[IEEE International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet

Closed-loop Identification of Hammerstein Systems Using Hybrid Neural

Model Identified by Genetic Algorithms

O.M. Mohamed vall*, M. Radhi**

*Laboratoire d’Analyse et de Commande des Systèmes (LACS), Dept. Génie électrique,

Ecole Nationale d’Ingénieurs de Tunis,

B.P 37, le Belvedere, 1002 Tunis Cedex, TUNISIA.

**Unit of research (RME), INSAT, TUNISIA.

Centre urbain Nord, B.P. N°676, 1080 Tunis Cedex, Tunisia

E-mail: [email protected].

[email protected]

Abstract

In this paper we present an approach for the closed loop

identification of Hammerstein systems. In this approach we

propose modelling the system to be identified by a Hybrid Neural Model, which is composed of a Neural Network (NN)

connected in series with a linear model. To optimize the

proposed model, Genetic Algorithms are used. The system to be identified is in closed-loop with Variable Structure

Controller (CSV) in order to have a command signal rich in

commutations and consequently a good identification. A simulation example is given in order to show the effectiveness

of the proposed approach.

Keywords-Closed loop identification, Hammerstein system, Neural Network, Genetic Algorithms, Variable Structure

Controller.

1. Introduction

For many systems, when there is a wide operating area

rather than a unique operating point, a linear model cannot be

used. In this case non-linear models such as multi-models,

Volterra series, neural networks, fuzzy logic, Hammerstein

and Wiener-type models can be used. In addition it has been

shown that nonlinear effects encountered in some industrial

processes, such as distillation columns, pHneutralization

processes, heat-exchangers, or electro-mechanical systems can

be effectively modelled as a combination of a nonlinear static

element and a linear dynamic part (i.e. can be regarded as

Hammerstein models). Several approaches and techniques

were proposed for the open loop identification of the

Hammerstein models [10],[9],[7], and [11].

In [8] an approach for the open loop identification of the

Hammerstein models was proposed. In this approach, the

authors use the Genetic Algorithms to approximate nonlinear

term and linear model order. Here, we propose an approach in

which the Genetic Algorithms are used to approximate the

nonlinear term (i.e. the weights of the neural network

modelling it) and to estimate the coefficients of the transfer

function representing the system linear part. The approach that

we propose is applied in closed loop context.

The goal of closed loop identification is to estimate a

process model while the process is still under feedback control

(i.e., in closed loop) [4]. For some reasons, performing

identification under output feedback is necessary. Hence,

when performing identification experiments on unstable

system it is necessary to do this in closed loop with a

stabilizing controller [6]. An other situation where the closed

loop identification is necessary is that of many industrial

production processes, where safety and production restrictions

are often strong reasons for not allowing identification

experiments in open-loop. In literature, a great variety of open

and closed loop identification approaches for the nonlinear

systems are available. These approaches can be categorized

into three main groups: the direct approach, the indirect

approach, and the joint input-output approach.

1) The direct approach: apply a prediction error method

and identify the open-loop system using measurement

of the input and the output, ignoring possible feedback.

This approach gives consistency and optimal accuracy,

given that the true noise characteristics are correctly

modelled. A drawback of the direct approach is that we

need good noise model. In practice this means that we

must include a sufficiently flexible, parameterized noise

model (which out-rules output error models). In case a

fixed, or too ‘’small’’, noise model is used the results

will be biased. The reason for this bias error is that

there is correlation between the output noise and the

input. This is also why other methods, like instrumental

variables, spectral analysis and subspace methods, fail

when applied directly to closed loop data.

2) The indirect approach: identify the closed-loop system

using measurements of the reference signal and the

output and use this estimate to solve the open-loop

system parameters using the knowledge of the

Proceedings of the 2005 International Conference on Computational Intelligence for Modelling, Control and Automation, and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA-IAWTIC’05) 0-7695-2504-0/05 $20.00 © 2005 IEEE

controller. For this approach the feedback structure

must be know (and linear), and it is also required that

an external reference signal is used and that this

measurable.

3) The joint Input-output approach: identify the transfer

function from the reference signal and the output and

from the reference signal and the input and use them

to compute an estimate of the open-loop system.

In this work we use the first approach such as the direct

approach where we use a prediction error method based on

genetic programming for identify the Hybrid neural model

that we propose for modelling the Hammerstein system to be

identified. The effectiveness of this approach is shown on a

Hammerstein stochastic system in Closed-loop with Variable

Structure Controller (VSC).

2. Statement of the problem

This paper deals with the problem of closed-loop

identification of the nonlinear systems whose models are of

Hammerstein type. Recall that a nonlinear system of

Hammerstein type is composed of a static nonlinear model

connected in series with a dynamic linear model. Fig. 1.

shows the structure of a Hammerstein stochastic system in

closed-loop with Variable Structure Controller.

Fig. 1. Closed-loop Hammerstein system with VSC

Where G0 and f(u) represent respectively the linear part and

the nonlinear part of the true process to be identified (f(u) is a

nonlinear function of u), u(t) describes the process input

signal (the variable structure controller signal), y(t) represent

the process output signal, e(t) an unmeasurable noise and r(t)

is the reference signal.

With this notation, the system output is given by:

)t(eH))t(u(f)q(G)t(y 00 (1)

the input u(t) is given by:

)S(Sgn.K)t(u (2)

K is a constant and it is the maximal value of the controller

output. S is called switching function. S is defined as:

)t(...)t()t()t(S )n(n

)( 11

11 (3)

where )t(y)t(r)t( , i is a constant and (i)(t) is the ith

derivative of (t) for i =1..n-1. n is the true system order.

Sgn(S) is a sign function, which is defined as:

01

01

Sif

Sif)S(Sgn

(4)

Despite the simplified structure, the identification of the

Hammerstein models is a challenging task. The identification

methods of the Hammerstein models can be divided into the

three following classes [10]:

1) Two-step approaches. In this approach the linear and

nonlinear parts of the Hammerstein model are identified

separately. In a first stage one identifies the dynamic

linear part. Then, and in a second stage, one identifies

the static nonlinear part. It should be noted that the

identification results in the two stages depend mainly

on the quality of the intermediate signal approximation

(i.e. of the block output approximation)

2) One-step, non-iterative solutions. This approach uses

predefined basis function to approximate the

nonlinearity by a model that is linear in its parameters

and uses a standard linear technique for estimate those.

3) One-step, iterative solutions. This class includes

techniques that alternately refine the estimate of the

dynamic linear and the static nonlinearity.

This paper focuses on one-step iterative solutions, where the

system to be identified is modelled by hybrid Neural model,

whose parameters are optimized by Genetic algorithms.

The proposed Hybrid Neural model is composed in Neural

Network connected in series with a linear model (ARMA).

Fig. 2. shows the structure of the proposed model.

Fig. 2. Structure of the HN model proposed

3. Identification by Genetic Algorithms

Genetic Algorithms (GAs)[15] are efficient search methods

based on principles of natural selection and genetics. They

work with a population of individuals, each representing a

possible solution to a given problem. Each individual has

assigned a fitness score according to how good solution to the

problem it is.

Any GA starts with a population of randomly generated

solutions, chromosomes, and advances toward better solutions

by applying genetics operators, modelled on the genetic

processes occurring in nature. The Most usual operators are

as follows:

1) Selection: The main goal is selecting the chromosomes

with the best qualities for integration in the next

generation (these would depend on the cost function for

each individual).

2) Crossover: By combining the chromosomes of two

individuals, new chromosomes are generated and

integrated into the population.

3) Mutation: Random variations of parts of the

chromosome of an individual in the population for

generate new individuals.


A simple genetic algorithm is described as a flowchart in

Fig. 3.

Fig. 3. Simple Genetic Algorithm

In this research work, we propose to use the Genetic

Algorithms to optimize the parameters of a Hybrid Neural

model. This model is composed of a Neural Network

connected in series with a linear model (ARMA). Thus,

certain genes of each chromosome are devoted to Neural

Network weights. The remainders are devoted to linear model

parameters.

Let us consider that the linear model can be described by the

following equation:

)nk(uc...)k(uc)nk(ya...)k(ya)k(y nn 11 11 (5)

and that the NN is same that is shown in the Fig. 2.

The application of the GA to optimize the model proposed

can be reformulated as follows:

1) Starting with an initial population randomly

generated (N vectors).each vector is form depicted in

Fig. 4.

2) Construct the intermediate output (the NN output),

evaluate estimated system output, produce error

between the actual and the estimated system output

and calculate the fitness function (that used here is

the Mean Square Error) value, for each individual

(vector).

3) Selection of the best individuals (we chose a

probability of selection equal to 0.9).

4) Creation of a new population (from the old one) by

the application of the operators :

- Crossover (with a Crossover probability PC = 0.95)

- Mutation (with a Mutation probability Pm= 0.09)

5) While the termination condition is not met, return at

step 2.

Fig. 4. Structure of the chromosome.

4. Simulation Example

In this section simulation results are included to show the

applicability of the proposed closed loop identification

method. Consider a nonlinear stochastic system described by

the following Hammerstein model:1 2 1

1 2 1

0.3787 0.2852 0.12( ) ( ) ( )

1 1.36 0.5274 1 0.58

q q qy t x t e t

q q q

+= +

+ +

2 3( ) 0.602u(t) 0.38u (t)+0.15u ( )x t t

Where y(t) is the output system, e(t) is a zero mean Gaussian

noise with variance 0.1, x(t) is the intermediate

signal(supposed immeasurable) and u(t) is the input of the

global system. Consider also that the system is in closed loop

with Variable Structure Controller, where:

)S(Sgn.)t(u 52 , )t(.)t(S )(1250 , )t(y)t( 13 .

For the identification of this system we propose to use a

Hybrid Neural model. The parmeters of this model are

optimized by a Genetic Algorithm.

The parameters for the GA simulation are set as follows:

1) Initial population size: 130;

2) Maximum number of generation: 700;

3) Crossover: Uniform crossover with probability 0.95;

4) Mutation probability: 0.09.

The fitness function was calculated by :

5) The mean of the squared errors between the output

of the system and the output of proposed model :

N

))it(y)it(y(

)t(F

N

i 1

2

(6)

Where N is the number of the time steps for which the fitness

is measured, y is the system output and y is the model output.


From the low results (Fig. 5., Fig. 6. and Fig. 7.), it can be

seen that the GA converged with very high precision. The

experiment was repeated with several different values of N, as

well as different GA parameters (population size, Crossover

probability, Mutation probability etc.). All gave qualitatively

similar results.

The Fig. 5 presents the Fitness Function evolution during the

optimization operation.

0 100 200 300 400 500 600 700

0

0.5

1

1.5

2

2.5

3

3.5

4

Generation

Fitn

ess

Fun

ctio

n

Fig 5. Fitness Function evolution during the optimization

operation.

The Fig. 6. And Fig. 7. shown respectively, the Error

between true output signal and estimated and Evolution of

Actual and Estimated output signal.

0 5 10 15 20 25 30-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time(sec)

Error between Actual and Estimated Output

Fig. 6. Error between the Actual and estimated output.

0 5 10 15 20 25 300

5

10

15

Time(sec)

Act

ual a

nd E

stim

ated

Out

put

ActualEstimated

Fig. 7. Evolution of Actual and estimated output signal

From Fig. 7. we remark that the estimated output model

reproduces the actual output very well.

5. Conclusion

In this research work, an approach to closed loop

identification of the Hammerstein system has been presented.

This approach proposes a Hybrid neural model to modelling

the system to be identified. The Genetic Algorithms were

used to optimize the parameters of this model. The system to

be identified is supposed under feedback with Variable

Structure Controller. The simulation results could confirm

the validity of the proposed approach.

6. References

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1993, p. 2-22.

[2] Feng Zhao and Vadim I. Utkin, “Adaptive Simulation and

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1042.

[3] V.I. Utkin, “Sliding Mode Control Design Principles and

Applications to Electric Drives”, IEEE Transactions

Industrials Electronics, vol. 40, n° 1, Feb 1993, p. 23-36.

[4] Forssell U and L. Ljung, “Closed-loop Identification

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[5] Van den Hof, P.M.J, “Closed-loop Issues in System

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[6] Ljung and U. Forssell, “A alternative Motivation for The

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[7] A. Akramizadeh, A. Farjami, H. Khaloozadeh, “Nonlinear

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[8] A. Akramizadeh, M. Hakimi-M, H. Khaloozadeh, “An

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[9] J. Madar, J. Abonyi, F. Szeifert, “Genetic Programming for the

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[11] José Vieira, Alexandre Mota, “Parameter Estimation of Non-

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[12] L. Ferariu, T. Marcu, “Evolutionary Design of Dynamic

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[IEEE International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet