58 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY

2.4.7 Analysis with both Real and Complex Perturbations

The above results only apply to the case where ∆ is considered as a constant complex
valued matrix at each frequency. In many engineering applications restricting certain of
the ∆ blocks to be real valued may result in a less conservative model. Analysis with
such restrictions is referredto as the “mixed” µproblem.
Fore xample,consider the LFT form of the engine combustion model developed in
Section 2.2.4 (Equation 2.10). The block structure contains both r eal and complex
perturbations. A closed-loop model will also include these perturbations and the robust
stability and robust performance analyses will involve calculation of µwith respect to
both real and complex valuedp erturbations. We couldsimply assume that all
perturbations were complex; this would certainly cover the situation. However, such an
assumption may be too conservative to be useful. Calculation of mixed µwill give a
more accurate result in this case.
Efficient computation of µin the mixed case is discussed by Doyle, Fan,Young, Dahleh
and others [73, 74, 75, 76]. Accurate mixed µanalysis software will be available in the
near future. Unlike the complex µcase, this will not directly lead to a compatible
synthesis procedure. Significantly more work is required in this direction.
2.5 µSynthesis and D-KIteration

2.5.1 µ-Synthesis

Wenow lo ok at the problem of designing a controller to achievea perfor mance
specification for all plants, P(s), in a set of plants, P. The pr evious sections havedealt
with the questions of performance and robust stability in depth and the same framework
is considered for the synthesis problem. Figure 2.11 illustrates the generic synthesis
interconnection structure.
The lower half of this figure is the same as that for the H∞and H2design procedure.
The controllermeasurements are y, and the controller actuation inputs to the system are
u. The configuration differs from the standard H∞or H2case in that Fu(P(s),∆)
(rather than the nominal plant, P22(s)) is used as the design interconnection structure.
The problem is to find K(s) such that for all∆ ∈B∆,K(s) stabilizes Fu(P(s),∆) and