# MOR toolbox

MOR is a MATLAB^{®} based toolbox gathering algorithms for

- reduction of large-scale linear dynamical models,
- creation of linear dynamical models from input-output frequency data.

The algorithms gathered in the MOR toolbox generate a linear state-space model which input-output behaviour is close to the initial model. Several tools for analysis are also provided to evaluate the relevance of the reduced-order model.

Several methods and options are available, allowing for a fine tuning of the approximation process to suit your needs. For instance, one can perform an approximation over a bounded frequency interval, preserve the stability, the static gain or some eigenvalues.
Some methods take advantage of the sparse nature of the models and can therefore be applied to very large-scale models with several thousands of states. Such models arise very often in physics, biology, etc.

# Features

# Reduction of linear model given as state-space

Obtain a reduced-order model from a large-scale dynamical model described by a state-space representation.

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# Reduction over a bounded frequency interval

Obtain a reduced-order model that is very accurate in a specific frequency interval of interest.

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# Preserve some user-defined eigenvalues

Obtain a reduced-order model that preserves some of the eigenvalues of the initial model to retrieve some physical properties.

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# Approximation from data obtained with a frequency-domain simulator

Obtain a dynamical model of appropriate complexity from data collected with a frequency-domain simulator of your system.

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# Approximation from experimental data

Obtain a dynamical model from experimental data, collected during real experiments.

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# Reduction of linear model given as state-space

Given a state-space representation of a continuous linear multiple-inputs multiple-outputs model represented by the matrices `A`

, `B`

, `C`

, `D`

and `E`

and a target approximation order `r`

, the MOR toolbox contains a simple user interface that enables to approximate the initial state-space model. This can be done with the dedicated MATLAB^{®} interface `mor.lti`

as follows:

```
Hr = mor.lti({A,B,C,D,E},r)
```

The call above creates `Hr`

, a state-space model of order `r`

that approximate the input-ouput behaviour of the initial model.

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# Reduction over a bounded frequency interval

For control design purpose, it is often consistent to have an accurate model over a given frequency range of interest. Indeed, in most of the cases, due to actuator limitations and disturances acting in a given frequency range, a model representing the system's behaviour over this range is sufficient enough to design controllers, observers or to analyse the performances.

Being given a continuous multi-inputs multi-outputs linear state-space model described by the `A`

, `B`

, `C`

, `D`

and `E`

matrices, a reduction objective `r`

and a frequency range of interest `[fmin fmax]`

, the MOR toolbox offers a simple user interface, allowing approximating the original representation with a reduced order one, in a faithfull manner over the considered frequency range. This is done through the main dedicated MATLAB^{®} interface `mor.lti`

, as:

```
opt.freqBand = [fmin fmax]
```

Hr = mor.lti({A,B,C,D,E},r,opt)

The above function generates `Hr`

, a `r`

-th order dynamical model that restitutes the original model input/output behaviour over the range `[fmin fmax]`

.

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# Preserve some user-defined eigenvalues

In some cases, engineers have a physical interpretation of the modal content of their application (eigenvalues/vectors). For these reasons, and for a comfort use, it might be relevant to preserve some eigen-content.

Being given a linear multi-inputs multi-outputs state-space representation described by the `A`

, `B`

, `C`

, `D`

and `E`

matrices, a reduction objective `r`

and `eigenvalues`

, the eigenvalues to be preserved, the MOR toolbox then offers a simple but yet effective interface allowing approximating the initial realisation by a reduced representation. This is done through the main MATLAB^{®} proposed by `mor.lti`

:

```
opt.eigen = eigenvalues
```

Hr = mor.lti({A,B,C,D,E},r)

The above function generates `Hr`

, a state-space model of order `r`

which well restitutes the input/output original model behaviour and which eigenspace includes the user-defined eigenvalues.

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# Approximation from frequency-domain data extracted from a simulator

When a complex simulator is available, it can be interesting to approximate it by a rational function (for control, optimisation, analysis pusrpose). As in the test phases, by collecting the time-domain input and output signals, one is then able to transform them in the frequency-domain (e.g. using the FFT). Then, it is possible to define the frequency response `Hi`

representing the transfer from the inputs to outputs, at each frequency `wi`

.

From these input data `{wi,Hi}`

, and by fixing an approximation objective `r`

, the MOR toolbox offers a simple user interface allowing constructing a user-defined reduced order model (one should also notice that the algorithm is also able to find the lowest order that exactly matches the data). This is done through the MATLAB^{®} dedicated interface `mor.lti`

, as follows:

```
Hr = mor.lti({wi,Hi},r)
```

The above function generates `Hr`

, a state-space model of order `r`

which well restitutes the input/output original model behaviour. This function provides a solution to the frequency-domain multi-inputs multi-outputs identification.

The MOR toolbox also offers functionalities allowing approximating over a limited frequency range, but also to guarantee the stability of the reduced order model (making its time-domain simulation feasible).

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# Approximation from experimental data

During experimental campains on a test-bed, engineers usually collect input and output signals from the system. Once transformed in the frequency-domain (e.g. using an FFT), it is posisble to definine a tranfer response `Hi`

representing the ratio between output and input signals, for each frequency `wi`

.

From these input data `{wi,Hi}`

, and by fixing an approximation objective `r`

, the MOR toolbox offers a simple user interface allowing constructing a user-defined reduced order model (one should also notice that the algorithm is also able to find the lowest order that exactly matches the data). This is done through the MATLAB^{®} dedicated interface `mor.lti`

, as follows:

```
Hr = mor.lti({wi,Hi},r)
```

The above function generates `Hr`

, a state-space model of order `r`

which well restitutes the input/output original model behaviour. This function provides a solution to the frequency-domain multi-inputs multi-outputs identification.

The MOR toolbox also offers functionalities allowing dealing with noise by an adequate digital filtering, or focussing on a frequency-limited range, but also to enforce the reduced order model stability.

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# Try it or buy it

The MOR toolbox is free for an academic use.

Licenses are available for professionals.

Any question or specific needs? Contact us.