44 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
Figure 2.9: Perturbed closed loop system for stability analysis
the induced norms, the reader is referred to Doyle [2]. The major advantage of choosing
BP or BL2is that the test forthe p erformancecan be considered in terms of the same
norm as stability. This has significant advantages when we are considering performance
and stability in the presence of perturbations, ∆.
2.4.2 Robust Stability and µNow we will consider the stability of a closed loop system under perturbations. In
Figure 2.9, G(s), is a perturbation model of a closed loop system. In the following
robust stability and robust performance analyses we will assume that ∆ is linear and
time-invariant.
Wewill a lsoassume that the interconnection structure G(s) consists of stable transfer
function matrices, where stability is taken to mean that the system has no poles in the
closed right half plane. In practice this amounts to assumingthat G22 (s) (the nominal
closed loop system model) is stable as the other elements, G11(s), G12(s), and G21(s),
are weighting functions and can be chosen to be stable. The no minal closed loop system,
G22(s), often arises from a standard design procedure (H2,H∞, or loopshaping for
example) and will be stable.