Small gain versus positive real modeling of real parameter uncertainty

The authors shown by means of a lightly damped oscillator example with uncertain stiffness that small gain modeling of constant real parameter uncertainty can be extremely conservative. This example was chosen to highlight inherent drawbacks of small-gain principles applied to the analysis of robust controllers for lightly damped flexible structures. If, for this class of problems, uncertainty in the stiffness operator, which may be relatively large, is modeled as a small gain H_∞ block, then the arbitrary phase characteristics of the uncertainty block can contribute to a significant perturbation of the damping operator, which may be relatively small. An alternative uncertainty modeling approach involving positive real transfer functions and the positivity theorem is shown to be significantly less conservative