Sensitivity shaping in feedback control and analytic interpolation theory∗

The problem of stabilizing a linear control system leads to several important considerations. In particular, there are certain basic requirements which need to satisfied in any practical feedback system [6, 7, 11, 18]. First, the closed-loop system needs to be internally stable, i.e., the transfer function between any two points in the loop should be stable. Internal stability guarantees that all signals in the system remain bounded when a bounded signal is injected at any location. Secondly, the absolute value of the closed-loop transfer function needs to be bounded in the right half-plane. Prescribing a fixed uniform bound, the design of an internally stable feedback system leads to a Nevanlinna-Pick interpolation problem [12,13,16]. In many situations, we require in addition that the closed-loop transfer function has bounded degree, often chosen to be equal to the number of interpolation conditions. In general, this ensures not only low degree of the controller but also that the feedback system behaves like a low-order system, a common specification in many applications. In general, there are infinitely many transfer functions satisfying these these conditions, and one would like to select one which best satisfied some additional specifications. However, classical theory does not provide procedures for determining an arbitrary such solution, but only a particular one, known as the central solution. Recently, however, a new theory has been developed [1,2,5,9,10], which provides a complete parametrization of all solutions and a procedure for determining each of them. This is a modification of a theory previously developed for the Caratheodory extension problem with degree constraint [3, 4, 8]. In this paper, we shall apply this theory to the design problem described above. In Section 2 we set up and motivate the problem, in Section 3 we review pertinent fact about the theory of Nevanlinna-Pick interpolation with degree constraint, and in Section 4 we discuss various desighn strategies and give some examples. The sensitivity shaping problem considered here is a special case of the model matching problem [6,7, 11], and the theory can also be appled to this problem. Only the single-input-single-output case is considered here. For such problems good results can be obtained by manual loop shaping. It is therefore desirable to extend