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Mixed-μ-Analysis for Flexible Systems. Part I: Theory1
MIXED-~-ANAL YSIS
FOR FLEXIBLE SYSTEMS. PART I: THE ...
14th World Congress oflFAC
P-8a-03-3
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
MIXED-/-I-ANALYSIS FOR FLEXIBLE SYSTEMS. PART I: THEORY 1 Jean-Franc;ois Magni ~ Carsten Dull· Caroline Chiappa Benolt Frapard ** * Bimedicte Girouart· u
• ONERA-CERT, DCSD, B.P. 4025, F31055 Toulouse Cedex 04 , Prance, Fax: -{·33 5 62 15 15 64, E-mail: [email protected] ** SUPAERO, B.P. 4025, F31055 Toulouse Cedex 04, France • •• l'vJatra Marconi Space, 3 1 av. des Cosmonaut es, F31402 Toulouse Cedex 04, France, Fax: + 33 5 62 19 78 97, E-mail: [email protected].!r· Abstract: This paper presents new tools for computing upper and lower bounds of /-I wit.hout. frequency gridding. The proposed techniques for computing lower b o unds of the peaks of the /-I-c urve, a re divided into two steps. The first. one consists of finding the p erturbation wit h minimum Frobenius norm that leads to the limit of stahility. Using this result as a ll initializa tion, the second algorithm finds the p e rturbation 'with minimum sigma-max norm such that the system remains at th e limit of stability. The limit of stability is considered bot h from a state space and from a transfer m at.rix point of view, which leads to two classes of techniques. The lower bounds are validated by using an upper-bound analysis t.echnique that consider:-. standard scalings over a range of frequencies instead of at a n isolated frequency. Copyrig ht © 1999 IFAC Keywords: Sta bility analy sis, robustness , satellite contro l, fr e4uency swe ep. 1. INTRODUCTIOI\
This paper is divid ed into two parts. Part I d escribes and justifies several Ilew algorit.hms. P a rt II consists of an evaluation of these tools in an indust.rial setting conside ring a satellite having a flexible flol a.r genera.t.or. St.ability analysis of this kind of highly fl exible structures is a challenging problem. Standard tt-analysis is not efficient for this class o f problerns for two reasons. First, if frequ ency gridding is used, p eak va lues of I~ are generall y missed. S econd, uncertainties are usually real, in t.his case t h e lower bound of /-I cannot b e computed wit.h s t anda rd numerical tools . All available lower bound comput.ation techniques consis t of finding a perturba tion which corresponds t.o the limit of stability. All techniques are more or les s heuristic. A fixed point algorithm is proposed in (Packard and Doyle , 1993; Young a nd Doy le , 1990). This algorithm is very efficient for mixed uncertainties in which the complex par t is large enough . lTnfort.uua t ely, when uncertainties are mode led as real para meter variatio ns this 311 This wor k was supported in p art by the Fren c h ~Ministry of Defense and for the second a uthor, by the HSP HI grant program of the German Ac ad emic EXChange Service. INe would like to t hallk Dr. G . F e rre res a nd Dr. J .Iv1. Biao u;c for h el pful discussions.
gorithm does not conve rge well enough. The npper boun ds are defined as conve x optimizatio n problems after frequency gridding, see for example (Fan et al., 1991; B eck and Doyle, 1992). These techniques are r eliable when t he tt- curve does n ot present sharp peaks. In orde r to treat the problem of sharp p eaks , frequen cy swe eping is used (see (Sideris, 1992)) , the frequency is considered as an additional real u ncertainty which should b e repeated as many times as the orde r o f the system. Therefore, computationally, this approach rapidly becomes too demanding for r eaJ-word applications. Our m ain contribution is de tailed now. Concerning the lower bound co mputation, nmv algorithms which compute a lower hound of the peaks of the f1, curve are proposed. These algori t hms are efficient for mixed or pure real uncer t ainties. Since frequen cy gridding is not used, the proposed algorithms are very fa~t . This is quit.e useful in a design cycle in which it is necessary to dete.-.t won'. cases, see (Magni et al., 1998). The idea beyond theses techniques consists of shifting the cigcnvalJ.es toward t1 e imaginary axis with a minimum perturba tion. The proposed algorithms are rlivid e d into t wo s teps. The first step is used to reach the limit of stability with a perturbation of minimum Frobenius norm . For that purpose the algorithm proposed in (De M oor, 1993) was t.est ed but th e
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MIXED-Il-ANALYSIS FOR FLEXIBLE SYSTEMS. PART I: THE ...
convergence was very erratic. The fixed point algorithm of (Young and Doyle, 1990) was used, the convergence propeI:ties are improved on account of the good initialization of st.ep 1 (see (Magni and Doll, 1998) for more details). But the alternative technique detailed in this paper is more reliable. Concerning the upper bound, t.he technique proposed here takes advantage, over a frequency interval, of sealings t.hat. are computed at a given frequency. For an alternative approach with a different purpose refer to (Ferreres and Biannic, 1998). The algorithm proposed here checks whether a given ('est value, for example the maximum value of the lower bound augmented of a given percentage, is larger than the upper bound of Jl. «(Deck and Doyle, 1992; Fan Et al., 1991; Packard and Doylc, 1993; Young Et al., 1995)) over all freqllencie;;. At each step of t.he proposed algorithm, scalings are computed at a given frequency. The frequency intervals for which these scalings permit us to conclude that the test value is larger than Jl. are eliminated. The frequency intervals that can be eliminated are characterized analytically (see Lemma 4). If all frequencies can be eliminated, the test value is an upper bound of the peak values of p, therefore the lower bound is validated. 2. NOTATION AND PRELIlVIINARIES Let us consider a quadruple in state space form (A, B, C, D) and in t.ransfer matrix form iVIes) = C(sI _A)-1 B+D. The poles of kI(s) wiII be called "open-loop poles". A class of uncenainties ~ (which act as a feedback on the systerIl, (M -~)-forIIl) has t.he following structure:
(1) in which .c.i might be a diagonal matrix of the form ~i = oi1ni with 6 i E 1R (real repeated scalar block) or Oi E ( (complex repeated scalar block) or a matrix in (71., xn, (full complex block). In t.he part relative to the lower bound Ai might also be a matrix in 1R. n,xn, (full real block). Such a matrix ~ will be called an "admissible perturbation". The real and imaginary part of a complex number are denoted R(.) andG(.), the maximum singular value a{.) the cigenvaluc with maximum modulus )."(.). First let us recall the well known result satated in Lemma 1. Form this result, two equivalent definitions will be proposed.
Lemma 1. If Li is such that the interconnection (!vI, Li) remains well-posed and if So does not belong to the spectrum of A: dct(I - M(so)Li) = 0 ~ So E spectrum(A + BLi(I - D.c.)-lC)
(2)
Definition 2.1. The S.S.v. at point At is defined as Ija(~) where.6. is an admissible perturbation such
14th World Congress ofIFAC
that: (1) At belongs t.o the dosed-loop spectrum of (A, B, C, D) with feedback ~, (2) 0'(.6.) is minimum.
Definition 2.2. The s.s.v. at. point At. is. defined as IjO'(Li) where .6.. is an admissible perturbation such that: (I) det(I - lvI(Atl.6.) = 0, (2) O'(Li) is minimum. In view of the above definitions, computing p- at point At is the problem of assigning the pole At with minimum norm of the "feedback". So, finding in one shoot the ma.ximum value of f-t over all point;; on the imaginary axis, is the problem of assigning the value of the imaginary axis for which the required "feedback" is minimum. The following result (see (Magni and Manouan, 1994» will be used for shifting the open-loop poles towards the imaginary axis.
Lemma 2. The first order approximation d)" of the motion of an eigenvalue ,vith multiplicity equal to one of the matrix Aa, induced by a gain variation d~ of Ao is
riJl. = (uB
+ tD)d.6.(Cv + Dw)
(3)
where v is the right eigenvector of Ao, u is the left eigenvector of Aa corresponding to the eigenvalue A and w = .6. o (I - D.6. 0 )-lCv, t = uB~o(I D~o)-l.
3. MINIMUM FROBENIUS NORM CASE Finding a lower bound of the peaks of the pcurve (worst case) consists of applying Algorithm 1 (state-space approach, see §3.1) or Algorit.hm l' (transfer matrix approach, see §3.2) and then Algorithm 2 (see §4) can be used to find the best candidate with minimum "sigma-max norm". In §3.3 is considered a possible pathology of Algorithms 1 and 1'.
3.1 State-space approach Principle of the algorithm. Let ), denote one of the eigenvalues of A + B.6. o (/ - D.6. o )-l C. It is intended to find d.6. (a variation of Ao) that shifts (first order approximation) A to a vertical line which is distant of a small aIIlount denoted ~i' A motion from A to the vertical line defined by Ri is performed as follows. Equation (3) is a linear constraint on d~
iJi«uB
+ tD)d.6.(Cv +
Dw» = Ri
(4)
d~
satisfying (4) will be computed such that the Frobenius norm
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Copyright 1999 IF AC
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MIXED-Il-ANALYSIS FOR FLEXIBLE SYSTEMS. PART I: THE ...
is mmlmum. This is a problem of quadratic optimization under linear constraints. Such a problem will be solved at each iteration of the proposed algorithm. The details on the way the structure of .6. is taken into account in order to solve (4), minimizing J 1 in (5), are given in (Magni and DoH, 1998). In order to avoid initialization problems due to the fact that, when the criterion Jr is minimum for large values of dA, the first order approximation of Lemma 2 is no longer valid, we shall minimize a combination of criteria J 1 and J o , where Jois defined by
(6) At the beginning of t.he algorithm, J o is considered, then a combination of J o and J 1 that becomes equal to J 1 (see (7)). AlgorithIll 1. Step 1 - Initialization. Choose the initial open-loop eigenvalueR that are to he moved t.owards the imaginary axis (this choice is made by considering a controllability measure). Choose also the expected number of steps (say N) that will be used in order to shift. each initial eigenvalue t.o the target. For each initial eigenvalue >., perform the following steps. Set i = -1 and .6.0 = O. Step 2 - Compute u, v, w, t as in Lemma 2. Then solve (4) for dA having the admissible structure, the variation of 1R(>') being given by 1/(i = -R(>'l/(N - i) and t.he following combination of criteria being minimized:
(N - i - l)Jo
+ (i + l)Jd/N
(7)
Step 3 - Set i = min(i + 1,N -1), Aa = Aa +dA. After t:..o is updated, select the new closed loop cigcnvalue that is the closest to .\ + Ri that will becorne the new),.. If ),. is close enough to the imaginary axis, stop, otherwise go to step 2.
COIllInCnts relative to AIgorithlll 1. Measure of controllability. In order to reduce the computing time it. is useful to apply the algorithm only to a subset of the poles of ,Ij,[(s). The llse of the bandwidth knowledge of the system behavior can help. It is also possible to apply a controllability measure. From Lemma 2, Cont.(>..) = IluB + tDIIIICv+Dwll can be consjden~d as such a measure. The number of poles to be considered varies depending on the system, so it must be fixed by trial and errors. (It is also possible to measure the global controllability by analysing the poles "assigned" by random trials of t:...) Details on computation involved at Step 2. Equat.ion (4) is a linear constraint relative to the entries of dA and (7) is a quadratic criterion. An intermediate step which consists of writing the entries of t:..o and d.6. as real vectors ';0 and dE, (in which nncerta,inties are nnt repeated) is necessary, see
14th World Congress ofIFAC
(Magni and Doll, 1998). Equation (4) can be written
Ad(= b
(8)
(A being a row vect.or) and J a and J 1 in (7) become respectively .10 = dl;T Ho cl!; and J 1 = dl;T HI dl:, + 2Cl dt;. The computation of the matrices A, Ho (= H d and Cl is treated in (Magni and Doll, 1998). Finally the problem to be solved at each step is a least square problem. Numbe. of steps, precision. At the first step, i = -1 so J o is optimized. \Vhen i is saturated at N - 1 the criterion is exactly equal to Jl. Usually, N = 20 is enough. After these initial iterat.ions it is worth adding some iterations (about 10) for improving the assignment on the imaginary axis. 3.2 Tmnsfer matrix approach
This sect.ion describes a technique similar to the one presented in §3.1, but it is the transfer matrix that is used instead of the state space representation. In fact, all lower bounds computation techniques are based on iterative techniques without guarantee of global convergence. So, it is very useful to have as many complementary tools as possible. The main difference is that, instead of assigning a pole on the imaginary axis (see Definition 2.1) we want det(I - 111(s)A) to become equal to zero for some value of .'i 011 the iIIlaginary axis (see Definition 2.2). It is well known that the loop in which 1\,-[(8) is in closed loop via A (M - A)-form) is equivalent to a similar loop where <'vI (oS) and t:.. are replaced respectively by AI and .c.. where:
-AI = [A B] CD
As we are interested in poles on the imaginary axis s takes the form l/.iw. We shall denote <5 = l/w moving the complex number .i inside M, we have -I -I to consider i\1 and the interconnectioll A : -I _
M
-
[-jA B] -jC D
. /),.' = ,
[li
I 0 ]
0
t:..
At this step, the following property is clearly sat.isfied: det(I - kI'A') = 0 <=? det(I - M(i).c..) = O. Therefore, the problem to bp treated consists of shifting a pole of the system (I,-Af',I,O) from 1 t.o the origin. But the difference with the algorithm of §3.1 is that it is not the norm of A' + dA' that must be considered, because thi" artificial perturbation contains the frequency (6). The norm to be considered is the norm of the matrix
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ISBN: 0 08 0432484
MIXED-Il-ANALYSIS FOR FLEXIBLE SYSTEMS. PART I: THE ...
The Frobcnius norm of the above matrix is not a positive definite criterion because the weighting relative to t5 is zero. In order to obtain a regular problem, an additional step is required. This step consists of eliminating t5 from the constraints and criteria. Elitnination of b. Applying Lemma 2 to the system (I,-M',I,O) (instead of (A,B,C,D)) for a variat.ion dJ.... we have: -v.j'v!' d,~' v = d>... As for (8) this equation can be written (see (Magni and Doll, 1998)) for do and the entries of d~ in vector form dE.
(9) Let A = [a.,. + ja; Ar + jAi J and dJ... = b,. + jb i , considering the imaginary and real parts a r do + A,. dt;, = by" and a; do + Ai do; b i , after elimination of db, only one equation remains
=
(IQ)
14th World Congress ofIFAC
Principle of the proposed algorithm. Considering the singular value decomposition (s.v.d.) of Llo = U SV' and denoting VI the first column vector of F, the maximum singular value of ~n is a(~o) = ,jVt ~o~o Vi For minimizing the maximum singular value, a new criterion
is defined in which Vl is relative to ~o. As it is expected to consider small va.riat.ions of 6. 0 we shall have J 2 ~ If(b. o + db.). It is this approximation J 2 of the maximum singular value that will be eonsidereu. Algorithm 2. Step 1 - Initialization. Perform Algorithm 1 or l' (or, if relevant, consider root locus as in §3.3). Let ~o denote the resulting admissible perturbation and), the eigenvalue which is approximatively on the imaginary axis. Choose the number of itera.tions IV. Set i = O. Step 2 - Compute n, v, W, t as in Lernma 2. Then, for dLl having the admissible structure, solve thA following LQ problem:
Finally, the frequency (or 0) is eliminated from the considered variable, therefore the criterion relative to the Frobenius norm is again positive definite. AlgorithIll 1'. This algorithm differs somewhat from Algorithm 1. The details can be found in (Magni and Doll, 1998). 3.3 Use of mot locus in some pathological cases For some systems, some open-loop poles might be sensitive to a single real block of ~. In this case there 1S only one degree of freedom for pole motion: "root locus" is the best solution to find the limit of stability.
4. MINIMUM SIG:MA-MAX NORM CASE After having used the algorithms of the previous sections, we have at our disposal a matrix Llo which assigns a pole At on the imaginary axis. But the norm that was minimized for obtaining this result is not the right one. So, a second algorithm is proposed: the assignment of lli(J.d is preserved while the convergence towards a matrix "6." with minimum sigma-max norm is performed. The power algorithm of (Young and Doyle, 1990) was also considered. Its convergence is very good for mixed or complex uncertainties. In the pure real case, convergence is erratic and fails in most cases. It was expected that a good initialization, as for example the results of Algorithm 1 (§3.1)' would improve eonvergence in the pure real case. Briefly, when the power algorithm is initialized by the result of Algorithm 1 it is faster that Algorithm 2 pI'esented below. But Algorithm 2 is luuch more reliable as convergence is almost insured, in addition computing time is still reasonable.
!R((uB
+ tD)d.6.(Cv + Dw)) J = (N - i).h
= -lR('\)
(12)
+ iJ2 )/N
(13)
Step 3 - Set i = min(i + 1, N - 1), .6.. 0 = .6.. 0 + d~. After .6.. 0 is updated, select the new closed loop eigenvalue that is the closest to 'J(.\) that will become the new .\. If the value of h becomes stationary, stop, otherwise go to step 2. COIllIllents relative to Algorithm 2. Details on computation involved at Step 2. Equation (12) is a linear constraint relative to the entries of dA and (13) is a quadratic criterion. An intermediate step which consists of writing the entries of 6. n and db.. as real vectors t;,o and dt;, (in which uncertainties are not repeated) is necessary, sec (Magni and Doll, 1998). Equation (12) can be written as Equation (8) and J 1 and J 2 in (13) become respectively Jl = dE;.T H 1 dt;, + 2Cl dE;, and h = dt;,T H 2 dE;. + 2C2 dE.. See (Magni and Doll, 1998) for details. Finally the problem to be solved at each step is a least square problem. Coalescence of singular values at the optimum. In order to al1eviate the presentation of the algorithm we ignored a very important fact. If the algorit.hm is run a.s above, the optimization will end when the two leading singular values of 6. become equal. It is worth noting that Algorithm 2 tends to tune the Frobenius optimal perturbation by reducing the leading singular values. Coalescence of two or more leading singular values is therefore usual. In order to optimize further after the two leading singular values are equal, a simple improvenlent is required. The details are given in (Magni and Diill, 1998). This improvement reduces to testing that some
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Copyright 1999 IF AC
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MIXED-f!-ANAL YSTS FOR FLEXIBLE SYSTEMS. PART T: THE ...
singular values become equal and then to adding some rows to the linear equation that corresponds to Equa tion (12).
Evolution oJ the criterion, number of iterations. When i increases, the weight of J',!, increases. \Vhen i is saturated at 1'>' - 1 the criterion is equal to (ljN)J 1 + (l\' - 1)jN)J2 • It is bette r to transform "smoothly" the criterion from J 1 to Jz, beca.use if (ljN)JJ + «(N - l)jN)Jz were used abruptly, for some systems the variation of ~ would be too large taking into account the first order approximation behind Equation (12). Usually, N = 20 is enough. After t.hese initial iterations it is worth adding some it.e rations in order to treat. singular values coalescence. The numb er of additional iterations depends on the number of singular values which become e qual. Globally, 80 iterat.ions arc enough for most systems . 5. UPPER BOUND
As for all lmver bound of p" the lower bound of the peaks presented in the previous sections has an h e uristic base. There is no guarantee that the proposed algorithms will lead to all the peaks of f1 with much accuracy. All computation of a lowe r bound must be va.lidated by an upper bound. Here we take advantage of the knowledge of the lower bounds of peak values of /.L. Let P,max be the highest lowe r b o und available. In order to validate this result say with 10 % accUl:acy, it suffices to consider a test value PT = 1.1 /-I.max and to check that P is lower than this value for all frequencies. First is considered the problem of finding the interval of freqnencies for which a given pair of scalings Do and Go enables to conclude that p. is less than /.LT (graphic point of view in §5.1, analytic in §5.2). Afterward, this result will b e organized as an algorithm,
14th World Congress ofIFAC
known as a generalized eigenvalue problem. If /3 is fixed a priori, it is also possible to find D and G satisfying (14) by solving an LMI.
Definition 5.1. A pair of sealings D et G being given, the mapping VD,G(W) is defined as being the minimum value of /3 satisfying (14). Note that, regardless of the optimality of the scalings, if VD. G (W) is less than flT therefore /.l(NlCiw)) is less tha.n /.LT. Algorithm 3 proposed in §5.3 aims at eliminating the interva ls of frequencies for which it is known that PT is larger than p,(Al(jw). Plotting VD,a(W) is useful for illustrating the elimina.tion procedure. After analysis of several plots of I'D,G(W), it turns out t.ha,t opt.ima.l scalings, i.e. scaJings that satisfy (14) at s = jwo fo r the minimum value of /3, correspond very often to mappings VD,G(W) which are a lmost vertical lines at Wo. In this case, the intervals that can be eliminated are very narrow. The use of non-optimal scalings permits us to improve con siderably the efficiency of the proposed algorithm by increasing the size of the eliminated intervals. In orde r to compute sub-optimal scalings, it is suggested to consider together three frequencies, say Wo, (1 +E)WO aud (1I')wo instead of the single frequency Wo and to find the scalings corresponding to the minimuIIl value of f3 in the syst.em of three LMls corresponding to (14) at the three frequencies. Figure 1 illustrates the mapping VD,a(W) and the corresponding eliminated frequencies for E = 0,0.1,0.2,0.4. The vertical linp. corresponds to VD.a(W) with optimal scalings (~ = 0), it. is clear that the size of elirninat.ed frequencies is very small.
\
5.1 Po int-wise upper-bound
Two sets of HeaIings having the same structure than .6. (see (1)) are considered. (1): V : D i = Dj > 0 and Di E <[n, xn, for real and complex repeated block,. D·i = dJ,.", and d i > 0 for full complex blocks. (2): Q : G; = Gi and G i E <[n, Xn; for real repeated blocks. G i = 0 for other block!'!. \Ve shall use t.he following well known result (see (Fan rot aI., 1991))
Lemma 3. The Laplace variable s being fixed, let D E V, G E 9 and f3 be such that lVI(s)* DJ.\,f(s)
+
j(GAl(s) - M'(s)G) ~ {32 D (14) then p(M(s»::; fJ
(15)
The proble m of finding the best upper bound !3 of p,(A1 (s») i~ equivale nt to the problem of minimizing fJ under the LMI condition of Equation (14). I t is
Fig.
1. Illustration of e liminated frequencies; VD,G(W) around Wf) 0.4 for E = 0,0.1,0.2,0.4.
=
5.2 Elimination of fr'equ ency intervals
The computation of eliminated intervals can be made graphically as above by considering; plots of VD,G(W). Now this problem is considered from an analytic point of view. Let us consider th e scalings Do and Go that satisfy (14) for s = jwo and
/3
= IJ.r·
Lemma 4. (Magni and DOll, 1998)) L e t (A, B,C, D) be a state space reali zation of M(s). For a given value fLr and twO scalings Go and Do
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MIXED-Il-ANALYSIS FOR FLEXIBLE SYSTEMS. PART I: THE ...
such tbat. M(jwO)H DoMUwo) + j(GoM(jwo) Ail! (jwo)G o ) < fl}Do and 1 liE spec((DT DoD + D- 1 j(GoD - DTGo))=:g-) we have ILT
1l,(M(jw) ::::
{J.T
A=
+ 73(f/.~Do
- D)-le
(16)
[CT~oC _~T J ;73= [CTDoD~jeTGoJ C
= [DT DoC + jGoe _ET]
D = (j(GoD - DTG O )
+
w
w,
for wE [w- w+]
where jw- and jw+ are the eigenvalues of the matrix H the closest to jwo (w- < Wo and Wo < w+) where: H = A
Step 3. Consider the first interval that is not already eliminated. • 3.1. Let denote the frequency in the center of the above interval. Then compute the scalings D et G relative to (1 - C3) wand (1 + <'"3) w. If the corresponding value of j3 is larger than ).LT go to 3.3 .. Otherwise go to 3.2. • Compute the newly eliminat.ed interval. If all frequencies are eliminated: end -+ Pr is validated. Ot.herwise go to 3.1. • 3.3. Compute the pair of optimal scalings at WOi· If the corresponding value of (3 is larger than !lr, end -+ f.1.T is not validated. Otherwise, compute the newly eliminated interval. If all frequencies are eliminated : end -+ /-IT is validated, Otherwise go t.o 3.1.
DTDlID)
5.3 Proposed algorithm AlgorithzIl 3. Step 1. Initialization. Compute lower bounds of the peal~ values of fl" for example using Algorithm 1 + Algorithm 2 or Algorithm l' 1- Algorithm 2. So wc have at our disposal q pairs (Wll;, !la;) lmch that 11 (AI CjwOi)) > 110i. Define the test value I-I.T and the interval of frequencies [1,;£ w] to be treated by the algorithm. Initialize also El (for example El = 0.1) and £2, £3 (for example ('2 = £3 = 0.2). Initialize t.he set. of eliminated intervals to [-00 ~l u [w + 00).
Step 2. This step consists of a preliminary elimination of t.he frequencies, .~tarting elimination from the peak 1fall~eS given by (wo" !lOi). The following "sub-steps" are to be performed for i = 1, ... ,q. • 2.1. Consider one of the q pairs, compute the "scaiings" D et G (sub-optimal at WOi) for three frequencies (1- I'd WD;, WOi and (1 + El) Wo;. If the corresponding minimum value of j3 is larger than PT, go to 2.4. Otherwise go to 2.2. • 2.2. Comput.e the newly eliminated iuterval [w- w+} using the techniqlle of §5.2. If all frequencies are eliminated: end -+ IlT is validated. Otherwise go t.o 2.3. • 2.3. If Iw+ - wod > 1"-'0; - "-'-I compute the scalings D and G relative to two frequencies (1 ("2) u.,'- and :....0-. Otherwise if Iw+ - wOil < IWoi w-I consider the two frequencies (1 + 1"2) w+ and w+. In uath cases compute the eliminated intervals using the technique of §5.2. If all frequencies are eliminated : end -+ PT is validated. Otherwise go to 2.1 in order to treat a new pair (w'Oi, 110;)' If all these pairs are already treated, go to Step 3. • 2.4. Comput.e t.he pair of optiTna/ scalings at Wo;. If the corresponding value of f3 is larger tha.n pr, end -+ f.1.T is not validated. Otherwise, Compute the newly eliminated interval. If all frequencies are eliminat.ed: end --'> P,T is validated. Otherwise go to 2.1 in order to treat a new pair (wo;, f.1.0;)' If all these pairs are already treated, go to Step 3.
COIllm.ents relative to Algorithm. 3. Concerning Step 2.3: The intervals that are eliminated at steps 2.1 & 2.2 are not centered around WOi because most of the elimination is done on one side of WOi. This remark justifies step 2.3 which aims at eliminating on the other side of wo,.
jA) l' h1_/ \i
:1
V
V
----::c------;:;~
: •• LI
Fig. 2. Illustration of the proposed algorithm : 5 curves VD,G(W), the 1st & ,1th correspond to step 2.2, the 3rd to step 2.3, the 2nd & 5th to step 3.2.
6. REFERENCES De 1vloor, B. (1993). Structured total least sqllare and 12 approximation problems. Linear Algebra and its Applications 188, 163-205. Magni, J.F. and A. Manouan (1994). In Proc. of the IFAC Sympm.ium on Robust Control, Rio de Janeiro, Brasil pp. 388-393. Magni, J.F. and C. Doll (1998). p-analysis for flexible systems. LAAS Toulouse, France pp. 1-34. Magni, J.F., Y. Le Gorrcc and C. Chiappa (1998). A multimodel-based approach to robust and self-scheduled control design. In Proc. IEEE CDC. Packard, A. and J.C. Doyle (1993). The complex structured singular value. Automatica. Young, P.M. and ,l.C. Doyle (1990). Computation of f.l with real and complex uncertainties. in Proc. IEEE GDG pp. 1230-1235. Young, P.M., M.P. Newlin and J.C. Doyle (1995). Computing bounds for the mixed f.1. problem. International Journal of RolJust and lVonlinear Control 5(6), 573-590. Missing references: see bibliography of Part H.
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14th World Congress oflFAC
P-8a-03-3
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
MIXED-/-I-ANALYSIS FOR FLEXIBLE SYSTEMS. PART I: THEORY 1 Jean-Franc;ois Magni ~ Carsten Dull· Caroline Chiappa Benolt Frapard ** * Bimedicte Girouart· u
• ONERA-CERT, DCSD, B.P. 4025, F31055 Toulouse Cedex 04 , Prance, Fax: -{·33 5 62 15 15 64, E-mail: [email protected] ** SUPAERO, B.P. 4025, F31055 Toulouse Cedex 04, France • •• l'vJatra Marconi Space, 3 1 av. des Cosmonaut es, F31402 Toulouse Cedex 04, France, Fax: + 33 5 62 19 78 97, E-mail: [email protected].!r· Abstract: This paper presents new tools for computing upper and lower bounds of /-I wit.hout. frequency gridding. The proposed techniques for computing lower b o unds of the peaks of the /-I-c urve, a re divided into two steps. The first. one consists of finding the p erturbation wit h minimum Frobenius norm that leads to the limit of stahility. Using this result as a ll initializa tion, the second algorithm finds the p e rturbation 'with minimum sigma-max norm such that the system remains at th e limit of stability. The limit of stability is considered bot h from a state space and from a transfer m at.rix point of view, which leads to two classes of techniques. The lower bounds are validated by using an upper-bound analysis t.echnique that consider:-. standard scalings over a range of frequencies instead of at a n isolated frequency. Copyrig ht © 1999 IFAC Keywords: Sta bility analy sis, robustness , satellite contro l, fr e4uency swe ep. 1. INTRODUCTIOI\
This paper is divid ed into two parts. Part I d escribes and justifies several Ilew algorit.hms. P a rt II consists of an evaluation of these tools in an indust.rial setting conside ring a satellite having a flexible flol a.r genera.t.or. St.ability analysis of this kind of highly fl exible structures is a challenging problem. Standard tt-analysis is not efficient for this class o f problerns for two reasons. First, if frequ ency gridding is used, p eak va lues of I~ are generall y missed. S econd, uncertainties are usually real, in t.his case t h e lower bound of /-I cannot b e computed wit.h s t anda rd numerical tools . All available lower bound comput.ation techniques consis t of finding a perturba tion which corresponds t.o the limit of stability. All techniques are more or les s heuristic. A fixed point algorithm is proposed in (Packard and Doyle , 1993; Young a nd Doy le , 1990). This algorithm is very efficient for mixed uncertainties in which the complex par t is large enough . lTnfort.uua t ely, when uncertainties are mode led as real para meter variatio ns this 311 This wor k was supported in p art by the Fren c h ~Ministry of Defense and for the second a uthor, by the HSP HI grant program of the German Ac ad emic EXChange Service. INe would like to t hallk Dr. G . F e rre res a nd Dr. J .Iv1. Biao u;c for h el pful discussions.
gorithm does not conve rge well enough. The npper boun ds are defined as conve x optimizatio n problems after frequency gridding, see for example (Fan et al., 1991; B eck and Doyle, 1992). These techniques are r eliable when t he tt- curve does n ot present sharp peaks. In orde r to treat the problem of sharp p eaks , frequen cy swe eping is used (see (Sideris, 1992)) , the frequency is considered as an additional real u ncertainty which should b e repeated as many times as the orde r o f the system. Therefore, computationally, this approach rapidly becomes too demanding for r eaJ-word applications. Our m ain contribution is de tailed now. Concerning the lower bound co mputation, nmv algorithms which compute a lower hound of the peaks of the f1, curve are proposed. These algori t hms are efficient for mixed or pure real uncer t ainties. Since frequen cy gridding is not used, the proposed algorithms are very fa~t . This is quit.e useful in a design cycle in which it is necessary to dete.-.t won'. cases, see (Magni et al., 1998). The idea beyond theses techniques consists of shifting the cigcnvalJ.es toward t1 e imaginary axis with a minimum perturba tion. The proposed algorithms are rlivid e d into t wo s teps. The first step is used to reach the limit of stability with a perturbation of minimum Frobenius norm . For that purpose the algorithm proposed in (De M oor, 1993) was t.est ed but th e
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convergence was very erratic. The fixed point algorithm of (Young and Doyle, 1990) was used, the convergence propeI:ties are improved on account of the good initialization of st.ep 1 (see (Magni and Doll, 1998) for more details). But the alternative technique detailed in this paper is more reliable. Concerning the upper bound, t.he technique proposed here takes advantage, over a frequency interval, of sealings t.hat. are computed at a given frequency. For an alternative approach with a different purpose refer to (Ferreres and Biannic, 1998). The algorithm proposed here checks whether a given ('est value, for example the maximum value of the lower bound augmented of a given percentage, is larger than the upper bound of Jl. «(Deck and Doyle, 1992; Fan Et al., 1991; Packard and Doylc, 1993; Young Et al., 1995)) over all freqllencie;;. At each step of t.he proposed algorithm, scalings are computed at a given frequency. The frequency intervals for which these scalings permit us to conclude that the test value is larger than Jl. are eliminated. The frequency intervals that can be eliminated are characterized analytically (see Lemma 4). If all frequencies can be eliminated, the test value is an upper bound of the peak values of p, therefore the lower bound is validated. 2. NOTATION AND PRELIlVIINARIES Let us consider a quadruple in state space form (A, B, C, D) and in t.ransfer matrix form iVIes) = C(sI _A)-1 B+D. The poles of kI(s) wiII be called "open-loop poles". A class of uncenainties ~ (which act as a feedback on the systerIl, (M -~)-forIIl) has t.he following structure:
(1) in which .c.i might be a diagonal matrix of the form ~i = oi1ni with 6 i E 1R (real repeated scalar block) or Oi E ( (complex repeated scalar block) or a matrix in (71., xn, (full complex block). In t.he part relative to the lower bound Ai might also be a matrix in 1R. n,xn, (full real block). Such a matrix ~ will be called an "admissible perturbation". The real and imaginary part of a complex number are denoted R(.) andG(.), the maximum singular value a{.) the cigenvaluc with maximum modulus )."(.). First let us recall the well known result satated in Lemma 1. Form this result, two equivalent definitions will be proposed.
Lemma 1. If Li is such that the interconnection (!vI, Li) remains well-posed and if So does not belong to the spectrum of A: dct(I - M(so)Li) = 0 ~ So E spectrum(A + BLi(I - D.c.)-lC)
(2)
Definition 2.1. The S.S.v. at point At is defined as Ija(~) where.6. is an admissible perturbation such
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that: (1) At belongs t.o the dosed-loop spectrum of (A, B, C, D) with feedback ~, (2) 0'(.6.) is minimum.
Definition 2.2. The s.s.v. at. point At. is. defined as IjO'(Li) where .6.. is an admissible perturbation such that: (I) det(I - lvI(Atl.6.) = 0, (2) O'(Li) is minimum. In view of the above definitions, computing p- at point At is the problem of assigning the pole At with minimum norm of the "feedback". So, finding in one shoot the ma.ximum value of f-t over all point;; on the imaginary axis, is the problem of assigning the value of the imaginary axis for which the required "feedback" is minimum. The following result (see (Magni and Manouan, 1994» will be used for shifting the open-loop poles towards the imaginary axis.
Lemma 2. The first order approximation d)" of the motion of an eigenvalue ,vith multiplicity equal to one of the matrix Aa, induced by a gain variation d~ of Ao is
riJl. = (uB
+ tD)d.6.(Cv + Dw)
(3)
where v is the right eigenvector of Ao, u is the left eigenvector of Aa corresponding to the eigenvalue A and w = .6. o (I - D.6. 0 )-lCv, t = uB~o(I D~o)-l.
3. MINIMUM FROBENIUS NORM CASE Finding a lower bound of the peaks of the pcurve (worst case) consists of applying Algorithm 1 (state-space approach, see §3.1) or Algorit.hm l' (transfer matrix approach, see §3.2) and then Algorithm 2 (see §4) can be used to find the best candidate with minimum "sigma-max norm". In §3.3 is considered a possible pathology of Algorithms 1 and 1'.
3.1 State-space approach Principle of the algorithm. Let ), denote one of the eigenvalues of A + B.6. o (/ - D.6. o )-l C. It is intended to find d.6. (a variation of Ao) that shifts (first order approximation) A to a vertical line which is distant of a small aIIlount denoted ~i' A motion from A to the vertical line defined by Ri is performed as follows. Equation (3) is a linear constraint on d~
iJi«uB
+ tD)d.6.(Cv +
Dw» = Ri
(4)
d~
satisfying (4) will be computed such that the Frobenius norm
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is mmlmum. This is a problem of quadratic optimization under linear constraints. Such a problem will be solved at each iteration of the proposed algorithm. The details on the way the structure of .6. is taken into account in order to solve (4), minimizing J 1 in (5), are given in (Magni and DoH, 1998). In order to avoid initialization problems due to the fact that, when the criterion Jr is minimum for large values of dA, the first order approximation of Lemma 2 is no longer valid, we shall minimize a combination of criteria J 1 and J o , where Jois defined by
(6) At the beginning of t.he algorithm, J o is considered, then a combination of J o and J 1 that becomes equal to J 1 (see (7)). AlgorithIll 1. Step 1 - Initialization. Choose the initial open-loop eigenvalueR that are to he moved t.owards the imaginary axis (this choice is made by considering a controllability measure). Choose also the expected number of steps (say N) that will be used in order to shift. each initial eigenvalue t.o the target. For each initial eigenvalue >., perform the following steps. Set i = -1 and .6.0 = O. Step 2 - Compute u, v, w, t as in Lemma 2. Then solve (4) for dA having the admissible structure, the variation of 1R(>') being given by 1/(i = -R(>'l/(N - i) and t.he following combination of criteria being minimized:
(N - i - l)Jo
+ (i + l)Jd/N
(7)
Step 3 - Set i = min(i + 1,N -1), Aa = Aa +dA. After t:..o is updated, select the new closed loop cigcnvalue that is the closest to .\ + Ri that will becorne the new),.. If ),. is close enough to the imaginary axis, stop, otherwise go to step 2.
COIllInCnts relative to AIgorithlll 1. Measure of controllability. In order to reduce the computing time it. is useful to apply the algorithm only to a subset of the poles of ,Ij,[(s). The llse of the bandwidth knowledge of the system behavior can help. It is also possible to apply a controllability measure. From Lemma 2, Cont.(>..) = IluB + tDIIIICv+Dwll can be consjden~d as such a measure. The number of poles to be considered varies depending on the system, so it must be fixed by trial and errors. (It is also possible to measure the global controllability by analysing the poles "assigned" by random trials of t:...) Details on computation involved at Step 2. Equat.ion (4) is a linear constraint relative to the entries of dA and (7) is a quadratic criterion. An intermediate step which consists of writing the entries of t:..o and d.6. as real vectors ';0 and dE, (in which nncerta,inties are nnt repeated) is necessary, see
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(Magni and Doll, 1998). Equation (4) can be written
Ad(= b
(8)
(A being a row vect.or) and J a and J 1 in (7) become respectively .10 = dl;T Ho cl!; and J 1 = dl;T HI dl:, + 2Cl dt;. The computation of the matrices A, Ho (= H d and Cl is treated in (Magni and Doll, 1998). Finally the problem to be solved at each step is a least square problem. Numbe. of steps, precision. At the first step, i = -1 so J o is optimized. \Vhen i is saturated at N - 1 the criterion is exactly equal to Jl. Usually, N = 20 is enough. After these initial iterat.ions it is worth adding some iterations (about 10) for improving the assignment on the imaginary axis. 3.2 Tmnsfer matrix approach
This sect.ion describes a technique similar to the one presented in §3.1, but it is the transfer matrix that is used instead of the state space representation. In fact, all lower bounds computation techniques are based on iterative techniques without guarantee of global convergence. So, it is very useful to have as many complementary tools as possible. The main difference is that, instead of assigning a pole on the imaginary axis (see Definition 2.1) we want det(I - 111(s)A) to become equal to zero for some value of .'i 011 the iIIlaginary axis (see Definition 2.2). It is well known that the loop in which 1\,-[(8) is in closed loop via A (M - A)-form) is equivalent to a similar loop where <'vI (oS) and t:.. are replaced respectively by AI and .c.. where:
-AI = [A B] CD
As we are interested in poles on the imaginary axis s takes the form l/.iw. We shall denote <5 = l/w moving the complex number .i inside M, we have -I -I to consider i\1 and the interconnectioll A : -I _
M
-
[-jA B] -jC D
. /),.' = ,
[li
I 0 ]
0
t:..
At this step, the following property is clearly sat.isfied: det(I - kI'A') = 0 <=? det(I - M(i).c..) = O. Therefore, the problem to bp treated consists of shifting a pole of the system (I,-Af',I,O) from 1 t.o the origin. But the difference with the algorithm of §3.1 is that it is not the norm of A' + dA' that must be considered, because thi" artificial perturbation contains the frequency (6). The norm to be considered is the norm of the matrix
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The Frobcnius norm of the above matrix is not a positive definite criterion because the weighting relative to t5 is zero. In order to obtain a regular problem, an additional step is required. This step consists of eliminating t5 from the constraints and criteria. Elitnination of b. Applying Lemma 2 to the system (I,-M',I,O) (instead of (A,B,C,D)) for a variat.ion dJ.... we have: -v.j'v!' d,~' v = d>... As for (8) this equation can be written (see (Magni and Doll, 1998)) for do and the entries of d~ in vector form dE.
(9) Let A = [a.,. + ja; Ar + jAi J and dJ... = b,. + jb i , considering the imaginary and real parts a r do + A,. dt;, = by" and a; do + Ai do; b i , after elimination of db, only one equation remains
=
(IQ)
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Principle of the proposed algorithm. Considering the singular value decomposition (s.v.d.) of Llo = U SV' and denoting VI the first column vector of F, the maximum singular value of ~n is a(~o) = ,jVt ~o~o Vi For minimizing the maximum singular value, a new criterion
is defined in which Vl is relative to ~o. As it is expected to consider small va.riat.ions of 6. 0 we shall have J 2 ~ If(b. o + db.). It is this approximation J 2 of the maximum singular value that will be eonsidereu. Algorithm 2. Step 1 - Initialization. Perform Algorithm 1 or l' (or, if relevant, consider root locus as in §3.3). Let ~o denote the resulting admissible perturbation and), the eigenvalue which is approximatively on the imaginary axis. Choose the number of itera.tions IV. Set i = O. Step 2 - Compute n, v, W, t as in Lernma 2. Then, for dLl having the admissible structure, solve thA following LQ problem:
Finally, the frequency (or 0) is eliminated from the considered variable, therefore the criterion relative to the Frobenius norm is again positive definite. AlgorithIll 1'. This algorithm differs somewhat from Algorithm 1. The details can be found in (Magni and Doll, 1998). 3.3 Use of mot locus in some pathological cases For some systems, some open-loop poles might be sensitive to a single real block of ~. In this case there 1S only one degree of freedom for pole motion: "root locus" is the best solution to find the limit of stability.
4. MINIMUM SIG:MA-MAX NORM CASE After having used the algorithms of the previous sections, we have at our disposal a matrix Llo which assigns a pole At on the imaginary axis. But the norm that was minimized for obtaining this result is not the right one. So, a second algorithm is proposed: the assignment of lli(J.d is preserved while the convergence towards a matrix "6." with minimum sigma-max norm is performed. The power algorithm of (Young and Doyle, 1990) was also considered. Its convergence is very good for mixed or complex uncertainties. In the pure real case, convergence is erratic and fails in most cases. It was expected that a good initialization, as for example the results of Algorithm 1 (§3.1)' would improve eonvergence in the pure real case. Briefly, when the power algorithm is initialized by the result of Algorithm 1 it is faster that Algorithm 2 pI'esented below. But Algorithm 2 is luuch more reliable as convergence is almost insured, in addition computing time is still reasonable.
!R((uB
+ tD)d.6.(Cv + Dw)) J = (N - i).h
= -lR('\)
(12)
+ iJ2 )/N
(13)
Step 3 - Set i = min(i + 1, N - 1), .6.. 0 = .6.. 0 + d~. After .6.. 0 is updated, select the new closed loop eigenvalue that is the closest to 'J(.\) that will become the new .\. If the value of h becomes stationary, stop, otherwise go to step 2. COIllIllents relative to Algorithm 2. Details on computation involved at Step 2. Equation (12) is a linear constraint relative to the entries of dA and (13) is a quadratic criterion. An intermediate step which consists of writing the entries of 6. n and db.. as real vectors t;,o and dt;, (in which uncertainties are not repeated) is necessary, sec (Magni and Doll, 1998). Equation (12) can be written as Equation (8) and J 1 and J 2 in (13) become respectively Jl = dE;.T H 1 dt;, + 2Cl dE;, and h = dt;,T H 2 dE;. + 2C2 dE.. See (Magni and Doll, 1998) for details. Finally the problem to be solved at each step is a least square problem. Coalescence of singular values at the optimum. In order to al1eviate the presentation of the algorithm we ignored a very important fact. If the algorit.hm is run a.s above, the optimization will end when the two leading singular values of 6. become equal. It is worth noting that Algorithm 2 tends to tune the Frobenius optimal perturbation by reducing the leading singular values. Coalescence of two or more leading singular values is therefore usual. In order to optimize further after the two leading singular values are equal, a simple improvenlent is required. The details are given in (Magni and Diill, 1998). This improvement reduces to testing that some
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singular values become equal and then to adding some rows to the linear equation that corresponds to Equa tion (12).
Evolution oJ the criterion, number of iterations. When i increases, the weight of J',!, increases. \Vhen i is saturated at 1'>' - 1 the criterion is equal to (ljN)J 1 + (l\' - 1)jN)J2 • It is bette r to transform "smoothly" the criterion from J 1 to Jz, beca.use if (ljN)JJ + «(N - l)jN)Jz were used abruptly, for some systems the variation of ~ would be too large taking into account the first order approximation behind Equation (12). Usually, N = 20 is enough. After t.hese initial iterations it is worth adding some it.e rations in order to treat. singular values coalescence. The numb er of additional iterations depends on the number of singular values which become e qual. Globally, 80 iterat.ions arc enough for most systems . 5. UPPER BOUND
As for all lmver bound of p" the lower bound of the peaks presented in the previous sections has an h e uristic base. There is no guarantee that the proposed algorithms will lead to all the peaks of f1 with much accuracy. All computation of a lowe r bound must be va.lidated by an upper bound. Here we take advantage of the knowledge of the lower bounds of peak values of /.L. Let P,max be the highest lowe r b o und available. In order to validate this result say with 10 % accUl:acy, it suffices to consider a test value PT = 1.1 /-I.max and to check that P is lower than this value for all frequencies. First is considered the problem of finding the interval of freqnencies for which a given pair of scalings Do and Go enables to conclude that p. is less than /.LT (graphic point of view in §5.1, analytic in §5.2). Afterward, this result will b e organized as an algorithm,
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known as a generalized eigenvalue problem. If /3 is fixed a priori, it is also possible to find D and G satisfying (14) by solving an LMI.
Definition 5.1. A pair of sealings D et G being given, the mapping VD,G(W) is defined as being the minimum value of /3 satisfying (14). Note that, regardless of the optimality of the scalings, if VD. G (W) is less than flT therefore /.l(NlCiw)) is less tha.n /.LT. Algorithm 3 proposed in §5.3 aims at eliminating the interva ls of frequencies for which it is known that PT is larger than p,(Al(jw). Plotting VD,a(W) is useful for illustrating the elimina.tion procedure. After analysis of several plots of I'D,G(W), it turns out t.ha,t opt.ima.l scalings, i.e. scaJings that satisfy (14) at s = jwo fo r the minimum value of /3, correspond very often to mappings VD,G(W) which are a lmost vertical lines at Wo. In this case, the intervals that can be eliminated are very narrow. The use of non-optimal scalings permits us to improve con siderably the efficiency of the proposed algorithm by increasing the size of the eliminated intervals. In orde r to compute sub-optimal scalings, it is suggested to consider together three frequencies, say Wo, (1 +E)WO aud (1I')wo instead of the single frequency Wo and to find the scalings corresponding to the minimuIIl value of f3 in the syst.em of three LMls corresponding to (14) at the three frequencies. Figure 1 illustrates the mapping VD,a(W) and the corresponding eliminated frequencies for E = 0,0.1,0.2,0.4. The vertical linp. corresponds to VD.a(W) with optimal scalings (~ = 0), it. is clear that the size of elirninat.ed frequencies is very small.
\
5.1 Po int-wise upper-bound
Two sets of HeaIings having the same structure than .6. (see (1)) are considered. (1): V : D i = Dj > 0 and Di E <[n, xn, for real and complex repeated block,. D·i = dJ,.", and d i > 0 for full complex blocks. (2): Q : G; = Gi and G i E <[n, Xn; for real repeated blocks. G i = 0 for other block!'!. \Ve shall use t.he following well known result (see (Fan rot aI., 1991))
Lemma 3. The Laplace variable s being fixed, let D E V, G E 9 and f3 be such that lVI(s)* DJ.\,f(s)
+
j(GAl(s) - M'(s)G) ~ {32 D (14) then p(M(s»::; fJ
(15)
The proble m of finding the best upper bound !3 of p,(A1 (s») i~ equivale nt to the problem of minimizing fJ under the LMI condition of Equation (14). I t is
Fig.
1. Illustration of e liminated frequencies; VD,G(W) around Wf) 0.4 for E = 0,0.1,0.2,0.4.
=
5.2 Elimination of fr'equ ency intervals
The computation of eliminated intervals can be made graphically as above by considering; plots of VD,G(W). Now this problem is considered from an analytic point of view. Let us consider th e scalings Do and Go that satisfy (14) for s = jwo and
/3
= IJ.r·
Lemma 4. (Magni and DOll, 1998)) L e t (A, B,C, D) be a state space reali zation of M(s). For a given value fLr and twO scalings Go and Do
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such tbat. M(jwO)H DoMUwo) + j(GoM(jwo) Ail! (jwo)G o ) < fl}Do and 1 liE spec((DT DoD + D- 1 j(GoD - DTGo))=:g-) we have ILT
1l,(M(jw) ::::
{J.T
A=
+ 73(f/.~Do
- D)-le
(16)
[CT~oC _~T J ;73= [CTDoD~jeTGoJ C
= [DT DoC + jGoe _ET]
D = (j(GoD - DTG O )
+
w
w,
for wE [w- w+]
where jw- and jw+ are the eigenvalues of the matrix H the closest to jwo (w- < Wo and Wo < w+) where: H = A
Step 3. Consider the first interval that is not already eliminated. • 3.1. Let denote the frequency in the center of the above interval. Then compute the scalings D et G relative to (1 - C3) wand (1 + <'"3) w. If the corresponding value of j3 is larger than ).LT go to 3.3 .. Otherwise go to 3.2. • Compute the newly eliminat.ed interval. If all frequencies are eliminated: end -+ Pr is validated. Ot.herwise go to 3.1. • 3.3. Compute the pair of optimal scalings at WOi· If the corresponding value of (3 is larger than !lr, end -+ f.1.T is not validated. Otherwise, compute the newly eliminated interval. If all frequencies are eliminated : end -+ /-IT is validated, Otherwise go t.o 3.1.
DTDlID)
5.3 Proposed algorithm AlgorithzIl 3. Step 1. Initialization. Compute lower bounds of the peal~ values of fl" for example using Algorithm 1 + Algorithm 2 or Algorithm l' 1- Algorithm 2. So wc have at our disposal q pairs (Wll;, !la;) lmch that 11 (AI CjwOi)) > 110i. Define the test value I-I.T and the interval of frequencies [1,;£ w] to be treated by the algorithm. Initialize also El (for example El = 0.1) and £2, £3 (for example ('2 = £3 = 0.2). Initialize t.he set. of eliminated intervals to [-00 ~l u [w + 00).
Step 2. This step consists of a preliminary elimination of t.he frequencies, .~tarting elimination from the peak 1fall~eS given by (wo" !lOi). The following "sub-steps" are to be performed for i = 1, ... ,q. • 2.1. Consider one of the q pairs, compute the "scaiings" D et G (sub-optimal at WOi) for three frequencies (1- I'd WD;, WOi and (1 + El) Wo;. If the corresponding minimum value of j3 is larger than PT, go to 2.4. Otherwise go to 2.2. • 2.2. Comput.e the newly eliminated iuterval [w- w+} using the techniqlle of §5.2. If all frequencies are eliminated: end -+ IlT is validated. Otherwise go t.o 2.3. • 2.3. If Iw+ - wod > 1"-'0; - "-'-I compute the scalings D and G relative to two frequencies (1 ("2) u.,'- and :....0-. Otherwise if Iw+ - wOil < IWoi w-I consider the two frequencies (1 + 1"2) w+ and w+. In uath cases compute the eliminated intervals using the technique of §5.2. If all frequencies are eliminated : end -+ PT is validated. Otherwise go to 2.1 in order to treat a new pair (w'Oi, 110;)' If all these pairs are already treated, go to Step 3. • 2.4. Comput.e t.he pair of optiTna/ scalings at Wo;. If the corresponding value of f3 is larger tha.n pr, end -+ f.1.T is not validated. Otherwise, Compute the newly eliminated interval. If all frequencies are eliminat.ed: end --'> P,T is validated. Otherwise go to 2.1 in order to treat a new pair (wo;, f.1.0;)' If all these pairs are already treated, go to Step 3.
COIllm.ents relative to Algorithm. 3. Concerning Step 2.3: The intervals that are eliminated at steps 2.1 & 2.2 are not centered around WOi because most of the elimination is done on one side of WOi. This remark justifies step 2.3 which aims at eliminating on the other side of wo,.
jA) l' h1_/ \i
:1
V
V
----::c------;:;~
: •• LI
Fig. 2. Illustration of the proposed algorithm : 5 curves VD,G(W), the 1st & ,1th correspond to step 2.2, the 3rd to step 2.3, the 2nd & 5th to step 3.2.
6. REFERENCES De 1vloor, B. (1993). Structured total least sqllare and 12 approximation problems. Linear Algebra and its Applications 188, 163-205. Magni, J.F. and A. Manouan (1994). In Proc. of the IFAC Sympm.ium on Robust Control, Rio de Janeiro, Brasil pp. 388-393. Magni, J.F. and C. Doll (1998). p-analysis for flexible systems. LAAS Toulouse, France pp. 1-34. Magni, J.F., Y. Le Gorrcc and C. Chiappa (1998). A multimodel-based approach to robust and self-scheduled control design. In Proc. IEEE CDC. Packard, A. and J.C. Doyle (1993). The complex structured singular value. Automatica. Young, P.M. and ,l.C. Doyle (1990). Computation of f.l with real and complex uncertainties. in Proc. IEEE GDG pp. 1230-1235. Young, P.M., M.P. Newlin and J.C. Doyle (1995). Computing bounds for the mixed f.1. problem. International Journal of RolJust and lVonlinear Control 5(6), 573-590. Missing references: see bibliography of Part H.
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Copyright 1999 IF AC
ISBN: 008 0432484