Automatic Semiqualitative Analysis: Application to a Biometallurgical System

Automatic Semiqualitative Analysis: Application to a Biometallurgical System R. M. Gasca, J. A. Ortega, M. Taro Departamento de Lenguajes y Sistemas Informaticos, Facultad de Informatica y Estadistica Avda. Reina Mercedes s/n, Sevilla (Espana) e-mail:{gasca,jaortega,mtoro }@lsi.us.es Abstract. The aim of this work is the representation and analysis of semiqualitative models. Their qualitative knowledge is represented by means of qualitative operators and envelope functions. A semiqualitative model is transformed into a family of quantitative models. In this paper the analysis of a model is proposed as a constraint satisfaction problem. Constraint satisfaction is an umbrella term for a variety of techniques of Artificial Intelligence and related disciplines. In this paper attention is focused on intervals consistency techniques. The semiqualitative analysis is automatically made by means of consistency techniques. The presented method is applied to a industrial biometallurgical system in order to show how increase the capacity of production. 1 Introduction In engineering and science, the models made up for the study of dynamical systems are normally composed of quantitative and qualitative knowledge. This knowledge is composed by both of them. It is known as semiqualitative knowledge. Real models contain quantitative, qualitative and semiqualitative knowledge. All this knowledge must be considered when these models are studied. The techniques developed to analyze and simulate quantitative models are well known. A great variety of techniques has been studied for the representation and the manipulation of qualitative knowledge, such as algebra of signs, interval arithmetic, fuzzy sets, and order of magnitude reasoning. In order to analyze industrial models, it is necessary sometimes to solve conflicts on the request of accuracy and flexibility. The models of dynamical systems should provide different levels of numerical abstraction for their elements. These levels may be a purely qualitative description [8), semiqualitative [2), [6), numerical based on intervals [11), quantitative and mixed of all levels [7]. On other hand, the systems dynamics obtains the differential equations of a system from its structure. This technique could obtain different qualitative behaviors of a given structure. The analysis of these behaviors constitutes the qualitative analysis of dynamical systems. The mathematical qualitative theory of dynamical systems involved studying qualitatively the behaviour (e.g. asymptotic behaviour) of time evolving systems. 322 In order to automate the qualitative analysis of dynamic systems several applications have been developed. They combine techniques of numerical methods with symbolic computation, and methods proceeding from the knowledge of the science and the mathematics. These applications begin with the top-level specifications of physical model. They prepare simulation experiments, and accomplish them. Also they interpret the numerical results, and they formulate the results in qualitative terms. Among them, we can cite PLR [9], bifurcation interpreter [1], KAM [12], POINCARE [10], and MAPS [13]. In this paper, a method to carry out the analysis of dynamical systems automatically is shown. The semiqualitative analysis is proposed as a set of interval constraint satisfaction problems. They are solved applying consistency techniques [5]. 2 Semiqualitative models A dynamical system can be considered as the constraints <P(x, x,p), x(to) = xo, <Po(p,xo) ( 1) being x the state variables of the system, p the parameters, x the variation of the state variables with the time, <Po the constraints among parameters and initial conditions, and <P the constraints on x, x and p. The dynamical system represented in (1) can symbolically be transformed into a set of contraints with variables, parameters and intervals. In this paper, we only study systems that can be transformed as x = l(x,p), x(to) = Xo, <Po(p, xo) (2) The vector field f may be composed of quantitative and qualitative variables, constants, arithmetic operators, functions and envelope functions, expressed as it is indicated in our previous paper [4], where qualitative variables and envelope functions are transformed to interval expressions. If we take into account the stablished concepts in that paper, the dynamical system (2) is transformed in x = f(x,r,p), x(t 0) = xo, <Po (p, r, xo) (3) where r E II are new parameters, p E II, xo E II, and f does not contain envelope functions, being II the set of closed intervals of JR. These functions represent a dynamical systems family depending on p, x0 and r. It is denoted as semiqualitative model and it is represented further on x = f(x,p), x(to) = xo, <Po(p,xo) (4) where p and r have been joined in an unique parameters vector p. 323 3 Semiqualitative analysis Qualitative analysis of a dynamical system intends to analyze the phase portrait or phase space of the system. The phase space of the dynamical system is constituted by the variables of state x, and the extended phase space by variables and parameters x, p. The phase portrait is formed by the projection of the trajectories of the dynamical system in the extended phase space. The phase portrait is interpreted as a correspondence between the differential equations and the vector field. In this paper semiqualitative systems that they are stable structurally are studied. In them, little perturbations keep their qualitative behaviors. The first step of semiqualitative analysis of a dynamical system ( 4) is the determination of the equilibrium regions. They are defined by the constraints Equilibrium(x,p)::: { f(x,p) = 0, (5) The study of solutions of (5) let us know the structure of the phase portrait. Each stable equilibrium region is an attractor region. The stability of each equilibrium region is related to the real part of the eigenvalues of the Jacobian of the system. It has been demonstrated in the bibliography that in the stable fixed points the real part of the eigenvalues is negative. In order to apply the stability criteria, it is necessary to construct the following determinants. They are formed with the coefficients of the characteristic polynomial Pn of the Jacobian matrix A of the dynamical system. The Jacobian matrix of (4) is A= Dxf(x,p), and Pn is defined as Pn(,\) = det(A- ,\I)= aoAn + a1,\n1 + ... + an-1,\ +an (6) In order to determine the stability conditions, the matrices are defined ( a1 a3 as ... a2i-1) . _ d t ao a2 a4 ... a2;- 2 b . . _ 1 g, -e emg z - , ... , n ... ... ... ... . .. 0 0 0 0 a; (7) The elements ak of g; are the coefficients of Pn for k > n, and 0 for k ::S n. Both ak and g; are symbolic expressions dependent on x, p. We can apply two stability criteria. First is the Routh-Hourwitz criterion. For this criterion the predicate Stable_Pol is defined as Stable_Pol(Pn(,\)) =: { g1 > 0, ... ,gn > 0 (8) Second is the Lineard-Chipard criterion. It defines the predicate Stable_Pol as Stable_Pol(Pn(,\)) := { a 1 > O, .... ,an> O, gn-1 > O,gn-3 > 0, ... Therefore the constraints that define the stable equilibrium regions are Stable(x ) = { Equilibrium(x,p), A= Dxf(x,p), ,p - Pn = Pc(A), Stable_Pol(Pn(,\)) (9) (10) 324 where Dx stands for the Jacobian, and Pc stands for the set of constraint of characteristic polynomial. If constraints (10) are satisfied by an equilibrium region, it is stable. Otherwise it is not stable. The study of the bifurcations points of a system intends to divide the parameters space in regions. The system has the same number and type of attractors in these regions. The frontiers of these regions are formed by bifurcation points. An at tractor appears, disappears or changes of type, when we cross a determined frontier. The most elemental classification of bifurcation points distinguishes them into statics and dynamics. The statics bifurcation points are the simplest. They appear in those points where the number of attractors points varies. The determinant of the Jacobian matrix is annuled in them, that is, the characteristic polynomial has a null root. The dynamic bifurcation points involve limit cycles or strange attractors. We study the Hopf bifurcation, where an attractor point is converted into a limit cycle or vice versa. In these bifurcation points the characteristic polynomial of the Jacobian matrix has a pair of roots with real part equal to zero. { E.quilibrimn(x,p), { l~quilibrium(x,p), A= Dxf(x,p), A= Dxf(x,p), Sta_Bif(x,p) := Pn = Pc(A), Din_Bif(x,p) := Pn = Pc(A), (11) Pn = ,\ Qn-1, Pn = (,\ 2 + w2) Qn-2, Stable_Pol(Qn-1) Stable_Pol(Qn-2) It is interesting to notice that all the predicates defined between (5) and (11) are formuled as interval constraint satisfaction problems. They are solved by adequate consistency techniques [5]. 4 A biometallurgical system 4.1 Description and determination of the model For a long time, it has been observed natural transformations of the sulphur and iron compounds. They are originated from the dissolution of minerals. Presence of iron-oxiding bacteria in mining areas and their acid drainages has been reported repeatedly. Thiobacillus Ferrooxidans is considered to be the most important organism for the bacterial leaching of minerals. In indirect leaching the bacteria generate ferric iron by oxidizing soluble ferrous iron. The global reaction is (11) This method for production of acidified ferric solutions is used because ferric iron in turn oxidizes other metals in mineral, transforming them in the soluble form, and because it avoids ecological contamination problem of industrial extraction of metals from the rocks. 325 If the equation of Michaelis-Mention is applied to the reaction ( 11), then oxidation rate V is calculated as follows [5] V = Vmax km + [5] (12) where Vmax is maximum rate that it can be reached by increasing in the substrate concentration, [5] is substrate concentration, and km is Michaelis constant. This constant stands for the concentration which the reaction rate is half of the maximum rate. This equation has two problems: the concentration bacterian is not constant and it cannot be applied to the bacterian growth because it is exponencial. Due to the complexity of the factors that take part in the bacteria oxidation of Fe(II) in Rotating Biological Contactors ( RBC), as shown in figure (1). It has not been possible to determine a general mathematical model for this process. However, it has been proved that the bioxidation reaction continues a kinetic of first order with respect to the substrate concentration. In the experimentation there are two interconnected RBC. In them it is introduced a flow Q with an ferrous iron concentration. Disk Divisiof ~--~---------.---. Drive shaft Outfluent +-C: :J +- Influent L-~--~--~--~--~ Lateral Raised Ground Fig. 1. A Rotating Biological Contactors ( RBC) The equations of the model of this dynamical system are (13) 4.2 Experimental data According to the experimental results [3] it has been determined the quasiequilibrium points for the system. They have been obtained studying different influent flows p1 and values of iron concentration in such flows P2. 326 According to the data supplied by the experts P5 is similar to p9 and their order of absolute magnitude is moderately positive, P7 is very positive, and p3 is slightly greater that P7. Therefore, if it is associated the corresponding intervals to the previously expressed qualitative operators, it is obtained P1 = 0.61ljh,p2 = 3.96gjl, P3 = [5.6,5.8),p4 = 0.741, P5 = [0.4,0.5), P6 = 0.015, P7 = [5.3, 5.6), Ps = 0.781, pg = [0.4, 0.5), Plo = 0.01 Using these data and applying the exposed techniques, we carry out the semiqualitative analysis of these dynamical systems. 4.3 Semiqualitative analysis The semiqualitative analysis of this system is carried out to study how to increase the capacity of production, when systems parameters are varied. The equilibrium regions of the system are determined solving the network of contraints ! (P1P2P3P4x,"'+psP1x1) P6 = 0, (Pl X1P7 Ps x2 ".[:P• - P1X2) PlD = 0, Equilibrium(x,p) ::::= 0.4:::; p5 :::; 0.5, 0.4:::; p9 :::; 0.5, 5.6 :=:; P3 :=:; 5.8, 5.3 :S P7 :S 5.6, P1 = 0.61, P2 = 3.96, P4 = 0.74, P6 = 0.015, Ps = 0.78, P10 = 0.01 If it is applied interval arithmetic the results obtained are too wide. Nevertheless it is applied interval consistency techniques developed in [5] and we will obtain a narrowing equilibrium region Equilibrium(x,p) = {[0.397,0.49], x [0.0198,0.034]} This solution includes all experimental results obtained from different experience data. The Jacobian matrix of this model is The characteristic polynomial of A is Pn(..\) = ao..\ 2 + a1..\ + a2 = ..\ 2 + (-aua22)..\ + aua22a12a21 and according to the Lienard-Chipard criterion, Stable_Pol(Pn(..\)) ::::= {(-aua22) > 0, aua22a12a21 > 0 Substituting Pi for their values and simplifying, the constraints that define the stability are 327 These constraints are satisfied with the obtained equilibrium region and therefore it is conclude that the region is stable. The constraints that define the bifurcations are St B .f( ) = {Equilibrium(x,p), D. B"f( ) = {Equilibrium(x,p), a_ z x, p _ 0 0 zn_ z x, p _ 0 0 a1 > , a2 = a1 = , a2 > When it is applied constraint satisfaction techniques to these constraints, there are no solutions, and hence the system has no bifurcations. 5 Conclusions This paper proposes a method to carry out automatically the semiqualitative analysis of dynamical systems by interval consistency techniques. Qualitative knowledge is represented by intervals, and they are qualitative operators and envelope functions. It has been applied the proposed approach to systems appeared in the bibliography and the obtained results are quite similar to them. In this paper, it has been studied a real biometallurgic system. The achieved results have allowed to know how to increase the capacity of production. In the future, we are going to apply the previous techniques to other real problems. 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