Parametric Synthesis of a Robust Controller on a Base of Interval
Characteristic Polynomial Coefficients
Sergey An. Gayvoronskiy, Tatiana Ezangina and Ivan Khozhaev
Department of Automatics and Computer Systems, National Research Tomsk Polytechnic University,
Lenin Avenue 30, Tomsk, Russia
Keywords: Acceptable Stability Degree, Synthesizing a Controller, Root Quality Indexes, Aperiodic Transient Process,
Interval Characteristic Polynomial, Interval Control System, Autonomous Underwater Vehicle.
Abstract: The paper is dedicated to deriving sufficient conditions, connecting root quality indexes of the control system
with interval coefficients of its characteristic polynomial, on the base of interval expansion of the coefficient
method. With the help of these conditions, a method of synthesizing a controller, providing an aperiodic
transient process and acceptable stability degree, was developed. The method is applied to a problem of
synthesizing a controller of an autonomous underwater vehicle submerging control system with interval
parameters.
1 INTRODUCTION
Modern level of industrial automation development
allows raising a quality of technological objects
control with the help of automated control systems.
Parameters of control object in such systems may
vary slowly or rapidly in some intervals of values
randomly or accordingly to known mathematical
laws. In both cases such parameters can be considered
and interval-uncertain parameters with deterministic
uncertainty.
Nowadays, there are two common approaches to
manipulating objects with parametric uncertainty:
adaptive control, robust control and their
combinations.
Adaptive control is based on a use of parametric
identifiers or ideal model of control process and
requires real-time tuning of adaptation channel [1, 2].
The main restrictions of adaptive approach
application are implementation difficulties and lack
of adaptation channel operating speed.
Robust control has no disadvantages, mentioned
above [3, 4]. It enables the system to operate with
desired control quality despite parametric
uncertainties, unmodeled dynamics, inaccuracy of
parametric identification, external disturbances, etc.
Robust controllers also are known for simple
implementation as their parameters are constant. It
should be noticed, that there is a vast variety of robust
controller parametric synthesis methods, providing a
desired control quality in interval systems [5] – [9].
One of the most important characteristics of a
control systems is a type of a transient process. In
most of control systems an aperiodic transient process
or transient process of similar type is required. In the
figure 1, variants of aperiodic-shaped transient
processes are shown. All of these transient processes
must have no oscillation around the steady-state
value.
Figure 1: Different types of aperiodic step responses.
In order to provide a desired operating speed for a
control system with interval parameters, it is
proposed to use typical linear controllers. To provide
an aperiodic transient process with the help of linear
controller, it is necessary to place a real pole of the
Gayvoronskiy, S., Ezangina, T. and Khozhaev, I.
Parametric Synthesis of a Robust Controller on a Base of Interval Characteristic Polynomial Coefficients.
DOI: 10.5220/0006442204110416
In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 2, pages 411-416
ISBN: Not Available
Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
411

system nearby imaginary axis and to make it dominant
to other poles allocation areas. The problem of
synthesizing an interval control system by a criteria of
the desired system poles allocation is relatively new.
Solutions of this problem can be found in [10, 11].
Another relevant synthesis problem is a
parametric synthesis of robust controllers, based on
interval expansion of the coefficient method [12, 13].
It directly links root quality indexes of a control
system to coefficients of its interval characteristic
polynomial, which are linear functions of controller
parameters.
In the paper an interval expansion of the
coefficient method is applied for synthesizing a
robust PID-control, manipulating an autonomous
underwater vehicle (AUV) submerging. The AUV
operates in conditions of interval parametric
uncertainty: hydrodynamic characteristics, angles of
attack and drift, etc. Considering this, a problem of
synthesizing a PID-controller for an AUV motion
control system consists in calculating its parameters,
providing an aperiodic transient process with desired
setting time.
2 COEFFICIENT CONDITIONS
FOR ROBUST CONTROLLER
PARAMETRIC SYNTHESIS
Let us describe a control object with a transfer
function:
()
00
()/ () [ ] / [ ]
с w
hq
co h q
hq
Ws BsAs bs as
==
==

,
where
aaa
qqq
≤≤
,
bbb
hhh
≤≤
– interval
coefficients of polynomials А(s) и and B(s), s –
Laplace operator. The traffic function of an astatic
controller can be written as follows:
()
,(,)/
c
Wsk Fsk s=

, where
k

– is a vector of
controller parameters. Considering this, an interval
characteristic polynomial (ICP) can be written as
follows:
()
1
1
0
0
, [ ()] [ ()] [ ()]
[()]
n
in n
inn
i
Dsk dks d ks d ks
dk
−
−
=
==+ +
+

 



(1)
[()]
i
dk

– interval coefficients of D(s).
In the considered paper it is proposed to derive
sufficient conditions for controller parameters,
providing necessary setting time of aperiodic
transient process and based on interval coefficient
stability
[]
i
λ
and oscillability
[]
p
δ
indexes. On the
base of interval stability index
[]
i
λ
, sufficient
conditions of providing a accepted robust stability
degree
d
η
can be derived.
Statement 1. In order to provide acceptable
robust stability degree, which is determined by a line
(,0)
d
j
η −
on a complex plane, it is enough to set
such parameters of controller
k

, which comply with
following conditions:
**
0
(, ) , 0.465, 1, 2 ;
(, ) 0, 1, 1;
(, ) 0.
id
gd
d
kin
Fk g n
Fk
<=







∀∈ −
≥∀∈ −
≥



λη λλ
η
η
(2)
where
()
()
()
()
12
112
() ()
(, )
() () 1 () () 2
ii
id
idii id
kd k
k
kkni kkn
d
dd d d i
λη
η
η
−+
+++
=
−−− −−−


 
1
(, ) () ()( 1)
gd g g d
Fk d k d kn g
+
=− −−


η
η
;
2
0012
(, ) () () 2 ()
3
d
dd
Fk dk dk dk=− +


η
ηη
.
Proof. Let us rewrite (2) with interval coefficients
*
0
[(, )] , 1, 2 ;
[(,)]0, 1, 1;
[(, )]0.
id
gd
d
kin
Fk g n
Fk
<∀∈−
≥∀∈−
≥










λη λ
η
η
(3)
where
()
()
()
()
11 12
112
( )] [ ( )]
[(, )]
()] [ ()] 1 ()] [[)
[
[(]2
ii
id
idii Дi
kd k
k
kkni
d
dd d dkkni
λη
ηη
−+
+++
=
−−− −−−


 
()
1
[(,)] ()][ ] 1[()
gd ggd
Fk k kddng
+
=− −−


η
η ;
2
0012
[ (, )] [ ()] [ ()] 2[ ()]
3
d
dd
Fk d k dk d k=− +


η
ηη
.
It is necessary to find such limit values of ICP
coefficients, which will provide the satisfaction of
these inequations with every other set of values.
Considering this and (3), functions
(, )
id
k λη

must
have their maximal values
(, )
id
k λη

, functions
(, )
g
d
Fk
η

and
0
(, )
d
Fk
η

– minimal values
(, )
g
d
Fk
η

and
0
(, )
d
Fk
η

. To calculate these values, ICP
coefficients values must be determined according to
(3).
On the base of interval oscillability indexes
[],
p
δ
a condition of having an aperiodic transient process
in interval system was formulated.
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
412

Statement 2. In order to provide an aperiodic
transient process in an interval system, it is enough to
set such values of controller parameters
k

, which
comply the following condition
()
()
() ()
1
2
1
,4 1,1,
p
pp
p
dd
k
kpn
kk
d
dd
−+
=∀≥≥∈−



δδδ . (4)
Proof.
According to [13], to provide an aperiodic
transient process it is enough to satisfy following
conditions:
11
2
/, 4, 1, 1
ppppp
pddd n
−+
=≥∀∈− δδ
.
Let us apply an interval expansion to these conditions
and determine a highest oscillability index of an
interval control system. It is obvious, that oscillability
index value is maximal if
p
min δ →
(5)
If
2
p
num d=
and
11 pp
den d d
−+
=
, then, to satisfy
(5) following conditions must be satisfied:
minnum → maxden →
(6)
Conditions (6) require following limits of interval
coefficients of ICP:
p
num d=
,
11 pp
den d d
−+
=
. If
(6) are satisfied with these values, then they are
satisfied with any other value from considered
intervals.
Acceptable minimal (robust) merit index
v
D
of
the interval control system can be calculated by the
following formula:
00 0
/
v
Dkba=
(7)
On the base of (2), (4) and (7), conditions of
providing a desired robust stability degree,
oscillability degree and merit index can be written as
follows:
()
*
0
00 0
(, )< , 1, 2 ;
(, ) 0, 1, 1;
(
,4,
/
,)0.
1,
.
1;
a
.
id
dd
v
gd
d
p
kin
Fk g n
Fk
kpn
Dkb





∀



≥≥


=

∈−
≥∀∈ −
≥
∀= −




ληλ
η
η
δδδ
(8)
In order to synthesize a PI- or PID-controller,
controller parameters functions of aperiodic stability
degree
d
η
, oscillability degree
d
δ
, merit index
v
D
and coefficients of systems transfer function can be
derived from the first, fifth and sixth conditions of (6).
Such functions are derived in [13].
Considered research resulted in the method of
synthesizing a PI- and PID-controllers, which
includes following steps:
1.
Determine initial data: controller type,
acceptable robust oscillability degree 4
d
δ ≥ ,
acceptable robust stability degree, acceptable
merit index, limits of interval coefficients of
control object transfer function.
2.
Calculate interval coefficients of an ICP of the
considered closed-loop system.
3.
According to initial data, choose functions of
controller parameters from [13].
4.
Derive the system of inequations (8).
5.
Solve (8) and calculate controller parameters.
3 MATHEMATICAL MODEL OF
THE SUBMERGING
VELOCITY CONTROL
LOOPCONCLUSIONS
Considered AUV is shown in the figure 2.
Figure 2: Hull of the considered AUV.
Submerging control is performed with the help of
two vertical thrusters. The structure of depth control
channel is shown in the figure 3.
Figure 3: Structure of the submerging control channel.
In the fig. 3 following designations are accepted:
0
h – depth setpoint;
h
– actual value of depth;
1
ε – a
difference between a setpoint and an actual value of
the depth; P – P-controller of the outer control loop;
1
u – output of the P-controller;
2
ε – a difference
between a P-controller output signal and a
submerging velocity signal
y
υ
; PID – PID-controller
of an inner control loop;
11
,TK
– transfer coefficient
and time constant of the thruster;
y
T
– thrust of
vertical steering thrusters;
m
– AUV mass;
22
λ –
additional mass of water;
y
c
– hydrodynamic lift
Parametric Synthesis of a Robust Controller on a Base of Interval Characteristic Polynomial Coefficients
413

force coefficient;
k
– interval linearization
coefficient;
y
υ
– submerging velocity;
VS
K –
transfer coefficient of the velocity sensor;
DS
K –
transfer coefficient of the depth sensor.
The problem of synthesizing a P-controller of the
outer control loop will not be considered in the paper.
Let us now consider the mathematical model of the
inner control loop of the submerging control channel,
which controls submerging velocity.
Considering previous designations, transfer
function of the submerging velocity control loop can
be written as follows:
()
() .
;2
;2
;2
2
)(
2213
11222
11
10
01
2
2
3
3
2
1
λ
λ
+⋅=
⋅⋅⋅+⋅⋅++=
⋅⋅⋅+⋅=
⋅⋅⋅=
+⋅+⋅+⋅
+⋅+⋅⋅⋅
=
mTa
KKKcTkma
KKKcka
KKKa
asasasa
KsKsKK
sW
VSDy
VSPy
VSI
IPD
where
DIP
KKK ,,
– proportional, integral and
differential coefficients of the PID-controller.
The model considers an interval uncertainty of the
AUV hydrodynamic parameters and interval
uncertainty of the transfer coefficient of its thrusters.
4 PID-CONTROLLER
PARAMETRIC SYNTHESIS
With the help of developed method, let us synthesize
a PID-controller for AUV submerging control
system. To do this, let determine an acceptable
oscillability degree
4
d
δ ≥ , acceptable degree of
robust aperiodic stability
0.65
d
η =
and acceptable
merit index
47
v
D =
. Then, with the help of
expression
00
/
Iv
KDpb=
let us calculate 20
I
K = .
According to [13], a proportional coefficient can be
calculated as follows
2
(15.1982+4.80 ) 7.866
()
19.25
d
Pd
K
KK
−
=
. Then, a
d
K
function of
d
η
can be derived:
33
1() () 1() ()
2 108 2 108
dddd
d
QDisQDis
K
η
ηη η
=− +− +− +−
,
32
() 4() 271(),
dd d
Dis e Q ηη η =− −
3
2
() ()
e( ) ;
()
3( )
dd
d
d
d
ηη
η
η
η
γυ
ν
ν
=−
3
3
(2 ( )) ( ) ( ) ( )
Q1( ) ,
3( ) ( )
27 ( )
dddd
d
dd
d
γυγηηηη
η
η
ϑ
νν
ν
η
η
=− +
-13.3550475572147 8;3786
ν
=
( ) 126.2578 11.625
dd
γ
η
η =− −
;
( ) 388.559 72.57
dd
η
η
υ =− +
;
( ) 234.833 113.261
dd
ϑ
η
η =− +
.
By
I
K
,
(())
Pdd
KK
η
and ()
dd
K
η
substituting a
into (8), an inequations system can be derived:
()
()
()
1
2
001
2
2
1
2
1
02
, ( ()), () ( ()), ()
(
()()
()()
, ( ()), () ( ()), ()
1) 0;
()()()
2( )()
( ( )), ( )
,(()),(
0
)
;
3
(
IPdd dd Pdd dd
dd
I P dd dd I P dd dd
dd
Pdd dd
IPdd d
g
d
d
d
Id
d
Fa
dn
KKK K KK K
K
KKKK KKKK
K
dKK K
i
Fdd
KKK
dKdK
d
K
ηη ηη
η
ηη ηη
η
ηη
ηη
η
η
η
η
δ
=−
−−−≥
=− +
+≥
=
()
.
)
4
d













≥
By solving the system, proportional and integral
coefficients can be calculated:
( ) 3.402
dd
K
η
=
and
( ( )) 31.859
Pdd
KK
η
= . Step responses of the
synthesized system and allocation of its poles are
shown in the figure 4. The figure 4 (a) shows, that
setting time of the system varies from 0.386 to 2.86
seconds; overshoot varies from 3.61% to 9.6%. The
figure 4 (b) shows, that robust aperiodic stability
degree α= 0.68.
5 CONCLUSIONS
Following results were achieved during the research:
1.
Formulas for robust parametric synthesis by
coefficient quality indexes, considering minimal
accuracy and aperiodic stability of the
synthesized system, were derived.
2.
General expressions for synthesizing a PI- and
PID-controller, providing an aperiodic transient
process in interval systems, were derived.
3.
The method of interval-parametric synthesis of
PI- and PID-controllers of interval systems was
developed.
4. Considered method is tested on a problem of
synthesizing an AUV submerging control
system.
ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics
414

(a)
(b)
Figure 4. (a) Step responses, (b) ICP root allocation areas.
ACKNOWLEDGEMENTS
The reported study is supported by the Ministry of
Education and Science of Russian Federation (project
#2.3649.2017/PCh).
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