Much work over several years has been directed toward the development and application of the H _? design method (see Refs. 1 13), as the historical sketch provided in Fig. 1.1 shows. This method has several very promising characteristics: the ability to specify the closed-loop system's frequency response via requirements on the sensitivity S and complementary sensitivity T (see Ref. 14), and the ability to directly minimize an undesirable resonance or peak in the frequency response15 while simultaneously closing all loops from the input to the output vector in essentially one single design step. ^{16 19}

Figure 1.1: History of H _? optimal control theory.

The H _? design algorithm is a linear, point design methodology whose success depends on two attributes of the model.20 ^, 21 First, the system must be linearized about the design point, requiring the nonlinear system to exhibit behavior close to that of the linear approximation at this trim point. Second, the system must behave in a linear manner throughout its operational envelope. If not, the point-designed system will not be robust enough to operate properly everywhere within this envelope. In this case the controller must adapt for operation off this trim point. The question then is how to modify it. Also, if several different controllers are generated for different conditions, how are the control gains scheduled between these conditions?

One method used to model nonlinear systems is the describing function approach. ^{22 27} The advantages to this approach are that it...