Cheap and singular controls for linear quadratic regulators

A general not necessarily square and not necessarily invertible linear system with quadratic cost function where a small parameterμ²multiplies the control cost is considered. Due to the cheapness of control, a strong control action in the form of high-gain feedback forces the given system to have slow and fast transients of a hierarchy of time-scales coupled with high- and low-amplitude interactions. By appropriate amplitude and time scaling of variables, these interactions are normalized and the problem is decomposed into several nonsingular subproblems of minimal order, each pertaining to only one time-scale. Complete results characterizing the limiting behavior of the optimal performance index, eigenvalues, trajectory, and control variables as are given. All the different ways by which nonuniqueness can occur into singular control are discussed. More importantly, the method developed here allows the design of a high-gain feedback system in terms of the design of several lower order subproblems. Thus, it gives a strong impetus for practical implementation of the theory developed.