generic bi-layered net, as the natural computational … · 2005. 5. 10. ·...

B.CSUKÁS and GY.BÁNKUTI: GENERAL BI-LAYER NET

I.J. of SIMULATION, Vol. 6, No. 6 ISSN:1473-804x online, 1473-8031 print 10

GENERIC BI-LAYERED NET, AS THE NATURAL COMPUTATIONAL MODEL OF CONSERVATION AND

INFORMATION PROCESSES

BÉLA CSUKÁS and GYÖNGYI BÁNKUTI

Institute of Mathematics and Information Technology, University of Kaposvár Guba S. u. !0. Kaposvár, 7400, Hungary

[email protected], [email protected]

Abstract: In Direct Computer Mapping the simple building blocks of the conservational and informational processes are mapped onto the generic “active” and “passive” elements of an executable program. The recently developed Generic Bi-layered Net model provides a common framework for the simulation of the hybrid (continuous and discrete, quantitative and qualitative) balance-based and rule-based processes. The common features of the process models are represented by a bi-layered net of variable structure that also determines the network (ring) structures of the influence routes and flux routes, as well as the Gantt Chart view of the process. The advantage of the new methodology is that the computational model is specified by the very structures and building elements of the process to be modeled. The software implementation of Generic Bi-layered Net model onto an executable program or dynamic database has been applied for the solution of various difficult practical problems. Keywords: generic simulation, hybrid processes, direct computer mapping, Generic Bi-layered Net 1. INTRODUCTION 1.1 ‘A Priori’, ‘a Posteriori’ and knowledge based Models of Processes The computer modeling of the continuous and discrete processes have been evolving in three different ways. The processes are usually described by a set of algebraic, differential and/or integral equations IPDAE (Pantelides, 2001). The ‘a priori’ (white box) models are derived from the simple first principle primitives, and then they are transformed into various sophisticated mathematical constructs. The ‘a posteriori’ (black box) models differ only in the origin, but the numerical solution of the identified mathematical equations is similar. Artificial Intelligence developed various knowledge-based methods without the explicit consideration of domain specific structures and of fundamental conservation laws. The qualitative and heuristic models, described by expert rules, often cannot be linked to any of the previously mentioned methods. Expert systems deal with an abstract set of signs and rules that should somehow be actualized in the engineering applications. The attempts to bridge this gap, with various kinds of qualitative models were not successful enough, because the qualitative knowledge representation evolved on its own, without effective connection to the quantitative modeling. The tools of Computational Intelligence (neural networks, genetic algorithm, fuzzy sets) were derived from the simplified pattern of the biological systems and from the human reasoning.

They cannot be combined easily with the quantitative engineering knowledge. The execution of the hybrid, discrete / continuous models is a difficult question, because the usual integrators do not tolerate the discrete events, while the usual representation of the continuous processes cannot be embedded into the discrete models conveniently. 1.2 General Formal Models of the Processes The general formal models of the systems had be developed before the powerful computers appeared. According to the Kalman’s approach [Kalman et al. 1969], the state space model of the continuous processes

t,g,f),0(x,Y,X,U (1) is described by the transition (f) and output (g) functions in the continuous time t:

( )( ) )t),t(u),t(x(gty

)t),t(u),t(x(ftx==

•

(2)

where, U)t(u ∈ = the input variables, X)t(x ∈ = the state variables and Y)t(y ∈ = the output variables

of the process.



The abstract automaton representation of the discrete processes describes the same in the discrete time k, as follows:

( )( )

x k 1 f (x(k), u(k),k)

y k 1 g(x(k),u(k),k)

+ =

+ = (3)

1.3 General Net Theory and Special Net Structures (Petri Nets) The General Net Theory [Brauer, 1980] proposed a generalized net model for the description of the structures. Accordingly the triplet

N , , F= Π Θ (4)

is a net iff

F ( ) ( )

dom(F) codom(F)

Π∩Θ =∅⊆ Π×Θ ∪ Θ×Π

Π∪Θ ≠ ∅∪ = Π∪Θ

(5)

where Π, Θ and F represent the sets of the states, of the transitions and of the functional relations (links), respectively. Many net models, like the early appeared and very innovative Petri Net [Petri, 1962], as well as the various State-Transition Nets belong to the above family. 1.4 Idea of Direct Computer Mapping of the Process Models Many recently used engineering methods had been established before the onset of powerful computers. Modeling starts either from the consideration of changes in characteristic measures, or from the rules and signs. Next this is transformed into mathematical construct. It usually cannot be solved, should be discretisized, and finally, the computer executes simple arithmetical steps. Figure 1. The idea of the Direct Computer Mapping In Direct Computer Mapping DCM (Csukás and Bánkuti, 2003a), we can map the simple building blocks of the conservational and informational

processes onto the generic “active” and “passive” elements of an executable program (see Fig.1.). The balance elements and the signs, as well as the elementary transitions and the rules can be described by brief uniform programs, executed by the same kernel algorithm. Direct Computer Mapping of process models allows the computer to know explicitly about the very structures and bounds of the physical world. In this knowledge representation, the model is organized rather by the transitions, than by the state. The key issue is that the computational software (and hardware) can copy the natural structure and building elements of the investigated problem. 2. GENERIC BI-LAYERED NET MODEL OF COMPLEX PROCESSES The recently developed Generic Bi-layered Net model (Csukás and Bánkuti, 2003b) is a theoretically established and practically validated powerful realization of the Direct Computer Mapping. It is a special case of the General Net Theory on the one hand, as well as an explicitly structured, generic combination of the state space model and of the abstract automaton. The generic, bi-layered net model can be defined by the ten-tuplet of

t,r,,,Y,X,G,B,A,P ΨΦ (6) where GB,A,P ∪ is a net. The communication channels B and G determine the passive→active

( ) ( ) ( )τ×τ⊂τ APB (7) ( ) ( ) ( )( ) ( )τ∈ττ=τ Ba,pb ijij (8) ( ) ( ) ( )( ) ( )τ∈ττ∀τ∃ ba,p|b jij

ij (9)

( ) ( ) ( )( ) ( )τ∈ττ∀τ∃ ba,p|b jiji

j (10)

and active→passive data flows

( ) ( ) ( )τ×τ⊂τ PAG (11) ( ) ( ) ( )( ) ( )τ∈ττ=τ Gp,ag jiji (12) ( ) ( ) ( )( ) ( )τ∈ττ∀τ∃ gp,a|g ijiji (13) ( ) ( ) ( )( ) ( )τ∈ττ∀τ∃ jjiij gp,a|g (14)

respectively. Index j designates the ordered sets of the existing output ( bj (9)) and input ( jg (14))

connections for the j-th passive element. Similarly, index i defines the ordered sets of the existing output ( gi (18)) and input ( ib (10)) connections for the i-th active element.

Problemtobe solved

Simple Eqs. or rules

Mathematical construct

Decomposition

Numerical method

Executableprogram

Directmappingof Bi-layeredGeneric Net

Abstraction

Problemtobe solved

Simple Eqs. or rules

Mathematical construct

Decomposition

Numerical method

Executableprogram

Directmappingof Bi-layeredGeneric Net

Abstraction



Variable τ denotes the optional points or intervals of the continuous or discrete time t, declaring the existence of the respective elements and relations. The passive elements P are associated with state variables jX and with an operator, describing the change of the state:

[ ][ ]⎥⎦

⎤⎢⎣

⎡=ψΨ∈ψ∈→∀

bxgy

;;XXpjj

jjijjj

j (15)

where jX contains any structured data set, and

operator jψ describes how jy changes the state via

the channels jg .

The active elements A are characterized by the operator iϕ , providing a mapping. This determines how the output changes yy iji ∈ , carried by gi are calculated from the coordinated input readings

iij xx ∈ that comes from the passive elements

through the channels ib : [ ][ ]⎥⎦

⎤⎢⎣

⎡=ϕΦ∈ϕ→∀

gybx

;aii

iiiii

i (16)

The operators Ψ∈ψ i and Φ∈ϕi may be anything from a simple input/output mapping to a brief program, calculating the elementary process or the rule. Variable r designates the geometrical and/or the property coordinates of the distributed systems and/or population balances. As an example, the GBN implementation of a simple hybrid automaton see the so-called Single Switch Server problem ([Perkins and Kumar, 1989]; [Agarwal et al., 2002]) in Fig. 2.

Figure 2. The Single Switch Server problem

This is one of the simplest discrete / continuous, switched, hybrid dynamic system. The suppliers send materials in the buffer tanks A, B and C at constant flow rates of VA, VB and Vc, respectively.

Machine can process any one material at a time at rate VMA, VMB and VMC. The level of materials changes in the buffers, however there are determined minimal and maximal levels. A material specific setup time is incurred each time machine switches to different material. The goal is to design a switching strategy, which satisfies various single or multiple objectives, e.g. maximal production, minimal setup and waiting time, minimal buffer levels, etc. There is an obvious interaction between the continuous and discrete (logic) components, i.e. it is a simple prototype of the hybrid dynamic systems. The GBN representation of the given problem can be seen in Fig. 3.

Figure 3. The GBN model of the SSS problem Two examples for the passive elements and one example for an active element are the followings:

∑+=∆

⎥⎦

⎤⎢⎣

⎡===∆=∆

Ψ→

i1iAA

A71A11

144A111A11

yM:M

MxMxyMyM

:p (17)

Wait Setup C Prod B Prod A Prod St

StxyStySt

:pi10i

101010111010

∨∨∨∨=⎥⎥⎦

⎤

⎢⎢⎣

⎡=∀

==Ψ→

(18) In the network view of the net we can interpret the alternating, connected, ordered set of the communication channels { }

1nnnn32222111 jiijjiijjiij g,b,g,b,g,b + (19) They are called influence routes, which determine a special network structure. The influence routes carry the influence. E.g. the perturbation of the content

1jX of the element 1jp affects the content 1njX + of

the element 1njp + , according to the influence:

a9

p6p9

fC cC

a8

p5p8

fB cB

a4 a6a5a10

p10

ST

a1 a2 a3

p7

a7

fA cA

p4M ΣP

p12

Σx

p13

a12

Σt

p14

A B C

p1 p2 p3

a11

p11

TS

p15

a9

p6p9

fC cC

a9

p6p9

fC cC

a9

p6p9

fC cC

p6p9

fC cC

a8

p5p8

fB cB

a8

p5p8

fB cB

a8

p5p8

fB cB

p5p8

fB cB

a4 a6a5a4 a6a5a10

p10

ST

p10

STSTSTST

a1 a2 a3a1 a2 a3a1 a2 a3

p7

a7

fA cAfA cAfA cA

p4MM ΣP

p12

ΣP

p12

Σx

p13

Σx

p13

a12

Σt

p14

A B C

p1 p2 p3

A B CA B C

p1 p2 p3p1 p2 p3

a11

p11

TS

p11

TSTSTS

p15

MA MB MC

VA VB VC

VMAVMB VMC TCl

Process

LNLP

UPUN

MA MB MC

VA VB VC

VMAVMB VMC TCl

Process

LNLPLNLP

UPUNUPUPUNUN



( ) ( ){ }1nn222111 jii2jji1ij

y,...,tx,y,tx+

∆∆∆∆

(20) where ij x∆ and ji y∆ refer to the perturbation of the state and the change, respectively. The sensitivity and its special forms, such as observability and controllability can be studied by means of the influence route network. The minimal (generating) influence routes are the basic edges. The maximal influence routes are the transferring routes and the complete loops. The simplified structure of the influence routes is a special ring,

where the two algebraic operations are the concatenation and the common part. As we have emphasized, the existence of elements A, P, B and G, as well as the contents X and Y of the communication channels depend on the time while τ denotes the well-defined points or intervals of the continuous or discrete time t, when the given element, channel or sign does exist. This temporal behavior of the system of variable structure can be seen from the Gantt Chart view (see Fig. 4).

Figure 4. The Gantt Chart view of a hybrid model

3. BALANCE PROCESSES AND CONSER-VATIONAL PROCESSES, AS A SPECIAL CASE 3.1 Balance Processes An important special case of net (6) is the class of balance processes, where the basic part of the state

jX is a measure and, the operator jψ summarizes the simultaneous rates. Depending on the discrete or continuous time, operator jψ generates also the appropriate

( ) [ ] [ ] ∑=∆∆

≈=ψi

jijjjj

jj yt

pX

td

pdXy (21)

difference or differential equations, called balance equations. In the balance model the descendent of the mappings, iϕ can be divided into two disjunct

parts, corresponding to the increases (+) and decreases (-) of the characteristic measures:

[ ] [ ] [ ]−−++ ∪= gygygy iiiiii (22) The active elements of the balance process models describe the various transportations and transformations. 3.2 Conservational Processes Conservational process is a special case of the balance process, where there are constant measures C determined by the model specific conservation laws. Simultaneously all of the measures M can be combined from these constant measures, according to the respective stoichiometry S, i.e.:

CSM|SC ⋅=∃∃ (23) where S is the stoichiometry matrix of the passive elements.

t

Gannt ChartView

t

Gannt ChartView

t

Gannt ChartView

Gantt Chart View



To get a sophisticated definition let C be a measure in the space of the geometric co-ordinates x,y,z and of other parameters (dimensions) 1 2 nd ,d ,...,d . This measure can change in continuous or discrete time t. Let us mark the finite, closed region in the above space with v and its volume with V. The not necessarily finite and closed ‘environment’ of region v, which will be indicated with u\v, is called universal complement. Let [ t ti j, ] be a finite time interval. vC (t) and u\vC (t) mark the measure C associated with the region v and with its universal complement at a time. The constant measures are characterised by the following axiomatic properties: Axiom 1. The change in the constant measure in any finite and closed region v during any time interval [t ti j, ] is accompanied by the identical change of the same measure in the universal complement with an opposite sign, i.e.

( ) ( )( ) ( ) ( )( ){ }i j \ i \ ji, ji j

C t C t C t C tν ν µ ν µ νν

<

∀ ∀ − = − −

(24) Axiom 2.The constant measures are bounded in any finite and closed region, i.e.

( ) ( )( ),t

C ,C | C C t Cν ν ν ν νν∀ ∃ −∞ < ≤ ≤ < +∞ (25)

As a consequence, the constant measures can be transformed simply into the

( ) ( )*ˆ0 C t C tν ν≤ ≤ (26) domain by subtracting the absolute lower bound from each value. Constant measures correspond to the quantities obeying to the existing conservation laws of the investigated system, within the given model hypothesis and regardless of the physical meaning. For example within the model hypothesis of chemistry the number and the mass of the atoms are constant measures that correspond to the conservation law of the given model hypothesis. Although all of this model hypotheses have a limited validity, they give a constructive, sound basis of the problem solving within the scope of the given model. Thinking about the above example, many chemical compounds can be built from the 120 atoms. These number and the mass of these molecules do not satisfy the axioms of the above definition, because these amounts can change within the region by

chemical reaction. However we can write the balance equations of the respective chemical reactions with the help of the stoichiometry that derives these secondary measures from the primary ones obeying the conservation law of the existing model hypothesis. On the other hand there might be a lot of measures that cannot be derived from the constant measures so simply or cannot be deduced from them all. In the following the measures that can be derived as the homogeneous linear function of the constant measures (such as components from atoms in chemistry) are called conservational measures. To define the conservational measures consider a finite closed region v, containing a set of the constant measures C { }1 2, mC ,C ,...,C= . Designate

{ }a b qM M ,M ,..., M= a set of measures in the same region. Measures M are called conservational measures if for any t there is a matrix ( )S t of coefficients ( )i, jr t that satisfies the

( ) ( ) ( )M , t S t C tνν = . (27) equation, where M(v,t) is the quantity of a conservational measure in region v at a time t. If the operator iϕ can be determined by a well-defined single rate iv , then the change of the conservational measures the expression of

TM(t) S v(t) V•

Γ= ⋅ ⋅ (28)

can be written for (where V is the reference measure, e.g. the volume). In the steady state:

T0 S v(t) VΓ

= ⋅ ⋅ (29)

As an example consider in Fig. 5 the GBN model of a simple enzyme reaction of

fumarate (S) + water (I) → malate (W)

where S=C4H4O4 I=H2O W=C4H6O5 while the active elements correspond to the elementary processes of the a1:E+S → ES; a2:ES → E+S; a3:ES+I → ESI; a4:ESI → EW; a5:EW → E+W and a6:E+W → EW reactions, as well as of the a7, a8, a9 = transportations.



Figure 5. GBN representation of an enzyme reaction where e.g.

VMc;yyyM

c:x,c:xy,y,y

:;Sp

SS121118S

S11S81

12111811

∆=++=∆

⎥⎦

⎤⎢⎣

⎡==

ψ→ (30)

tVcckvvy,vy,vykx,cx,cx

:;va

IS33

343333323

335I33ES32333

∆⋅⋅⋅⋅=

⎥⎥

⎤⎢⎢

⎡==−====

ϕ→

(31) The state of the model can be described by the vectors of the constant and conservational measures. The stoichiometry of the changes is characterized by the so called process stoichiometry matrix

S 4 4 4 0ES 4 4 4 0

CI 0 2 1 0

HC M SESI 4 6 5 1

OE 0 0 0 1

EEW 4 6 5 1W 4 6 5 0

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥= = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(32)

(33)

(34) 3.3 The Ring of the Balance Routes In the balance process models the alternating, connected, ordered set of the communication channels { }+−+−+−

+1nnnn32222111 jiijjiijjiijg,g,...,g,g,g,g

(35) is called flux route. Flux routes determine another network structure, which carry the constant and conservational measures. For example, if we modify the value of measure

1jX in element 1jp , then this

change effects on measure 1njX + of the element

1njp + , i.e.:

( ) ( ) ( ) ( ){ }nji1ij2ji1ij ty,...,ty,ty,ty n1n222111 +−+− ∆∆∆ − (36)

The −+∆ /jiy values refer to the dispersion of the

changes in the rate of subsequent processes (multiplied by the stoichiometric coefficients). The minimal (generating) flux routes are the basic edges. The maximal flux routes are transferring routes and the complete loops. The simplified structure of the flux routes is a special ring, where the two algebraic operations are the concatenation and the common part. In steady state the flux routes describe a Kirchoffian network. 4. INFORMATIONAL PROCESS, AS A SPECIAL PART OF CONSERVATIONAL PROCESS It is to be noted that a part of the above conservational model, responsible for the enzymatic control, can be replaced *for a simplified model of rules and signs, respectively. This is illustrated in Fig. 6. where the signs are symbolized by circles and the rules are represented by bar nodes.

GGB

EWS I WEES ESI

v7 v8 v1 v3v2 v4 v5 v6 v9

p1 p2 p3 p4 p5 p7p6

a7 a8 a1 a2 a4 a5 a9

S I W

v7 v8 v3 v9a6

GGB

EWS I WEES ESI

v7 v8 v1 v3v2 v4 v5 v6 v9

p1 p2 p3 p4 p5 p7p6p1 p2 p3 p4 p5 p7p6

a7 a8 a1 a2 a4 a5 a9

S I W

v7 v8 v3 v9a6

T

1 1 0 0 0 0 1 0 01 1 1 0 0 0 0 0 00 0 1 0 0 0 0 1 0

S 0 0 1 1 0 0 0 0 01 1 0 0 1 1 0 0 0

0 0 0 1 1 1 0 0 00 0 0 1 1 1 0 0 1

Γ

−⎡ ⎤⎢ ⎥− −⎢ ⎥⎢ ⎥−⎢ ⎥

= −⎢ ⎥⎢ ⎥− −⎢ ⎥

−⎢ ⎥⎢ ⎥− − −⎣ ⎦

t∆V

)t(v)t(v)t(v)t(v)t(v)t(v)t(v)t(v)t(v

100111000000111000000110011000001100010000100000000111001000011

)t(M∆

9

8

7

6

5

4

3

2

1

⋅⋅

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−

−−−

−−−

−

=



Figure 6. An example for the conservation based information process

Examples for the ψ and ϕ mappings of a sign (p6, evaluation of W), and of a rule (a5, calculation of the reaction rate) are as follows:

V/Mccx

y:;cp

WW

W56

666W6

=

⎥⎦

⎤⎢⎣

⎡=

ψ→

)c,c,c(Fkky

cx,cx,cx:;ra

WIS3

355

W56I54S52555

=

⎥⎥

⎤⎢⎢

⎡=

===ϕ→

(37

) 5. INFORMATIONAL PROCESS, AS A SPECIAL SUPPLEMENT OF THE CONSERVATION PROCESS The human made “artificial” processes often do not have the above described self-determined control, but they can be supplied by an informational process, determining the control signs and rules. Fig. 7 AND 8 show an example for the GBN model of a controlled heat exchanger. In the Figures V and W, as well as H and Q refer to the volume and the enthalpy of the inside and outside liquid, respectively. The inlet and outlet flows of the inside and outside agents are signed by the symbols Vb and Vk as well as Wb and Wk respectively. The heat transfer is symbolized by the elementary process Ht.

Figure 7. A conservational / informational process

Figure 8. Plausible visualization of a conservation based information process

GGB

cWS I Wk3cS cI

v7 v8 r1 v3r2 r4 r5 r6 v9

p1 p2 p3 p4 p5 p7p6

a3 a6a7 a8 a1 a2 a4 a5 a9

GGB

cWS I Wk3cS cI

v7 v8 r1 v3r2 r4 r5 r6 v9

p1 p2 p3 p4 p5 p7p6p1 p2 p3 p4 p5 p7p6

a3 a6a7 a8 a1 a2 a4 a5 a9a7 a8 a1 a2 a4 a5 a9

∞

V H Q W

Vb

Vk

Wb

WkHt

u T

m PID

aVb

WbWk

Vk

Ta

XPID

m

uVH

W,Q

∞

V H Q W

Vb

Vk

Wb

WkHt

u T

m PID

aVb

WbWk

Vk

Ta

XPID

m

uVH

W,Q

∞Vk

V H Q W

Vb Wb

WkHt

mb

m

mk

Vb

WbWk

Vk

m

mb

mkT

u

uT

VH

W,Q

∞Vk

V H Q W

Vb Wb

WkHt

mb

m

mk

Vb

WbWk

Vk

m

mb

mkT

u

uT

VH

W,Q



The temperature T is measured by the thermometer m, and it is compared with the set point a. With the knowledge of this difference, the PID controller calculates the control action u. In the right hand side the Generic Bi-layered Net model of the heat exchanger and the controller are represented by a connected pair of a conservational and an informational process. In the practical realization the informational process is carried out by another physical (electronic / electric, hydraulic or pneumatic) process (i.e. another conservational process). The apparently paradox, but meaningful notion of conservation based informational processes is illustrated in Fig. 8 more plausibly. Here the liquid flow through the jacket of the heat exchanger is controlled by another liquid flow through another vessel. By decreasing the size of the heat exchanger and by the simultaneous increase of the upper right hand side volume, the exciting question appears whether the parts changed their relative position. It means beyond a certain point we recognize that the heat exchanger controls the other unit. From theoretical point of views, this results a new interpretation of the informational process. Accordingly, a given part of the conservational process behaves as an informational process with respect to its complementary part, if this special part consumes and produces significantly less conservational measures, than the complementary process, while, along the feedback influence loops

and transferring influence routes the informational process exerts more influence on the complement, than the completing part on it. The informational process can be a special part of the self-determined natural processes (e.g. neural system, enzyme regulation), or it can be a supplied part of the non-self-determined artificial one (e.g. control systems). The essential feature of the informational process is that it •transports negligible amount of conservational measures with the complementing part and with the environment, •while it has a greater influence on the operation of the complementing part than vice versa. If the above criteria are fulfilled, then it is not necessary to describe the conservational processes for this special subsystem. Instead, we can read, calculate and overwrite the appropriate signs simply. Accordingly, we neglect the conservational process carrying these signs, and deal only with the informational process carried by the vehicle conservation process. 6. CLASSIFICATION OF THE PROCESS MODELS The above described relation between the conservational and informational processes can be overviewed by the classification of the processes according to Fig.9.

Figure 9. The classification of the GBN processes

GBN processes

Informational processes

Balance processes

Conservational processes

Fictitious processes

SupplementedInformational part

Transformed part

Special part

Special part

Primary inform. process

GBN processes

Informational processes

Balance processesBalance processes

Conservational processesConservational processes

Fictitious processes

SupplementedInformational part

Transformed part

Special partSpecial partSpecial part

Special part

Primary inform. process



The set of balance processes is a subset of the Generic Bi-layered Net processes. Conservational processes are in a subset of the balance processes. Both of the balance and conservational processes might have a special part that consumes and produces less additive measures, but exerts more influence on the completing part. These special parts can be transformed into the respective informational processes. Marginally the whole balance or conservational process can be mapped into an informational process. Another case is, when the balance or conservational process is supplied with an informational process. In addition there are also primary informational processes. On the other hand, all of the above described informational processes must have a vehicle conservational process that carries the signs and

executes the rules. The brain and the computers can work as conservational processes, themselves. It is to be noted that there might be also fictitious processes, outside of the set of the informational processes. 7. PRACTICAL APPLICATIONS The methodology has been applying for the solution of various difficult practical problems. A couple of practical examples and the difficulties to be overcome are summarized in Table I. In Table II. typical examples for the passive and active elements, describing the various models are illustrated. It is to be noted, that the general kernel program is the same in every application.

Table I. Overview of the simulation based identification, control and design problems, solved with the method

Problem to be solved

Special difficulties

Simulated Moving Bed prep. Chromatography

Cyclically changing initial and boundary conditions

Batch polymerization of directed structures

Partially mixed volume, multiple discrete feeds

Identification of metabolic network models

Large models with roughly estimated initial data set

Optimization of supply / demand chains

Dynamic simulation of hybrid automatons

Macro level simulation of a chicken poultry

Common use of models coming from various fields

Planning and scheduling of an agricultural farm

Dynamic cost analysis and year long cash flow calc.

Quantitative health risk analysis of batch plant

Changing allocation of dan-gerous sources and workers

Table II. Typical examples for the passive and active elements, describing the various applications

Task

Passive and active elements

Preparative chromatography [e. g. Temesvari et al. 2004]

Components of phases and in volume Component transfer, input/output flows

Batch polymerization [e. g. Csukás and Balogh 1998]

Populations in the compartments Feed, reactions, dissolving, heat transfer

Metabolic network [e. g. Csukas and al. 2003]

Metabolites, enzymes, cofactors Reactions, transport processes

Optimization of supply / demand chains [e. g. Csukás and Bánkuti 2003b]

Storage levels, fuzzy constraints, various signs Purchases, processing, sales, rules

Chicken poultry

Parts of the animal, air, feeds, walls Eating, metabolization, transportations

Agricultural farm [ongoing analysis of an agricultural plant]

Plants, feeds, pig populations Harvest, mixing, fattening, purchase, sale

Health risk analysis [ongoing analysis of a pharmaceutical plant]

Workers and components in the space Ventilation, pollution, work



In Figs. 10-12 the results obtained for the above mentioned hybrid automaton (see Figs. 2 and 3) are summarized. As it can be seen, controlled by the built-in rules a periodic solution evolves rapidly. As a more sophisticated example, in Fig. 13 see a typical part of a generic metabolic model. The conventional models are built from the previously derived “aggregated” blocks (like the various versions of the Michaelis-Menten model), and we are not able to add any new mechanism for the same enzymes correctly. The reasons are that we do not have any access to the enzyme concentration on one hand and, the supplied models can contradict the assumptions, made in the determination of the original models, on the other. As a rough solution, we could determine the effect of the new agents on the parameters of the existing kinetic equations

however it is, in principle, an incorrect way of thinking.

Buffer levels

0

50

100

150

200

250

300

0 50 100 150 200Time, h

Leve

l, kg

Supp_A MSupp_B MSupp_C M

Figure 10. Change of the level in the storage

volumes

Cumulated production

0

1000

2000

3000

4000

5000

6000

7000

8000

0 50 100 150 200Time, h

Prod

uct,

kg

Prod_A MProd_B MProd_C M

Figure 11. Cumulated productions

Evaluation parameter

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200

Time, h

Para

met

er

Supp_A MEASSupp_B MEASSupp_C MEAS

Figure 12. Change of the evaluating parameter

In addition the practical realization of this identification can also be questioned. Using published experimental data, we built two different GBN models, as follows [Csukás et al., 2003]: • First, we generated a “simplified, gross” GBN

model (GBN1), based on the published individual enzyme kinetic model, as well as on the published model parameters. In this case the GBN framework played the role of a simple simulating tool.



Figure 13. Measured and simulated changes of metabolite concentration in glycolysis

•Next, we synthesized a more detailed model

(GBN2), consisting of each elementary processes, and of each individual enzyme complexes, playing any role in the process. In this version we can exploit all of the capabilities of the GBN model, however, there is an enormous number of parameters to be identified. As an example, consider one of the reactions of the metabolic network. In the usual metabolic models, the transformation

AcAld ⇔ EtOH, (38) ADH catalyzed by the enzyme alcohol dehydrogenase can be modeled by an ordered bimolecular kinetics, with the cofactor binding first. According to the usual approach of the metabolic simulation we are advised to use the rate equation

ivbiaiubiaivuivbia

a

ivuia

v

biaivivu

v

bia

a

ia

ivubiaADH

KKKbuv

KKKabu

KKuv

KKKbvK

KKKapK

KKab

Kv

KKpK

KKbK

Ka1

KKuvV

KKabV

v

++++

+++++++

−−+=

−+

(39) In this rate equation a=EtOH, b=NAD, u=AcAld and v=NADH. V+ and V- are the main (Michaelis-Menten like) rate parameters and there are many equilibrium parameters Kxy. The gross Generic Bi-layered Net model (GBN1) of the reaction (38) contains a single active element, with a program that calculates the rate expression (39). Similarly, we had to build many case specific prototypes for the various types of metabolic reactions. In the detailed Generic Bi-layered Net model (GBN2) of the gross reaction (38) we have seven passive elements for the participating components:

GlcoutGlcinATP

F6P

F16bP

GAP

Glycogen

BPG

P3GA

P2GA

TransHK

PGI

PFKNAD

NADH

ADP ATPADP

ALD

TPINAD

ATP

GAPDH

PGM

ENO

G6P

NADH

NADHPGK

EtOH

PEP

PYR

PDC NADH NAD4ATP3NAD

SuccinateADH

4ADP3NADH

AcAld

ADP

ATPPYK

CO2

DHAP

NADHNAD

G3PDHGlycerol

Glucose uptake

0

20

40

60

80

100

120

0 1000 2000 3000 4000Time, s

Qua

ntity

, mM

Meas_GlcGBN1_GlcGBN2_Glc

Glycerol production

0

5

10

15

20

25

0 1000 2000 3000 4000Time, s

Qua

ntity

, mM

M_Glycerol

GBN1_Glycerol

GBN2_Glycerol

GlcoutGlcinATP

F6P

F16bP

GAP

Glycogen

BPG

P3GA

P2GA

TransHK

PGI

PFKNAD

NADH

ADP ATPADP

ALD

TPINAD

ATP

GAPDH

PGM

ENO

G6P

NADH

NADHPGK

EtOH

PEP

PYR


SuccinateADH

4ADP3NADH

AcAld

ADP

ATPPYK

CO2

DHAP

NADHNAD

G3PDHGlycerol

GlcoutGlcinATP

F6P

F16bP

GAP

Glycogen

BPG

P3GA

P2GA

TransHK

PGI

PFKNAD

NADH

ADP ATPADP

ALD

TPINAD

ATP

GAPDH

PGM

ENO

G6P

NADH

NADHPGK

EtOH

PEP

PYR


SuccinateADH

4ADP3NADH

AcAld

ADP

ATPPYK

CO2

DHAP

NADHNAD

G3PDHGlycerol DHAP

NADHNAD

G3PDHGlycerol

Glucose uptake

0

20

40

60

80

100

120

0 1000 2000 3000 4000Time, s

Qua

ntity

, mM

Meas_GlcGBN1_GlcGBN2_Glc

Glycerol production

0

5

10

15

20

25

0 1000 2000 3000 4000Time, s

Qua

ntity

, mM

M_Glycerol

GBN1_Glycerol

GBN2_Glycerol



ADH enzyme, AcAld and EtOH compounds, ADH_NADH, ADH_NADH_ AcAld, ADH_NAD_EtOH and ADH_NAD complexes. There are ten (or five combined) active elements for the ten reactions. The full system of equilibrium chemical reactions is the following:

ADH + NADH ⇔ ADH_NADH ADH_NADH + AcAld ⇔ ADH_NADH_AcAld

ADH_NADH_AcAld⇔ADH_NAD_EtOH (40)

ADH_NAD_EtOH ⇔ ADH_NAD + EtOH ADH_NAD ⇔ ADH + NAD In the GBN2 model the majority of the elementary processes can be described by three simple prototypes corresponding to the 2→1, 1→1 and 1→2 simple bilinear/linear, linear/linear and linear/bilinear equilibrium kinetics, respectively. The general program for all of these active elements can be calculated by the

iii

iiiii vuK

kbakv −= (41)

rate expressions, where k is the kinetic parameter, K is the equilibrium constant, a, b, and u, v are the concentrations of input and output components, respectively. In the GBN1 representation of the conventional model we used the published concentrations, rate constants and equilibrium parameters. Regardless to the known parameters of GBN1 model, it could be tuned with serious difficulties. This is caused by the rigid connection of previously and independently prepared, “canned” building blocks. The GBN2 representation of the detailed elementary model was identified with a special multistage identification methodology. First, we started from a broader possibility space, containing ranges for the enzyme concentrations, equilibrium constants, kinetic parameters, initial concentrations, etc., and then the search space was narrowed stepwise. The consecutive steps of the identification were the following:

• Heuristic rules and manual adjustment; • Pre-screening with the help of a multicriteria

genetic algorithm; • Manual refinement with trial and error

method; • Genetic search in a narrow possibility space;

Although model GBN2 has an enormously great number of unknown parameters, it can be identified easier. The most important parameters are the enzyme concentrations, while the roughly estimated equilibrium and kinetic parameters can compensate each other’s error somehow.

As it can be seen from the example shown in Fig. 13, the two models gave similar result that fit the experiments. The case study proved that the identification of the detailed elementary model is not hopeless. It is an interesting observation that the detailed model runs faster, because the built-in natural control helps to avoid the stiff behavior. The independently developed greater building blocks of the conventional model do not connect to each other smoothly and this causes more difficulties in the implementation and in the use. The result is straightforward also for the solution of the drug discovery or agent discrimination problems which cannot be solved by the conventional modeling approach neither in principle, nor in practice. Similarly it is not possible to supplement the usual mathematical constructs and numerical algorithms with the new elements and interactions. Consequently, the feasibility of generating and using the detailed generic models is promising for the future developments. 9. CONCLUSIONS In Direct Computer Mapping we map the simple building blocks of the conservational and informational processes onto the generic “active” and “passive” elements of an executable program, directly. Direct Computer Mapping of process models allows the computer to know explicitly about the very structures and bounds of the physical world. The recently developed Generic Bi-layered Net model is a theoretically established realization of the Direct Computer Mapping. It is a special case of the General Net Theory on the one hand, as well as an explicitly structured, generic combination of the state space model and of the abstract automaton. The Generic Bi-layered Net model provides a common framework for the simulation of the hybrid (continuous and discrete, quantitative and qualitative) balance-based and rule-based processes. The common features of the process models are represented by a bi-layered net of variable structure that also determines the network (ring) structures of the influence routes and flux routes, as well as the Gantt Chart view of the process. The advantage of the new methodology is that the computational model is specified by the very structures and building elements of the process to be modeled. The Generic Bi-layered Net model makes possible an interesting, new representation of the informational processes. Accordingly the



informational process is a special part of the self-determined natural processes, or it can be a supplied part of the non-self-determined artificial process. The essential feature of the informational process is that it • transports negligible amount of conservational

measures with the complementing part and with the environment,

• while it has a greater influence on the operation of the complementing part than vice versa.

If the above criteria are fulfilled, then it is not necessary to describe the conservational processes for this special subsystem. Instead, we can read, calculate and overwrite the appropriate signs simply. Accordingly, we neglect the conservational process carrying these signs, and deal only with the informational process carried by the vehicle conservation process. The method offers robust solution for the hard, hybrid, multidimensional and non-linear problems, as well as supports parallel programming. The software implementation of Generic Bi-layered Net model onto an executable program or dynamic database has been applied for the solution of many difficult practical problems. REFERENCES Brauer, W. Ed. 1980. “Net Theory and

Applications”. Springer Lecture Notes in Computer Science, (84).

Csukás B. and Bánkuti Gy. 2003a. “Direct computer mapping of process models”. In: Foundations of Computer Assisted Process Operations, I. E. Grossmann and C. M. MacDonald, Eds., AIChE INFORMS, pp. 577-581.

Csukás B. and Balogh S.: “Combining Genetic Programming with Generic Simulation Models in Evolutionary Synthesis”. Computers in Industry, 36, 181-197.

Csukás B. and Bánkuti Gy. 2003b. “Generic Bi-layered Net model of conservational and informational processes”. In: C. H. Dagli, A. et al. Eds. Intelligent Engineering Systems through Artificial Neural Networks, 13, ASME Press, New York, pp. 769-774.

Csukás B., Debelak, K. A., Prokop, A, Balcarcel, R. R., Tanner, R. D., Bánkuti Gy, Balogh S. 2003. “Generic Bi-layered Net Model Based Discrimination of Chemical and Biological Warfare Agents”, AIChE Annual Meeting, San Francisco, Manuscript 474f.

Kalman, R., Falb, P. and Arbib, M.. 1969. “Topics in Mathematical System Theory”, McGraw Hill.

Pantelides, C. C. 2001. “New Challenges and Opportunities in Process Modeling”. Proceedings of ESCAPE-11, Coppenhagen, Elsevier.

Petri, C. A. 1962. “Kommunikation mit Automaten”, Schriften des Institut für Instrumentelle Mathematik, 2, Bonn.

Temesvári K., Aranyi A., Csukás B., and Balogh S. 2004. “Simulated Moving Bed Separation of a Two Components Steroid Mixture”. Chromatographia,

ACKNOWLEDGEMENT The work was supported in part by Hungarian Scientific Research Fund T 037-297. BIOGRAPHY Béla Csukás was born in Keszthely, Hungary. He studied chemical engineering and process control at the University of Veszprém and obtained CSc/PhD degree in 1985. He worked for research institutes, industrial R&D and universities. Now he is Professor of Information Technology, leading the Institute of Mathematics and Information Technology at the University of Kaposvar, Hungary. Gyöngyi Bánkuti was born in Nagybajom, Hungary. She studied mathematics and mechanical engineering at the Technical University of Budapest and obtained PhD degree in 1990. She worked for various firms before moving to the University of Kaposvár, Hungary, where she is Associate Professor of Applied Mathematics and Chair of Department of Applied Mathematics and Physics.

generic bi-layered net, as the natural computational … · 2005. 5. 10. ·...

Documents