42 CHAPTER.UNCERTAINTY AND ROBUSTNESS where the gain k is uncertain but is known to lie in the interval [04110.This plant too can be embedded in a family consisting of multiplicative perturbations of a nominal plant P(S= k4 52 The weight W2 must satisfy P(jw) W2(w)1,w1k1 /P(jw) that is, 4i441sw01,w4 The lethand side is minimized by k4505,for which the left-hand side quals 445.545.In this way we get the nominal plant P(S= 505 and constant weight W2(S=405.545. The multiplicative perturbation model is not suitable for every application because the disk covering the uncertainty set is sometimes too coarse an approximation.In this case a controller designed for the multiplicativeuncertainty model would probably be too conservative for the original uncertain ty mo del. The discussion above illustrates an important point.In mo deling a plant we may arrive at a certain plant set.This set may be too awkward to cope with mathematically,so we may embed it in a larger set that is easier to handle.Conceivably,the achievable performance for the larger set may not be as good as the achievable performance for the smaller;that is,there may exist-even though we cannot find it-a controller that is better for the smaller set than the controller we design for the larger set.In this sense the latter controller is conservative for the smaller set. In this book we stick with plant uncertainty that is disk-like.It will be conservative for some problems,but the payoff is that we obtain some very nice theoretical results.The resulting theory is remarkably practical as well. Other Perturbations Other uncertainty mo dels are possible besides multiplicative perturbations,as illustrated by the following example Example 4 As at the start of this section,consider the family of plant transfer functions S+S,104≤a084 Thus 806+0里△1-1≤△≤11 so the family can be expressed as P(S 1+△W2SPS1-1≤A≤11 CHAPTER UNCERTAINTY AND ROBUSTNESS where the gain k is uncertain but is known to lie in the interval   This plant too can be embedded in a family consisting of multiplicative perturbations of a nominal plant P s  k s The weight W￾ must satisfy P j P j  jW￾jj  k that is max k k k  jW￾jj  The lefthand side is minimized by k   for which the lefthand side equals  In this way we get the nominal plant P s   s and constant weight W￾s The multiplicative perturbation model is not suitable for every application because the disk covering the uncertainty set is sometimes too coarse an approximation In this case a controller designed for the multiplicative uncertainty model would probably be too conservative for the original uncertainty model The discussion above illustrates an important point In modeling a plant we may arrive at a certain plant set This set may be too awkward to cope with mathematically so we may embed it in a larger set that is easier to handle Conceivably the achievable performance for the larger set may not be as good as the achievable performance for the smaller that is there may existeven though we cannot nd ita controller that is better for the smaller set than the controller we design for the larger set In this sense the latter controller is conservative for the smaller set In this book we stick with plant uncertainty that is disklike It will be conservative for some problems but the payo is that we obtain some very nice theoretical results The resulting theory is remarkably practical as well Other Perturbations Other uncertainty models are possible besides multiplicative perturbations as illustrated by the following example Example  As at the start of this section consider the family of plant transfer functions  s￾  as     a  Thus a       so the family can be expressed as P s W￾sP s