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2011 17: 1964 originally published online 21 December 2010Journal of Vibration and ControlAlfonso Baños, Joaquín Cervera, Patrick Lanusse and Jocelyn Sabatier

Bode optimal loop shaping with CRONE compensators  

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Article

Bode optimal loop shaping withCRONE compensators

Alfonso Banos1, Joaquın Cervera1, Patrick Lanusse2 andJocelyn Sabatier2

Abstract

Ideal Bode characteristics give a classical answer to optimal loop design for linear time invariant feedback control systems

in the frequency domain. This work recovers eight-parameter Bode optimal loop gains, providing a useful and simple

theoretical reference for the best possible loop shaping from a practical point of view. The main result of the paper is to

use CRONE compensators to make a good approximation and in addition a way for the synthesis of the Bode optimal

loop. For that purpose, a special loop structure based on second and third generation CRONE compensators is used. As

a result, simple design relationships will be obtained for tuning the proposed CRONE compensator.

Keywords

Fractional systems, feedback control, linear systems, frequency domain

Received: 16 February 2009; accepted: 28 September 2010

1. Introduction

As is well known, Bode (1945) first solved thefollowing optimal feedback control problem for linearand time invariant control systems: for an operationalbandwidth 0�!�!0 in which it is desired thatWL(j!) W¼M0� 1, and a given gain crossing frequency!c (defined as WL(j!c) W¼ 0 dB), compute the open loopfunction L(j!) that maximizes !0. The solution of theproblem is to decrease WL(j!) W as fast as possible, beingthe only limitation closed loop stability. For the specialcase in which WL(j!) W has the structure

jLð j!Þj ¼M0, 05!5!0,ffLð j!Þ ¼ ���, !4!0,

�ð1Þ

known as the ideal Bode characteristic, the optimumsolution for a given feedback level M0 is obtained bymaximizing the phase lag �� or minimizing the phasemargin given by (1� �)�.

The ideal Bode characteristic corresponds to thetransfer function

LðsÞ ¼M0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þs2

!20

sþ

s

!0

!2�: ð2Þ

In general, the problem is well defined as a functionof the four parameters M0, �, !0, and !c, and theoperational bandwidth !0 is given as a function of theother three parameters. For !�!0, and in particularfor !¼!c, the loop magnitude is given by

log jLð j!cÞj ¼ logM0 � 2� log!c

!0þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!c

!0

� �2

�1

s8<:

9=;ð3Þ

from where finally

!0

!c¼

2M1=2�0

M1=�0 þ 1

: ð4Þ

As a result, maximizing !0 is obtained by maximiz-ing the phase lag of the open loop function over all thefrequency range, which in this case will correspond to aminimum phase margin.

1Dpt. Informatica y Sistemas, Univ. Murcia, Murcia, Spain2Universite de Bordeaux, Laboratoire IMS, Talence, France

Corresponding author:

Joaquın Cervera, Dpt. Informatica y Sistemas, Univ. Murcia, 30071 Murcia,

Spain

Email: jcervera@um.es

Journal of Vibration and Control

17(13) 1964–1974

! The Author(s) 2010

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DOI: 10.1177/1077546310388002

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In practice, simplifications of (4) are usually used toestimate the maximum available feedback M0 for givenvalues of the operational bandwidth !0 and the gaincrossing frequency !c. For example, for a typicalphase margin of 30� a good approximation, provided

that M0� 1 and thus M1�

0 þ 1 �M1�

0, is (see, for exam-ple, Lurie and Enright (2000))

M0,dB :¼ 20 log10 M0 � 10 log2!c

!0

� �þ 1

� �: ð5Þ

This optimum design must be modified to cope withthe sensor noise amplification problem (Horowitz,1993). Also, in practice it is not realistic to considerthat the designer has a good control over WL(j!) W forhigh frequencies; thus the best that it can be done isto decrease WL(j!) W as fast as possible at high frequen-cies. A common approach (Horowitz, 1973, 1993; Lurieand Enright, 2000) is to modify the ideal Bode charac-teristic by choosing a high frequency !2 and shapingthe loop with ffLð j!Þ ¼ �epz

�2 for !>!2 and epz2N.

The connection between low and high frequencies ismade by introducing some gain margin M1 at thephase crossing frequency !1, and connecting themwith a constant open loop gain of value M1 (Figures 1and 2). As a result, this modified ideal characteristic isdefined by seven parameters (not all independent): M0,!0, !c, �, M1, !1, and epz. Since the gain or phase of theopen loop function is defined over different frequency

intervals, it is possible to obtain the whole open loopfunction (gain and phase) for every frequency by meansof Bode integrals, and thus the optimal loop is comple-tely defined.

It can be shown (Horowitz, 1973; Lurie and Enright,2000) that the feedback level M0 or the operationalbandwidth !0 are maximized by maximizing thephase lag of the open loop function over the entirefrequency axis. This well-known fact is based on theBode phase integral (Bode, 1945; Gera and Horowitz,1980; Horowitz, 1973; Lurie and Enright, 2000), andmakes clear the key relationship between the availablefeedback over the operational bandwidth and the loopphase lag. In Horowitz (1993) the following relation-ships between the seven parameters are given:

!2

!1¼

0:5epz�

!2

!0¼

0:5epz�

� �2

M0,dBþM1,dB12� �1

8>><>>: ð6Þ

As has been pointed out by Horowitz (1993),the (seven-parameter) Bode optimal loop providesa useful, simple, and good approximation to the fastestcut-off possible, giving a theoretical reference forthe best possible loop shaping from a practical pointof view.

Note that originally the Bode optimal loop has beendefined for an open loop with no poles at the origin.

Figure 1. Bode plot of the Bode optimal loop frequency response.

Banos et al. 1965

In general, a number of poles at the origin, n, is com-monly used in practice to achieve zero error in thesteady response. As a result, the Bode optimal loop ofFigure 2 has to be modified to exhibit a different low-frequency structure (see Figure 3), and has now eightparameters, including the number of poles at theorigin n.

On the other hand, there exist an important body ofliterature related with the applications of fractional cal-culus in control (Oustaloup, 1991; Tenreiro Machado,1997; Podlubny, 1999). In particular, CRONE (‘com-mande robuste d’ordre non entier’, non-integer orderrobust control) will be used as the basis for the develop-ment of a Bode optimal loop shaping method. Morespecifically, the goal of this work is to derive relation-ships between the eight parameters of the Bode optimalloop and the parameters of a modified CRONE frac-tional system, taking as starting point the work ofOustaloup et al. (2000). Other more recent worksabout fractional control are those of Monje et al.(2004), Chen and Moore (2005) and Monje et al. (2008).

This work is a revised version of the conferencepaper Banos et al. (2008). Related work about interplayof frequency domain control systems design and frac-tional system may be found for example in Barbosaet al. (2004), Cervera and Banos (2008), Chen et al.(2006), and Nataraj and Tharewal (2007). The worksBarbosa et al. (2004) and Chen et al. (2006) approachthe design problem using as basis the four-parameter Bode optimal loop, while Cervera andBanos (2008) and Nataraj and Tharewal (2007) givesdesign methods in the framework of quantitative feed-back theory.

The main contribution of this work will be the defi-nition a new CRONE-based loop that allows the shap-ing of an eight-parameter optimal Bode loop. The rest

of the work is organized as follows. In Section 2,second/third CRONE compensators are introduced,jointly with the proposed CRONE loop. Then, thestructure and properties of CRONE components areused to derive simple and efficient design rules to give(close to) optimal Bode loops. Finally, in Section 3 adesign example is given.

2. Loop shaping with

crone compensators

In this Section, a special loop structure based onCRONE compensators will be postulated to efficientlyshape (close to) Bode optimal loops. The plant is con-sidered to be stable and minimum phase, so thus theloop is synthesized by using only CRONE compensa-tors. For this purpose, loop shaping can be efficientlydone by using a combination of second and third gen-eration CRONE compensators.

From a historical perspective, CRONE has beendeveloped following three generations since 1975.The first generation consists of a order n integro-

differentiator loop that allows a constant phase n�

2around the gain crossing frequency, thus resulting ina robust control against loop gain uncertainty.Second generation CRONE modifies the loop to beband-limited and includes integral and low-pass effects.Finally, third generation CRONE includes animaginary order integro-differentiator that resultin robust control against loop gain and phaseuncertainty.

An important property of second/third generationCRONE compensators is that their tuning is very intui-tive and is based on a relatively small number of

Figure 3. Nichols plot of the Bode optimal loop frequency

response (Type I loop).Figure 2. Bode optimal loop Nichols plot.

1966 Journal of Vibration and Control 17(13)

parameters. Specifically, for the second/third genera-tion compensator, given by (Oustaloup et al., 2000)

Dr ¼ C0

1þs

!l

1þs

!h

0B@

1CA

a

cos �b log C0

1þs

!l

1þs

!h

0B@

1CA

264

375, ð7Þ

the values and slopes of the open loop frequencyresponse at frequency !u are expressed by

jDrð j!uÞj ¼ chð�bð�l ð!uÞ � �hð!uÞÞÞ

argDrð j!uÞ ¼ að�l ð!uÞ � �hð!uÞÞ

d jDrð j!Þj

d log!

� �!¼!u

¼ 20a!2u

!2u þ !

2l

�!2u

!2u þ !

2h

� �dargDrð j!Þ

d log!

� �!¼!u

¼ 2:3b!2u

!2u þ !

2l

�!2u

!2u þ !

2h

� � tanhð�bð�l ð!uÞ � �hð!uÞÞÞ

8>>>>>>>><>>>>>>>>:

ð8Þ

where �l ¼ arctan!u

!l

� �, �h ¼ arctan

!u

!h

� �, and the

constant C0 is given by C0¼!u/!h, being !u¼ (!h!l)1/2.

Second generation terms are obtained simply bymaking b¼ 0.

It is important to note that values of the gains andslopes of the phases only depend on the imaginaryorder of differentiation b, while values of phases andslopes of gains only depend on the real order of differ-entiation a. This fact makes it very appealing to usethese types of compensators for Bode optimal loopshaping where, as has been seen, an optimal designwill have in the medium frequency range a frequencyinterval [!l, !h] with constant phase followed by a fre-quency interval ½!0l,!

0h with constant gain. In these two

intervals, the loop can be ideally shaped with a second-order CRONE term corresponding to b¼ 0, withoperative frequency band in the interval [!l, !h], fol-lowed by a third-order generation term with a¼ 0 (cor-responding to an imaginary order) with operative band½!0l,!

0h.

If, in addition, two additional terms are included toshape the low and high frequencies, the proposedCRONE compensator will be

LðsÞ ¼ k!l

sþ 1

� nlC0

1þs

!l

1þs

!h

0B@

1CA

a

cos �b log C00

1þs

!0l

1þs

!0h

0B@

1CA

264

375 1

s

!hþ 1

� �nh ð9Þ

where corner frequencies for the low- and high-fre-quency terms are equal to !l and !h. In addition, theconstants C0 and C00 are given by C0¼!u/!h, andC00 ¼ !

0u=!0h, being !u¼ (!h!l)

1/2 and !0u ¼ ð!0h!0l Þ

1=2.As a result, the proposed compensator has nine para-meters: k,!l,!h,!

0l,!0h, a, b, nl, nh.

2.1. Frequency-band real differentiator term

In this section, the term corresponding to the frequencyband [!l, !h] will be studied, in order to analyze itspotential for Bode optimal loop shaping. It is a fre-quency-band differentiator with real order a, given bythe transfer function

L2ðsÞ ¼ C0

1þs

!l

1þs

!h

0B@

1CA

a

: ð10Þ

In terms of loop shaping, in principle this term will bein charge of contacting the stability boundary at somefrequency, and giving a vertical straight line that fitsas close as possible to the stability limit given bythe phase margin (1� �)� (see Figure 4). Thefrequency band in which it operates will contain thegain crossing frequency !c. This is a similar role tothe original use in CRONE to obtain loops withconstant stability margins for a given gain uncertaintyrange.

Note that, according to equation 8, the real orderdifferentiator, corresponding to b¼ 0, will have a zerophase slope at !u (that is a vertical tangent line in theNichols plane), and a phase value at !u that onlydepends on the real order a.

As a result, using equation 8 a first design relation is

a�

2� 2�l ð!uÞ

� ¼ ð1� �Þ�: ð11Þ

On the other hand, the length of the vertical line withphase (1� �)� of the Bode optimal loop is given byM0,dBþM1,dB. If the frequency interval of theCRONE2 term [!l, !h] is wide enough, that is!h/!l� 1, then a second design relation is

!h

2!l

� ��a�M0M1: ð12Þ

2.2. Adding low- and high-frequency terms

The structure of the postulated loop in equation 9 atlow and high frequencies will be considered nowin addition to the real order differentiator term.

Banos et al. 1967

The three-term composition is in fact a second-ordergeneration CRONE compensator given as

LCRONE2ðsÞ ¼ k!l

sþ 1

� nlC0

1þs

!l

1þs

!h

0B@

1CA

a

1

s

!hþ 1

� �nh :

ð13Þ

The low-frequency structure will be responsible forthe loop fitting to the corresponding boundaries (typi-cally tracking and/or disturbance rejection boundaries).On the other hand, the high-frequency term will be incharge of decreasing as fast as possible the loop mag-nitude or, in other words, of maximizing the loop phaselag without violating the stability boundary. Obviously,by closed loop stability and in order for the feedbackcompensator to be physically realizable, it is clear that

nl � n,nh � eppzð� nÞ,

ð14Þ

where eppz is the poles/zeros excess of the plant, and n isthe number of plant integrators.

In Figure 5, several Nichols plots of the secondgeneration CRONE are shown. Here nl¼ 1, nh¼ 3,a¼�1.5, and !l¼ 0.1 rad/s have been fixed, the pointof interest being the influence of the frequency-bandwidth. Note that although second generation CRONEis good to contact the stability boundary from low fre-quencies, and to give a frequency band with constantphase (this is its main goal), it is unable to make a goodfitting at the bottom of the stability boundary.

Two more design relationships, affecting low- andhigh-frequency terms, are given by

jLCRONE2ð j!cÞj ¼ 1 ð15Þ

and

jLCRONE2ð j!uÞj ¼M0,dB �M1,dB

2: ð16Þ

2.3. Frequency-band imaginary orderdifferentiator term

The role of the frequency-band imaginary order term inthe proposed loop (9) is to give a good fitting to the

Figure 4. Nichols plot of frequency-band order differentiator (!h/!l¼ 104), for a¼�2, �1.5, �1. Triangle and square marks stands

for !l and !h, respectively.

1968 Journal of Vibration and Control 17(13)

stability boundaries at frequencies around the phasecrossing frequency !1. The term, as given by

L3ðsÞ ¼ cos �b log C00

1þs

!0l

1þs

!0h

0B@

1CA

264

375, ð17Þ

is defined on the basis of three parameters: !0l, !0h, and

b. L3 is a simplified version, for the particular case ofb� 0, chosen without any loss of generality, of the moregeneral expression (Oustaloup et al., 2000)

L3ðsÞ ¼ cos �b log C00

1þs

!0l

1þs

!0h

0B@

1CA

264

375

0B@

1CA

signðbÞ

ð18Þ

in which phase and magnitude are not affected byb sign.

Regarding Bode optimal loop shaping, this termwill be used as a complement to the second-ordergeneration CRONE term in equation 13, specificallyto increase phase lag as fast as possible in afrequency band ½!0l,!

0h containing the frequency !h,

which is the corner frequency of the second

generation CRONE term. However, a detailed analysisof this term is first needed, since its transfer functioncan be of non-minimum phase for some parametercombinations.

2.4. Third generation CRONE compensators canbe non-minimum phase

In general, the third generation CRONE compensatoris stable but can exhibit non-minimum phase behaviorfor particular combinations of its parameters. Forexample, for the transfer function

LðsÞ ¼ cos �0:75 log 0:11þ

s

10

1þs

1000

0B@

1CA

264

375 ð19Þ

it can be shown after some computation that it has tworight-half-plane zeros at s¼ 2.34 and s¼ 4267.43.In general, it can be shown that third generationCRONE compensators, and thus the compensator inequation 9, are stable and minimum phase if and only if

b log!0h!0l

� �5�: ð20Þ

Figure 5. Nichols plot of CRONE2 loop for different values of !h.

Banos et al. 1969

2.5. Maximizing loop phase lag

An important point in order to tune the parameter b isthat it has to be restricted to be in the interval (it will besupposed that b� 0 and !0h 4!0l)

b5 bmax ¼�

logð!0h=!0l Þ: ð21Þ

Note that since the phase lag (given by (8)) is amonotonically increasing function of the imaginaryorder b, a maximum phase lag increment will beobtained at the frequency !0u ¼ ð!

0h!0l Þ

1=2 for b¼ bmax.However, these adjustments should be made withsome care, since the maximum phase lag at that fre-quency is made at the cost of losing phase lag atlower frequencies. In practice, some lower value willhave to be used, balancing between the phase lagwaste at frequencies !5!0u and the phase lag incre-ments at !4!0u.

For example, consider the third generation CRONEterm given by

LðsÞ ¼ cos �b log 0:11þ

s

10

1þs

1000

0B@

1CA

264

375: ð22Þ

Following the above reasoning, the min value of b forthe transfer function to be minimum phase is

bmax ¼�

logð1000=10Þ¼ 0:68. In Figure 6, the phases of

the terms corresponding to different values of the para-meter b are shown. Also, Figure 7 shows the maximumloop phase lag slope obtained at !0u for the valueb¼ bmax.

A simple and effective election of the frequency band½!0l,!

0h, and the imaginary order b can be made as a

complement to the second generation CRONE looppreviously designed. Since the goal is to introducephase lag in the neighborhood of the corner frequency!h, a simple good (but not necessarily the best) electionis to choose the center frequency !0u to be equal to thatcorner frequency, and locate that frequency around thephase crossing frequency !1, and in addition to choosea b close to its maximum value:

!0u ¼ !h � !1,b � bmax,

�ð23Þ

In this way (see Figure 8) the phase characteristic of theimaginary order term (as shown in Figure 6) can beused to make a loop shaping close to the stabilityboundary. Note that it is more convenient to use asprevious second generation term a loop that violates

10–4 10–3 10–2 10–1

Figure 6. Phase of the frequency-band imaginary order differentiator (!0h=!0l ¼ 100).

1970 Journal of Vibration and Control 17(13)

Figure 7. Maximum slope of frequency-band imaginary order differentiator phase as a function of (!h/!l).

Figure 8. Loop shaping with the proposed compensator L(j!), making the center frequency !0u ¼ !h.

Banos et al. 1971

the stability boundary (corresponding in Figure 8 to thevalue !h¼ 100 rad/s), and then to use the imaginaryorder term to introduce phase lead at frequencies!5!0u ¼ !h, and phase lag at frequencies !4!0u,maximizing phase lag as much as possible. As aresult, comparison of two loops that do not violatethe stability boundary, L and LCRONE2 (!h¼ 1000rad/s), gives a significant phase lag waste of LCRONE2

in relation to the proposed loop L.

2.6. Loop shaping method

A summary of the loop shaping method is given asfollows.

1. Define the control problem in terms of a specifica-tions set given by:a. operational bandwidth: M0, !0;b. gain crossing frequency and phase margin: !c, �;c. phase crossing frequency and gain margin: !1,M1;d. plant poles/zeros excess: eppz;e. number of plant integrators: n.

2. Design a second-order generation CRONE term asgiven by equation 13, and with parameters !l, !h, a,k, n, and epz:a. use equations 11 and 12 to determine �¼!u/!l

and a (note that !u ¼ffiffiffiffiffiffiffiffiffiffi!l!hp

);b. determine nh¼ epz and n by observing

equation 14;c. determine k by using equation 16 and the value

of � obtained above (here use the identity

C0 ¼!u

!h¼

1

�);

d. finally, compute !u using equation 15 and then

!l ¼!u

�and !h¼!u�).

3. Design a third-order generation CRONE term asgiven by equation 17, and with parameters !0l, !

0h,

and b.a. determine the value

!0h!0l

and thus bmax by equa-

tion 21 as a function of the phase lead needed;

b. determine b to be close to its maximum value bmax,and the pivoting frequency!u by using equation 23(see Figure 8 and the discussion in Section 2.5).

3. Design example

A design example will now be developed according tothe loop shaping method given in Section 2. The start-ing point (the first step in Section 2.6) will be a set ofdesign specifications: M0,dB¼ 30 dB, !0¼ 0.4 rad/s,!c¼ 4 rad/s, �¼ 0.22 (corresponding to a phasemargin of 40�), M1,dB¼ 30 dB, eppz¼ 3, n¼ 2, !1¼ 40rad/s and !2¼ 80 rad/s.

Following the second step in Section 2.6, the second-order generation CRONE term is designed.

1. First solve equations 11 and 12 for the design speci-fications. In particular, since equation 12 is anapproximate relation some iteration is needed toarrive at a good parameter election. In particular,the values of M0 and M1 are increased to 40 dB toobtain a second-order term with a vertical line widthlarge enough at the phase (1� �)� (see Figure 4).The result is �¼!u/!l¼ 10 and a¼�1.45.

2. The inequalities given in equation 14 are used to setup the loop poles/zeros excess nh¼ epz¼ 3, and thenumber of loop integrators nl¼ 2.

3. The value k¼ 1 is obtained by using �¼ 10 in equa-tion 16.

4. Finally, by using equation 15 the value !u¼ 4 rad/sis obtained. Thus !l¼ 0.4 rad/s, !h¼ 40 rad/s, andC0¼ 0.1.

The result of this design step is the second generationCRONE term LCRONE2,

LCRONE2ðsÞ ¼0:4

sþ 1

� �2

0:11þ

s

0:4

1þs

40

0B@

1CA�1:45

1

s

40þ 1

� 3ð24Þ

with parameters k¼ 1, !l¼ 0.4 rad/s, !h¼ 40 rad/s,a¼�1.45, nl¼ 2, and nh¼ 3 (see Figure 9).

The last design step is the third step in Section 2.6,which is to design the third-order generationCRONE term.

1. First it is observed (Figure 9) that a phase lead upto 20 degrees over the range [1, 50] rad/s is needed.After some iteration, analyzing the phase of thethird-order generation CRONE term (Figure 6) itis determined that !0h=!

0l ¼ 3:5, and then bmax¼ 2.5.

2. Finally, a feasible design is obtained with theneeded phase lead for the pivoting frequency!0u ¼ 86:6 rad/s, and b¼ 1.4. As a result !0l ¼ 50rad/s and !0h ¼ 175 rad/s.

As a result, the loop final design L(s) is given by

LðsÞ ¼0:4

sþ 1

� �2

0:11þ

s

0:4

1þs

40

0B@

1CA�1:45

cos �1:4 log 0:11þ

s

50

1þs

175

0B@

1CA

264

375 1

s

40þ 1

� 3 :ð25Þ

1972 Journal of Vibration and Control 17(13)

Figure 9 shows the loop frequency responses L(j!)and LCRONE2

(j!). Note how the effect of the third gen-eration CRONE term described in Figure 8 can beclearly observed in Figure 9. Finally, in Figure 10 aBode plot of the design CRONE compensator isgiven, showing how the proposed compensator anddesign rules can be used to efficiently synthesize eight-parameter Bode ideal loops.

−350 −300 −250 −200 −150 −100 −50 0−80

−60

−40

−20

0

20

40

60

Arg(L(jω)) (degrees)

|L(jω

)| (

dB)

ω = 0.1 rad/s

ω = 1 rad/s

ω = 10 rad/s

ω = 100 rad/s

Figure 9. L(s) (thick line) and LCRONE2(s) (thin line) Nichols plots.

Figure 10. L(s) (solid) and LCRONE2(s) (dashed) Bode plots.

4. Conclusions

After introducing the ideal Bode characteristics as opti-mal loops in the frequency domain, a new CRONE-based compensator, obtained by decoupling the fre-quency bands corresponding to the real order and theimaginary order terms, has been postulated to efficientlyapproximate this Bode optimal loop. The optimal loop

Banos et al. 1973

has been defined based on a number of parameters, andthen simple design rules have been obtained for tuningthe proposed compensators. These rules can be used fora first manual solution of a rather hard problem.However, fine tuning may require the use of some auto-matic loop shaping procedure.

Funding

This work was supported in part by Ministerio de Ciencia e

Innovacion (Spanish Government) under grant DPI2010-20466-C02-02.

List of symbols

!0 operational bandwidthM0 loop magnitude level in the operational

bandwidth!c gain crossing frequency� (1��)� is the frequency margin in rad

M1 gain margin!1 phase crossing frequencyepz poles/zeros excess!2 frequency of high-frequency asymptoten number of loop integrators

CRONE ‘Commande Robuste d’Ordre Non Entier’(non-integer order robust control)

!l low frequency of the second generationCRONE term

!u central frequency of the second generationCRONE term

!h high frequency of the second generationCRONE term

a real order of differentiation!0l low frequency of the third generation

CRONE term!0u central frequency of the third generation

CRONE term!0h high frequency of the third generation

CRONE termb imaginary order of differentiation

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