Extended multi-interval Legendre-Gauss-Radau pseudospectral method for mixed-integer optimal control problem in engineering

Abstract

Many engineering optimisation problems can be summarised as mixed-integer optimal control problems (MIOCPs) owing to the needs for mixed-integer dynamic control decisions. However, the convergence theory of Legendre-Gauss-Radau (LGR) approximation fails to apply to such non-smooth and discontinuous optimal control problems. Therefore, this paper develops an extended multi-interval LGR pseudospectral method (EMLGR), which has the following features: (i) the mixed-integer controls at the end of each interval and the interval intersections are added as two new controls to avoid the unrestrained control and shorten the switching time of integer control, and (ii) a smart adaptive collocation monitor (SACM) is provided to optimise the polynomial order and interval structure for further reducing computational complexity and improving approximation precision. The detailed solution procedure of EMLGR is given in this study, and experimental studies including five challenging practical engineering MIOCPs are taken to verify the superiorities of the proposed EMLGR in efficiency, accuracy and stability.