Minimum-time/energy output-transitions in linear systems

This article addresses the optimal (minimum-time/energy) trajectory design for rapid output transitions, i.e. changing the output from one value to another, in linear systems. Furthermore, the output is required to be maintained constant (e.g. without vibration) outside the transition time interval. The output-transition problem is posed as a linear quadratic minimum-time (LQMT) optimal control problem, which avoids bang-bang controllers that result from solving the traditional time-optimal problem. Additionally, the LQMT approach allows the time-optimal requirement to be traded off with the energy requirement by selecting appropriated weighting factors. Current methods transform this LQMT output-transition problem into a state-transition problem by constraining the initial and final state of the output-transition interval. However, the choice of the initial and final states can be ad hoc and the resulting control law may not be optimal. In contrast, the proposed approach directly solves the LQMT output-transition problem by optimally choosing the initial and final states to minimize the output-transition cost. The novelty of the proposed approach is that inputs are not applied just during the output-transition time interval; rather, inputs are also applied before the beginning of and after the end of the output-transition time interval (these inputs are called pre- and post-actuation). The method is illustrated using a flexible structure model, and simulation results show substantial reduction in output-transition cost when compared with the cost of standard state-transition-based approaches, which do not use pre-and post-actuation.