40 CHAPTER 4.UNCERTAINTY AND ROBUSTNESS Multiplicative Perturbation Suppose that the nominal plant transfer function is P and consider perturbed plant transfer func tions of the form P =(1+AW2)P.Here W2 is a fixed stable transfer function,the weight,and A is a variable stable transfer function satisfying Alo<1.Furthermore,it is assumed that no unstable poles of P are canceled in forming P.(Thus,P and P have the same unstable poles.) Such a perturbation A is said to be allowable. The idea behind this uncertainty model is that Aw2 is the normalized plant perturbat ion away from 1: 元-1=AW2. Hence if‖△‖lo≤l,then P(jw) -1≤IW2(0w儿，w, P(jw) soW2(jw)provides the uncertainty profile.This inequality describes a disk in the complex plane: At each frequency the point P/P lies in the disk with center 1,radius W2.Typically,W2(jw) is a (roughly)increasing function of w:Uncertainty increases with increasing frequency.The main purpose of A is to account for phase uncertainty and to act as a scaling factor on the magnitude of the perturbation (i.e.,A varies between 0 and 1). Thus,this uncertainty model is characterized by a nominal plant P together with a weighting function W2.How does one get the weighting funct ion W2 in practice?This is illustrated by a few examples. Example 1 Suppose that the plant is stable and its transfer function is arrived at by means of frequency-response experiments:Magnitude and phase are measured at a number of frequencies, wi,i=1,...,m,and this experiment is repeated several,say n,times.Let the magnitude-phase measurement for frequency wi and experiment k be denoted (Mik,).Based on these data select nominal magnit ude-phase pairs (M,)for each frequency wi,and fit a nominal transfer funct ion P(s)to these data.Then fit a weighting function W2(s)so that Mikeloik |W2(0)儿，i=1,.,m;k=1,.,n. Example 2 Assume that the nominal plant transfer function is a double integrator: 1 P(s)=8 For example,a dc motor with negligible viscous damping could have such a transfer function.You can think of ot her phy sical systems with only inertia,no damping.Suppose that a more detailed model has a time delay,y ielding the transfer function P(s)=e-rs1 2 and suppose that the time delay is known only to the extent that it lies in the interval 0<<0.1. This time-delay factor exp(-Ts)can be treated as a multiplicative perturbation of the nominal plant by embedding P in the family {(1+△W2)P:‖△lo≤1. CHAPTER UNCERTAINTY AND ROBUSTNESS Multiplicative Perturbation Suppose that the nominal plant transfer function is P and consider perturbed plant transfer func tions of the form P    W￾P Here W￾ is a xed stable transfer function the weight and  is a variable stable transfer function satisfying kk￾   Furthermore it is assumed that no unstable poles of P are canceled in forming P Thus P and P have the same unstable poles Such a perturbation  is said to be al lowable The idea behind this uncertainty model is that W￾ is the normalized plant perturbation away from  P P W￾ Hence if kk￾  then P j P j  jW￾jj  so jW￾jj provides the uncertainty prole This inequality describes a disk in the complex plane At each frequency the point P P  lies in the disk with center  radius jW￾j Typically jW￾jj is a roughly increasing function of  Uncertainty increases with increasing frequency The main purpose of  is to account for phase uncertainty and to act as a scaling factor on the magnitude of the perturbation ie jj varies between  and  Thus this uncertainty model is characterized by a nominal plant P together with a weighting function W￾ How does one get the weighting function W￾ in practice This is illustrated by a few examples Example Suppose that the plant is stable and its transfer function is arrived at by means of frequencyresponse experiments Magnitude and phase are measured at a number of frequencies i i  m and this experiment is repeated several say n times Let the magnitudephase measurement for frequency i and experiment k be denoted Mik ik  Based on these data select nominal magnitudephase pairs Mi  i for each frequency i and t a nominal transfer function P s to these data Then t a weighting function W￾s so that Mike jik Mieji  jW￾jij i  m k    n Example Assume that the nominal plant transfer function is a double integrator P s   s￾ For example a dc motor with negligible viscous damping could have such a transfer function You can think of other physical systems with only inertia no damping Suppose that a more detailed model has a time delay yielding the transfer function P se s  s￾  and suppose that the time delay is known only to the extent that it lies in the interval    This timedelay factor exp s can be treated as a multiplicative perturbation of the nominal plant by embedding P in the family f  W￾P  kk￾ g