Vector-valued versus scalar-valued figures of merit in H^∞-feedback system design

In this paper, we adopt the point of view that the real figure of merit in H^∞-feedback systems design is the vector-valued performance measure (∥W₁S∥_∞, ∥W₂T∥_∞), where W₁S is the frequency-weighted sensitivity function and W₂T:= W₂ (I-S) the weighted complementary sensitivity function. A compensator C₀ is "optimal" if its induced performance (∥W₁S(C₀)∥, ∥W₂T(C₀)∥) is a minimal element of the set of achievable performances in the (∥W₁S∥, ∥W₂T∥)-plane. This set is shown to be convex, and the "fundamental limitations on achievable feedback performance" take the geometric interpretation of a polygon bounding from below this convex set. The H^∞-theory deals with feedback system performance tradeoffs by lumping the two conflicting objective functions S and T into a scalar-valued criterion of the form (α^p_p∥W₁S∥^p + β^p_p∥W₂T∥^p) ^{1 p}, where α_p, β_p, are (scalar) weighting factors and W₁, W₂ are frequency-dependent weighting functions. In this paper, we develop strategies for weighting functions and weighting factors selections, so as to direct the resulting scalar-valued criterion design to a minimal element of the set of achievable performances in the (∥W₁S,∥W₂T∥)-plane, if this is possible. It appears that a scalar-valued criterion is most likely to direct the design toward a performance acceptable from the vector-valued criterion point of view if p = 2 and W₁, W₂ are nonoverlapping.