^{1}

^{2}

^{1}

^{2}

^{1}

^{2}

This work presents a straightforward methodology based on neural networks (NN) which allows to obtain relevant dynamic information of unknown nonlinear systems. It provides an approach for cases in which the complex task of analyzing the dynamic behaviour of nonlinear systems makes it excessively challenging to obtain an accurate mathematical model. After reviewing the suitability of multilayer perceptrons (MLPs) as universal approximators to replace a mathematical model, the first part of this work presents a system representation using a model formulated with state variables which can be exported to a NN structure. Considering the linearization of the NN model in a mesh of operating points, the second part of this work presents the study of equilibrium states in such points by calculating the Jacobian matrix of the system through the NN model. The results analyzed in three case studies provide representative examples of the strengths of the proposed method. Conclusively, it is feasible to study the system behaviour based on MLPs, which enables the analysis of the local stability of the equilibrium points, as well as the system dynamics in its environment, therefore obtaining valuable information of the system dynamic behaviour.

The innumerable strategies and new proposals in the control system area are generally based on the knowledge of the system to be controlled. In some cases, its model is achieved by mathematical and analytical procedures, because such model is a simplified representation of the system, or the problem to study is highly restricted to a specific operating point. Although the modelling is often approached by the means of mathematical and analytical methods, with the growing complexity involved in the systems and the requirements of accurate representation, such approaches turn into an increasingly challenging solution.

Consequently, considering the maturity of computational intelligence (CI) techniques and new enabler technologies, CI methods represent an attractive alternative to develop accurate modelling and control solutions for highly complex models [

Taking into account the need to obtain precise models of complex systems which are applicable in a wide operating range, the CI techniques are an ideal method to reproduce the behaviour of a complex system [

When studying a process in engineering, in biomedical field, in natural sciences, and even in social systems, approaching the analysis from a dynamic point of view can be very attractive and convenient, depending on the focus of such study. A dynamic analysis of the system can provide wide and very rich information related to how the system will respond under certain inputs. Moreover, it can allow to study its dynamic behaviour through the analysis of the stability in open-loop, both locally and globally. In addition, it will be possible to study whether certain nonlinear phenomena affect the system, for example bifurcations, saddle, and limit cycles [

The methods traditionally applied in control engineering are based on linear approximations around several operating points of the system. This is suitable when problems are studied and solved in a local domain. However, there is a trend to approach bigger problems with a more abstracted and global perspective, leading to the use of nonlinear methods [

One of the main reasons for the use of nonlinear models is based on the dynamics of linear systems, since conventional mathematical formulations are not rich enough to reproduce a series of phenomena that usually appear in the real life [

The typically appropriate initial approach to analyze nonlinear systems is to use a representation of the system by means of a mathematical model, generally represented in state variables. This is possible assuming that sufficient information and knowledge of the system is available to generate its state equations, provided that the system dynamic is not extremely complex. In many applications, current research deals with the study of unknown complex systems, whether due to a complex dynamic, high dimensionality, or lack of information about the physical relationships that govern the behaviour of the system. In such situations, the techniques from the field of intelligent control can help to improve these studies, as Barragán et al. present in [

During the analysis and design of control solutions, knowing the equilibrium states of a system, as well as the stability of such states, is an aspect of great interest. When the model of the system is completely unknown, this information could help to clarify how the system works, even to ease the design of an appropriate control. It should be noted that despite the existence of recent works that present formal analysis methodologies based on Fuzzy logic [

This work presents a straightforward and easy to use methodology for extracting information from unknown systems using NNs. The main objective of this proposal is to develop a method that allows obtaining information on the dynamics of nonlinear systems, when there is no mathematical model, neither accurate nor approximate, to analyze them. In these situations, any additional information reached by new methods is significant, especially when this information is related to the analysis of the presence of equilibrium states and their local stability, as presented below. In this work, an MLP neural network is trained with a set of measured values of inputs and outputs of supposedly unknown systems, in order to reproduce the behaviour of these systems. For this purpose, taking into account that the dynamic study aims at analyzing the behaviour of each nonlinear system in their corresponding equilibrium states, the dataset of examples to train the NN will be obtained from its entire operating range. More specifically, the equilibrium states of three nonlinear systems will be studied through their NN models, which reproduce their corresponding state variable models. The equilibrium states are reached by a precise linearization in a grid of operating points extracted from the NN models, and subsequently performing a study of local stability. Using this information, the local stability of equilibrium states is obtained, as well as the system dynamics in the vicinity of the studied points, achieving valuable information about the dynamic behaviour of the nonlinear system.

This paper is organized as follows: Section

A generic continuous dynamic system will be considered, represented by state variables

By selecting a simple MLP structure that consists of one hidden layer of

Taking into account that a NN will reproduce the evolution of each state variable

From the above representation, in order to simplify the methodology in studying the obtained NN models, each state variable will be modeled by a different NN.

After obtaining an accurate model of a system, it is a fact that this model can be used to obtain system information through well-known techniques. In this section, a very important technique is presented to study nonlinear systems in two phases, as required by the methodology of this work. Firstly, the linearization of a neural state model will be exposed in detail. Secondly, the study of the equilibrium states of an unknown nonlinear system will be presented. This study is carried out through a NN model that reproduces the behaviour of the aforementioned nonlinear system. The study of the equilibrium states from the NN model, together with the study of their local stability from the linearization, allows to analyze the operational behaviour of a system from a qualitative point of view.

Linearization is one of the most commonly used techniques in solving design problems in the field of nonlinear control systems, even though it is necessary to point out that this is a technique not ideal in many situations where the effects of nonlinearities are not negligible. It is a very convenient technique for the control of not excessively complex systems or in situations when the dynamics of the system is approximately known in regions where the system behaviour is close to a linear one, basically around equilibrium states.

Thereby, apart from being a method that aims at the control of systems, linearization could be a powerful resource to obtain information from a nonlinear system. It could be considered that, except in some situations, the behaviour of a nonlinear system around an equilibrium state is analogous to the one observed after linearization of the system in such state [

The generic state model, obtained from a nonlinear system, is represented by

The first-order simplification of the Taylor series of the nonlinear system, in the domain of the state

Being

For the rest of the presented work, the time

If

When the system (

By extending the previous (

Subsequently, based on the works of Pirabakaran and Becerra [

The first partial derivative of (

As the activation function

Then, the overall solution for

For the second derivative

Substituting (

In order to perform an exhaustive analysis of the nonlinear system, it is first necessary to obtain an appropriate neural state model of that system, as presented in (

For nonlinear dynamics, the equilibrium states could be very difficult to solve analytically, so it is necessary to use numerical methods [

These equations represent the

The L-M algorithm requires the Jacobian matrix of the system, in order to accelerate its convergence. This matrix can be obtained, either with explicit calculation or with some technique to approximate it. In the previous Section

Furthermore, the Jacobian matrix can be used, both for solving and finding the equilibrium states and for the linearization of the system in each of the solutions obtained. In this way, it is also possible to study the characteristics of the located equilibrium states, from the eigenvalues of the dynamic matrix of the linearized system. This analysis could improve the interpretation of system dynamics; it could help to study the local stability, even to observe more complex behaviours, such as bifurcations, saddle points, or limit cycles.

In this section, three different examples are presented. These examples come from different areas, being nonlinear electrical, mechanical, and biological systems, which initially will be considered as unknown. The algorithms have been implemented with the tools of MATLAB R, both for the MLP neural network training and for the calculation of the NN model linearization.

Let the tunnel-diode circuit shown in Figure

Tunnel-diode circuit.

Supposing that

Phase portrait of the tunnel-diode circuit.

Considering that the system dynamics is unknown, the first stage in the work plan will be to obtain a NN model of the system from input-output data. These data will come from an exhaustive selection of examples that will cover the whole universe of the discourse of operating points of the system. Subsequently, after analyzing the NN model and its Jacobian matrix, relevant conclusions will be drawn regarding the dynamics of the system.

In the case of the tunnel-diode, a set of 2000 points is created to model the behaviour of the nonlinear relation

In order to keep an appropriate methodological procedure that provides the most adequate MLP neural network, a batch of training experiments has been prepared where the number of neurons in the hidden layer changes, being this the parameter to be studied in relation to the precise response of the NN output. All training experiments have been initialized with an extension of an advanced algorithm to assure the initial local stability of the NN [_{1}, _{2}, and _{1} and ˙_{2}. The training process was performed with a second order algorithm based on the L-M approach [

To solve the set of equations shown in (

Equilibrium states of the tunnel-diode NN model.

Comprehensively using the linearization proposed in this work, it has been possible to find linearized models in each of the equilibrium states shown in Figure

The second example is an inverted pendulum with friction presented in Figure ^{2}.

Inverted pendulum with friction.

Let be the state vector given by

Then, the system of inverted pendulum can be defined by the state model as

As a starting point, we consider that the system dynamics is unknown with the aim of validating the techniques developed in this paper. Moreover, we assume that it is possible to collect enough input-output data to obtain the neural model of the system. Thus, the first stage of this work will be to extract a MLP neural model of the pendulum system to be studied.

According to the previous example, a dataset has been prepared considering the universes of discourse of the state variables and input signal, respectively,

From these discourse universes, 2000 input-output data samples have been extracted, proportionally distributed, taking 1600 data samples (80%) for neural network training and 400 data samples (20%) for the validation. As was the case of the above study, the pendulum system will be considered as unknown.

Also as the previous case, the MLP neural network has been trained using the L-M algorithm [_{1}, _{2}, and ^{−6} rad/s and 0

Taking into account the obtained NN model of the system, the equilibrium states have been calculated as presented in Section

Equilibrium states of the inverted pendulum with friction.

For the real system, its equilibrium states are located on the line

Nature is capable of providing diverse real-life examples in which animal behaviour presents multiple stable equilibrium states. This is the case of a predator-prey system with adaptation of prey behaviour to changing environmental conditions [

Then, the system of a biological prey-predator can be defined by the state model.

In response to an external challenge, each individual into the prey community may modify its behaviour between risky and safe conduct. On the basis of the vulnerability of individuals in risky mode, the first expression in (

In (

Taking into account that

Functional response: probability of hunted prey

Moreover, to reject biologically unreachable equilibrium states, it is necessary to preserve the condition.

As Pimenov et al. argue [

Considering again that the system dynamics are unknown, an extensive set of input-output data have been obtained from the state model in (

Afterwards, the set of equations in (

Equilibrium states of the predator-prey NN model.

After the linearization developed in this work, different equilibrium states have been found in the real system, as shown in Figure

This work has presented a straightforward methodological procedure based on neural network models, to analyze the dynamical behaviour of unknown nonlinear systems. This implies presenting the linearization of a neural model to be compared with the proper linearization of the real system, and finally explaining the calculation of the equilibrium states of the real system through the NN model and its relation with the study of the local stability.

This work initiates a research line where NNs are the basis of a nonlinear system model, allowing to calculate its corresponding Jacobian matrix to analyze the local stability of the equilibrium states. Since it is assumed that there is no approximate mathematical model of the nonlinear system, the accuracy of the study of equilibrium states is related to the NN training process. In this process, many experiments have been developed to ensure the representativeness of the entire operating range that will be considered in the analysis of the nonlinear system. This has involved keeping the system under control and acquiring the appropriate amount of examples of variables and states.

The proposed method has been validated with three nonlinear systems coming from different fields (electrical, mechanical, and biological). This approach has been developed with multilayer perceptron neural network structures, one of the most simple and feasible in training and application. The training processes have been based on widely used Levenberg-Marquardt algorithm and developed with exhaustive input-output data of the above systems, guaranteeing the representativeness of each universe of discourse. Taking into account that after the training the errors have been within permissible limits, it is necessary to emphasize that the extracted information, with this neural approach, through the analysis of the neural structure is a powerful method to obtain information from an unknown system.

The results have shown that it is possible to obtain dynamical information of nonlinear systems uniquely using the corresponding NN model. This information is directly related to the equilibrium states of the real system, enabling to study the local stability around each equilibrium state.

Due to the interest of this research line, more additional works could be developed in the future considering other NN structures different to that presented in this paper, the multilayer perceptron with sigmoidal activation function in its unique hidden layer. The study of nonlinear systems of more complicated dynamic features (including systems with memory and with noncontinuous operating ranges), together with the modification of the NN structures (recurrent, radial basis function, deep neural networks, etc.), could establish a promising and extensive field of study.

The authors declare that they have no conflicts of interest.

The authors would like to thank the Ministry of Economy, Industry and Competitiveness of Spain that has funded this work under the project DPI2017-85540-R (H2SMART-GRID).