Planning a haemodialysis process by minimum time control of hybrid systems with sliding motion

The aim of the paper is to provide a computational tool for planning a haemodialysis process. It is shown that optimization methods can be used to obtain the most effective treatment focused on removing both urea and phosphorus during the process. Our approach to the planning process is based on a model which takes into account a rebound phenomenon and that results in a hybrid model in which sliding motion is likely to occur. For such a model we construct an optimization problem and a computational method for solving it. The presented approach to optimal control problems with hybrid systems is different from the others in several aspects. First of all, it is assumed that a hybrid system can exhibit sliding modes. Secondly, the system's motion on the switching surface is described by index 2 differential-algebraic equations and that guarantees accurate tracking of the sliding motion surface. Thirdly, the gradients of the problem's functionals are evaluated with the help of adjoint equations. The adjoint equations presented in the paper take into account sliding motion and exhibit jump conditions at transition times. The stated adjoint equations are used in a globally convergent algorithm which generates a sequence of controls whose accumulation points satisfy the weak maximum principle for optimal control problems with hybrid systems. The paper presents numerical results of solving haemodialysis planning problem.