Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 19–22___________________________________________________________________________________________________________Parameter Optimization with Input/Output Data via DE
for Adaptive Control System with Neural Network
Taro Takagi

Department of Control Engineering, Natl. Inst. of Tech., Maizuru College,
234 Shiroya, Maizuru, Kyoto 625-8511, Japan
Ikuro Mizumoto

Graduate School of Science and Technology, Kumamoto University,
2-39-1 Kurokami Chuo-ku, Kumamoto 860-8555, Japan
E-mail: t.takagi@maizuru-ct.ac.jp, ikuro@kumamoto-u.ac.jp

Abstract
In this paper, adaptive control system with neural network (NN) will be designed. At the beginning, parallel
feedforward compensator (PFC) will be designed by using one-shot experimental data of controlled system via
differential evolution (DE). From the obtained PFC and the ideal almost strictly positive real (ASPR) model, nominalmodel of controlled system can be obtained. Then, parameters of adjust law for NN will be optimized by using
obtained nominal model via DE.
Keywords: Adaptive Control, ASPR, PFC, Neural network, Differential evolution.

if the conditions are satisfied. However, those parameters
will directly affect control performance. Therefore, it hasto be appropriate value in actual control systems. Since
there are no decision method for those parameters,
designer has been decided it by trial and error ofnumerical simulations. Therefore, it is quite difficult todecide appropriate value.
In this paper, the parameter optimization method of
PFC and NN by using input/output data via differential
evolution (DE) will be proposed. DE is one of theevolutionary computation methods and it has a good
convergence and uniqueness of solution8.

1. Introduction
In spite of their simple structure, Almost strictly positive
real (ASPR) based adaptive control systems have highrobustness with respect to disturbances and uncertainties1,2. However, the ASPR conditions which controlled
system has to satisfy are very severe conditions for actualsystems. To overcome this problem, introduction ofparallel feedforward compensator (PFC) has been
proposed3. Several design methods of PFC which arebased on mathematical model of controlled system suchas relative degree have been proposed4. However, toobtain the mathematical model, iterative of identification
experiments have to be done and it cost time and money.
Therefore, a design method which is using system’s
input/output data has been proposed5, 6. For ASPR system,an adaptive control system with feedforward input
generated by neural network (NN) has been proposed7.
This input is applied to avoid tracking error caused byPFC. In theory, PFC and NN parameters can be any value

2. Adaptive Control System with Adaptive NN
Feedforward Input
Consider the controlled system is described in equation
state model as follows.

.

Copyright © 2018, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
19(1)

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 19–22___________________________________________________________________________________________________________For this system, a PFC described as follows is introduced.
(2).
Then, supposed that the augmented system:(3)
where,Fig. 2. Block diagram of adaptive control system with NN
feedforward input.

,,is ASPR. Fig. 1. shows the block diagram of augmentedsystem. Then, for this augmented system, design anadaptive output feedback control system with adaptive
NN as follows.

Here, the problems are how to design the PFC andhow to decide the parameters of adjusting laws.
3. Design of PFC by Input/Output Data
Define the ideal output of the ideal augmented systemwith the input data as follows.
(9).
Then, the ideal output of PFC can be described by using
the output of controlled system as follows.

(4)

(10)
.
Here, suppose that the ideal PFC given by the following:

where
(5)

(11)

(6)
and

then the ideal output of PFC also can be given as

is the output of PFC obtained by(7)By introducing a stable filter:.
Also,
is the estimated value ofideal weight vector of NN and l is the number of NN
nodes. In addition,is aradial basis function vector generally designed by theGaussian functions such as

(12)

.

,

Eq. (12) can be represented as

(13)

(14)
By defining the ideal parameters as

(8)

(15)

where is the center of receptive field and is the widthof Gaussian function. Fig. 2. shows the block diagram ofadaptive control system which is shown in this section.

and the filtering signal as(16)then the ideal output of PFC can be represented as(17).
From eq. (9), for any parameters, the output of PFC canbe given asFig. 1. Block diagram of augmented system.

.

20

(18)

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 19–22___________________________________________________________________________________________________________

Finally, consider minimizing the following evaluation
function.

Select the next generation as follows.
(24)

(19)
.
Although the order of PFC and parameters of filter affectto the output of PFC, there are no theorical method todecide these parameters. Therefore, it will be optimizedby DE later. Remark that the ideal output of PFC isknown but parameters of ideal PFC and filter areunknown.

.

The shown DE algorithm is so-called DE/best/1/binstrategy9.
5. Control System Design via Differential
Evolution
Here, concrete designing method of control system via
DE will be shown.
(i) PFC design(a) Obtain the input/output data set of controlledsystem.
(b) Creation of initial population
Randomly create an initial N sets of population
as follows

4. Differential Evolution
In this section, the basic algorithm of DE which is appliedin this study will be shown.
(i) Creation of initial population
Randomly create an initial N sets of population
as follows.

(25)

(20)

.
The superscript means a generation number ofthe vector, and D is the order of decision variables.
(ii) Extraction of base vector
Extract the best vector from nth generation
vectorwhich minimizing the evaluation
function. This is, set the base vector as follows when
the lth target vector minimize the evaluation function.
.

(c) Extraction of base vector
Extract the best vector from nth generation
vector as Eq. (21) which minimizing theevaluation function Eq. (19) by using obtained
input/output data.
(d) Mutation
Generate a mutated vector from randomly
chosen two vectors as Eq. (22).
(e) Crossover
Generate a trial vector from the target vector
and the mutated vector as Eq. (23).
(f) Selection
Select the next generation as Eq. (24).
(ii) Adaptive controller design(a) Obtain an approximation model of controlled
systemFrom ASPR modeland obtained PFC
in above step as(21)(iii) Mutation
Generate a mutated vector from randomly
chosen two vectors
asfollows.
(22).
Here, F is a scaling factor for scaling thedifference vector.
(iv) Crossover
Generate a trial vector from the target vector andthe mutated vector as follows.

(26)(b) Creation of initial population
Randomly create an initial N sets of population
as follows

(23)

(27)

.
Here, C is a crossover rate, randi,j is a random
uniform numbers in the range [0, 1] and jr is arandomly chosen integer number in the range [1, D].
(v) Selection(c) Extraction of base vector
Extract the best vector from nth generation
vector as Eq. (21) which minimizing theevaluation function as follows.

21

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 19–22___________________________________________________________________________________________________________.

(28)(d) Mutation
Generate a mutated vector from randomly
chosen two vectors as Eq. (22).
(e) Crossover
Generate a trial vector from the target vector
and the mutated vector as Eq. (23).
(f) Selection
Select the next generation as Eq. (24).

Fig. 3. Control input/output data.

6. Numerical Simulation
Assume that input/output data shown in Fig. 3 areobtained. From these data, by setting the parameters F=
0.8 and C=0.9 with ASPR model

third order PFC was obtained as

Fig. 4. Control result by initial parameters.

,.
Then the adaptive controller parameters were optimized.
The initial value of design parameters wereThen the optimized parameters wereFig. 5. Control result by optimized parameters.

Fig.4 and Fig. 5 show the control result by initial
adaptive controller parameters and optimized parametersrespectively. It can be seen that the control performance
became better after optimizing the parameters.

4. Z. Iwai, I. Mizumoto and H. Ohtsuka, Simple Adaptive
Control SAC, (Morikita Publishing, 2008)
5. I. Mizumoto and T. Takagi, PFC design via FRIT
approach and adaptive PID control system design for
discrete-time systems, In proc. of IFAC Conference on
Advances in PID control, (2012) 64–69.
6. T. Takagi and I. Mizumoto, Adaptive output feedback
control with adaptive PFC and its stability analysis, ICIC
Express Letters, Part B: Applications, 6(12) (2015) 31893195.
7. I. Mizumoto, H. Tanaka and T. Tokimatsu, Adaptive
output regulation of a class of discrete-time nonlinear
systems based on output feedback and NN feedforward
control, In proc. of 49th IEEE Conference on Decision andControl, (2010) 3631-3636.
8. M. Ito and M. Tanaka, A comparative study of particle
swarm optimization, differential evolution, real-coded
genetic algorithm on function optimization, Konan
University Bulletin. Science and Technology, ed., 52(1)(2005) 125-135. (in Japanese)
9. G. C. Onwubolu, B. V. Babu, New Optimization
Techniques in Engineering (Springer Science & BusinessMedia, 2004).

7. Conclusions
In this paper, the parameter optimization method whichusing input/output data via DE was proposed. The
numerical simulation shows the usefulness of the method.
References
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strictly passivity of discrete-time nonlinear systems andadaptive control system design with a PFC, Automatica,
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IEEJ Transactions on Electronics, Information andSystems, 134(9) (2014) 1175-1181 (in Japanese)
3. H. Kaufman, I. Barkana and K. Sobel, Direct Adaptive
Control Algorithms 2nd Ed., (Springer-Verlag New York,1997)22