Category A PRACTICAL. APPROACH TO. ROBUSTNESS. ANALYSIS

• When comparing the two^ upper bounds in figures 9.1 and 9.2, the second и upper bound appears more conservative (at least in this ex­ample). Nevertheless, this conservatism remains quite reasonable: note especially that the two upper bounds are nearly identical around the peaks on the и plots, so that the robustness margin obtained with both upper bounds is nearly the same. This justifies to a large extent this second v upper bound, since it is easier to implement than the first v upper bound.

• In the example of Figure 9.2, the model perturbationAi (which is to be maintained inside its unit ball) contained two one-dimensional complex blocks, namely a fictitious performance block and a block of neglected dynamics. The scaling matrix G was consequently chosen as G — 0, while the scaling matrix D i was computed with the Perron eigenvector approach.

To further evaluate the methods of section 2.4 (chapter 8), anew mixed model perturbation Ai is chosen, which contains the fictitious perform­ance block, the neglected dynamics and two parametric uncertainties. Remember there are 4 parametric uncertainties in Ma, Za, Mg and Zg. In the lower subfigure of Figure 9.3, Ma and Za are included in the model perturbation Дi, i. e. the associated parametric uncertainties are 1This problem is very close to the problem of subsection 1.3 (chapter 6), the only difference being the full complex block which corresponds to the high frequency bending mode and which is not taken into account in chapter 6 (subsection 1.3).

to be maintained within ± 5 %. Conversely, in the upper subfigure of Figure 9.3, Ms and Z$ are included in

о ‘ і

Г’ о.

frequency m red » frequency n rad s

Figure 9.3. New problem of robust performance – Ms and Zs are expanded in the upper subhgure, where as Mn and Zn are expanded m the lower subhgure – the dashed dotted plot represents the first v upper bound, where as the dotted plot represents the second v upper bound without scaling matrices D and G (not shown in the upper subngure) – the two plots of the second v upper bound with scaling matrices

Dі and Gі are represented in solid line.

The following quantities are computed:

■ The first mixed и upper bound: this one is used as a reference.

■ The second mixed v upper bound with scaling matrices D = I and Gі = 0: the structure of the model perturbation Z i is not accoun­ted for. This upper bound is not presented in the upper subfigure of Figure 9.3, because of the bad results which were obtained: the assumption Z( Mi ) < 1 was not satisfied on many points of the fre-

quency gridding, so that the value of the upper bound was chosen as +oo on these frequency points.

The second mixed v upper bound with suboptimal scaling matrices D and G\ matrix G was computed with the suboptimal method by (Young et al., 1995), whereas matrix D i was computed using either the Perron eigenvector approach or Osborne’s method. The results provided by both methods are most generally equivalent. When com­paring with the first mixed v upper bound, the second mixed supper bound with suboptimal scaling matrices D and G appears more conservative, but this additional conservatism is quite reasonable.

frequency in rad’s

Irequency in rad/s

Figure 9.2. Robust performance – upper subhgure: // analysis, mixed // upper and lower bounds m solid line, complex /і upper bound in dashed line – lower subhgure: и analysis, first and second mixed v upper bounds in solid and dashed lines, mixed и lower bound in solid line.

Robust performance is analyzed. A fictitious performance block is added to the model perturbation, which thus contains 4 non repeated real scalars and two full complex blocks. The template on the sensitivity function S is the same as in chapter 6 (subsection 1.3).

When applying the fj, tools to this robust performance problem ‘, the maximal s. s.v. is 0.86 at и = 10.8 rad/s (see Figure 9.2 – the result is nearly non conservative). The corresponding uncertainty in the stability derivatives is thus 5/0.86 « 5.8%. However, if the performance block and the bending mode are maintained inside their unit balls, the maximal uncertainty in the stability derivatives becomes 5/0.27 « 18.5% (see the bottom diagram of Figure 9.2). This illustrates that a brute application of d analysis can provide an overly conservative result in the context of a robust performance problem.

The skewed s. s.v. presents two peaks at ш = 2.5 rad/s and u; = 12 rad/s. When comparing the d upper and lower bounds at these frequencies, the result is nearly non conservative for the first peak and reasonably conservative (6 % between the two bounds) for the second one.

Robust stability is first analyzed with the ц tool (see Figure 9.1). The maximal value of the mixed fi upper bound is 0.24 at ui = 228 rad/s, so that the maximal uncertainty in the stability derivatives is 5/0.24 « 20%. However, in the context of (Balas and Packard, 1992), the controller must tolerate a given amount of uncertainty in the bending mode, as defined

by the template. It is thus logical to maintain this uncertainty inside its unit ball in the analysis problem. The maximal value of the first mixed v upper bound is obtained as 0.13 at ш = 0.1 rad/s, so that the maximal uncertainty in the stability derivatives becomes 5/0.13 « 40% (see Figure 9.1).

D 15

frequency m rad’s

frequency in rad/s

Remarks:

(i) In figures 9.1 to 9.3, "mixed fj, upper bound" should be understood as the /л upper bound by (Fan et al., 1991), whereas the ‘mixed // lower bound" should be understood as the ц lower bound by (Young and Doyle, 1990). The "complex /x upper bound" corresponds to the specialization of the mixed ц upper bound to the case of complex uncertainties (see subsection 2.2 of chapter 5 – in the context of the missile example, the real nature of the parametric uncertainties is not accounted for in the

computation of the complex /r upper bound). The "first mixed и up­per bound", the "second mixed у upper bound" and the "mixed ^ lower bound" are defined in chapter 8.

(ii) Neither the mixed p lower bound, nor its skewed version are presen­ted in Figure 9.1, since the power algorithms did not converge. The p problem at low and medium frequencies is indeed too close to a real p problem.

The aim of this chapter is threefold. The first one is to analyze the robust stability and performance properties of the missile autopilot in the presence of aerodynamic uncertainties and high frequency bending modes: see also (Ferreres and Fromion, 1999). The second purpose is to illustrate the usefulness of the skewed ц tool in the context of a realistic application. The applicability of the computational methods, which were presented in chapter 8 (see section 4. for a summary), is finally evaluated trough this example.

1. INTRODUCTION

The reader is first referred to chapters 2 (subsection 2.2) and 4 (section

1. ) for the description of the linearized missile model and the building of the interconnection structure. The local stability and performance properties of the Я,*, autopilot are to be analyzed in the presence of parametric uncertainties (in the 4 stability derivatives Ma> Ms, Za, Zs) and neglected dynamics (a high frequency bending mode). The model perturbation consequently contains 4 non repeated real scalars and a single full complex block. The weights in the stability derivatives are chosen as 5 %.

Two mixed v upper bounds and a mixed slower bound were proposed in this chapter. All these v bounds can be computed in polynomial time.

The exact value of v can be obtained by computing recursively the exact value of /x: in the same way, the first mixed v upper bound can be computed either directly, or recursively using the mixed и upper bound of (Fan et al., 1991). A first solution for computing this mixed supper bound is thus to recursively compute the classical mixed supper bound of (Fan et al., 1991): this one is available in e. g. the ц Analysis and Syn­thesis Toolbox or in the LMI Control Toolbox. A more computationally efficient solution is to solve the quasi-convex LMI problem associated to this и upper bound (using e. g. the LMI Control Toolbox). In this context, when comparing the LMI problems associated to the mixed //and sup­per bounds, it is worth pointing out that the computational complexity is the same.

Concerning the second v upper bound, the idea is to transform the v problem into an augmented fi problem. When splitting the model per­turbation as Д = diag(Ai, Д2), remember as apreliminary that Ді is to be maintained inside its unit ball while the size of Д2 is free. The main advantage of this second n upper bound is that it is easier to implement than the first one. Its computation is indeed done in two steps. The first one consists in computing D and G scaling matrices for the model perturbation Ді: in the special but practically important case of a full complex block Ді, D = 1 and G = 0 are simply chosen. Otherwise, simple suboptimal methods for computing D and G were proposed. The second step consists in a single computation of the ц upper bound of (Fan et al., 1991), applied to an augmented д problem: this second step is straightforward, since the /x upper bound of (Fan et al., 1991) is directly available in standard Matlab softwares.

The main drawback of this second v upper bound is that Д2 can only contain real (possibly repeated) scalars (unlike the first mixed v upper bound, which can be applied to a generic problem of robustness ana­lysis) . Skewed n problems with such a specific structure are nevertheless encountered in practice. Chapter 7 especially illustrated the practical interest of the problem of checking a small gain condition despite para­metric uncertainties (i. e. Дi is a full complex block while Д2 contains the parametric uncertainties). Such a problem can be especially encountered when analyzing the robust performance properties of a closed loop in the presence of parametric uncertainties. As an other physical example, in a linear closed loop containing nonlinearities (e. g. saturations), a sufficient condition for the absence of limit-cycles corresponds to a small gain con­dition, which is to be satisfied despite LTI parametric uncertainties in the linear part of the closed loop.

A power algorithm in the spirit of (Packard et al., 1988; Young and Doyle, 1990) is finally proposed for the computation of a mixed Д lower bound.

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Like the exact value of p, the exact value of v can be obtained as the global maximum of a non convex optimization problem. In the spirit of (Packard et al., 1988; Young and Doyle, 1990), a power algorithm is proposed for solving this problem. The necessary conditions of optimality are rewritten as x = f(x), and a lower bound of и corresponds to a limit of the sequence x^+i = /(a;*). Even if the sequence does not necessarily converge and if the result depends on the initial value xo, this algorithm usually exhibits good convergence properties and a low computational burden.

The optimization problem is first introduced in Proposition 3..1, which is an extension of Proposition 6.2.c of (Fan and Tits, 1992). The power algorithm is then proposed in propositions 3..2 (see appendix B for the proof) and 3..3.

PROPOSITION 3..1

v{M) – maxpr(QM — Р,Рї)

where (A = diag(Aі, Д2) and mi is the dimension of Ai):

pr(A, B) = sup(|A| / det(A — XB) = 0 and A Є R) (8.23)

Q = {A / 6ІЄ [-1,1], SfSI = 1 and A°HAC = 1} (8.24)

pr(A, B) is called the generalized spectral radius of matrices A and B. The rest of the exposition is simplified by considering a block structure with just a real repeated scalar, a complex repeated scalar and a full complex block. The general case of a mixed uncertainty is easily obtained by duplicating the appropriate formulae for each block.

PROPOSITION 3..2 Let A = diag(6[Ikl, 6flk2, A^)vith А? Є Скз’кзА

lower bound P of v(M) must satisfy:

(8.25)

a*2w 2 a3

bi = qai, b2 = j-;—ra2, 63 = ,—r а^гі to3

for some real scalar q Є [—1,1] with:

|.| denotes the Euclidean norm. The complex vectors b, a, z and w are partitioned compatibly with the uncertainty structure as:

with e. g. bi Є Cki.

Remark: the above Proposition is an extension of Theorem 4 of (Young and Doyle, 1990). It is applicable under technical non-degeneracy con­ditions exposed in this reference.

A solution to this system of equations can be found via a power iter­ation method.

PROPOSITION 3..3 Consider the following power iteration:

– w2 ,ka2,k + l |u>3,fc|

zi, k+i = qk+iwi’k, z-2,k+i = і—;—————– ;W2,k, Z3,k+i = і———– гОзл+і

w2ka2,k+i |оз,*.+і|

Pk+lWk+l = ^ ^q1 і ^ M*Zk+

iff 1 > f I > 1 f then qk+f > ff else qk+i = otk+i.

14*1

lal, fc+ll

iff >> 1 I > 1 i then >> 1 > ff else qk+i = аш.

and(3k+i, j3k+i are chosen positive real so that |afc+i| = wk+\ = 1.

If the algorithm converges to some equilibrium point, then тахфф) is a lower bound for u{M).

Remarks:

(/’) An alternative to equation (8.33) is to use a first order filter 7*;+i = xjк + (1 — х)тахфкфк) where x = e~l/N. If necessary, a large value of N can be chosen to recover the convergence properties of the standard power algorithm of (Young and Doyle, 1990).

(ii) The power algorithms of (Packard et al., 1988; Young and Doyle,
1990) usually exhibit good convergence properties and a low computa­tional burden. Nevertheless, when considering complex model perturba­tions, "limit-cycles can occur, and seem to occur more often when there are large repeated scalar blocks" (Packard et al., 1988). In the same way, "significantly poorer convergence properties" are obtained in the case of a real model perturbation (Young and Doyle, 1990). In this last case, note that the extension to skewed ц problems of the method of chapter 5 (section 3.) is essentially straightforward.

The main advantage of the v upper bound above is its ease of im­plementation, so that we look for a simple method of computation of (sub) optimal scaling matrices D and G1, which minimize to some ex­tent a(FfllbiMuD((l – jGi)Ff1/4), with Fi = I + G-

■ In the special but significant case of a full complex block Ді, the scaling matrices are simply D — I and Gi = 0. This is especially the case of a robust performance problem in the face of parametric uncertainties.

■ Concerning the computation of a suboptimal diagonal scaling matrix D, which minimizes to some extent o(DiMib(l), a first method uses the Perron eigenvector approach (Safonov, 1982). An associated routine is available in the Robust Control Toolbox of Matlab. The method is computationally efficient.

■ A second method for computing a suboptimal diagonal scaling mat­rix D is to minimize the Frobenius norm instead of the 2-norm of DiMnDf1: see the classical Osborne’s method and its variations: see especially (Beck and Doyle, 1992), which proposes an efficient im­plementation of the method, and included references. Here again, an algorithm is available in the Robust Control Toolbox of Matlab.

■ Concerning the computation of a not necessarily diagonal scaling mat­rix Du see also the routines in the p Analysis and Synthesis Toolbox.

■ Concerning the computation of a scaling matrix G i, a simple subop­timal method is proposed in (Young et al., 1995). Loosely speaking, the idea is simply to cancel with G the skewed hermitian part of the blocks of DMnDfl, which correspond to the real parametric uncertainties.

See also chapter 10 (subsection 5.2) for an other treatment of the same problem. See finally chapter 9 (section 4.) for a first evaluation of these methods on the missile example.

• As a preliminary, using equation (8.6), it is first easy to prove that:

Ff1/4(D1Fl(M, A2)D];1 – = Ft(H, A2) (8.14)

with:

‘ #n #12 ‘ #21 #22 .

Ff1/4DiM12

M2lD^Ffl/4

M22

Note especially that H22 = M22 • Lemma 2..4, which is inspired by a previous work in (Sideris and Pena, 1990), transforms the original skewed p problem into an augmented p problem (see appendix B for the proof).

LEMMA 2..4 Let:

(8.16)

Assume that ст(#ц) < 1 and let к < 1//ід2(#2г)- Then: а(ЩН, А2)) < 1 УД2 Є kBA2

if and only if:

det(I – AG) ф 0 MA2 Є kBA2

Remarks:

(i) Matrix X is invertible in the above Proposition because of the as­sumption <x(#n) < I-

(ii) Lemmas 2..1, 2..2 and 2..3 do not require the assumption of a real
diagonal model perturbation Д2. This one is however necessary in the above Lemma.

• The mixed и upper bound is presented in the following proposition. Its proof is a straightforward application of Lemmas 2..3 and 2..4.

PROPOSITION 2..5 Let M a complex matrix. Let Dgnd Gsome (D, G) scaling matrices associated to the model perturbation A. Let F = I + G and H the complex matrix of equation (8.15). Let ma(G) the s. s.v. of equations (8.16) and (8.17). Assume that Мд(£) > ^Д2№г) and< Hi < ) < 1 . Then << ( G) is an upper bound of v(M).

Remarks:

(i) The following subsection presents simple methods for computing scal­ing matrices D and Gi.

(ii) For fixed values of scaling matrices D and G, the simplest solu­tion for obtaining a и upper bound with Proposition 2..5 is to compute an upper bound of p^(G), typically the mixed p upper bound of (Fan et al., 1991). Since this one is directly available in the < Analysis and Synthesis Toolbox or in the LMI Control Toolbox, the implementation of a computational algorithm is rather easy.

(Hi) Nevertheless, a technical difficulty is to check the condition p^(G) > Pa? (-/VT22)і when p bounds are computed instead of the exact values. A solution is to check that pLB&(G) > puв, д2(ЛТ22), where рів and рив mean p lower and upper bounds.

Ді is now a mixed model perturbation. The following Lemma is es­sentially a combination of Lemmas 2..1 (chapter 5) and 2..1.

LEMMA 2..3 Let Dand Gsome (D, G) scaling matrices associated to Ai (see equation (5.16)). Let Fi = 7 + Gf. Т/тхд^Мгг) < a and:

max a(Ff1/4(AF,(M, Д2)А-1 ~ jGi)Ffl/4) < 1 (8.12)

дзЄІВД2

then v{M) < a.

Proof of Lemma 2..3: Using Lemma 2..1 of chapter 5, simply note that equation (8.12) implies:

(8.13)

An alternative mixed v upper bound is proposed in this section (Fer­reres and Fromion, 1999). The approach, which consists in transforming the v problem into an augmented p problem, basically uses the Main Loop Theorem.

2.1 CHECKING A SMALL GAIN CONDITION DESPITE MODEL UNCERTAINTIES IS A SKEWED n PROBLEM

• First reshape the standard interconnection structure of Figure 8.1.a (with A = diag(A{, A2)j into the structure of Figure 8.1.b and partition M compatibly with the Aj’s as:

Remember finally in Figure 8.1.b that the LFT transfer Fi(M, Д2) between w and z (with Ді = 0) can be computed as:

Fi(M, Д2) = Mn + Ml2A2(I – M22A2)~lM2l (8.6)

Ді is to be maintained inside its unit ball, while the size of Д2Іs free.

Д2 is moreover supposed to contain only real (repeated) scalars.

• Lemma 2..1 is essentially a skewed version of the Main Loop Theorem (see subsection 4.4 of chapter 1).

LEMMA 2.

Proof of Lemma 2..1: the Lemma is essentially a scaled version of the Main Loop Theorem:

with:

kmax — ШШ шє[0,оо]

The rest of this subsection considers the case of a full complex block Ді: the general case of a mixed model perturbation Ді will be considered in the following subsection. The following Lemma is essentially a restate­ment of Lemma 2..1, since цаі = о in the case of a full complex block

Ді-

LEMMA 2..2 If v(M) < a, then ^а2(^22) < and:

max a(Fi{M, Д2)) < 1 (8-11)

A2£^BA 2

Checking a small gain condition despite model uncertainties Д 2 is thus a skewed ц problem. It suffices indeed to compute an upper bound of v(M) so as to compute an upper bound of й(М):

v(M) = 1 /тіп(к / ЗД2 Є кВД2 with a(Fi(M, A2)) > 1)

іщу represents the maximal size of the model uncertainty Д2, for which the small gain condition Д Д ( Д, Д 2 ) ) < 1 is satisfied.