Stability Criterion for General Proper Systems with Constrained Control Structure

The focus of this paper is directed towards decentralized stabilizability of interconnected systems. It is assumed that the modes of the system are nonzero and distinct. The notion of quotient fixed mode (QFM), which was introduced in the literature for the class of strictly proper systems, is extended to comprise the class of general proper systems. It is to be noted that this extension is not trivial at all, even for simple two-input two-output systems. It is then shown that a mode of the system is fixed by means of any type of decentralized control law (i.e., nonlinear and time-varying) if and only if it is a QFM of the system. Two different approaches are proposed to determine the QFMs of a system, each one having its own advantages. It is also proved that any distinct and nonzero mode which is not a QFM, can be eliminated by means of sampling. Then, the important problem of placing all modes of the system simultaneously at some desired locations is discussed and an efficient procedure is given to solve it accordingly. A numerical example is presented to thoroughly illustrate the underlying results.