the resulting
generated model. Default value is
0.1, for \(10\%\)
mismatch error objective.
target defines an
approximation order.target defines
a relative reduction error in terms of
(frequency-limited) \(\mathcal{H}_2\)-norm.freq_band:
frequency-band of interest. Default is
[0,inf].normalise: a boolean to
activate input/output normalisation.
Normalisation is useful in the MIMO case
to improve the look-alike with the
initial model when different i/o
channels have very different magnitudes.
Default is false.ls_fit: a boolean to
activate a posteriori residues
adjustment. Default is
true.status: integer
indicating the status of the algorithm:
0: no error.1: the model cannot be
diagonalised.2: target reduction
order cannot be reached due to the
number of un-reductible modes.r: dimension of the
reduced stable subsystem.k: number of
un-reductible poles (marginally
stable/unstable and infinite) that have
been kept.err: relative
(frequency-limited) \(\mathcal{H}_2\)-norm
of the error (excluding direct
feedthrough, if any).In this example, modal truncation is applied to reduce a \(48\)-th order dynamical model. By default, the approximation order is automatically selected so that the relative approximation error falls below \(10\)%.
% Loading test model: 48-th order LAH model
load build
% Perform modal truncation
[Hr, info] = mdspack.modal(G);
% Check the resulting approximation error
[info.err , norm(G-Hr,2)/norm(G,2)]
% Plot the Bode response
w = logspace(0,2,200);
figure,
subplot(211), hold on
mdspack.bodemag(G,w,'-')
mdspack.bodemag(Hr,w,'--')
subplot(212), hold on
mdspack.bodephase(G,w,'-')
mdspack.bodephase(Hr,w,'--')
legend('Original model','Reduced order model')Set of routines for data and model analysis. First evaluation and frequency responses and then functions for norms and stability estimations.
Interface that computes the (parametric) MIMO LTI transfer function evaluation. It is used as follows,
[H, info] = mdspack.eval(G, s, [p])where the inputs are
G: a MIMO LTI model
with \(n_u\)
inputs and \(n_y\)
outputs provided in as a MATLAB
handle function and reading
as
G = @(s) ... in the
nonparametric case, one evaluates a
transfer function \(G(s)\).G = @(s,p) ... in the
parametric case, one evaluates a
transfer function \(G(s,p)\)
where \(p\) is the
parameter vector given as a real \(n_p\)
vector. It describes the dependency
along \(p(1)\),
\(p(2)\),
…s: the evaluation grid
points of the dynamical part of the
complex function, i.e. along \(s\)
parameter (complex \(n_s\)
vector).and the outputs are
H: the evaluation of
G being
info: additional
informations
ny, nu: number of
inputs and outputs (integers).combi_idx: parameters
combinations index table (dimension
\((n_{p_1}
n_{p_2} ...)\times n_p\)).p: the parametric
gridding points given as a cell of size
\(n_p\)
where p{1} is the value
vector for p(1) (real \(n_{p_1}\)
vector) p{2} is the value
vector for p(2) (real \(n_{p_2}\)
vector)…The variable s denotes
the complex point where the function is
evaluated. Usually one evaluates along
the imaginary axis for continuous-time
models and along the unit circle for
sampled-time ones. However, the function
allows evaluating at any complex
point.
Here one aims at evaluating the
transfer function \[
G(s,p_1,p_2) =
\frac{p_2^2s^4}{2s^3+3p_1^3+sp_2-1}
\] along
s=[1,...,100]i, for
p1=[2,...,5] and
p2=[1,...,5]. The code
below leads to a complex-valued
evaluation of dimension \(1\times 1\times
106\times 4\times 3\).
% Test model G(s,p1,p2)
G = @(s,p) [p(2)^2*s^4]/(2*s^3+3*p(1)^3+s*p(2)-1);
% Complex-domain samples 's'
w = linspace(1,100,106);
s = 1i*w;
% Real-domain parametric samples 'p1' and 'p2'
p1 = linspace(2,5,4);
p2 = linspace(4,5,3);
% Evaluation of H, the response of G(s,[p1 p2]) along 's' and 'p1', 'p2' (notice that H is a 1x1x106x4x3 complex-valued matrix)
[H,info] = mdspack.eval(G,s,{p1,p2});To convince, one may compare the responses using the frequency sigma plot diagram as follows.
kk = 0;
figure, hold on
for jj = 1:numel(p2)
for ii = 1:numel(p1)
kk = kk + 1;
mdspack.sigma(G(tf('s'),[p1(ii),p2(jj)]),w,'-')
mdspack.sigma(H(:,:,:,ii,jj),w,'k--')
mdspack.sigma(H(:,:,:,info.combi_idx(kk,1), ...
info.combi_idx(kk,2)),w,'r:')
drawnow
end
endInterface that computes the MIMO LTI frequency response. It is used as follows,
F = mdspack.freqresp(H, w)where the inputs are:
H: a LTI model given in
one of the following forms
ss, dss or
tf continuous or
sampled-time model.{A,B,C,D,E,dt}
continuous or sampled-time model of the
form (if dt<=0 or if
dt=[]) \[
E\dot x(t) = Ax(t) + Bu(t), y(t) = Cx(t)
+ Du(t)
\] or if dt>0,
\[
Ex(k+dt) = Ax(k) + Bu(k), y(k) = Cx(k) +
Du(k).
\]freqresp response
provided by mdspack.freqresp
or the MATLAB freqresp
routine.handle function, given
as H = @(s) ... (in
continuous time).structure with
the following fields
fun: a handle function
given as
H.fun = @(s) ...dt: the sampling time
H.dt being \(\leq 0\)
for continuous-time and \(>0\) in
the sampled-time case.w: the frequency
evaluation gridding points (real \(1 \times
n_w\) vector).And the outputs are:
F: a complex frequency
response of dimension \(n_y\times n_u
\times n_w\).plotOptions: classical
MATLAB plot function
arguments% Associated frequency-domain grid
w = logspace(-2,2,200);
% Test model
Htf = tf(1, [1/100, 0.2/10, 1]);
Hhan = mdspack.freqresp(@(s) 1/(s^2/100+0.2*s/10+1),w);
Hfreqresp = mdspack.freqresp(Htf,w);
% mdspack Bode diagram
figure, hold on
mdspack.bodemag(Htf,w,'-')
mdspack.bodemag(Hhan,w,'--')
mdspack.bodemag(Hfreqresp,w,'.','Color','b')Interface that computes the MIMO LTI norms associated with frequency-domain data. It is used as follows,
N = mdspack.fdnorm(H, [w, ntype, freq_band])where the inputs are
H: a complex \(n_y \times n_u
\times n_\omega\) matrix
containing the response of the
system.w (optional): a real
vector containing the associated
frequencies.ntype (optional): the
norm to be computed (default is
'FRO')
'L2': trapezoidal
approximation of the integral defining
the \(2\)-norm
\[
\textstyle
\sqrt{\frac{1}{2\pi}\sum_{k=1}^{n_\omega-1}
(\omega_{k+1} - \omega_k)(\Vert H_{k+1}
\Vert_F^2 +\Vert H_k \Vert_F^2)}.
\]'FRO': average sum of
squared Frobenius norms'LINF': data-driven
approximation of the \(\infty\)
norm, given by \(\max_{k} \Vert H_k
\Vert_2 % max_k || H_k ||_2\). It
is a lower bound for the real \(\infty\)
norm.'ABS': maximum
pointwise data \(\max_{k,i,j}
|H_{k}(i,j)| % max_k,i,j | H_k(i,j)
|\).freq_band (optional):
the frequency band of interest, default
is [0,inf].and the outputs is:
N: is the value of the
norm.Note that the frequency \(\omega\) is not required by several of these norms which can thus be used on data not associated with axis data.
% Test model
H = tf(1, [1/100, 0.2/10, 1]);
w = logspace(-2, 2, 200);
h = freqresp(H, w);
% Comparison of 2 and inf norms
[norm(H, 2), mdspack.fdnorm(h, w, 'L2')]
[norm(H, inf, 1e-8), mdspack.fdnorm(h, w, 'LINF')]Interface that estimates the LTI dynamical model stability from frequency-domain data. It is used as follows,
[stab, Hr] = mdspack.fdstabest(w, H)where the inputs are
w: a real vector
containing the associated
frequencies.H: a complex \(n_y \times n_u
\times n_\omega\) matrix
containing the response of the
system.And the outputs are
stab: indicator of
whether the data have been generated by
a stable system.Hr: a candidate stable
linear model.Beware that this routine gives an estimate of the stability but should not be considered a formal certificate.
% Stable test model
Hs = tf(1, [1/100, 0.2/10, 1]);
% Associated frequency-domain data
w = logspace(-2,2,100);
hs = freqresp(Hs, w);
[stab, ~] = mdspack.fdstabest(w, hs);
[isstable(Hs), stab]
% Unstable test model
Hu = tf(1, [1/100, -0.2/10, 1]);
% Associated frequency-domain data
w = logspace(-2,2,100);
hu = freqresp(Hu, w);
[stab, ~] = mdspack.fdstabest(w, hu);
[isstable(Hu), stab]The algorithm implements the idea developed in [5] and evaluates the stability of frequency-domain data based on the proximity of the stable interpolant model that can be built with the Loewner algorithm and a stable rational approximation.
Set of routines for transformation and manipulation of dynamical models.
Interface that computes the additive decomposition of a MIMO LTI model into finite stable/unstable/marginally-stable and infinite parts. It is used as follows,
[Hfs, Hfu, Hfa, Hi] = mdspack.split(H, [opt])where the inputs are
H: a (descriptor)
state-space modelopt: a structure
describing the options of the method,
which may be
tola: tolerance for the
separation of marginally stable
eigenvalues. Default value is
1e-11.and the outputs are
Hfs: state-space
component containing the finite stable
eigenvalues.Hfu: state-space
component containing the finite unstable
eigenvalues.Hfa: state-space
component containing the finite
marginally stable eigenvalues.Hi: descriptor
state-space component containing the
infinite eigenvalues, including the
direct feedthrough.% Test model
s = tf('s');
H = ss(1/(s + 1) + 1/(1 + 1e-13/10 * s + 1/100 * s^2) + s^2 + 1 + 1/(s-2));
% Additive decomposition
[Hfs, Hfu, Hfa, Hi] = mdspack.split(H);Interface that computes the projection of MIMO LTI model onto some stable subspace. It is used as follows,
[Hp, info] = mdspack.sproj(H, [opt])where the inputs are
H: a (descriptor)
state-space model.opt (optional): a
structure containing the options,
space: target space for
projection: 'I' for the
\(H_\infty\)
space (default) or '2' for
the \(H_2\)
space.and the outputs are
Hp: the projected
model. Note that infinite and marginally
stable poles cannot be projected and are
kept in Hp.info: a structure
containing the informations about the
projection process,
status: integer
indicating the status of the algorithm:
0: no error.>0: number of
infinite/marginally stable poles that
could not be projected.err: the projection
error accordingly to the projection
space.% Test model containing one stable, one marginally stable and one unstable pole
s = tf('s');
H = ss(1/(s+1) + 1/s + 1/(s-1));
% Projection onto Hinf space
[Hp, info] = mdspack.sproj(H);
% Check eigenvalues of initial and projected system
eig(H.A), eig(Hp.A)
% Error induced by the projection
[norm(minreal(H - Hp), inf), info.err]The algorithm largely depends on the selected projection space:
Note that in both cases, components
that are not in \(L_2\) or
\(L_\infty\)
such as infinite or marginally stable
poles, cannot be projected and are kept
in Hp.
Interface that evaluates a parametric MIMO LTI model at a frozen parametric configuration. It is used as follows,
Hp = mdspack.usubs(Hr, p)where the input data are:
Hr: set of matrices
describing a parametric descriptor
state-space model, as generated by mdspack.ploewner.p: a real parametric
value at which Hr should be
evaluated.and the output is:
Hp: a state-space
model, describing Hr at a
p value, given as a
dss MATLAB format.See example given in mdspack.ploewner.
Interface that constructs a single
parameter MIMO LTI uncertain MATLAB
uss model. This routine
requires the MATLAB Robust Control
Toolbox. It is used as follows,
Hu = mdspack.dssu2uss(Hr, p_range)where the input data are:
Hr: set of matrices
describing a parametric descriptor
state-space model, as generated by mdspack.ploewner.p_range: a real \(1\times 2\)
parametric range (lower and upper bound)
at which Hr should be
evaluated.and the output is:
Hu: an uncertain
state-space model, given as a
uss MATLAB format.See example given in mdspack.ploewner.
Routine for signals pre-prcessing.
Interface that performs a digital filtering of data using the Savitzky-Golay algorithm. It is used as follows,
Hf = mdspack.filter_sg(H, nl, [r])where the inputs are
H: frequency response
data (complex \(n_y \times n_u
\times n_w\) matrix e.g. from
mdspack.freqresp) or vector
(complex \(1
\times n_w\)).nl: forward/backward
steps to be smoothed (integer).and the outputs are
Hf: smoothed frequency
response (same format as
H).r: ratio defining the
order d of the polynomial
fitting the data, given as
d=floor(nl*(1-r)). Default
is r = 0.9 (real scalar
between 0 and
1).By choosing r close to 1
generally smoothes a lot the data. This
is typically useful if the signal is
very noisy. You may also increase
nl to smooth even more.
When performing any identification
technique, it is often interesting to
pre-process the raw with such a filter.
It avoids overfitting.
Being given a frequency response with
random noise,
mdspack.filter_sg may be
used to smooth the data without phase
distortion.
% Test model G(s)
G = rss(100,2,3);
% Evaluation of H, the response of G(s) along '1i w'
w = linspace(.1,100,1e4);
[H,info] = freqresp(G,w);
% Add noise
Hn = H.*(1 + .1*randn(size(H)));
% Filter
Hf = mdspack.filter_sg(Hn,10,1);
% Compare Nyquist responses
figure, hold on
mdspack.nyquist(Hn,w,'.')
mdspack.nyquist(H,w,'-','LineWidth',2)
mdspack.nyquist(Hf,w,'--','LineWidth',2)The mdspack.filter_sg
implements a digital (and non-causal)
Savitzky-Golay filter. It may be applied
to a set of digital data points for the
purpose of smoothing the data, e.g. to
increase the signal-to-noise ratio
without greatly distorting the signal.
This is performed by a convolution
fitting successive sub-sets of adjacent
data points with a low-degree polynomial
(with linear least squares).
Set of routines for plotting MIMO LTI dynamical model / data responses and properties.
Plots one single MIMO LTI system Bode
gain. Works for LTI systems given is
dss, ss,
tf, handle or
freqresp forms.
mdspack.bodemag(H, w, [plotOptions])where the inputs are:
H: a LTI model given in
one of the following forms
ss, dss or
tf continuous or
sampled-time model.{A,B,C,D,E,dt}
continuous or sampled-time model of the
form (if dt<=0 or if
dt=[]) \[
\textstyle
E\dot x(t) = Ax(t) + Bu(t), y(t) = Cx(t)
+ Du(t)
\] or if dt>0,
\[
\textstyle
Ex(k+dt) = Ax(k) + Bu(k), y(k) = Cx(k) +
Du(k).
\]freqresp response
provided by mdspack.freqresp
or the MATLAB freqresp
routine.handle function, given
as H = @(s) ... (in
continuous time).structure with
the following fields
fun: a handle function
given as
H.fun = @(s) ...dt: the sampling time
H.dt being \(\leq 0\)
for continuous-time and \(>0\) in
the sampled-time case.w: the frequency
evaluation gridding points (real \(1 \times
n_w\) vector)plotOptions: same as
the MATLAB plot function
arguments% Associated frequency-domain grid
w1 = logspace(-2,2,200);
w2 = logspace(-2,2,20);
% Test model
Htf = tf(1, [1/100, 0.2/10, 1]);
Hss = ss(Htf); I = eye(length(Hss.a));
Hhan = @(s) Hss.c*((I*s-Hss.a)\Hss.b) + Hss.d;
Hfreqresp = freqresp(Htf,w2);
% mdspack Bode diagram
figure, hold on
mdspack.bodemag(Htf,w1,'-','LineWidth',2)
mdspack.bodemag(Hhan,w1,'--','LineWidth',2)
mdspack.bodemag(Hfreqresp,w2,'x', 'Color', 'b','MarkerSize', 10, 'LineWidth', 2)Same as mdspack.bodemag but plotting the Bode phase.
% Associated frequency-domain grid
w = logspace(-2,2,200);
% Test model
Hss = rss(10,1,2);
Htf = tf(Hss);
Hhan = @(s) Hss.c*( (s*eye(length(Hss.a)) - Hss.a)\Hss.b ) + Hss.d;
Hfreqresp = freqresp(Hss,w);
% mdspack Bode diagram
figure, hold on
mdspack.bodephase(Hss,w,'-','LineWidth',2)
mdspack.bodephase(Hhan,w,'--','Color',[1 .2 .2])
mdspack.bodephase(Hfreqresp,w,':','LineWidth',2)Same as mdspack.bodemag but plotting the Nyquist.
% Associated frequency-domain grid
w = logspace(-2,2,400);
% Test model
Htf = tf(1, [1/100, 0.2/10, 1]);
Hss = ss(Htf);
Hhan = @(s) 1/(s^2/100+0.2*s/10+1);
Hfreqresp = freqresp(Htf,w);
% mdspack Bode diagram
figure, hold on
mdspack.nyquist(Hss,w,'-','LineWidth',2)
mdspack.nyquist(Hhan,w,'--','Color',[1 .2 .2])
mdspack.nyquist(Hfreqresp,w,':','LineWidth',2)Same as mdspack.bodemag but plotting the Sigma-plot.
% Associated frequency-domain grid
w = logspace(-2,2,200);
% Test model
Hss = rss(10,1,2);
Htf = tf(Hss);
Hhan = @(s) Hss.c*( (s*eye(length(Hss.a)) - Hss.a)\Hss.b ) + Hss.d;
Hfreqresp = freqresp(Hss,w);
% mdspack Bode diagram
figure, hold on
mdspack.sigma(Hss,w,'-','LineWidth',2)
mdspack.sigma(Hhan,w,'--','Color',[1 .2 .2])
mdspack.sigma(Hfreqresp,w,':','LineWidth',2)The open-source softwares used by MDSPACK are listed in this section together with their licenses.
MDSPACK uses BLAS and LAPACK for some linear algebra tasks.
BLAS: from https://netlib.org/blas/, it is licensed as follows,
The reference BLAS is a freely-available software package. It is available from netlib via anonymous ftp and the World Wide Web. Thus, it can be included in commercial software packages (and has been). We only ask that proper credit be given to the authors.
Like all software, it is copyrighted. It is not trademarked, but we do ask the following:
If you modify the source for these routines we ask that you change the name of the routine and comment the changes made to the original.
We will gladly answer any questions regarding the software. If a modification is done, however, it is the responsibility of the person who modified the routine to provide support.
LAPACK: from https://netlib.org/lapack, it is licensed as follows,
Copyright (c) 1992-2013 The University of Tennessee and The University
of Tennessee Research Foundation. All rights
reserved.
Copyright (c) 2000-2013 The University of California Berkeley. All
rights reserved.
Copyright (c) 2006-2013 The University of Colorado Denver. All rights
reserved.
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