A toolbox for LPV system indentification using the Regularized State-space Model Interpolation of Local Estimates (ReSMILE) method.

For a 10 minute introduction, watch the talk about this project here:

• MathWorks MATLAB (Tested versions: R2019b on GNU/Linux, R2020a on Windows 10)
• MathWorks Control System Toolbox for MATLAB
• OptiSpline (release v0.2) package contains both OptiSpline and CasADi, and is used to formulate and solve optimization problems involving B-splines
• YALMIP (commit db31920) is used by OptiSpline to parse optimization problems involving B-splines
• Mosek (version 8.0.0.60) is used as the solver for SOCP problems
• (optional) LC Toolbox can be used to work with the results of the ReSMILE, contains LPV control design routines. It contains the resmile repo as a Git submodule.

Download all prerequisites and unpack them to a folder.
It is recommended to get the ReSMILE as part of the LC Toolbox:

git clone https://github.com/meco-group/lc_toolbox
git submodule update --init --recursive

A script that you can use to add prequisites to path and check that they work:

addpath(genpath('yalmip'))                          % Add YALMIP to path
addpath('C:\Program Files\Mosek\8\toolbox\r2014a')  % Add Mosek to path
addpath('optispline')                               % Add optispline to path
addpath(genpath('lc_toolbox'))                      % Add LCToolbox (with the ReSMILE) to path
ss                                                  % Test for Control System Toolbox, should print "Empty state-space model."
mosekdiag                                           % Test for Mosek, should print "mosekopt is working correctly." in the last line
casadi.MX                                           % Test for CasADi, should print "0x0" if CasADi is working correctly
sdpvar                                              % Test for YALMIP, should print "Linear scalar (real, 1 variable)" in the first line
IOSystem                                            % Test for LCToolbox, should not print anything, should run without an error.
Resmile                                             % Test for Resmile, should print "Resmile with properties:"

We want to create a B-spline interpolated model of an overhead crane with a low amount of knots.

Load state-space models coresponding to an overhead crane:

rsm = Resmile(Resmile.dataset_oc); 

Bring these state-space models to the same basis:

rsm.make_coherent

Create a B-spline interpolated model from the state-space matrices, and figure out which knots to remove:

rsm.fit_resmile

Create simplified model, prepare for plotting (analyze step) and plot the splines:

rsm.simplify
rsm.analyze
rsm.plot(Resmile.PL_TABS_SPLINES)

Plot the FRFs:

rsm.plot(Resmile.PL_TABS_FRF)

We have published a paper on the topic, which is available in the ISMA 2020 proceedings.
You can find the related example code in examples/isma2020_examples.m.
The paper also references to other scripts in the examples directory.

Some things explained in the paper: why to use B-spline based LPV models, background on B-splines (formal definition and how to imagine them), background on the formulas behind ReSMILE and how they were developed.

If you this project in your academic work, please cite the following paper:

@inproceedings{resmile_isma2020,
    journal = {Proceedings of ISMA2020},
    pages = {2203--2217},
    publisher = {KU Leuven; Leuven},
    isbn = {9789082893113},
    year = {2020},
    title = {ReSMILE: trading off model accuracy and complexity for linear parameter-varying systems},
    author = {Retzler, András and Swevers, Jan and Gillis, Joris and Kollár, Zsolt},
}

1. Estimate state-space LTI models for a selected set of scheduling parameter values (using either LCToolbox or the MathWorks System Identification Toolbox).
2. Supply the input data to the LCToolbox in the appropriate format using set input or the Resmile class constructor.
3. In case of a black box model, apply make_coherent to bring the state-space models into a coherent basis. Skip this step for a white box model.
4. Apply fit_resmile with the default parameters.
5. Apply simplify to remove knots and to create a simplified LPV model.
6. Check errors (analyze) and graphs (plot) to evaluate the quality of the result.
7. Check if the correct splines are auto-detected as straight lines using spline_remover_info.
8. If not satisfied with model accuracy/simplicity: change gamma, and rerun from step 4 until satisfied.
9. If still not satisfied, adjust epsilon and knot_removal_threshold, and rerun from step 4.
10. If still not satisfied, change the initial distribution of knots (knots_distribute_mode), and rerun from step 4.
11. Export model to LCToolbox LPVDSSmod object using simplified_ssmod, to use for control design.

This project is licensed under LGPLv3, except for lib/gif.m which is available under the 2-BSD license.