Sensitivity analysis for the optimization of switched dynamical processes with state-dependent switching conditions and its application

Abstract

This paper considers an optimization problem of switched dynamical processes, which is essentially a mixed integer optimization problem. Unlike the existing optimization problem, the state-dependent switching strategy is adopted to design the switching rules. Our main contributions are as follows. Firstly, by introducing a discrete-valued function, a control vector parameterization technique, and a novel relaxation technique, the switched dynamical process optimization problem is approximated by using a more tractable nonlinear constrained parameter optimization problem. Unlike the existing works, the number of locally optimal solution does not increase due to the relaxation transformation. Next, in order to obtain a numerical solution of this parameter optimization problem, a novel penalty function approach is developed. Unlike existing penalty functions, this novel penalty function is neither quadratic nor linear, and has exponential convergence rates. Furthermore, a sensitivity analysis method is developed for investigating the effect of small perturbations in constraints and switching conditions on the objective function. Finally, a dynamic optimization problem in cancer chemotherapy process is presented to illustrate the effectiveness of the approach provided by this paper. Numerical simulation results show that the approach provided by this paper is low time-consuming, has faster convergence speed, can obtain a better objective function value than the existing methods, and is robust for the small perturbations in constraints and switching conditions.