Input/output structure of the infinite horizon LQ bumpless transfer and its implications for transfer operator synthesis

The steady-state input/output structure of the infinite horizon LQ bumpless transfer topologies for strictly proper controllers is analyzed. It is found that in these topologies (1) the steady-state gain of the transfer function from the output of the online controller to that of the offline one is unity/asymptotically unity, and (2) the steady-state gain of the transfer function from the input of the online controller to the output of the offline one is zero/asymptotically zero, for essentially all strictly proper controllers with the input dimension being no less than the output one. These facts (1) reveal the steady-state gain structure of the closed-loop transfer matrix in the standard LQ tracking problems, (2) demonstrate that the LQ bumpless transfer technique solves the ideal bumpless transfer problem for controllers with an integrator in each of their output channels, (3) reveal the structure of unavoidable signal discontinuities in the controller input/output upon transfer between controllers with non-integrating channels, and (4) provide guidance for minimizing the above signal discontinuities and the resulting bumps in the plant output in bumpless transfer synthesis.