ON THE PARAMETRIC UNCERTAINTY OF WEAKLY REVERSIBLE REALIZATIONS OF KINETIC SYSTEMS

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1 HUNGARIAN JOURNAL OF INDUSTRY AND CHEMISTRY VESZPRÉM Vol. 42(2) pp. 3 7 (24) ON THE PARAMETRIC UNCERTAINTY OF WEAKLY REVERSIBLE REALIZATIONS OF KINETIC SYSTEMS GYÖRGY LIPTÁK, GÁBOR SZEDERKÉNYI,2 AND KATALIN M. HANGOS,3 Process Control Research Group, MTA SZTAKI, Kende u. 3-7, Budapest,, HUNGARY 2 Faculty of Information Technology, Péter Pázmány Catholic University, Práter u. 5/a, Budapest, 83, HUNGARY 3 Department of Electrical Engineering and Information Systems, University of Pannonia, Egyetem u., Veszprém, 82, HUNGARY lipgyorgy@gmail.com The existence of wealy reversible realizations within a given convex domain is investigated. It is shown that the domain of wealy reversible realizations is convex in the parameter space. A LP-based method of testing if every element of a convex domain admits wealy reversible realizations is proposed. A linear programming method is also presented to compute a stabilizing inetic feedbac controller for polynomial systems with parametric uncertainty. The proposed methods are illustrated using simple examples. Keywords: parametric uncertainty; computational methods; optimization; inetic systems Introduction The notion of parametric robustness is well-nown and central in linear and nonlinear systems and control theory [. It is used for ensuring a desirable property, such as stability, in a given domain in the parameter space around a nominal realization having the desired property. The aim of the paper is to extend the notions and tools of parametric robustness for a class of positive polynomial systems, namely a class of inetic systems. Only the very first steps are reported here that offer a computationally efficient method for checing one of the many important properties of inetic systems, their wea reversibility. Basic Notions and Methods The basic notions and tools related to reaction inetic systems and their realizations are briefly summarized in this section. Kinetic Systems, their Dynamics and Structure Deterministic inetic systems with mass action inetics or simply chemical reaction networs (CRNs) form a wide class of non-negative polynomial systems, that are able to produce all the important qualitative phenomena (e.g. stable/unstable equilibria, oscillations, limit cycles, multiplicity of equilibrium points and even chaotic behaviour) present in the dynamics of nonlinear processes [2. The general form of dynamic models studied in this paper is the following ẋ = M ψ(x), () where x R n is the state variable and M R n m. The monomial vector function ψ : R n R m is defined as ψ j (x) = n i= x Yij i, j =,..., m (2) where Y N n m. The system Eq.() is inetic if and only if the matrix M has a factorization M = Y A. (3) The Kirchhoff-matrix A has non-positive diagonal and non-negative off-diagonal elements and zero column sums. The matrix pair (Y, A ) is called the realization of the system Eq.(). The chemically originated notions: The chemically originated notions of inetic systems are as follows: the species of the system are denoted by X,..., X n, and the concentrations of the species are the state variables of Eq.(), i.e. x i = [X i for i =,..., n. The structure of inetic systems is given in terms of its complexes C i, i =,..., m that are non-negative linear combinations of the species i.e. C i = n j= [Y jix j for i =,..., m, and therefore Y is also called the complex composition matrix. The reaction graph: The weighted directed graph (or reaction graph) of inetic systems is G = (V, E), where

2 4 V = {C, C 2,..., C m } and E denote the set of vertices and directed edges, respectively. The directed edge (C i, C j ) (also denoted by C i C j ) belongs to the reaction graph if and only if [A j,i >. In this case, the weight assigned to the directed edge is C i C j is [A j,i. Stoichiometric subspace: Stoichiometric subspace S is given by the span of the reaction vectors S = {[Y i [Y j [A ij > }. (4) The stoichiometric compatibility classes of a inetic system are the affine translations of the stoichiometric subspace: (x + S) R n. Structural Properties and Dynamical Behaviour It is possible to utilize certain structural properties of inetic systems that enable us to effectively analyze the stability of the system. Deficiency: There are several equivalent ways to define deficiency. We will use the following definition δ = dim(er(y ) Im(B G )), (5) where B G is the incidence matrix of the reaction graph. It is easy to see that deficiency is zero if er(y ) = {} or equivalently ran(y ) = m. Wea reversibility: A CRN is called wealy reversible if whenever there exists a directed path from C i to C j in its reaction graph, then there exists a directed path from C j to C i. In graph theoretic terms, this means that all components of the reaction graph are strongly connected components. Deficiency zero theorem: A wealy reversible inetic system with zero deficiency has precisely one equilibrium point in each positive stoichiometric compatibility class that is locally asymptotically stable (conjecture: globally asymptotically stable). Computing Wealy Reversible Realizations Formulated as an Optimization Problem In this section, first a method for computing wealy reversible realization based on Ref.[3 is briefly presented. We assume that we have a inetic polynomial system of the form Eq.(). We use the fact nown from the literature that a realization of a CRN is wealy reversible if and only if there exists a vector with strictly positive elements in the ernel of A, i.e. there exists b R n + such that A b = [4. Since b is unnown, too, this condition in this form is not linear. Therefore, we introduce a scaled matrix à à = A diag(b) (6) where diag(b) is a diagonal matrix with elements of b. It is clear from Eq.(6) that à is also a Kirchhoff matrix and that R m (the m-dimensional vector containing only ones) lies in ernel of Ã. Moreover, it is easy to see that à defines a wealy reversible networ if and only if A corresponds to a wealy reversible networ. Then, the wea reversibility and the Kirchhoff property of à can be expressed using the following linear constraints m [à =, j =,..., m i= m [à i= [à [à ij ji ij ii =, j =,..., m, i, j =,..., m, i j, i =,..., m. (7) Moreover, the equation Eq.(3) is transformed by diag(b) (we can do this, because diag(b) is invertible): M diag(b) = Y A diag(b) }{{} à (8) Finally, by choosing an arbitrary linear objective function of the decision variables à and b, wealy reversible realizations of the studied inetic system can be computed (if any exist) in a LP framewor using the linear constraints Eq.(7) and (8). Wealy Reversible CRN Realizations In this section, first the convexity of the wealy reversible Kirchhoff matrix will be shown. After that the practical benefits of this property will be demonstrated in the field of system analysis and robust feedbac design. Convexity of the Wea Reversibility in the Parameter Space Theorem. Let A () and A (2) be m m wealy reversible Kirchhoff matrices. Then the convex combination of the two matrices remains wealy reversible. Proof. The idea behind the proof is based on Ref.[5. A Kirchhoff matrix is wealy reversible if and only if there is a strictly positive vector in its ernel. Therefore strictly positive vectors p, p 2 exist such as A () p = and A (2) p 2 =. Let us define the following scaled Kirchhoff matrix:  () = A () diag(p ) and Â(2) diag(p 2 ). These scaled matrices have identical A (2) = structures to the original ones. Moreover, Â() (m) = and Â(2) (m) = where the vector (m) denotes the m dimensional column vector composed of ones. For that (λâ() + ( λ)â(2) ) (m) =. (9) for any λ [,. Therefore the convex combination of the original two realizations has to be wealy reversible.

3 5 (a) (b) (c) Figure : A wealy reversible reaction graphs of the three realizations (Y, A () ), (Y, A(2) ) and (Y, A(3) ) Wea Reversibility of CRN Realizations with Parametric Uncertainty We assume that a CRN with parametric uncertainty is given as ẋ = M ψ(x), () where x R n is the state variable, ψ R n R m contains the monomials and the matrix M R n m is an element of the following set M = { l α i M i ( i : α i ) i= l α i = }. i= () The goal is to find a method for checing the wea reversibility of the system Eq.() for all matrices M M. When all vertices M i have a wealy reversible realization (Y, A (i) ) then any element of the set M has a realization (Y, A ) such that A is the convex combination of the Kirchhoff matrices A (i). The obtained realization A will be wealy reversible due to Theorem. Therefore, it is enough to compute a wealy reversible realization for each matrix M i by using the previously presented LP-based method. A Simple Example Let us consider the following polynomial system [ ẋ = M x x ẋ 2, (2) 2 x x 2 where M is an arbitrary convex combination of the following three matrices [ M =, M 2 = M 3 = [ [, and. Figure 2: A wealy reversible realization of the convex combination M =.2M +.4M 2 +.4M 3 In order to show wea reversibility for all possible convex combinations, we have to find a wealy reversible realization for each matrix M, M 2 and M 3. The resulting wealy reversible reaction graphs are depicted in Fig., while Fig.2 illustrates an inner point realization which is wealy reversible too. Computing Kinetic Feedbac for a Polynomial System with Parametric Uncertainty Besides the possible application of the above described LP-based method for robust stability analysis, it can also be used for stabilizing feedbac controller design. For this purpose, a generalized version of our preliminary wor on inetic feedbac computation for polynomial systems to achieve wea reversibility and minimal deficiency [6 is used here. The Feedbac Design Problem We assume that the equation of the open-loop polynomial system with linear constant parameter input structure is given as ẋ = M ψ(x) + Bu, (3) where x R n is the state vector, u R p is the input and ψ R n R m contains the monomials of the open-loop system. The input matrix is B R n p, the corresponding complex composition matrix is Y with ran m, and M R n m is an element of the following set { l M = α i M i ( i : α i ) i= } l α i =. i= (4) Moreover, a positive vector x R n > being the desired equilibrium point is given as a design parameter. Note that the above polynomial system is not necessarily inetic, i.e. not necessarily positive, and may not have a positive equilibrium point at all.

4 6 The aim of the feedbac is to set a region in the state space R R n where x is (at least) a locally asymptotically stable equilibrium point of the closed-loop system for all M M. For this purpose we are looing for a feedbac in the form u = Kψ(x) (5) which transforms the open-loop system into a wealy reversible inetic system with zero deficiency for all M M with the given equilibrium point x. Feedbac Computation Similarly to the realization computation, the matrix K will be determined by solving an LP problem. The convexity result shows that it is enough to compute one wealy reversible realization (Y, A (r) ) in each vertex M r to ensure wea reversibility for all possible closed-loop systems. All realizations will have zero deficiency, because of the ran condition ran(y ) = m [7. First we note, that the realization (Y, A (r) ) that corresponds to the closed-loop system is M r + B K = Y A (r). (6) where the matrix A (r) should be Kirchhoff m [Ã ij =, j =,..., m i= [Ã ij, i, j =,..., m, i j [Ã ii, i =,..., m. (7) In order to obtain a wealy reversible closed-loop system with an equilibrium point x, the matrix A (r) should be wealy reversible and has to have the vector ψ(x) in its right ernel, i.e. A (r) ψ(x) =. (8) Finally, by choosing an arbitrary linear objective function of the decision variables A (),..., A(l) and K, the feedbac gain K can be computed (if it exists) in a LP framewor using the linear constraints Eqs.(6-8). With the resulting feedbac gain K, the point x will be an equilibrium point of all possible closed-loop systems, and x will be locally asymptotically stable in the region S = (x + S) R n, where S is the stoichiometric subspace of the closed-loop system. Example Let the open-loop system be given as ẋ = M x x 2 x 2 x 3 + x u (9) Figure 3: Wealy reversible realization of the closed-loop system, where M =.6M +.2M 2 +.2M 3 where M is an arbitrary convex combination of the following three matrices: M = M 2 = M 3 = ,, and The desired equilibrium point x = [ T. We are looing for a feedbac law with gain K which transforms the matrices M i into wealy reversible inetic systems with the given equilibrium point. By solving the feedbac design LP optimization problem using the linear constraints Eqs.(6-8), the computed feedbac is in the following form:. u = [ 2 2 ψ(x). (2) Fig.3 depicts a wealy reversible realization of the closed-loop system. The obtained closed-loop system in an inner point of the convex set M has the following stoichiometric subspace: S = span,. (2) Therefore, the equilibrium point x will be asymptotically stable with the region S = (x + S) R n. Note, that one should choose the initial value of the state variables from S. Fig.4 shows the time dependent behaviour of the closed-loop solutions started from different initial points in S.

5 7 REFERENCES Figure 4: Time-domain simulation of the closed-loop system Conclusion It is shown in this paper that the domain of wealy reversible realizations is convex in the parameter space. This property is utilized for developing methods in system analysis and robust control design. An LP-based optimization method is proposed for testing if every element of a convex domain given by its extremal matrices admits a wealy reversible realization. An LP-based feedbac design method is also proposed that guarantees stability with a desired equilibrium point. The proposed methods are illustrated with simple examples. [ SILJAK D.: Parameter space methods for robust control design: a guided tour, Aut. Contr., IEEE Transactions on, 989, 34(7), [2 ANGELI D.: A tutorial on chemical networ dynamics, Eur. J. Contr., 29, 5, [3 SZEDERKÉNYI G., HANGOS K. M., TUZA Z.: Finding wealy reversible realizations of chemical reaction networs using optimization, MATCH Commun. Math. Comp. Chem., 22, 67, [4 FEINBERG M., HORN F.: Chemical mechanism structure and the coincidence of the stoichiometric and inetic subspaces, Arch. Rational Mechanics and Analysis, 977, 66(), [5 RUDAN J., SZEDERKÉNYI G., HANGOS K.M., PÉNI T.: Polynomial time algorithms to determine wealy reversible realizations of chemical reaction networs, J. Math. Chem., 24, 52(5), [6 LIPTÁK G., SZEDERKÉNYI G., HANGOS K.M.: Kinetic feedbac computation for polynomial systems to achieve wea reversibility and minimal deficiency, 3th Eur. Control Conf. ECC, Strasbourg, France, 24, [7 ANGELI D., LENHEE, P.D., SONTAG E.D.: On the structural monotonicity of chemical reaction networs, Proc. 45th IEEE Conf. Decision and Control, 26, 7 2