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Satisfaction of gain and phase margin constraints usingproportional controllersNevra Bayhan a & Mehmet Turan Söylemez ba Department of Electrical and Electronics Engineering , Istanbul University , Avcilar,Istanbul 34850, Turkeyb Department of Control Engineering , Istanbul Technical University , 34469 Maslak,Istanbul, TurkeyPublished online: 08 Jun 2010.

To cite this article: Nevra Bayhan & Mehmet Turan Söylemez (2010) Satisfaction of gain and phase margin constraints usingproportional controllers, International Journal of Systems Science, 41:7, 853-864, DOI: 10.1080/00207720903470114

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International Journal of Systems ScienceVol. 41, No. 7, July 2010, 853–864

Satisfaction of gain and phase margin constraints using proportional controllers

Nevra Bayhana and Mehmet Turan Soylemezb*

aDepartment of Electrical and Electronics Engineering, Istanbul University, Avcilar, Istanbul 34850, Turkey; bDepartment ofControl Engineering, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey

(Received 11 December 2007; final version received 29 October 2009)

Gain and phase margins (GM and PM) are frequently used as robustness indicators for linear time invariantsystems. In many cases, the design problem is reduced to determination of a loop gain to satisfy several designcriteria including gain and PM specifications. Therefore finding the set of fixed order compensators that satisfygiven gain and PM specifications attracted the attention of a considerable part of the scientific communityrecently. It is possible to show that it may not be possible to satisfy the required gain and/or PM specificationsusing only a proportional controller. As a result, finding the limits of the proportional controllers for a givenplant (i.e. finding the maximum achievable gain and PMs, MAGM and MAPM) is also important. Afterproviding alternative methods (to those that already exist in the literature) for determining all stabilisingproportional controllers that satisfy gain and PM constraints, a new method for calculating MAGM and MAPMusing proportional controllers is given in this article. A formulation to calculate maximum gain that results inMAGM is also provided.

Keywords: stabilisation; gain margin; phase margin; maximum achievable gain and phase margins;proportional control

1. Introduction

In this article, linear time invariant systems that canbe represented by a continuous-time rational transferfunction are considered. A possible approach to designcontrollers for such systems is to find an optimalcontroller (such as an H1 or H2 optimal controller) fora given specification (Sinha 2007). Such an approach,however, usually results in a compensator that is nearthe border of region of allowable compensators for thegiven specification. This property of optimal control-lers forbids playing with controller parameters tosatisfy further design criteria that may appear inimplementation stage (Kim, Keel, and Bhattacharyya2007). Actually, it has been shown that it is evenpossible to obtain fragile controllers at the end of acareless optimal controller design (Keel andBhattacharyya 1997; Makila 1998; Ho, Datta, andBhattacharyya 2001).

An alternative approach in design is to find the set ofall allowable controller parameters for a fixed structureand order of the controller. To this extent, extensiveresearch has been pursued to determine all stabilisinglow-order compensators (Ho, Datta, andBhattacharyya 1997a, 1997b; Keel and Bhattacharyya1999; Munro 1999; Munro, Soylemez, and Baki 1999;Munro and Soylemez 2000; Soylemez,Munro, and Baki2003; Ackermann and Kaesbauer 2001; Tan 2003;

Bajcinca 2006; Tan, Kaya, Yeroglu, and Atherton

2006; Tan and Atherton 2006). When considered

categorically, it is possible to see that methods for

finding stabilising low-order compensators can be

considered in three main categories: methods based on

Nyquist theorem (Keel andBhattacharyya 1999;Munro

1999; Munro et al. 1999; Munro and Soylemez 2000;

Soylemez et al. 2003), methods based on a generalised

version of the Hermite–Biehler theorem (Ho et al.

1997a, 1997b; Tan 2003) and methods based on param-

eter space and the concept of singular frequencies

(Ackermann and Kaesbauer 2001; Bajcinca 2006; Tan

and Atherton 2006; Tan, Kaya, Yeroglu, and Atherton

2006). Many of these methods have already been

extended to cover the digital control case (Ackermann,

Kaesbauer, and Bajcinca 2002; Ho, Silva, Datta, and

Bhattacharyya 2004; Bayhan and Soylemez 2006; Kiani

and Bozorg 2006) and/or imposition of further design

criteria, such as time domain specifications (Ho, Datta,

and Bhattacharyya 1999), H1 specifications (Blanchini,

Lepeschy, Miani, and Viara 2004; Ho et al. 2004), gain

margin (GM) and phase margin (PM) specifications

(Tantaris, Keel and Bhattacharyya 2003; Ho et al. 2004;

Bayhan and Soylemez 2007). Computer aided control

system design toolboxes have already started to appear

in this context (Kim et al. 2007, Mitra and

Bhattacharyya 2007).

*Corresponding author. Email: soylemezm@itu.edu.tr

ISSN 0020–7721 print/ISSN 1464–5319 online

� 2010 Taylor & Francis

DOI: 10.1080/00207720903470114

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A GM, which can be defined as the maximum gainuncertainty that can be tolerated in the open-loopsystem without losing stability in the closed-loopsystem, and PM, which can be defined similarly forphase uncertainty are measures of relative stability,and are frequently used in the analysis and design ofthe control systems in the context of robust stabilisa-tion. Therefore, finding all low-order controllers thatachieve specified gain or PMs is very useful.

A graphical approach for finding regions ofspecified gain and PMs in the parameter space of asingle-input single-output control system with adjust-able parameters is presented by Shenton and Shafiei(1994). Alternative methods for computation ofstabilising PID controllers that achieve specified gainand PMs are also given in Tantaris et al. (2003), Hoet al. (2004) and Tan et al. (2006). Although thetechniques presented in these references are usefulwhen controllers with two or three free parameters areused, they do not present an efficient algorithm if onlya proportional controller is to be used, since theseapproaches require gridding of the third parameter andsweeping over the frequency to obtain stabilityboundaries. However, sweeping over frequency isunacceptable in calculation of stabilising gains, sincestability boundaries occur at so called singularfrequencies.

Actually, many practical controllers are of propor-tional type. Moreover, the design problem can beconverted to finding a suitable gain in many cases (seee.g. Soylemez and Ustoglu (2006)). The ideas presentedin Ho et al. (2004) can be used to find proportionalcontrollers that satisfy gain and PM specificationsusing a generalised version of the Hermite–Biehlertheorem to polynomials with complex coefficients.However, such an approach has an exponentiallyincreasing computational complexity with respect tothe system order as indicated in the literature(Soylemez, Munro, and Baki 2003). An alternativeapproach that is based on Nyquist stability criterion ispresented in Bayhan and Soylemez (2007). This articlepursues the approach given in Bayhan and Soylemez(2007) to provide all stabilising proportional control-lers that satisfy certain gain and PMs. It has beenparticularly shown here that the whole set of gains thatsatisfy given gain and PM specifications can be foundrather easily by finding the roots of a polynomial withreal coefficients.

It may not be possible to satisfy the required gainand/or PM specifications using only a proportionalcontroller. An important question that needs to beanswered at this point is the following: what are themaximum achievable gain (MAGM) and phase mar-gins (MAPM) that can be achieved using proportionalcontrollers? The designer can understand the limits of

the current setup of the system by finding an answer to

this question, and may decide to make some changes

to allow achievement of larger gain and PMs in the

closed-loop system. We remark that maximising the

GM and/or PM does not always give the best results.

However, knowing the limits speeds up the design

process considerably. This article extends the results of

Bayhan and Soylemez (2007) to provide a practical

answer to this important question.After presenting the concept of relative stability

and the notions of GM and PM in the next section,

the method of Munro, Soylemez and Baki (1999) for

calculation of stabilising controllers is revised in

Section 3. The extension of this method to systems

with gain or phase uncertainties is given in Section 4.

Section 5 presents methods for calculating MAGM

and MAPM with a proportional controller. Several

numerical examples are given in Section 6, which is

followed by the conclusions in Section 7.

2. Relative stability and notions of GM and PM

In the time domain, relative stability of a linear time

invariant single-input single-output control system is

measured by parameters, such as the maximum over-

shoot and settling time. In the frequency domain, the

proximity of the Nyquist plot of the open-loop system

transfer function GðsÞ to the critical point (�1, j0)

yields an indication of the closed-loop system’s degree

of stability.The size of gain (phase) uncertainty that can be

tolerated without losing stability is referred to as the

gain (phase) margin of a system. It is possible to

determine the GM and PM of a given system using a

frequency independent complex gain (referred to as

gain-phase tester) Kej� (Figure 1). Here, we assume

that the closed-loop system is stable for K ¼ 1 and

� ¼ 0. GM is then defined as

GM ¼D

minGð j!p Þ¼

�1K

K�1

ðK Þ, ð1Þ

where !p 2Rþ is called the phase crossover frequency.

It is assumed that no phase uncertainty exists ð� ¼ 0Þ in

the definition given by (1). A common practice is to

G(s)

–

r yKe−jθ

Figure 1. Gain-phase tester and the closed-loop system.

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give the GM in decibels:

GMDB ¼ 20 logðGMÞ ¼ �20 log Gð j!pÞ�� ��: ð2Þ

Note that GM defined as in (1) considers only gainuncertainties that are larger than unity, and thereforethe resulting GMDB is always a nonnegative numberðGMDB � 0Þ. This can be deceiving in some cases, sinceGM does not provide any information on how muchgain uncertainty can be tolerated for K � 1. In order toconsider gain uncertainties that are less than unityð05K � 1Þ, we define the ‘negative gain margin’ðGM�Þ as follows:

GM� ¼D

maxGð j!�p Þ¼

�1K

05K�1

ðKÞ: ð3Þ

We remark that the resulting GM in decibelsðGM�DB ¼

D20 logðGM�ÞÞ is always nonpositive for

this case. As a result, it is possible to state that theclosed-loop system in Figure 1 is stable forGM�5K5GM for � ¼ 0. It is also possible todefine a ‘symmetric gain margin’ ðGM�Þ to indicate thesize of gain uncertainty that can be tolerated in bothpositive and negative direction in logarithmic scalesuch that

GM�DB ¼Dminð�GM�DB,GMDBÞ ¼ 20 logðGM�Þ ð4Þ

For example, GM�DB ¼ 20 dB means that theclosed-loop system can tolerate gain uncertainties bya factor of 10 (i.e. 0:15K5 10).

It is possible to interpret GM from a Nyquistcriterion perspective as the minimum value of K suchthat the Nyquist plot of KGðsÞ crosses the real axis atthe critical point ð�1, 0j Þ. Another way of interpretingthe GM is as the minimum value of K such that theNyquist plot of GðsÞ crosses the real axis at the criticalpoint ð�1=K, 0j Þ.

PM can be defined similarly as the maximum phaseuncertainty that can be tolerated without getting anunstable closed-loop system, and is formulated as

PM ¼D

minGð j!g Þ¼ej ð180þ�Þ

0���180�

ð�Þ, ð5Þ

where !g 2Rþ is called as the gain crossover frequency.

A possible Nyquist criterion interpretation of PM isthe maximum rotation of the Nyquist plot that can betolerated before the plot crosses the real axis at thecritical point (�1, 0j). Note that PM covers bothpositive and negative phase uncertainties, since theNyquist plot is symmetric around real axis for systemswith real rational transfer functions, and therefore, wedo not need to define a notion of negative PM. It ispossible to state that the closed-loop system in Figure 1is stable for K ¼ 1 and �PM5 �5PM.

The notions of GM and PM and their Nyquist plotinterpretations are used frequently in the rest of thearticle. First, however, let us present a method to findall stabilising gains for a linear time invariant system.

3. Calculation of stabilising gains

A fast approach to compute the entire set of stabilisinglow-order compensators is given by Munro et al.(1999), Munro (1999), Munro and Soylemez (2000),Soylemez et al. (2003). This approach is based on theuse of the Nyquist plot and suggests calculating thenumber of the unstable poles for certain gain intervalsby determining the location and direction of thecrossings of the Nyquist plot of the real axis (withoutactually drawing the plot).

Consider the single-input single-output controlsystem of Figure 2 where

GðsÞ ¼NðsÞ

DðsÞ¼

amsm þ am�1s

m�1 þ � � � þ a1sþ a0sn þ bn�1sn�1 þ � � � þ b1sþ b0

ð6Þ

is the plant to be controlled (with ai, bi 2R) and Kp isconstant gain controller. The problem is to compute allKp controllers that stabilise the closed-loop system ofFigure 2.

Decomposing the numerator and the denominatorpolynomials of (6) into their even and odd parts andsubstituting s ¼ jw gives

Gð jwÞ ¼Nð jwÞ

Dð jwÞ¼

Nre þ jNim

Dre þ jDim, ð7Þ

where Dre ¼DRe Dð jwÞ� �

, Dim ¼DIm Dð jwÞ� �

and Nre

and Nim are defined similarly. By noting that

Dre ¼ Deð�w2Þ, Dim ¼ Doð�w

2Þw, ð8Þ

Nre ¼ Neð�w2Þ, Nim ¼ Noð�w

2Þw: ð9Þ

It is possible to write

Gð jwÞ ¼Ne þ jwNo

De þ jwDo¼

DeNe þDoNow2

D2e þD2

ow2

þ jwDeNo �DoNe

D2e þD2

ow2

� �, ð10Þ

Gð jwÞ ¼Xðw2Þ

Zðw2Þþ jw

Yðw2Þ

Zðw2Þ, ð11Þ

G(s)

–

r yKp

Figure 2. Closed-loop control system with constant gain.

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where

Xðw2Þ ¼DDeNe þDoNow

2,

Yðw2Þ ¼DDeNo �DoNe, Zðw2Þ ¼

DD2

e þD2ow

2 ð12Þ

and where for notation purposes De,Do, Ne and No

are used instead of Deð�w2Þ, Doð�w

2Þ, Neð�w2Þ and

Noð�w2Þ, respectively. By denoting v ¼

Dw2 and the

positive real roots of Y(v) as v�1, v�2, . . . , v�� it is obvious

that the Nyquist plot of Gð jwÞ crosses the real axisonly if w ¼ 0, w ¼ 1, or w ¼ �

ffiffiffiffiffiv�i

pfor i ¼ 1, 2, . . . , �.

Denoting v��þ1 ¼ 0 and v��þ2 ¼ 1, the real axis crossingpoints are found as xi ¼ Xðv�i Þ=Zðv

�i Þ for

i ¼ 1, 2, . . . , � þ 2. Relabelling the pairs ðxi, v�i Þ (for

i ¼ 1, 2, . . . , � þ 2) as ðxi, v�i, jÞ (for i ¼ 1, 2, . . . , q) such

that xi 5 xiþ1 and xi ¼ Xðv�i, jÞ=Zðv�i, jÞ (for all

j ¼ 1, 2, . . . , pi), it is possible to state the followingtheorem.

Theorem 3.1 [Munro (1999), Munro et al. (1999),Munro and Soylemez (2000), Soylemez et al.(2003)]: Consider a linear time-invariant system givenby a proper rational transfer function GðsÞ ¼ NðsÞ=DðsÞgiven as in (6), and assume that D(s) has no roots onthe imaginary axis. Let Xðw2Þ, Yðw2Þ and Zðw2Þ bepolynomials as defined in (12), and the pairs ðxi, v

�i, jÞ

ði ¼ 1, 2, . . . , qÞ be as defined above. Furthermore,denote the first coefficient of YðvÞ as y1, and the lastnonzero coefficient of Y(v) as y0. Then, for a given gaink2 IKi ¼

Dð�1=xi�1, � 1=xiÞ, the number of unstable

poles of the closed-loop system ðuiÞ is given by

ui ¼ u0 þXi�1t¼1

rt, ð13Þ

where u0 is the number of unstable poles of GðsÞ,

ri ¼Xpij¼1

di, j ð14Þ

and the direction of the crossings of the Nyquist plot withthe real axis is calculated as

di, j ¼ð1� ð�1Þl ÞSgnðYðl Þðv�i, jÞÞ if 05 v�i, j51Sgnðy0Þ if v�i, j ¼ 0�Sgnðy1Þ if v�i, j ¼1

8<: ð15Þ

in which Yðl Þðv�i, jÞ is the first nonzero derivative of YðvÞ atthe point v�i, j. Note that the stabilising intervals are thosefor which ui ¼ 0. Note also that if Sgnðxi�1Þ 6¼ SgnðxiÞthe corresponding gain interval is divided into two parts:

IKi ¼ ð�1=xi�1,1Þ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}Kia

[ ð�1, � 1=xiÞ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}Kib

:

It needs to be remarked that it is possible to extendTheorem 3.1 to cover systems with imaginary axispoles.

4. Calculation of all stabilising gains for providing

desired GM and PM

It is possible to use the complex gain-phase tester

introduced in Section 2 (Figure 3) to provide a

modified version of Theorem 3.1 to determine all

possible values of Kp satisfying a gain (or phase)

margin specification. We consider GM and PM cases

separately in the following.

4.1. Calculation of all stabilising constant gains withGM constraints

It is possible to use the following lemmas to determine

all stabilising gains that provide a given GM

specification.

Lemma 4.1: Let the entire set of stabilising gains for

the given system to be IK ¼ K1,K2, . . . ,Ktf g where the

gain intervals IKi ¼DðKimin,KimaxÞ (for i¼ 1, 2, . . . , t) are

found via Theorem 3.1. All stabilising constant gains

that satisfy a given minimum gain margin ðGMÞmin

constraint are given by K ¼ K1,K2, . . . ,Kt

� �where

Ki ¼1 if Kimin 4

Kimax

ðGMÞmin

Kimin,Kimax

ðGMÞmin

Otherwise

8<:

9=;, ð16Þ

where 1 denotes the empty set. When all gain intervals

Ki are empty, there are no stabilising constant gains that

satisfy the desired GM specification.

Proof of the lemma is straightforward from the

discussions above and is not given here. Note that

similar lemmas can be given for the ‘GM�’ and ‘GM�’

defined in Section 2.

Lemma 4.2: All stabilising constant gains that satisfy

a given maximum negative gain margin ðGM�Þmax

constraint are given by �K ¼ f �K1, �K2, . . . , �Ktg where

Ki ¼1 if Kimax 5

Kimin

ðGM�Þmax

Kimin

ðGM�Þmax,Kimax

Otherwise

8<:

9=;: ð17Þ

Lemma 4.3: All stabilising constant gains that satisfy

a given minimum symmetric gain margin ðGM�Þmin

G(s)

–

r yKpKejθ

Figure 3. Proportional control of a system with gain and/orphase uncertainties.

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constraint are given by �K ¼ f �K1, �K2, . . . , �Ktg where

Ki ¼1 if ðGM�Þ2min5

Kimax

Kimin

KiminðGM�Þmin,Kimax

ðGM�Þmin

Otherwise

( ):

ð18Þ

4.2. Calculation of all stabilising constant gains withPM constraints

In order to determine all gain intervals for which the

system has a PM larger than a given value, �, the phaseshifted system

GðsÞ ¼e�j�NðsÞ

DðsÞ¼ðcos � � j sin �ÞNðsÞ

DðsÞ¼D NðsÞ

DðsÞð19Þ

is considered and all stabilising controllers for this

system are found by the help of Theorem 3.1. Note that

the coefficients of the numerator polynomial of (19)

become complex in this case. Since the Nyquist

criterion is general enough to cover systems with

complex coefficients Theorem 3.1 can easily be

extended to cover the system given in (19). To this

extent, substituting s¼ jw into (19), it is possible

to write

Gð jwÞ ¼Nð jwÞ

Dð jwÞ¼ðcos � � j sin �ÞNð jwÞ

Dð jwÞ

¼ðXðw2Þ cos � þ wYðw2Þ sin �Þ

Zðw2Þ

þ jðwYðw2Þ cos � � Xðw2Þ sin �Þ

Zðw2Þ: ð20Þ

Now, defining two polynomials

XðwÞ ¼DXðw2Þ cosð�Þ þ wYðw2Þ sinð�Þ

YðwÞ ¼DwYðw2Þ cosð�Þ � Xðw2Þ sinð�Þ,

ð21Þ

where Xðw2Þ and Yðw2Þ are polynomials as defined in

(12), denoting all (not only positive) real roots of YðwÞ

as w�1, w�2, . . . , w�� , the real axis crossing points are

found as xi ¼ Xðw�i Þ=Zðw�2

i Þ for i ¼ 1, 2, . . . , �. If the

difference between the order of denominator and

numerator polynomials of G(s) is odd (i.e. n�m is

odd, (6)) add w�þ1 ¼ 1 and x�þ1 ¼ 0 pair to this list.

Now, the following theorem can be stated:

Theorem 4.1: Consider a linear time-invariant system

given by a proper rational transfer function G(s)¼N(s)/

D(s) given as in (6), and assume that D(s) has no roots on

the imaginary axis. Let XðwÞ and YðwÞ be polynomials

as defined in (21), and the pairs ðxi, w�i, jÞ ði ¼ 1, 2, . . . ,

� þ 1Þ be as defined above. For a given gaink2 Ki ¼

Dð�1=xi�1, � 1=xiÞ, define the numbers ðuiÞ as

ui ¼Du0 þ

Xi�1t¼1

rt, ð22Þ

where u0 is the number of unstable poles of G(s),

ri ¼Xpij¼1

di ð23Þ

di is defined as

di ¼1�ð�1Þlð Þ

2 SgnðYðl Þðw�i ÞÞ, if 05 w�i 51�Sgnðy1Þ, if w�i ¼ 1

( ), ð24Þ

where Yðl Þðw�i Þ is the first nonzero derivative of YðwÞ atthe point w�i , and y1 denotes the first coefficient of YðwÞ.All stabilising gain intervals that satisfy a given PMspecification PM ¼ �4 0 are those with ui ¼ 0.

The proof of the theorem is straightforward afterobserving the fact that Gð jwÞ ¼ Nð jwÞ=Dð jwÞ ¼

Xðw2Þ=Zðw2Þ þ jwYðw2Þ=Zðw2Þ, Gð jwÞ ¼ XðwÞ=Zðw2Þ þ

jYðwÞ=Zðw2Þ, and real axis crossings do not need to bein pairs since the rotated Nyquist plot is not necessarilysymmetric with respect to the real axis anymore, andalso noting that there do not exist any real axiscrossings at infinite frequency ðw ¼ 1Þ when thedifference between the order of denominator andnumerator polynomials of G(s) is even.

Remark 4.1: Theorem 4.1 requires computation ofroots of real polynomials, and therefore has severaladvantages over existing methods. Another importantadvantage of the method proposed by Theorem 4.1 isthat there is no need for a search on an exponentiallygrowing set of possible solutions as in Ho et al. (2004).

5. Calculation of MAGM and MAPM

In the previous section, a method is proposed tocalculate all stabilising gains (Kp) that satisfy a givenGM or PM constraint. At this point, a naturalquestion that comes into mind is the following: ‘whatis the maximum gain (or phase) margin that can beachieved by a proportional controller’. These valuesare known as MAGM and MAPM. Methods forcalculating these values are given below by the help ofthe development given in the previous section.

5.1. Calculation of MAGM

To calculate MAGM, the lower and upper bounds ofstabilising gain intervals computed via Theorem 3.1 areused. The following theorem utilises the idea that the

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ratio of maximum stabilising gain to minimum

stabilising gain gives us the MAGM over a set of

stabilising controllers:

Lemma 5.1 (Bayhan and Soylemez 2007): For each of

the stabilising gain intervals IKi ¼DðKimin,KimaxÞ for the

system, GðsÞ, if the corresponding GMs are defined as

GMi ¼D max Kiminj j, Kimaxj jð Þ

min Kiminj j, Kimaxj jð Þ, ð25Þ

MAGM is calculated as

MAGM ¼ maxi

GMif g: ð26Þ

We remark that MAGM is achieved at a gain

Kp ¼ Kimin for some i2 1, 2, . . . , t½ . However, GM�

becomes 0 dB, if Kp ¼ Kimin. In parallel with the

discussions in Sections 2 and 4, it is possible to talk

about a minimum achievable negative gain margin

ðMAGM�Þ.

Lemma 5.2: For each of the stabilising gain intervals

IKi ¼DðKimin,KimaxÞ for the system, G(s), if the corre-

sponding negative gain margins are defined as

GM�i ¼D min Kiminj j, Kimaxj jð Þ

max Kiminj j, Kimaxj jð Þ, ð27Þ

MAGM� is calculated as

MAGM� ¼ mini

GM�

i

n o¼

1

MAGM: ð28Þ

Similar to the MAGM case, MAGM� is achieved

at a gain Kp ¼ Kimax for some i2 1, 2, . . . , t½ , and for

this gain the GM becomes 0 dB. In many practical

cases, the controller gain Kp is required to be selected

such that the closed-loop system has a satisfactory GM

and GM�, knowing the maximum achievable symmet-

ric gain margin ðMAGM�Þ can be useful. The follow-

ing lemma provides a formulation for this purpose.

Lemma 5.3: For each of the stabilising gain intervals

IKi ¼DðKimin,KimaxÞ for the system, G(s), if the corre-

sponding symmetric gain margins are defined as

GM�i¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimax Kiminj j, Kimaxj jð Þ

min Kiminj j, Kimaxj jð Þ

s, ð29Þ

the MAGM� is calculated as

MAGM� ¼ maxi

GM�i

n o¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMAGMp

: ð30Þ

Note that this GM is achieved if the proportional

controller is set to Kp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKiminKimax

p.

Remark 5.1: From Lemma 5.1–5.3, we conclude that

for an open-loop stable system or a high gain

closed-loop stable system (stable for Kp!1),MAGM and MAGM� are 1, and MAGM� is 0.

5.2. Calculation of MAPM

Three different cases are examined in the following,namely: open-loop stable systems (stable for Kp! 0),high gain closed-loop stable systems (stable forKp!1), and open-loop and high gain closed-loopunstable systems.

5.2.1. MAPM for open-loop stable systems

In this case, MAPM is 180� as stated by the followinglemma.

Lemma 5.4 (Bayhan and Soylemez 2007): MAPM is180� for an open-loop stable system G(s) with a constantgain controller.

Proof of Lemma 5.4: The result can be seen directlyby observing the fact that it is always possible to finda small enough gain Kp in this case such that theclosed-loop system is stable and Nyquist plot ofKpGð jwÞ is entirely inside the unit circle.

Obviously MAPM is obtained for small gains.However, using small gains may not be practical insome applications, and therefore knowing the largestgain that provides MAPM is important. The followinglemma helps us in this direction:

Lemma 5.5: For an open-loop stable system, thelargest constant gain Kp ¼ Kp180max that provides a180� PM can be found as

Kp180max ¼ mini

1

Gð jwiÞ�� �� , ð31Þ

where the frequencies wi are the nonnegative real rootsof the following polynomial equation:

PðwÞ ¼DZðw2Þ Xðw2Þ

@Xðw2Þ

@wþ YwðwÞ

@YwðwÞ

@w

� �

� X2ðw2Þ þ Y2wðwÞ

� @Zðw2Þ

@w¼ 0 ð32Þ

in which

YwðwÞ ¼DwYðw2Þ ð33Þ

and X(w2), Y(w2) and Z(w2) are as defined in (12).

Proof of Lemma 5.5: As Kp! 0, the Nyquist plot ofKpGð jwÞ moves toward inside the unit circle andbecomes tangent to the unit circle at a critical valueof Kp (Figure 4). This critical value of Kp is calledKp 180max.

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From Figure 4, it is obvious that this critical valuecorresponds to the frequency with the largest magni-tude of G (jw). The following equation can be solvedto find this critical frequency.

@

@wGð jwÞ�� �� ¼ 0: ð34Þ

Let us denote the nonnegative real solutions of (34)as wi. In order to ensure that the whole Nyquist plot ofKpGð jwÞ is covered by the unit circle, we consider onlythe frequency wi that corresponds to the largestmagnitude of Gð jwÞ. Since Kp ¼

1Gð jwÞj j

for this criticalvalue, Equation (31) can be written. Furthermore,using Gð jwÞ ¼ ðXðw2Þ þ jwYðw2ÞÞ=Zðw2Þ ¼ ðXðw2Þ þ

jYwðwÞÞZðw2Þ in Equation (34), we obtain

@

@wGð jwÞ�� �� ¼ PðwÞ

Z2ðw2ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2ðw2Þ þ Y2

wðwÞp ð35Þ

where P(w) is as defined in (32). It is then easy to showthat (34) is equivalent to (32).

5.2.2. MAPM for open-loop unstable and high-gainclosed-loop stable systems

This is the case where at least one pole of G(s) is onRHP whereas all finite or infinite zeros of G(s) are onLHP. The following lemma can be used for this case:

Lemma 5.6: For an unstable system G(s) which isknown to be closed-loop stable as Kp!1, MAPM iscalculated as follows:

(a) If n�m ¼ 0, MAPM is 180�.(b) If n�m ¼ ð2kþ 1Þ ð for k ¼ 0, 1, 2, . . .Þ,

MAPM is 90�.(c) If n�m ¼ 2 k ð for k ¼ 1, 2, . . .Þ, MAPM is

computed by Lemma 5.8 given in the nextsubsection.

Proof of Lemma 5.6: The proof is straightforward

after considering the fact that the Nyquist plot of

the open-loop system intersects the unit circle at the

imaginary axis as Kp!1, if G(s) is strictly proper

and n�m is odd, and does not cross the unit circle

for high gains, if G(s) is biproper ði:e: n ¼ mÞ.If G(s) is biproper, a further question that can be

asked is what is the minimum proportional gain Kp

that results in a PM of 180�. The following lemma is

given to answer this question.

Lemma 5.7: For an unstable biproper system G(s)

which is known to be closed-loop stable as Kp !1, the

minimum gain Kp ¼ Kp180min that provides a 180� PM

can be found as

Kp180min ¼ maxi

1

Gð jwiÞ�� �� , ð36Þ

where the frequencies wi are the nonnegative real roots

of the following polynomial Equation (32).

The proof of the lemma is similar to that of Lemma

5.6 with the difference that we seek for the gain value

for which the Nyquist plot of KpGð jwÞ is completely

outside the unit circle except for one point (at which

it is touching the unit circle).

5.2.3. MAPM for open-loop unstable and high-gainclosed-loop unstable systems

For an open-loop unstable system with at least one

(finite or infinite) RHP zero, MAPM can be found

using the following lemma:

Lemma 5.8: Define the frequency intervals

Wi ¼D

wi�1,wif g corresponding to stabilising gain inter-

vals IKi ¼ ð�1=xi�1, � 1=xiÞ, then find the real frequen-

cies w�k 2Wi that satisfy the equation

Xðw2Þ@YwðwÞ

@w� YwðwÞ

@Xðw2Þ

@w¼ 0 ð37Þ

where Xðw2Þ and YwðwÞ are as defined in (12) and (33).

Assume that there are r4 0 such frequencies. For each

of the frequencies found, calculate the corresponding

phase angle �k ¼DffGð jw�kÞ ðk ¼ 1, 2, . . . , rÞ. The maxi-

mum PM that can be achieved using proportional control

is given as

MAPM ¼ max1�k�rð�kÞ: ð38Þ

Proof of Lemma 5.8: MAPM can be found by

determining circles of radius Gð jw�Þ�� �� that are tangent

to the Nyquist curve at a frequency w� (Figure 4).It is possible to write that

tanff@Gð jwÞ

@w¼ tanð��90�Þ ¼ tan�¼ tanffGð jwÞ: ð39Þ

Figure 4. Maximum PM is achieved at a frequency where theNyquist plot of KpGð jwÞ is tangent to the unit circle. Such again ðKpÞ can always be found when a circle of radius jGð jwÞjis tangent to the Nyquist curve of Gð jwÞ.

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Therefore at the frequency for which MAPM is

found the following equation must be satisfied:

Im @Gð jwÞ@w

n oRe @Gð jwÞ

@w

n o ¼ Im Gð jwÞ� �

Re Gð jwÞ� � : ð40Þ

The first derivative of (11) with respect to w is

given by

@Gð jwÞ

@w¼

X0ðw2ÞZðw2Þ � Z0ðw2ÞXðw2Þ

Z2ðw2Þ

þ jY0wðwÞZðw

2Þ � Z0ðw2ÞYwðwÞ

Z2ðw2Þ

� �ð41Þ

in which X 0ðw2Þ, Y0

wðwÞ and Z 0ðw2Þ are the first

derivatives of Xðw2Þ, wYðw2Þ and Zðw2Þ with respect

to w. Substituting (11) and (41) into (40) and

rearranging results in the following equation:

Zðw2Þ Xðw2ÞY0wðwÞ � YwðwÞX0ðw2Þ

�¼ 0: ð42Þ

Note that since Zðw2Þ4 0, 8w (42) is equivalent to

(37). Considering the fact that only positive real roots

of the above equation that are in one of the stabilising

frequency ranges ðWiÞ are meaningful Lemma 5.8

immediately follows.

6. Numerical examples

Example 6.1: Consider the system given as

GðsÞ ¼ NðsÞ=DðsÞ, where

NðsÞ ¼ s3 � 1:4s2 � 1:6sþ 100, ð43Þ

DðsÞ ¼ s5 þ 15s4 þ 48s3 þ 289:8s2 þ 210s� 300: ð44Þ

It is possible to find the polynomials X, Y and Z

from (10) as

XðvÞ ¼ �v4 þ 67:4v3 þ 961:08v2 � 29736v� 30000,

ð45Þ

YðvÞ ¼ �16:4v3 þ 233v2 þ 5269:68v� 20520, ð46Þ

ZðvÞ ¼ v5 þ 129v4 � 5970v3 þ 54824v2

þ 217980vþ 90000: ð47Þ

When we compute the roots of D (s), we see thatthere is only one unstable root. Hence, u0 is 1. Thepositive real roots of Y(v) are v�1 ¼ 3:48808 andv�2 ¼ 25:043. Adding v�3 ¼ 0 and v�4 ¼ 1, there existfour crossing frequencies. The crossing points ðxiÞcorresponding to these frequencies are given byx1 ¼ �0:333333, x2 ¼ �0:092953 and x3 ¼ 0:0730122.Relabelling the pairs ðxi, v

�i Þ, and noting that yðl Þðv�1Þ ¼

6296:53, yðl Þðv�2Þ ¼ �13916:2, y0 ¼ Yð0Þ ¼ �20520, andy1 ¼ �1, the net crossing counts, and the gainintervals IKi are given in Table 1.

From Table 1, it is possible to observe that thesystem is open-loop and high gain closed-loopunstable. Therefore, MAGM and MAPM are tobe calculated using Lemma 5.1 and Lemma 5.8,respectively. Nyquist plot of the system is shown inFigure 5(a) and (b).

6.1. Computation MAGM

An examination of Table 1 reveals that the closed-loopsystem is stable for gains ði ¼ 2Þ

IK2 ¼ ðK2min,K2maxÞ ¼ ð3, 10:7581Þ: ð48Þ

From Lemma 5.1, the MAGM is calculated as

MAGM ¼max K2minj j, K2maxj jð Þ

min K2minj j, K2maxj jð Þ¼

K2max

K2min¼ 3:586:

ð49Þ

It is also possible to show that MAGM� ¼ 0:2789and MAGM� ¼ 1:893.

Now, let us determine all stabilising gains thatprovide a 10 dB GM ðGMDB ¼ 20 logðGMÞminÞ.

From Lemma 4.1, the set of Hurwitz stabilisinggain compensators for 10 dB GM are obtained asfollows.

K2 ¼ K2min,K2max

ðGMÞmin

� �¼ 3, 3:4020096ð Þ: ð50Þ

6.2. Computation MAPM

From Theorem 3.1, the frequency rangeW2 2 w1,w2f g corresponding to the stabilising gain

Table 1. Calculation of ri, ui, and the stabilising gains for Example 6.1.

i v�i w�i xi ri ui IKi

1 0 0 �0.333333 �1 1 05Kp532 3.48808 1.86764 �0.092953 2 0 35Kp510.75813 1 1 0 1 2 10:75815Kp 514 25.043 5.0043 0.0730122 �2 3 �15Kp 5�13:69635 – – 1 – 1 �13:69635Kp 5 0

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interval is found as 05w5 1:86764. Since the system

is open-loop and high gain closed-loop unstable,

MAPM are to be calculated using Lemma 5.8. The

real frequency satisfying (38) is then found using

Lemma 5.8 as w�i ¼ 0:728805. Substituting this real

frequency w�i into G(jw), we find MAPM ¼ 15:7884�.Hence, it is not possible to achieve a PM larger than

15:7884� using a proportional controller.Now, let us examine whether or not there are gain

intervals stabilising the closed-loop system for a PM

close to but less than calculated MAPM. For example,

let us find all stabilising gains that achieve at least a

15.7� PM. From (21), we have

XðwÞ ¼�28880:7�5552:71w�28626:5w2þ1425:98w3

þ925:22w4þ63:0498w5þ64:8854w6

�4:43784w7�0:96269w8, ð51Þ

YðwÞ ¼ 8118� 19754:4wþ 8046:56w2 þ 5073:07w3

� 260:068w4 þ 224:307w5 � 18:2384w6

� 15:7881w7 þ 0:2706w8: ð52Þ

From Theorem 4.1, the real roots of YðwÞ

are w�1 ¼ 59:2493, w�2 ¼ �5:22795, w�3 ¼ 4:83082,w�4 ¼ �2:52611, w�5 ¼ 0:791948 and w�6 ¼ 0:66813.

Since n�m ¼ 2 is even there does not exist a real axiscrossing at w�i ¼ 1. Thus, we do not need to addw�i ¼ 1 to the list of crossing frequencies. When thecrossing frequencies are substituted into xi ¼Xðw�i Þ=Zðw

�2

i Þ, the crossing points ðxiÞ are found asx1 ¼ �0:00028004, x2 ¼ 0:058313, x3 ¼ 0:086, x4 ¼�0:0717107, x5 ¼ �0:20308 and x6 ¼ �0:226327,respectively. Then, it is possible to obtain Table 2using Theorem 4.1. As it can be observed from Table 2,the stabilising gain interval that provides a 15.7� PM is4:4183865Kp 5 4:924168.

Example 6.2: Consider a system as in Figure 2 with

NðsÞ ¼ s4 þ 5s3 þ 14:75s2 þ 84:5sþ 2, ð53Þ

DðsÞ ¼ s5 þ 25s4 þ 48s3 þ 19:8s2 þ 21sþ 3, ð54Þ

The set of stabilising gain compensators arecalculated using Theorem 3.1 as given in Table 3.

An examination of Table 3 reveals that theclosed-loop system is stable for the set of gains,Kp 2 ð�0:125431, 0:133872Þ. Since the open-loopsystem is stable, it is possible to state thatMAGM ¼ 1 and MAPM ¼ 180� by the help ofRemark 5.1 and Lemma 5.4, respectively.

Using Lemma 5.5, let us compute Kp180max. Thenonnegative real frequencies (wi) satisfying (32) are 0,

–0.3 –0.2 –0.1ReG

–0.1

–0.05

0.05

0.1ImG(a) (b)

–0.004 –0.003 –0.002 0.001 0.001ReG

0.01

–0.005

0.005

0.01

ImG

Figure 5. (a), (b): Nyquist plot of the system in Example 6.1.

Table 2. Calculation of ri, ui and the gains that satisfy PM¼ 15.7� for Example 6.1.

i w�i xi ri ui IKi

1 0.66813 �0.226327 �1 1 05Kp 5 4:4183862 0.791948 �0.20308 1 0 4:4183865Kp 5 4:9241683 �2.52611 �0.0717107 1 1 4:9241685Kp 5 13:944924 59.2493 �0.00028004 1 2 13:944925Kp 5 3570:91845 �5.22795 0.058313 �1 3 3570:9185Kp,Kp 5�17:14886 4.83082 0.086 �1 2 �17:14885Kp 5 � 11:627917 – 1 – 1 �11:627915Kp 5 0

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0.666701, 4.01239 and 10.3581. If these frequencies aresubstituted into Kpi ¼ 1= Gð jwiÞ

�� ��, we have the constantgains given as

Kp1 ¼ 1:5,Kp2 ¼ 0:0157469,

Kp3 ¼ 225:936,Kp4 ¼ 26:7069:ð55Þ

From (31), the largest constant gain that provides180� PM is Kp2. Thus, Kp180max ¼ 0:0157469.

Example 6.3: Consider the system given as inFigure 2 with

NðsÞ ¼ 2s4 þ 2s3 þ 7s2 þ 5sþ 1, ð56Þ

DðsÞ ¼ s4 þ 3s3 þ 5s2 � 6sþ 5: ð57Þ

The set of stabilising gain compensators arecalculated using Theorem 3.1 as given in Table 4. Ascan be seen from this table, the closed-loop system isstable for Kp!1, whereas the open-loop system isunstable. From Remark 5.1, it is possible to state thatMAGM ¼ 1. From Lemma 5.6,MAPM ¼ 180�. Inorder to determine the smallest gain that provides aPM of 180�, the nonnegative real roots of (32) arefound as w1 ¼ 0, w2 ¼ 0:80848 and w3 ¼ 1:77542.Then, from Lemma 5.7, Kp180min ¼ 1= Gð jw3Þ

�� �� ¼10:5394. This means that for any gain larger that10:5394 the system has a PM of 180�.

7. Conclusions

In this article, an earlier result has been extended tocalculate all the gains that achieve given GM and

PM specifications. The main point of departure fromprevious approaches was due to a generalisation of the

Nyquist stability criterion, where it is shown that thenumber of unstable closed-loop system poles for a

given constant gain compensator can be found byexamining the real axis crossings of the Nyquist plot.The proposed method requires solution of a polyno-

mial equation with real coefficients and does not needa search over an exponentially growing set of possible

solutions, and as a result, has computational advan-tages over existing methods. It should be remarked

that it is possible to determine gain intervals thatsimultaneously satisfy given GM and PM specifica-

tions by finding the intersection of intervals for whichGM and PM specifications are met separately. As faras the GM is concerned, it is discussed that the classical

definition of GM is inadequate in some cases, andconsequently the notions of GM� and GM� are

introduced. Formulations to determine gain intervalsthat satisfy given GM� or GM� specifications are also

developed as a result of these discussions.It may not be possible to achieve required GM and

PM using only proportional controllers in some cases.Classical approaches involve a trial and error process

to determine what is achievable for the given system.In this article, new formulations have been proposed

to compute MAGM and MAPM using proportionalcontrol. These formulations involve calculation of

roots of real polynomials, and therefore present astraight way to find the limits for the given system as

far as GM and PM are concerned. Formulations todetermine MAGM�, MAGM�, the maximum gainthat provides a 180�PM for open-loop stable systems,

Table 3. Calculation of ri, ui and the stabilising gains for Example 6.2.

i w�i xi ri ui IKi

1 0.820503 �7.46982 2 0 05Kp 5 0:1338722 1 0 1 2 0:1338725Kp 513 4.19901 0.00711159 �2 3 �15Kp 5 � 140:615534 0 0.666667 1 1 �140:615535Kp 5 � 1:49999935 0.469665 7.97252 �2 2 �1:49999935Kp 5 � 0:1254316 – 1 – 0 �0:1254315Kp 5 0

Table 4. Calculation of ri, ui and the stabilising gains for Example 6.4.

i w�i xi ri ui IKi

1 0.66267 �0.563279 �2 2 05Kp 5 1:77532 1.82808 0.105066 2 0 1:77535Kp and Kp 5 � 9:517823 0 0.2 1 2 �9:517825Kp 5 � 54 1 2 �1 3 �55Kp 5 � 0:56 – 1 – 2 �0:55Kp 5 0

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and the minimum gain that provides a 180�PM foropen-loop unstable and high gain closed-loop stablesystems are also given as byproducts of the discussionsdeveloped.

Future studies should focus on extending theresults found in this article to more complex controllerstructures and time-delay systems.

Notes on contributors

Nevra Bayhan was born in Istanbul,Turkey, in 1977. She received her BScand MSc degrees in Electrical andElectronics Engineering from IstanbulUniversity (IU), Turkey, in 1997 and2001, respectively. She completed herPhD in Control and AutomationEngineering in the Electrical Engineer-ing Department of Istanbul Technical

University (ITU), Turkey, in 2008. Since 1998, she has beenworking at the Electrical and Electronics EngineeringDepartment of Istanbul University. Her research interestsare automatic control systems, control systems design, robustcontrol, time-delay systems, digital control systems,low-order controller design and control of systems withparameter uncertainties.

Mehmet Turan Soylemez received theBSc degree in Control and ComputerEngineering in 1991 from IstanbulTechnical University (ITU), Turkey,and the MSc degree in control engi-neering and information technologyfrom the University of ManchesterInstitute of Science and Technology(UMIST), UK, in 1994. He completed

his PhD in Control Engineering in Control Systems Center,UMIST in 1999. He has been working at the ControlEngineering Department of ITU as an Associate Professor.Dr Soylemez is the author of one book and has publishedover 30 articles in journals and conference proceedings. Hisresearch interests include inverse eigenvalue problems (poleassignment), multivariable systems, robust control, computeralgebra, numerical analysis, genetic algorithms, PID con-trollers, low-order controller design, simulation of powertraction systems and railway signalling.

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