Complexity

Volume 2017, Article ID 3125073, 11 pages

https://doi.org/10.1155/2017/3125073

## Feedforward Nonlinear Control Using Neural Gas Network

Departamento de Ingeniería Eléctrica, Electrónica de Computadores y Sistemas, Universidad de Oviedo, Edificio Departamental 2, Zona Oeste, Campus de Viesques s/n, 33204 Gijón/Xixón, Spain

Correspondence should be addressed to Iván Machón-González; se.ivoinu@navinohcam

Received 19 July 2016; Accepted 17 November 2016; Published 15 January 2017

Academic Editor: Francisco Gordillo

Copyright © 2017 Iván Machón-González and Hilario López-García. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Nonlinear systems control is a main issue in control theory. Many developed applications suffer from a mathematical foundation not as general as the theory of linear systems. This paper proposes a control strategy of nonlinear systems with unknown dynamics by means of a set of local linear models obtained by a supervised neural gas network. The proposed approach takes advantage of the neural gas feature by which the algorithm yields a very robust clustering procedure. The direct model of the plant constitutes a piece-wise linear approximation of the nonlinear system and each neuron represents a local linear model for which a linear controller is designed. The neural gas model works as an observer and a controller at the same time. A state feedback control is implemented by estimation of the state variables based on the local transfer function that was provided by the local linear model. The gradient vectors obtained by the supervised neural gas algorithm provide a robust procedure for feedforward nonlinear control, that is, supposing the inexistence of disturbances.

#### 1. Introduction

Although some physical systems can be approximated as a linear model, almost all real plants actually have a nonlinear functioning. A wide understanding of the behavior of nonlinear processes is available but it is sometimes difficult to choose the appropriate control method. Lyapunov theory is a classic method for nonlinear system control. If and only if there is a positive definite continuous function whose derivative is negative under the suitable conditions of the control design, then the control asymptotic stability is guaranteed. However, this method is unfortunate because obtaining the Lyapunov function is difficult. This problem is even worse when dealing with unknown plants that are not defined mathematically. Therefore, it is usually not easy to guarantee the stability of a complex nonlinear system [1]. However, if the local linear system corresponding to an equilibrium point is controllable, then sufficient conditions can be stated for local stability [2].

Hartman-Grobman theorem states that the behavior of a nonlinear system in the neighborhood of an equilibrium point can be approximated by its linearized model. The systems theory is based on many mathematical procedures about stability, controllability, and observability regarding linear systems. The stability and, to a great extent, the dynamic response of a linear system can be described in terms of eigenvalues of the system matrix in state space design or poles of the transfer function. No such method exists for nonlinear systems. For this reason, industrial control processes are still usually designed using this linear control theory. After linearization, the typical approach is to design a linear controller such as PID with fixed parameters.

The classical approach to get local linear models can be achieved with RLS (Recursive Least Squares) method. However, sometimes this method throws up unfavorable results due to the intrinsic nonlinearities of the process to be controlled. The problem is to establish the different operating points for a nonlinear system. At this point, the proposed algorithm can establish each operating point as a cluster centre of the neural gas network. It is for such reason that artificial intelligence techniques improve the control performance.

Research into identification and control of nonlinear systems by means of neural networks (NN) began over two decades ago [3]. One of the major advantages of control by NN is that precise knowledge of the plant such as a mathematical model is not needed. Initially, control applications using NN were based on a trial-and-error approach. Research efforts have improved the control algorithms and several journals have published special issues with a strong mathematical foundation [4]. Many applications are based on a combination between feedforward and recurrent NN. Recurrency, also known as dynamic backpropagation, is necessary due to the dependency of the output on the previous values of the same output, which are also functions of the weights [5]. Zhang and Wang [6] proposed a pole assignment control using recurrent NN.

The typical design procedure is to carry out the system identification in order to model the plant and, secondly, to obtain the controller. Traditional methods rely heavily on models extracted from physical principles, whereas approaches based on NN theory usually create black-box models as function approximators using data obtained from the plant. Knowledge about the mathematical model of the plant or any other physical principle is not necessary.

Neural gas (NG) is an unsupervised prototype-based method [7] in which the prototype vectors are the weights and carry out a partition of the training data space. It considers the cooperation-competition computation, allowing the algorithm to be prevented from local minima problems. In addition, the batch NG allows fast training so that the convergence is achieved in a small number of epochs [8]. Supervised versions of NG have also been developed, specially for classification [9, 10]. The algorithm has a great robustness for clustering tasks but has also been proven to be robust to obtain direct models of plants [11].

After years of works in identification and control of dynamical systems by means of NN, there is agreement among researchers that linear identifiers and controllers should be used as first attempt, as stated in Chen and Narendra [2]. If a set of local linear models corresponding to several equilibrium points can approximate with certain accuracy a nonlinear system, then linear controllers can be designed for each model and the global control is related to control by switched linear models.

This divide-and-conquer approach is applied in this work. The resulting model is a set of local linear maps. Each neuron of the NG model corresponds to one local model. These local models are obtained after NG training. In this way, a direct model of the plant is obtained. After obtaining this NG model, the design of the local linear controller is simpler than that of the global nonlinear controller. Local linear mapping using another prototype-based algorithm such as SOM was successfully tested at the NASA facilities [12].

This paper aims to apply the robustness modeling capability of NG to control a nonlinear plant such as a typical robot manipulator.

The paper contains the learning rules of the considered NG algorithm in Section 2, the model of the plant and the control strategy are explained in Sections 3 and 4, respectively, and the proposed technique is tested in Section 5.

#### 2. Neural Gas Approach

The unsupervised version of the NG algorithm is based on energy cost function (1) according to the Euclidean metric. The notation used for the squared Euclidean distance is given in (2). Moreover,

A neighborhood function (3) is needed to implement the algorithm. The rank function represents the rank distance between prototype and data vector . The minimum distance takes the value and the rank for the maximum distance is equal to , where is the number of neurons or prototypes and is the neighborhood radius:

The neighborhood radius is usually chosen to decrease exponentially according to (4). The decrease goes from an initial positive value, , to a smaller final positive value, :where is the epoch step, is the maximum number of epochs, and was chosen as half the number of map units (), as in Arnonkijpanich et al. [13]. In addition, in order to minimize the quantization error at the end of the training.

The learning rule of the batch version is obtained in Cottrell et al. [8]. The batch algorithm can be obtained by means of Newton’s method using the Jacobian and Hessian matrices, and , respectively, of the cost function . The adaptation of the prototype is formulated accordingly based on this method

Kernel function can be considered locally constant [8]. In this way, the Jacobian and Hessian matrices are

Substituting (6) into (5), the increment can be obtained

Finally, the updating rule for each prototype vector appears in

##### 2.1. Supervised Learning

Supervised learning with NG is possible by means of local linear mapping over each Voronoi region defined by prototype vector . A constant and a vector with the same dimension as are assigned to each neuron . The goal is to approximate the function from to , where is the number of training variables, that is, the dimension of data vector . The training thus becomes supervised and the dataset contains input-output pairs of data vector and variable as the objective function. The estimation is carried out by where is the estimated output value, is the reference value learned for , is the gradient of the approximated function obtained in the th Voronoi region defined by , and is the neuron with its closest to data vector , that is, the best matching unit (BMU). The asterisk super index denotes the winning neuron for input data vector .

The probability distribution of the input data is represented by prototype vectors which are previously updated according to the typical rule of the unsupervised version of the algorithm [14] using (8). Each prototype vector can be considered as the centroid of the th Voronoi region. After unsupervised training, regions are well defined by these vectors. At this point, local models will be created in each of these regions so that local models will represent the whole data distribution.

The energy cost function of the supervised version of the algorithm is based on the mean square error of the output variable estimation averaged over each Voronoi region [15] according to (10). Prototypes are already obtained in (8), whereas the adaptation rules of and are calculated considering Newton’s method for energy cost (10). The learning rules for and are shown in (11) and (12), respectively:

#### 3. Plant Model

After NG training, the plant is modeled as a set of linear systems whose output depends on the previous values of both output and input . The Nonlinear Autoregressive-Moving Average (NARMA) model has been proven for nonlinear identification [16, 17] and can be expressed aswhere is the system output at the sampling instant , is the system input at instant , and is the system delay.

Considering zero delay system and substituting (13) for (9) remains

Hereafter, the gradients will be denoted as coefficients and .

, , , .

And the following terms will be gathered to form variable :

Denoting the polynomials with backward shift operator by and which reminds one of an ARMAX model, is not only a zero mean independent identically distributed white noise process but also a known disturbance calculated according to (15), and it depends on the input and output values since it is obtained by BMU . The internal noise of the system can be included in .

Using the -transform, is the system output and is the system input where the controller must be connected.

#### 4. Local Linear Control by State Feedback

If the system is linear (locally), then the superposition theorem can be applied remaining the linear transfer function between the system output and the control input as follows:

Defineand choose the following relationship between the state variables: , .

Transfer function (17) can be expressed in control canonical form for linear state space design as

Assuming that the system is controllable, the purpose of the control by state feedback via pole placement is to assign a set of pole locations for the closed-loop system that will correspond to satisfactory dynamic response in terms of rise time, settling time, and overshoot of the transient response. The control law is a linear combination of the state variables which are estimated in (19) by way of local transfer function (17).

The characteristic polynomial of the closed-loop system depending on system matrix , input matrix , and gain vector iswhereas the characteristic polynomial of the desired pole locations is

For an th-order system, the gain vector for state feedback is obtained by matching coefficients in (21) and (22) forcing the closed-loop poles to be placed at the desired locations:

It is possible that there are enough degrees of freedom to choose arbitrarily any desired root location by selecting the proper values . This is an inexact procedure that may require some iteration by the designer. The solution of the local linear model lies in finding the matrix or the regulator coefficients that implement the state feedback control. The stability condition for linear discrete-time systems is that all the eigenvalues must be inside the unit circle. Obviously, this criterion is not valid for nonlinear systems but there is a region inside the stable linear area where the asymptotic stability of the switched linear systems is achieved [18]. Thus, not only a desired dynamic response can be designed, but also stability criteria will be accomplished. In this work, this stability region was found by means of trial-and-error with different eigenvalues.

The proposed control strategy scheme is shown in Figure 1. Gain vector is calculated to fulfill the dynamics according to (22) depending on the local linear model defined by the current winning neuron or BMU. State variables are also obtained by the local linear model of the NG in (19). Tracking of the setpoint reference is possible using the inverse static gain of the feedback loop. In addition, since disturbance is known (it is included by the model), it can be compensated as . The transfer function of the prefilter has been chosen as and determines the switching rate of the local linear models. Although the pole assignment method does not affect the zeros of the plant, the prefilter can be optionally designed in order to cancel dominant zeros located inside the unit circle.