44 CHAPTEPL INCEERINTY AND RE SINESS =k5k2/≤ Thus if kSk2-1,the Nyquist plot comes close to the critical point,and the feedback system is nearly unstable.However,as a measure of stability margin this distance is not entirely adequate because it contains no frequency information.More precisely,if the nominal plant P is perturbed to P,having the same number of unstable poles as has P and satisfying the inequality P(jw)C(jw)1 P(jw)C(jw)j<kSki/,;w, then internal stability is preserved(the number of encirclements of the critical point by the Nyquist plot does not change).But this is usually very conservative;for instance,larger perturbations could be allowed at frequencies where P(jw)C(jw)is far from the critical point. Better stability margins are obtained by taking explicit frequency-dependent perturbat ion mod- els:for example,the mult iplicative perturbation model,P=(1+AW2)P.Fix a positive number B and consider the family of plants fp:△is stable and k△k24Bg≤ Now a controller C that achieves internal stability for the nominal plant P will stabilize this entire family if B is small enough.Denote by Baup the least upper bound on B such that C achieves internal stability for the entire family.Then Baup is a stability margin(with respect to this uncertainty model).Analogous stability margins could be defined for the other uncertainty models. We turn now to two classical measures of stability margin,gain and phase margin.Assume that the feedback system is internally stable with plant P and controller C.Now perturb the plant to kP,with k a positive real number.The Upper gain margin,denoted kmax,is the first value of k greater than 1 when the feedback system is not internally stable;that is kmax is the maximum number such that internal stability holds for 1<mo.If there is no such number,then set kmox:=fi.Similarly,the lOwer gain margin,kmin,is the least nonnegative number such that internal stability holds for min<1.These two numbers can be read off the Nyquist plot of L;for example,1 1-kmax is the point where the Nyquist plot intersects the segment (1 1,0)of the real axis,the closest point to I 1 if there are several points of intersection. Now perturb the plant to eiP,with a positive real number.The phaSe m argin,max,is the maximum number(usually expressed in degrees)such that internal stability holds for 0< You can see that mox is the angle through which the Ny quist plot must be rotated until it passes through the critical point,1 1;or,in radians,mo equals the arc length along the unit circle from the Ny quist plot to the critical point. Thus gain and phase margins measure the distance from the critical point to the Nyquist plot in certain specific direct ions.Gain and phase margins have tradit ionally been important measures of stability robustness:if either is small,the system is close to instability.Notice,however,that the gain and phase margins can be relatively large and yet the Nyquist plot of L can pass close to the critical point;that is,Sm Utan eOUs small changes in gain and phase could cause internal instability.We return to these margins in Chapter 11. Now we look at a typical robust stability test,one for the multiplicative perturbation model. Assume that the nominal feedback system (i.e.,with A =0)is internally stable for controller C. Bring in again the complementary sensitivity function L PC T=11S=1+乙=1+PC≤ Theorem 1 -Multiph cat ve Uh certain tym Olel)C provi des rchust sta blityi2 kW2Tk2 1
CHAPTER UNCERTAINTY AND ROBUSTNESS kSk Thus if kSk the Nyquist plot comes close to the critical point and the feedback system is nearly unstable However as a measure of stability margin this distance is not entirely adequate because it contains no frequency information More precisely if the nominal plant P is perturbed to P having the same number of unstable poles as has P and satisfying the inequality jP jCj P jCjj kSk then internal stability is preserved the number of encirclements of the critical point by the Nyquist plot does not change But this is usually very conservative for instance larger perturbations could be allowed at frequencies where P jCj is far from the critical point Better stability margins are obtained by taking explicit frequencydependent perturbation mod els for example the multiplicative perturbation model P WP Fix a positive number and consider the family of plants fP is stable and kk g Now a controller C that achieves internal stability for the nominal plant P will stabilize this entire family if is small enough Denote by sup the least upper bound on such that C achieves internal stability for the entire family Then sup is a stability margin with respect to this uncertainty model Analogous stability margins could be dened for the other uncertainty models We turn now to two classical measures of stability margin gain and phase margin Assume that the feedback system is internally stable with plant P and controller C Now perturb the plant to kP with k a positive real number The upper gain margin denoted kmax is the rst value of k greater than when the feedback system is not internally stable that is kmax is the maximum number such that internal stability holds for k kmax If there is no such number then set kmax Similarly the lower gain margin kmin is the least nonnegative number such that internal stability holds for kmin k These two numbers can be read o the Nyquist plot of L for example kmax is the point where the Nyquist plot intersects the segment of the real axis the closest point to if there are several points of intersection Now perturb the plant to ejP with a positive real number The phase margin max is the maximum number usually expressed in degrees such that internal stability holds for max You can see that max is the angle through which the Nyquist plot must be rotated until it passes through the critical point or in radians max equals the arc length along the unit circle from the Nyquist plot to the critical point Thus gain and phase margins measure the distance from the critical point to the Nyquist plot in certain specic directions Gain and phase margins have traditionally been important measures of stability robustness if either is small the system is close to instability Notice however that the gain and phase margins can be relatively large and yet the Nyquist plot of L can pass close to the critical point that is simultaneous small changes in gain and phase could cause internal instability We return to these margins in Chapter Now we look at a typical robust stability test one for the multiplicative perturbation model Assume that the nominal feedback system ie with is internally stable for controller C Bring in again the complementary sensitivity function T S L L P C P C Theorem Multiplicative uncertainty model C provides robust stability i kWT k
4.2.ROBUST STABILITY 45 Proof (Assume that W2Tlloo 1.Construct the Nyquist plot of L,indenting D to the left around poles on the imaginary axis.Since the nominal feedback system is internally stable, we know this from the Nyquist criterion:The Nyquist plot of L does not pass through-1 and its number of counterclockwise encirclements equals the number of poles of P in Res >0 plus the number of poles of C in Res >0. Fix an allowable A.Construct the Nyquist plot of PC =(1+AW2)L.No additional inden tations are required since AW2 introduces no additional imaginary axis poles.We have to show that the Nyquist plot of (1+AW2)L does not pass through-1 and its number of counterclockwise encirclements equals the number of poles of(1+AW2)P in Re s >0 plus the number of poles of C in Re s >0;equivalently,the Nyquist plot of(1+AW2)L does not pass through-1 and encircles it exactly as many times as does the Nyquist plot of L.We must show,in other words,that the perturbation does not change the number of encirclements. The key equation is 1+(1+△W2)L=(1+)(1+△W2T) (4.1) Since ‖△W2TI‖o≤IW2Tlo<1, the point 1+AW2T always lies in some closed disk with center 1,radius 1,for all points s on D. Thus from (4.1),as s goes once around D,the net change in the angle of 1+(1+AW2)L equals the net change in the angle of 1+L.This gives the desired result. (→)Suppose t hat‖W2T‖o≥l.We will construct an allowable△t hat destabilizes the feedback system.Since T is strictly proper,at some frequency w, W2(0w)T(jw)1=1. (4.2) Suppose that w=0.Then W2(0)T(0)is a real number,either +1 or-1.If A =-W2(0)T(0),then △is allowable and 1+△W2(0)T(0)=0. From (4.1)the Nyquist plot of (1+AW2)L passes through the critical point,so the perturbed feedback sy stem is not internally stable. If w >0,constructing an admissible A takes a little more work;the details are omitted. The theorem can be used effectively to find the stability margin Bsup defined previously.The simple scaling technique {P=(1+△W2)P:‖△I‖l≤3}={P=(1+B-1△6W2)P:IB-1△‖≤1} ={P=(1+△13W2)P:‖△l≤1} toget her with the theorem shows that Fsup=sup{B:IBW2Tl‖lo<1}=1/川W2Tlo The condition W2T<1 also has a nice graphical interpretat ion.Note that lW2Tlo<1÷ W2(jw)L(jw) <1,w 1+L(w) 台IW2(w)L(w)川<|1+L(w),w
ROBUST STABILITY Proof Assume that kWT k Construct the Nyquist plot of L indenting D to the left around poles on the imaginary axis Since the nominal feedback system is internally stable we know this from the Nyquist criterion The Nyquist plot of L does not pass through and its number of counterclockwise encirclements equals the number of poles of P in Res plus the number of poles of C in Res Fix an allowable Construct the Nyquist plot of P C WL No additional inden tations are required since W introduces no additional imaginary axis poles We have to show that the Nyquist plot of WL does not pass through and its number of counterclockwise encirclements equals the number of poles of WP in Re s plus the number of poles of C in Re s equivalently the Nyquist plot of WL does not pass through and encircles it exactly as many times as does the Nyquist plot of L We must show in other words that the perturbation does not change the number of encirclements The key equation is WL L WT Since kWT k kWT k the point WT always lies in some closed disk with center radius for all points s on D Thus from as s goes once around D the net change in the angle of WL equals the net change in the angle of L This gives the desired result Suppose that kWT k We will construct an allowable that destabilizes the feedback system Since T is strictly proper at some frequency jWjT jj Suppose that Then WT is a real number either or If WT then is allowable and WT From the Nyquist plot of WL passes through the critical point so the perturbed feedback system is not internally stable If constructing an admissible takes a little more work the details are omitted The theorem can be used eectively to nd the stability margin sup dened previously The simple scaling technique fP WP kk g fP WP k k g fP WP kk g together with the theorem shows that sup supf k WT k g kWT k The condition kWT k also has a nice graphical interpretation Note that kWT k WjLj Lj jWjLjj j Ljj