Possibility of Quenching of Limit Cycles in Multi Variable Nonlinear
Systems with Special Attention to 3X3 Systems
KARTIK CHANDRA PATRAa,*, ASUTOSH PATNAIK
Department of Electrical Engineering,
C. V. Raman Global University, Bhubaneswar, Odisha 752054,
INDIA
aORCiD: 0000-0002-4693-4883
*Corresponding Author
Abstract: - The present work proposes novel methods of Quenching self-sustained oscillations in the event of
the existence of limit cycles (LC) in 3x3 non-linear systems. It explores the possibility of Stabilising/Quenching
the LC by way of signal stabilization using high frequency dither signals both deterministic and random when
3X3 systems exhibit such self-sustained nonlinear oscillations under autonomous state. The present work also
explores the suppression limit cycles of 3X3 systems using state feedback by either arbitrary pole placement or
optimal selection of pole placement. The complexity involved, in implicit non-memory type nonlinearity for
memory type nonlinearities, it is extremely difficult to formulate the problem. Under this circumstance, the
harmonic linearization/harmonic balance reduces the complexity considerably. Furthermore, the method is
made simpler assuming the whole 3X3 system exhibits the LC predominantly at a single frequency. It is
equally a formidable task to make an attempt to suppress the limit cycles for 3X3 systems with memory type
nonlinearity in particular. Backlash is one of the nonlinearities commonly occurring in physical systems that
limit the performance of speed and position control in robotics, the automation industry, and other occasions of
modern applications. The proposed methods are well illustrated through examples and substantiated by digital
simulation (a program developed using MATLAB CODES) and the use of the SIMULINK Toolbox of
MATLAB software.
Key-Words: - Describing Function, Pole Placement, 3x3 nonlinear systems, limit cycles, harmonic
linearization, signal stabilization, Random Input, Gaussian Signals, suppression limit cycles,
Ricatti Equation.
Received: April 22, 2023. Revised: December 23, 2023. Accepted: December 28, 2023. Published: December 31, 2023.
1 Introduction
In the present scenario, nonlinear self-sustained
oscillations or LC are the basic features of
instability. The importance and weight-age of this
problem were felt among the researchers, [1], [2],
[3], [4], [5] in the decades past, where they were
mostly focussing on single input and single output
(SISO) systems. However, for the last five to six
decades, the analysis of 2X2 Multi Input and Multi
Output (MIMO) Nonlinear Systems gained
importance and quite a good amount of literature is
available, [6], [7], [8], [9], [10], [11], [12], [13],
[14], [15], [16], [17], [18], [19], [20], [21], [22],
[23], [24], [25], [26], [27], [28], [29], [30], [31],
[32], [33], [34], [35], [36], [37], [38], [39], [40],
[41], [42], [43], [44], [45], [46], [47], [48],
addressing this area of research. The analysis and
prediction of limit cycles in both SISO and MIMO
systems, a means of increasing the reliability of the
describing function (DF) are well established, [4],
[5], [10], [13], [16], [23], [49], [50] and others used
harmonic linearization/harmonic balance, [29], [33],
[51].
In several cases in physical 2X2 nonlinear
systems limit cycles are observed such as a couple
Core Reactor [12], Pressurised Water Reactor
(PWR) nuclear Reactor System [20], Radar Antenna
pointing system, [11], and Inter Connected power
system [39], which can fit the structure, [1], [24] of
a MIMO two-dimensional nonlinear system.
Backlash is a most remarkable nonlinearity,
commonly existing in physical systems that limit the
performance of speed and positions, this has been
extensively discussed for 2X2 MIMO systems, [7],
[8], [35], [39], [43], [44], [45], [46], [48]. The recent
literature describes some facts of multidisciplinary
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applications where limit cycle oscillations have been
discussed. The authors, [52], presented three
possible scenarios, namely, stable limit cycles and
chaos arise naturally in the flow and thermal
dynamics of the device. The researchers, [53],
formulated/initialized the cell model to the limit
cycle, running one-dimensional simulations of 500
stimuli at a BCL of 300ms. In [54], the dynamic
nature of the nonlinear system switches between a
stable equilibrium point and a stable limit cycle has
been presented. In [55], the stable limit cycle has
been observed in an autocatalytic system through
the characteristics of the Hopf bifurcation. In [56],
the existence of limit cycling oscillations has been
observed in Biological Oscillators having both
positive and negative feedback. The researchers,
[57], have observed in natural systems a closed loop
as in a stable limit cycle by reviewing empirical
dynamic modeling.
Scanty literature is available, which addresses
3X3 nonlinear MIMO systems in the last two
decades only [6], [38], [40], [41], [42].
However, several industrial problems with two
or more higher dimensional configurations, [14],
and prediction of limit cycles via the describing
function method prove to be quite essential, [4], [5],
[10], [13], [14], [16], [23], [49]. Hence the
prediction of limit cycles in three dimensional
nonlinear multivariable systems which can fit the
structure of general 3X3, [6], [27], nonlinear
systems has been addressed.
In the event of the existence of limit cycling
oscillations, the possibility, of quenching the
sustained oscillations using the method of signal
stabilization has been investigated, [5], [30], [31],
[49], [50], in 2X2 nonlinear systems with non-
memory type nonlinear elements and memory type
nonlinear elements in [46] using deterministic
signals. Until, [47] signal stabilization with random
signal for memory type multivariable nonlinear
systems was not available even for 2X2 systems.
The authors in [47], focused on robust and non-
fragile stabilization of nonlinear systems described
by the multivariable Hammerstein model. The
method illustrates a general procedure that addresses
the general multi-variable nonlinear systems. Of
course, the method considers uncertainties and most
importantly control is adopted for stabilization
which fails to project insight into the problem.
However, the present work shows a simple method
to quench the limit cycling oscillations exhibited in
a class of 3x3 nonlinear systems and stabilize the
systems using random signals in particular a
Gaussian Signal. The signal stabilization refers to
the possibility of quenching the self-sustained
oscillations by injecting a suitable high frequency
preferably more than ten times of s (the frequency
of LC) signal at any point of the system, [5]. The
random signal having Gaussian distribution contains
infinite components of frequency. The Gaussian
Signals are passed through a high pass filter so that
the high frequency signal quenches the limit cycles
and stabilizes the system. This has been illustrated
through examples 1 and 2 revisited. This method
projects a clear and lucid insight into the problem.
Prediction and suppression of limit cycles
oscillations in 2X2 memory type nonlinear systems
using arbitrary pole placement has been discussed in
[8], [32], [43], [44], [58] and pole placement by
optimal selection using Riccati equation, [48], [59].
The suppression of limit cycle oscillations using the
state feedback approach has been dealt with to an
extent [8].
The proposed work follows the dynamics of
general 3X3 nonlinear systems shown in Figure 2,
Figure 3, [6], which can also be taken as an
equivalent representation of the general
multivariable system considered in [42].
Having realized the importance of
quenching/suppression of limit cycle oscillations the
proposed work first establishes the exhibitions of
limit cycles in 3X3 nonlinear systems following a
similar procedure as depicted/illustrated, [6].
2 Prediction of Limit Cycles in a
General 3X3 Nonlinear Systems
To avoid the complexity involved in the structure,
[6], [49], a graphical method is opted for
investigation of the existence of limit cycling
oscillations in 3X3 nonlinear systems.
2.1 Graphical Method
A graphical method has been adapted, [6], for the
prediction of limit cycling oscillations in a 3x3
nonlinear system. The steps as depicted in [6], have
been followed for the establishment of existing limit
cycling oscillations in a 3X3 nonlinear system
which has been; illustrated through numerical
examples and validated by (i) digital simulation, (ii)
by use of SIMULINK Toolbox of MATLAB
software.
Consider a system of Figure 1, a class of 3X3
nonlinear systems for simplicity it is assumed that
the whole 3X3 system exhibits the LC
predominantly of a single frequency sinusoid and
harmonic linearization/harmonic balance leading to
the use of describing function methods have been
opted.
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The normalized phase diagrams, [28], [46], are
drawn with three combinations such as:
Combination 1: For subsystems S1 & S2: C1 (+ve),
C2 (-ve) and C3 (+ve)
Combination 2: For subsystems S3 & S2: C2 (+ve),
C3 (-ve) and C1 (+ve)
Combination 3: For subsystems S1 & S3: C3 (+ve),
C1 (-ve) and C2 (+ve)
Example 1 and 2 are used for illustration of
procedures of Normalized phase diagrams.
The linear elements are represented by
; ;
= and Nonlinear elements are taken, Ideal
relays as shown in Figure 2(a) and ideal saturations
as shown in Figure 2(b) for Example 1 and Example
2 respectively
Fig. 1: An equivalent 3X3 multivariable nonlinear
system of Figure 2 in [6]
Fig. 2(a): All Ideal Relays
Fig. 2(b): All Ideal Saturation type nonlinear
elements (with slopes k1, k2, k3)
In examples 1 and 2 non-memory type nonlinear
elements are used. Assuming harmonic linearization
these nonlinear elements can be equivalently
represented by their describing functions, [28],
which are real functions in these two examples and
do not contribute any phase angles to the system.
Hence the phase angles of the system are due to
linear functions) G1(s), G2(s), G3(s) which are
complex functions of complex variable s, the
Laplace operator. It may be noted that for frequency
response, input is sinusoidal and outputs are steady
state values considered, so that s (Laplace
Operator) is replaced by j
, [6].
X1, X2 & X3 are the amplitudes of respective
sinusoidal inputs to the nonlinear elements. C1, C2 &
C3 are the amplitudes of sinusoidal output of
subsystems S1, S2 & S3 respectively. G1, G2 & G3
are the magnitudes/absolute values of linear
elements represented by their transfer functions of
subsystems S1, S2 & S3 respectively. N1, N2 & N3
are the magnitudes/absolute values of linear
elements represented by their describing functions
of subsystems S1, S2 & S3 respectively.
θL1 = Arg. ( (j ω)) = -90 - (ω):
θL2 = Arg. ( (j ω)) = -90 - ( ):
θL3 = Arg. ( (j ω)) = -90 - ( ):
N2=(11-3 ) ±
(1)
N1= N2 + (2)
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= (3)
= = (4)
For a fixed value of ω the Combinations of
Subsystems 1, 2, and 3, Normalised Phase Diagrams
are shown in Figure 3(a), (b), and (c) respectively.
However, any one of these combinations can be
used for the determination of limit cycling
conditions and the related quantities of interest.
Fig. 3(a): Normalised Phase Diagram with C1, C2 &
C3 for the combination 1, where C1 (+ve), C2 (-ve)
and C3 (+ve)
Fig. 3(b): Normalised Phase Diagram with C1, C2 &
C3 for the combination 2, where C2 (+ve), C3 (-ve)
and C1 (+ve)
Fig. 3(c): Normalised Phase Diagram with C1, C2 &
C3 for the combination 3, C3 (+ve), C1 (-ve) and C2
(+ve)
With reference to a normalized phase diagram
[28], [46], the phase representing X2 would lie along
a straight line drawn at an angle θL2 with the phase
C2 (C2 = - R1). The intersections of this straight line
with the circle drawn concerning θL1 would
represent possible self-oscillations [28]. The concept
has been extended for 3 x 3 as:
(i) Consider Figure 3(a) the phase representing
X2 and X3 would lie along straight lines drawn at
angles θL2 and ϴL3 with the phase C2 (C2 = - R1) and
C3 (C3=R1) respectively. The intersections of these
straight lines with the circle drawn concerning θL1
would represent possible self-oscillations.
(ii) Consider Figure 3(b), the phase representing
X3 and X1 would lie along straight lines drawn at
angles
θL3 and θL1 with the phase C3 (C3= - R2) and C1
(C1=R2) respectively. The intersections of these
straight lines with the circle drawn concerning θL2
would represent possible self-oscillations.
(iii) Consider Figure 3(c) the phase X1 and X2
would lie along straight lines drawn at angles θL1
and θL2 with phase C1(C1=-R3) and C2(C2 = R3)
respectively. The intersections of these straight lines
with the circle drawn concerning θL3 would
represent possible self-oscillations.
Table 1 shows the θL1, θL2, θL3, r (radius), and
the intersection points of the straight lines and circle
for combination 1 corresponding to example 1. It
may be noted that Table 1 Contains obtained
from Eqn.3 and Eqn.4 are matched at a limit cycling
frequency.
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Table 1. θL1, θL2,θL3, r (radius), and the intersection points of the straight lines and circles for combination 1
corresponding to Example 1
θL1
θL2
θL3
r
X1/X2
from eqn.
3
X1/X2
from eqn. 4
Normalized Phase Diagrams
Remark
0.600
-151.93
-98.531
-106.7
-0.55257
-
-
No intersection
of straight lines
and circle
0.650
-156.05
-99.23
-108
0.58256
-
-
No intersection
of straight lines
and circle
0.700
-159.98
-99.926
-109.29
-2.128
-
-
No intersection
of straight lines
and circle
0.701
-160.06
-99.94
-109.32
-3.1323
1.0
1.02
(matched)
The intersection
of st. lines &
circle found:
Confirms the
occurrence of
limit cycles
=0.701,
C1 = OD2 = 6
C2 = 1
C3 = 1
X1=BD2=6.08
X2=AD2=6.08
X3=B’D2= 6.32
0.750
-163.74
-100.62
-110.56
-1.3583
-
-
No intersection
of straight lines
and circle
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Fig. 4(a): Equivalent Canonical form of Figure 1 for
Ex1 & 2
2.2 Digital Simulation
Example 1 and Example 2 are revisited.
A program has been developed [6] with the use of
MATLAB code for digital simulation.
The equivalent canonical form of Figure 1 for
example 1 and 2 is shown in Figure 4(a) and digital
representation is shown in Figure 4(b).
Numerical results obtained from different
methods are compared in Table 2(a) and Table 2(b)
for example 1 and 2 respectively.
The results/images for example 1 and 2 are
shown in Figure 5 and Figure 6 respectively. These
are also compared with those of obtained using the
SIMULINK Toolbox of MATLAB software.
Table 2(a). Results obtained using different methods
corresponding to Ideal Relay Example-1
Sl.
No
Methods
C1
C2
C3
X1
X2
X3
1
Graphical
6.0
1.0
1.0
6.08
6.08
6.32
0.701
2
Digital
Simulation
4.83
0.74
0.95
4.72
4.91
5.23
0.70
3
Using
SIMULINK
TOOL BOX
OF
MATLAB
5.95
1.01
0.96
4.84
5.12
5.62
0.70
Table 2(b). Results obtained using different methods
corresponding to Example2: (Saturation)
Fig. 4(b): The Digital representation of Figure 1 for
Ex.1 & 2
Sl.
No
Methods
C1
C2
C3
X1
X2
X3
1
Digital
Simulation
4.345
1.06
1.06
4.464
4.581
4.762
0.628
2
Use of
SIMULINK
TOOL
BOX OF
MATLAB
4.30
1.05
1.05
4.425
4.534
4.74
0.6283
S1
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Fig. 5: Results/Images from digital simulation and SIMULINK for C1, C2, C3, X1, X2 and X3 of Example 1
(relay type nonlinearities)
Fig. 6: Results/Images from digital simulation and SIMULINK for C1, C2, C3, X1, X2, and X3 of Example 2
(saturation type nonlinearities)
30 40 50 60
-6
-4
-2
0
2
4
6
8
C1 = 5.95, w = 0.69
Result / Image from Simulink Application
Result / Image from Digital Simulation
C1 = 4.51, w = 0.70
(a)
30 40 50 60
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
(c)
Result / Image from Digital Simulation
C2 = 0.74, w = 0.70
Result / Image from Simulink Application
C2 = 1.01, w = 0.69
30 40 50 60
-1.0
-0.5
0.0
0.5
1.0
(e)
Result / Image from Digital Simulation
C3 = 0.94, w = 0.70
Result / Image from Simulink Application
C3 = 0.95, w = 0.69
30 40 50 60
-6
-4
-2
0
2
4
6
(b)
Result / Image from Digital Simulation
X1 = 4.84, w = 0.70
Result / Image from Simulink Application
X1 = 4.72, w = 0.70
30 40 50 60
-6
-4
-2
0
2
4
6
(d)
Result / Image from Digital Simulation
X2 = 5.12, w = 0.70
Result / Image from Simulink Application
X2 = 4.91, w = 0.70
30 40 50 60
-6
-4
-2
0
2
4
6
(f)
Result / Image from Digital Simulation
X3 = 5.24, w = 0.70
Result / Image from Simulink Application
X3 = 5.62, w = 0.70
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3 Signal Stabilization in a 3X3
Nonlinear System
3.1 Using Deterministic Signal
In the case of 2x2 nonlinear systems the concept of
Signal Stabilisation is as under:
If a 2x2 Nonlinear system exhibits limit cycles
(L.C.), in the autonomous state, the possibility of
quenching the limit cycling oscillations by injecting
a suitable high frequency signal, preferably at least
10 times of limit cycling frequency [5], [30], [46],
[49], [50]. The process is also termed forced
oscillations which have also been extensively
discussed by several researchers [30], [33]. Under
the forced oscillation process the phenomena of
Synchronization and De-synchronization have been
addressed thoroughly in [46] for 2x2 nonlinear
systems. Such phenomena have been realized
/observed by injecting a sinusoidal input B1 sin f t
/B2 sin f t at any one or both input points of two
subsystems S1 and S2 respectively. When the
amplitude B1 of forcing signal B1 sin f t gradually
increased keeping the amplitude B2 of forcing signal
B2 sin f t fixed, the system would continue to
exhibit a limit cycle. The variables at various points
in the system would be composed of signals of the
input frequency (f), the frequency of self-
oscillations (s), and the combination of
frequencies, k1f ± k2s where k1, k2 assume various
integer values. At this condition the system exhibits
complex oscillations. In the process of gradual
increase of B1, the frequency of oscillations s
would also gradually change and for a certain value
of B1, the synchronization would occur, the self-
oscillation would be quenched and the system would
exhibit forced-oscillation at frequency f. On the
other hand, if subsequently the magnitude B1 is
reduced gradually, a point would arrive at which the
self-oscillations would reappear which is termed a
de-synchronization phenomenon. It may be noted
that the synchronization value of B1 is larger than
the De-synchronization value of B1 [46].
Similar facts have been observed in 3x3
nonlinear systems. The forced oscillation can be
realized by feeding deterministic or random signals
of high frequency, at least greater than 10 times the
limit cycling frequency at any one / all input points
of subsystems S1, S2 and S3.
If the amplitude B of the high frequency signal
is gradually increased, the system would exhibit
complex oscillations before the synchronization
takes place. On the reverse operation, if the
amplitude B is gradually reduced at a certain value
of B the self-oscillations i.e. the Limit cycle would
reappear and the system would reappear and the
system would exhibit complex oscillations again
which can be called de-synchronisation. The
phenomena of synchronization and de-
synchronization can be observed/identified
analytically using the Incremental Input Describing
function (IDF).
However, the forced oscillation can also be
analyzed using the Equivalent Gain/Dual input
Describing Function (DIDF), [33], [49], in the case
of a deterministic forcing signal in particular with a
sinusoidal signal. Similarly, Equivalent Gain
(Random input Describing Function-RIDF), [60],
[61], [62] in case of random forcing signal, in
particular with Gaussian Signal.
The complexity arises in the structures, [6],
particularly for the implicit non-memory type or
memory type nonlinearities, it may be extremely
difficult to formulate and simplify the expressions
even using the harmonic linearization method [48].
Hence an attempt has been made to develop a
graphical technique using the harmonic linearization
/ harmonic balance method for prediction of limit
cycles in 3x3 nonlinear systems by extension of the
procedure as presented in [28]. The method uses the
simultaneous intersection of two straight lines and
one circle in three combinations.
The analytical/mathematical observation of
synchronization and de-synchronization of complex
oscillation in the process of signal stabilization
would be quite involved and time-consuming.
Hence the digital simulation (Using our developed
program) opted for the demonstration of signal
stabilization with deterministic/random (Gaussian)
signals which have been validated through the use
of the SIMULINK Toolbox of MATLAB Software.
It is established that the system shown in Figure
1 with Numerical Example 1 and Example 2
exhibits a limit cycle in the autonomous state. The
possibility of quenching the self-sustained
oscillations has been explored by injecting suitable
high frequency preferably more than ten times
of s signals, [5], at any one/all three input points
(U1, U2, U3) .
However taking the second option i.e. all three
inputs are the same as B sinf t at 3 input points U1,
U2, & U3, shown in Figure 7. Amplitude B is
gradually increased, the frequency of self-
oscillation, s would gradually change, the system
will synchronize to forcing frequency i.e. the self-
oscillation would be quenched and the system would
exhibit forced oscillations at frequency f.
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Fig. 7(a): Equivalent System of Figure 1 for forced
oscillations (Signal Stabilization) with deterministic
signal for Example 1.
Fig 7(b): Equivalent System of Figure 1 for forced
oscillations (Signal Stabilization) with deterministic
signal for Example 2
The results/ images from digital simulation for
signal stabilization with deterministic signals in
Examples 1&2 are shown in Figure 8 & Figure 9
respectively.
The steady state values are represented as C1ss,
C2ss, C3ss and X1ss, X2ss, and X3ss with their
frequencies,, which are almost equal to f.
Fig. 8: Forced Oscillations by Signal Stabilization
with deterministic signal for Example 1
Forcing Signal U = 5sinf t (f = 7.5 rad / sec)
3.2 Using Gaussian Signal
The concept of signal stabilization with random
inputs for SISO nonlinear systems was discussed,
[60], [61], [62]. Current research gives importance
to robust design and analysis which considers
uncertainty/ randomness. Until [47], signal
stabilization with random signal for multivariable
nonlinear systems was not available even for 2X2
systems.
The authors in [47], focused on robust and non-
fragile stabilization of nonlinear systems described
by the multivariable Hammerstein model. The
method illustrates a general procedure that addresses
the general multi variable nonlinear systems. Of
course, the method considers uncertainties and most
importantly control is adopted for stabilization
which fails to project insight into the problem.
However, the present work shows a simple method
to quench the limit cycling oscillations exhibited in
a class of 3x3 nonlinear systems and stabilize the
systems using random signals in particular a
Gaussian Signal. The signal stabilization refers to
the possibility of quenching the self-sustained
oscillations by injecting a suitable high frequency
preferably more than ten times of s (the frequency
of LC) signal at any point of the system, [5]. The
random signal having Gaussian distribution contains
infinite components of frequency. The Gaussian
Signals are passed through a high pass filter so that
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the high frequency signal quenches the limit cycles
and stabilizes the system. This has been illustrated
through examples 1 and 2 revisited. This method
projects a clear and lucid insight into the problem.
Consider the Examples 1 and 2 again. The
system is exhibiting LC under an autonomous state,
A Gaussian signal with specified mean and variance
is injected at U1, U2 & U3 of subsystems for
stabilizing the system / quenching the self-sustained
oscillations. At a suitable value of mean () and
variance (), the self-sustained oscillations vanish /
the system is synchronised to high frequency forcing
input.
The results/ images are shown in Figure 10 and
Figure 11 for Examples 1 & 2 respectively, which
are obtained from digital simulation by signal
stabilization with Gaussian signals in examples 1
and 2 replacing B sinf t using a suitable random
signals in Figure 7(a) & Figure 7(b).
Fig. 9: Forced Oscillations by Signal Stabilization
with deterministic signal for Example 2
Forcing signal U = 5 sin ft, (f = 8 rad/sec)
Fig. 10: Forced Oscillations by signal stabilization
with Gaussian Signal of mean 50 and variance 0.05
for Example 1
Fig. 11: Forced Oscillations by signal stabilization
with Gaussian Signal of mean 300 and variance
0.025 for Example 2
Example 3: Consider a system where linear
elements are represented by their transfer functions
G(s) and the nonlinear elements are dead zone with
saturation whose input output characteristics is
shown in Figure 12 and represented by their
describing functions N: where ;
; =
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Fig. 12: The nonlinear elements used in the system
of example 3 (the same nonlinear elements are used
for three sub systems, S1, S2, and S3.)
The simulation diagram of Ex 3 is shown in
Figure 13(a).
Fig. 13(a): Equivalent System of Figure 1 for forced
oscillations (Signal Stabilization) with
random/Gaussian signal for Example 3
The result / images obtained from digital
simulation, using Gaussian signals are shown in
Figure 13 (a).
Figure 13 (b) shows the limit cycling oscillation
in the absence of the forcing signal and Figure 13(c)
shows the forced oscillation with a Gaussian signal
of mean 70 and variance 0.025 for Example 3.
Fig. 13(b): The limit cycle oscillations in the
absence of forcing signals for Example 3. (s=0.647
rad/sec)
Fig. 13(c): Forced oscillations with Gaussian signal
of mean 70 and variance 0.025 for the Example 3
4 Suppression of Limit Cycle in 3x3
Nonlinear System using Pole
Placement Technique
Limit cycles or self-sustained oscillations of a 2X2
system can be suppressed by pole placement
technique, [8]. The problem of placing the closed
loop poles or Eigen values of the closed loop
systems at the desired location using state feedback
through an appropriate state feedback gain matrix K
[k1, k2, k3]. Necessary and sufficient condition for
arbitrary pole placement is that the system be
completely state controllable, [58]. This can also be
Y
-0.1
K
-M=-0.1
M
S
0.1
1
K1 = 1
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done by optimal selection of feedback gain matrix K
using Riccati Equation, [48], [59].
4.1 Suppression of Limit Cycles in 3X3
Nonlinear System using Arbitrary Pole
Placement by State Feedback
Pole placement technique by state feedback is done
by determining the Eigen values or poles of the
system. These Eigen values cause the limit cycles in
the system, and as the complete removal of these
self-oscillations may not be possible, the location of
the poles must be changed from its original position
to bring about suppression of the limit cycle. The
most general multivariable nonlinear system, [6], is
shown in Figure 14(a). For the existence of limit
cycles, an autonomous system (input U=0) Figure
12(a) can be represented in a simplified form as
shown in Figure 14(b). Making use of the first
harmonic linearization of the nonlinear elements, the
matrix equation for the system of Figure 14(b) can
be expressed as
X = -HC, where C = GN(x) X. Hence,
X = -HGN(x) = AX (5)
Where, A = -HGN(x)
Fig. 14(a): Block diagram representation of a most
general nonlinear multivariable system
Fig. 14(b): Equivalent of the system of Figure 12 (a)
with input U= 0
Realizing Eqn. (5) As a transformation of the
vector X onto itself, it is noted that for a limit cycle
to exist the following two conditions should be
satisfied, [6]. For every non-trivial solution of X, the
matrix A must have an Eigen value λ equal to unity,
and
(i) The Eigen vector of “A” corresponding to
this unity Eigen value must be
coincident with X.
4.1.1 Arbitrary Pole Placement for Suppression
of Limit Cycles in Example 1 with All Ideal
Relays
To suppress the limit cycles, arbitrary pole
placements may be possible if the system is
completely state controllable [58].
The controllability matrix
(6)
Where,
;
;
From Table 1 for Example 1,
X1= 6.08, X2= 6.08, X3
=6.32
N1(X1) = = = 0.419; N2(X2) = =
= 0.314, N3(X3) = = = 0.202
= = 1.913
= = = 0.351
= 0.673
On substitution of the numerical values:
= -0.419 X 1.913 = - 0.802,
= - 0.314 X 0.351 = -0.110,
= -0.202 X 0.673 = -0.136
; AB =
= ;
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B = =
Hence S = = 0.0215≠0
(The system is completely state controllable)
Hence arbitrary pole placement is possible [58]
= (7)
The system under autonomous state is represented
as shown in Figure 15.
Fig. 15: A system with state feedback
Consider Figure 15:
The control law u = -KX (8)
Where K= [ ] is the feedback matrix.
Replacing K in Eqn. (7) by Eqn. (8), we get:
= (A-BK) X (9)
Substituting the values of A, B and K, we get: The
Characteristic Equation as:
Hence,
=
=
={( ( +
( }
( -
( + )}
( -
( + )}
={ + - + + -
+ + + +
-
+ + -
+ + +{
+ -
+ -
-
+ ( - + -
)
= ( + + + )+ ( +
+ + + -
+ + + -
)+ -
+ + -
+ - -
- + -
= + ( + + + )+ (2 +2
+2 + + +
+)+(4 +
+=0 (10)
(Ch. Equation)
On substitution of the values of , , , and
, in Eqn. (10), we get,
+ (0.136+0.11+0.802+ + 0.177+0.218+0.0
30+ + +
+ }+(0.048+0.522+ =0 Or
+ (1.048+ + 0.136
(0.57 x0.03)=0 (11)
If the poles are selected arbitrarily at
respectively, the
characteristic equation becomes:
( ( ( = +4 +5 +2=0
(12)
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Comparing Eq. (12) with Eq. (11), and equating the
coefficients of like powers of we get:
4 = 1.048 (13)
2 = (0.57 x 0.03), whence (14)
5 = (0.425 x 0.136 + x 0.136 x 0.91)
5= (0.425 0.136 + x 0.136
x 0.91) or
5=(0.425 + x 0.136 ), whence
(15)
Hence K = =
(16)
From Eqn. (9), (A – BK) = A1, with shifted poles for
Example 1. Or
A1 = =
(17)
The images/response in the
autonomous state obtained from digital simulation
for A1 of Example 1, are shown in Figure 16.
Fig. 16: Suppression of Limit Cycles by State
Feedback with arbitrarily selection of feedback gain
matrix for Example 1
4.1.2 Optimal Selection of Feedback gain Matrix
using Riccati Equation for Example 1
The Riccati Equation is A′P+PA- B′P+Q=0
(18)
And K = Feedback gain matrix = B′P (19)
Assuming R = 1, B= , Q =
Let P= , considering P to be
symmetric matrix: = ,
Hence P =
P =
=
(20)
PA=
=
(21)
B′P= ,
=
=
=
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=
(22)
On substitution of numerical values, Eqn. 20 can be
written as:
(23)
On substitution of numerical values, Eqn. 21 can be
written as:
(24)
On substitution of these values of Eqns. (22), (23),
(24) and the assumed value of Q in Riccati Eqn. 18
yields:
(-1.604 -1.604 )+1-p213 = 0
(25)
(-0.912 +0.802 -0.802 -0.11 +0.11 -
p13 p23=0 (26)
(-0.938 +0.802 p23-0.802 p33+0.136 p11-0.136 p12-
p13 p33=0 (27)
(-0.22 p12-0.22 p22+0.22 p23- p223 = 0 (28)
(-0.11 p13-0.246 p23+0.11 p33+0.136 p12-0.136 p22-
p23 p33=0 (29)
(0.272 p13-0.272 p23-0.272 p33- p33 p23) = 0 (30)
Further, subtracting Eqn. (29) from Eqn. (30), we
get,
0.382 p13 – 0.026 p23 – 0.382 p33 – 0.136 +
0.136 p22 (31)
The solution of these simultaneous Eqns.
(26),(27),(28),(29),(30) & (31) yields :
= -116.68, = -110.48, =6.58, = -
93.24, p23 = -6.58, p33 = 0
From Eqn. (19), K = B′P = 1
Or =
Or =
= ,
Whence, = 6.58, = -6.58 and = 0 (32)
Hence, A – BK =
A2 =
On substitution of numerical values for Example 1,
A2 becomes:
A2 =
= (33)
The images/ responses C = and = in the
autonomous state, obtained from digital simulation
for Example 1, are shown in Figure 17.
Fig. 17: Suppression of Limit Cycles by State
Feedback with optimal selection of feedback gain
matrix for Example 1
5 Conclusion
In today’s scenario, nonlinear self-sustained
oscillations or Limit Cycles are the basic feature of
instability. The existence /exhibition of such
phenomena limit the performance of most of the
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physical systems such as the speed and position
control in robotics, the automation industry in
particular. Quenching / complete extinction of such
LC has been severe headache among researchers for
several decades. There are some methods, seen in
the available literatures which suggest the solution
to this problem occurring in SISO or 2x2 systems.
However, the present work explores the solution for
3x3 systems in the event of the existence of an LC
problem and establishes the result graphically &
validated by digital simulation. The novelty of the
work claims in (i) Quenching of LC exhibited in
nonlinear systems by Signal Stabilization with
deterministic as well as random (Gaussian) signals,
(ii) Suppression of limit cycles in 3x3 nonlinear
systems by Pole Placement using State feedback
with arbitrary selection as well as optimal selection
of feedback gain matrix K.
More importantly, the poles of such 3x3 systems
are shifted or placed suitably by State feedback so
that the system does not exhibit limit cycles. This
pole placement is done either by arbitrary selection
satisfying the complete state controllability
condition or by optimal selection of feedback gain
matrix K using the Riccati equation which has not
been attempted earlier.
The present work has the brighter future scope of
adopting techniques like signal stabilization [46],
[47] and suppression of limit cycles [48], in the
event of the existence of limit cycling oscillations
for 3x3 or higher dimensional systems through an
exhaustive analysis.
Analytical/Mathematical procedures may also be
developed for signal stabilization using both
deterministic and random signals applying DIDF
and RIDF respectively.
Backlash is one of the nonlinearities commonly
occurring in physical systems which are an inherent
characteristic of Governor, more popularly used for
load frequency control (LFC) in power systems. The
LFC shows poor performance due to the backlash
characteristic of the governor. Similarly, the
backlash characteristic limits the performance of
speed and position control in the robotics, and
automation industry. The poor performance of LFC,
speed, and position control in robotics and in
automation industries is happening since these
systems exhibit limit cycles due to their backlash
type of nonlinear characteristics. The proposed
method of suppression of L.C. can be extended and
developed for backlash type nonlinearity in 3x3
systems and used to eliminate the limit cycle to
mitigate such problems.
The phenomena of synchronization and de-
synchronization can be observed / identified
analytically using the Incremental Input Describing
function (IDF).
Acknowledgments:
The Authors wish to thank the C.V Raman Global
University, Bhubaneswar – 752054, Odisha, India
for providing the computer facilities for carrying out
the research and preparation of this paper.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Kartik Chandra Patra has formulated the problem,
methodology of analysis adopted and algorithm of
computation presented.
- Asutosh Patnaik has made the validation of the
results using the geometric tools and SIMULINK
toolbox of MATLAB software.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The C.V. Raman Global University has provided all
computer facilities with relevant software for the
research work and also for the preparation of the
paper.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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