INT J COMPUT COMMUN, ISSN 1841-9836
Vol.7 (2012), No. 3 (September), pp. 540-549

Neural Network Model Predictive Control of Nonlinear Systems Using
Genetic Algorithms

V. Ranković, J. Radulović, N. Grujović, D. Divac

Vesna Ranković, Jasna Radulović, Nenad Grujović
Faculty of Mechanical Engineering, University of Kragujevac
Department for Applied Mechanics and Automatic Control
Serbia, 34000 Kragujevac; Sestre Janjić 6
E-mail: vesnar@kg.ac.rs, jasna@kg.ac.rs, gruja@kg.ac.rs

Dejan Divac
Institute for Development of Water Resources "Jaroslav Černi"
Serbia, 11000 Belgrade
Jaroslava Černog St., 11226 Beli Potok
E-mail: ddivac@eunet.rs

Abstract: In this paper the synthesis of the predictive controller for control of the
nonlinear object is considered. It is supposed that the object model is not known.
The method is based on a digital recurrent network (DRN) model of the system to be
controlled, which is used for predicting the future behavior of the output variables.
The cost function which minimizes the difference between the future object outputs
and the desired values of the outputs is formulated. The function ga of the Matlab’s
Genetic Algorithm Optimization Toolbox is used for obtaining the optimum values
of the control signals. Controller synthesis is illustrated for plants often referred to
in the literature. Results of simulations show effectiveness of the proposed control
system.
Keywords: model predictive control, nonlinear system, identification, digital recur-
rent network, genetic algorithm.

1 Introduction

The predictive controllers are based on the mathematical model of the object, which is being con-
trolled. Nonlinear system identification and prediction is a complex task. All the processes in nature are
nonlinear. In large number of processes, the nonlinearities are not prominent, so their behavior can be
described by the linear model. In the linear systems theory there exist a large number of methods that can
be applied for obtaining the linear model of processes. The nonlinear model must be chosen when the
nonlinearity is strongly exhibited. In the identification process, the parameters of the mathematical model
are being determined as such that the difference between the system response and its mathematical model
is as least as possible, both in the transient regime and in stationary state. The general model of linear
processes is ARX (Auto Regressive eXogenous), while for the nonlinear ones it is NARX (Nonlinear
Auto Regressive eXogenous). The NARX model structure enables application of the neural networks,
the fuzzy systems and the neuro-fuzzy systems for approximation of the nonlinear function.

Neural networks have been applied to the identification of nonlinear dynamical systems. The most of
the works are based on multilayer feedforward neural networks with backpropagation learning algorithm.
However, the conventional back-propagation algorithm has the problems of local minima and slow rate
of convergence. A novel multilayer discrete-time neural network is presented for the identification of
nonlinear dynamical systems, [1]. In [2] a new scheme for on-line states and parameters estimation
of a large class of nonlinear systems using radial basis function neural network has been designed. A
new approach to control nonlinear discrete dynamic systems, which relies on the identification of a

Copyright c⃝ 2006-2012 by CCC Publications



Neural Network Model Predictive Control of Nonlinear Systems Using Genetic Algorithms 541

discrete model of the system by a feedforward neural network with one hidden layer, is presented in [3].
Nonlinear system identification via discrete-time recurrent single layer and multilayer neural networks
are studied in [4]. In [5] an identification method for nonlinear models in the form of fuzzy-neural
networks is introduced. The fuzzy-neural networks combine fuzzy if-then rules with neural networks.
The adaptive time delay neural network is used for the identification of nonlinear systems [6], and four
architectures are proposed for identifying different classes of nonlinear systems. The identification of
nonlinear systems by feedforward neural networks, radial basis function neural networks, Runge-Kutta
neural networks and adaptive neuro-fuzzy inference systems is investigated in [7]. Result of simulation
indicates that adaptive neuro fuzzy inference systems are a good candidate for identification purposes.
However, neural networks are the simplest approaches in the sense of computational complexity. In [8]
nonlinear system identification via feedforward neural network and digital recurrent network is studied.

Model predictive control (MPC) is applied to a large number of nonlinear industrial process, [9,10,11,12].
The methodology to design and implement neural predictive controllers for nonlinear system has been
developed in [13,14,15].

In [13] feedforward neural networks to estimate the nonlinear process are applied. Also, for the
minimization of the cost function, the Matlab′s Optimal Toolbox functions fminunc and fmincon were
used. The design methodology for predictive control of industrial processes via recurrent fuzzy neural
networks is presented in [14]. In [15] the multilayer perceptron is used to identification of the nonlinear
object and genetic algorithm is applied to solve the multi-criteria optimization problem.

In this paper the control of the nonlinear object is studied and it is identified by the DRN. Recurrent
networks are more powerful than nonrecurrent networks and have important uses in control and signal
processing applications, [16]. They have been shown to be more efficient than feedforward neural net-
works in terms of the number of neurons required to model a dynamic system [17,18]. Models with
recurrent networks are shown to have the capability of capturing various plant nonlinearities, [19,20].

The major objective of the study presented in this paper is to take advantages of the recurrent neural
networks for modelling and genetic algorithms for optimization. The proposed method formulates a
dynamic nonlinear optimization problem, where the cost function consists of two terms: the differences
between the DRN model predictions and the desired output trajectory over a prediction horizon and the
control energy over a control horizon. For the solution of nonlinear optimization problem, a genetic
algorithm is used, which can approximate the optimum solution very fast, compared to conventional
optimization techniques. In this paper, the genetic algorithm has been successfully used in combinations
with digital recurrent network.

In the second section the identification of the nonlinear object by application of the DRN is explained.
In section three the principle of work of the predictive controllers is analyzed. In this paper for obtaining
the optimum values of the control signals, the genetic algorithm (GA) was used. Results of simulations
are given in section four, while section five presents the concluding remarks.

2 Neural network for identification of nonlinear dynamic

Different methods have been developed in the literature for nonlinear system identification. These
methods use a parameterized model. The parameters are updated to minimize an output identification
error.

A wide class of nonlinear dynamic systems with an input and an output can be described by the
model:

ym(k) = fm(φ(k),θ), (1)

where ym(k) is the output of the model, φ(k) is the regression vector and θ is the parameter vector.
Depending on the choice of the regressors in φ(k), different models can be derived:



542 V. Ranković, J. Radulović, N. Grujović, D. Divac

• NFIR (Nonlinear Finite Impulse Response) model -

φ(k) = (u(k − 1),u(k − 2), . . . ,u(k − nu)),

where nu denotes the maximum lag of the input.

• NARX (Nonlinear AutoRegressive with eXogenous inputs) model -

φ(k) = (u(k − 1),u(k − 2), . . . ,u(k − nu),y(k − 1),y(k − 2), . . . ,y(k − ny)),

where ny denotes the maximum lag of the output.

• NARMAX (Nonlinear AutoRegressive Moving Average with eXogenous inputs) model -

φ(k) = (u(k − 1),u(k − 2), . . . ,u(k − nu),y(k − 1),y(k − 2), . . . ,y(k − ny),

e(k − 1),e(k − 2), . . . ,e(k − ne))

where e(k) is the prediction error and ne is the maximum lag of the error.

• NOE (Nonlinear Output Error) model -

φ(k) = (u(k − 1),u(k − 2), . . . ,u(k − nu),ym(k − 1),ym(k − 2), . . . ,ym(k − ny)).

• NBJ (Nonlinear Box-Jenkins) model - uses all four regressor types.
The NARX and NOE are the most important representations of nonlinear systems. The block scheme of
the DRN model which corresponded to the NOE model is shown in Figure 1.

Figure 1: The block scheme of the neural network model

Figure 2 is an example of a DRN. The output of the network is feedback to its input. The output of
the network is a function not only of the weights, biases, and network input, but also of the outputs of the
network at previous points in time. In [16] dynamic backpropagation algorithm is used to adapt weights
and biases.

DRN network is composed of a nonlinear hidden layer and a linear output layer. The inputs u(k −
1),u(k − 2), . . . ,u(k − nu) are multiplied by weights ωyi j outputs ym(k − 1), ym(k − 2), . . . ,ym(k − ny) are



Neural Network Model Predictive Control of Nonlinear Systems Using Genetic Algorithms 543

multiplied by weights ωyi j and summed at each hidden node. Then the summed signal at a node activates
a nonlinear function. The hidden neurons activation function is the hyperbolic tangent sigmoid function.
In Figure 2, ωi represents the weight that connects the node i in the hidden layer and the output node; bi
represents the biased weight for i-th hidden neuron and is a biased weight for the output neuron.

The output of the network is:

ym(k) =
nH∑
i=1

ωiνi + b (2)

Figure 2: Digital Recurrent Network

where nH is the number of hidden nodes and:

νi =
eni − e−ni
eni + e−ni

(3)

ni =
nu∑
i=1

u(k − j)ωui j +
ny∑
j=1

ym(k − j)ωyi j + bi (4)

This error is used to adjust the weights and biases in the network via the minimization of the following
function:

ε=
1
2

[y(k) −ym(k)]2 (5)

Using the gradient decent, the weight and bias updating rules can be described as:

ωui j (k + 1) =ωui j (k) −η
∂ε

∂ωui j
(6)

ωyi j (k + 1) =ωyi j (k) −η
∂ε

∂ωyi j
(7)

bi(k + 1) = bi(k) −η
∂ε

∂bi
(8)



544 V. Ranković, J. Radulović, N. Grujović, D. Divac

b(k + 1) = b(k) −η
∂ε

∂b
(9)

where:
∂ε

∂ωui j
=
∂eε

∂ym

∂ym
∂ωui j

;
∂ε

∂ωyi j
=
∂eε

∂ym

∂ym
∂ωyi j

;
∂ε

∂bi
=
∂eε

∂ym

∂ym
∂bi

;
∂ε

∂b
=
∂eε

∂ym

∂ym
∂b

;

where the superscript e indicates an explicit derivative, not accounting for indirect effects through time.

The terms
∂ym
∂ωui j

,
∂ym
∂ωyi j

,
∂ym
∂bi

and
∂ym
∂b

must be propagated forward through time, [16].

3 Model predictive control

Here the principle of operation of the predictive controllers will be briefly presented. Let us suppose
that the mathematical model of the process is known. Based on the model, it is possible to determine the
future outputs from the object y(k + 1), l = 1,2, . . . , NH , where NH is the prediction horizon. The future
outputs depend on the object current states and future control signals, u(k+ l), l = 1,2, . . . , Nc, where Nc
is the control horizon and Nc ≤ NH . The predictive controllers compute potential future control signals
such that the future outputs will be as close as possible to the desired values r(k + l), l = 1,2, . . . , NH .
Figure 3 shows the basic concept of the model predictive control.

Figure 3: Basic concept of a model predictive control method

There are several types of the predictive controllers. In this work the GPC (General Predictive Con-
trol) controller is used, where the cost function is calculated as:

J(k) =
NH∑
l=1

[
r(k + l) −ym(k + l)

]2
+α

Nc∑
l=1

∆u2(k + l − 1), (10)

where:
∆u2(k + l − 1) = u(k + l − 1) − u(k + l − 2),

and α is the weight factor of the control signal.



Neural Network Model Predictive Control of Nonlinear Systems Using Genetic Algorithms 545

Real processes are subject to constraints. We consider constraints which limit the range of the control
signal, the gradient of the control signal and the future model predictions:

umin ≤ u(k + l) ≤ umax

| u(k + l) − u(k + l − 1) |≤∆umax

ymin ≤ ym(k + l) ≤ ymax.

Model output ym(k + l) is calculated based on (2):

ym(k + l) =
nH∑
i=1

ωi ·
1

1 + e−(
∑nu

j=1 u(k+l− j)ωui j+
∑ny

j=1 u(k+l− j)ωyi j+bi)
+ b. (11)

Taking into account the term α
∑Nc

m=1 ∆u
2(k + l − 1) in the cost function (10) prevents the control

signals to be too big such that the executive organs would not be able to realize.

The control signals in the k-th step: [u(k),u(k + 1), . . . ,u(k + Nc − 1)]T , is possible to solve both
numerically and analytically. Analytically it is solved in such a manner that from the system of algebraic
equations:

∂J
∂u(k)

= 0,
∂J

∂u(k + 1)
= 0, . . . ,

∂J
∂u(k + Nc − 1)

= 0

one obtains [u(k),u(k + 1), . . . ,u(k + Nc − 1)]T .
For analytical solution, it is necessary for the model to be linear. The neural network model is

nonlinear, its linearization should be performed first, or apply the numerical methods for solving the
optimization problem. In this work for obtaining the optimum values of the control signals, the genetic
algorithm is used (the function ga of the Matlab’s Genetic Algorithm Toolbox). The genetic algorithms
represent the global optimization technique. In the k-th step Nc −1 control signals are obtained. The first
calculated signal is being sent to the controller output.

The GA used in this study is simple genetic algorithm. The elements of the populations are encoded
into bit-strings. The chromosome selection for reproduction is performed using the Roulette selection
method. The multi-point crossover operator was used. The uniform mutation operator is applied in this
study.

4 Simulation results

Example 1

For the simulation example 1, we consider the nonlinear plant, which is described by the following
nonlinear difference equation:

y(k) = 0.5y(k − 1) + u(k − 1)
(
1 + 0.2y2(k − 1)

)
+ u3(k − 1), (12)

where y is the output of the plant and u is the plant input.
We assume that structure of the model is known, nu = 1,ny = 1. The inputs and output of the neural

network model are u(k − 1),ym(k − 1) and ym(k) respectively. The data set are obtained by applying
random input signal uniformly distributed in the interval [-0.5 0.5]. The plant output is bounded within
region [-2.2 2.2]. In this example, 4500 training patterns are generated to train the DRN and 1500 to test
the obtained DRN model.



546 V. Ranković, J. Radulović, N. Grujović, D. Divac

Selection of an appropriate number of neurons in the hidden layer is very important. The optimal
network size was selected from the one which resulted in maximum correlation coefficient for the training
and test sets, Table 1.

Based on Table 1, it was concluded that the optimal number of hidden neurons is 12.

Table 1.Correlation coefficient for the training and test sets
DRN-structure 2-10-1 2-12-1 2-15-1 2-17-1
Training 0.9878 0.9987 0.9856 0.9879
Test 0.9845 0.9921 0.9795 0.9812

Since the object model is formed, it is necessary to define the cost function parameters, (10). These
parameters are selected by the trial and error method. By increasing the variance of the control signal
is decreasing, but simultaneously the difference between the set and real value of the object output is
increasing. In the considered example, the satisfactory results are obtained for the following values of
parameters: NH = 3, NC = 3,α= 0.05.

The optimal predictive controller in the k-th step computes the control signals:
[u(k),u(k + 1)]T .
In Figure 4, the reference signal is shown. The difference between the reference signal and the object

output is presented in Figure 5. From the Figure 5, it is obvious that the tracking error is small. The
obtained solution is good for practical realization.

Example 2

The plant is given by:

y(k) = 0.35


y(k − 1)y(k − 2)

[
y(k − 1) + 2.5

]
1 +y2(k − 1) +y2(k − 2)

+ u(k − 1)

 , (13)
where y is the output of the plant and u is the plant input. The dynamical system used in the simulation
example is given in [21]. It is assumed that structure of the model is known, nu = 1,ny = 2. The inputs
and output of the neural network are u(k − 1),ym(k − 1),ym(k − 2) and ym(k), respectively.

The input-output patterns are generated randomly. The data set included 4000 data samples. In the
training process of the DRN, 3000 samples were used. The DRN model was tested using 1000 selected
data. In this example, it is found that the optimal number of hidden neurons is 15 (Table 2).

Table 2.Correlation coefficient for the training and test sets
DRN-structure 3-10-1 3-12-1 3-15-1 3-17-1
Training 0.9811 0.9899 0.9945 0.9921
Test 0.9799 0.9831 0.9897 0.9895

The control and predictive horizons are chosen as respectively, NH = 3, and NC = 3. The choice of
these values determines the complexity of the optimization problem. The values of NH and NC may not
be too large, because the computation time would become also too large. These values should provide a
good compromise between performance and computational load. It is found that the weight factor of the
control signal α= 0.045 by the trial and error method.

The optimal predictive controller in the k-th step computes the control signals:
[u(k),u(k + 1)]T . The reference signal is shown in Figure 6. The difference between the reference signal
and the object output is presented in Figure 7.

From the Figure 7, it is obvious that the control is quite effective. The obtained solution is good for
practical realization.



Neural Network Model Predictive Control of Nonlinear Systems Using Genetic Algorithms 547

Figure 4: The reference signal

Figure 5: Difference between the reference signal and the object output

Figure 6: The reference signal



548 V. Ranković, J. Radulović, N. Grujović, D. Divac

Figure 7: Difference between the reference signal and the object output

5 Conclusions

In this paper the synthesis of the predictive controller for control of the nonlinear object is considered.
The object is modeled by the digital recurrent network. In the designing of neural network model, the
problem is how to determine an optimal architecture of network. The determination of the values of nu
and ny is an open question. Large time lags result in better prediction of the NN. However, large nu and
ny also result in large number of parameters (weights and biases) that need to be adapted.

The simulation results, given in Section 4, show that the predictive controllers can successfully be
applied for control of the prominently nonlinear object. The optimum values of the control signals are
obtained by the genetic algorithm which represents the global optimization technique. The given trajec-
tory tracking error is small. The proposed structure can be applied in control of the linear objects that are
modeled by the neural network. Recurrent networks are more powerful than nonrecurrent networks and
have important uses in control and signal processing applications.

Simulation results on the examples selected from literature provide good results and that encourages
our future plan for real time control system implementation.

Acknowledgement

Parts of this research were supported by the Ministry of Sciences, Technologies and Development of
Republic of Serbia.

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