10.1.P-STABLE 157 (1+c)2<6. (10.4) Since P is strictly proper,so is N.This fact together with the equation NX+MY=1 shows t hat M(joo)Y (joo)=1. Since M(jw)Y(jw)is a cont inuous function of w,it is possible to choose w3>w2 such that |M(0w)Y(jw)川≤1+c,w≥w3. (10.5) The assumpt ion on P implies that N-is stable (but not proper).Choose a function V in S with the following three properties: 1.VN-is proper. 2.|1-V(0w川≤C,w≤w3. 3.l1-Vo≤1+c The idea behind the choice of V can be explained in terms of its Nyquist plot:It should lie in the disk with center 1,radius c up to frequency w3(property 2)and in the disk with center 1,radius 1+c thereafter (property 3).In addition,V should roll off fast enough so that VN-is proper.It is left as an exercise to conv ince yourself that such a V exists-a function of the form (T8+1)(T2s+1)k will work. Finally,take Q to be Q:-VN-Y. Substit ution into (10.2)gives S=MY(1-V). Thus for w≤w3 IS(jw)l≤clMY‖o from proprty2 e from(10.3) and for w>w3 lS(jw)川≤(1+cM(jw)Y(jw)from property3 ≤(1+c)2from(10.5) <6from(10.4).■ P ￾ STABLE    c    Since P is strictly proper so is N This fact together with the equation NX  M Y  shows that MjY j Since jMjY jj is a continuous function of  it is possible to choose   such that jMjY jj   c     The assumption on P implies that N￾ is stable but not proper Choose a function V in S with the following three properties  V N￾ is proper  j V jj  c     k V k￾   c The idea behind the choice of V can be explained in terms of its Nyquist plot It should lie in the disk with center  radius c up to frequency  property  and in the disk with center  radius  c thereafter property  In addition V should roll o fast enough so that V N￾ is proper It is left as an exercise to convince yourself that such a V existsa function of the form ￾s  s  k will work Finally take Q to be Q  V N￾Y  Substitution into   gives S  M Y  V  Thus for    jSjj  ckM Y k￾ from proprty   from   and for   jSjj    cjMjY jj from property    c from    from   ￾