Multi-objective optimal trajectory design and tracking with non-periodic tracking-transition switching for non-minimum phase linear systems

In this paper, the problem of trajectory design and tracking of non-periodic tracking-transition switching for non-minimum phase linear systems is considered. Such a problem exists in various applications, where the output trajectory consists of application-dependent tracking sessions and to-be-designed transition sessions. The challenge arises when multiple control objectives are considered, including the smooth transition from one output tracking session to the next one without large oscillations during the transition, smooth tracking-transition switching without inducing pre- and/or post-switching oscillations, input-energy minimisation without saturation under amplitude constraint, and furthermore, minimisation of the overall transition time. The proposed approach extends the previous work that attained smooth output transition and smooth tracking-transition switching to further achieve amplitude-constrained input-energy minimisation and transition time minimisation. First, the constrained input optimisation problem is converted to an unconstrained input minimisation problem. Then, the optimal output and input are obtained by using an improved conjugate gradient method. Finally, the total transition time is further minimised via one-dimensional search. The proposed approach is illustrated through a simulation example in probe-based nanomanipulation utilising a piezoelectric actuator.