Chapter 4 Uncertainty and Robustness No mat hematical system can exactly model a physical system.For this reason we must be aware of how modeling errors might adversely affect the performance of a control system.This chapter begins with a treatment of various models of plant uncertainty.Then robust stability,stability in the face of plant uncertainty,is studied using the small-gain theorem.The final topic is robust performance,guaranteed tracking in the face of plant uncertainty. 4.1 Plant Uncertainty The basic technique is to model the plant as belonging to a set P.The reasons for doing this were presented in Chapter 1.Such a set can be either structured or unstructured. For an example of a structured set consider the plant model s2+a8+1 This is a standard second-order transfer funct ion with nat ural frequency 1 rad/s and damping ratio a/2-it could represent,for example,a mass-spring-damper setup or an R-L-C circuit.Suppose that the constant a is known only to the extent that it lies in some interval [amin,amax].Then the plant belongs to the struct ured set 1 p={32+as+:amm≤a≤om Thus one type of structured set is parametrized by a finite number of scalar parameters (one parameter,a,in this example).Another type of structured uncertainty is a discrete set of plants, not necessarily parametrized explicit ly. For us,unstructured sets are more important,for two reasons.First,we believe that all models used in feedback design should include some unstruct ured uncertainty to cover unmodeled dynam- ics,particularly at high frequency.Other types of uncertainty,though important,may or may not arise naturally in a given problem.Second,for a specific type of unstructured uncertainty,disk uncertainty,we can develop simple,general analysis met hods.Thus the basic starting point for an unstructured set is that of disk-like uncertainty.In what follows,multiplicative disk uncertainty is chosen for detailed study.This is only one type of unstructured perturbation.The important point is that we use disk uncertainty instead of a more complicated description.We do this because it greatly simplifies our analysis and lets us say some fairly precise things.The price we pay is conservativeness. 39Chapter Uncertainty and Robustness No mathematical system can exactly model a physical system For this reason we must be aware of how modeling errors might adversely aect the performance of a control system This chapter begins with a treatment of various models of plant uncertainty Then robust stability stability in the face of plant uncertainty is studied using the smallgain theorem The nal topic is robust performance guaranteed tracking in the face of plant uncertainty ￾ Plant Uncertainty The basic technique is to model the plant as belonging to a set P The reasons for doing this were presented in Chapter  Such a set can be either structured or unstructured For an example of a structured set consider the plant model  s￾  as   This is a standard secondorder transfer function with natural frequency  rad s and damping ratio a it could represent for example a massspringdamper setup or an RLC circuit Suppose that the constant a is known only to the extent that it lies in some interval amin amax Then the plant belongs to the structured set P  ￾  s￾  as    amin a amax Thus one type of structured set is parametrized by a nite number of scalar parameters one parameter a in this example Another type of structured uncertainty is a discrete set of plants not necessarily parametrized explicitly For us unstructured sets are more important for two reasons First we believe that all models used in feedback design should include some unstructured uncertainty to cover unmodeled dynam ics particularly at high frequency Other types of uncertainty though important may or may not arise naturally in a given problem Second for a specic type of unstructured uncertainty disk uncertainty we can develop simple general analysis methods Thus the basic starting point for an unstructured set is that of disklike uncertainty In what follows multiplicative disk uncertainty is chosen for detailed study This is only one type of unstructured perturbation The important point is that we use disk uncertainty instead of a more complicated description We do this because it greatly simplies our analysis and lets us say some fairly precise things The price we pay is conservativeness