Stabilization of a stock‐loan valuation PDE process using differential flatness theory

Abstract

The article proposes flatness-based control for stabilization of a stock-loan valuation process that is described by a partial differential equation. By applying semi-discretization and the finite differences method, the state-space model of the stock loan has been obtained. It has been proven that the individual rows of this state-space model are nonlinear ODEs which can be viewed as differentially flat subsystems. For the local subsystems, into which the stock-loan PDE is decomposed, it becomes possible to apply boundary feedback control. The controller design proceeds by showing that the state-space model of the stock loan PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem's dynamics and can eliminate the subsystem's tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the stock loan PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space description. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the stock loan PDE system so as to assure that all its state variables will converge to the desirable setpoints.By showing the feasibility of such a control method it is also proven that through selected modification of the PDE boundary conditions the value of the stock loan can be made to converge and stabilize at specific reference values.