# (651b) Modeling and Optimization of Biological Reactive Systems By Using Fractional Calculus Theory

Authors:
Tecnologico Nacional de Mexico en Celaya
Instituto Tecnológico de Celaya
Instituto Tecnologico de Celaya
Vishwamitra Research Institute

Recent work in science and engineering have demonstrated that the dynamics of some systems are described more accurately by using fractional order differential equations (FDE). For example, it has been illustrated that materials with memory effects and dynamic diffusion processes, such as gas diffusion and heat conduction in fractal porous media, can be more adequately modeled by fractional order models than integer order models.  This paper focuses on the use of fractional calculus as a novel mathematical tool in process systems engineering. First we show that the dynamics of some reactive systems displaying anomalous behavior, such as some biological processes, can be represented by fractional order differential equations. Then, when the resulting fractional system is incorporated into an optimization framework, the resulting problem is a fractional optimal control problem (FOCP). Therefore, this work is concerned with both the modeling and optimization of systems whose governing equations contain at least one fractional derivative operator. A fermentation system and a thermal hydrolysis reaction are the two biological processes which are used as the bases for our analysis.

Further, three main issues are involved in our approach. The first issue is to formulate a consistent fractional dynamic system. For the case of the fermentation system, where an empirical ordinary model obtained from experimental data was originally reported in the literature, we use nonlinear numerical fitting techniques to obtain the fractional orders and the kinetics parameters for the fractional dynamic system. The resulting fractional model is clearly simpler than the ordinary model. Our second example, however, is a thermal hydrolysis system where an ordinary model including several ordinary differential equations was derived from fundamental principles. In that case, the formulation of a fractional reactive model is not as simple as changing the order of the ordinary derivatives of the left-hand side of the differential equations, since that approach may produce inconsistent systems which violate mass balances. A consistent fractionalization approach was used to formulate a fractional system which simultaneously satisfies mass balances and allows different fractional orders. Once the fractional dynamic models were available, the second issue includes the definition of a performance index and the definition of the control variables so that consistent FOCPs are formulated. The last issue is the implementation of the numerical solution approach to the FOCPs. For the fermentation model, we formulated the Euler-Lagrange optimality conditions of the FOCP, which resulted in a fractional boundary value problem. Such a problem was solved through a numerical iterative procedure based on the gradient method developed for conventional optimal control problems. The case of the thermal hydrolysis is interesting. In that case, the solution approach involves the combined use of Laplace transformation, discretization of the fractional dynamic system and a nonlinear programming technique. Moreover, a fractional order integration technique was need as an additional tool for the numerical fitting and the optimization techniques. We used a generalization of the Adams-Bashforth-Moulton method for its application to fractional order equations. The results are used to show the potential advantages of fractional calculus as a modeling tool and to analyze the differences among the solutions obtained for the FOCP with those obtained for the integer case (conventional optimal control problem).