Trajectory planning and tracking control for multiple-input multiple-output,linear time-invariant, sampled data systems in continuous time

In applications, usually sampled data controllers are employed. If the sampling time is sufficiently small, the sampled data structure may be neglected and the system is treated quasi-continuously. If the sampling time is longer, the system is treated in discrete time neglecting the behaviour in between the sample instances and the z-transform is used. This article combines these two approaches. The plant is driven by an input which consists of a sequence of values and a function forming the actuating variable. By use of the modified z-transform, the plant is modelled by a parametric transfer function matrix, whose additional parameter discloses the behaviour between sampling instances. Thus, the output signal is calculated not only at the points of sampling. A right co-prime matrix-fraction description is derived. Building on that description, basis variables are defined in the z-domain. The corresponding basis sequences can be chosen arbitrarily and with them the input sequences and the output functions are fixed and can be calculated without solving a differential or difference equation. This mathematical fact is applied to plan trajectories in continuous time. Hence, the entire output trajectory in continuous time can be taken into account. A tracking controller may be added to ensure that the disturbed system complies with the plan.