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58 CHAPTER 2. OVERVIEW OF 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 referred to as the “mixed” µ problem.
For example, consider the LFT form of the engine combustion model developed in
Section 2.2.4 (Equation 2.10). The block structure contains both real and complex
perturbations. A closed-loop model will also include these perturbations and the robust
stability and robust performance analyses will involve calculation of µ with respect to
both real and complex valued perturbations. We could simply 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 µ will give a
more accurate result in this case.
Efficient computation of µ in the mixed case is discussed by Doyle, Fan, Young, Dahleh
and others [73, 74, 75, 76]. Accurate mixed µ analysis software will be available in the
near future. Unlike the complex µ case, this will not directly lead to a compatible
synthesis procedure. Significantly more work is required in this direction.
2.5 µ Synthesis and D-K Iteration
2.5.1 µ-Synthesis
We now look at the problem of designing a controller to achieve a performance
specification for all plants, P(s), in a set of plants, P. The previous sections have dealt
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 H
2
design procedure.
The controller measurements are y, and the controller actuation inputs to the system are
u. The configuration differs from the standard H
∞
or H
2
case in that F
u
(P(s), ∆)
(rather than the nominal plant, P
22
(s)) is used as the design interconnection structure.
The problem is to find K(s) such that for all ∆ ∈ B∆, K(s) stabilizes F
u
(P(s), ∆) and