Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59
Yazdan Bavafa-Toosi
No. 2/71, Abuzar 11 St., Ahmadabad Ave., Mashhad, Iran ybavafat@yahoo.com
AbstractThe classical notions of gain margin and phase margin ( Gm and Pm ) which are among the fundamental topics in undergraduate control education have some confusing subtleties especially for students. These subtleties are in part due to the inherent shortcomings of the traditional definitions, and in part due to false guidelines provided by some classical texts. In this work these problems are discussed and settled. In particular, a new definition of Gm is proposed to better capture the underlying philosophy of being a measure of robustness. The false guidelines and claims are also revisited and rectified. The arguments are supported by several instructive NMP examples which are missing in the classical texts.
Keywords- Control Engineering Education; Gain Margin; Phase Margin; MATLAB
I.
INTRODUCTION
The traditional gain and phase margin concepts probably were first introduced around 1921 by the German physicist
Heinrich Barkhausen as margins to oscillations in the context of electric circuit design [1]. Subsequently they were polished and adopted by the control community. Whilst their usage is widespread and experts know how to interpret them, they have turned out to be somewhat problematic for students, even for part of academicians especially junior ones. This is partly because the traditional definitions of these concepts have some shortcomings, and partly because the educational literature does not teach them properly. On the other hand, the implementation of the underlying philosophy of these concepts in the ubiquitously used software MATLAB is done correctly. So when one uses MATLAB, one encounters answers which are not consistent with the lessons learned from some classical texts. This inconsistency of expectations and the computed results is highlighted in the following cases: systems, either minimum phase (MP) or non-minimum phase (NMP), whose stability domains are discontinuous in the parameter and have multiple crossover frequencies, and NMP systems. These are intricate issues which are not well addressed in standard texts. Indeed, some texts provide false guidelines for them, while others do not cover them, see e.g.
[2-10].
In this article which is also of pedagogical value the above problems are addressed. Understanding of the control community and students in particular, of these issues is enhanced and thus they can use the powerful tools of Gm / Pm more effectively. To this end the conventional Gm / Pm concepts are revisited. A redefinition of the Gm is offered which better captures the underlying notion of being a measure of robustness. The discussion is motivated and supported by the analysis of several instructive NMP examples. More precisely, we define two Gm ‟s for a system: one positive (in the increasing direction) and one negative (in the decreasing direction). For stable systems these are the destabilizing Gm ‟s and for unstable ones these are the stabilizing Gm ‟s. A false theorem and a false interpretation of the Gm concept for unstable systems are also rectified. This is done in Section II-A. As for the Pm , there cannot be more than one Pm associated with a system, with the explanation that a multiple of 360 degrees can be added to it. A false claim about Pm is also rectified. More precisely, unlike what some texts claim Pm is not necessarily measured with respect to the line -180 degrees but to either of the lines

180

2 k

which results in the smallest Pm (in absolute value). This value of Pm is between -180 and 180 degrees. This is presented in Section II-B. In Section II-C we investigate the stability/instability properties of NMP systems in terms of their Gm / Pm signs. This is a critical and important issue for which some sources provide false guidelines, while others do not cover it. More precisely, some standard texts and course notes claim that stable NMP systems (necessarily and sufficiently) have negative Gm and Pm . Some others say that Bode diagrams should be used with care in case of NMP systems, suggesting that it is possible to decide the stability/instability of the system from its Bode diagram. In this article, through some instructive examples it is shown that mere signs of Gm and Pm cannot be indicative of the stability of NMP systems – any combination of the signs of Gm and Pm is possible, depending on the system. Thus if the objective of deciding the stability properties of NMP systems from the Bode diagram is possible at all, higher level of analysis is needed and some other factors have to be considered. What those factors are, however, is not known to us at the moment. In Section II-D we discuss another confusing issue – the case of multiple gain crossover frequencies. It has been claimed by some texts that if there are two or more gain crossover frequencies, the phase margin is measured at the largest one.
We rebut this claim with two nice counter examples. A similar discussion for multiple phase crossover frequencies follows in
Section II-E. It should be stressed that the materials of this paper may be known to some experts – some might have discovered for themselves these ideas before, but to the best of our knowledge they have not yet appeared in any published work. Especially they are missing in the standard textbooks and course notes around the globe.
The organization of the rest of this article is as follows. Section II embodies the main ideas and arguments of the paper as mentioned above. Several examples are included to support and expound the ideas. Whilst the examples can be discussed in the
Nyquist context equally well, due to lack of space we suffice to use the root locus method only. Finally, Section III draws the
- 51 -
DOI: 10.18005/JCSE0301004

Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59 conclusions. It is hoped that the contents of this paper, being of particular pedagogical and instructive merit, are signed by the control community and enter the classical textbooks and the software MATLAB, thus relieving and enhancing the use of these pervasively used classical tools.
II.
THE GM / PM CONCEPTS REVISITED
The traditional Gm/Pm concepts are revisited in this Section. A new definition of Gm is presented in the ensuing Subsection.
A.
A New Definition for the Gm Concept
Recall that the traditional definition of Gm is the amount gain can be increased or decreased before the system changes its stability condition. As such, it is one directional and only one Gm is defined for every system. This definition is not good – it does not address the issue of „changing stability condition by gain variations (which is the conceptual meaning of gain margin)‟ completely, as elaborated in the following. To motivate and highlight the discussion we present some nice and instructive examples which are missing in the classical texts.
Example 2.1: Consider the NMP system
L

( s

K

0 .
002
0 .
0005 )( s


0 .
( s

001
0 .
)(
02 )( s

0 .
s

0 .
05 )( s

01 )( s

5 )( s

10
0 .
2 )( s

1 )( s
)

100 )
2
.
The root locus of the system is depicted in Fig. 1. It is observed that the stability range is discontinuous – the union of some regions: K

( A B )  ( C D )  ( E F ) , where A through F are the corresponding values of K at the j -axis crossings. Note that only the upper part of the root locus is shown, and that the picture is not drawn to scale. The gain and phase margin thus depend on the specific point (of the stable or unstable region) at which the system resides. It should be noted that if the pole at
0 .
0005 is replaced by

0 .
0005 (an MP system) the shape of the root locus remains unchanged.
An important lesson should be learned from this system. Suppose we are in any of the stable regions. Then by either increasing or decreasing the gain beyond the borders, the system becomes unstable. That is, any point is actually associated with two Gm ‟s: one in the increasing direction of gain ( Gm
 
0 ) and one in the decreasing direction of gain ( Gm
 
0 ). Similarly, suppose we are in either the region ( B C ) or ( D E ) . Then in both directions of increasing and decreasing gain the system
( becomes stable. That is to say, again any point is actually associated with two Gm ‟s: one in the increasing direction of gain
Gm
 
0 ) and one in the decreasing direction of gain ( Gm
 
0 ). In (

A ) and ( F
 
) the system becomes stable by increasing and decreasing the gain, respectively. This argument is valid for systems with NMP zeros as well, as discussed in the next example.
F
60
E
D
Some stable real poles and zeros:
Stable part of the root locus
C
B
A
Fig. 1 The root locus of Example 2.1
Example 2.2: As an example consider the above system with two NMP zeros added to it at 1

20 j . Then the general shape of the root locus remains unchanged, that is to say the stability range is again the union of three disjoint regions as before. The above discussion for the Gm concept thus applies.
Thus we can formally present the following definition.
Definition of Gm: It is the amount gain can be increased or decreased before the system becomes unstable, if stable, or stable if unstable. Associated with a system, either MP or NMP, are one positive ( Gm

) and one negative ( Gm

) Gm . For stable systems these are the destabilizing Gm ‟s, whereas for unstable systems these are the stabilizing Gm ‟s.
- 52 -
DOI: 10.18005/JCSE0301004

Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59
Paraphrased in other words, the definition provides a better measure of robustness for the system. The reason is that in practice perturbations in gain appear in both increasing and decreasing directions of gain, and not just in one direction. It should also be stressed that in case any of these quantities ( Gm

and Gm

) is infinite, it means that the objective is not achievable. For instance, stabilizing the system of Example 2.1 when in the unstable region K

(

A ) by decreasing the gain is impossible.
That is, Gm

Thus Gm
 
 

.
. Similarly, stabilizing the unstable system in the region ( F
 
) by increasing the gain is impossible.
The question that rises at this point is that for a given system (that is, for a given K ) which one of the aforementioned two
Gm ‟s is more important than the other and is in some sense the “main” one?
The answer is the smaller one in absolute value, because the system is more sensitive to perturbation in gain in that direction.
More precisely, if Gm
 
| Gm

| then Gm

is the main Gm , and if | Gm

|

Gm

then Gm

is the main Gm . Paraphrased in simple words: If the gain is nearer to the larger border then the main Gm is Gm

, and if the gain is nearer to the smaller border then the main Gm is Gm

.
D
As a numerical example, first note that in the system of Example 2.1,

2 .
29896720

10 unstable in the region
4
, E
( D E

)
1 .
59390084

10
7
, F

7 .
24790829

10
A
8

0 .
1 , B

3 .
31861923 ,
. Now, suppose that
. The two stabilizing gain margins associated with the system are
K
Gm



C

1 .
44475946
3

10
4

10
2
,
. The system is
20 log( D / K )

0 and
Gm
 
20 log( E / K )

0 . Because K / D

E / K (or | Gm

|

Gm

),

Gm is the main gain margin. Next suppose
K


Gm
10
7

for which the system is unstable in the same region. The two stabilizing gain margins associated with the system are
20 log( D / K )

0 and Gm
 
20 log( E / K )

0 . Because E / K

K / D (or Gm
 
| Gm

| ), Gm

is the main gain margin. The two cases considered are both unstable cases. Let us now discuss two stable cases. With K system is stable in the region ( C D )

2

10
2
, the
. The two destabilizing gain margins associated with the system are
Gm
 
20 log( C / K )

0 and Gm
 
20 log( D / K )

0 . Because K / C

D / K (or | Gm

|

Gm

),

Gm is the main gain margin. Next consider K
 associated with the system are
10
4
for which the system is stable in the same region. The two destabilizing gain margins
Gm
 
20 log( C / K )

0 and Gm
 
20 log( D / K )

0 . Because D / K

K / C (or
Gm
 
| Gm

| ), Gm

is the main gain margin.
In the following remark we discuss two other problems in the existing literature.
Remark 2.1: Especially for students a source of confusion over Gm is the following: (i) In some standard texts it is found that for unstable systems Gm is indicative of how much the gain should be decreased to make the system stable. Nevertheless, users encounter both positive and negative Gm ‟s for unstable systems when they use MATLAB. (ii) In some standard texts it is said that MP systems are stable if and only if both Gm and Pm are positive. Nonetheless, users sometimes come across stable MP systems with negative Gm when they use MATLAB. The explanation is that what MATLAB does is correct. With regard to (i) what the sources say is correct only in certain cases – in cases that the main Gm is Gm

. In case the main Gm is Gm

(as with
K

10 7
discussed above) it indicates how much the gain must be increased to make the system stable. As for (ii) first note that an example for it is the system of Example 2.1 if the pole at 0 .
0005 is replaced by

0 .
0005 . Recall that as we said before this
MP system has an (almost) identical root locus shape as of Fig. 1. Now if we are e.g. in the stable region K

( C D ) with
K / C

D / K then Gm
 
0 is the main Gm , which is correctly outputted by MATLAB.
The above observations can be made in simpler systems as well. For the sake of clarity and brevity in the following we work with simpler but still general and instructive NMP systems. We will come back to the system of Example 2.1 in Remark 2.3 and
Example 2.9. Also, we will reconsider the system of Example 2.2 in Example 2.4. Interested reader can find more sophisticated examples in the closing part of the article as well.
Example 2.3: Consider the NMP system L

K (( s

0 .
1 ) 2

1 )( s

0 .
1 ) /(( s

0 .
1 ) 2

4 )( s

1 ) . The root locus of this system is shown in Fig. 2.
The stability borders approximately: A

0 .
47 ,
A , B ,
B

C
11 .
are the values of the gain K at the corresponding crossing points. These values are
02 , C

39 .
70 . The stability range thus has the form K

( 0 A )  ( B C ) . Here again it is observed that a point should be associated with two Gm ‟s, one positive and one negative, with the following explanation: if the system is stable with K

( 0 A ) then it becomes unstable in the decreasing direction of gain (with negative gain) due to the real pole becoming unstable – imagine its root locus for negative gain. (If K
 
1 the system is unstable but with

1

K

0 the
- 53 -
DOI: 10.18005/JCSE0301004

Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59 system is stable.) In the increasing direction of gain it becomes unstable. If it is unstable in the part K

( A B ) then it becomes stable in both decreasing and increasing directions of gain. If it is stable with decreasing and increasing directions of gain. If it is unstable in the region K
K

( B C )

( C

)
then it becomes unstable in both
then it becomes stable only in the decreasing direction of gain.
Fig. 2 The root locus of Example 2.3
As a numerical example suppose K

1 . Then Gm

is the main gain margin and is stabilizing. If K

10 main gain margin and is stabilizing. On the other hand if if K

35 then
K

13 then

Gm is the main gain margin and is destabilizing.
then Gm

is the

Gm is the main gain margin and it is destabilizing, and
The Pm concept is revisited in the following Subsection.
B.
Comments on the Pm Concept
Unlike the Gm we have discussed in Subsection II-A, in case of phase margin the situation is different. By definition it does not make sense and in fact it is impossible that a system become stable/unstable in both directions of increasing and decreasing phase. Only one Pm is defined for every system. For stable systems it is the destabilizing Pm and for unstable systems it is the stabilizing Pm . Formally,
Definition of Pm: It is the amount phase can be increased or decreased (at the gain crossover frequency) before the system becomes unstable, if stable, or stable if unstable. Associated with a system, either MP or NMP, is one Pm which may be positive or negative. For stable systems this is the destabilizing Pm , while for unstable systems this is the stabilizing Pm .
In case changing the stability condition of the system by addition or subtraction of phase is impossible, infinity by MATLAB. As a numerical example consider the system of Example 2.3. With system has Pm
 
, Pm
 
11 .
8

0 , and Pm

10 .
6

0 , respectively.
K

0 .
1 , K

Pm is outputted as
0 .
3 , and K

1 the unique up to addition of multiples of 360 degrees. That is, a phase margin Pm is equivalent to a phase margin k

Remark 2.2: There are some flaws in the traditional definition of Pm which need to be explained. (i) For a system Pm is
1 , 2 ,...
Pm

2 k

,
. (ii) From among the Pm ‟s just mentioned the one smallest in absolute value is the correct answer, because the system is most sensitive to phase variations in that direction. (Note that this means that the correct phase margin is between -180 and 180 degrees.) In other words unlike the claim made by some standard textbooks that Pm is measured with respect to the line -180 degrees, it is measured with respect to either of the lines

180

2 k

which results in the smallest Pm (in absolute value). This is shown in Examples 2.4 and 2.5 below.
| 90
Example 2.4: Consider the system of Example 2.2 with
.
5 |

|

269 .
5 |
K

20000 and K

1 . The Bode diagrams are shown in Fig. 3, left and right panels, respectively. For the system on the left phase margin is 90.5 not 90 .
5

360
 
269 .
5 , because
. Note that here Pm is measured with respect to line -180 degrees, not the line 180 degrees. Similarly, for the system on the right phase margin is -24.3 and not

24 .
3

360 k , because |

24 .
3 |

|

24 .
3

360 k | . But in this case Pm is measured with respect to the line 180 degrees.
- 54 -
DOI: 10.18005/JCSE0301004

Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59
Fig. 3 Bode diagrams of the of Example 2.4: Left with K

20000 , Right K

1
Example 2.5: Consider the system Ks /( s
2 
2 s

2 )
4
. The interested reader is encouraged to refer to [11] for the correct way of computing the phase of a system, e.g. this system. In this paper with the computations of the 2015a release of MATLAB we present our arguments. The Bode diagrams of this system with
Pm
 
127

K

100 and
which is measured with respect to the line 180 degrees; With respect to the line -540. From among the lines

180
 the systems. (Similarly, it is easy to verify that with
2
K k

K
K


10000
10000 ,
are given in Fig. 4. With
Pm

39 .
1
K

100
which is measured with
these lines are the nearest (in absolute value) to the phase angle of
50 , Pm
 
100 which is measured with respect to the line -180
, degrees. This case is not shown in the figure.)
Fig. 4 Bode diagrams of the system of Example 2.5: Left with K

100 , Right K

10000
Remark 2.2: It would be instructive and helpful for the users, i.e., control community and students in particular, if these quantities are produced by a MATLAB command. Just some minor changes in the command „allmargin‟ are needed to produce these quantities.
In the next subsection we consider stability of NMP systems in the Bode diagram context. As we stated before, this is a critical point with wrong analysis and guidelines in some classical texts.
C.
Stability of NMP Systems
In this subsection we show that it is not possible to decide the stability of NMP systems by merely considering the signs of the
Gm / Pm . We start with a motivating example.
L
2
Example

K (( s

0 .
1 ) 2
2.6:

1 ) /( s

Consider
0 .
3 )(( s

0 .
2 ) 2 the NMP systems L
1

K (( s

0 .
1 ) 2

1 ) /( s

0 .
1 )(( s

0 .
2 ) 2

4 ) and

4 ) , whose root loci are given in Fig. 5. The designated symbols A , B , and C are the values of K at the corresponding crossing points.
- 55 -
DOI: 10.18005/JCSE0301004

Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59
C

Read from the picture by clicking on the crossing points, for L
1
:
1 .
2 . The stability range for L
2
is thus K

( C B ) .
A

0 .
54 , B

14 .
7 , and for L
2
: A

0 .
53 , B

15 .
9 ,
Four different cases are considered: (i) i.e., destabilizing Gm / Pm , (iv) K

20
K

0 .
3 , i.e., stabilizing Gm / Pm , (ii)
, i.e., stabilizing Gm / Pm .
K

2 , i.e., destabilizing Gm / Pm , (iii) K

13 ,
Gm
For

0 ,
L
1
: (i)
Pm

K
0 .

0 .
3 : Gm

0 , Pm
 inf , (ii) K

2 : Gm

0 , Pm

0 , (iii) K

13 : Gm

0 , Pm

0 , (iv) K

20 :
Gm
For

0 ,
L
2
: (i)
Pm

K
0 .

0 .
3 : Gm

0 , Pm
 inf , (ii) K

2 : Gm

0 , Pm

0 , (iii) K

13 : Gm

0 , Pm

0 , (iv) K

20 :
An important point is noteworthy: Considering the stable cases (ii) and (iii) together, it is observed that all possible combinations of signs of Gm and Pm are present. This refutes the claim made by some sources that stable NMP systems necessarily have their Gm / Pm both negative. In fact this observation proves that by merely considering the Gm / Pm signs, it is not possible to decide the stability of NMP systems. To determine the stability of NMP systems from the Bode diagram, if possible at all, requires higher level of analysis – other factors should also be taken into account. However, at present we do not know those factors.
Remark 2.3: In passing it is worth adding the side comment that cases (iv) of the above argument also reveal that in some special situations delay is useful: some unstable NMP systems – like the ones in this example – become stable by adding delay to them. On the other hand, some unstable NMP systems become stable by adding a negative delay to them, i.e., by adding a negative phase, Pm

0 . This is the case e.g. for the system of Example 2.1 in the regions ( B C ) , ( D E ) , and ( F

) . The former observation is not new, it can be found e.g. in [10], however the latter is new to the best of our knowledge.
Fig. 5 The root loci of systems of Example 2.3: L
1
on the left, L on the right
2
In the sequel subsection the issue of multiple gain crossover frequencies is discussed.
D.
Case of Multiple Gain Crossover Frequencies
Another source of confusion which has made the use of Gm / Pm somewhat problematic especially for students is that some classical references claim that if there are two or more gain crossover frequencies, the phase margin is measured at the largest one.
This is not necessarily correct. The correct answer is that the one that results in the smallest Pm (in absolute value) matters, because the system is most sensitive to perturbation in phase in that direction. This is illustrated in the following examples.
Example 2.7: Consider systems L
1
and L
2
of Example 2.6. They are stable with K

4 and have the Bode diagrams of Fig.
6.
Both systems have three gain crossover frequencies and thus three phase margins or Pm ‟s. The gain crossover frequencies are the frequencies at which the transfer function of the system is zero dB in magnitude, as marked with circles in Fig. 6. The crossover frequencies and corresponding Pm ‟s are as in the following (

, Pm ) pairs: For L
1
: (0.65,90.2), (1.32,-52.9),
(4.66,87.7), and for L
2
: (0.62,56.9), (1.32,-70.1), (4.65,82.8). It is observed that for system L
1
the smallest (in absolute value)
- 56 -
DOI: 10.18005/JCSE0301004

Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59
Pm is -52.9 which is at the at the second largest gain crossover frequency (not the largest one), i.e., 1.32 rad/s, whereas for system
L
2
the smallest (in absolute value) Pm is 56.9 which is at the smallest gain crossover frequency, 0.62 rad/s (not the largest one).
Fig. 6 Bode diagram of systems of Example 2.7 with K

4 : L
1
on the left, L on the right
2
Example 2.8: Consider the system of Example 2.3 with K in Fig. 7, on its left and right panels, respectively.

0 .
5 and K

15 . Bode diagrams of these systems are depicted
Fig. 7 Bode diagram of systems of Example 2.8 with K

0 .
5 on the left and K

15 on the right
The system on the left has two gain crossover frequencies and thus two Pm ‟s. These crossover frequencies and their associated Pm ‟s are in the sequel (

, Pm ) pairs: (1.77,1.10) and (2.56,-150.32). The smallest (in absolute value) Pm is 1.1 which is at the smallest gain crossover frequency (not the largest one). As for the system on the right there are three gain crossover frequencies and hence three Pm ‟s. The (

, Pm ) pairs are (0.27,-82.54), (0.84,-96.36), (1.07,-11.98). Here the smallest (in absolute value) Pm is -11.98 which appears to be at the largest gain crossover frequency.
In the ensuing subsection we study the issue of multiple phase crossover frequencies which is an intricate and problematic issue especially for students.
E.
Case of Multiple Phase Crossover Frequencies
This is another reason of confusion in particular because some textbooks give false guidelines for it. Similar to the case of multiple gain crossover frequencies they say that gain margin should be measured at the largest crossover frequency. However, the correct answer is that in case of multiple phase crossover frequencies, Gm is measured at the frequency which results in the smallest Gm in absolute value, because the system is most sensitive to gain variations in that direction. This is evinced in the subsequent examples.

Example 2.9: The Bode diagram of the system of Example 2.1 with phase crossover frequencies and thus six Gm ‟s. The phase crossover frequencies are frequencies at which transfer function is
180

2 k

K

1 is shown in Fig. 8. For this system there are six
in phase, as designated by circles in Fig. 8. The crossover frequencies and the respective Gm ‟s are in the following
- 57 -
DOI: 10.18005/JCSE0301004

Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59
(

, Gm ) pairs: (0,-20), (0.0038,10.41), (0.0219,43.19), (0.4403,87.23), (6.8626,144.04), (84.9333,177.20). It is observed that the smallest (in absolute value) Gm is 10.41 which corresponds to the second phase crossover frequency (not the largest one).
Fig. 8 Bode diagram of the NMP system of Example 2.9 with K

1
Example 2.10: For the systems of Example 2.3 with K

2 and the system L
2
of Example 2.6 with K

8 the Bode diagrams are given on the left and right panels of Fig. 9, respectively.
Both systems have three phase crossover frequencies and thus three G m ‟s. The crossover frequencies and the corresponding
Gm ‟s are in the following (

, Gm ) pairs. Left panel: (0,25.95), (1.12,14.82), (1.78,-12.53). It is seen that the smallest (in absolute value) Gm is -12.53 which appears to correspond to the largest phase crossover frequency. For the right panel (

, Gm ) pairs are: (0,-16.47), (0.98,5.97), (2.01,-23.51). It is observed that the smallest (in absolute value) Gm is 5.97 which corresponds to the second largest phase crossover frequency (not the largest one).
Fig. 9 Bode diagrams of Example 2.10: Left: system of Example 2.3 with K

2 , Right: system L
2
of Example 2.6 with K

8
III.
CONCLUSIONS
The Gm and Pm concepts which are amongst the first tools developed for quantifying robustness against uncertainty have some confusing subtleties especially for students. One reason for this is the inherent shortcoming of their definition provided in classical texts and the other is some false claims related to Gm / Pm made in some standard texts for systems with multiple crossover frequencies and NMP systems. In this article which is also of pedagogical interest we have provided a redefinition of the Gm concept which better captures the concept of being a measure of robustness: every system is associated with two Gm ‟s, one in the increasing direction ( Gm
 
0 ), and one in the decreasing direction ( Gm
 
0 ) of gain. In case the system is stable these are the destabilizing Gm ‟s and in case the system is unstable these are the stabilizing Gm ‟s. Moreover, we have discussed the subtle points and refuted the false claims. We have shown that: (i) by merely considering the signs of Gm/Pm it is not possible to determine the stability of NMP systems as all combinations of their signs is possible, (ii) some unstable NMP systems become
- 58 -
DOI: 10.18005/JCSE0301004

Journal of Control and Systems Engineering Dec. 2015, Vol. 3 Iss. 1, PP. 51-59 stable by adding delay to them ( Pm

0 ) whereas others by adding negative delay to them ( Pm

0 ) . The arguments have been supported by several instructive NMP examples which are missing in the classical texts. It is hoped that the ideas of this article enter the standard texts and the software MATLAB, and help relieve and enhance the utility of these important concepts. Among the topics for future research are: (i) Development of a methodology – if possible at all – for determining the stability/instability properties of NMP systems from the Bode diagram; (ii) Addressing the Gm / Pm concepts in the time domain. This is an interesting topic which has attracted the attention of the control community since the introduction of state-space methods in the
1960s. Only for certain design methods – like the classical LQR and some others – the answer is already known, see e.g. [10].
Finally we should mention that for the sake of brevity we have sufficed to the presented examples. The interested reader might like to study the following systems as well. The phase margin of the system K ( s

1 ) /( s
2 
2 s

2 )
8
is measured with respect to the lines -540, -900, and -1260 degrees, depending on the value of gain. The phase margin of the system
K / s 2 ( s

1 ) 8
is measured with respect to the lines -180, -540, -900, -1260, and -1580 degrees, depending on the value of gain.
The system L

K (( s

0 .
1 ) 2

1 )(( s

0 .
1 ) 2

9 ) /(( s

0 .
2 ) 2

4 )(( s

0 .
2 ) 2

25 )( s

1 ) with K

10 has five gain crossover frequencies (thus five Pm ‟s) and four phase crossover frequencies (hence four Gm ‟s). The system
L

K (( s

0 .
1 )
2

1 )(( s

0 .
1 )
2

9 )( s

0 .
1 ) /(( s

0 .
1 )
2

4 )(( s

0 .
1 )
2

25 )( s

1 ) with K

5 has three gain crossover frequencies (hence three Pm ‟s) and five phase crossover frequencies (therefore five Gm ‟s). The system of Example 2.2 with
K

1000 has three gain crossover frequencies (thus three Pm
‟s) and six phase crossover frequencies (and so six
Gm
‟s).
REFERENCES
[1] H. Barkhausen, Lehrbuch der Elektronen-Röhren und ihrer technischen Anwendungen , 4th ed., Leipzig: S. Hirzel, 1935.
[2] F. Golnaraghi and B. Kuo, Automatic control systems , 9th ed., John Wiley & Sons, 2011.
[3] R. C. Dorf and R. H. Bishop, Modern control systems , 12th ed., Prentice Hall, 2010.
[4] G. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback control of dynamic systems , 6th ed., Prentice-Hall, 2010.
[5] G. C. Goodwin, S. F. Graebe, and M. E. Salgado, Control system design , Prentice-Hall, 2005.
[6] K. Ogata, Modern control engineering , 5th ed., Prentice-Hall, 2009.
[7] K. J. Astrom and R. Murray, Feedback systems: An introduction for scientists and engineers , Princeton University Press, 2008.
[8] J. J. D‟Azzo, C.H. Houpis, and S.N. Sheldon, Linear control system analysis and design with MATLAB , 5th ed., Marcel Dekker, 2003.
[9] P. Bélanger, Control engineering: A modern approach , Saunders College Publications, 1995.
[10] S. Skogestand and I. Postlethwaite, Multivariable feedback control: Analysis and design , 2nd ed., John Wiley, 2005.
[11] Y. Bavafa-Toosi, “How to draw the Nyquist plot,” Int. Jr. Automation and Control , 2015 (under review).
Yazdan Bavafa-Toosi was born in Mashhad, Iran, on Sept. 22, 1974. He received B.Eng. and M.Eng. degrees in electrical power and control engineering from Ferdowsi University, Mashhad, and K.N. Toosi University of
Technology, Tehran, Iran, in 1997 and 2000, respectively. He earned his Ph.D. degree in system design engineering
(also known as systems and control) from Keio University, Yokohama, Japan, in 2006.
His multi-disciplinary research spans systems and control theory and applications. Between and after his educations he has held various research and teaching positions in Germany, Japan, and Iran, and co-authored about 40 technical contributions. Currently he is self-employed.
Dr. Bavafa-Toosi has been a reviewer of some journals in the field of systems and control theory and applications in the past.
DOI: 10.18005/JCSE0301004
- 59 -