Chapter 10 Design for Performance The performance criterion WiS<1 was introduced in Section 3.4.The associated design problem is to find a proper C for which the feedback system is internally stable and WiS<1. When does such a C exist and how can it be computed?These questions are easy when the inverse of the plant transfer function is stable.When the inverse is unstable,the questions are more interest ing.The solut ions presented in this chapter use model-mat ching theory. 10.1 P-1 Stable We assume in this section that P has no zeros in Res >0,or in ot her words,P-is stable.The weighting function Wi is assumed to be stable and strictly proper.The latter condition is not too serious a loss of generality.We will see that under these conditions it is always possible,indeed quite easy,to design a proper C which is internally stabilizing and makes WiSo<1. Let k be a positive integer and T a positive real number,and consider the transfer function 1 J)=8+1 Sketch the Bode plot ofJ:The magnitude starts out at 1,is relatively flat out to the corner frequency w=1/T,and then rolls off to-oo with slope-k;the phase starts out at 0,is relatively flat up to,say,w =0.1/T,and then rolls off to-k/2 radians.So for low frequency,J(jw)1. This function has the useful property that it approximates 1 beside a strictly proper function. Lemma 1 If G is stable and strictly proper,then 1imG(1-J)川o=0. T3 Proof Let e>0 and w>0.By the argument above regarding the Bode plot of J,if r is sufficiently small,then the Nyquist plot of J lies in the disk with center 1,radius e for w<w,and in the disk with center 0,radius 1 for w>w1.Now G(1-J)oo equals the maximum of max G(jw)[1-J(jw)] 1 and G(j)[1-J(jwl. 153
Chapter Design for Performance The performance criterion kWSk was introduced in Section The associated design problem is to nd a proper C for which the feedback system is internally stable and kWSk When does such a C exist and how can it be computed These questions are easy when the inverse of the plant transfer function is stable When the inverse is unstable the questions are more interesting The solutions presented in this chapter use model matching theory P Stable We assume in this section that P has no zeros in Res or in other words P is stable The weighting function W is assumed to be stable and strictly proper The latter condition is not too serious a loss of generality We will see that under these conditions it is always possible indeed quite easy to design a proper C which is internally stabilizing and makes kWSk Let k be a positive integer and a positive real number and consider the transfer function J s s k Sketch the Bode plot of J The magnitude starts out at is relatively at out to the corner frequency and then rolls o to with slope k the phase starts out at is relatively at up to say and then rolls o to k radians So for low frequency J j This function has the useful property that it approximates beside a strictly proper function Lemma If G is stable and strictly proper then lim kG J k Proof Let and By the argument above regarding the Bode plot of J if is suciently small then the Nyquist plot of J lies in the disk with center radius for and in the disk with center radius for Now kG J k equals the maximum of max jG j J j j and max jG j J j j
154 CHAPTER 10.DESIGN FOR PERFORMANCE The first of these is bounded above by eGloo,and the second by l1-J‖xmax|G(jw)小. idil Since 1-J川∞≤1l+JI川o=2, we have IG1-∞≤mx{Gle,2nx1G(joj川. This holds for T sufficiently small.But the right-hand side can be made arbitrarily small by suitable choice of e and w because 1 im max|Gw)川=lGGo)川=0. 1→00Ww1 We conclude that for every 6>0,if r is small enough,then G(1-J)川≤d. This is the desired conclusion. We'll develop the design procedure first with the additional assumption that P is stable.By Theorem 5.1 the set of all internally stabilizing Cs is parametrized by the formula Q C=1-PQ° Q∈S. Then WiS is given in terms of Q by W1S=W1(1-PQ). To make WiSlloo<1 we are prompted to set Q=P-1.This is indeed stable,by assumption,but not proper,hence not in S.So let's try Q=P-1J with J as above and the integer k just large enough to make p-J proper (ie.,k equals the relative degree of P).Then W1S=W(1-J), whose oo-norm is 1 for sufficiently small T,by Lemma 1. In summary,the design procedure is as follows. Procedure:P and P-!Stable Input:P,Wi Step 1 Set k =the relative degree of P. Step 2 Choose r so small that ‖W(1-J)川o<1, where 1 J(s)=8+1
CHAPTER DESIGN FOR PERFORMANCE The rst of these is bounded above by kGk and the second by k J k max jG j j Since k J k kk kJ k we have kG J k max kGk max jG j j This holds for suciently small But the right hand side can be made arbitrarily small by suitable choice of and because lim max jG j j jG j j We conclude that for every if is small enough then kG J k This is the desired conclusion Well develop the design procedure rst with the additional assumption that P is stable By Theorem the set of all internally stabilizing Cs is parametrized by the formula C Q P Q Q S Then WS is given in terms of Q by WS W P Q To make kWSk we are prompted to set Q P This is indeed stable by assumption but not proper hence not in S So lets try Q P J with J as above and the integer k just large enough to make P J proper ie k equals the relative degree of P Then WS W J whose norm is for suciently small by Lemma In summary the design procedure is as follows Procedure P and P Stable Input P W Step Set k the relative degree of P Step Choose so small that kW J k where J s s k
10.1.P11 STABLE 155 Step 3 Set Q=Pl1J> Step 4 Set C=Q<1-PQ)> When P is unstable,the parametrization in Theorem 52 is used> Procedure P11 Stable Input:P,Wi Step-Do a coprime factorization of P:Find four functions in S satisfying the equations N P二M NX +MY 100 Step 2 Set=the relative degree of P> Step 3 Choose-so small that WiMY(1-J)loo<1, where 1 J):=8+10 Step 4 Set Q=YN11J> Step 5 Set C=(X +MQ)<Y-NQ)> Example Consider the unstable plant and weighting function P)=a-W(回=10x 8+7o This weight has bandwidth 1 rad/s,so it might be used to get good tracking (iez approximately 1%tracking error,up to this frequency)>The prev ious procedure for these data goes as follows: Step-First,do a coprime factorization of P over S: 1 N(s)= (8+1)7 M(s)= (s-2)7 (s+1)7 X(s)= 2781 8+1: Y(s)= 8+7 8+700 Step2‖=2
P STABLE Step Set Q P J Step Set C Q P Q When P is unstable the parametrization in Theorem is used Procedure P Stable Input P W Step Do a coprime factorization of P Find four functions in S satisfying the equations P N M NX M Y Step Set k the relative degree of P Step Choose so small that kWM Y J k where J s s k Step Set Q Y NJ Step Set C X MQ Y NQ Example Consider the unstable plant and weighting function P s s W s s This weight has bandwidth rads so it might be used to get good tracking ie approximately tracking error up to this frequency The previous procedure for these data goes as follows Step First do a coprime factorization of P over S N s s M s s s X s s s Y s s s Step k
156 CHAPTER 10.DESIGN FOR PERFORMANCE Step 3 Choose r so that the oo-norm of 100(s-2)2(s+7) (s+1)A (Ts+1)2 is 1.The norm is computed for decreasing values of r: oo-Norm 10- 199.0 10-2 19.97 1.997 10 0.1997 So take T 10-4. Step 4 Q(s)= (s+1)(s+7) (10-4s+1)2 Step 5 C(s)=104 (s+1)3 s(s+7)(10-4s+2) This section concludes with a result stated but not proved in Section 6.2.It concerns the performance problem where the weight Wi sat isfies w1≤w≤w2 else Thus the criterion WiSo<1 is equivalent to the conditions |S(jw)‖<e,w1≤w≤w2 S(jw)<6,else. (10.1) Lemma 2 If P-is stable,then for every e>0 and 6>1,there erists a proper C such that the feedback system is internally stable and (10.1)holds.I Proof The idea is to approximately invert P over the frequency range [0,w2]while rolling off fast enough at higher frequencies.From Theorem 5.2 again,the formula for all internally stabilizing proper controllers is C X+MQ Q∈S. Y-NQ' For such C S=M(Y-NQ). (10.2) Now fix e>0 and 6>1.We may as well suppose that e<1.Choose c>0 so small that clMY‖o<e, (10.3) The assumption on P in Lemma 2 is slightly stronger than necessary;see the statement in Section 6.2
CHAPTER DESIGN FOR PERFORMANCE Step Choose so that the norm of s s s s is The norm is computed for decreasing values of Norm So take Step Q s s s s Step C s s s s s This section concludes with a result stated but not proved in Section It concerns the performance problem where the weight W satises jW j j else Thus the criterion kWSk is equivalent to the conditions jS j j jS j j else Lemma If P is stable then for every and there exists a proper C such that the feedback system is internal ly stable and holds Proof The idea is to approximately invert P over the frequency range while rolling o fast enough at higher frequencies From Theorem again the formula for all internally stabilizing proper controllers is C X MQ Y NQ Q S For such C S M Y NQ Now x and We may as well suppose that Choose c so small that ckM Y k The assumption on P in Lemma is slightly stronger than necessary see the statement in Section
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 M j Y j Since jM j Y j j is a continuous function of it is possible to choose such that jM j 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 exists a function of the form s s k will work Finally take Q to be Q V NY Substitution into gives S M Y V Thus for jS j j ckM Y k from proprty from and for jS j j c jM j Y j j from property c from from