International Conference on Information Technologies (InfoTech-2010) 16 th – 17 th September 2010 Varna – St. St. Constantine and Elena resort, Bulgaria The forum is organized in the frame of “Dais of the Science of the Technical University-Sofia, 2010” and unites the joint events: 24 th International Conference on Systems for Automation of Engineering and Research (SAER-2010) 6 th International Workshop on Technological Aspects of e-Governance and Data Protection (eG&DP-2010) 2 nd International Seminar with Discussion on Security Policy (Security-2010) PROCEEDINGS Edited by Prof. Dr. Radi Romansky Sofia, 2010
Transcript
International Conference on Information Technologies
(InfoTech-2010)
16th – 17th September 2010 Varna – St. St. Constantine and Elena resort, Bulgaria
The forum is organized in the frame of “Dais of the Science of the Technical University-Sofia, 2010”
and unites the joint events:
24th International Conference on Systems for Automation of Engineering and Research
(SAER-2010)
6th International Workshop on Technological Aspects of e-Governance and Data Protection
(eG&DP-2010)
2nd International Seminar with Discussion on Security Policy (Security-2010)
PROCEEDINGS
Edited by Prof. Dr. Radi Romansky
Sofia, 2010
PROCEEDINGS of the Int’l Conference InfoTech-2010 4
International Program Committee
Luís BARROSO (Portugal) Dencho BATANOV (Cyprus) Francesco BERGADANO (Italy) Pino CABALLERO-GIL (Spain) Ed F. DEPRETTERE (The Netherlands) Vassil FOURNADJIEV (Ghana) Georgi GAYDADJIEV (The Netherlands) Iliya GEORGIEV (USA) Adam GRZECH (Poland) Luis HERNANDEZ-ENCINAS (Spain) Ivan JELINEK (Czech Republic) Karl O. JONES (UK) Nikola KASABOV (New Zealand) Nikola KLEM (Serbia) Todor KOBUROV (Bulgaria) Emil KONSTANTINOV (Bulgaria)
Oleg KRAVETS (Russia) Gwendal LE GRAND (France) Karol MATIAŠKO (Slovakia) Irina NONINSKA (Bulgaria) Dimitri PERRIN (Ireland) Angel POPOV (Bulgaria) Radi ROMANSKY (Bulgaria) Giancarlo RUFFO (Italy) Heather RUSKIN (Ireland) Radomir STANKOVIĆ (Serbia) Anastassios TAGARIS (Greece) Ivan TASHEV (USA) Aristotel TENTOV (Macedonia) Dimitar TSANEV (Bulgaria) Michael VRAHATIS (Greece) Vasilios ZORKADIS (Greece)
National Organizing Committee
Honorary Chairmen:
Parvan RUSINOV (Deputy-Minister of Transport, IT and Communications) Prof. Kamen VESELINOV (Rector of TU-Sofia)
(C205) Evaluation of Single Server Queueing System with Polya 203 Arrival Process and Constant Service Time Seferin Mirtchev, Rossitza Goleva, Velko Alexiev (Bulgaria)
(C206) Application of Sine Regression for Modelling of Hall Sensors 213 Ventseslav Shopov, Vanya Markova (Bulgaria)
(C207) RAID as a Method for Improving PC Parameters 221 Sergey Nedev (Bulgaria)
(C208) Decision Planning of System Identification 229 Kaloyan Yankov (Bulgaria)
(C209) Lumped Dynamic Model of Vibrating Tunable Energy 239 Harvester with Serial Capacitive Feed-back Roumen Nikolov, Todor Todorov (Bulgaria)
Automation of System Design (C301) Sensitivity Analysis of the Discrete-Time LMI-Based H∞ 245 Quadratic Stability Problem Andrey S. Yonchev, Mihail M. Konstantinov, Petko H. Petkov (Bulgaria)
(C302) Minimal Diagnosis Set Generation Based on 255 the Deterministic Approach Pavlinka Radoyska (Bulgaria)
(C303) Virtual Research Laboratories in the Field of Electronic 263 and Computer Technologies Pavlinka Radojska, Nadezhda Spasova (Bulgaria)
(C304) Optimal Control of Heat Integrated Batch Reactors 271 Boyan Ivanov, Desislava Nikolova, Dragomir Dobrudzhaliev (Bulgaria)
(C305) An Approach for Automation of Research Using Wireless 279 TriLink Logger and LabVIEW Tsvetozar Georgiev (Bulgaria)
Intelligent Systems and Knowledge-Based Applications (C401) Customer Worth Evaluation Using Data Mining Prediction 285 Galina Ilieva (Bulgaria)
(C402) Comparative Analysis’ of Sequential and Fixed Length Tests 293 for Evaluation of Agent’s Behaviour Vanya Markova (Bulgaria)
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Proceedings of the International Conference on Information Technologies (InfoTech-2010),
16th – 17 September 2010, Bulgaria
DECISION PLANNING OF SYSTEM IDENTIFICATION
Kaloyan Yankov
Medical Faculty, Trakia University, Armeiska str., 11, Stara Zagora 6000,
Abstract. System identification in software Korelia-Dynamics is carried out using Cyclic coordinate descent method. Its main drawback is the possibility of obtaining the local extrema instead of the global. The choice of suitable sufficiently narrow domain of identification parameters is a prerequisite for convergence of the optimization procedure at the global minimum. The work is aimed to improve the identification algorithms determining the parameter’s domain depending on the type of identification function. Key words: system identification, identification planning, coordinate descent method, mathematical model 1. INTRODUCTION System identification is a powerful tool for better understanding, learning,
modeling and simulating of processes. Because of its complexity and the need for specialized skills, identification is difficult to use by humanitarian professionals. The goal to make this powerful scientific tool closer to humanities scholars has motivated much of the work of the author on the realization of the software Korelia-Dynamics.
The algorithm for system identification (SI) of experimental data is described in (Yankov, 2006). The identification goal is translated into an optimization problem and Cyclic Coordinate Descent (CCD) method is applied as optimization procedure. The proposed domain specific language in (Yankov, 2008) allows creating and editing of identification models described with arbitrary arithmetical expressions and N-th order ordinary differential equations (ODE). The next level in the development of identification software is the creation of a system for recognition of experimental data and their association with an appropriate function (Yankov, 2009). A training set of functions and quality feature differences between them are formulated. The experimental data are normalized in the square Cnorm=[0,1]x[0,1] and the procedure
PROCEEDINGS of the International Conference InfoTech-2010 230
for SI is applied to recognize the most suitable function. Normalization reduces the number of identification parameters and limits the domain of each parameter. The program offers extended analytical model with more parameters for the final identification. The task remaining for the user is to make the final identification in world coordinates based on the identification in Cnorm.
As mentioned, the optimization is carried out using CCD. CCD is a simple to realize, relatively robust method. Some disadvantages of this method are the locality and the fact that it may take a lot of calculations. The obtained local solution depends on the initial values and the selected domain of the identification parameters. Defining a broad domain leads to increased number of iterations and is a necessary condition for convergence to local minimum. This problem arises when CCD is used as an optimization method. The shortcoming of the method requires identification to be made in several different ranges of unknown parameters. The process of identification would facilitate and speed up if the algorithm determines the closest possible range of meanings for each parameter. Therefore it is necessary to complement the CCD with rules for an initial choice of domain for parameters identification. The process of determining of parameters domain that leads to finding a global extreme will be called planning of identification process.
An appropriate algorithm to avoid local extremes and get the global minimum for solving the inverse kinematics problem in robotics using CCD is proposed by the author in (Yankov, 1989). A finite set of local solutions is estimated from the finite set of sub-domains and the global minimum is searched in this set of solutions. The characteristic equations of motion of a robot are trigonometric functions with periodic solutions, which were used as a criterion for determining the sub-domains.
Korelia Dynamics was established to identify a large number of algebraic functions and ODE. The class of all ODE is enormous and it is impossible to create a general approach to decision planning. The best approach is to consider a class of ODE that is sensible enough to describe a wide spectrum of phenomena - namely the class of first and second order ordinary differential equations.
In this work algorithms are developed for overcoming the locality of the CCD, based on planning of the identification process to reach a global minimum for classes of functions considered in (Yankov, 2009), namely ODE of order I and II.
2. DATA PREPROCESSING The data acquisition unit produces experimental data vector in world coordinates
described by: Y(t) = [(t0,y0), (t1,y1),... (t N-1,yN-1)] (1)
N is the number of experimental samples collected for acquisition time tp = tN-1 - t0 . The vector Y(t) is used in the recognition process and subsequently for identification. Because these data are collected over time they can be called a ‘time
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series’. tp may be longer than is necessary for the experiment itself and the process of identification - an excessive amount of data after a certain time may not be sufficiently informative, yet it increases the difficulty of the calculations. After some time ts , called settling time, the observed signal enters and remains within a specified tolerance band ε related to the amplitude of expected final value Ys.
|)()(:| sis tytyt
|)(:| sis YtyY
i=s,s+1,…N-1 ε is an tolerance (error) band
(2)
To facilitate the identification process, the data received after time ts can be removed. The settling time is of cardinal significance for data acquisition systems because it is the primary factor that defines the data rate for a given error level. The expected final value Ys for converging time series is its accumulation point and will be called ‘Steady state level’.
The algorithm for evaluation of the settling time and the steady state level is on Fig.1. This procedure is important because it reduces dramatically the number of necessary samples.
procedure AccumulationPoint(const Y: array of real;//experimental data: time and amplitude
const Epsilon : real; // tolerance band var T_s : real; // Settling time var Y_s : real); // Steady state level
var Y_Sort, // sorted experimental data in ascending order Y_Diff : array of real; // discrete derivatives
Indices, Diff_Ind : array of integer; II : integer; begin
Y_Sort := sort(Y); // sort the samples in ascending order for II:=low(Y_Sort) to high(Y_Sort)-1 do begin Y_Diff[II] := Y_Sort[II+1] – Y_Sort[II] // discrete derivatives of the sorted list Indices[II] := II*ord(Y_Diff[II] > Epsilon); // indices where repeated values change Diff_Ind[II]:= Indices[II] - Indices[II-1]; // find repeated successive indices end; II := longest_persistence(Diff_Ind); // longest persistence length of repeated values Y_s:= Y(Indices(II)); // Steady state level
T_s:= X(Indices(II)); // Settling time end;
Figure 1. Evaluation of settling time and steady state level Other operations on time series, such as truncation, transformation, filtration and
so on, to facilitate and accelerate the subsequent data manipulations are discussed in (Yankov, 2010).
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3. IDENTIFICATION PLANNING Let Q=[q1,q2,…,qm] be an identification vector (Yankov, 2006). For
identification classes treated in (Yankov, 2009) an algorithm for finding the global minimum in the domain Dom(qi) of each identification parameter must be determined:
Dom(qi): [dimin, di
max], i=1..m where di
min, dimax are the lower and the upper boundary of the parameter qi.
Convergence depends on the Dom(qi).The wider the domain is, the more the time for identification and the time necessary for the probability to fall into local minimum increases. The aim is to find an approximate global solution for each identification parameter. Then a domain is chosen in the neighborhood of this solution. In this domain optimization procedures are applied for reaching a final decision with a preset error.
3.1. Exponential model – cases C1.1-C1.4 On Fig.2 are presented the models recognized by the program Korelia-
Dynamics. The identification vector is Q=[C0 , C∞ , r].
Case C1 r < 0 r > 0 С1.1
С1.2
)()()(
tkUtrydt
tdy (3)
y(t0) = C0
Model for identification: trtr eCeCty .
0. )1()(
(4)
identification vector Q=[C∞,C0,r]
С1.3
С1.4
Figure 2. – Exponential model
Initial value C0. It is searched in the neighborhood of y(0). If assume relative deviation δy(0):
Dom(C0): [(1- δy(0)).y(0), (1+ δy(0)).y(0)] (5)Infinite asymptote C∞ is calculated numerically using the procedure
AccumulationPoint on Fig.1. The procedure returns a settling time ts and a steady state level Ys for a time series. If the relative deviation for C∞ is δC∞ , then
Dom(C∞): [(1- δC∞). Ys, (1+ δC∞). Ys] (6)Rate constant r. In the model presented by Eq.(4) the identification parameters
C0 and C∞ are substituted their analogues in the time series from Eq.(1). The signal amplitude in settling time ts is:
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ss rrss eyeYy t.t. ).0()1.()t(
After equivalent transformations the following expression is obtained:
)0(
)t(t.
yY
yYe
s
ssr s
The numerator is the tolerance error in (2), the denominator is the amplitude range of time series. Therefore the right side of equation is relative deviation δC∞. The approximate value ra for rate constant is:
sT
Cra _
)ln(
(7)
The recommended r-domain with a relative deviation δr is: Dom(r): [(1-δr).ra, (1+δr).ra] (8)
3.2. Model of systems with limited capacity - cases C1.5-C1.6
).(
0
1)(
treC
KK
ty
(9)
y(t0) = C0
identification vector Q=[C0, K, r] C1.5. r < 0 C1.6. r > 0 Figure 3 – Model of systems with limited capacity
The graphs of the solutions are shown on Fig.3. Identification parameters C0 and
K are calculated in a fashion similar to the one for the analogous parameters in Eq.(8) of the exponential model:
The parameter r presents the slope and the growth of the S-curve. The first derivative of y(t) (Eq.(9)) will be used to find the tangent in inflexion point:
0
2
.
0
.2
1
)(
CeC
K
reK
dt
tdy
tr
tr
The second derivative of y(t) is:
0
)(
03
0.
0.
02
2
2
CKe
CKeCreK
dt
tydtr
trrt
(12)
Solving the last equation for t the abscissa tinfl of inflexion point is obtained:
rK
C
t l
0
inf
ln
PROCEEDINGS of the International Conference InfoTech-2010 234
Figure 4
Accordingly the tangent in the inflexion point is:
Kre
CK
Kty
ltrl 4
1
1)(
).(
0
infinf
Using the geometric relations shown on Fig.4:
Krtt
Ktg
s 4
1
0
The approximate value ra of the slope is:
sTra _
4
If assume that δr is an admissible relative deviation, then r –domain is: Dom(r): [(1-δr).ra, (1+δr).ra] (13)
3.3. Second order ordinary differential model Second order systems are the simplest systems that exhibit oscillations and
overshooting. Second order behavior is part of the behavior of higher order systems and understanding and investigation of second order systems helps to understand higher order systems.
A general second order ODE is:
)()()(
2)( 22
2
2
tUKtydt
tdy
dt
tydu
y(t0),y’(t0) – initial conditions
(14)
where U(t) is the input force both for impulse function and step function. The possible recognized cases with identification vector Q = [, ώ, Ku] are
presented on Fig. 5. Damping ratio ζ - defines the type of oscillation. Case 2.1: Overdamping, when ≥ 1. The increasing of decreases the
slope of the oscillation front. Preferable approximate value is:
norma Cnormnorm ,
where norm is the identified value in normalized coordinates during the data recognition. In practice the values are between 1 and 2, rarely larger.
Case 2.2: Underdamping when 0<<1. Approximate value ζa is proposed in (Alciatore and Michael, 2007) using logarithmic decrement ln(p0/p1):
22
1
0
1
0 2lnln
p
p
p
pa (15)
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where p0 is the first peak amplitude and p1 is the next peak amplitude. They are numerically calculated in the procedure ExtremePoints (Yankov, 2009).
If the relative deviation is δζ, then ζ–domain is: Dom(ζ): [(1-δζ)ζa , (1+δζ)ζa ] (16)
Case 2.3. Undamping, when = 0. This case is unambiguous.
Input force U(t) Case C2
Impulse response Step response
Case 2.1: ≥ 1
Overdamping. C2.1.1
C2.1.2 ≡
C1.2
Case 2.2: 0< < 1
Underdamping decaying
oscillations
C2.2.1
C2.2.2
Case 2.3: = 0
Undamping C2.3.1
C2.3.2
Figure 5 – Second Order Differential Models
Natural frequency ω. The identification value of ω must be commensurate with the period of identification. Let M is a number of periods, then the frequency is approximately.
ωa=2πM/ts (17)Dom(ω):[(1-δω)ωa , (1+δω)ωa ] (18)
For case C2.1 the domain is calculated assuming M=0.5. Gain of the system Ku. Because it determines the steady state response when
the input settles out to the steady state level (cases C2.x.2), the approximate value for Ku is obtained dividing the steady state value Ys by the value of the input step U:
UYK sa
u (19)
The calculations in the case of impulse response are more difficult (cases C2.x.1).
For ζ > 1 the solution is in form (Boyce and Diprima, 2001):
ttu eeK
ty .2.12
21)(
(20)
,11 2 12 2
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The peak of time series will be used to appreciate the Ku. The first derivative of Eq.(20) is used to find this peak and to compare with the peak Ymax of time series:
0.2.121
)( .2.12
ttu eeK
dt
tdy
21
)2/1ln(
peakt
(21)
max.2.1
2
21)( Yee
Kty peakpeak ttu
peak
Accordingly the gain of the system can be approximately calculated by:
peakpeak ttau
eeYK .2.12max
)21(
(22)
For ζ < 1 we start with a solution in the form:
1.sin,1
)( 2..
2
te
Kty tu (23)
using the first derivative, the first time peak and amplitude are:
1
1tan
2
2
a
tpeak
max
1tan.
1
2
2
.)( YeKtya
upeak
and the approximate gain is:
2
2
1tan.
1max .a
au e
YK
(24)
The domain of system gain with relative deviation δKu is: Dom(Ku):[(1-δKu) K
au , (1+δKu) K
au ] (25)
4. EXAMPLE A contraction of smooth muscle strips of rat urinary bladder after treatment with
the hormone Angiotensin II (Fig.6) is studied in (Georgiev et al., 2009). After numerical filtering a recognition procedure (Yankov, 2009) is applied. The muscle reaction is recognized as case C2.1.1 and the results after parameter identification in normalized space Cnorm are:
Iterations: 3 adaptive steps: 114 identification time: 0.125 s
Fig.8. Planning and system identification in world coordinates
The comparison shows that the planning of the identification process improves the computational process.
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238
4. CONCLUSIONS This paper describes an approach for planning of the identification process when
the CCD method is applied as an optimization algorithm. Thus a global minimum of a target function is reached. To guarantee this it is necessary to search the values of the identification parameters in a narrow domain of possible values. Planning involves creation of an identification algorithm to calculate the approximate value and to search for extrema in her neighborhood. Domains for first and second order differential equation models are discussed.
The CCD method is certainly not a general-purpose optimization approach. But when there is a partitioning of the variables, it is worthy of consideration. Simulation results show that the proposed algorithm has better ability for system identification of time series.
The delimitation of the domain of calculable parameters is a requirement in the implementation of optimization algorithms. Therefore, the approach can be used generally to determine the domain of the unknown parameters in numerical methods.
REFERENCES
Alciatore, D.G., B. Michael (2007). Introduction to Mechatronics and Measurement Systems (3rd
ed.). McGraw Hill.p. 466. Boyce, W., R.Diprima (2001). Elementary Differential Equations and Boundary Value Problems.
270, John Wiley & Sons, Inc. 7-th ed., New York. Georgiev Tz,, P. Hadzhibozheva, R. Iliev. Reporting, analysis and conversion of signals, obtained
in experiments of isolated tissues. Proc. 18-th Int.conf. Young Sci., Univ. of Forestry, Sofia, 2009, pp.256-264.
Yankov, K. (1989) Manipulator Motion Planning. Proc. International Workshop on Sensorial Integration for Industrial Robots, nov.22-24, 1989, Zaragoza, Spain, , pp.114-118.
Yankov, K., (1998) Software Utilities for Investigation of Regulating Systems, Proc. Ninth Nat. Conf. "Modern Tendencies in The Development of Fundamental and Applied Sciences". June, 5-6, 1998, Stara Zagora, Bulgaria, pp.401-408.
Yankov, K. (2006). System Identification of Biological Processes. Proc. 20-th Int.Conf. "Systems for Automation of Engineering and Research (SAER-2006). St.St. Constantine and Elena resort, sept.23-24, Varna, Bulgaria, pp 144-149.
Yankov, K. (2008). Simple Expression Language for Model Identificaton. Proc. Int. Conference on Information Technologies (InfoTech-2008). St.St. Constantine and Elena resort, sept.19-21, 2008, Varna, Bulgaria, vol.2, pp.259-266.
Yankov, K (2009). Recognition and Function Association of Experimental Data. Proc. of the Int. Conference on Information Technologies (InfoTech-2009). Constantine and Elena resort, sept.17-20, 2009, Varna, Bulgaria, pp.131-140.
Yankov, K (2010). Preprocessing of Experimental Data in Korelia Software. Jubilee scientific conference with international participation ‘15 years Trakia University’., may 21, Stara Zagora, Bulgaria.
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International Conference on Information Technologies (InfoTech-2010) 16 – 17 September 2010
St. St. Constantine and Elena resort, Varna, BULGARIA
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