FPGA Based Wireless Multi-Node Transceiver and

Journal of Mathematics and System Science Volume 2, Number 1, January 2012 (Serial Number 2)

David Publishing

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Publication Information: Journal of Mathematics and System Science is published monthly in hard copy and online (Print: ISSN 2159-5291; Online: ISSN 2159-5305) by David Publishing Company located at 9460 TELSTAR AVE SUITE 5, EL MONTE, CA 91713, USA. Aims and Scope: Journal of Mathematics and System Science, a monthly professional academic and peer-reviewed journal, particularly emphasizes new research results in theory and methodology, in realm of pure mathematics, applied mathematics, computational mathematics, system theory, system control, system engineering, system biology, operations research and management, probability theory, statistics, information processing, etc.. Articles interpreting practical application of up-to-date technology are also welcome. Editorial Board Members: Wattanavadee Sriwattanapongse (Thailand) Zoubir Dahmani (Algeria) Claudio Cuevas (Brazil) Baha ŞEN (Turkey) Shelly SHEN (China) Emily TIAN (China) Manuscripts and correspondence are invited for publication. You can submit your papers via web submission, or E-mail to [email protected], [email protected], [email protected]. Submission guidelines and web submission system are available at http://www.davidpublishing.org. Editorial Office: 9460 TELSTAR AVE SUITE 5, EL MONTE, CA 91713, USA Tel: 1-323-9847526, 1-302-5977046 Fax: 1-323-9847374 E-mail: [email protected], [email protected], [email protected] Copyright©2012 by David Publishing Company and individual contributors. All rights reserved. David Publishing Company holds the exclusive copyright of all the contents of this journal. In accordance with the international convention, no part of this journal may be reproduced or transmitted by any media or publishing organs (including various websites) without the written permission of the copyright holder. Otherwise, any conduct would be considered as the violation of the copyright. The contents of this journal are available for any citation. However, all the citations should be clearly indicated with the title of this journal, serial number and the name of the author. Abstracted / Indexed in: Database of EBSCO, Massachusetts, USA Ulrich’s Periodicals Directory CSA Technology Research Database Summon Serials Solutions Chinese Database of CEPS, Airiti Inc. & OCLC Subscription Information: Price (per year): Print $450; Online $300; Print and Online $560 David Publishing Company 9460 TELSTAR AVE SUITE 5, EL MONTE, CA 91713, USA Tel: 1-323-9847526, 1-302-5977046 Fax: 1-323-9847374 E-mail: [email protected]

D

DAVID PUBLISHING

David Publishing Company www.davidpublishing.org

Journal of Mathematics and System Science Volume 2, Number 1, January 2012 (Serial Number 2)

Contents Pure Mathematics 1

Estimation of Ordered Means of Two Normal Distributions with Ordered Variances Yuan-Tsung Chang and Nobuo Shinozaki

Applied Mathematics 8

Teaching—A Way of Implementing Statistical Methods for Ordinal Data to Researchers Elisabeth Svensson

13

Modelling of Wage and Income Distributions Using the Method of L-Moments Diana Bílková

20

The Influence of Exchange Rate on the Volume of Japanese Manufacturing Export Hitomi Okamura, Yumi Asahi and Toshikazu Yamaguchi

26

The Strict Swedish vs. the Loose Dutch System for Regulations on Prostitution and Drug Use Peter Sarkany

36

Reducing of Seismic Vulnerability & Short-Time Earthquake Prediction: Methods and Instruments of Nonlinear Seismology Oleg Borisovich Khavroshkin and Vladislav Vladimirovich Tsyplakov

45

Controlling Nonlinear Dynamics in Continuous Crystallizers Victoria Gámez-García, Eusebio Bolaños-Reynoso, Oscar Velazquez-Camilo and Héctor Puebla

System Science 53

FPGA Based Wireless Multi-Node Transceiver and Monitoring System Özkan Akin, İlker Başaran, Radosveta Sokullu, İrfan Alan and Kemal Büyükkabasakal

58

Linguo-Combinatorial Simulation of Complex Systems Mikhail B. Ignatyev

D

Journal of Mathematics and System Science 2 (2012) 1-7

DAVID

PUBLISHING

Estimation of Ordered Means of Two Normal Distributions with Ordered Variances Yuan-Tsung Chang1 and Nobuo Shinozaki2 1. Department of Social Information, Faculty of Studies on Contemporary Society, Mejiro University, Tokyo 161-8539, Japan 2. Department of Administration Engineering, Faculty of Science and Technology, Keio University, Yokohama 223-8522, Japan

Received: May 28, 2011 / Accepted: August 12, 2011 / Published: January 25, 2012. Abstract: The authors consider the problem of estimating the ordered means of two normal distributions with unknown ordered variances. The authors discuss the estimation of two ordered means, individually, in terms of stochastic domination and MSE (mean squared error). The authors show that in estimating the mean with larger variance, the usual estimator under order restriction on means can be improved upon. However, in estimating the mean with smaller variance, the usual estimator can’t be improved upon even under MSE. The authors also discuss simultaneous estimation problem of two ordered means when unknown variances are ordered. Key words: Restricted MLE (maximum likelihood estimator), unbiased estimator, Graybill-Deal estimator, stochastic dominance.

1. Introduction In many practical situations, statistical inference under restrictions on parameters is quite important. Most earlier works related to statistical inference under restrictions are reviewed by Barlow et al. [1], and Robertson and Wright [2]. Recent years, two monographs, accomplished by Silvapulle and Sen [3] and Van Eeden [4], reviewed literatures on restricted statistical inference after 1988. Many authors have considered inference of parameters under simple order restriction or tree order restriction on parameters and proposed some improved estimators upon the unbiased estimator, especially for normal distribution. Here we consider the estimation of two ordered normal means when unknown variances are ordered. Such a situation can occur, if there are two kinds of new motor engines, one is developed by a new method, which has higher power than the one developed by a standard method, but has larger variation than the one Corresponding author: Yuan-Tsung Chang, Ph.D., professor, research field: mathematical statistics. E-mail: [email protected].

developed by a standard method. We are interested in estimating their powers, individually and/or simultaneously. X ij  i  1 2  j  1 ... n i

Let

be

independent

observations from normal distribution with mean i and variance

 i2 , where both i and  i2 are

unknown. Also let ni

ni

j 1

j 1

2 2 X i   X ij  ni  si   ( X ij  X i )  (ni  1), (1)

be the unbiased estimators of respectively.

i and  i2 ,

We first state some fundamental results on the estimation of the common mean. For the case when two variances are unknown and there is no order restriction between two variances, the estimator of the common mean, 1   2   ,

ˆ GD 

n1s22 n2 s12  X1 X2 n1s22  n2 s12 n1s22  n2 s12

was proposed by Graybill and Deal [5] and they gave a necessary and sufficient condition on n1 and n2 for

ˆ GD to have a smaller variance than both X 1 and 2 X 2 . For the case when it is known that  1 is smaller

Estimation of Ordered Means of Two Normal Distributions with Ordered Variances

2

 22 ,  12   22 , Nair [6] modified ˆ GD and proposed an estimator of  ,

than

 ˆ GD  ˆ Nair   n1 n2  n  n X 1 n  n X 2 1 2  1 2

if s12  s22 if s12  s22 

has smaller variance than ˆ . and showed that ˆ 2 For the case when  i ’s are known and k normal means satisfy the simple order restriction, 1  …  k , the restricted maximum likelihood estimator of i is given by Nair

GD

t

t

ˆ iRMLE  min max[  j 2 n j ]1[  j 2 n j X j ] , i  1 ... k  t i

s i

js

j s

ˆ

1  2 , ˆ OS uniformly improves upon X i i GD if and only if the risk of the ˆ i is not larger than the when

risk of X i under squared error loss, and applied the results of Graybill and Deal [5] to give a necessary and sufficient condition on n1 and n2 . When order restrictions are given on both means and variances, Shi [12] has discussed some properties of

i and  i2 and has given an algorithm for 2 finding the MLE of i and  i . Ma and Shi [13]

MLE of

have suggested an iterative algorithm for computing the two order restricted MLE of

i and  i2 . When

sample sizes are equal and we estimate the mean with

uniformly improves

larger variance they gave a sufficient condition for the

upon X i under MSE. Kelly [8] and Hwang and

restricted MLE to dominate the sample mean under

Lee [7] showed that

RMLE i

universally

squared error loss. For other distributions see two monographs,

dominates X i . In estimating two ordered means when the variances

accomplished by Silvapulle and Sen [3] and Van

 ,

Here we consider the estimation of two ordered

Peddada

[9]

proved

ˆ iRMLE

that

are unknown, Garren [10] used the MLE of

2 i

s   j 1 ( X ij  X i )  ni , and proposed a plug-in 2 i

ni

2

Eeden [4] for details. means with unknown ordered variances. Based on the estimators given by Nair [6] and Oono and Shinozaki [11], we consider the estimators

estimator  min max[  s j 2n j ]1[ s j 2n j X j ] , i  1 2 ˆ Garren i t i

s i

t

t

js

js

They showed that neither the unrestricted MLE of

i , X i , nor the plug-in estimator, ˆ Garren , dominates i the other in general under MSE when k  2 , considering the case when 1  2 . Motived by Garren [10], Oono and Shinozaki [11] proposed a truncated estimator of i  i  1 2 ,

ˆ GD } ˆ 1OS  min{ X 1 ˆ GD } ˆ OS 2  max{ X 2  2 2 using the unbiased estimator of  i , si , in Eq. (1). They showed that 1   2 is the critical point for to improve upon the unrestricted MLE of i , ˆ OS i OS X i , that is, ˆ i uniformly improves upon X i , if OS and only if the risk of ˆ i is not larger than that of

X i when 1  2 . Furthermore, they showed that

OS  if s12  s22 ˆ 1   (2) ˆ 1   n1 n2   2 2 min  X 1 n  n X 1  n  n X 2  if s1  s2  1 2 1 2   

and OS  ˆ 2   ˆ 2   n1 n2    max  X 2 n  n X 1  n  n X 2  1 2 1 2   

if s12  s22 if s12  s22 

(3)

We may expect that ˆ i is better than ˆ i , since the order restriction on variances is taken into account. OS

In Section 2, we consider the estimation of

i  i  1 2 individually and show that when estimating the mean with larger variance, ˆ 2 OS stochastically dominates ˆ 2 . We give the definition of stochastic dominance later. However, when estimating the mean with smaller variance, we show that ˆ 1 has smaller MSE than ˆ 1 for certain values of parameters. In Section 3, we consider the OS


simultaneous estimation of ( 1   2 ) and show that

( ˆ 1 ˆ 2) dominates ( ˆ 1OS  ˆ OS 2 ) under the criterion 2 2 ˆ  ( i  i )   d  P  (4) 2 i  i 1  ˆ where is an estimator of  , i

i

P{ ˆ 2  2  d  X 1  X 2} P  ˆ 2   2  d  X 1  X 2 s12  s22  P{s12  s22 }

as

 P  ˆ 2   2  d  X 1  X 2 s12  s22  P{s12  s22 }.

 i2   i2  ni  i  1 2 , and d  0 is an arbitrary constant. In Section 4, we make a conclusion.

To show that ˆ 2 dominates ˆ 2

OS

P  ˆ 2   2  d  X 1  X 2 s12  s22 

dominance and its implications. Let ˆ and  be two estimators of an unknown parameter  . ˆ stochastically dominates  if for all d  0 and for

,

P{ ˆ    d }  P{     d } with strict inequality for some d  0 and for some  . Hwang [14] has shown that ˆ stochastically dominates  if and only if ˆ universally dominates  , that is, for any no decreasing function L() E {L( ˆ   )}  E {L(    )} 



 P  ˆ 2   2  d  X 1  X 2 s12  s22   OS

.

(5)

Put c  n1  (n1  n2 ) , the left-hand side of Eq. (5) is P{ c ( X 1   2 )  (1  c )( X 2   2 )  d  X 1  X 2 s12  s22 } (6)

Next we make the variable transformation

Wi  X i  2  i  1 2 Then W1 and W2 are independently distributed

as N (  1 ) and N (0 2 ) , respectively, where 2

with strict inequality for at least one nondecreasing loss function and some

stochastically,

we need only to show that

Finally, we give the definition of stochastic

all

3

2

  2  1 and  i2   i2  ni , I = 1,2. Then Eq. (6) becomes

2. Estimation of Individual Means

P{d  cW1  (1  c)W2  d W1  W2 , s12  s12 }

In this section we discuss the estimation of

i  i  1 2 individually. In subsection 2.1, for the estimation of the mean,  2 , of the population with larger variance, we will show that ˆ 2 in Eq. (3), OS stochastically dominates ˆ 2 . In subsection 2.2, for the estimation of the mean, 1 , of the population with smaller variance we will show that ˆ 1 in Eq. (2), OS does not dominates ˆ 1 even in terms of mean squared error. 2.1 Stochastic Dominance of ˆ 2 over ˆ 2

OS OS

in Theorem 2.1.

ˆ 2 stochastically 2 2 dominates ˆ , i.e., for all 1   2   1   2 , Theorem 2.1. The estimator





P  ˆ 2  2  d  P  ˆ  2  d   d  0 Proof: We first express

OS 2

V1  W1  W2  V2  ( 22   12 )W1  W2  Then

V1

V2

and

are

independent

and

respectively distributed as

N ( 12   22 ) , N (( 22   12 ) 22 ( 12   22 )   12 ) Since W1   12 (V1  V2 ) / ( 12   22 ),W2  ( 12V2   22V1 ) / ( 12   22 ) fix

We show that ˆ 2 stochastically dominates ˆ 2

OS 2

We further make the variable transformations

s12  s22 ,

for

convenience,

we

have,

P   d  cW1  (1  c)W2  d  W1  W2    (c 2  (1  c) 22 )V1   12V2  P d  1  d  V1  0  2 2 1   2  

 P (c 12  (1  c) 22 )V1   12V2   d ( 12   22 )  V1  0  P (c 12  (1  c) 22 )V1   12V2  d ( 12   22 )  V1  0 

We have


4

P (c 12  (1  c) 22 )V1   12V2   d ( 12   22 )  V1  0

Theorem 2.3. ˆ 1

has smaller MSE than ˆ 1 for

OS

 P (c 12  (1  c) 22 )V1   12V2  d ( 12   22 )  V1  0

sufficiently large    2  1 .

 E{g (c V1 )  V1  0}

c  n1  (n1  n2 ) again, we evaluate the conditional

where

Proof:

 0  n1s22  (n1s22  n2 s12 )

Putting

and

expectation of the difference between squared error

g (c v1 ) 

losses of two estimators given

s s 2 1

E{[( 0 X 1  (1   0 ) X 2  1 ) 2

 ((1  c ) 2  c 2 )v   2   d ( 2   2 )  2 1 2 1 2 1   2 2        1 2   1 2 

 

2 ((1  c ) 2

 

2  c 1 )v1

2 2

2  d ( 1

2 2 )

 1 2  1   2 2

2

(c X 1  (1  c) X 2  1 ) 2 ]  X 1  X 2 s12  s22 }  E{[( 0 ( X 1  1 )  (1   0 )( X 2  1 )) 2 

   

2 2 P  ˆ OS 2   2  d  X 1  X 2 s1  s2 

Z i  X i  1  i  1 2 Then, Z1 and Z 2 are mutually independent, and 2 2 distributed as N (0 1 ) , N (  2 ) , respectively, where    2  1  0 . Then Eq. (7) becomes

E{[( 02  c2 )Z12  2( 0 (1   0 )  c(1  c))Z1Z2

 E{g ( 0 , V1 ) V1  0, s12  s22 }

 ((1   0 )2  (1  c)2 )Z 22 ]Z1  Z2  s12  s22}(8)

 0  n1s22  (n1s22  n2 s12 ) .

We can easily verify that for fixed v1  0 ,

We further make the variable transformation

g (u v1 ) is an increasing function of 0  u  1 and 2 2 that c   0 for s1  s2 . Then we have P  ˆ 2  2  d  X 1  X 2 s12  s22    

 P  ˆ

OS 2

(7)

(c ( X 1  1 )  (1  c)( X 2  1 )) 2 ]  X 1  X 2 s12  s22 } We make the variable transformation

Similarly, the right-hand side of Eq. (5) can be evaluated as

where

X 1  X 2 and

2 2.

Then Y1 and Y2 are independent and distributed

N ( 12   22 )

as

 2  d  X 1  X 2 s  s  2 1

Y1  Z1  Z 2  Y2  Z1  ( 12   22 ) Z 2 

2  2 

 12 ( 12   22 )   22 ) respectively. We can easily see

this completes the proof. OS Remark: Since ˆ 2 stochastically dominates ˆ 2 , under squared error loss. ˆ 2 dominates ˆ OS 2

that Z1  ( 12Y1   22Y2 ) / ( 12   22 ),Z 2   22 (Y2  Y1 ) / ( 12   22 ) Since

( 0  c 2 ) 12  ( 0 (1   0 )  c(1  c))( 12   22 ) 2

2.2 Non-dominance of ˆ 1 over ˆ 1

OS

In this subsection, we will show that MSE of ˆ

OS 1

is smaller than that of ˆ 1 when the difference of two

  2  1 , is sufficiently large. This implies in particular that ˆ 1 does not stochastically OS dominate ˆ 1 . means,

We need the following Lemma. Lemma 2.2. Let   0 , and let Y is distributed as

E (Y 2 IY 0 ) N (  2 ) , then lim  0  E (YI Y 0 ) It is obvious that we will omit the proof.

N (( 12   22 )

((1   0 ) 2  (1  c) 2 ) 22  ( 0  c)( 12   22 ) Then Eq. (8)  ( 2 1 2 )2 becomes 1

E

2

[ ( 012  (1   0 ) 22 )2  (c12  (1  c) 22 )2  Y12

2 22  ( 0  c)( 12   22 )  Y1Y2

 Y1  0 s12  s22 (9)

Since Y1 and Y2 are mutually independent and

E (Y2 )  ( 12   22 ) , Eq. (9) becomes E

[ ( 012  (1   0 ) 22 )2  (c12  (1  c) 22 )2  Y12 2  (  c)(   )  Y  Y  0 s  s   (10) 0

2 1

2 2

2 1 1

1

2 1

2 2

Estimation of Ordered Means of Two Normal Distributions with Ordered Variances 2

The coefficient of Y1 is

( 0 12  (1   0 ) 22 ) 2  (c 12  (1  c) 22 ) 2 =   

( 0  c)12  ((1   0 )  (1  c)) 22  ( 0  c)(12   22 )

 12   22 , we have

Since s1  s2 and 2

2



n1s22 n1s22  n2 s12





1 n2

 n1 n1n2





12 n1





n2 s12 n1s22  n2 s12

 n1n2n2

  2     i 1 



  2     i 1 

 22 n2

 c(   )  (1  c)(   )  2 X1 1 X2 1 = P   1  

 c(   )  (1  c)( X 2  2 )   X 1 2  2  

The coefficient of Y1 is ( 0  c)( 1   2 ) 0 indicates a depreciation of the yen, εt 4) during two years. As presented in Fig. 2, in period prior to earthquake the increase of SAE level in narrow frequency band which is associated with rebuilding of the noise spectrum occurs. The important promising peculiarity of SAE shown in Fig. 2 are always present the oscillation or abrupt decrease of SAE intensity within 2-3 hours prior to the event while the common background of SAE noise is increasing. There is another SAE feature. Because of tectonic stress unloading the SAE intensity level sharply drops after

39

the seismic event. For example, the decrease of the SAE level was more than 100 for Tadjikistan (Fig. 2). The relations between seismicity and other geophysical media properties under month averaging such as SAE level variations, elastic wave velocity variations, AE intensity in a deep bore hole in Ashkhabad’s region are shown in Fig. 4 [1]. It follows from the experimental data that the maximum of seismic activity lags behind the noise activity maximum of SAE in time. It is known that the P-wave velocity change is determined by the crack generation processes and therefore by stress rebuilding before earthquake [8]. The observed behaviour may be caused by creep process in the region. One can see that the seismic wave velocity gradients correlate variations of high frequency SAE. We suppose that both the increasing of SAE activity and the P-wave velocity change relate to the crack generation processes which accompany creep in fault zone and the activation of weak seismicity on creep final stage. The tentative conclusion from experimental observations can be made that increasing of month averaging level of SAE envelope leads the local and region seismicity by 1.5-2 months in time. That is, this parameter of SAE (month averaging level of SAE envelope) may be used for middle-term regional seismic prediction.

Fig. 3 Seismo-acoustic emission (SAE) and seismicity. Turkmenistan’s region. Local earthquakes of K classes, K—the value which is linearly dependent on M (K = 1.8M + 4.0); J—intensity of SAE under Δf = 0.1 Hz: f = 15 Hz (a); f = 30 Hz (b, c, d); The arrows mark the earthquakes of local seismicity of tested region.

40

Reducing of Seismic Vulnerability & Short-Time Earthquake Prediction: Methods and Instruments of Nonlinear Seismology

Fig. 4 Comparison of temporal relation under month averaging for Ashkhabad’s region: 1-intensity of acoustic noise Ia ; 2-intensity Ic of seismo-acoustic noise, Δf = 0.1 Hz; 3-seismicity L; 4-travel times tp of P-wave.

The results of the present investigation indicate that the seismo-acoustic emission is very sensitive to processes of earthquake preparation. We suppose that the most part of information about seismic hazards is contained in SAE noise, it should be recorded in dangerous areas permanently and may become deciding prognosis feature.

3. Operative Prediction for Private User Regular deformation fields and waves in the Earth crust, for example from Sun-Moon tides, have the elements of chaos. These conclusions have been derived from the analysis of a lot of data on regular deformation fields and waves recorded in the South Tadjikistan and Turkmenia during 15-20 years [1, 8, 9].

It is likely that the crust deformation is trigger for many earthquakes, as a consequence the seismicity behaviour may have the elements of chaos also. The results of the other scientists at 90th years testify to it [10-12]. Thus, the problem of the reliable middle-term earthquake prediction has no exact solution in principle [13]. The government is interested in existing seismic networks as the instrument for investigations of the region seismicity and structure of geophysical media. It is our opinion that it is logic to add these networks to ones for operative prognosis which might be developed by individual users. Private user is interested in permanent significant operative prognosis information on local variation of the strained state of a media at the point in question. The method of SAE noise observation and analysis allows to realizing and developing the simple scheme of short-term operative prediction for private user (OPPU). Processing and analyzing of data of SAE noise in real time may give him necessary information and thus a possibility to take decision for providing the object’s safety. In the future private users may take part in foundation of the system of the emission indication of seismic hazard (EMISH). EMISH should consist of elements as follows: user device module, regional EMISH system and central station. The module contains the SAE recorder, which is isolated in the bore hole to depth about 100 m, indication and notification systems and system of information transmission (for example, personal computer). The regional station EMISH should include the computer network consisting of the system of EMISH modules and should summarize information within the region. The central server EMISH should be organized as a government service. The probability (P) of the earthquake prediction will be determined by density of EMISH module network and will increase with the system utilization. Even in initial period without necessary statistics P > 60%. The individual module will ensure P > 80%. The approximate cost of module in conditions of Russia is 2000-8000$.

41


4. Operative Prognosis Has No Alternative Because of Chaotic Deformation Fields of the Earth [1, 8, 9, 12] In succeeding years the investigations showed that the geophysical media as irreversible nonlinear system has stochastic properties (features) [1, 8] and thus probability character of manifestation of these properties, the seismicity among them. According to concepts of nonlinear seismology such complex but determinant signal as lunar-solar tides results in the general features of wave dynamics which should be observed in behavior of waves or zones of deformations in the Earth crust. We took Puason’s factor as investigated media parameter indirectly reflecting variations of the Earth crust structure. The spectrum-time analysis (STA) of time variations of Puasson’s factor for three seismic active areas of Tadjikistan (Dushanbe’s, Nurec’s, Rogun’s regions) was made. We estimated the effective values of factors on time series of average month values of relation of Vp/Vs received on the seismological data in the period from 1960 to 1985. The sliding time window has duration of 6 years. The 8-th degree polynomial trend was preliminary eliminated from time series to reduce its masking influence. For Dushanbe’s, Nurec’s, Rogun’s regions the time series have commensurable maximal scope and an average level of variations. Dominant cyclic components and components with the periods earlier observed and the moments of spectrum-time structure reorganization show up. So, for Dushanbe’s area (Fig. 5) since 1963 it is observed migration of the periods of Puasson’s factor variations from Т = 0.26-0.27, 0.37 up to Т = 0.45, 0.7, 1.25-1.4 (1966); i.e. for 3 years energy of deformation through first or second subsequent doubles of Т concentrates on Т = 0.47, 0.7, 1.4 (hereinafter T has dimensions of time per years). Last three periods are in the ratio of 1:2:3. Further till 1968 there is only one strongly pronounced group of Т = 0.25, 0.27, 0.3 years; Т = 0.3 years is steadiest and has duration of existence

of 6.0-7.0 years (1966-1973) etc.. Similar features are observed at analysis STA of Nurec’s and Rogun’s areas. Taking into account high activity of researched regions we accept that the original cause or the physical mechanism of seismicity is the tectonic current of rock masses, considering it as chaotizating (self-organizing) of dynamic nonlinear system. We accept also that the value of Puasson’s factor is the parameter which describes occurrence of certain structure (for example, existential periodicity or a wave of deformation) at the homogeneous media. If the ratio between the periods is 1:2:4, 1:3, 1:5 the features of occurrence and time development of the periods of variations of Puasson’s factor are known respectively as the features of Feigenbau’s script; Ruelle’s, Takens’s script and script of chaos through intermittent. Ruelle’s,

Takens’s

script

is

appropriate

to

development of only three dominant periods from a wide set. Transition of one strongly pronounced frequency of process to the noise spectrum corresponds to script of chaos through intermittent. All specified scripts

are

realized

(non-synchronization)

through

synchronization

and

three-frequency

disintegrated instability. These processes earlier found out as properties of seismic fields. Deformational waves which are trigger for earthquakes are similiarity to long period seismic waves. As seismic waves have a chaotic propertis that deformation waves (for example tides waves) have these property too [14]. Despite of a generality of the processes for all regions, we should point to the fact that their uniform synchronism is absent, i.e. each observed structure can be considered as

independent

structure.

If

nevertheless

synchronization will arise, amplitudes of variations of Puasson’s factor will sharply increase for common region and a seismic mode connected to it will change. At the International Symposium in Erevan (1989) after Spitak earthquake on the basis of further research O.B. Khavroshkin has reported that only short-term prediction in time is realistic due to stochastic properties

42

Reducing of Seismic Vulnerability & Short-Time Earthquake Prediction: Methods and Instruments of Nonlinear Seismology 6.0

3.0 2.0 1.5

1.0 0.7

0.5 0.4 0.3 0.25

0.2 0

1964

1966

1968

1970

1972

1974

1976

0.041

1978

1980

Годы

Fig. 5 Spectrum-time structure of variations of Puasson’s factor for Dushanbe’s area. The reorganization of spectrum-time structure of variations consists in change of one periodicity by others, their merge or transition each other.

of media [1, 8]. He has granted the appropriate interview to regional journal “Armenia” with the great edition and suggested to apply SAE level control for short-term prediction [15]. The device based on this prediction method has been compared with barometer for its operative simplicity and low cost. However, specialists’ response did not take place. The failure could be traced to two main courses: (1) SAE concept was incomprehensible to some experimenters; (2) nonlinear seismology had an imperfect development as a solo division of seismology.

5. Inevitability of Destructions and Victims: Seismic Caustics (Isoseismic Amplitude Curve) Are the Consequences of Wave Front Catastrophes [1, 16-18] In seismically muddy media seismic caustics inevitably occur if ℓ > λ (ℓ—the characteristic size of heterogeneity or indignation of border, λ—length of a seismic wave) [1]. Concentration of seismic wave energy on caustic can far exceed background energy. In specific media with the certain structure and for seismic waves of a narrow frequency range seismo-caustics can form the original mesh structure

on the day surface similar to optical systems. In the case in question the day surface with the formed structure of seismic caustic has elevated magnitude and we can interpret the border or envelope of caustics as isoseismic amplitude curve [16-18]. The observable range of seismic energy changes is evident from a simple ratio: 2

K 

Ai  A0

2 ф 2





2 ф

;



 , 2 L

where A0—initial amplitude value at a source, ф—focussing index of the beams, —the wave length, L—scale of media heterogeneity or perturbation of border in the Earth crust. Value ф grows with density of the beams focused on seismic caustic. We analyzed the maps of seismic caustic (isoseismic amplitude curve) for earthquakes of various regions (Tadjikistan, Italy, California). It is revealed that on cases isoseismic amplitude curves are formed at the terrestrial surface into caustics of elementary catastrophes which as the set of the lines which connect the points with the same magnitude. The observed results are contradictory to the simplified representations. The further studying demands numerical approximation of parameters of


seismo-caustics observed and the detailed computer analysis. We believe that algorithms of the automatic analysis of these data are necessary because of problem of subjectivity of visual analytical allocation of isoseismic amplitude curves. The elementary variants were developed [1]. Thus, under earthquake the amplitudes of seismic waves due to focusing effect repeatedly grow and considerably exceed the average and rating ones. The inevitable destruction and victims result.

6. Conclusions Since recording Sultanabad’s earthquake in 1980 it has been immediately obvious that the control of SAE intensity level gives the information about seismicity and seismicity danger. It appeared that the anomaly behaviour of SAE level is the reliable prognosis evidence which has been searching for a long time. Two years of investigations (1982-1983) carried out in Turkmenia have confirmed the data and idea. Thereafter these findings have been published and reported at all consultations with specialists in prognosis field. In spite of positive comments the proposed approach of prediction has not been called for. It was due to novelty of concept of seismo-acoustic emission. By now there are some publications which indirectly confirm the possibility of stable registration of SAE as source of information about deformation processes in media and potentiality for new approach of short-term prognosis. The authors have mastered the method of a narrow band filtration and singling out an enveloping curve for recording seismic noise and applied it to the research. Unfortunately, the authors do not understand that the short-term prediction problem is closely bound up with nonlinear properties of real media that tend to stochastic behaviour. They offer to insert the method of SAE level control in routine methods of seismic networks. But the validity of SAE method is possible only in the scheme: local information flow of SAE level plus its application by private users. Another line

43

of attack rules out its efficiency in principle. What’s more, the authors have underestimated the importance of the following: higher sensitivity SAE in comparison with AE in the high frequency region. They have explained this fact by structural geological influence only. This regularity follows in our point of view from nonlinear (secondary) nature of SАE and means that SAE is more informative than AE. Actually, the impulse AE is the result of microdestruction of an element of a deformed media. The SAE impulse is more complex. The SAE wave field derives from evolution of AE impulses propagated in nonlinear media (sometimes in a seismically active media) and from their complicated interaction. Each SAE impulse reflects a change of the energy states of media along the whole propagation path. The impulse interaction shifts the spectrum maximum to lower frequencies and thus it allows to making the monitoring of larger region. That is the reason why SAE has high informative implication for prognosis. On the other hand, if the observation point of the region tested rests on the rocks AE recording by special channel is represented quite proved according with several reasons: high protection of this channel against seismic and industry interferences and existence of advanced fractality of the geological media of the region [19-21].

References [1]

[2]

[3]

[4]

O.B. Khavroshkin, Some Problems of Nonlinear Seismology, United Institute of Physics of the Earth Press, Moscow, 1999, p. 286. L.N. Rykunov, O.B. Khavroshkin, V.V. Tsyplakov, The effect of modulation of high frequency noise of the Earth, Discovery Diploma No. 282, Goskomizobreteniy USSR, Moscow, 1983, p. 1. L.N. Rykunov, O.B. Khavroshkin, V.V. Tsyplakov, The modulation of high frequency microseism, J. Dokl. Science Section, Translated from: Reports of the Academy of Sciences USSR, 1978, pp. 303-306. O.B. Khavroshkin, A.V. Nikolaev, L.N. Rykunov, V.V. Tsyplakov, Methods, results and perspectives of the high-frequency seismic noise and vibrosignals study, in: Proceedings of XVII General Assembly IUGG, Hamburg,

44

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12] [13] [14]

Reducing of Seismic Vulnerability & Short-Time Earthquake Prediction: Methods and Instruments of Nonlinear Seismology Aug. 15-27, 1983, 6th Reports of Microseisms Commission, 1985, pp. 123-149. B.P. Diakonov, B.S. Karryev, O.B. Khavroshkin, A.V. Nikolaev, L.N Rykunov, R.R. Seroglasov, et al., Manifestation on earth deformation processes by high frequency seismic noise characteristics, Physics of the Earth and Planetary Interior 63 (1990) 151-162. L.N. Rykunov, O.B. Khavroshkin, V.V. Tsyplakov, Methods and some results of statistic investigation of high frequency microseism, Volcanology and Seismology, Izvestya Academy of Sciences USSR 1 (1981) 64-69. O.B. Khavroshkin, V.V. Tsyplakov, R.I. Urdukhanov, Pressure resistant deeply-dipped geophysical equipment (GGS): Tests of a working model, Seismic Instruments 36 (2001) 38-44. (in Russian) S.I. Alexandrov, A.G. Gamburtsev, O.B. Khavroshkin, V.V Tsyplakov, Elements of chaos and self-organisation of determinated deformation waves in the Earth crust, in: International Symposium “Geodesy-Seismology: Deformations and Prognosis”, Erevan, USSR, Oct. 2-6, 1989, p. 6. S.Kh. Negmatullaev, O.B. Khavroshkin, The Peculiarities of the Spatial-Temporal Structure of Seismicity of Tadjikistan, Reports of the Academy of Sciences Tadjik. USSR, 1988, pp. 43-48. C. Berge, A very broadband stochastic source model used for near source strong motion prediction, J. Geophys. Res. Lett. 25 (1998) 1063-1066. M. Heimpel, Earthquake size-frequency relations from an analytical stochastic rupture model, J. Geophys. Res. 101 (1996) 22435-22448. D.L. Turcotte, Chaos, fractals, nonlinear phenomena in Earth sciences, J. Rev. Geophys 33 (1995) 341-344. R.J Geller, Shake-up for earthquake prediction, Nature 352 (1991) 275-283. O.B. Khavroshkin, V.V. Tsyplakov, Modeling, Simulation

[15]

[16]

[17]

[18]

[19]

[20]

[21]

and Application, World Scientific Publishing Co. Pte. Ltd., pp. 14-20. O.B. Khavroshkin, Do you want a seismo-barometer? J. Industry, Construction and Architecture of Armenia, Erevan 4 (1990) 14-16. O.B. Khavroshkin, V.V. Tsyplakov, Theory of catastrophes in seismology: Ray method, narrow band filtration, interference, Prognosis of earthquake, MSSSS, AN SSSR, TISSS AN Tadj.SSR, Mingeo SSSR Dushambe-Moskva 10 (1988) 137-155. O.B. Khavroshkin, V.V. Tsyplakov, Seismic fields and theory of catastrophes, J. Dokl. Science Section, Translated from: Reports of the Academy of Sciences USSR, 1987, pp. 313-317. S.Kh. Negmatullaev, S.L. Mikhailov, O.B. Khavroshkin, Norsar: Search of Caustics and Solitons in Seismic Wave Fields, Reports of the Academy of Sciences Tadjik, USSR 1987, pp. 717-721. J. Paparo, S. Guarniere, A. Bottari, A. Marino, A. Teramo, G.P. Gregori, Acoustic emission for diagnosing the incipient generation of flaws within solid materials, the natural environment and manmade structures, in: 1st International Conference on Applied Geophysics for Engineering, Osservatorio Sismologico, Universita Degli Studi Di Messina, Messina, Italy, Oct. 13-15, 2004, p. 15. G.P. Gregori, J. Paparo, I. Marson, M. Poscolieri, Crystal stress and acoustic emission, in: 1st International Conference on Applied Geophysics for Engineering, Osservatorio Sismologico, Universita Degli Studi Di Messina, Messina, Italy, Oct. 13-15, 2004, p. 34. J. Paparo, G.P. Gregori, Endogenous fluids and acoustic emission volcanoes and heat flow, in: 1st International Conference on Applied Geophysics for Engineering, Osservatorio Sismologico, Universita Degli Studi Di Messina, Messina, Italy, Oct. 13-15, 2004, p. 73.

D


DAVID

PUBLISHING

Controlling Nonlinear Dynamics in Continuous Crystallizers Victoria Gámez-García1, Eusebio Bolaños-Reynoso2, Oscar Velazquez-Camilo3 and Héctor Puebla1 1. Universidad Autónoma Metropolitana-Azcapotzalco. Av. San Pablo No. 180, Reynosa-Tamaulipas, Azcapotzalco D.F. 02200, Mexico 2. Facultad de Ciencias Químicas, Instituto Tecnológico de Orizaba, Orizaba, Veracruz 94340, Mexico 3. Facultad de Ingeniería, Universidad Veracruzana Región Boca de Rio, Veracruz 94340, Mexico Received: May 28, 2011 / Accepted: August 12, 2011 / Published: January 25, 2012. Abstract: Crystallization is used to produce vast quantities of materials. For several applications, continuous crystallization is often the best operation mode because it is able to reproduce better crystal size distributions than other operation modes. Nonlinear oscillation in continuous industrial crystallization processes is a well-known phenomenon leading to practical difficulties such that control actions are necessary. Nonlinear oscillation is a consequence of the highly nonlinear kinetics, different feedbacks between the variables and elementary processes taking place in crystallizers units, and the non-equilibrium thermodynamic operation. In this paper the control of a continuous crystallizer model that displays oscillatory behavior is addressed via two practical robust control approaches: (i) modeling error compensation, and (ii) integral high order sliding mode control. The controller designs are based on the reduced-order model representation of the population balance equations resulting after the application of the method of moments. Numerical simulations show good closed-loop performance and robustness properties. Key words: Crystallization, continuous crystallization, population balances, nonlinear dynamics, robust model-based control.

1. Introduction Crystallization is a solid-liquid separation technique used to produce vast quantities of materials for chemical, petrochemical, pharmaceutical, food, metals, electronics, and other industries, for instance, sodium chloride, sodium and aluminum sulphates, sucrose, and proteins [1, 2]. Crystallization can be carried out in batch, fed-batch, and continuous crystallizers. Batch crystallization is widely practiced in pharmaceutical processing. Evaporative and cooling crystallization from solution are generally carried out in continuous crystallizers. Advantages of the continuous crystallizers have the built-in flexibility for control of temperature, supersaturation, nucleation, crystal growth and all the Corresponding author: Héctor Puebla, professor, researcher, Ph.D., research field: process systems engineering. E-mail: [email protected].

other process parameters that influence crystal size distribution [2]. All particulate processes are nonlinear in nature. From a purely physical point of view, nonlinearities arise from mechanical, thermal, and chemical interactions of the dispersed phase with the continuous phase [3]. Another source of the nonlinearity is the multiple interactions among particles with different attributes. Continuous crystallizers are desired to operate at a steady state and the product quality is determined by the steady state crystal size distribution (CSD) [3, 4]. Non-linearity, as well as the interactions between the kinetics, fluid dynamics and CSD may give rise to different complexities in both the steady state and dynamic behavior of continuous crystallizers. Indeed, continuous crystallizers are known to exhibit sustained oscillations in solute concentration and size distribution [5-9]. Nonlinear oscillation in continuous industrial crystallization processes is a well-known phenomenon

46

Controlling Nonlinear Dynamics in Continuous Crystallizers

leading to practical difficulties in design and operation. In particular, this undesired behavior negatively affects downstream process units such as filtration and drying and the product also often fails to meet customer quality specifications. The control of continuous crystallizers has been shown to be feasible and can result in a considerable improvement of the process performance [4, 8, 10]. Control policies are required to maintain the specific product quality, to suppress the influence of process disturbances, to switch from one product quality to other at minimal production loss, to stabilize the crystallizer in case of cyclic behavior and optimize the economic process performance [8, 10]. Challenges in controlling crystallization include significant uncertainties associated with their kinetics, phenomenological effects are difficult to characterize, and the fact that crystallization processes are highly nonlinear, and are modeled by coupled nonlinear algebraic integro-differential partial equations. Robust feedback control seems to be essential for crystallization processes as it permits to deal with model uncertainties related to model parameters and model reduction for control design purposes [8, 10, 11]. Modeling error compensation and high-order sliding mode controllers provides two frameworks for the systematic consideration of such robustness issues and also leads to simple control structure that can be implemented in practice [12, 13]. The aim of this contribution is to design robust control approaches based on the above robust techniques for the suppression of nonlinear dynamics in continuous crystallization processes. This work is organized as follows: In Section 2 the class of continuous crystallizers is presented. For the sake of completeness some fundamental issues on crystallization processes are also presented. In Section 3 robust control designs are presented. Numerical simulations are provided and discussed in Section 4. Finally in Section 5 some concluding remarks are presented.

2. Continuous Crystallizers 2.1 Crystallization Phenomenology Crystallization is a separation process that brings about the formation of solid crystals from a fluid phase, namely vapor, solution or melt, wherein the solubility characteristics of certain materials are exploited to produce particles of a very high purity [2]. Main elements for crystallization are: (i) the driving force, (ii) identification of metastable and labile zones, (iii) nucleation, and (iv) growth [2]. The fundamental thermodynamic driving force for a crystallization process is the change in molar Gibbs free energy, which for first-order phase transition is given by the difference in chemical potential between the crystallizing substance in the crystal and in the solution phase, and it is common practice to use the supersaturation as the driving force for the process. Labile (unstable) and metastable supersaturation refers to supersaturated solutions in which the spontaneous deposition of the solid phase, in the absence of crystallizing solid material, will or will not occur, respectively [1]. A supersaturated solution is not in equilibrium. In order to relieve the supersaturation and move towards equilibrium, the solution crystallizes. Once crystallization starts, the supersaturation can be relieved by a combination of nucleation and crystal growth. In any crystallization process, supersaturation is the necessary driving force for crystal formation from solution, but at the same time, nuclei are necessary for the deposition of solute material on the crystal lattice surface (growth). The process of nucleation involves the formation of new crystals in a crystallizing environment. The next stage of the crystallization process is for these nuclei to grow larger by the addition of solute molecules from the supersaturated solution. This process is called crystal growth [1, 2]. Crystallization is applied in continuous operation on a large scale, for the bulk production of inorganic materials like potassium chloride (a fertilizer), sodium


chloride, and ammonium sulphate (a fertilizer), and organic materials like adipic acid (a raw material for nylon), paraxylene (a raw material for polyester), and pentaerythritol (used for coatings) [1]. 2.2 Mathematical Model A typical mathematical model of a crystallizer consists of the population balance equation for crystals and the balance equations for solvent and crystallizing substance, the enthalpy balance, and the equations describing the variation of the equilibrium saturation concentration. The population balance is in fact a distributed mass balance for a solid or dispersed phase, and is linked to the liquid or continuous phase component mass balances via the crystallization kinetics [3]. Consider a simple isothermal crystallizer. Under the assumptions of isothermal operation, constant volume, mixed suspension, nucleation of crystals of infinitesimal size, and mixed product removal, the following dynamical model consisting of a population balance for the particle phase and a mass balance for the solute concentration was presented in Ref. [9],

n G (t ) n n     ( r  0) B (t ) t r  (1) dc ( c0   ) (   c ) (   c ) d     dt dt    where n(r, t) is the number density of crystals of radius r at time t in the suspension, τ is the residence time, c is the solute concentration in the crystallizer, c0 is the solute concentration in the feed, ε is the volume of liquid per unit volume of suspension, G(t) is the growth rate, δ(r−0) is the standard Dirac function, ρ is the density of crystals and B(t) is the nucleation rate. The last term in first equation in model (1) accounts for the production of crystals of infinitesimal (zero) size via nucleation. Rates G(t) and B(t) are assumed to follow McCabe’s growth law and Volmer’s nucleation law, respectively, i.e.,

G (t )  k1 (c  cs ) B (t )   k2 exp 

 k3 (c / cs  1) 2



(2)

47

where k1, k2, and k3 are positive constants and cs is the concentration of the solute at saturation. The population balance model is not appropriate for synthesizing simple model-based, low-order feedback control laws due to its distributed parameter nature. Following the moments reduction technique in Ref. [3], the following reduced-order moments model is obtained,

   k3 d 0   4    0  1  3  k2 exp  2  dt   3   (c / cs  1)  d 1    1  vk1 (c  cs ) 0 dt  (3) d 2    2  vk1 (c  cs ) 1 dt  d 3 3    vk1 (c  cs ) 2 dt  dc 1 (u  c  4 k1 (c  cs ) 2 (   c))  dc  4   1  3   3  where μi(i = 0, 1, 2, 3) are dimensionless moments of the crystal size distribution, c is the dimensionless concentration of the solute in the crystallizer and u is a dimensionless concentration of the solute in the feed. The stability of an open loop continuous crystallizer has been the subject of many studies [9]. For model (3), it has been proved that the stationary state of this system is always unique, but it exhibits sustained oscillations in a certain range of kinetic and process parameters. The oscillatory behavior is the result of the interplay between growth and nucleation caused by the relative nonlinearity of the nucleation rate as compared to the growth rate. However, the crystallizer is often characterized by high average performance in the regions of sustained oscillations thus it seems to be advisable to operate the crystallizer in such operating states, simultaneously controlling the oscillations of the system appropriately. Sustained oscillations in crystallization processes lead to variations of the CSD and the mean particle size over time. Since mean particle size and CSD are key factors to the product quality, cost intensive reworking of the product is often necessary. The dynamics in the CSD can be also a

48


result of disturbances (feed changes, blockages, utility failures), and changes in operating conditions (start-up, shutdown, grade changes).

3. Robust Crystallizers

Control

of

Continuous

3.1 Control Problem Control of crystallization processes has been an area of active research in the last two decades [8, 10]. From the point of view of controlling a crystallizer the main quality criteria are the properties of the produced crystals, first of all the PSD and the mean size. However, due to the complexity of the measurement and control of the PSD, most authors have been addressed the control of crystallization processes via the regulation of related variables to the PSD such as the second and third moment, solute concentration and crystallizer temperature. Here, the control problem consists of the suppression of nonlinear dynamics of the output solute concentration y = c to a desired reference yref (t) = cref via manipulation of the control input u = c0. The control problem description is completed by the following assumptions: A1—The measurement of the variable to be controlled y is available for control design purposes; A2—Nonlinear functions are uncertain, and can be available rough estimates of these terms; A3—The reduced approximation model (3) is affected by unmodeled dynamics ψ(y, z) and external perturbations Ԅ(t). 3.2 Modeling Error Compensation Approach Sun et al. in Ref. [14] proposed a robust controller design method for single-input/single-output (SISO) minimum-phase linear systems. The design approach consists of a modeling error compensator (MEC). The central idea is to compensate the error due to uncertainty by determining the modeling error via plant input and output signals and uses this information in the design. In addition to a nominal feedback, another

feedback loop is introduced using the modeling error and this feedback action is explicitly proportional to the parametric error which is the source of uncertainty. The MEC approach was extended for a class of linear time-varying and nonlinear linearizable lumped parameter systems with uncertain and unknown terms by Alvarez-Ramirez [12], where instead of designing a robust state feedback to dominate the uncertain term, the uncertain term is viewed as an extra state that is estimated using a high-gain observer. The estimation of the uncertain term gives the control system some degree of adaptability. The extension of the MEC approach to distributed parameter systems has been applied by Puebla [15] and Puebla et al. [16] for a class of biological distributed parameter systems. The underlying idea behind MEC control designs is to lump the input-output uncertainties into a term, which is estimated and compensated via a suitable algorithm. For control design purposes, the class of continuous crystallizers described in Section 2 can be written as, dy  f1 ( y , z )  g ( y , z ) u   ( y , z )   ( t ) dt (4) dy  f2 ( y, z ) dt where f1(y, z)ℝ , f2(y, z)ℝ n-1 and g(y, z)ℝ are smooth functions of their arguments, y is the measured and controlled variable, z is the internal state, and u is the control input. Define the modeling error function η as, (5)   f 1 ( y , z )   ( y , z )   (t ) System (3) can be written as,

dy    g ( y , z )u dt

(6)

Let e = y – yref be the regulation error and consider the inverse dynamics control law,

u   g ( y, z )1 ( 

dyref dt

  c 1e)

(7)

where τc> 0 is a closed-loop time constant. Under the inverse-dynamics control law (7), the closed-loop system dynamics is de/dt = −τc−1e, so that the error dynamic behavior is given as e(t) = e(0)exp(−t/τc). In this way, the asymptotic convergence e(t) → 0, and so

49


y → yref , is guaranteed. In order to implement the control input an estimate of the real uncertain term is computed using a high-gain reduced-order observer, which after some direct algebraic manipulations can be written as, dw   g ( y , z )u   (8) dt    e 1 ( w  e )

the sign function and this structure provides several advantages as simplification of the control law, higher accuracy and chattering prevention [11, 13, 17]. In this section we present some ideas of the integral high order sliding mode control (IHOSMC).

where τe> 0 is the estimation time constant.The final form of the controller is given by the feedback function (7) and the modeling error estimator (8). The resulting feedback controlled depends only on the measure y and on estimated values of uncertain term g (y, z). The above model-based control approach has only two control design parameters, i.e., τc and τe. The closed-loop parameter τc can be chosen as the inverse of the dominant frequency of the open-loop dynamics. On the other hand, the estimation parameter τe > 0, which determines the smoothness of the modeling error, can be chosen as τe < 1/2τc.

controlled to move along the sliding surface (sliding

Sliding mode control design consists of two phases. In the first phase the sliding surface is to be reached (reaching mode), while in the second the system is mode). In fact, these two phases can be designed independently from each other. Reaching the sliding surface can be realized by appropriate switching elements [11]. Defining,

 (e)  e  y  yref

(9)

as the sliding surface, we have that the continuous part of the sliding mode controller is given by, dy ueq   g ( y , z ) 1 ( f1 ( y , z )   ( y , z )   (t )  ref ) (10) dt Once on the surface, the dynamic response of the

3.3 High-Order Sliding Mode Control

system is governed by de/dt = 0. To force the system

Sliding mode control techniques have long been recognized as a powerful robust control method in Refs. [11, 13, 17]. Sliding-mode control schemes have shown several advantages like allowing the presence of matched model uncertainties and convergence speed over others existing techniques as Lyapunov-based techniques, feedback linearization and extended linearization. However, standard sliding-mode controllers have some drawbacks: the closed-loop trajectory of the designed solution is not robust even with respect to the matched disturbances on a time interval preceding the sliding motion, the classical sliding-mode controllers are robust in the case of matched disturbances only, the designed controller ensures the optimality only after the entrance point into the sliding mode. To try to avoid the above a relatively new kind of sliding-mode structures have been proposed as the named high-order sliding-mode technique, these techniques consider a fractional power of the absolute value of the tracking error coupled with

trajectory to converge to the sliding surface in the presence of both model uncertainties and disturbances, with chattering minimization and finite-time convergence, the sliding trajectory is proposed as [11, 18], t

udis   g ( y , z ) 1[1e   2  sign(e) e 0

1

p

d ] (11)

where δ1 and δ2 are control design parameters. The final IHOSMC is given by,

u  ueq  udis

(12)

The synthesis of the above control law requires accurate knowledge of terms f1, Ԅ(y, z), ψ(t) and dyref/dt to be realizable. To enhance the robust performance of the above control laws, the uncertain terms are lumped in single terms and compensated with a reduced-order observer. However, by exploiting the properties of the sliding part of the sliding-mode type controllers to compensate uncertain nonlinear terms, the knowledge of the unknown terms f1, ψ(y, z), φ(t) can be avoided.

50


Summarizing, the IHOSMC is composed by a proportional action, which has stabilizing effects on the control performance, and a high order sliding surface, which compensates the uncertain nonlinear terms to provide robustness to the closed-loop system. This behavior is exhibited because, once on the sliding surface, system trajectories remain on that surface, so the sliding condition is taken and make the surface and invariant set. This implies that some disturbances or dynamic uncertainties can be compensated while still keeping the surface an invariant set.

0.04

0.035

0.03

x1 0.025

0.02

0.015 0.7 0.65

0.019 0.018 0.6

0.017 0.016 0.55

0.015 0.014 0.5

0.013

y

Fig. 1

x2

Phase-plane of model (3).

0.8

4. Results and Discussion

0.7

4.1 Linear Stability Analysis In order to gain some insights about the stabilization properties of nonlinear model (3), the linearization of model (3) is determined and its stability is evaluated with the eigenvalues of matrix A of the linear model [19]. Matrix A of the linearized model (3), in the single steady-state of model (3), x= (0.065 0.040 0.024 0.015 0.612), is given by,

0 0  0.06 1.71   1 0.61  1 0 0 0.06   A   0 0.61 0 0.04  1   0 0.61 0.02  1  0  0 0  24.47  0.17  1.96 with eigenvalues, -2.58 ± 1.42i, 0.1073 ± 1.49i, -1. Thus, the local stability analysis shows that the unique equilibrium point is unstable. 4.2 Non-linear Stability Analysis Nonlinear stability analysis was performed with the nonlinear model (3) via both phase-plane and bifurcation diagrams. Phase-plan analysis is presented in Fig. 1. It can be seen that a periodic unstable orbit converges to a stable cycle limit. The bifurcation diagram using as a bifurcation parameter the control input u = c0 is presented in Fig. 2. The bifurcation diagram shows a pitchfork bifurcation for u > -0.35.

0.6

0.5

y

0.4

0.3

0.2

0.1

0

-1

-0.8

-0.6

-0.4

-0.2

0 u

0.2

0.4

0.6

0.8

1

Fig. 2 Bifurcation diagram of the model (3) for bifurcation parameter u.

4.3 Suppression of Nonlinear Dynamics Consider the crystallizer model (3) with the MEC scheme (7) and (8) and the high-order sliding mode control (12). The control objective is the suppression of the natural oscillatory behavior to the open-loop unstable equilibrium state of solute concentration, i.e., yref = 0.612. Control laws are turned on at t = 40 time units. We have set the controller parameters as τc = 0.5 and τe = 0.1 for the MEC control and δ1 = 0.5, δ2 = 0.5, p = 3 for the sliding mode control. Initial conditions are set as xi (0) = (0.066 0.041 0.025 0.015 0.056)T. Figs. 3 and 4 show the control performance for both control approaches. It is noted that both controllers can successfully suppress the oscillatory behavior. Figs. 3 and 4 show that closed-loop system provides a stable transition from the open-loop oscillations to the desired reference. It can be seen from above figures that in order to obtain the desired steady-state, the control

Controlled variable, y


0.65

0.6

0.55

0.5 0

10

20

30

10

20

30

40

50

60

70

80

40

50

60

70

80

Control input, u

0.4

0.3

0.2

0.1 0

Time

Fig. 3

MEC control performance.

Controlled variable, y

0.7

0.65

0.6

Control input, u

In order to analyze zero dynamics we have used results from Maya-Yescas and Aguilar [20] that are based on the convergence to zero of time-derivatives of uncontrolled states for affine systems with relative-degree one. Fig. 5 shows the time-derivatives of uncontrolled states. It can be seen that the time-derivatives converges to zero such shows that if we stabilize the solute concentration of the crystallizer (3) then the overall system will be stable.

5. Conclusions

0.55

0.5 0 0.4

10

20

30

10

20

30

40

50

60

70

80

40

50

60

70

80

0.3

0.2

0.1

0 0

Time

Fig. 4 IHOSMC control performance. 0.2

dx /dt 1

0.15

dx /dt 2

dx /dt 3

Time derivatives, dx/dt

51

dx /dt 4

0.1

0.05

0

Two practical robust controllers are designed for a non-lineal model of a continuous crystallizer that displays oscillatory behavior. The oscillatory behavior is a consequence of nonlinear terms, feedback loops, and the continuous interchange of mass in the crystallizer unit. Linear and nonlinear analysis of stability is used to investigate open-loop system properties. Zero dynamics is also analyzed to investigate closed-loop stability properties. The robust controllers using output feedback of the solute concentration and an estimate of nonlinear terms is able to suppress the nonlinear dynamics of the continuous crystallizer.

-0.05

Acknowledgments

-0.1

Victoria Gámez-García acknowledges financial support from CONACyT through a scholarship grant.

-0.15

0

Fig. 5

1

2

3 Time

4

5

6 4 x 10

Zero-dynamics of the closed-loop system.

input reaches a moderate peak with respect to the nominal value and after some transient behavior the control input converges near of the original nominal value of the control input. 4.4 Zero-Dynamics Zero dynamics of nonlinear systems indicates the stability properties of the closed system when the output is forced to be zero. Thus, it is essential in order to assure that uncontrolled states go to a stable behavior.

References [1] [2] [3] [4]

[5]

[6]

A.G. Jones, Crystallization Process Systems, Butteroworth-Heinemann, Woburn, Massachusetts, 2002. J.W. Mullin, Cristallization, Butterworth-Heinemann, Englewood Cliffs and New Jersey, 2001. A.D. Randolph, M.A. Larson, Theory of Particulate Process, Academic Press, San Diego, 1988. J.D. Ward, C.C. Yu, Population balance modeling in Simulink: PCSS, Computers and Chemical Engineering 32 (2008) 2233-2242. S.J. Lei, R. Shinnar, S. Katz, The stability and dynamic behavior of a continuous crystallizer with fines trap, AIChE J. 17 (1971) 1459-1470. G.R. Jerauld, Y. Vasatis, M.F. Doherty, Simple conditions

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[7]

[8] [9]

[10]

[11]

[12]

[13]

Controlling Nonlinear Dynamics in Continuous Crystallizers for the appearance of sustained oscillations in continuous crystallizers, Chemical Engineering Science 38 (1983) 1675-1681. P.K. Pathath, A. Kienle, A numerical bifurcation analysis of nonlinear oscillations in crystallization processes, Chemical Engineering Science 57 (2002) 4391-4399. P.D. Christofides, Model-based Control of Particulates, Kluwer, Dordrecht, The Netherlands, 2002. D. Shi, N.H. El-Farra, M. Li, P. Mhaskar, P.D. Christofides, Predictive control of particle size distribution in particulate processes, Chemical Engineering Science 61 (2006) 268-281. J.B. Rawlings, S.M. Miller, W.R. Witkowski, Model identification and control of solution crystallization process, Ind. Eng. Chem. Res. 32 (1993) 1275-1296. K.M. Hangos, J. Bokor, G. Szederkényi, Analysis and Control of Nonlinear Process Systems, Springer-Verlag, London, 2004. J. Alvarez-Ramirez, Adaptive control of feedback linearizable systems: A modeling error compensation approach, Int. J. Robust Nonlinear Control 9 (1999) 361-371. A. Levant, Universal single-input-single-output sliding mode controllers with finite-time convergence, IEEE

Automatic Control 46 (2001) 1447-1451. [14] J. Sun, A.W. Olbrot, M.P. Polis, Robust stabilization and robust performance using model reference control and modeling error compensation, IEEE Transactions Automatic Control 39 (1994) 630-635. [15] H. Puebla, Controlling intracellular calcium oscillations and waves, J. Biol. Sys. 13 (2005) 173. [16] H. Puebla, R. Martin, J. Alvarez-Ramirez, R. Aguilar-Lopez, Controlling nonlinear waves in excitable media, Chaos Solitons Fractals 39 (2009) 971-980. [17] H. Sira-Ramirez, Dynamic second order sliding-mode control of the Hovercraft vessel, IEEE Trans. Control Syst. Tech. 10 (2002) 860-865. [18] R. Aguilar-Lopez, R. Martinez-Guerra, H. Puebla, R. Hernandez-Suarez, High order sliding-mode dynamic control for chaotic intracellular calcium oscillations, Nonlinear Analysis-B: Real World Applications 11 (2010) 217-231. [19] B.A. Ogunnaike, W.H. Ray, Process Dynamics, Modeling and Control, Oxford University Press, New York, 2004. [20] R. Maya-Yescas, R. Aguilar, Controllability assessment approach for chemical reactors: Nonlinear control affine systems, Chemical Engineering Journal 92 (2003) 69-79.

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DAVID

PUBLISHING

FPGA Based Wireless Multi-Node Transceiver and Monitoring System Özkan Akin1, İlker Başaran1, Radosveta Sokullu1, İrfan Alan2 and Kemal Büyükkabasakal1 1. Electrical & Electronics Engineering Faculty, Ege University, Izmir 35100, Turkey 2. Electrical & Electronics Engineering Department, Engineering Faculty, Abdullah Gul University, Kayseri, 38039, Turkey Received: June 17, 2011 / Accepted: September 12, 2011 / Published: January 25, 2012. Abstract: In recent years the variety and complexity of Wireless Sensor Network (WSN) applications, the nodes and the functions they are expected to perform have increased immensely. This poses the question of reducing the time from initial design of WSN applications to their implementation as a major research topic. RF communication programs for WSN nodes are generally written on microcontroller units (MCUs) for universal asynchronous receiver/transmitter (UART) data communication, however nowadays radio frequency (RF) designs based on field-programmable gate array (FPGA) have emerged as a very powerful alternative, due to their parallel data processing ability and software reconfigurability. In this paper, the authors present a prototype of a flexible multi-node transceiver and monitoring system. The prototype is designed for time-critical applications and can be also reconfigured for other applications like event tracking. The processing power of FPGA is combined with a simple communication protocol. The system consists of three major parts: wireless nodes, the FPGA and display used for visualization of data processing. The transmission protocol is based on preamble and synchronous data transmission, where the receiver adjusts the receiving baud rate in the range from min. 300 to max. 2400 bps. The most important contribution of this work is using the virtual PicoBlaze Soft-Core Processor for controlling the data transmission through the RF modules. The proposed system has been evaluated based on logic utilization, in terms of the number of slice flip flops, the number of 4 input LUTs (Look-Up Tables) and the number of bonded IOBs (Input Output Blocks). The results for capacity usage are very promising as compared to other similar research. Key words: FPGA, PicoBlaze, RF, multi-node, wireless, transceiver.

1. Introduction RF communication programs for wireless sensor nodes are generally written on microcontroller units (MCUs) for UART data communication, but the data processing capacity and the number of I/O ports of MCUs are limited, so they are not convenient for simultaneous multiple parallel data processing that would be needed in simultaneous multi-node wireless communication and monitoring. Nowadays, FPGA based designs present powerful alternative to MCU designs due to their parallel data processing capacities and software reconfigurable structures. Nowadays, as the single chip capacity of an Corresponding author: Özkan Akin, M.Sc., research assistant, research fields: field programmable gate arrays, power electronics, electrical machines. E-mail: [email protected].

FPGA that accommodates desirable properties increases, its power consumption decreases. Because of these features, high performance FPGA based systems have simple circuits, flexible designs and reconfigurable structures. Besides, depending on the capacity of the I/O ports of an FPGA selected, the communication can be established simultaneously with wireless multi-nodes which are connected to a single FPGA chip in parallel. This provides more processing capacity with lower power consumption. PicoBlaze is an efficient, compact, capable, and cost-effective fully embedded 8-bit reduced instruction set computing (RISC) microcontroller core which can be synthesized in Spartan 3 FPGAs (also in Virtex II and Virtex-4) and can be used for processing data in the implementation of a flexible multi-node transceiver

54

FPGA Based Wireless Multi-Node Transceiver and Monitoring System

and monitoring system for wireless sensor nodes. PicoBlaze is similar to many microcontroller architectures but it is specifically designed and optimized for Xilinx FPGAs. PicoBlaze provides cost-effective microcontroller based control and simple data processing and consumes considerably less resources than a comparable 8-bit microcontroller architectures. It occupies just 96 FPGA slices and delivers 44 to 100 million instructions per second (MIPS) depending on the target FPGA family and the speed grade–which is many times faster than commercially available microcontroller devices [1]. In this paper, the authors present a prototype of flexible multi-node transceiver and monitoring system. The prototype is designed for time-critical applications and can be also reconfigured for other applications like event tracking. From here on, the paper is organized as follows: in Section 2 an overview of the related work is presented; Section 3 gives details on the architecture of the proposed system; in Section 4 experimental results are discussed followed by conclusion and future work in Section 5.

2. Previous Work In recent years the variety and complexity of Wireless Sensor Network (WSN) applications, the nodes and the functions they are expected to perform have greatly increased. This poses the question of reducing the time from initial design of WSN applications to their implementation as a major research topic. In this respect, the power of FPGA as a system design has become even more important. So there are a number of studies in this direction, which examine the different aspects of integrating FPGA into WSN applications. In Ref. [2] the authors define an FPGA based system for wireless communication. The main objective is to provide a detailed control of the nRF2401 RF chip using FPGA. As a result, the design of a low price, simple and easy to control NIOS II system embedded in FPGA is proposed which can be adapted to monitoring WSN.

In Ref. [3] the authors focus on a strategy to design a hardware platform for enabling research in massive wireless sensor networks for ambient systems. A prototype modular sensor node with 25mm form factor is described. This miniature module comprises 4 different layers: FPGA layer, communication layer, interface and power layers. The work in Ref. [4] presents a wireless image processing sensor embedded in FPGA including a NIOS II processor and a microcontroller. Using the capacity of FPGA for image processing, different experiments have been carried out and the results are presented in the paper. In most works so far, the reconfigurability advantage of an FPGA is generally not considered. The conventional wireless multi-node monitoring platforms include a microcontroller and an FPGA to implement all the processes [3-7]. In some studies, instead of using both a microcontroller and an FPGA, a single FPGA chip has been used [2, 8, 9] and they stress on the benefits and advantages of the 32-bit NIOS II soft core processor. Furthermore, in Ref. [10] the authors provide some insides on achieving better power consumption performance using FPGA. The aim of this project is to demonstrate the feasibility of using PicoBlaze soft core instead of NIOS II, where PicoBlaze provides greater programming simplicity and easiness, and requires much less resources because it is an 8-bit platform. Furthermore, a virtual PicoBlaze soft core processor is used inside the FPGA in this project instead of using classical independent microcontrollers used in the works mentioned above.

3. Architecture of the Proposed System The overall system consists of three major parts. These are: wireless nodes, the FPGA and display used for visualization of data processing (Fig. 1). The number of wireless nodes that can be connected depends on the available I/O ports of the FPGA. Each node is communicating with a single dedicated PicoBlaze


Fig. 1 System overview block diagram; system consists of many PicoBlaze cores that run simultaneously and each one is connected to a transceiver module while one drives the display.

core embedded in the FPGA platform. Simultaneous data processing from all nodes is done in parallel by the FPGA which allows intensive digital signal processing (DSP) tasks to be implemented in time critical applications. 3.1 RF Modules In this work 433.920 MHz UHF band, ARX 34-X and ATX 34-S RF (Fig. 2, (1, 3)) receiver and transmitter chips have been used. They use ASK (Amplitude-Shift Keying) modulation technique in a low data rate for short range industrial, science and medical (ISM) band. The transmission protocol is based on preamble and synchronous data transmission. The receiver adjusts the receiving baud rate in the range from min. 300 to max. 2400 bps. The frame format is defined as: preamble + sencron + data1+.....+dataX, where the preamble is 5 bytes of 0xAA or 0x55. Synchronization is 5 bytes 0x00 or 5 bytes 0xFF. The algorithm for start/stop has been embedded in software in the PicoBlaze. To start receiving data, an acknowledgement in the form of 3 bytes of 0x3A is sent by the receiver side. Both the sender and the receiver side include a simple voltage regulator and logic level max232. (Fig. 2, (2, 4)).

55

Fig. 2 Prototype of wireless transceiver node and transmission protocol; ATX 34-S transmitter chip (1) and transmitter circuit (2), ATX 34-S receiver chip (3) and receiver circuit (4).

3.2 PicoBlaze Soft-Core Processor PicoBlaze is an efficient, compact, capable, and cost-effective fully embedded 8-bit RISC microcontroller core [1]. PicoBlaze soft-core processors are used in the XC3S1200E FPGA chip in this system instead of using classical independent microcontrollers and 32-bit soft-cores like NIOS II which have been used in other researches as mentioned in Section 2. The major advantage of using PicoBlaze is that its code is very small and also the fact that a great number of cores can operate simultaneously in parallel. This processing advantage can be very useful in a number of time critical applications, or WSN applications that require simultaneously processing of the data from a large number of nodes. Another aspect of the FPGA platform that so far has been little touched in relation to WSN applications is its reconfigurability, which can greatly shorten the design-implementation cycle for new applications. The soft-core PicoBlaze prototype discussed in this paper is schematically presented in Fig. 3.

4. Experimental Results Wireless transceiver circuit has analog devices like

56


Table 1 Source code reports results for 1 RF module. Logic utilization Number of slice flip flops Number of 4 input LUTs Number of bonded IOBs

Used 171 278 3

Available 17,344 17,344 250

Utilization 1% 1% 1%

Table 2 Source code reports for 50 PicoBlaze core control modules for RF and LCD screen. Logic utilization Number of slice flip flops Number of 4 input LUTs Number of bonded IOBs

Fig. 3 Block diagram of one PicoBlaze soft-core that includes UART module and LCD interface.

voltage regulator and RF module and no microcontroller. Serial data processing from wireless transceiver nodes is realized in PicoBlaze soft core units embedded in the FPGA. UART communication standard is used for this application. The data is collected from the multiple nodes in parallel into a single FPGA chip by means of many PicoBlaze cores, embedded into the FPGA. Multiple nodes can be monitored simultaneously, not serially. If desired, the data collected is processed at one point with high speed and minimum resources. The prototype is designed to show the feasibility of the system with a liquid crystal display (LCD) and wireless data transceiver nodes. The proposed system has been evaluated based on logic utilization. In Table 1, the results for one RF transceiver module from the FPGA PicoBlaze core source report are presented. These include results for number of slice flip flops, number of 4 input LUTs (Look-Up Tables) and number of bounded IOBs (Input Output Blocks). As it can be seen from these results a max of 62 RF transceivers can be connected. The available pins are also sufficient. If a higher capacity FPGA chip is selected an even higher number of RF transceivers can be simultaneously connected. A certain percent of the resources has to be put aside for the DSP processing in

Used 8,559 13,921 111

Available 17,344 17,344 250

Utilization 49% 80% 44%

the FPGA. That is why, for the XILINX XC3S1200E FPGA that is used in our work, 50 PicoBlaze core control modules for the RF modules have been embedded in the FPGA thus providing around 20% of the resources of the FPGA for DSP. In a similar way, in Table 2 the results from the source report for 50 PicoBlaze core control modules for the RF and one LCD screen have been presented.

5. Conclusions In this paper the authors have presented the initial results from their work related to the use of powerful FPGA platform and the Virtual PicoBlaze soft-core processor for wireless sensor nodes and applications. These preliminary results are very optimistic and in the future the authors plan to test the software with a greater number of RF modules and incorporate more complicated wireless communication protocols. It is also interesting to experiment in real environments, for example in industrial settings for collecting information from a number of nodes used for monitoring a specific production cycle. In such situations both the advantages of FPGA DSP processing power and the adverse effects of the environment (number of machines in a closed area) will be better estimated.

References [1]

Picoblaze 8-bit embedded microcontroller user guide, for spartan-3, virtex-ii and virtex-ii Pro FPGAs. UG29 [online], Xilinx Company, 2008, v1.1.2, http://www-users.york.ac.uk/~cdh3/_dnl/PicoBlaze/ug12


[2]

[3]

[4]

[5]

9.pdf. M. Zhang, H. Zhang, Design and implementation of wireless transceiver system based on FPGA, in: International Conference on Innovative Computing and Communication and Asia-Pacific Conference on Information Technology and Ocean Engineering, 2010, p. 355. S.J. Bellis, K. Delaney, B. O’Flynn, J. Barton, K.M. Razeeb, C. O’Mathuna, Development of field programmable modular wireless sensor network nodes for ambient systems, Computer Communications 28 (2005) 1531-1544. C.H. Zhiyong, L.Y. Pan, Z. Zeng, M.Q.H. Meng, A novel FPGA-based wireless vision sensor node, in: IEEE International Conference on Automation and Logistics, 2009, p.841. B. O’Flynn, S.J. Bellis, K. Mahmood, M. Morris, G. Duffy, K. Delaney, et al., A 3D miniaturised programmable transceiver, Microelectronics International 22 (2005)

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8-12. Y.E. Krasteva, J. Portilla, J.M. Carnicer, E. de la Torre, T. Riesgo, Remote HW-SW reconfigurable wireless sensor nodes, in: 34th IEEE Annual Conference of Industrial Electronics, 2008, p. 2483. [7] J. Portilla, T.R.A. de Castro, A reconfigurable FPGA-based architecture for modular nodes in wireless sensor networks, in: 3rd Southern Conference on Programmable Logic, 2007, p. 203. [8] K. Arshak, E. Jafer, C.S. Ibala, FPGA based system design suitable for wireless health monitoring employing intelligent RF module, IEEE Sensors 10 (2007) 276-279. [9] G. Chalivendra, R. Srinivasan, N.S. Murthy, FPGA based re-configurable wireless sensor network protocol, in: International Conference on Electronic Design, 2008, p. 1. [10] P.P. Czapski, A. Sluzek, Power optimization techniques in FPGA devices: A combination of system-and low-levels, International Journal of Electrical, Computer and Systems Engineering 1 (2007) 148-154.

[6]

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PUBLISHING

Linguo-Combinatorial Simulation of Complex Systems Mikhail B. Ignatyev St-Petersburg State University of Aerospace Instrumentation, St-Petersburg 190000, Russia Received: July 22, 2011 / Accepted: September 25, 2011 / Published: January 25, 2012. Abstract: Contemporary world is developing system and we must have the new models. Any complex system interacts with its changing environment and its viability depends on its adaptability. The number of arbitrary coefficients in the structure of equivalent equations of complex system changes in the process of learning. In systems with more than six variables, the number of arbitrary coefficients increases first, and then, passing through the maximum, begins to decrease. This phenomenon makes it possible to explain the processes of system growth, complication and death in biological, economical and physical-engineering systems. The author uses the Linguo-combinatorial method of investigation of complex systems, in taking key words for building equivalent equations. This phenomenon is able to increase the adaptability of different systems. Key words: Adaptability, combinatorial simulation, uncertainty, appearance, essence, general systems theory, physics, biology, social-economics.

1. Introduction The natural language is the main intellectual product of mankind. The structure of the natural intellect is reflected in natural language that is accessible for investigation. Some scientific experiments can be expensive and dangerous. The simulation techniques permit to decrease the cost for investigating these systems. The simulation must accurately reflect the characteristics of the real world. Combinatorial simulation allows studying the full set of system variants including uncertainty. Any system contains some types of uncertainty, which are determined by their existence in real world. Humans interact with both physical objects and their descriptions in terms of natural language, mathematics or tables. Descriptions often only partially represent the essence of real processes. The inaccuracy of description introduces uncertainty. More often the uncertainty of systems is, however, inherent to the real world. This study is aimed toward such types of uncertainty in mental processes. Corresponding author: Mikhail B. Ignatyev, Dr., professor, research fields: cybernetics, informatics, robotics, simulation & modeling, complex systems. E-mail: [email protected].

Physical laws, the balance of energy and matter, and information limit the systems behavior. Within these limits, systems interact and adapt to other systems and environment, and undergo destructive actions.

2. Linguo-Combinatorial Simulation Frequently we use the natural language to describe systems. We propose to transfer this natural language description to mathematical equations. For example, we have a sentence WORD1 + WORD2 + WORD3

(1)

where we assign words and only imply meaning of words, the meaning (sense) is ordinary implied but not designated. We propose to assign meaning in the following form (WORD1)*(SENSE1) + (WORD2)*(SENSE2) + (WORD3)*(SENSE3) = 0

(2)

This Eq. (2) can be represented in the following form A1*E1 + A2*E2 + A3*E3 = 0

(3)

where Ai, i = 1, 2, 3, will denote words from English Appearance and Ei will denote senses from English Essence. The Eqs. (2) and (3) are the model of the sentence (1). This model is an algebraic ring and we

Linguo-Combinatorial Simulation of Complex Systems

can resolve this equation with respect to the appearances Ai or the essences Ei [1-3]: A1 = U1*E2 + U2*E3 A2 = – U1*E1 + U3*E3

(4)

A3 = – U2*E1 – U3*E2 or E1 = U1*A2 + U2*A3 E2 = – U1*A1 + U3*A3

(5)

E3 = – U2*A1 – U3*A2 where U1, U2, U3 are arbitrary coefficients, which can be used for solution of different tasks on the initial manifold (2) or (3). In general if we have n variables in our system and m manifolds restrictions, then the number of arbitrary coefficients S will be defined as the number of combinations from n to m + 1 [1], as shown in Table 1, m1

S = Cn

,n  m

(6)

The Eq. (6) is the basic law of cybernetics, informatics and synergetics for complex systems. The number of arbitrary coefficients is the measure of uncertainty. Usually, when solving mathematical systems, the number of variables is equal to the number of equations. In practice we frequently do not know how many constraints there are on our variables. Combinatorial simulation makes it possible to simulate and study the systems with uncertainty on the base of incomplete information. The problem of simulation of condition, guaranteeing the existence of maximum adaptability is investigated. Table 1 The number of arbitrary coefficients depending on the number of variables n and the number of restriction m. m N 2 3 4 5 6 7 8 9

1

2

3

4

5

6

7

8

1 3 6 10 15 21 28 36

1 4 10 20 35 56 84

1 5 15 35 70 126

1 6 21 56 126

1 7 28 84

1 8 36

1 9

1

59

It is supposed that the behavior of a system with n variables is given with an accuracy of m intersecting manifolds, n > m. If the system is considered as a multidimensional generator in Fig. 1 where at least a part of the variables interact with environment variables, and if the objective of the system is to decrease the functional of discoordination between them (  1…  k ), the system control unit has two instruments of impact, a and b, upon the system. First, this is the tuning—the changing of uncertain coefficients in the structure of the differential equations of the system, taking account that the greater number of these coefficients implies more accurate system response to changing environment. Second, this is the learning—the imposing new restrictions on the system behavior. The number of arbitrary coefficients, in the structure of equivalent equations, changes in the process of learning, of consecutive imposing new and new restrictions on the system behavior. In the systems with more than six variables the number of arbitrary coefficients increases first, and then, passing through the maximum begins to decrease. This phenomenon makes it possible to explain the processes of system growth, complication and death. The existence of maximum adaptability phenomenon is observed in and proved by numerous biological, economical and physical-engineering systems. Fig. 1 shows the interaction between system and environment. It is important that we describe a system with a full sum of combinations and have all the variants of decisions. The Linguo-combinatorial simulation is a useful heuristic approach for investigation of complex, poorly formalized systems. Natural language is the main intellectual product of mankind; the structure of natural language reflects the structure of natural intellect of mankind and its separate representatives on the level of consciousness and unconscious. Linguo-combinatorial simulation is the calculation, which permits to extract the senses from texts. Morick and Ignatiev wanted to have the calculation of senses [4, 5]. In our calculation we have the three groups of variables: the first group—the words

60


X1

Y1

a Yk

b Xk

1 control unit

k FEEDBACK

E N V I R O N M E N T

SYSTEM Fig. 1 Model of “system-environment”.

of natural language Ai, the second group—the essences Ei, which can be the internal language of brain [6]; we can have the different natural languages, but we have only one internal language of brain; this hypothesis opens a new way for experimental investigation; the third group of variables—the arbitrary coefficients, uncertainty in our model, which we can use for adaptation in translation processes and etc.. Each complex system interacts with environment, which is changing, and the life of complex system depends on the adaptational possibility of our system. The problem of simulation of condition of guarantee to the adaptational maximum is investigated. It is suggested that the behavior of system with n variables is given to an approximation of m intersecting manifolds, n > m. If the system is considered as a multidimentional generator where at least a part of variable interact with environment’s variables, and if the objective of system is to decrease the functional of discoordination between them, the system control unit has two instruments of influence of the system. First, this is the tuning—the change of underdeterminated coefficients in the structure of the differential equations of system taking account that the more is these coefficients the more accurate are the responses of the system to the change of environment. Second, this is the learning—the imposition of new restriction on the systems behavior. The amount of arbitrary coefficients

in the structure of equivalent equations is changing in the process of learning, of consecutive imposition of new and new restrictions on the system behavior. In the systems with the number of variables more than six the amount of arbitrary coefficients increases first and then going through the maximum begins to decrease. This phenomenon permits to explain the processes of growth, complication and death of a system. The existence of adaptational maximum phenomenon is proved by numerous biological, economical and physical-technical systems. We use the Linguo-combinatorial method of investigation of the poorly formalized complex system, then we use the key words for creation of equivalent equations. The study of adaptational phenomenon in complex systems permits to increase the adaptational possibility in different systems and to resolve a lot of paradoxes.

3. Combinatorial Model of Atoms For example we consider the problem of atom simulation. For hydrogen we have the key words Atom + Proton + Electron (7) Then the equivalent equation will be Eqs. (4) and (5), where A1—characteristic of hydrogen atom in particular his spectral characteristic, E1—variation of this characteristic, A2—characteristic of proton, E2—variation of this characteristic, A3—characteristic


of electron, E3—variation of this characteristic. For simulation of deuterium we will have the key words Atom + proton + electron + neutron (8) Then equivalent equations will be E1 = U1*A2 + U2*A3 + U3*A4 E2 = – U1*A1 + U4*A3 + U5*A4 E3 = – U2*A1 – U4*A2 + U6*A4 (9) E4 = – U3*A1 – U5*A2 – U6*A3 where U1, U2, U3, U4, U5, U6—the arbitrary coefficients, A1—characteristic of deuterium atom, E1—variation of this characteristic, A2—characteristic of proton, E2—variation of this characteristic, A3—characteristic of electron, E3—variation of this characteristic, A4—characteristic of neutron, E4—variation of this characteristic. In case of nuclear reaction it is possible to conversion of deuterium in hydrogen by means of transformation of Eq. (10) to Eq. (5). In the same way it is possible to create the combinatorial models of all atoms from Mendeleev table and molecules. The superconductivity zone is the zone of adaptational maximum. It is way for computer simulation & modeling of chemical reactions, anticipation of the new property of substance and nanorobots synthesis [7, 8].

4. Structure of General Model of Organism Now the medical treatment is determination of illness symptom and generation of corresponding actions. This methodology is based on the physician education and support by means of telemedicine. Computer has big possibilities for complex system simulation, but physicians do not use these possibilities now. The main idea of global computer model of organism has three parts: (1) It is necessary to create the integral model of generalized organism of man on the basis of biology and medical science; (2) Physician must have the possibility to tune the generalized model of organism on the concrete

61

parameters of patient; (3) Physician must have the possibility to simulate the different variants of treatment and to select the best treatment way by means of model. Since Aristoteles, there have been a lot of attempts in this direction, but now we have computer for investigations of complex systems and can use the combinatorial simulation method [1, 9-11]. We have different levels of description of organism—organ level, cell level, molecular level, but for physician the organ level is useful and suitable. We can use the traditional system of organs: (1) The system of motion organs (bones, muscles, fasciae); (2) The digestive system; (3) The respiratory system; (4) The urogenital system; (5) The blood vascular and limphatic systems; (6) The central nervous system; (7) The peripheral nervous system; (8) The ductless glands; (9) The skin and sensory organs. We can increase the number of organ systems, but for illustration of our approach we will use nine systems, which interact among themselves. The organism equation will consist nine variables: A1*E1 + A2*E2 + . . . + A9*E9 = 0 (10) where: A1—characteristic of motion organs, E1—variation of this characteristic; A2—characteristic of digestive system, E2—variation of this characteristic; A3—characteristic of respiratory system, E3—variation of this characteristic; A4—characteristic of urogenital system, E4—variation of this characteristic; A5—characteristic of blood vascular and limphatic systems, E5—variation of this characteristic; A6—characteristic of central nervous system, E6—variation of this characteristic; A7—characteristic of peripheral nervous system,

62


E7—variation of this characteristic; A8—characteristic of dustless glands, E8—variation of this characteristic; A9—characteristic of skin and sensory organs, E9—variation of this characteristic. The structure of equivalent equations of organism model will be Eq. (11): Е1 = U1*A2 + U2*A3 + U3*A4 + U4*A5 + U5*A6 + U6*A7 + U7*A8 + U8*A9 E2 = – U1*A1 + U9*A3 + U10*A4 + U11*A5 + U12*A6 + U13*A7 + U14*A8 + U15*A9 E3 = – U2*A1 – U9*A2 + U16*A4 + U17*A5 + U18*A6 + U19*A7 + U20*A8 + U21*A9 E4 = – U3*A1 – U10*A2 – U16*A3 + U22*A5 + U23*A6 + U24*A7 + U25*A8 + U26*A9 (11) E5 = – U4*A1 – U11*A2 – U17*A3 – U22*A4 + U27*A6 + U28*A7 + U29*A8 + U30*A9 E6 = – U5*A1 – U12*A2 – U18*A3 – U23*A4 – U27*A5 + U31*A7+U32*A8 + U33*A9 E7 = – U6*A1 – U13*A2 – U19*A3 –U24*A4 – U28*A5 – U31*A6 + U34*A8 + U35*A9 E8 = – U7*A1 – U14*A2 – U20*A3 – U25*A4 – U29*A5 – U32*A6 – U34*A7 + U36*A9

University 1

E9 = – U8*A1 – U15*A2 – U21*A3 – U26*A4 – U30*A5 – U33*A6 – U35*A7 – U36*A8 where U1, U2, . . . ,U36—the arbitrary coefficients, which can be used for tuning of the model. System of Eq. (11) is full, this system covers all combination of interaction between different organs of organism. In general we have the representative point of organism in parameters space, each organism has the zone of health, where the parameters correspond the health of concrete man. During illness the representative point of organism is found in another zone of parameters—in illness zone. The process of treatment is the movement of the representative point from illness zone to health zone. In our example the equation of illness organism will be (X1 – X10)2 + (X2 – X11)2 + (X3 – X12)2 + (X4 – X13)2 + (X5 – X14)2 + (X6 – X15)2 + (X7 – X16)2 + (X8 – X17)2 + (X9 – X18)2 = (X19)2 (12) where X1, X2, . . , X9—characteristics of health organism, X10, X11, . . , X18—characteristics of illness organism, X19—the distance between health zone and illness zone. For system (12) we can create the

University N

General Model of Organism

Model of Patient 1

Physician M

Model of Patient K

Physician L

Fig. 2 Interaction between scientific organization (University 1,..., University N) and general model of organism.


equivalent equations system and can use the arbitrary coefficients for simulation of physician actions. The physician actions must decrease the variable X19 and return the representative point from illness zone to health zone. Interaction between models of particular patients and physicians is shown in Fig. 2. There is a good experience in creation of the international classification of diseases by means of WHO. It is necessary to organize the next step of scientific cooperation to create of general model of organism and to provide the possibilities of each physician to access the general model of organism by means of telecommunications. It is a very difficult problem, but successful implementation of it can be based on recent simulation technology advances. It is necessary to decrease the number of mistakes made by physicians.

5. Simulation of Solar System For simulation of solar system we can use 10 keys words—Sun, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto. In our equivalent equations system we will have 10 variables—10 appearances, 10 essences and 45 arbitrary coefficients, which are the structured uncertainty: E1 = U1*A2 + U2*A3 +U3*A4 + U4*A5 +U5*A6 + U6*A7 + U7*A8 + U8*A9 + U9*A10 E2 = – U1*A1 + U10*A3 + U11*A4 + U12*A5 + U13*A6 + U14*A7 + U15*A8 + U16*A9 + U17*A10 E3 = – U2*A1 – U10*A2 + U18*A4 + U19*A5 + U20*A6 + U21*A7 + U22*A8 + U23*A9 + U24*A10 E4 = – U3*A1 – U11*A2 – U18*A3 + U25*A5 + U26*A6 + U27*A7* + U28*A8 + U29*A9 + U30*A10 E5 = – U4*A1 – U12*A2 – U19*A3–U25*A4 + U31* A6 + U32*A7 + U33*A8 + U34*A9 + U35*A10 E6 = – U5*A1 – U13*A2 – U20*A3 – U26*A4 –

63

U31*A5 + U36*A7 + U37*A8 + U38*A9 + U39*A10 E7 = – U6*A1 – U14*A2 – U21*A3 – U27*A4 – U32*A5 – U36*A6 + U40*A8 + U41*A9 + U42*A10 E8 = – U7*A1 – U15*A2 – U22*A3 – U28*A4 – U33*A5 – U37*A6 – U40*A7 + U43*A9 + U44*A10 E9 = – U8*A1 – U16*A2 – U23*A3 – U29*A4 – U34*A5 – U38*A6 – U41*A7 – U43*A8 + U45*A10 E10 = – U9*A1 – U17*A2 – U24*A3 – U30*A4 – U35*A5 – U39*A6 – U42*A7 – U44*A8 – U45*A9 In this system of equations A1—Sun characteristic, E1—variations of this characteristic, A2—characteristic of Mercury, E2—variations of this characteristic,..., U1, U2…U45—arbitrary coefficients, which carry the control possibility. In the same way we can create the equivalent equations system for Galaxy and Universe. Ordinary Newton laws had been used for decision of direct task about the planet movements, but when we decide the reverse task, we have a lot variants of decision by means of arbitrary coefficients. Linguo-combinatorial simulation is the way for everything simulation. The structure of this model consists of 3 blocks of variables: Appearances block, Essences block and block of the structural uncertainty. The structural uncertainty is the substance U, which is apeiron, which penetrate to different structures—atoms, minds, organisms, planets, galaxy etc.. Apeiron hypothesis was expressed by Anaximander a lot of years ago. This hypothesis together the dark energy and the dark matter investigation permit to create the hierarchical control system of Universe [8, 11].

6. Structure of General Model of City If we have the key words—Population, Passionarity, Territory, Production, Ecology and Safety, Finance and External Relation for simulation of city [12], then the equivalent equation of our model will be

64


this characteristics, A2—a characteristics of “passionarity”, intentions of social groups of population, E2—a variation of this characteristics, A3—a characteristics of territory, E3—a variation of this characteristics, A4—a characteristics of production (industrial, agricultural, science, service etc.), E4—a variation of this characteristics, A5—a characteristics of ecology and safety, E5—a variation of this characteristics, A6—a characteristics of finance, banking, individual finance etc., E6—a variation of this characteristics, A7—a characteristics of external relation, input and output flows of material, energy, information, finance, population, E7—a variation of this characteristics, U1, U2, …, U21—arbitrary coefficients, which compose the block of control in our city structure (Fig. 3).

E1 = U1*A2+ U2*A3 + U3*A4 + U4*A5 + U5*A6 + U6*A7 E2 = – U1*A1 + U7*A3 + U8*A4 + U9*A5 + U10*A6 + U11*A7 E3 = – U2*A1 – U7*A2 + U12*A4 + U13*A5 + U14*A6 + U15*A7 E4 = U3*A1 – U8*A2 – U12*A3 + U16*A5 + U17*A6 + U18*A7 (13) E5 = – U4*A1 – U9*A2 – U13*A3 – U16*A4 + U19*A6 + U20*A7 E6 = – U5*A1 – U10*A2 – U14*A3 – U17*A4 – U19*A5 + U21*A7 E7 = – U6*A1 – U11*A2 – U15*A3 – U18*A4 – U20*A5 – U21*A6 where A1 is a characteristics of population (health, education, employment and etc.), E1—a variation of Feedback Town-Hall,

Town mapping

decision makers

in the large

Interaction between blocks Model of town INFORMATION REALTOWN Fig. 3 Simulation of a town for decisions making support.

RELATIONS

EXTERNAL

FINANCE

ECOLOGY

TERRITORY

PASSIONARITY

POPULATION

INFORMATION

DECISIONS

PRODUCTION

Consequence of decision

Project of decision

65


By means of these models it is possible to resolve the paradox in urban development and social interaction.

7. Maximum Adaptability Phenomenon The equivalent equations of any system contain arbitrary coefficients, which can be used for controlling it. The control may be internal or external. The behavior of any system with an environment contact will be determined by means of Eq. (6), which is the main law of cybernetics. Each organism has a maximum adaptability zone. Table 2 shows the mortality depending on the age as a result of the census in Russia in different times. The

minimum of mortality is observed within 10-14 ages in different historical periods. The minimum of mortality is identified with the maximum adaptability. Having passed through the maximum adaptability zone, the organism has got the possibility of reproduction. Fig. 4 shows the evolution of system, the cycle of development begins in point 1, passes the maximum of the arbitrary coefficients number, and finishes in point 2, where the system must have the transformation, forgetting old restrictions, after new cycle begin in point 3 and etc.. Between points 2-3 and 4-5 we have the creative processes. Between points 1-2, 3-4 and 5-6 we have the adaptation processes. Maximum adaptability

Table 2 The mortality depending on the age as a result of the census in Russia in different times. Years Ages 0-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54

1896-1897

1958-1959

1969-1970

1978-1980

1982-1983

1984-1985

133.0 12.9 5.4 5.8 7.6 8.2 8.7 10.3 11.8 15.7 18.5

11.9 1.1 0.8 1.3 1.8 2.2 2.6 3.1 4.0 5.4 7.9

6.9 0.7 0.6 1.0 1.6 2.2 2.8 3.7 4.7 6.0 8.7

8.1 0.7 0.5 1.0 1.7 2.3 2.9 4.3 5.4 7.8 10.3

7.9 0.6 0.5 1.0 1.6 2.2 2.9 3.8 5.6 7.4 10.9

7.7 0.6 0.5 0.9 1.5 2.0 2.8 3.6 5.7 7.3 11.3

S n3

t n2 5

6

n1 1

4

3 2

m

Fig. 4 Transformation of developing system, n1 < n2 < n3, trajectory of system: 1-2-3-4-5-6-…, dotted lines–creative processes, compact lines–evolutionary processes.

66


phenomenon makes it possible to explain different cycles in biological and socio-economical systems, for example, Kondratiev cycles. Each enterprise must be within maximum adaptability zone if we would like to retain this enterprise in changes flow. The sustainable development of systems can be only within maximum adaptability zone. The sustainable thermonuclear reaction is possible only within this zone. For retaining the system within maximum adaptability zone, we have the different instruments—increasing the variables number, imposing new restrictions or removing the old ones etc.. For example, we can joint different systems in an integral system to increase or decrease the adaptability of systems. So, from the two following systems m11

S1 = Cn1

and

m 21

S2 = Cn 2

(14)

we can join them in imposing new restrictions, Scol, in view of obtaining the new collective system m1m 2 mcol

Scol = Cn1n 2

(15)

where the adaptability of this new system can be either Scol > S1 + S2 or Scol < S1+S2 depending upon concrete parameters. We can only see the collective, total effect.

8. Conclusions The combinatorial simulation is a universal method for simulation and modeling. With it, it is possible to create a new model in different areas—in physics, chemistry, biology, psychology, etc.. The linguistic basement of the simulation determines the universality of this method: the natural language is the universal sign system and the Linguo-combinatorial simulation is thus the simulation method, perhaps, of everything. We have tried to show different levels of models. For reliability, each system must be then within maximum

adaptability zone. It is necessary to carry out the verification of these models, but their structure is interesting for understanding complex systems.

References [1]

M.B. Ignatiev, F.M. Kulakov, A.M. Pokrovskij, Robots-manipulators control algorithms, Maschinostroenie, 1972, p. 248. [2] M.B. Ignatiev, Simulation of adaptational maximim phenomenon in developing systems, in: Proceedings of the SIMTEC 93-1993 International Simulation Technology Conference, San Francisco, USA, 1993. [3] M.B. Ignatyev, D.M. Makina, N.N. Petrischev, I.V. Poliakov, E.V. Ulrich, A.V. Gubin, Global model of organism for decision making support, in: Proceedings of the High Performance Computing Symposium, A. Tentner (Ed.), Advanced Simulation Technologies Conference, Washington D.C., USA, 2000, pp. 66-71. [4] Wittgenstein and the Problem of Other Mind, H. Morick (Ed.), McGraw Hill, N.Y., 1967. [5] M.B. Ignatiev, Golonomical Automatic Systems, Monograph, Publ. AN USSR, Moscow-Leningrad, 1963, p. 204. [6] A. Sanctus, Opera Omnia. T., 1864, pp. 1-11. [7] M.B. Ignatyev, Information technology in micro-nano-and optoelectronics, Monograph, St-Petersburg, 2008, p. 200. [8] M.B. Ignatyev, Necessary and sufficient conditions of nanorobot synthesis, Dokladi Akademii Nauk 433 (2010) 613-617. [9] M.B. Ignatyev, Linguo-combinatorial method for complex systems simulation, in: Proceedings of the 6th World Multiconference on Systemics, Cybernetics and Informatics, Computer Science II, Orlando, USA, 2002, pp. 224-227. [10] M.B. Ignatyev, Seven-block model of city for decisions making support, in: Proceedings of the Seminar Computer Models of Urban Development, St-Petersburg, Russia, 2003, pp. 40-45. [11] M.B. Ignatyev, The study of adaptation phenomenon in complex systems, in: AIP Conference Proceedings, Melville, New York, 2006, pp. 322-330. [12] M.B. Ignatyev, Cybernetical picture of world, Monograph, St-Petersburg, 2010, p. 416.

FPGA Based Wireless Multi-Node Transceiver and

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