NisI: Non-Ideal System Identification

Overview

This repository contains the source code NisI: Non-Ideal System Identification using Particle Swarm Optimization (PSO) for the identification of systems with high parametric sensitivity in regions of chaotic operation.

In experimental studies of non-ideal dynamic models, typically due to physical or sensor limitations, only one state of the system is observed. In this context, NisI can perform the parametric identification of N unknown variables. The algorithm operates with each PSO particle acting as a fourth-order Runge-Kutta integrator, seeking the best approximation of the simulated model to the experimental data.

USAGE

NisI provides an installation script to simplify its installation and usage. Some prerequisites are essential before preparing the environment.

Requirements

• make
• python

Next, the script setting up the environment can be utilized.

$ make prepare-env
$ source venv/bin/activate

Now the environment with the nisi library is available. An example can be executed using the generic command provided below:

(venv)$ python examples/$(DYN_SYS_EXAMPLE)

Please replace the name $(DYN_SYS_EXAMPLE) with one of the examples or with a new script written by the user."

Examples

To demonstrate the parametric identification of a system in a chaotic regime, we will present the dynamic system of the Duffing oscillator.

So, the Non-Ideal model is described below:

$$\begin{align*} \dot{x}_0 =& x_1 \\\ \dot{x}_1 =& -\alpha x_1 -\delta x_0 - \beta x_0^3 + F\cos(x_2 + a_0\sin[b_0x_2 + c_0]) \\\ \dot{x}_2 =& \omega \end{align*}$$

where:

model parameter	value
$$a_0$$	non-ideal term
$$b_0$$	non-ideal term
$$\alpha$$	0.5
$$\beta$$	1.0
$$\delta$$	-1.0
$$\omega$$	unknown value
$$F$$	unknown value

Dynamic Non-Ideal Model is define by the EqSystem class:

class EqSystem(Model):
    def __init__(self, params=None):
        super().__init__(params)
        self._params = params

    def model(self, t, x, *args):
        k = self.unknown_const
        alpha = 0.5
        beta  = 1
        delta = -1
        omega = k[0]
        F     = k[1]

        dx = np.zeros(len(self.x0),)
        dx[0] = x[1]
        dy[1] = -alpha*y[1] -delta*y[0] -beta*y[0]**3 + F*np.cos(y[2] + a_0*np.sin(b_0*y[2]+c_0))
        dx[2] = omega
        return dx

Next the parameter configurations need to provide:

params = {'optmizer': {'lowBound': [0.1 , 0.1],
                        'upBound': [5.0,  0.5],
                        'maxVelocity':  2, 
                        'minVelocity': -2,
                        'nPop': 10,
                        'nVar': 2,
                        'social_weight': 2.0,
                        'cognitive_weight': 1.0,
                        'w': 0.9,
                        'beta': 0.1,
                        'w_damping': 0.999,
                        'escape_min_vel_percent': 0.0005,
                       'escape_min_error': 2e-3},
            'dyn_system': {'model_path': '',
                            'external': None,
                            'state_mask' : [True, False, False],
                           'loss': 'rmse',
                            'x0': [0., 0., 0.],
                            't': [0,50,500]
                            }
            }

Note, only one state was observed of system:

#            x_0    x_1    x_2
state_mask: [True,  False, False]

So, to run this example, please follow the steps above:

$(venv)$ python ./examples/duffing_oscilator_two_unknown_variables_one_state_observed.py

Expected Result

The simulation of the experimental system has the parameters $\omega = 1.0$ and $F = 0.385$. The algorithm found the parameters with a significantly small error as shown below

e: [8.7e-06], predict: [100000017 0.38499979]

An modified_oscillator model has been added to the examples/ folder.

Citation

Please cite our work if you use it.

@article{Lima2024,
  title = {NisI: A Tool for Non-Ideal System Identification},
  DOI = {10.36227/techrxiv.170630655.56990506/v1},
  publisher = {Institute of Electrical and Electronics Engineers (IEEE)},
  author = {Lima,  Jeferson José de and Kaster,  Mauricio S and Martins,  Marcella S R and Ribeiro,  Mauricio A and Tusset,  Angelo M and Balthazar,  José M},
  year = {2024},
  month = jan 
}

Social

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References

Bugs & Feature Requests

Please report bugs and request features using the issues