The steady-state invertibility and feedforward control of linear time-invariant systems

The notion of steady-state invertibility of a system is introduced, which is concerned about the problem of being able to find an input so that the output of a stable system is asymptotically equal to a specified output of a certain class of functions. Necessary and sufficient conditions are found for a linear time-invariant system to be steady-state invertible. Application of these results is then made to find necessary and sufficient conditions for a feedforward controller to exist for a general linear time-invariant system, so that asymptotic tracking, in the presence of a general class of measurable disturbances occurs. Explicit feedforward controllers which will accomplish this are obtained. Properties of the steady-state invertibility condition are then obtained; in particular, it is shown that a system is "almost always" steady-state invertible if the number of plant inputs is not less than the number of outputs; if the number of plant inputs is less than the number of outputs, then a system is "almost never" steady-state invertible. It is then shown that a system which is minimum phase and which has at least the same number of inputs as outputs is always steady-state invertible.