Output detectability, steady-state invertibility and the general servomechanism problem

The notion of output detectability of a system is introduced which is concerned about the problem of being able to detect the effect of disturbances on the output of a system using different measurable outputs. Necessary and sufficient conditions are found for a linear time-invariant system to be output-detectable. Properties of the output-detectability condition are then obtained; in particular, it is shown that a system is "almost always" output-detectable, if the number of measurable outputs is not less than the number of disturbances; if the number of measurable outputs is less than the number of disturbances, then a system is "almost never" output detectable. Application of these results is then made to find necessary and sufficient conditions for a solution to exist to a general servomechanism problem, in which the measured outputs of the system are, in general, different from the outputs to be regulated. Explicit controllers which solve this general servomechanism problem are obtained. It is then shown that there is "almost always" a solution to this general servomechanism problem if and only if the number of control inputs is not less than the number of outputs to be regulated and the number of measured outputs is not less than the number of disturbances. A frequency-domain interpretation of output-detectability and the solvability of the general servomechanism problem is then made and it is shown, in particular, that for stable, minimum phase systems which have sufficient inputs and measurable outputs, there always exists a solution to the servomechanism problem.