Cheap optimal control of discrete single input single output systems

This paper presents a solution to the linear discrete quadratic optimal control problem for a single input single output plant with a control weight-ing that tends to zero. Explicit expressions are derived for the optimal gain vector in terms of the first significant powers of the control weighting, and the limiting values of these expressions arc considered. The optimal index of performance is obtained both for the minimum and non-minimum phase plants. The clear distinction between these two types of plants is discussed and the non-vanishing value of the index of performance in the latter case is expressed in terms of the zero directions of the system. Extensions to the single input multi output case and the case where the index of performance is a function of dynamic operators are introduced.