Effect of stability indices on robustness and system response in coefficient diagram method

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 04 Issue: 10 | OCT-2015, Available @ http://www.ijret.org 282
EFFECT OF STABILITY INDICES ON ROBUSTNESS AND SYSTEM
RESPONSE IN COEFFICIENT DIAGRAM METHOD
Surekha Bhusnur1
1
Professor, Department of Electrical and Electronics Engineering, BIT, Durg, C.G., India
Abstract
In Control systems, designing a robust controller such that a desired system response is obtained despite plant parameter
variations is ubiquitous problem. In this context, Coefficient diagram method is an effective method and one of the recent design
methods based on the polynomial approach introduced by Shunji Manabe. In CDM, stability indices, stability limits and time
constant are the main design parameters. The stability indices and stability limits are indicative of stability and equivalent time
constant is indicative of speed of system response. A semi-log diagram known as coefficient diagram is the design tool using
which one can analyse the important features of a design such as stability, speed of response and robustness, all in one diagram.
The right choice of the stability indices is of paramount importance in the controller design. This paper deals with the effect of
variation in the stability indices upon the system response and robustness. A type 2 fourth order plant has been considered as an
example to analyse the effects of stability indices. The stability indices are varied one by one relative to the standard Manabe form
and in each case response is observed. The transient response of the system is sensitive to lower order indices. Also, robustness in
the design is analysed by coefficient diagrams of the perturbed plant.
Keywords: Coefficient Diagram Method (CDM), robustness; Stability Indices, Coefficient Diagram
--------------------------------------------------------------------***----------------------------------------------------------------------
1. INTRODUCTION
In CDM [1, 2], firstly, the type and degree of the controller
polynomial and the closed loop transfer function are
partially specified and later other parameters are obtained by
design. The key features of the method are: adaptation of
the polynomial representation for both the plant and the
controller, this avoids pole-zero cancellations; use of two
degree of freedom control structure; almost no overshoot in
the step response of the closed loop system; the
determination of the settling time at the start, good
robustness with respect to parameter changes [3, 5, 7]. A
brief survey on CDM is presented in [9]. Preliminaries of
CDM have been briefly introduced in many contributions
based on CDM [8, 10, 11, 12, 15, 16, 17]. The comparative
study of other control design methods with CDM is found in
[11].
In CDM the design parameters are Stability index γi
Stability limit γi
*
and Equivalent time constant τ. The
stability indices and stability limits are indicative of system
stability; and the equivalent time constant is indicative of
speed of response. In CDM design, the given design
specifications are rewritten in terms of γi and τ. These
parameters are related to controller parameters algebraically
and specify the target characteristic equation that gives
desired performance. A special diagram known as
Coefficient Diagram is used as a design tool using which the
coefficients of the characteristic polynomial can be re-tuned
to get the required system performance. CDM is one such
algebraic method which gives the most proper results with
the easiest procedure.
Rest of the paper is organized as follows:
In section 2, the control structure used in CDM is described
in brief. It includes a pre-filter to adjust steady state gain [1,
3, 6, 7]. In section 3, the stability and instability conditions
used in CDM are discussed and relevant mathematical
relations are listed in terms of stability indices. These
conditions have been presented in terms of design
parameters used in CDM [3]. In section 4, the standard
Manabe form of CDM design and its features are briefed [3,
5]. Coefficient diagram, the design tool of CDM [7] is
explained with an example in Section 5. Also, the design
steps involved in controller design are enlisted in Section 5.
In section 6 the effect of variation in stability indices on the
system response is discussed considering a fourth order
plant transfer function taken from existing literature [4].
Also, simulation results are discussed focusing the role of
stability indices upon the transient response of the closed
loop system and robustness. Section 7 is the concluding part.
2. CONTROL STRUCTURE
Fig-1: Block diagram of CDM control

_______________________________________________________________________________________
The standard block diagram for CDM [2, 7, 8, 12] is shown
in Fig. 1. Here, R(s) is the reference input, Y(s )is the output,
u is the control and Z(s) is the external disturbance signal.
The effect of external disturbance has not been considered in
this paper as the objective is to analyze the utility of stability
indices. Bp(s) and Ap(s) are the numerator and denominator
polynomials of the transfer function of the plant
respectively. F(s) and Bc(s) are the reference numerator and
the feedback numerator polynomials while Ac(s) is the
forward denominator polynomial of the controller transfer
function. Bc(s) and Ac(s) are designed to meet the desired
transient response and the pre-filter F(s) is used to provide
the steady state gain.
The output of this closed loop system is
)(
)(
)()(
)(
)(
)()(
)( sZ
s
cl
A
s
p
Bs
c
A
sR
s
cl
A
sFs
p
B
sY  (1)
Where Acl(s) is the characteristic polynomial and is given by
)()()()()( sBsBsAsAs
cl
A cpcp  (2)
01............ asasa n
n 


n
i
i
i sa
0
(3)
The design parameters of CDM the equivalent time constant
 , the stability index γi, and stability limit 
i are defined as
0
1
a
a
 (4)
)1..(,.........2,1,
11
2


ni
aa
a
ii
i
i



0
11
,),1,.......(2,1,
11


 n
ii
i ni (5)
The equivalent time constant specifies the speed of time
response, the stability indices and limits specify stability,
nature of time response; and the variation of stability indices
due to plant perturbations indicates the robustness [2, 3].
3. STABILITY CONDITIONS
The stability conditions suitable to CDM are based on
Lipatov conditions and are stated in Theorem 4 in [2, 3] as
follows: “The system is stable if all the partial fourth order
polynomials of the characteristic equation are stable with a
margin of 1.12. The system is unstable if some partial third
order polynomial is unstable”.
 Thus the sufficient condition for stability is given as
)2,.......(2,1,12.1  
niii  (6)
 The sufficient condition for instability is given by
( ),11  ii  forsome )2,.......(2,1  ni (7)
4. STANDARD FORM
Shunji Manabe suggested values of stability indices such
that a response with no overshoot is obtained. This new
form is known as “Standard Manabe Form” of CDM and is
given by [1]
2..........,5.2 211    n (8)
The advantages of standard form are enlisted below [7]
 for system type 1, overshoot is almost zero
 the Manabe form has shortest settling time among the
systems with the same equivalent time constant τ and
the value is about 2.5τ ∼ 3τ
 The step responses show almost equal wave forms
irrespective of the order of the characteristic
polynomials
 the lower order poles are aligned almost on a vertical
line
 The values of stability indices in Manabe Form are
easy to remember
5. COEFFICIENT DIAGRAM AND
CONTROLLER DESIGN
Fig-2: Coefficient diagram
The Coefficient Diagram (CD) is a semi-log diagram of the
coefficients ai of the polynomial in logarithmic scale on left
hand side vertical axis and the corresponding powers of s
are placed in decreasing order in linear scale on horizontal
axis, also the stability index, stability limit and the
equivalent time constant are read on right hand side vertical
axis. The equivalent time constant is expressed by a line
connecting 1 to τ. A Coefficient Diagram for the
characteristic polynomial given by

_______________________________________________________________________________________
2.022.25.0)( 2345
 ssssssAcl
(9)
is shown in Fig. 2. Greater the curvature of the coefficient
curve the more the system stability. The smaller the value of
τ, the steeper is the left end part of the coefficient curve and
the faster is the system response. The change in the shape of
coefficient curve due to changes in parameters is an
indicative of robustness and sensitivity to variation in
parameters.
The controller design steps [2, 7] are summarized as
follows:
 Express the plant transfer function as ratio of
numerator and denominator polynomials.
 Translate the given performance specifications into
design specifications for CDM.
 Assume a suitable controller configuration and
express in the polynomial form.
 Solve for the unknown parameters using Diophantine
equation.
 Draw Coefficient Diagram, visualize and make
adjustments in the relevant coefficients, if necessary
to satisfy the performance specifications.
6. EFFECT OF VARIATION IN STABILITY
INDICES
Simulation Example
A fourth order plant transfer function [4] has been
considered in this section to analyse the effect of variation in
stability indices γi.
The real plant is represented as
]4,2[],5.1,5.0[],12,8[
,
))(()(
)(
)( 2



baK
bsass
K
sA
sB
sP
m
m
p
p
(10)
A third order controller is designed using CDM as described
in Section 5. The controller transfer function is assumed as
01
2
2
3
3
01
2
2
3
3
)(
)(
lslslsl
ksksksk
sA
sB
c
c


 (11)
From (2) and (10) we see that the characteristic polynomial
is of order 7. Using (6) and (8) the stability indices are
assumed to have six sets of values as given in Table 1 to
observe the effect in system response.
Table 1. Assumed values of stability indices
Se
t
γ1 γ2 γ2 γ4 γ5 γ6
1 2.5,4,6,
8
2 2 2 2 2
2 2.5 2,4,6,
8
2 2 2 2
3 2.5 2 2,4,6,
8
2 2 2
4 2.5 2 2 2,4,6,
8
2 2
5 2.5 2 2 2 2,4,6,
8
2
6 2.5 2 2 2 2 2,4,6,
8
A settling time specification of ts ≤ 5 sec has been assumed
in the controller design. For every set, four controllers are
designed and system responses are plotted using MATLAB
and SIMULINK environment, the controller parameters are
enlisted in Table 2. In each case the responses obtained are
shown in Fig. 3 to Fig. 8. It is observed that γ1 and γ2 have
greater influence on the system response, whereas increase
in higher order indices increases robustness. Also, the
controller parameters deviate less for higher order indices as
depicted in Table 2.
Further to observe robustness coefficient diagrams are
plotted for the eight plants of the family and are shown in
Fig. 9. There is less deviation in the wave shape of the
coefficient curves for the perturbed family of plants and
indicates robustness. Also, stability is ensured by stability
conditions given in Section 3 and the ratio of stability
indices to stability limits is enlisted for all the eight plants in
Table 3.
Table 2. Controller parameters
l3 l2 l1 l0 k3 k2 k1 k0
S
e
t
1
1.60
00e-
005
5.76
00e-
004
0.01
04
0.0
845
0.0
271
0.1
347
0.2
000
0.1
000
9.53
67e-
007
5.72
20e-
005
0.00
17
0.0
242
0.0
148
0.0
927
0.2
000
0.1
000
8.37
24e-
008
7.70
27e-
006
0.00
04
0.0
078
0.0
079
0.0
643
0.2
000
0.1
000
1.49
01e-
008
1.84
77e-
006
1.14
63e-
004
0.0
034
0.0
048
0.0
490
0.2
000
0.1
000
S
e
t
2
1.60
00e-
005
0.00
06
0.01
04
0.0
845
0.0
271
0.1
34
0.2
000
0.1
000
5.00
00e-
007
3.80
00e-
005
0.00
14
0.0
261
0.0
211
0.1
522
0.2
000
0.1
000
6.58
44e-
008
7.63
79e-
006
4.43
33e-
004
0.0
124
0.0
162
0.1
563
0.2
000
0.1
000
1.56
25e-
008
2.43
75e-
006
1.90
20e-
004
0.0
072
0.0
131
0.1
578
0.2
000
0.1
000
S
e
t
3
1.60
00e-
005
0.00
06
0.01
04
0.0
845
0.0
271
0.1
347
0.2
000
0.1
000
1.00
00e-
006
0.00
01
0.00
29
0.0
522
0.0
423
0.1
443
0.2
000
0.1
000
1.97
53e-
007
2.29
14e-
005
0.00
13
0.0
373
0.0
487
0.1
488
0.2
000
0.1
000
6.25
00e-
008
9.75
00e-
006
7.60
81e-
004
0.0
289
0.0
522
0.1
513
0.2
000
0.1
000
S
e
t
4
1.60
00e-
005
5.76
00e-
004
0.01
04
0.0
845
0.0
271
0.1
347
0.2
000
0.1
000
2.00
00e-
006
0.00
02
0.00
58
0.1
044
0.0
205
0.1
287
0.2
000
0.1
000
5.92
59e-
007
0.00
01
0.00
40
0.1
118
0.0
181
0.1
264
0.2
000
0.1
000

_______________________________________________________________________________________
2.50
00e-
007
3.90
00e-
005
0.00
30
0.1
157
0.0
168
0.1
253
0.2
000
0.1
000
S
e
t
5
1.60
00e-
005
0.00
06
0.01
04
0.0
845
0.0
271
0.1
347
0.2
000
0.1
000
4.00
00e-
006
0.00
03
0.01
16
0.0
808
0.0
282
0.1
358
0.2
000
0.1
000
1.77
78e-
006
0.00
02
0.01
20
0.0
795
0.0
286
0.1
361
0.2
000
0.1
000
1.00
00e-
006
0.00
02
0.01
22
0.0
788
0.0
288
0.1
363
0.2
000
0.1
000
S
e
t
6
1.60
00e-
005
0.00
06
0.01
04
0.0
845
0.0
271
0.1
347
0.2
000
0.1
000
8.00
00e-
006
0.00
06
0.01
03
0.0
848
0.0
270
0.1
346
0.2
000
0.1
000
5.33
33e-
006
0.00
06
0.01
03
0.0
849
0.0
269
0.1
345
0.2
000
0.1
000
4.00
00e-
006
0.00
06
0.01
03
0.0
850
0.0
269
0.1
345
0.2
000
0.1
000
In Fig. 3, nominal plant is considered and the results show
that the transient response deviates from that of Standard
form and settling time increases, transient response is
affected by the lower order indices, γ1 and γ2. Also, it is seen
that increase in higher order indices γ3, γ4, γ5 and γ6 least
affects the transient response and the responses are similar
to that of the response corresponding to the standard
Manabe form.
In Fig. 9, perturbed plant, with the parameters set to extreme
values of the interval box given in (10) is considered and
coefficient diagrams for all the eight plants of the family are
plotted, the results show that the standard Manabe form
based controllers give robust performance despite plant
parameter variations.
Fig-3: Effect of γ1

_______________________________________________________________________________________
Fig-9 Coefficient Diagrams
7. CONCLUSION
In this paper, the behavior of a simple fourth order plant
subjected to variation in stability indices in CDM based
design has been described. Emphasis has been given to have
an insight to the way, transient response and robustness
features are affected. An ideal linear nominal model of the
plant and a CDM based controller has been employed to
generate output corresponding to standard form. The effects
of variation in stability indices and robust stability
considering the perturbed plant have been analyzed. It is
observed that the lower order indices influence the transient
behavior the most and the higher order indices can be
relaxed from the standard form values to improve
robustness. As the role of stability indices has been focused
disturbance rejection and the other properties have been
ignored.
Table 3: Quantitative Robust stability analysis
Sl.
No.
γi/γi
*
>1.12
1 5.0256 2.0709 1.5804 1.8604 5.3009
2 4.6822 2.4210 2.1065 1.9118 4.0353
3 4.7011 2.1640 1.9208 2.1031 4.8960
4 4.4596 2.3526 2.5123 2.3151 4.2600
5 4.0294 1.7202 1.6134 2.2350 6.2207
6 3.9834 2.0239 2.0479 2.2290 4.9168
7 3.9197 1.8274 1.8741 2.4046 5.7346
8 3.8924 2.0333 2.3388 2.5421 4.9825
REFERENCES
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problem by coefficient diagram method”, in
Proceedings of 2nd Asian Control Conference, July
22-25, Seoul, II-135-138.
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Proc. 14th IFAC Symposium on Automatic Control
in Aerospace, Seoul, Korea, August.
[3] S. Manabe, (1999)“Sufficient condition for stability
and instability by Lipatov and its application to the
coefficient diagram method”, in Proceedings of 9th
Workshop on Astrodynamics and Flight Mechanics,
ISAS, Sagamihara, pp.440–449.
[4] J.J.D. Azzo and C. H. Houpis,( 1988) “ Linear
Control System Analysis and Design:Conventional
and Modern”, McGraw-Hill, third edition.
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SSSC'01, Prague.
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[15] P. Pattanavij,(2006) “Simplified design of pi
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special issue on Robust Design, 5(3/4):250-269.

_______________________________________________________________________________________
BIOGRAPHY
Surekha Bhusnur is a Professor in the
Department of Electrical and
Electronics Engineering at Bhilai
Institute of Technology, Durg,
Chhattisgarh Swami Vivekanand
Technical University, India. Her
research interests include robust control
and instrumentation.

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