58 CHAPTER 5 STABILIZATION &tt品a.风知.Q, C Q:=1+P元5 Then,s aa (cCrse,supp to,s aade Q C:= 1-PQ5 (5.1) AceGail tOte den liticintimn2.te reedka systm i te rin rIster fuld 1 1+PC tftan Jainti 「1-PQ-P(1-PQ)-(1-PQ)1 Q 1-PQ -Q 5 PQ P(1-PQ) 1-PQ clay,tese rinertis batos. Nct ttalritater untaao a aingnaotrpaaetrt ecdhek The fen T5+T.Q fs Ghe T5 T.I.IIatcua the se IWI alac p emeIay se IWI fuldIsae S=1-PQ- T-PQ5 toaiu!apiaiotonac cttda ystm Iastt as arupkey taks apr (when=0).Pachecam the tecem.Thgn aympka tas atp It the belter fultGhr (le.,s)ha a3rCts=0,ti丛， t6a益Linzwr从aadii P(0)Q(0)=15 c=90Q,sQ0= P而5 oRrve tcte iatp atac.ALOy G caned ttacrcer tetr rGn naa c ts =0,aItmusthThe crem 3 (Cho.ter 3. Example Fte p P(S)=8+1)s+2 CHAPTER STABILIZATION Proof  Suppose that C achieves internal stability Let Q denote the transfer function from r to u that is Q  C   P C Then Q S and C  Q  ￾ P Q  Conversely suppose that Q S and dene C  Q  ￾ P Q  According to the denition in Section   the feedback system is internally stable i the nine transfer functions    P C   ￾P ￾ C  ￾C P C P    all are stable and proper After substitution from  and clearing of fractions this matrix becomes   ￾ P Q ￾P  ￾ P Q ￾ ￾ P Q Q  ￾ P Q ￾Q P Q P  ￾ P Q  ￾ P Q   Clearly these nine entries belong to S ￾ Note that all nine transfer functions above are ane functions of the free parameter Q that is each is of the form T￾  TQ for some T￾ T in S In particular the sensitivity and complementary sensitivity functions are S   ￾ P Q T  P Q Let us look at a simple application Suppose that we want to nd a C so that the feedback system is internally stable and y asymptotically tracks a step r when d    Parametrize C as in the theorem Then y asymptotically tracks a step i the transfer function from r to e ie S has a zero at s   that is P  Q   This equation has a solution Q in S i P    Conclusion The problem has a solution i P    when this holds the set of all solutions is ￾ C  Q  ￾ P Q  Q S Q   P  Observe that Q inverts P at dc Also you can check that a controller of the latter form has a pole at s   as it must by Theorem  of Chapter  Example For the plant P s   s   s