Slovak University of Technology in Bratislava Institute of Information Engineering, Automation, and Mathematics PROCEEDINGS

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1 lovak University of echnology in Bratislava Institute of Information Engineering, Automation, an athematics PROCEEDING 7 th International Conference on Process Control 009 Hotel Baník, Štrbské Pleso, lovakia, June 9, 009 IBN Eitors:. Fikar an. Kvasnica Šebek,., Hurák, Z.: An often isse Detail: Formula Relating Peek ensitivity with Gain argin Less han One, Eitors: Fikar,., Kvasnica,., In Proceeings of the 7th International Conference on Process Control 09, Štrbské Pleso, lovakia, 65 7, 009. Full paper online:

2 7th International Conference on Process Control 009 June 9, 009, Štrbské Pleso, lovakia Le-We-, 5.pf AN OFEN IED DEAIL: FORULA RELAING PEEK ENIIVIY WIH GAIN ARGIN LE HAN ONE. Šebek an Z. Hurák Department of Control Engineering, Faculty of Electrical Engineering Czech echnical University in Prague fax: , s: m.sebek@polyx.cz, hurak@fel.cvut.cz Abstract: An inequality relating gain margin with sensitivity peek value is presente in numerous basic control textbooks. In fact, this inequality fails to hol as soon as the open-loop Nyquist plot crosses the negative real axis on the left of the critical point. his opposite case is usually ignore by the textbook authors. A simple alternative inequality is erive in the paper to cover the not so popular opposite case. his fills a small gap one often encounters in basic control courses. Keywors: Gain argin. ensitivity. Complementary sensitivity. Nyquist Plot. INRODUCION everal moern control textbooks provie a simple inequality relating gain margin to the peak of sensitivity function in their sections evote to frequency omain esign specifications. Along with another inequality relating similarly phase margin an, this approach makes it possible to express traitional esign specs in a unifie manner using only. his inequality can be foun in moern control textbooks e.g. (kogesta et al. 005), (eborg et al. 004) etc. Unfortunately, it appears to work only in the case of > when the open-loop transfer function Nyquist plot crosses the negative real axis on the right of the critical point (-,0). In fact, it is invali in the opposite case, when the open-loop transfer function Nyquist plot crosses the negative real axes on the left of the critical point so that <. In such an opposite case, the stanar inequality must be replace by similar yet ifferent one. Its erivation is so simple that the authors consier their contribution or. On the other han, they believe that this or but often encountere gap shoul be fille. he authors are not aware of any paper or textbook presenting the opposite-case inequality but woul not be surprise to learn that it has been publishe anyway. GAIN ARGIN AND PEAK ENIIVIY As usually, we enote the open-loop transfer function by Ls () an the close loop sensitivity function by s () Ls ( ) (.) throughout the paper. For a particular frequency 0, the istance of the corresponing openloop Nyquist plot point L( j ) from the critical point,0 in the complex plain reas ist L( j), L( j) ( ) L( j) ( j) while the imum istance of the whole Nyquist plot of () well known to be Ls from the critical point,0 inf ist L( j), inf L( j) inf ( j) sup ( j) is (.) 65

3 7th International Conference on Process Control 009 June 9, 009, Štrbské Pleso, lovakia Here, as often, stans for the peek sensitivity function value or its H norm sup ( j) ( s) It is evient that for any particular frequency is. (.3) If only the open-loop gain happens to be uncertain, the frequency 80 at which the open-loop Nyquist plot L( j) crosses the negative real axis plays a crucial role. hen the istance of the negative real axis crossing point L( j80) from the critical point,0 enote here by 80 ist L( j80 ), gives rise to the classical concept of gain margin. Le-We-, 5.pf Putting together (.) an (.4) with (.3) yiels an finally (.5) 80 (.6) results. his is the inequality frequently encountere in textbooks. It turns out, however, that (.6) fails to hol as soon as well as in other more complex cases. 4 HE OPPOIE CAE Let us now investigate the opposite case when an the open-loop Nyquist plot crosses the negative real axis on the left from the critical point. If this happens, the inequality (.6) is not vali any longer. However, a similar yet ifferent formula can easily be erive. he situation is illustrate on Fig. Fig. : he case of frequently encountere in textbooks 3 HE CLAICAL CAE Only the classical case is encountere in textbooks when because the open-loop transfer function Nyquist plot crosses the negative real axis on the right of the critical point as in Fig. Note the gain margin epicte in Fig accoring to its stanar efinition. It such a situation, we enote the gain margin by to emphasize its physical meaning: A noally stable close loop remains stable even when the open-loop gain is multiplie by any factor k such that k he figure reveals immeiately that 80 which further implies 80 (.4) Fig. : he opposite case of In such a case, we enote the gain margin by to emphasize that a noally stable close loop remains stable even when the open-loop gain is multiplie by any factor k such that k By efinition, 0. It is clear from Fig, that now so that (.7) Combining (.) an (.7) with (.3) again implies (.8) 80 66

4 7th International Conference on Process Control 009 June 9, 009, Štrbské Pleso, lovakia his finally gives rise to the "opposite" formula (.9) his is the missing inequality to replace (.6) in the opposite case. Although (.9) is as simple an as practical as (.6), this inequality is missing in the textbooks. uch worse, even the concept of twosie margin as well as the fact of the two-sie crossing itself is ignore by the most of textbooks. One of notable exceptions is (Zhou, Doyle an Glover, 996). 5 HE WO-IDED CAE It is even possible that the open-loop Nyquist plot crosses the negative real axis both on the left form the critical point an on the right of it as in Fig 3. Hence one can replace (.0) by another, possibly narrower, interval escribe only by means of as follows k 6 HE GENERAL CAE Le-We-, 5.pf (.) houl the open-loop Nyquist plot cross the negative real axis several times on the left an/or right sie of the critical point (-,0), only one crossing on each sie counts (the closest one to the critical point, of course). he above reasoning hols true when applie to these closest neighbors. uch a situation is illustrate by Fig. 4. Fig. 4: ultiple crossings on the both sies Fig. 3: he two-sie case Whenever this happens, we must employ two ifferent gain margins at the same time: the imum gain margin as well as the imum gain margin. he case of two-sie crossing is illustrate by Fig. 3 where the both margins are inicate. Physical meaning of the two margins is as follows: A noally stable close loop remains stable even when the open-loop gain is multiplie by any factor k such that k (.0) However, the close loop stability is lost as soon as either k or k. We procee as above applying simultaneously (.6) for an (.9) for. Putting (.6), (.9) an (.0) together yiels k (.) 7 UING COPLEENARY ENIIVIY imilar formulae can be erive the peak value of complementary sensitivity Ls () s () (.3) Ls ( ) Comparing efinitions (.) an (.3), it is easy to see that L ( L) L L L (.4) o one can simply repeat all the erivations above, replacing s () by s () an, at the same time, exchanging the Nyquist plot of Ls () by the Nyquist plot of its reciprocal function Ls ( ). uch a way, it can be prove that (.5) 67

5 7th International Conference on Process Control 009 June 9, 009, Štrbské Pleso, lovakia Here, as usually, stans for the peek value of the complementary sensitivity function or for its H norm sup ( j) ( s) he inequality (.5) is encountere in textbooks as often as (.6). However, it again hols true only for. In the opposite case of, the inequality (.5) must be replace by is "opposite counterpart" (.6) while (.9) an (.6) o hol 3 his result in fact justifies of the current paper. Le-We-, 5.pf Finally, for the case of "two-sie crossing", a twosie margin applies similarly to (.0) giving rise to the "complementary version" of (.), which is k (.7) 8 EXAPLE Example : o prove that stanar inequalities (.6) an (.5) inee fail in the opposite case, just consier a trivial unstable open-loop transfer function Ls () s (.8) with Nyquist plot on Fig 5. he rawing reveals that Fig. 6: ensitivity an complementary sensitivity relate to (.8) Example : he case of two-sie crossing can be emonstrate by another quite elementary unstable open-loop transfer function s Ls () s (.9) Its Nyquist plot on Fig 6 inee crosses the negative real axis twice an this happens on the right as well as on the left of (-,0) giving rise to two-sie margin with an. As Fig. 5: he Nyquist plot of (.8) the negative axis crossing appears to be on the left of the critical point while. he corresponing close-loop sensitivity an complementary sensitivity functions are show on Fig 6 from which it is clear that an. It is easy to check that both (.6) an (.5) fail Fig. 7: he Nyquist plot of (.9) an (see Fig. 8), we can apply (.) to learn that the close-loop system remains stable even if the 68

6 7th International Conference on Process Control 009 June 9, 009, Štrbské Pleso, lovakia open-loop transfer function is multiplie by a factor k such that k. Alternatively, we can employ (.7) to conclue that it remains stable for any multiplicative factor k such that 3 k Note that a narrower interval for k results by chance when using in this example. point. o fin the two-sie gain margin, only one crossing point shoul be consiere on each sie. 9 HOW CAN WE BENEFI FRO ALL HE INEQUALIIE? At first, (kogesta et al. 005) propose using the peaks or to replace traitional measures for esign specifications (an P). For instance, requiring implies requiring (by (.6)) an P 30(by another formula not iscusse here). hank to the evelopment above, we can tweak this claim by aing that requiring implies requiring 3 as well (by (.9)). Hence by specifying one guarantees resulting robust stability gain interval to be at least (.), i.e. k Le-We-, 5.pf his interval may be narrower than the classical k Fig. 8: ensitivity an complementary sensitivity relate to (.9) Example 3: We en the section with example of a multiple negative real axis crossing. Nyquist plot of a complicate open loop transfer function s Ls () s 3 ( s s 0. 3 ( s 4.95s 7.30) has been plotte by atlab as follows 5) In rewar, it is more luci as it is using one variable only. In aition, it is easier to hanle an guarantee by moern loop-shaping techniques. As another outcome, the above iscussion shes more light on the relation between gain margins an sensitivity peaks. Given (as the esign has alreay been mae or for other reasons), what oes it mean for the two-sie gain margin? First the inequalities 0 (.0) imply that 0 (.) Fig. 9: ultiple crossing of the negative real axis It crosses negative real axis six times: there times on the left an three times on the right of the critical o put it in wors, once is given, the. In fact, it cannot be worse than can only range the interval 0, (.) where the left boun is a ream while the right one is the worst case. his, however, tells us nothing about where within the interval is actually locate. his eviently epens on other properties of the sensitivity function rather than just on its peak. o have an iea, look at the following plot on Fig 0. 69

7 7th International Conference on Process Control 009 June 9, 009, Štrbské Pleso, lovakia Le-We-, 5.pf, Here again (.4) provie the worst case for,,, this time the lower boun. For a given cannot be worse (lower) than hen it must range,. (.5) Fig 0: Depenence of the upper boun of on the sensitivity peak implies For example, 0,. his means that cannot be worse than /, but can be better, perhaps even 0. As another example, take a larger. his results in a larger interval 0, 3 allowing to rang up to 3, that is larger an hence worse than before. o summarize, for a given, the inequality provies the worst case boun for he inequalities,. (.3) (.4) For example, if, then even the lower boun reaches an is fixe to be. For a rising, the lower boun becomes smaller. It falls own to for large until the interval (.5) blows up to the efinition interval, an thereby, in fact, loses its purpose. 0 PEAK ENIIVIY AND NONLINEAR ACUAOR Yet another benefit from the inequalities evelope above is that they help to explain the impact of eventual actuator nonlinearity. hanks to them, one can apply the Circle criterion for stability of nonlinear systems, as we show in this section. can be investigate similarly. Just consier the plot on Fig, which is a hyperbola with an asymptote at. Fig : he circle efinitely avoie by Nyquist plot Fig : Depenence of the lower boun of on the sensitivity peak ince by efinition an since for any physical system, only the right branch of the, is always hyperbola counts. o for Regarless of where an how many times it crosses the negative real axis, the open-loop Nyquist plot never gets insie the circle aroun the critical point with raius. his is guarantee by the very efinition of an illustrate on Fig. Assug that the close loop is stable, we are sure that the open-loop Nyquist plot encircles the critical point properly as many times as the Nyquist stability criterion requires. hen, however, it properly encircles the whole circle as well. his enable us to apply the Circle criterion for stability of nonlinear systems, 70

8 7th International Conference on Process Control 009 June 9, 009, Štrbské Pleso, lovakia see e.g. (Gla at al. 00). Next, the criterion requires to fin conitions that the nonlinearity must satisfy. hey naturally epen on the position an circle size. o this en, we calculate points where the circle crosses the negative real axis. o be consistent with nonlinear control textbooks, we enote A, the left crossing point, by an B, the right crossing point, by yieling. hen it is clear from Fig that an an (.6) (.7) respectively. he smaller is the sensitivity peak, the more spacious is the region an the more complex nonlinearity fits. It is then less likely that a nonlinear actuator violates overall stability. "DO NO ENER" CIRCLE Le-We-, 5.pf As a byprouct, we win even better geometrical insight. It is crystal clear form Fig 4 what happens when applying the inequalities. In fact, we just trae the original interval given by the points where L( j ) crosses negative real axis for another interval efine by the points where the circle crosses negative real axis. he new interval may be smaller, but it is nicer, his brings our favorite inequalities into the game. Before stating the final result of this section, let us re what was proven so far. We alreay know that if the close loop is stable, then it remains stable even if the open-loop transfer function L( j ) is replace by kl( j ) with k such that by (.) k (.8) Now, using the Circle criterion (Gla at al. 00) we claim even more. If the close loop is stable, the it remains (globally asymptotically) stable even if we insert into the loop a nonlinearity f( x ) such that f (0) 0 an that for any x 0 satisfies f( x) x (.9) he geometric meaning of (.9) is clear from Fig 3 Fig 4: he final picture easier to express an esign. However, its main avantage is that not only the negative real axis segment is untouche by L( j) but the whole circle is intact. his is important as it guarantees true robust stability, regarless whether gain, phase or both are uncertain. Fig 3: Nonlinearity an its bouns he nonlinearity graph is confine to a cone shape region boun by two straight lines passing through the origin an having slopes,. Fig 5: High sensitivity peak Example 4: Consier first the case of a high sensitivity peak as on Fig 5 where two ifferent open-loop transfer functions are plotte having the same close-loop sensi- 7

9 7th International Conference on Process Control 009 June 9, 009, Štrbské Pleso, lovakia tivity peak. As the peak is high, the "o not enter" circle is small an the stability robustness is poor. Note that although L( ) j offers much larger gain margin than L ( j )(inee,, ) the same lower boun results from, (.6) leaing to exactly the same new interval (.5) or (.). Here one coul wrongly conclue that more is lost for L( j ) than for L ( ) j. However, just the reverse is true: Robustness is evenly poor for L( j ) an for L ( ) j an the larger gain margin really means nothing. he correct explanation for the ifference is simply that the gain margin is a shoy measure to evaluate the close-, loop stability robustness for L( j ). Example 5: For change, consier now a perfect esign with the smallest reasonable peak, for instance an LQR esign for a ouble integrator that results in.4s Ls () (.30) s plotte on Fig 6. As, the "o not enter" circle is large enough to touch the origin. In such a case, Example 6: For a non-physical system or a reuce moel or alike, it may well happen that. For instance s Ls () (.3) s gives rise to 3 an the "o not enter" circle becomes even larger having 3 in iameter. his is illustrate on the following Fig. 7. Fig 7: ensitivity peak less than one Le-We-, 5.pf When the circle goes beyon ten imaginary axis, however, gain margin woul be negative. his woul require moifying the efinitions. We postpone this to another paper an finish the exposition right here. ACKNOWLEDEN he work has been supporte by the by the Grant Agency of the Czech Republic grant No. 0/08/086 an by the inistry of Eucation of the Czech Republic uner contract No. LA REFERENCE Fig 6: Low sensitivity peak the inequalities (.9) an (.6) reveal goo robustness with, ince physical systems o not transfer infinitely high frequencies, their open-loop Nyquist plot always ens in the origin. Hence the "o not enter" circle can never cross the imaginary axis an the sensitivity peak never rops below for a physical system. Gla,. an Ljung, L (00). Control heory: ultivariable an Nonlinear ethos. aylor & Francis. Lonon. eborg, D.E., Egar.F. an ellichamp D.A. (004). Process Dynamics an Control. John Wiley & ons. New York. kogesta,. an Postlethwait, I. (005). ultivariable Feeback Control-Analysis an esign. John Wiley & ons. Chichester. Zhou, K., Doyle, J.C. an Glover, K. (996). Robust an Optimal Control. Prentice Hall. Upper ale River, NJ. 7