Stability Studies of Critical DC Power System Componentfor More Electric Aircraft using µ Sensitivity

M.R. KuhnGerman Aerospace Center (DLR)

Institute of Robotics and MechatronicsMnchner Str. 20, 82234 Wessling, Germany

martin.kuhn@dlr.de

Y. Ji, D. SchrderTechnical University of Munich

Institute of Electrical Drive SystemsArcisstr. 21, 80333 Munich, Germany

yang.ji@mytum.de;dierk.schroeder@tum.de

Abstract— In this paper an approach for stability analysisof power systems for ” More Electric Aircraft ” (MEA) usingµ sensitivity is proposed. The application of the proposedapproach is illustrated via a regulated buck converter as atypical critical component in a power system. Due to thenegative resistance at low frequencies the regulated buckconverter could be unstable in combination with the inputfilter. A typical regulated buck converter is first modelledin the simulation package Dymola. The model equationsare transferred to Maple or Matlab with a Dymola toolboxand symbolically linearized at steady state conditions. Forcomputing µ sensitivity it is necessary to get a LinearFractional Representation (LFR) of the buck converter. Thisis done by using the enhanced LFR Toolbox 2.0. Finally, theµ sensitivity is computed with the Robust Control Toolbox3.1.1 of Mathworks. Compared with other methods forsmall signal stability, e.g. Middlebrook criterion and ModalAnalysis, the µ sensitivity approach gives a much moreglobal and direct result for the influence of all componentson stability.

I. INTRODUCTION

Aiming at global energy optimization of aircraft, theconcept ” More Electric Aircraft ” (MEA) becomes moreinteresting for the aeronautical industry. Electric equip-ment is replacing hydraulic, pneumatic and mechanicalequipment in the domains of propulsion, control andauxiliary systems (see figure 1 [16]). Compared withthe conventional power distribution network, the MoreElectric Aircraft Architecture will demonstrate signif-icant weight gains, reduced maintenance requirementsand increased reliability and also increased passengercomfort. Simultaneously, more electric equipment bringsone key obstruction. Namely, the electric network tendsto instability because the system load can change swiftlyand without warning due to a large range of duty cyclesand unpredictable spikes [13], especially for modernpower optimized architectures using large high voltageDC networks [16]. Therefore, it is necessary to investigatethe stability of the whole electric network both at smallsignal level for steady state conditions and large signallevel for transients, impacts and network reconfiguration.The applied methods must be capable of handling systemsbecoming more and more extensive for future airplanes.One key problem is to find when and how the electricon-board system could become unstable. It is alreadyrecognized that the regulated DC/DC converter, e.g. thebuck converter, is one of the most critical components inon-board electric networks [3]. Switching-mode regula-

tors have a negative input resistance at low frequenciesand may become unstable by the addition of an inputfilter [15]. Due to the inherent character of the DCdistribution system, various methods for the investigationof its stability exist. Till now, the Middlebrook crite-rion, which examines small signal stability of averaged(=non switching) models, is one of the standard methodsbecause of its simplicity. Only the input impedance ofthe load subsystem and the output impedance of thesource subsystem and their Bode diagrams are needed[15]. But the Middlebrook criterion is restrictive and notappropriate for design and optimization goals [3]. Theanalytic calculation of the input/output impedances maybe extensive while there is no information on the sensi-tivity of the impedances to parameter variations availablefor an hardware measurement approach [5] or automatedcalculation with a simulation tool. As a result, electricon-board systems are increasingly analyzed by anothermethod for small signal stability. The Modal Analysisgives proof of stability via the region of the eigenvaluesand uses participation factors and eigenvalue sensitivities([12], [3], [8]) to retrieve dependencies between systemmodes and elements of the state matrix A of a lineartime invariant system. The LTI representation needed iscalculated by numerical linearization of the system atan equilibrium point. It is easy to see that nonlinearimpacts on the steady state condition cannot be regardedwith the Modal Analysis approach. For example, thesystem load has a nonlinear influence and must be treatedas constant at linearization which limits the validity ofthe model to a special loading case. Additionally, theModal Analysis approach is not appropriate for findingthe critical component which is most fragile to destabilizethe system. The drawbacks of Modal Analysis can beavoided with µ-analysis. In this paper a brief reviewon the theory is given in chapter II. In chapter III weproof the applicability of this approach to power networkstability analysis at a regulated buck converter (see figure2). The method is compared to the Middlebrook criterionand the Modal Analysis in chapter IV.

The present work was generated as part of ”VIVACE”project aiming at advanced simulation capabilities. Themethods proposed will also contribute to the EU project”MOET” launched in July 2006 which is explicitly tar-geting network stability.

Proceedings of the 15th Mediterranean Conference onControl & Automation, July 27 - 29, 2007, Athens - Greece

T26-009

Fig. 1. Future aircraft power distribution system block diagram

Modulator PI Controller

E

Rf fL

C f

L

CR Vc

reference Vref

Zi

Zs

Fig. 2. Time discrete buck converter with PWM controller blockdiagram

II. THEORETICAL PRELIMINARY

The Linear Fractional Representation of uncertain sys-tems is often used as the basis for stability analysis andcontroller synthesis in Robust Control. The structured sin-gular value µ, the skewed-structured singular value ν andµ sensitivity are some of the most important definitionsfor robust stability analysis and robust controller design.In this section, these basic concepts are briefly reviewed.

A. Linear Fractional Transformation (LFT)

Suppose P is the transfer matrix of a nominal systemwith P = [M P12; P21 P22]. K and ∆ represent thecontroller and system uncertainties of the parameters,respectively. The lower Linear Fractional TransformationFl(P,K) of P and K is defined as

Fl(P,K) = M + P12K(I − P22K)−1P21 (1)

The lower LFT represents the transfer function fromsignal w to z when closing the lower loop. In general,w can account all uncontrollable signals, e.g. disturbanceand noise and z denotes the signal that allows character-izing whether a controller has certain desired properties.z is called controlled variable e.g. z equals the controlerror [17](see figure 3.a). u and y include the controllerinputs and the system outputs. Similarly, the upper LFTFu(∆, P ) is defined as

Fu(∆, P ) = P22 + P21∆(I −M∆)−1P12 (2)

K

u y

zw12

21 22

M P

P P

(a) lower LFT

w z

u y

12

21 22

M P

P P

(b) upper LFT

Fig. 3. Linear Fractional Transformation framework

(see figure (3.b)). The upper LFT is the transfer functionfrom w to z, which define all uncertainty outputs andinputs in the system when closing the upper loop. Allsystem uncertainties are normalized to ±1 and pulled outinto the ∆ matrix. The row dimension of ∆ is calledthe order of the LFR. The above defined two LinearFractional Transformations play a very important role inRobust Control. The LFR-toolbox [11] for Matlab canbuild the LFR models ’∆ − P ’ (figure 3) automatically.Low order LFT-based uncertainty models can be obtainedwith the new version of the LFR-toolbox for Matlab usingsymbolic preprocessing and numeric reduction techniques[10]. This allows the utilization of the LFT based stabilityanalysis for industrial relevant complex systems [9].

B. Structured and skewed-structured singular value

Let the ∆ matrix in terms of a ’∆−P ’ model in figure3 be diagonal or block-diagonal. It includes all systemuncertainties and its norm remains less than 1. Then thestructured singular value µ∆(M) of P with respect to thestructured uncertainty matrix ∆ is defined as:

µ∆(M) =1

min∆δ(∆) : det(I −M∆) = 0(3)

where σ(∆) is the largest singular value of matrix ∆ i.e.||∆||∞ and µ∆(M) = 0 if there is no ∆ that satisfiesthe determinant condition. The structured singular valuedefines the reciprocal of the minimal H∞ norm of theuncertainty matrix which makes the term I −M∆ sin-gular. The nominal system M has must be stable. Thenthe linear formulation of an uncertain system given by(2) can only be unstable if the term I −M∆ is singular.Therefore,

• µ∆(M) ≤ 1 implies that I −M∆ is non-singularfor all uncertainties which satisfy ||∆||∞ ≤ 1

• µ∆(M) ≥ 1 implies that a system uncertainty existswhich makes I −M∆ singular

Robust Stability: The system described by figure 3.bis robustly stable against all uncertainties in the specifieduncertainty range iff

µ∆(M) < 1 for ||∆||∞ ≤ 1 (4)

A system tends to be unstable as the structured singularvalue increases [18]. The structured singular value isusually determined by calculating its upper and lowerbounds because the exact computing of µ∆(M) is ingeneral NP-hard [4]. The upper bound is used to calculatethe µ sensitivity, which will be introduced in the following

Proceedings of the 15th Mediterranean Conference onControl & Automation, July 27 - 29, 2007, Athens - Greece

T26-009

section. The lower bound of µ is suitable for determi-nation of the worstcase combination of the uncertainparameters. It is calculated by algorithms using localoptimization. Numerous methods for computing boundsof µ are introduced in [2] and [6].

If a subset ∆f of ∆ (in figure 3.b) may not vary freelyand ||∆f || < 1 when computing the µ, while the normof another subset ∆v in matrix ∆ may increase withoutrestriction then it is called the skewed Structured SingularValue ν∆s(M) of matrix M with respect to the uncertaintymatrix ∆s = diag(∆v,∆f ). It is defined as:

ν∆s(M) =1

min∆σ(∆v) : det(I −M∆s) = 0(5)

where ||∆f || ≤ 1. ν∆s(M) is zero if no ∆s exists thatmakes det(I−M∆s) = 0 [6]. The skewed Structured Sin-gular Value gives a solution for determining the maximumallowable range for a subset of uncertain parameters whileother parameters are fixed in a specified range. µ∆(M)is a special case of ν∆s(M).

C. µ sensitivity

In the last subsection the relationship between thestability degree of a system against uncertainties and µwas defined. Also important is the identification of themost critical parameters of a system. This is equivalentto the uncertain parameters which are most sensitive withrespect to µ. The most sensitive parameters are thoseparameters which lead to the greatest increase of µ bya variation of their normalized range. Consider there is a∆ with two parametric uncertainties ∆ = [∆1 0; 0 ∆2]of the nominal system M , δβ represents the perturbationof the first uncertain parameter (see figure 4.a) withβ = 1 + δβ. The uncertainty matrix ∆β perturbed withδβ becomes

∆β =

[∆1 00 ∆2

] [βI 00 I

](6)

It is easy to see that figure 4.b is equivalent to figure4.a. The perturbation in the uncertainty matrix ∆ can beabsorbed into Mβ with figure 4.a, so that the originaluncertain matrix ∆ remains. The new system matrix withabsorbed perturbation β∆1 becomes

Mβ =

[βI 00 I

] [βM11 βM12

M21 M22

](7)

The µ sensitivity with respect to the jth uncertaintyparameter in ∆1 can be defined as:

Senµpj

= limδβj→0

µ∆(M(βj) − µ(M(βj − δβj))

δβj

(8)

with j ∈ 1 . . . n. It is proofed that the skewed µ isequivalent to µ sensitivity [14]. Thus, the µ sensitivitycan be calculated easily in two ways. In the first way,the parametric perturbation is absorbed into the systemmatrix. µ sensitivity is computed using definition (8) bycomputation of the Structured Singular Value µ. The µsensitivity can also be calculated with the help of skewedµ. Both can be done with the Skew µ Toolbox [7].

M

00I

I1

2

00

(a)

M

1

2

00

00I

IM

(b)

Fig. 4. Transformation of parametric perturbation into system matrix

III. APPLICATION FOR BUCK CONVERTER

In this section a buck converter is used to demon-strate the µ sensitivity approach for stability analysis.A DC/DC converter is usually a critical component forsystem stability in an electric on-board network due toits inherent character: its controller exhibits a negativeinput impedance Zi (see figure 2). In combination withan unsuitable input filter, the DC/DC converter begins tooscillate and becomes unstable [15]

A. Average model of buck converter

µ analysis is only suitable for LTI systems. The buckconverter with time discrete parts in figure 2 must befirst transformed into a time continuous model usingthe average modelling technique [13] (see figure 6). Allnominal parameters except for the controller of the buckconverter model in figure 6 are taken from [3]. Theparameters including the PI controller parameter are givenin table I. The differential equations of the buck converterbecome

d

dtiLf =

1

LF

(E −Rf · iLf − Vc)

d

dtVCf =

1

Cf

(iLf − α · iL)

d

dtiL =

1

Lh

(α · VCf − Vc)

d

dtVC =

1

C(iL −

Vc

R)

d

dtα = −

Kp

C(iL −

Vc

R) +Ki(Vref − Vc) (9)

X = (x1, x2, x3, x4, x5) = (iLf , VCf , iL, VC , α)represent the five states of the buck converter systemwhere α is the duty circle. u includes all system inputs.The model was symbolically constructed with the objectoriented modelling and simulation tool Dymola (see figure6) 1 using the open source language Modelica 2. Weused a Dymola/Maple interface to extract the complexstructured model equations into a flat set of symbolic,Maple readable formulas. Although the tool still is underdevelopment at Dynasim and DLR, it is expected toovercome the significant difficulties in developing thesystem equations by hand. For small signal analysiswith µ sensitivity, the system must be linearized aroundequilibria points. The linear state space system

1www.dynasim.se2www.modelica.org

Proceedings of the 15th Mediterranean Conference onControl & Automation, July 27 - 29, 2007, Athens - Greece

T26-009

∆X = A∆X +B∆u

∆Y = C∆X +D∆u (10)

of the buck converter is obtained by symbolic lineariza-tion for all equilibria points

x1,eq

x2,eq

x3,eq

x4,eq

x5,eq

=

350−

q

3502−

4·282

R·Rf

2RfA

282

R· 1

x1,eqV

28

RA

28V28

x2,eq

(11)

The resulting state space matrices in (10) are

A =

−Rf

Lf− 1

Lf0 0 0

1

Cf0 −

x5,eq

Cf0 −

x3.eq

Cf

0x5,eq

L0 1

−L

x2,eq

L

0 0 1

C− 1

RC0

0 0 −Kp

C

Kp

RC−Ki 0

(12)B = [ 1

Lf0; 0 0; 0 0; 0 0; 0 Ki], C = [0 0 0 1 0], D =

[0 0].

B. Generation of suitable LFT-based model

All elements in the state space matrix must be constantor of polynomial type in order to use the LFR Toolbox forgenerating a LFR. As seen in (12), there are some itemswhich are not polynomials because of the root in the first

system equilibrium point x1,eq =350−

q

3502−

4·282

R·Rf

2Rf.

Thus, this expression has to be approximated. Utilizing afirst order Taylor series expansion [1] at the point x1,eq =14.27A, the first equilibrium point becomes

x1,eq =89.304 − 7.29e− 4 · (3502 − 4 · 282 ·

Rf

R)

Rf

(13)The reliability of the approximation is analyzed by plot-ting the error of the approximation (x1,eq − x1,eq)/x1,eq

with respect to R and Rf . Figure 5 shows that a first orderTaylor series expansion is surely valid when 0.086Ω ≤R ≤ 0.24Ω and Rf ≥ 0.14Ω (error of approximation≤ 1%). Otherwise, an approximation by higher orderTaylor series or a rationale function should be used.Inserting the equilibrium point approximation into thesystem matrix of the average value buck converter (12)yields the symbolic linearized model which is compatiblewith the LFR Toolbox 2.0.

IV. RESULTS OF ANALYSIS

A. Analysis with µ sensitivity

Figure 7 shows the variation of µ with regard tothe equal small perturbation of all parameters in thebuck converter in percent. According to the figure, thefour most critical parameters of the buck converter aredetermined. They are Rf , Lf , Cf in the input filter and Lin the load system of the buck converter with decreasingeffect for the system stability. This is a very important

00.2

0.40.6 0.1

0.20.3

0.4−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Rf [Ω]

R [Ω]

norm

aliz

ed e

rror

of a

ppro

xim

atio

n

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

equilibrium x1,eq

Fig. 5. Error of approximation (x1,eq − x1,eq)/x1,eq

SVE

+-

ground

L=100e-6

Lf

R=250e-3

Rf

C=399e-6

C

L=290e-6

L

R=0.1568

R

product

A

Sensor1

SV

+-

SCV

Sensor

V

Sensor2

-

FB

E

k=350

Vref

k=28

C=100e-6

Cf

Ki

b(s)

a(s)

k=0.06

Kpadd

+1

+1

add

++1

+1product1

Vc

Fig. 6. Average model of buck converter with PI controller blockdiagram

result since the input impedance seems to be crucial. In areal application this impedance could vary due to the DCline feeders and other interconnected components. Thus,the variation of those most critical parameters e.g. Rf

and Lf has to be considered with very high accuracyin the design and operation of the buck converter. Therestrictions for the relative unsensitive parameters e.g. Cin the load system could be relaxed. This means that fora design or study or control, the exact identification ora guaranteed range of a parameter is not crucial. Themaximum uncertainty range for each uncertain parameteror for a group of uncertain parameters can be calculatedby the skew µ toolbox [7]. The details for computing themaximum uncertainty range can be found in [14].

Parameter Value Unitvoltage source E 350 Volts

reference output voltage Vref 28 Voltsoutput power P 5000 Watts

output resistance R 0.1568 Ohmsinductance L 290e-6 Henryscapacitance C 399e-6 Farads

resistance of input filter Rf 250e-3 Ohmsinductance of input filter Lf 100e-6 Henryscapacitance of input filter Cf 100e-6 Farads

gain of I controller Ki 5 -gain of P controller Kp 0.05 -

TABLE INOMINAL PARAMETERS OF BUCK CONVERTER

Proceedings of the 15th Mediterranean Conference onControl & Automation, July 27 - 29, 2007, Athens - Greece

T26-009

1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

45

50

1:C 2:Cf 3:K

i 4:K

p 5:L 6:L

f 7:R 8:R

f

µ se

nsiti

vity

[%]

µ sensitivity of parameters in the buck converter

Fig. 7. µ sensitivity analysis

B. Comparison with Middlebrook Criterion

The Middlebrook criterion is intended originally forinvestigating the stability of a power converter systemincluding two subsystems. Let Zs and Zi be the outputimpedance of the source subsystem and input impedanceof the load subsystem, respectively. The system is stableif the amplitude of the output impedance Zs is smallerthan the amplitude of the input impedance Zi within thestudied frequency range. This condition is very restric-tive [15]. For the buck converter in figure 2 the outputimpedance of the input filter is

Zs =Rf + sLf

s2LfCf + sRfCf + 1(14)

The peak Zsmax (in figure 8) of the magnitude of theoutput impedance of the input filter is Lf/CfRf . Thefrequency ωs at which the maximum output impedanceZsmax appears equals 1/

√LfCf . Compared to the re-

sult of the µ sensitivity approach, Middlebrook criterionshows a similar result. Varying an input filter for agiven regulated buck converter, the system could becomeunstable from stable in two ways [15]:

• the peak Zsmax crosses over the input impedance ofthe buck converter Zi due to the variations of Rf

mostly• the frequency ωs moves to the left, so that the peakZsmax crosses over the input impedance Zi (Lf andCf are most responsible)

Middlebrook criterion basically shows a result for thecritical parameters similar to the µ analysis. In the de-termination of the critical parameter combination or theallowed range Middlebrook criterion is quite unhandy.The system engineer has to be well aware of the interpre-tation of the bode plot. The standard way to assess thestability bound is a time consuming parameter variation.The impedance criterion has to be repeatedly evaluated forall components of a network with special considerationsfor parallel circuits. Middlebrook criterion cannot givea quantitative analysis for the critical level of uncertainparameters.

101

102

103

104

105

−60

−40

−20

0

20

40

60

80

100

120Bode Diagram

Frequency [rad/sec]

Mag

nitu

de (d

B)

Zi Buck

Zs Input Filter

Zsmax

ωs

Fig. 8. Bode diagram of output impedance of source Zs and inputimpedance of load Zi

C. Comparison with Modal Analysis

Modal Analysis was used in [3] and [8] for design andstability studies of More Electric Aircraft architectures.For a LTI system with a state matrix An×n, φi and ψi arethe right and left eigenvector and λi is the correspondingeigenvalue. The sensitivity of the eigenvalue λi to theelement akj of the state matrix A is calculated by

∂λi

∂akj

= ΨikΦji (15)

with Φ =[φ1 φ2 · · · φn

]and Ψ =[

ψ1 ψ2 · · · ψn

].

The correlation between the kth state variable and theith eigenvalue can be found by calculating the participa-tion factor pki. It is actually equal to the sensitivity ofthe eigenvalue λi to the diagonal element akj of the statematrix A

pki =∂λi

∂akk

(16)

The matrix P =[

p1

p2

· · · pn

]is called partici-

pation matrix with

pi =

p1i

p2i

...pni

=

Φ1iΨi1

Φ2iΨi2

...ΦniΨin

(17)

[12]. Figure 9 shows all eigenvalues and the most corre-sponding states for the buck converter. With the resultsfrom [3]:

• λ1 is mostly sensitive to Kp, Ki and R• λ2,3 are mostly sensitive to Lf and Cf

• λ4,5 are mostly sensitive to L, C and Kp

The combined information from the participation fac-tors and the eigenvalue sensitivity can be used by asystem designer to detect suitable parameters for poledisplacement e.g. for a higher stability margin or betterdynamic performance. For example, the state variablesiLf and VCf can be significantly influenced by changingthe parameter Lf and Cf because of the great partic-ipation factor of the state variables iLf and VCf to

Proceedings of the 15th Mediterranean Conference onControl & Automation, July 27 - 29, 2007, Athens - Greece

T26-009

−8000 −7000 −6000 −5000 −4000 −3000 −2000 −1000 0−1.5

−1

−0.5

0

0.5

1

1.5x 10

4 Pole Map of Buck Converter

Real Axis

Imag

inar

y A

xis

iLf

, VCf V

c

iL, V

c, α

λ2=−1161+i9705

λ4=−8043+i11433

λ3=−1161−i9705

λ1=−76

λ5=−8043−i11433

Fig. 9. Poles and their corresponding state variables of buck converter

the eigenvalues λ2,3 and the great sensitivity from Lf

and Cf to the eigenvalues λ2,3. Although the ModalAnalysis is very suitable in design, it is not reliableto look for the most critical parameters with respect tosystem stability. It is easy to see in figure 9 that themost critical eigenvalue for stability in the nominal systemis now the first eigenvalue λ1, which is the nearest tothe imaginary axis. However, an equal perturbation ofall parameters will first let the eigenvalues λ4,5 crossover the imaginary axis, although the eigenvalues λ4,5

are the most uncritical ones in the nominal system. TheModal Analysis cannot explain the phenomenon at all.Once again, the standard way for doing global stabilityanalysis is exhaustive iterative parameter variation, e.g.by the Monte Carlo method, linearization at equilibriumand calculation of the eigenvalues. There is no guaranteeto find all critical parameter variations.

V. ACKNOWLEDGMENT

This work was in parts supported by the ”Vivaceproject” (http://www.vivaceproject.com/) under contractnumber 502917.

VI. CONCLUSION

In this paper we presented an overview of µ analysisand its application for small signal analysis and demon-strated a tool chain. The model equations were extractedfrom the graphical simulation environment Dymola andsymbolically linearized for a wide range of equilibriapoints. Nonlinearities had to be approximated by suit-able polynomials. A Linear Fractional Representation ofsmall order was generated by the DLR/ONERA LFR2.0 toolbox for Matlab. From this the Matlab RobustControl toolbox was capable of calculating the StructuredSingular Value µ which is an indicator for the systemrobustness. We also used this tool for the µ sensitivityto detect the system parameters most critical for stability.An alternative possibility would have been to calculatethe Structured Singular Value ν by the Skew mu toolbox(smt)[7].The methodology was demonstrated with a DC/DC buckconverter model which is a small but critical system

component in the electric on-board network of the MoreElectric Aircraft. Two standard methods for stability in-vestigations, Middlebrook criterion and Modal Analysis,were briefly shown and compared with the new approach.The µ analysis was shown to be superior in the charac-terization of the robustness level and identification of thecritical parameters. While the computation of µ is morecomplex than e.g. Modal Analysis, it is more reliable fora broader range of operating points due to the symboliclinearization.Future publications of DLR will include detailed investi-gations of the maximum uncertainty and demonstrate theapplication of the methodology to a representative on-board power network.

REFERENCES

[1] Rainer Ansorge and Hans Joachim Oberle. Mathematik fr Inge-nieure, volume 2. WILEY-VCH Verlag, 2000.

[2] Doyle J.C. Glover K. Packard A. Balas, G.J. and Smith R. µ-analysis and synthesis toolbox. June 1998.

[3] F. Barruel, N. Retire, J.L. Schanen, and A. Caisley. Stabilityapproach for vehicle dc power networks: Application to aircrafton-board system. Power Electronics Specialists, 2005 IEEE 36thConference, Brazil, 2005.

[4] Young P.M. Doyle J.C. Braatz, R.D. and M. Morari. Computationalcomplexity of mu calculation. IEEE Transactions on AutomaticControl, 39: 1000-1002, 1994.

[5] X Feng, J Liu, and F C Lee. Impedance specifications forstable dc distributed power systems. IEEE Transactions on PowerElectronics, 17(2):157–163, 2002.

[6] G. Ferreres. A Practical Approach to Robustness Analysis withAeronautical Applications. Kluwer Academic/Plenum Press, 1999.

[7] G Ferreres and J.M. Biannic. Skew mu toolbox (smt): a presen-tation. Technical Report, ONERA, department of Systems Controland Flight Dynamics, Toulouse, France, August, 2003.

[8] Liqiu Han, Jiabin Wang, and David Howe. Small-signal stabilitystudies of 270v dc power system for more electric aircraft em-ploying switched reluctance generator technology. 25TH INTER-NATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES,2006.

[9] S. Hecker. Generation of low order LFT Representations forRobust Control Applications. Fortschrittberichte VDI, series 8,Nr. 1114, VDI-Verlag (Dsseldorf), 2006.

[10] S. Hecker and A. Varga. Symbolic techniques for low order lft-modelling. Proceedings of IFAC World Congress, Prague, CzechRepublic, July 2005.

[11] S. Hecker, A. Varga, and Jean-Franois Magni. Enhanced lfr-toolbox for matlab. Aerospace Science and Technology, 2005.

[12] Prabha Kundur. Power System Stability and Control. McGraw-Hill, Inc., 1993.

[13] Konstantin P. Louganski. Modelling and analysis of a d.c. powerdistribution system in 21st century airlifters. Masters thesis,Virginia Polytechnic Institute and State University, BlacksburgVirginia, 1999.

[14] Andres Marcos, Declan Bates, and Ian Postlethwaite. Controloriented uncertainty modelling using µ sensitivities and skewedµ analysis tools. Proceedings of the 44th IEEE Conference onDecision and Control, and the European Control Conference 2005Seville, Spain, December 12-15, 2005, 2005.

[15] R. D. MIDDLEBROOK. Input filter consideration in designand application of switchung regulators. Proc. IEEE IndustryApplications Society Annual Meeting, 1976.

[16] C. Schallert, A. Pfeiffer, and J.Bals. Generator power optimi-sation for a more-electric aircraft by use of a virtual iron bird.25TH INTERNATIONAL CONGRESS OF THE AERONAUTICALSCIENCES, 2006.

[17] Carsten Scherer. Theory of Robust Control. Lecture Note, Mechan-ical Engineering Systems and Control Group, Delft University ofTechnology, The Netherlands, April 2001.

[18] Kemin Zhou and John C. Doyle. ESSENTIALS OF ROBUSTCONTROL. Prentice-Hall, Inc, 1998.

Proceedings of the 15th Mediterranean Conference onControl & Automation, July 27 - 29, 2007, Athens - Greece

T26-009