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A New Technique for Calculation of Maximum
Achievable Gain and Phase Margins with
Proportional Control
Nevra Bayhan*, Mehmet Turan Söylemez**
*
Istanbul University/Electrical and Electronics Engineering Department, Istanbul, Turkey
** Istanbul Technical University/Electrical Engineering Department, Istanbul, Turkey
* nevra@istanbul.edu.tr
** soylemez@elk.itu.edu.tr
Keywords: Gain margin, phase margin, Nyquist criterion
Abstract— In this paper, a new method is suggested to
compute maximum achievable gain and phase margins that
can be achieved using a proportional controller. The method
focuses on using the Nyquist stability criterion and then
applying a previously developed theorem which allows
determination of stabilizing gains. Methods for determining
all stabilizing proportional controllers that satisfy gain and
phase margin constraints are also given.
I. INTRODUCTION
A method for algebraically computing all stabilizing
gains for a given linear time invariant system is introduced
in [1]. The method is then generalized to that of finding
stabilizing controllers for PID controllers [2,3]. An
extension of this method to discrete time systems is given
recently by Bayhan and Söylemez [4].
Gain margin (GM), which can be defined as the
maximum gain uncertainty that can be tolerated in the
open loop system without loosing stability in the closedloop system, and phase margin (PM), which can be
defined similarly for phase uncertainty, are measures of
relative stability, and are frequently used in the analysis
and design of the control systems in the context of robust
stabilization. Therefore, finding all low order controllers
that achieve specified gain or phase margins is very
useful.
An approach for finding regions of specified gain and
phase margins in the parameter space of a single-input
single-output control system with adjustable parameters is
presented by Shenton and Shafiei [5]. Another method for
computation of stabilizing PI and PID controllers that
achieve specified gain and phase margins is given in [6].
Although the techniques presented in [5] and [6] are
useful when controllers with 2 or 3 free parameters are
used, they do not present an efficient algorithm if only a
proportional controller is to be used.
This paper extends the results of [1, 2, 3] to find all
stabilizing proportional controllers that achieve given gain
and phase margin specifications. A further question of
maximum achievable gain and phase margins with a
proportional controller is then answered in the paper for
the first time.
The method of [1] for calculation of stabilizing
controllers is revised in the following section. The
extension of this method to systems with gain or phase
uncertainties is given in Section 3. Section 4 presents
methods calculating maximum achievable gain and phase
margins with a proportional controller. Several numerical
examples are given in Section 5, which is followed by the
conclusions in Section 6.
II. CALCULATION OF ALL STABILIZING GAINS
A new and fast approach to computation of the entire
set of stabilizing all low-order compensators is given by
Munro and Söylemez [1,2,3]. This approach is based on
the use of the Nyquist plot.
Consider the single input single output control system
of Figure 1 where
G (s ) =
N ( s) a m s m + a m −1 s m −1 + " + a1 s + a 0
=
D ( s)
s n + bn −1 s n −1 + " + b1 s + b0
(1)
is the plant to be controlled (with ai , bi ∈ ℜ ) and C(s) is
a constant gain controller of the form
(2)
C(s) = KP
The problem is to compute all KP controllers that stabilize
the closed-loop system of Figure 1.
r +
y
C(s)
G(s)
-
Figure 1. Closed loop system with constant gain
Decomposing the numerator and the denominator
polynomials of (1) into their even and odd parts and
substituting s = jw , we have
G ( jw ) =
N ( jw ) N re + jN im
=
D ( jw ) D re + jD im
(3)
where D re =ˆ Re {D ( jw )} , Dim =ˆ Im{D( jw)} and N re
and N im are defined similarly. By noting that
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Dre = De (−w2 )
2
Nre = Ne (−w )
Dim = Do (−w2 )w
2
Nim = No (−w )w
coefficient of
(5)
k ∈ i =ˆ ( −1 / xi −1 ,−1 / xi ) , the number of unstable poles of
where for notation purposes De and Do denotes the even
and odd parts of D(s) , and N e and N o denotes those of
N (s) , respectively. It is possible to write
the closed-loop system ( ui ) is given by
i −1
u i = u 0 + ∑ rt
pi
 D N − Do N e 
De N e + Do N o w
+ jw  e 2 o
2
2 2
2 2 
De + Do w
 De + Do w 
ri = ∑ d i , j
2
G( jw) =
X (w2 )
Y (w 2 )
+ jw
2
Z (w )
Z (w2 )
( 6)
(7)
di , j
X (w2 ) =ˆ De Ne + Do No w2
Y (w2 ) =ˆ De No − Do N e
(8)
and where for notation purposes De , Do , N e and N o
are used instead of De (− w 2 ) , D o ( − w 2 ) , N e (−w 2 ) and
N o ( − w 2 ) , respectively. The imaginary part of G ( jw) is
given by
Y (w 2 )
Z ( w2 )
1 − (−1) l ) Sgn(Y (l ) (vi∗, j ))

= Sgn( y0 )

− Sgn( y1 )
if 0 < vi∗, j < ∞

if vi∗, j = 0


∗
if vi , j = ∞ 
(12)
in which Y (l ) (vi*, j ) is the first nonzero derivative
of
Y (v) at the point v i∗, j . Theorem 2.1 can easily be
extended to cover systems with imaginary axis poles
[1,2,3].
Z (w 2 ) =ˆ De2 + Do2 w2
Im{G( jw)} = w
(11)
j =1
and
where
(9)
2
By denoting v =
ˆ w and the positive real roots of
∗
∗
∗
Y(v) as v1 , v 2 ,..., vγ it is obvious that the Nyquist plot of
G ( jw) crosses the real axis only if w = 0 , w = ∞ , or
w = ± v i∗
(10)
t =1
where u 0 is the number of unstable poles of G(s) ,
N e + jwN o
=
De + jwD o
G ( jw ) =
Y (v) as y 0 . Then, for a given gain
(4)
for i = 1,2,..., γ . Denoting vγ∗+1 = 0 and
v γ∗+ 2 = ∞ , the real axis crossing points are found as
xi = X (vi∗ ) / Z (vi∗ ) for i = 1, 2,..., γ + 2 . Relabeling the
III. CALCULATION OF ALL STABILIZING GAINS FOR
PROVIDING DESIRED GAIN AND PHASE MARGINS
It is possible to represent gain and/or phase
uncertainties in a plant using a complex element Ke jθ as
illustrated in Fig. 2. Gain margin denotes the largest value
that K can assume for θ = 0 , and the phase margin
denotes the largest value that θ can assume for K = 1
without affecting closed-loop system stability. A modified
version of Theorem 2.1 can be used to determine all
possible values of K P that satisfies a gain margin (or
phase margin) specification.
r
+
-
KP
Ke jθ
G(s)
y
pairs ( xi , vi∗, j ) (for i = 1,2,..., γ + 2 ) as ( xi , vi∗, j ) (for
such
that
and
xi < xi +1
i = 1,2,..., q )
∗
∗
x i = X (v i , j ) / Z (v i , j ) (for all j = 1,2,..., p i ), it is possible
to state the following theorem [1,2,3].
A. Theorem 2.1 : (Munro and Söylemez)
Consider a linear time-invariant system given by a
proper rational transfer function G (s) = N ( s) / D( s)
given as in (1), and assume that D(s) has no roots on the
imaginary axis. Let X (w 2 ) , Y ( w 2 ) and Z (w 2 ) be
polynomials as defined (8), and the pairs ( x i , v i∗, j )
( i = 1,2,..., q ) be as defined above. Furthermore, denote the
first coefficient of Y (v) as y1 , and the last nonzero
Figure 2. Proportional control of a system with gain and/or
phase uncertainties
A. Calculation Of All Stabilizing Gains With Gain
Margin Constraints
Lemma 3.1:
Let the entire set of stabilizing gains for the given
system to be  = {K 1 , K 2 ,..., K t } where the gain
intervals i =ˆ ( K i min , K i max ) (for i=1,2,…,t) are found
via Theorem 2.1. All stabilizing constant gains that satisfy
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a given minimum gain margin (GM ) min constraint are
given by K = K1, K2 ,..., Kt where
{
}

∅


Ki = 
 K , Ki max 
 i min (GM ) 
min 

if
Ki max 
(GM )min 


Otherwise


Ki min >
(13)
where ∅ denotes the empty set. When all gain intervals
K i are empty, there are no stabilizing constant gains that
satisfy the desired gain margin specification.
Proof of the Lemma is straightforward from the
definition of gain margin and is not given here.
B. Calculation Of All Stabilizing Gains With Phase
Margin Constraints
Lemma 3.2:
In order to determine all gain intervals for which the
system has a phase margin larger than a given value, θ min ,
we propose to consider the phase shifted system
Gnew(s) =
e− jθ N (s) (cosθ − j sinθ ) N (s) Nnew(s)
=
=ˆ
D(s)
D(s)
D(s)
(14)
and try to find out all stabilizing controllers for this system
by the help of Theorem 2.1. Note that the coefficients of
the numerator polynomial of (14) become complex in this
case. In this case, if we substitute s =ˆ jw into (14), it is
possible to write
Gnew ( jw) =
N new ( jw) (cos θ − j sin θ ) N ( jw)
=
D ( jw)
D ( jw)
(15)
Multiplying the numerator and denominator of the last
equation by complex conjugate of the denominator,
( D ∗ ( jw) ) yield
Gnew ( jw) =
{
}
{
}
Re N new ( jw)D∗ ( jw) + j Im N new ( jw)D∗ ( jw)
D( jw)D∗ ( jw)
(16)
In the sprit of Theorem 2.1, the real roots ( wi∗ ) of
are
substituted
into
Im{N new ( jw) D ∗ ( jw)}
∗
Re{N new ( jw) D ( jw)} to find the intersections of the
Nyquist plot with the real axis. In order to find the
directions of crossings ( ri ), the first derivative of
Im{Gnew ( jw)} with respect to w is computed at the points
wi∗ . The gain intervals not changing the number of
unstable poles are then determined with the help of
Theorem 2.1. The gain intervals (i) with u i = 0
stabilize the closed-loop system and satisfy the given
phase margin specification.
CALCULATION OF MAXIMUM ACHIEVABLE GAIN
AND PHASE MARGINS
In the previous section, a method is proposed to
calculate all stabilizing gains ( K P ) that satisfy a given
gain margin constraint or phase margin constraint. At this
point, a natural question that comes into mind is the
following: “what is the maximum gain (or phase) margin
that can be achieved by a proportional controller”. These
values are called as maximum achievable gain margin
(MAGM) and maximum achievable phase margin
(MAPM). By the help of the development given in the
previous section, methods for calculating these values are
given below.
IV.
A. Calculation of Maximum Achievable Gain Margin
(MAGM)
To calculate maximum achievable gain margin, the
upper bound of stabilizing gain intervals computed via
Theorem 2.1 are used. The following theorem utilizes the
idea that the ratio of maximum stabilizing gain to
minimum stabilizing gain gives us the maximum
achievable gain margin (MAGM) over a set of stabilizing
controllers:
Theorem 4.1:
For each of the gain intervals i =ˆ ( K i min , K i max )
stabilizing system, G ( s ) , as defined in (1), if the gain
margins are defined as
GM i =
K i max
K i min
(17)
maximum achievable gain margin is calculated by the
following equation.
MAGM = max {GM i }
(18)
i
Remark 4.1: From Theorem 4.1, we conclude that for an
open-loop stable system (a closed-loop stable system for
K P → 0 ) or a high gain closed-loop stable system
(stable for K P → ∞ ), MAGM is ∞ .
B. Calculation of Maximum Achievable Phase Margin
(MAPM)
In this section we propose a method to calculate
MAPM. The upper bound and the lower bound of the gain
intervals which stabilize the closed-loop system and the
frequencies wi∗ corresponding to these bounds are used
for this purpose. Theorem 2.1 is extended for several
special cases. The cases of the closed-loop system being
stable as K P → ∞ and K P → 0 require special
examination. Consequently we examine three cases in the
following.
B1. Calculation of Maximum Achievable Phase Margin
for Open-Loop Stable Systems
In this case, maximum achievable phase margin is
determined using the following theorem.
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Theorem 4.2:
MAPM is 1800 for an open-loop stable system G(s)
with a constant gain controller.
Proof 4.2:
The result can be seen directly by observing the fact
that it is always possible to find a small enough gain K P
in this case such that the closed-loop system is stable and
Nyquist plot of K p G ( jw) is entirely inside the unit
circle (i.e. there is no gain crossover frequency).
MAPM = max (θ k )
1 ≤ k ≤ r
(20)
Proof 4.4:
Maximum achievable phase margin can be found by
determining circles of radius G ( jw* ) that are tangent to
the Nyquist curve at frequency w∗ (see Figure 3)
B2. Calculation of Maximum Achievable Phase Margin
for Open-Loop Unstable and High-Gain Closed-Loop
Stable Systems
This is the case where at least one pole of G (s ) is on
RHP where as all finite or infinite zeros of G (s ) are on
LHP. The following theorem can be used in this case :
Theorem 4.3:
For unstable system G (s ) , which is known to be closedloop stable as K P → ∞ the maximum achievable phase
margin is 900, if G (s ) is strictly proper, and is 1800, if
G (s ) is biproper.
Proof 4.3:
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 K P → ∞ , if G (s )
is strictly proper, and does not cross the unit circle for
high gains if G (s ) is biproper.
B3. Calculation of Maximum Achievable Phase Margin
for Open Loop Unstable and High Gain Closed-Loop
Unstable System
For an open-loop unstable system with at least one
(finite or infinite) RHP zero, maximum achievable phase
margin is usually less than 900 and can be found using the
following theorem :
Theorem 4.4:
frequency intervals i =
ˆ {wi −1 , wi }
to
stabilizing
gain
intervals
i =ˆ (−1 / xi −1 , −1 / xi ) , then find the real frequencies
Define the
corresponding
wk∗ that satisfy the equation
 ∂

Im 
G ( jw ) 
∂
w


Im {G ( jw )}
=
Re {G ( jw )}
 ∂

Re 
G ( jw ) 
∂
w


(19)
and wk∗ ∈ i . Assume that there are r > 0 such
frequencies. For each of the frequencies found, calculate
the corresponding phase angle θ k =ˆ ∠G ( jwk∗ )
( k = 1,2,..., r ). The maximum phase margin that can be
achieved using proportional control is given as
Figure 3. The Nyquist plot for proof 4.4
It is possible to write that
∠
∂G ( jw)
= α − 900
∂w
(21)
Taking the tangent in both sides
tan ∠
∂G ( jw)
= tan(α − 90 0 )
∂w
= tan α
= tan ∠G ( jw)
(22)
Therefore at the frequency for which MAPM is found
the following equation must be satisfied.
 ∂ G ( jw ) 
Re 

Re {G ( jw ) }
 ∂w 
=
Im {G ( jw )}
 ∂ G ( jw ) 
Im 

 ∂w 
(23)
Considering the fact that only positive real roots of
above equation that are in one of the stabilizing frequency
ranges (i) are meaningful Theorem 4.4 immediately
follows.
V.
NUMERICAL EXAMPLES
A. Example 1
Consider the control system given in [1], where
N ( s ) = 0.5 s 4 + 2 .5 s 3 + 5 s 2 + 24 . 375 s + 31 . 22
D ( s ) = 1.09 s 4 − 13 . 12 s 3 + 64 .23 s 2 − 151 .11s + 70 .89
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{
{
}
}
{
{
}
}
Applying the even-odd decompositions to N(s) and D(s),
and using equation (8) it is possible to write
Im{G( jw)} Im N ( jw) D ∗ ( jw) / D( jw) D ∗ ( jw) Im N ( jw) D ∗ ( jw)
=
=
Re{G( jw)} Re N ( jw) D ∗ ( jw) / D( jw) D ∗ ( jw) Re N ( jw) D ∗ ( jw)
X (v) = 0.545v 4 − 70 .365v 3 + 1088 .2v 2 − 6043 .02 v + 2213 .19
From Theorem 4.4, the real frequencies satisfying (19)
are
Y (v ) = −9 .285 v 3 + 328 .299 v 2 − 2907 .99 v + 6445 .6
Z (v ) = 1.188v 4 + 32 .11v 3 + 314 .91v 2 + 13727 .7 v + 5025 .39
When we compute the roots of D(s), we see that there
are four unstable roots. Hence u0 is 4. The positive real
roots of Y(v) are v1∗ = 3.38864 , v 2∗ = 8 . 86778 and
∗
v3∗ = 23.1016 . Adding v4∗ = 0 and v 5 = ∞ , there exist
five crossing frequencies for this problem. The crossing
points ( xi ) corresponding to these frequencies, the net
crossing counts, and the gain intervals (i) are shown in
Table I. Noting that u 0 = 4 , the number of unstable
closed-loop system poles, u i , are calculated from (10). An
examination of Table 1 reveals that the closed-loop
system is stable for 6.71 < K P < 15.75 .
ri , u i
, ± 2 . 48131
From
Theorem 2.1, the frequency range
i ∈ {wi −1 , wi } corresponding to each of the stabilizing
gain intervals i =ˆ (−1 / xi −1 ,−1 / xi ) are
1 . 84083 < w < 2 . 97788 or
− 2.97788 < w < −1.84083
The real frequencies satisfying (19) from inside i :
w i∗ = ± 2 . 48131
∗
Substituting these real frequencies wi into G ( jw) , we
find MAPM = 17 .342 0 . This means that it is not possible
TABLE I.
CALCULATION OF
w i = ± 3 . 62618
AND THE STABILIZING GAINS
to achieve a phase margin larger than this value using a
proportional controller.
vi∗
xi
ri
ui
i
23.1016
-0.219624
-2
4
0 < K P < 4.566
3.38864
-0.149116
-2
2
4.566 < K P < 6.71
Now, let us examine whether or not there are the gain
intervals stabilizing the closed-loop system for a phase
margin being close to 17.3420 but less than MAPM. For
example, let us find all stabilizing gains that achieve at
least a 17.30 phase margin.
8.86778
-0.063442
2
0
6.71 < K P < 15 .75
Setting e − j17..3 = 0.9548 − j 0.2974 in (14), we have
0
0.440402
1
2
K P > 15 .75, K P < −2.27
N new ( s ) = ( 29 .8089 − j 9 .28483 ) + ( 23 .2733 − j 7 .24913 ) s +
∞
0.457
1
3
− 2.27 < K P < −2.188
-
∞
-
4
− 2.188 < K P < 0
From Table I, it is possible to observe that the system
is an open-loop unstable and high gain closed-loop
unstable. Therefore, maximum achievable gain and phase
margins are to be calculated using Theorem 4.1 and
Theorem 4.4, respectively.
Computation of Maximum Achievable Gain Margin:
An examination of Table I reveals that the closed-loop
system is stable for gains
( 4 .774 − j1 .487 ) s 2 + ( 2 .387 − j 0 .7435 ) s 3 +
( 0 .4774 − j 0 .1487 ) s 4
From Lemma 3.2 and Theorem 2.1, the gain intervals
not changing the number of unstable poles are determined
as shown in Table II.
TABLE II.
CALCULATION OF
ri , u i
AND THE STABILIZING GAINS FOR PM=17.3
wi∗
xi
ri
ui
i
5.50314
-0.27334
-1
4
0 < K P < 3.658447
i = ( K i min , K i max ) = ( 6 .71, 15 . 75 )
-1.51471
-0.183384
-1
3
3.658447 < K P < 5.45304
From Theorem 4.1, the maximum achievable gain
margin is calculated as follows.
-4.09399
-0.150471
-1
2
5.45304 < K P < 6.645799
2.45496
-0.091003
-1
1
6.645799 < K P < 10 .98865
2.50726
-0.086682
1
0
10.98865 < K P < 11 .53647
-3.28855
-0.073789
1
1
11.53647 < K P < 13 .55199
0.104292
0.434699
1
2
-54.3686
0.457145
1
3
-
∞
-
4
MAGM = K i max / K i min = 2.3472
Computation of Maximum Achievable Phase Margin:
When we substitute s = jw into N(s) and D(s) and
then multiply N ( jw) and D( jw) by complex conjugate
of D( jw) ( D ∗ ( jw) ), the coefficients of the denominator
polynomial of G(jw) become real.
0
K P > 13.55199,
K P < −2.30044
− 2.30044 < K P < −2.18749
− 2.18749 < K P < 0
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As it can be observed from Table II, the gain interval
stabilizing the closed loop system for 17.30 phase margin
is 10.98865 < K P < 11.53647 .
Computation of Maximum Achievable Gain Margin:
From Remark 4.1, it is possible to state that
MAGM = ∞ .
B. Example.2:
Consider the example given in [7] , where
Computation of Maximum Achievable Phase Margin:
Since this control system is closed-loop stable as
0
K P → ∞ , from Theorem 4.3, MAPM = 90 .
D(s) = s 5 + 11s 4 + 22s 3 + 60s 2 + 47s + 25
N ( s ) = s 4 + 6 s 3 + 12s 2 + 54s + 16
The Nyquist plot of G ( s ) is given in Figure 5.
Since the open-loop system is stable it is possible to
state that MAGM = ∞ and MAPM = 1800 by the help
of theorems 4.1 and 4.2, respectively.
For instance, the Nyquist plot of K p G ( s ) for
K P = 0.01 is given in Figure 4. Note that phase margin
for K p G ( s ) is 1800 for this value of Kp, since there does
not exist any gain crossover.
Figure 5. The Nyquist plot of G(s)
VI.
Figure 4. The Nyquist plot for
K P = 0.01
C. Example.3:
Consider the system given as G ( s ) =
N (s )
, where
D( s)
N ( s ) = s 4 + 6s 3 + 12 s 2 + 54s + 16
In this paper, an earlier result has been extended to
calculate all the gains that achieve given gain and phase
margin specifications, and a new computational method
has been proposed to compute maximum achievable gain
and phase margins using proportional control. The main
point of departure was due to a generalization of the
Nyquist stability criterion, where it is shown that the
number of unstable closed-loop system poles for a given
constant gain compensator can be found by examining the
points where the Nyquist plot crosses the real axis of the
complex plane. Future work should focus on extending the
results found in this paper to more complex controller
structures and discrete-time systems.
D( s) = s5 + 9.402s 4 + 5.698s3 + 45.27 s 2 + 38.37s + 25
The set of stabilizing gain compensators are calculated
using Theorem 2.1 as shown in Table III.
TABLE III.
CALCULATION OF r , u AND THE STABILIZING GAINS
i
i
vi∗
xi
ri
9.26861
-0.0225509
∞
0
1
0.64
3.68284
1.03078
0.41405
1.42371
-
∞
-
2
ui
i
-2
2
1
0
0 < K P < 44.3441
44 .3441 < K P < ∞
1
1
− ∞ < K P < −1.5625
2
2
-2
4
− 1.5625 < K P < −0.97
− 0.97 < K P < −0.702
− 0.702 < K P < 0
It is possible to observe through Table III that the
closed-loop system is stable for K P ∈ ( 44 .3441 , ∞ ) .
Hence, this is an open-loop unstable and high-gain closedloop stable system.
CONCLUSIONS
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
N. Munro, M. T. Söylemez and H. Baki, “Computation of
D-Stabilizing Low-Order Compensators,” Control Systems Centre
Report 882, Umist, Manchester, 1999.
N. Munro and M. T. Söylemez, “Fast Calculation Of Stabilizing
PID Controllers For Uncertain Parameter Systems,” IFAC,
Rocond, Prague, Czech Republic, 2000.
M. T. Söylemez, N. Munro and H. Baki, “Fast Calculation Of
Stabilizing PID Controllers,” Automatica,, vol. 31, pp. 121-126,
2003.
N. Bayhan and M. T. Söylemez , “Fast Calculation of All
Stabilizing Gains For Discrete-Time Systems”, Istanbul
University-Journal of Electrical & Electronics Engineering,
vol. 6, No.1, 2006.
A. T. Shenton and Z. Shafiei, “Relative Stability For Control
Systems With Adjustable Parameters,” Journal of Guidance,
Control and Dynamics, vol. 17, No. 2, pp. 304 -310, 1994.
N. Tan, I. Kaya, C. Yeroğlu and D. P. Atherton, “Computation of
Stabilizing PI and PID Controllers Using The Stability Boundary
Locus”, Energy Conversion and Management, in press.
M. T. Ho, A. Datta and S. P. Bhattacharyya , “A New Approach
To Feedback Stabilization”, IEEE, Kobe, Japan, vol 12, pp. 46434648, 1996.