Stability and Robustness 1


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1 Lecture 2 Stability and Robustness This lecture discusses the role of stability in feedback design. The emphasis is notonyes/notestsforstability,butratheronhowtomeasurethedistanceto instability. The small gain theorem is introduced as a means to verify stability in presence of model uncertainty. 2. Stability of feedback systems Feedback and stability are closely connected issues. On one hand, the introduction of feedback may potentially create instability. On the other hand, properly applied feedbackisoftenthebestwaytogetridofinstabilities. There are several well known examples in the history. One is the construction ofairplanes.thefirstairplanesintheearly9sweredependingonthepilotto stabilize the dynamics manually using the controlstick. The modern fighter JAS Gripen was built unstable for the sake of maneuverability and relies on computer control for stabilization. Figure 2. Lawrence Sperry demonstrates a stabilizing gyroscopic controller. He waves his handintheair,whilehismechaniciswalkingonthewing. Another striking example was the construction of the first electronic feedback amplifiers, that were necessary to build long distance telephone connections in the93s.inthiscase,highgainfeedbackwasneededtoreducethenonlinear signal distorsion. Stability problems became a major issue, and the development of frequency domain stability criteria was critical for successful implementation. A modern example is the maneuver test for Mercesdes Aclass, the so called elktest, that created unstable oscillations severe enough to turn the car over. The problem was solved by introducing electronic feedback control. Recall from the previous lecture that a system is called inputoutput stable (orl 2 stable)ifitsl 2 gainisbounded.atransferfunctioniscalledstableifit corresponds to an inputoutput stable system. The following stability criterion is available for linear timeinvariant systems. THEOREM 2. ArationaltransferfunctionG(s)isstableifandonlyifallpolesofGhavenegative realpart.inparticular,if G(s) = C(sI A) B+D,itissufficientthatall eigenvalues of A have negative real part. WrittenbyA.RantzerwithcontributionsbyK.J.Åström
2 Lecture2. StabilityandRobustness Figure 2.2 The stability problems of Mercedes Aclass were solved by electronic feedback Moregenerally,asystemwithimpulseresponse (t)isstableprovidedthatg(iω)= e iωt (t)dtiswelldefinedandthereisacwith G(iω) cforall ω. Example Let us determine inputoutput stability of the following systems (I) [ ] [ ] 2 ẋ= + u 3 2 y=[ ]x+u (II) y(t)= t ) e (u(τ) y(τ) τ t dτ (I) The first system is inputoutput stable due to stable eigenvalues of the systemmatrix.atwobytwomatrixlikethisisstableifandonlyifthetraceis negative(here 3) and the determinant is positive(here 8). This is because thetraceisthesumoftheeigenvaluesandthedeterminantistheproduct. In general, eigenvalues can be computed by the matlab command eig(a): >> eig([ 2; 32]) ans = i i Note that the coefficients of the characteristic polynomial should not be computed, at least for high order systems, since this generally leads to numerical difficulties. (I I) The inputoutput relationship can be written y= (u y) where (t)=e t,t.afterlaplacetransformation,thisgives Y(s)= s+ [U(s) Y(s)] Y(s)= s+2 U(s) Thetransferfunction/(s+2)showsthatthesystemisstable,sincethe only pole 2 is negative. Our main objective is to study stability of feedback loops. From the basic course, we recall the Nyquist criterion, which supports understanding by graphical illustrations. 2
3 2. Stability of feedback systems Σ G(s) Im.5 ω Am m.5 ω c.5.5 Re Figure 2.3 TheclosedloopsystemremainsstableaslongastheNyquistdiagramdoes notencircle.theamplitudemargina m andphasemargin φ m measurethedistancefrom instability. THEOREM 2.2 THE NYQUIST CRITERION SupposethatG(s)isstableandthattheNyquistplotG(iω), ω Rdoesnot encircle.then(+g(s)) isalsostable. The following example illustrates the use of the theorem. Example 2 As motivating example, consider a position control in a mechanical system with damping coefficient c. The controller contains a time delay: ẍ+cẋ+x=u u(t)= k[x(t T)+ẋ(t T)] The feedback loop is illustrated in the figure below. Nominal values of the parametersarek=,c=andt=.letusinvestigatehowmuchmarginthereisin each of the parameters before the system becomes unstable? k(s+) s 2 +cs+ e st For this purpose, we plot the Nyquist and Bode diagrams of the nominal transfer function(s+)/(s 2 +s+).thematlabcommand margingivesnumericalvalues for the amplitude and phasemargins. Im.5 Nyquist diagram Re Magnitude Phase Gm = Inf, Pm = 9.47 deg (at.442 rad/sec) Frequency Figure2.4 Nyquistandbodeplotsforthenominaltransferfunction(s+)/(s 2 +s+) 3
4 Lecture2. StabilityandRobustness Fork=c=theopenlooptransferfunctionis s+ s 2 +s+ e st ForsmallvaluesofTtheNyquistplotwillnotencircle,sothesystemsremains stable.tofindoutexactlyhowlargevaluesoftareneededforinstability,notethat the phase margin 9 degrees(or 9π/8 radians) is obtained at the frequency 9π.4rad/sec.Henceatimedelayof 8.4 =.35secondscanbetoleratedfor k=c=. Toinvestigatetherobustnesstovariationsintheparameterscandk,wenote thattheclosedloopcharacteristicpolynomialfort=is (s 2 +cs+)+k(s+)=s 2 +(c+k)s++k The stability condition for a second degree polynomial is positive coefficients, so stabilityismaintainedaslongasc+k>and+k>. Having analyzed the example with respect to parametric uncertainty above, it is natural to ask about robustness to unmodelled dynamics. For example, this would be relevant to capture the difference between a rigid body and an elastic one.thisisalsothesubjectfortheremainingpartofthelecture. 2.2 Sensitivity Two transfer functions are of particular interest in the study of the feedback loop below. d n r e u Σ C(s) Σ P(s) x Σ y Figure 2.5 A simple control loop These are S(s)= +P(s)C(s) T(s)= P(s)C(s) +P(s)C(s) (the sensitivity function) (the complementary sensitivity function) ThetermcomplementaryreferstothefactthatS(s)+T(s).NotethatT(s) isthetransferfunctionfromreferencesignalrtothetheplantoutput x.another name for T(s) is therefore the closed loop transfer function of the system, while P(s)C(s) is called the open loop transfer function or just the loop transfer function. The name sensitivity function refers to the fact that S measures how small relativeerrorsin ParemappedintorelativeerrorsinT.Thisisverifiedbya simple calculation: dt dp = d dp ( ) = +PC C (+PC) 2=TS P dt/t dp/p =S 4
5 2.3 Norms of signals Im R =sup ω +C(iω)P(iω) R Re C(iω)P(iω) Figure 2.6 TheL 2 gainofthesensitivityfunctionistheinverseofthedistancefromthe Nyquist plot to. Notice that T is a nonlinear function of P. Hence the sensitivity calculation only says something about the response to small pertubations in P. For larger perturbations, the system could have a drastically different behaviour and even become unstable. Given the Nyquist criterion, it is natural to conjecture that the robustness to unmodelled dynamics should somehow be related to the distance from the Nyquist plottothepoint.itisthereforestrikingtonotethatthel 2 gainofthesensitivity function turns out to be exactly the inverse of this distance. Consequences are investigated in the next section. 2.3 Norms of signals InthepreviousChapterwelookedatsizesofsignals(norms)andgainsofsystems. RecallthattheL 2 gainofasisosystemwasthelargestmagnitudeinthebodediagram. Later in the course we will work with multivariable systems where the relation between vectors of inputs and outputs could be described as a matrix with transfer function elements, as for example [ y y 2 ] [ = 2 s+ s s 2 +s+ ] 4 [u 2s+ 3 u s+4 2 ] (2.) Forthatpurposewewillfirstintroducehowtomeasurethesizeofavector (vectornorm)andbasedonthistheinducednorm(gain)ofamatrix. Forx R n,weusethe L 2 norm x = x T x= x 2+ +x2 n YoumaythinkofitasPythagoras theoreminr n. FormatricesM R n n,weusethe L 2 inducednorm,thelargestratiobetween the output and the corresponding input M :=sup x Mx x =sup x xt M T Mx x T x = λ(m T M) Here λ(m T M)denotesthelargesteigenvalueof M T M.Thelargestgainis thus λ(m T M),alsocalledthelargestsingularvalueofM,denoted σ(m).the 5
6 Lecture2. StabilityandRobustness.5 Input u= [.39.95] T, u =.5 u u y=gu = [ ] T, y = y (red):eigenvectors ; (blue): V ; (green): U A=U*S*V T Figure 2.7 Matlabexample showing how the output vector y = Mu changes size and directionwhentheinputsignalwith u =arechangedindifferentdirectionsaroundtheunit circle.(note scaling of axes in lower plot). fraction Mx / x ismaximizedwhenxisacorrespondingeigenvectortom T M, butnotethatweingeneralhavedifferentdirectionsoftheinputvectorxandthe outputvectormx(asxingeneralisnotaeigenvectorofm). Example: [ y y 2 ] = [ ][ u u 2 ] Matlabcode for singular value decomposition of the matrix [ ] 2 4 M= 3 Singula Value Decomposition(SVD): M=U S V whereboththematricesu andv areunitary(i.e.haveorthonormalcolumnss.t. V V = I)and Sisthediagonal matrixwith(sorteddecreasing)singularvalues σ i. MultiplyingMwithainputvectoralongthefirstcolumnin Vgives M V (:,) =USV V (:,) = [ ] =US =U (:,) σ Thatis,wegetmaximalgain σ intheoutputdirectionu (:,) ifweuseaninputindirectionv (:,) (andminimalgain σ n = σ 2 ifweusethelastcolumnv (:,n) =V (:,2) ). >> M=[2 4 ; 3] M = >> [U,S,V]=svd(M) U = S = V = >> M*V(:,) ans = >> U(:,)*S(,) ans = Nowwhenweknowhowtocalculatetheinducednormforamatrix,wecan apply this to a transfer function matrix, where the elements depend on the frequency. The magnitude plot of the Bode diagram for singleinputsingleoutput systems has the multivariable correspondence of plotting all singular values for 6
7 2.3 Norms of signals Singular Values 2 System: G Frequency (rad/sec):.6 Singular Value (abs): 5.26 Singular Values (abs) System: G Frequency (rad/sec):..6 Singular Value (abs): Frequency (rad/sec) Figure 2.8 The singular values of the tranfer function matrix in Eq. 2.. Note that G()=[2,4 ;3]whichcorrespondstoMintheSVDexampleabove. G = thematrixtransferfunctionasafunctionofthefrequency.inmatlabthisisdone with the command sigma, see example below. ThegainofthesystemGisdefinedas G =max ω G(iω) and is the largest magnitude(largest singular value) when sweeping over all frequenciesfor G(iω),seeFig2.8. Example: Consider the transfer matrix G(iω) from Eq. 2. G(s)= 2 s+ s s 2 +.s+ 4 2s+ 3 s+ >> s=tf( s ) >> G=[ 2/(s+) 4/(2*s+); s/(s^2+.*s+) 3/(s+)]; >> sigma(g) % plot sigma values of G wrt fq >> grid on >> norm(g,inf) % infinity norm = system gain ans =
8 Lecture2. StabilityandRobustness 2.4 Robustness via the small gain theorem r e S e 2 r 2 S 2 In this section, we will investigate stability robustness using the following theorem,basedonthenotionofinputoutputl 2 gain.forsimplicity,calculations are done assuming zero initial conditions. THEOREM 2.3 THE SMALL GAIN THEOREM AssumethatS ands 2 areinputoutputstablesystemswith L 2 gain S and S 2.If S S 2 <,thenthel 2 gainfrom(r,r 2 )to(e,e 2 )intheclosedloop system is finite. Proof. Define y T = T e =S 2 (e 2 )+r e 2 =S (e )+r 2 y(t) 2 dt.then S(y) T S y T. e =r +S 2 (r 2 +S (e )) ) e T r T + S 2 ( r 2 T + S e T e T r T + S 2 r 2 T S S 2 Boundedgainfrom(r,r 2 )toe followsast.thegaintoe 2 isboundedin the same way. To demonstrate how the small gain theorem can be used for robustness analysis,considerthefeedbackloopinfigure2.4,wheretheplant P(s)hasbeen replaced by[ + (s)]p(s). v (s) w C(s) P(s) Figure 2.9 Loop diagram with perturbed plant[ + (s)]p(s) Thetransferfunctionfromwtovisequalto T(s),thecomplementarysensitivity function. Hence, by the small gain theorem, the feedback system remains stableaslongas T < 8
9 2.4 Robustness via the small gain theorem Note that the small gain theorem does not assume linearity or timeinvariance. HencetheclosedloopsystemwillremainstableforallplantsoftheformP(s)[+ (s)] where has L 2 gain smaller than[sup ω T(iω) ], even for that are nonlinear or timevarying. Asasecondexample,letusderiveastabilitycriterionforthecasethatthe perturbation appears additively, i.e. P(s) is replaced by P(s) + (s). v (s) w C(s) P(s) Figure 2. Loop diagram with perturbed plant P(s) + (s) ThenthetransferfunctionfromwtovisequaltoC(s)S(s),sothesmallgain theorem shows stability for all perturbations satisfying CS < For linear timeinvariant perturbations, this criterion can be nicely illustrated in the Nyquist diagram. Clearly, the condition C < +PC guaranteesthatthenyquistplotof(p+ )Cdoesnotencircle. +PC C 9
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