18 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
Figure 2.5: Generic LFT model structure including per turbations,∆
A generic model structure, referred to as a linear fractional transformation (LFT),
overcomes the difficulties outlined above. The LFT model is equivalent to the
relationship,
y=P21∆(I−P11∆)−1P12 +P22u, (2.6)
where the ∆ is the norm bounded perturbation. Figure 2.5 shows a block diagram
equivalentto the system descr ibed by Equation 2.6. Because this form of interconnection
is widely used, we will give it a specific notation. Equation2.6 is abbreviated to,
y=Fu(P,∆)u.
The subscript, u, indicates that the ∆ is closed in the upper loop. We will also use
Fl(.,.) when the lower loop is closed.
In this figure, the signals, u,y,zand vcan all be vector valued, meaning that the
partitioned parts of P,(P
11, etc.) can themselves be matrices of transfer functions.
Toma kethis clea rwe will lookat the perturbed system example, given in Equation 2.4,