This repository contains the Python code to reproduce the results of the paper Model structures and fitting criteria for system identification with neural networks by Marco Forgione and Dario Piga.

The following fitting methods for State-Space (SS) and Input-Output (IO) neural dynamical models are implemented

1. One-step prediction error minimization
2. Open-loop simulation error minimization
3. Multi-step simulation error minimization

The block diagram below illustrates the proposed multi-step simulation error minimization approach applied to a state-space model. Quantities in red are tunable optimization variable (so as the parameters of the state and output neural network mappings).

At each iteration of the gradient-based optimization loop:

1. A batch consisting of q length-m subsequences of measured input, measured output, and hidden state is extracted from the training dataset (and from the tunable hidden state sequence)
2. The system's simulated state and output subsequences are obtained by applying m-step-ahead simulation to the input subsequences. The initial condition is taken as the first element of the hidden state sequence
3. The fit loss is computed as the discrepancy between measured and simulated output; the consistency loss is computed as the discrepancy between hidden and simulated state; the total loss is a defined as a weighted sum of the fit and consistency loss
4. Derivatives of the total loss w.r.t. the hidden state and the neural network parameters are computed via back-propagation
5. Using the derivatives computed at the previous step, a gradient-based optimization step is performed. The hidden state and neural network parameters are updated in the negative gradient direction, aiming to minimize the total loss

• torchid: pytorch implementation of the fitting methods 1,2,3
• examples: examples of neural dynamical models fitting
• common: definition of performance index R-squared, etc.

The examples are:

• RLC_example: nonlinear RLC circuit thoroughly discussed in the paper
• CSTR_example: CSTR system from the DaISy dataset
• cartpole_example: cart-pole mechanical system. Equations are the same used here
• CTS_example: Cascated Tanks System from the Nonlinear System Identification Benchmark website

For the RLC example, the main scripts are:

• symbolic_RLC.m: Symbolic manipulation of the RLC model, constant definition
• RLC_generate_id.py: generate the identification dataset
• RLC_generate_val.py: generate the validation dataset
• RLC_SS_fit_1step.py: SS model, one-step prediction error minimization
• RLC_SS_fit_simerror.py: SS model, open-loop simulation error minimization
• RLC_SS_fit_multistep.py: SS model, multistep simulation error minimization
• RLC_SS_eval_sim.py: SS model, evaluate the simulation performance of the identified models, produce relevant plots and model statistics
• RLC_IO_fit_1step.py: IO model, one-step prediction error minimization
• RLC_IO_fit_multistep.py: IO model, multistep simulation error minimization
• RLC_IO_eval_sim.py: IO model, evaluate the simulation performance of the identified models, produce relevant plots and model statistics
• RLC_OE_comparison.m: Linear Output Error (OE) model fit in Matlab

Simulations were performed on a Python 3.7 conda environment with

• numpy
• scipy
• matplotlib
• pandas
• sympy
• numba
• pytorch (version 1.3)

These dependencies may be installed through the commands:

conda install numpy numba scipy sympy pandas matplotlib ipython
conda install pytorch torchvision cpuonly -c pytorch

If you find this project useful, we encourage you to

• Star this repository ⭐
• Cite the paper

@inproceedings{forgione2020model,
  title={Model structures and fitting criteria for system identification
  with neural networks},
  author={Forgione, Marco and Piga, Dario},
  booktitle={Proc. of the 14th IEEE International Conference Application
  of Information and Communication Technologies, Tashkent, Uzbekistan},
  year={2020}
}