Investigation of the Existence of Limit Cycles in Multi Variable
Nonlinear Systems with Special Attention to 3X3 Systems
KARTIK CHANDRA PATRA*, ASUTOSH PATNAIK
Department of Electrical Engineering,
C. V. Raman Global University, Bhubaneswar, Odisha 752054,
INDIA
*Corresponding Author: ORCID ID: 0000-0002-4693-4883
Abstract: - The proposed work addresses the dynamics of a general system and explores the existence of limit
cycles (LC) in multi-variable Non-linear systems with special attention to 3x3 nonlinear systems. It presents a
simple, systematic analytical procedure as well as a graphical technique that uses geometric tools and computer
graphics for the prediction of limit cycling oscillations in three-dimensional systems having both explicit and
implicit nonlinear functions. The developed graphical method uses the harmonic balance/harmonic linearization
for simplicity of discussion which provides a clear and lucid understanding of the problem and considers all
constraints, especially the simultaneous intersection of two straight lines & one circle for determination of limit
cycling conditions. The method of analysis is made simpler by assuming the whole system exhibits the limit
cycling oscillations predominantly at a single frequency. The discussions made either analytically/graphically
are substantiated by digital simulation by a developed program as well as by the use of the SIMULINK
Toolbox of MATLAB Software.
Key-Words: - Describing function, 3 x 3 nonlinear systems, limit cycles, harmonic linearization.
Received: September 12, 2022. Revised: May 26, 2023. Accepted: June 27, 2023. Published: July 19, 2023.
1 Introduction
Limit cycles/nonlinear oscillations are the modus
operandi of several physical systems and are often
the basic feature of instability. Therefore, our
objective is to predict the limit cycle, [1]. The
importance of this problem was well felt among the
researchers, [2], [3], [4], [5], [6], where the people
were mostly focusing on single input single output
systems. However, for the last five decades, the
analysis of 2x2 nonlinear multivariable systems
gained importance, particularly for the prediction of
limit cycles and quite a good amount of literature is
available addressing this area of research, [1], [2],
[3], [4], [5], [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], [37], [38], [39], [40], [41], [42],
[43], [44], [45], [46], [47], [48], [49], [50], [51].
Prediction of limit cycle and its analysis in both
single as well as two dimensional nonlinear systems,
a means of increasing the reliability of the
describing function (DF) are well realised, [5], [6],
[8], [11], [14], [22], and many others based on
harmonic linearization / harmonic balance, [11],
[28], [32], [36].
If there exit limit cycle oscillations, the
possibility of quenching the sustained oscillations
using the method of signal stabilization has been
investigated, [6], [16], [23], [29], [30], in 2x2
nonlinear systems with non-memory type nonlinear
elements. Prediction of limit cycling oscillations and
its quenching using signal stabilization technique by
a deterministic signal in memory type nonlinear
elements for 2x2 multivariable systems has been
discussed in [49], and the same has been addressed
with Gaussian signals, [50].
Prediction and Suppression of limit cycle
oscillations in 2x2 memory type nonlinear systems
using arbitrary pole placement has also been
discussed in some literature, [33], [45], [46], [48],
and pole placement by optimal selection using
Riccati Equation, [35], [51].
It has been realized that the exhibition of limit
cycles in two-dimensional multivariable nonlinear
systems on several occasions like Couple Core
Reactor, [10], Pressurised Water Reactor (PWR)
nuclear Reactor System, [19], Radar Antenna
Pointing System, [9], and Interconnected Power
System, [41], which can fit the structure, [24], [27],
of a general two-dimensional nonlinear system.
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Backlash is one of the most important
nonlinearities commonly occurring in physical
systems that limit the performance of speed and
positions, which has been extensively discussed for
two-dimensional multivariable systems, [34], [37],
[41], [45], [46], [47], [48], [49], [50], [51].
The recent literature depicts some instances of
multidisciplinary applications where limit cycle
oscillations have been discussed. The researchers,
[52], discussed three possible scenarios, namely,
stable, limit cycles and chaos arise naturally in the
flow and thermal dynamics of the device. The
authors, [53], formulated/initialized the cell model
to the limit cycle, running one-dimensional (OD)
simulations of 500 stimuli at a BCL of 300ms.In
[54], the dynamic behaviour of the nonlinear system
switches between a stable equilibrium point and a
stable limit cycle has been discussed. In [55], the
stable limit cycle has been observed in autocatalytic
systems through the characteristics of the Hopf
bifurcation. In, [56], the exhibition of limit cycling
oscillations has been observed in Biological
Oscillators having both positive and negative
feedbacks. The authors, [57], have observed in
natural systems a closed loop as in a stable limit
cycle through reviewing empirical dynamic
modelling.
Scanty literature is available addressing 3 x 3
nonlinear multivariable systems in the last decade
only, [27], [40], [42], [43], [44]. The researchers,
[27], have speculated for investigation of limit
cycles in 3x3 systems. It has been observed that the
exhibition of limit cycles in three-dimensional
multivariable nonlinear systems on several
occasions like a boiler turbine unit is a 3 x 3
multivariable process showing nonlinear dynamics
under a wide range of operating conditions, [40].
Most of the chemical process is multivariable,
considering a 3x3 model of nonlinear chemical
process, the limit cycling conditions have been
reported, [44]. Similarly the literature, [42], [43],
depicts limit cycles in 3x3 nonlinear models.
However detailed analysis, well-established
conclusions, and state forward techniques are still
lacking in the available literature on such 3 x 3
nonlinear multivariable systems.
However a number of industrial problems with
two or more higher dimensional configurations,
[12], and the prediction of limit cycles via the
describing function method prove to be quite
essential, [5], [6], [8], [11], [12], [14], [16], [22].
Hence the exhibition of limit cycles in three-
dimensional nonlinear multivariable systems which
can fit the structure of general 3 x 3, [16], [27],
nonlinear systems has been addressed in the present
work. In the event of the existence of limit cycling
oscillations, the possibility of quenching such
oscillations using the signal stabilization technique
by deterministic, [49], as well as Gaussian random
signals, [50], may be adopted. Alternatively,
suppression of limit cycles using the Pole placement
technique, [51], can also be adopted. All such
methods developed either by graphical or analytical
approach have been substantiated by digital
simulation with the developed program using
MATLAB Code and also with the use of
SIMULINK Toolbox of MATLAB Software.
The complexity involved in the structure, [16],
[27], in particular for implicit nonlinearity or the
system having the memory type nonlinearities, it
would be extremely difficult to formulate and
simplify the expressions even using harmonic
balance, [32], [51]. Hence in the present work to get
an alternative attempt has been made to develop a
graphical method for the prediction of limit cycles
in 3x3 non-linear systems by extension of the
procedure as depicted in [1]. The developed
method/procedure considers all constraints,
especially the simultaneous intersection of two
straight lines and one circle in three combinations.
The method can be used for the prediction of limit
cycles in non memory type explicit and implicit
nonlinear functions and also for memory type 3x3
nonlinear systems. Such a general graphical method
for 3x3 systems has never been developed before
and hence claims its novelty.
The proposed paper presents the dynamics of
general 3x3 nonlinear systems shown in Figure 2,
Figure 3, [27], which is an equivalent representation
of the general multivariable system of Figure 1,
[27]. The governing equation under limit cycling
condition, with reference to Figure 2 (autonomous
system i.e. U=0) in frequency response form is X =-
HC and C=GN(X)X: leading to X=-HGN(X)=AX,
where A=-HGN(X), [27], which facilitates the
determination of Eigen value of the multivariable
systems (illustrated in 2.1)
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X1, X2, X3 and C1, C2, C3 are Amplitudes of
respective Sinusoids. G1, G2, G3, and N1, N2, and N3
are magnitudes/absolute values of respective
functions. 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
.
2 Analysis of Nonlinear Self-
Oscillations (Limit Cycle)
The variables in Figure 1 are related by the
following equations:
Y = Y (x); X = G1 U – H G2 Y; C = G2 Y + G3 U
Where U, X, Y, and C are vectors of diminutions k,
l, m, and n respectively and G1, G2, G3, and H are
linear transfer function matrices of dimensions l x k,
n x m, n x k, and l x n respectively.
Fig. 1: Block Diagram Representation of a most
General Nonlinear Multivariable System, [27]
2.1 Analysis of self-oscillations in 3x3
nonlinear systems
Fig. 2: Equivalent of the system of Fig. 1for U = [0]
For the autonomous state (U = 0) the system of
Figure 1 can equivalently be represented as shown
in Figure 2.
Considering Figure 3 and making use of the
first-order harmonic linearization of the nonlinear
elements (nonlinear characteristics are replaced by
their respective Describing Functions: N1, N2, N3),
the matrix equations for the system of Figure 2 can
be expressed as:
Leading to X = - H G N (X) X = A X (i)
Where, A = -H G N (X)
Fig. 3: An equivalent 3X3 multivariable nonlinear
system of Figure 2
Visualising Eqn. (i) As a transformation of the
vector X onto itself, we note that for a limit cycle to
exist the following two conditions must be satisfied,
[16], [27],
(I) For every nontrivial solution X, the matrix A
must have an Eigen value λ, equal to unity; and
(II) The Eigen vector of A corresponding to this
unity Eigen value must be coincident with X.
“Comparing the systems of Examples 3 and 4 with
those of Examples 1 and 2 respectively, it is seen
that if all cross feedbacks are made negative then
the analysis of such system becomes trivial, [16].
Therefore, a standard three-dimensional nonlinear
feedback system is shown in Figure 3 in the form of
Figure 2” from which we get,
And
Hence
The characteristic Equation is
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For , from Eqn. (ii) we get,
Or
Where in general G1=G1(j), G2=G2(j),
G3=G3(j), and N1=N1 (X1, j), N2=N2 (X2, j),
N3=N3 (X3, j).
Four unknowns X1, X2, X3, and (frequency of L.
C) require four independent equations for their
evaluation. By separating the real and imaginary
parts of the characteristic equation (iii) only two
independent equations involving four unknown
quantities can be determined. Therefore, the
characteristic equation alone is not sufficient for the
analysis of the limit cycle in such systems.
However, by replacing the nonlinear elements with
their respective DF’s, ensuring harmonic balance,
we can explore for possible limit cycles, the
following conditions must be satisfied, [32].
(i) The Phase condition:
(ii) The Gain condition:
(Derived from-I: Eigen Value Condition):
characteristic equation
(iii) The Amplitude Ratio Condition (derived from
II: Eigen Vector condition), [16], [27], [32].
This has been applied in 2x2 nonlinear systems.
However, the logic can also be extended for 3x3
nonlinear systems as in the present case in the
following way:
A: Non-memory type nonlinearities:
(i) The Phase condition:
(ii) The Gain Condition:
Hence
(iii) The Amplitude Ratio Condition:
B: Memory type nonlinearities: N1=N(X1, j),
N2 = N2 (X2, j); N3 = N3 (X3, j);
Accordingly the changes in derivations in (i), (ii) &
(iii) are incorporated into numerical expressions
The Eigen vector V of A Matrix corresponding to
λ=1, also satisfies the equation: From (II)
Or
Addition of iv (a) & iv (b) yields:
Addition of iv (a) & iv (c) yields:
Similarly adding iv (b) and iv (c) we get,
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From Eqn. (i)
X = AX
Hence X =
Adding viii (a) and viii (b), we get,
Adding viii (a) and viii (c), we get,
Adding viii (b) and viii (c), we get,
Comparing (vi) & (b) we get
Comparing (vii) & ix (c) we get
Comparing (v) & ix (a) we get
2.2 Non-memory Type Nonlinear Elements
2.2.1 (A1): Explicit nonlinear functions (DFs)
(a)
(b) : are the functions of jω in
frequency response analysis.
(c) Eqn. (iii) can be separated into real and
imaginary parts for numerical examples:
Real (iii) = xi (a) and Imaginary (iii) = xi (b)
Alternatively using the phase condition and gain
condition, equations (4) and (5) are developed
respectively in place of xi (a) and xi (b).
(d) Eqn. x: Amplitude Ratio Conditions:
I. Numerical examples:
Example 1: Consider Figure 3 where the linear
elements are
the nonlinear elements are shown in
Figure 4.
Fig. 4: All Ideal Relays
These three nonlinear elements are represented by
their D.Fs as:
N1(X1) = ; N2(X2) = ; N3(X3) =
(Explicit nonlinear functions).
In the autonomous state L.C (assuming a single
frequency limit cycle oscillation) exists when the
following conditions are satisfied:
(i) The Phase condition:
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(ii)The Gain condition:
Where
(iii) Amplitude ratio conditions:
From (iii): (Amplitude Ratio conditions)
From (i): (Phase Condition)
Similarly,
Or tan
Or From (i):
Or
Hence tan [180
(
Putting the value of from A′ in the
above equation, we get,
From (ii): (Gain conditions)
(In terms of N1 N2 N3)
(In terms of X1, X2, X3): (Uses Gain Condition)
Summarizing: there are 4 variables: ω, X1, X2, X3:
To determine these 4 unknowns, 4 independent
equations are necessary: any two of Eqns. (1), (2) &
(3), and equations (4) and (5) can be used.
Use of amplitude ratio conditions:
From (1),
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From (2),
From (3),
Putting (6) in (4) we get, 0=1+
Putting Eqn. (8) in the above equation we get,
0=1+
Or multiplying both sided by in the above we
get,
.............(S)
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which can also be used for the determination of X3
for a fixed value of .
Eqn. (4) (Uses phase condition) can also be further
simplified as:
I. Equations (4-uses Phase Condition) or (9),
(5-uses Gain Condition) and any two of (6), (7), (8)
(use Amplitude Ratio Condition) can be used to
solve for ω, X1, X2 & X3: for explicit non-memory
type nonlinearities.
2.2.1 (A2): Implicit nonlinear functions (DFs)
(b) : Are the functions of j ω in
frequency response analysis.
(c) Eqn. (iii) can be separated into real and
imaginary parts for numerical examples:
Real (iii) = xi (a); Imaginary (iii) = xi (b)
Alternatively using the phase condition and gain
condition, equations (4) and (5) are developed
respectively in place of xi (a) and xi (b).
(d) Eqn. x: Amplitude Ratio Conditions:
Under autonomous state to determine limit cycle
in 3 X 3 nonmemory type system with implicit
nonlinear function there are four unknowns:
ω (Frequency of LC), N1 or X1, N2 or X2, N3 or X3:
There are five Eqns. xi(a), xi(b) and x(a), x (b) and x
(c) can be used. To determine the above four
unknowns Eqns. (4), (5) and any two of equations
x(a), x(b) & x(c) can be used.
II. Numerical problems:
Example 2: Consider Figure 3 where the linear
elements are
; And the nonlinear elements are
shown in Figure 5.
Fig. 5: All Ideal Saturation type nonlinear elements
(with slopes k1, k2 & k3)
These nonlinear elements are represented by their
D.Fs as shown in Eqn. (10).
(10)
In the autonomous state L.C (assuming a single
frequency limit cycle oscillation) exists when the
following conditions are satisfied:
(a) The Phase condition:
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(b) Gain Condition:
(c) Eqn. x: Amplitude Ratio Conditions:
Under autonomous state to determine limit cycle
in 3 X 3 non-memory type system, there are four
unknowns: ω (Frequency of LC), N1 or X1, N2 or X2,
N3 or X3: There are five Eqn. xi(a), xi(b) and x(a), x
(b) and x (c). To determine the above four
unknowns Eqns. (4), (5) and any two of equations x
(a), x (b) & x (c) can be used. It may be noted that in
this case X1, X2, and X3 are determined from N1, N2,
and N3 respectively using Newton Raphson method.
B: For Memory type nonlinear elements:
(a) ,
(b) (For frequency
response)
(c) Eqn. (iii) can be separated by Gain and Phase
conditions
(i) Gain of the loop should be 1
(ii) Phase of the Loop should be θ =1800 =
Where phase angles of G come from j ω but phase
angles of N come from the phase shift of concerned
D. F.
3 Methods of Determination of Limit
Cycles (L. C)
3.1 Analytical
(A1): For non-memory type nonlinear elements:
Explicit nonlinear functions (DFs).
Eqn. (iii) (characteristic equation) is separated
into real and imaginary parts.
(A2): For non-memory type nonlinear elements:
Implicit nonlinear functions (DFs).
(a)
(b)
(c) Use eqn. x (a), x (b) & x (c)
Solve for ω = 0.1, 0.2, 0.3.................3.0 with
relevant Eqns.
(B): For memory type nonlinearity.
Eqn. (iii) (characteristic equation) is separated
into real and imaginary parts.
(a)
(b) Conditions Loop Gain = 1
(c) Phase Loop
, where phase angles of G come from jω and
the phase of N comes from the phase shift of
DFs.
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Example 4, [16]: 3 X 3 systems:
H=
(All the cross feedbacks are made negative: the
analysis of such systems becomes trivial)
Hence Example 1: 2 X 2 systems:
Example 2: 3 X 3 system:
Simultaneous solution of Eqns. (4-Gain Condition),
(5-Phase Condition) and any two of Eqns. (6), (7) ,
(8) (Amplitude Ratio Condition), will yield the four
unknown parameters of possible self-
oscillations in the system considered.
3.2 Graphical
When a closed loop system exhibits limit cycles, the
signal at any point of the loop is transmitted around
the loop to that point without any change in
amplitude and phase. Thus, a system exhibits a limit
cycle when the loop gain is one and the loop phase
shift is ± 2nπ, [21], [51], when n is an integer.
For the proposed work the complexity involved
in the structure in particular for implicit or having
the memory type nonlinearities, it would be
extremely difficult to formulate and simplify the
expressions, even in the harmonic balance method,
[32], [51]. Hence it is felt necessary to develop a
graphical technique using the harmonic balance
method as discussed in the references, [16], [27],
[49], [51], for non-memory and memory type
nonlinearities respectively.
3.2.1 Graphical Method for 2X2 system
Fig. 6: A class of 2 x 2 nonlinear systems
Fig. 7: Phase diagram for the system of Figure 6
Consider the system of Figure 6. The self-
oscillation revealed by the system is shown in the
phase diagram in Figure 7. The sides of the triangle
OBD correspond to the quantities for subsystem S1
and those of the triangle OAD correspond to
subsystem S2 of Figure 8. For a fixed frequency ω,
the angles ϴ L1 (Arg. G1 (j ω)) and ϴL2 (Arg. G2 (j
ω)) are fixed.
As a result, the angle ODB must remain constant
at (180-ϴL1) constraining D to lie on a circle as
given in Figure 9. The phase representing X2 would
lie along the straight line drawn at an angle ϴL2
with the phase C2 (= - R1). The intersections of the
straight line with the circle shown in Figure 9 will
represent possible self-oscillations if the following
conditions are satisfied:
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Fig. 8: Linear equivalent for the system of Figure 6
Fig. 9: Locus of Di satisfying constant loop phase
shift condition
(a) The point Di lie on the intersection of segment
O Di B of a circle subtending an angle (1800-
ϴL1) on OB and a straight line A Di making an
angle ϴL2 with OA.
(b) O D i = C1=Y1G1 : i = 1 or 2
And B Di = X1, where Y1= N1X1 and
(c) OA = C2 = Y2G2
And A Di = X2, where Y2 = N2X2
The need to consider separate phase diagrams for
various values of R1 for checking possible self-
oscillations can be eliminated if all the quantities are
normalized with respect to the magnitude R1,
leading to a single phase diagram for a particular
value of ω as, shown in Figure 10, wherein, OB =
R1/R1 = 1; OA = C2/R1 = -1.0
O Di = C1/R1 = C1/-C2, i = 1 or 2
B Di = X1/R1; A Di = X2/R1
In Figure 10, ‘C’ is the center of the circle OC Di B
of radius r = OC.
In Figure 10, Figure 11 (a), and Figure 11(b),
selecting ‘O’ as the origin, the coordinates of the
point Di (i = 1, 2) can be determined in the
following manner:
Fig. 10 Normalized Phase Diagram for the system of
Figure 6
Fig. 11(a) Normalized Phase Diagram with General
Di
Fig. 11(b) Normalized Phase Diagram with D1 or D2
(Di)
Consider the Figure 11 (b), OCD2B is a
circumscribed circle where radius = r
Fundamental relation: sine Law:
Explanation: OCB is an isosceles triangle:
OCD2 is an isosceles triangle: CD2O = COD2
Hence D2OB = CBM + CD2O
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In BOD2 triangle: BD2O+ OBD2+ BOD2=1800
Or BD2O+ OBD2+BD2O+ OBD2 = 1800
∵ [ BOD2 = CD2O+ ]
Or BD2O+ OBD2 =1800
Or BOD 2 + OBD2 = 900 [BOD2= BOD2+ OBD2]
BOD 2 = 900 [Since BOD2=BD2O+ OBD2]
Hence from sine law,
Or r = = radius of the circle
The coordinates of the center of the circle:
X coordinate of C = x0=OM =
Y coordinate of C = y0 = CM=
=
=
If O is considered as the origin (0, 0): y0 of C =
Equation to the straight line A Di; y = m x where, X
coordinate of A = u +AO = u+1 and m = slope of
the line = tanϴL2
Hence the Equation to straight line ADi
Y = v = (u+1) tan ϴL2 or u = v cot ϴL2 – 1 (11)
The coordinates of the intersection point:
Consider the Figure 11 (a):
The x coordinate of Di (2) = +r
Y coordinate of Di (2) = y 0 + r sin ϴc
The equation for a circle can be written as:
(u - ) 2 + (v - ) 2 = ( ) 2 (12)
(We can determine the coordinate of Di (the point of
intersection of the straight line (A Di) with the
circle): and eliminating either u or v from
equation (12) using eqn. (11) :
(vcotϴL2 – 1 - ) 2 + (v - ϴ 5 cotϴL1)2 = ( ) 2 =
(0.5 cosecϴL1)2
Or v2 cot2ϴL2 – 2x v cotϴL2 + + v2 – 2 x 0.5
cotϴL1 + 0.25
=0.25 (1+ ) = 0.25 + 0.25
Or v2 (cot2ϴL2+1) – v (3cotϴL2 + cotϴL1) + - 0.25 = 0
Or v2cosec2ϴL2 – v (3cotϴL2 + cotϴL1) + 2 = 0 (13)
Hence =
(14)
And = – 1 (15)
It may be noted that the coordinates of the points
(i = 1, 2) in the normalized phase diagram are
functions only of the angles ϴL1 and ϴL2. For a
specific system the angles ϴL1 and ϴL2 are known
for an assumed value of ω, and, therefore, the
coordinates and can be evaluated and
subsequently we obtain:
N1(X1) = , (where = )
= = = (16)
N2(X2) = = = = (17)
= = (18)
Subsequently, the values of and
corresponding to the values of ( ) and )
obtained in Eqns. (16) & (17) are obtained from the
expressions for the DFs, and the ratio thus
obtained can be compared with that obtained from
Eqn. (18). The process is to be repeated for various
assumed values of ω. The values of ω at which the
ratio computed from Eqn. (18) and that from the
D.F expressions are equal determines the
frequencies of self Oscillations.
Once ω is determined the amplitudes of other
variables of interest are calculated directly from the
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different equations developed above or directly from
the normalized Phase Diagram drawn to scale
corresponding to the Limit Cycling (LC) frequency.
3.2.2 Graphical Method for 3 x 3 systems
The steps depicted and illustrated in section 3.2.1
are extended for 3x3 nonlinear systems. The
normalised phase diagrams are drawn with three
combinations such as:
Combination 1: For subsystems S1 & S2: C1 (+ve)
and C2 (-ve)
Combination 2: For subsystems S3 & S2: C2 (+ve)
and C3 (-ve).
Combination 3: For subsystems S1 & S3: C3 (+ve)
and C1 (-ve).
Example 1 & Example 2 are revisited:
Linear elements are represented by
; ; =
and Nonlinear elements are taken Ideal relays as
shown in Figure 4 and ideal saturations as shown in
Figure 5.
ϴ L1 = Arg. ( (j ω)) = -90 - (ω):
ϴ L2 = Arg. ( (j ω)) = -90 - ( ):
ϴ = Arg. ( (j ω)) = -90 - ( ):
For a fixed value of ω the Combinations of
Subsystems 1, 2, and 3, Normalised Phase Diagrams
are shown in Figure 12(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.
N2 = (11-3 ) ±
(1.10), [16]
N1= N2 + (1.8), [16]
= (1.11), [16]
= = (1.18), [16]
Fig. 12 (a): Normalised Phase Diagram with C1, C2
& C3 for the combination 1
Fig. 12 (b): Normalised Phase Diagram with C1, C2
& C3 for the combination 2
Fig. 12 (c): Normalised Phase Diagram with C1, C2
& C3 for the combination 3
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. 1.11 and Eqn. 18 are matched at a limited
cycling frequency.
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Table 1(a). Shows the 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. 18
X1/X2
from eqn.
1.11(A1)
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=6.0
8
X2=AD2=6.0
8
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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4 Digital Simulation
I. Numerical problems
Example 1 & Example 2 are revisited: A 3x3 system
represented by Figure 3 has three nonlinear elements
as shown in Figure 4 and Figure 5 for Ex.1 & Ex.2
respectively and three linear transfer functions are
Partial Fraction Expansion of G1(s), G2(s) and G3(s):
Or A=2, A+B=0: B=-A =-2, 2A+B+C=0: C=-2A-B=-
4+2=-2
For a very small value of the sampling period T,
TG(z) G(s). Figure 13 and Figure 14 represent the
canonical equivalent and Digital equivalent to Figure
3 for Examples 1 & 2 respectively.
Z-transfer functions from Laplace functions:
Fig. 13: Equivalent Canonical form of Fig. 3 for Ex.1
& 2
From Figure 14 the following algorithm has been
derived:
(5)
Taking inverse z-transform: OW1 (n T) = 2Y1 (n T) +
OW1 )
Taking inverse z-transform: OW2 (n T) = -2Y1 (n T)
+ OW2 )
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Fig. 14: The Digital representation of Fig. 3 for Ex. 1
& 2
Or -2T =OW3 (z) -2 OW3 (z)
+ OW3 (z)
Taking inverse z-transform: OW3 (n T) = -
2T +2 OW3 ) –
)
Taking inverse z-transform: TU1 (n T) = 0.25Y2 (n T)
+TU1 )
(5)
Taking inverse z-transform: TU2 (n T) = -0.25Y2 (n
T) + TU2 )
(6)
Taking inverse z-transform: TV1 (n T) =0.5Y3 (n T)
+TV1 )
(7)
Taking inverse z-transform: TV2 (n T) = -0.5Y3 (n T)
+ AK2* TV2 )
Let us take ) is the zeroth instant; nT is the first
instant, so we can write:
OW1 ) = OW1NOW1N;
OW1 (n T) = OW1N1
OW2 ) = OW2NOW2N;
OW2 (n T) = OW2N1
OW3 ) = OW3N (-1)OW3NN;
OW3 ) = OW3N OW3N ;
OW3 (nT)=OW3N1
Now C1 (nT) = OWN1 = T [OW1N1 + OW2N1 +
OW3N1]
= T [OW1 (nT) +OW2 (nT) + Ow3 (nT)]
=T [2Y1 (nT) + OW1N-2Y1 (n T) + AK1 OW2N –
2 T Y1N + 2 AK1 OW3N
- AK2 OW3 NN] =OWN1=C1
Similarly,
TU1 ) = TU1N TU1N; TU1 (nT) =
TU1N1, TU2 ) = TU2N= TU2N; TU2 (nT)
= TU2N1
Now C2 (nT) = TUN1 = T* [TU1N1 + TU2N1] = T*
[TU1 (nT) + TU2 (nT)] = T* [0.25 Y2 (n T) + TU1N
– 0.25Y2 (n T) + AK3 TU2N] = TUN1 = C2
Similarly,
TV1 ) = TV1N TV1N; TV1 (nT) =
TV1N1
TV2 ) = TV2N TV2N; TV2 (nT) =
TV2N1
Now C3 (n T) = TVN1 = T*[TV1N1 + TV2N1] =
T*[TV1 (n T) + TV2 (n T)]
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= T*[0.5Y3 (n T) + TV1N-0.5Y3 (n
T) +AK2 TV2N] = TVN1 = C3
Next Run:
R1=ORN1=C3 – C2 =TVN1 – TUN1
R2 = TRN1 = C1 – C3 = OWN1 – TVN1
R3 = THRN1= C2 – C1 = TUN1 – OWN1
X1 = OXN1 = ORN1 – OWN1, OYN1 = OF (OXN1)
X2 = TXN1 = TRN1 – TUN1, TYN1 = TF (TXN1)
X3 = THXN1 = THRN1 – TVN1, THYN1 = THF
(THXN1)
An appropriate program in MATLAB code
following the above algorithm produces the results.
The results/images of digital simulation along with
that of using SIMULINK Toolbox are presented in
Figure 16 and the numerical values obtained there are
shown in Table 2.
5 Use of SIMULINK Toolbox in
MATLAB
The SIMULINK Toolbox is used to determine X1, X2,
X3, C1, C2 & C3 for both Examples 1 and 2 and the
results so obtained are compared with those of
graphical method and digital simulation (Figure 15).
Figure 15 shows the SIMULINK representation for
the prediction of Limit Cycles,
Fig. 15 (a): Diagram used in SIMULINK Toolbox for
the solution of example 1.
Fig. 15(b): Diagram used in SIMULINK toolbox for
the solution of example 2.
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6 Comparison of Results
Figure 16 and Figure 17 show the Results/Images
obtained from Digital Simulation and with the use of
the SIMULINK Toolbox of Example-1 and Example-
2 respectively.
Tables 2(a) and 2(b) show the numerical results of
Example 1 and 2 for Ideal Relay and Saturation
respectively using different methods.
It has been observed that the graphical results are
almost matching with that obtained by digital
simulation as well as by use of SIMULINK Toolbox
both in images and also in Tables with numerical
values.
Fig. 16: Results/Images from digital simulation and SIMULINK for C1, C2, C3, X1, X2, and X3 of Example 1
(relay type nonlinearities)
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Fig. 17: Results/Images from digital simulation and SIMULINK for C1, C2, C3, X1, X2, and X3 of Example 2
(saturation type nonlinearities)
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)
7 Conclusion
The work presented, claims the novelty in the
following lines: It discusses multivariable (n x n)
nonlinear systems and proposes a general
formulation in matrix form applicable in both non-
memory and memory-type nonlinearities for
prediction of limit cycles. The complexity arises in
the formulation, particularly for implicit non-
memory type non-linearity or memory type
nonlinearities, it may be extremely difficult to
formulate and simplify the expressions even using
the harmonic linearization method, [32], [51].
Hence an attempt has been made to develop a
graphical technique using the harmonic
linearization/harmonic balance method for the
prediction of limit cycles in 3 x 3 nonlinear systems
which is not available elsewhere. This is an
opening and has the brighter future scope of
adopting the techniques like signal stabilization,
[49], [50], suppression of limit cycles, [51], in the
event of the existence of limit cycling oscillations
for 3 x 3 or higher dimensional nonlinear systems
which brings in development in the design of
nonlinear systems on several occasions.
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.
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
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References
[1] PATRA, K. C, SINGH, Y.P, Graphical
method of prediction of limit cycle for
multivariable nonlinear system. IEE Proc.
Control Theory Appl.: 143, 1996, pp. 423-
428.
[2] GELB, A, Limit cycles in symmetric
multiple nonlinear systems. IEEE Trans.
Autumn. Control: AC-8, 1963, pp. 177-178.
[3] JUD, H.G Limit cycle determination of
parallel linear and non- linear elements. IEEE
Trans. Autumn. Control: AC-9, 1964, pp.
183-184.
[4] GRAN, R., and RIMER, M Stability analysis
of systems with multiple nonlinearities. IEEE
Trans. Autumn. Control: 10, 1965, pp. 94-97.
[5] DAVISON, E.J., and CONSTANTINESCU,
D Describing function technique for multiple
nonlinearity in a single feedback system
IEEE Trans Autumn. Control: AC-16: 1971,
pp. 50-60.
[6] OLDENBURGER, R., T. NAKADA T
Signal stabilisation of self - oscillating
system IRE Trans. Automat Control. USA, 6,
1961, pp: 319-325.
[7] VISWANDHAM, N., and DEEKSHATULU,
B.L Stability analysis of nonlinear
multivariable systems. Int. J. Control, 5,
1966, pp. 369-375.
[8] GELB, A. and VADER-VELDE, W.E
Multiple-input describing functions and
nonlinear system design, McGraw- Hill, New
York, 1968
[9] NIKIFORUK, P.N., and WINTONYK,
B.L.M Frequency response analysis of two-
dimensional nonlinear symmetrical and non
symmetrical control systems. Int. J. Control,
7, 1968, pp.49- 62.
[10] RAJU, G.S., and JOSSELSON, R Stability
of reactor control systems in coupled core
reactors, IEEE Trans. Nuclear Science, NS-
18, 1971, pp. 388-394.
[11] ATHERTON, D.P Non-linear control
engineering - Describing function analysis
and design. Van Noslrand Reinhold, London,
1975
[12] ATHERTON, D.P., and DORRAH, H.T A
survey on nonlinear oscillations, Int. J.
Control, 31. (6), 1980, pp. 1041-1 105.
[13] GRAY, J. O. And NAKHALA, N.B
Prediction of limit cycles in multivariable
nonlinear systems. Proc. IEE, Part-D, 128,
1981 pp. 233-241.
[14] MEES, A.I Describing function: Ten years
on. IMA J. Appl. Math., 34, 1984 pp. 221-
233.
[15] SEBASTIAN, L the self-oscillation
determination to a category of nonlinear
closed loop systems, IEEE Trans. Autumn.
Control, AC-30, (7), 1985 pp. 700-704.
[16] PATRA, K.C Analysis of self oscillations
and signal stabilisation of two-dimensional
nonlinear systems. Ph. D Thesis
(Dissertation), IIT, Kharagpur. India, 1986
[17] COOK, P.A Nonlinear dynamical systems,
Prentice-Hall, Englewood ClilTs, NJ, 1986
[18] CHANG, H.C., PAN, C.T., HUANG, C.L.,
and WEI, C.C A general approach for
constructing the limit cycle loci of multiple
nonlinearity systems, IEEE Trans. Autumn.
Control, AC-32, (9), 1987, pp. 845-848.
[19] PARLOS, A.G., HENRY, A.F.,
SCHWEPPE, F.C., GOULD, L.A., and
LANNING, D.D Nonlinear multivariable
control of nuclear power plants based on the
unknown but bounded disturbance model,
IEEE Trans. Autumn. Control, AC-33, (2),
1988 pp. 130-134.
[20] PILLAI, V.K., and NELSON, H.D A new
algorithm for limit cycle analysis of
nonlinear systems, Trans. ASME, J. Dyn.
Syst. Meas. Control, 110, 1988, pp. 272-277.
[21] GENESIO, R., and TESI, A On limit cycles
of feedback polynomial systems, IEEE
Trans. Circuits Syst., 35, (12), 1988, pp.
1523-1528.
[22] FENDRICH, O.R Describing functions and
limit cycles, IEEE Trans. Autom. Control,
AC -31, (4), 1992, pp. 486487.
[23] PATRA, K.C., SWAIN, A.K., and MAJHI, S
Application of neural network in the
prediction of self-oscillations and signal
stabilisation in nonlinear multivariable
systems. Proceedings of ANZIIS-93, Western
Australia, 1993, pp. 585-589.
[24] PATRA, K.C., and SINGH, Y.P Structural
formulation and prediction of limit cycle for
multivariable nonlinear system. IETE, Tech.
Rev. India, 40, (5 & 6), 1994, pp. 253-260.
[25] ZHUANG, M., and ARTHERTON, D.P PID
controller design lor TITO system, TEE
Proc. Control Theory Appl. 141, (2), 1994,
pp. 111-120.
[26] LOH, A.P., and VASANU, V.V Necessary
conditions for limit cycles in multi loop relay
systems, IEE Proc., Control Theory Appl.,
141, 31, 1994, pp. 163-168.
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DOI: 10.37394/232026.2023.5.9
Kartik Chandra Patra, Asutosh Patnaik
E-ISSN: 2766-9823
Volume 5, 2023
[27] PATRA, K.C., et al Prediction of limit cycles
in nonlinear multivariable systems, Arch.
Control Sci. Poland, 4(XL) 1995, pp 281-
297.
[28] TESI, A, et al Harmonic balance analysis of
periodic doubling bifurcations with
implications for control of nonlinear
dynamics, Automatic, 32 (9), 1996, 1255,
1271.
[29] PATRA, K.C, et al Signal Stabilization of
two Dimensional Non Linear relay Control
Systems Archives of Control Sciences,
Volume 6(XLII), No. 1 – 2, 1997, pages 89 –
101.
[30] PATRA, K. C, PATI, B.B An investigation
of forced oscillation for signal stabilization of
two dimensional non linear system, Systems
& Control Letters, 35, 1998, pp. 229 – 236.
[31] LIN, C.H., HAN, K.W Prediction of Limit
cycle in Nonlinear two input two output
control system, ‘IEE Proc.-Control Theory
Appl. Vol.146, No.3 may. 1999.
[32] PATRA, K.C, et al Structural Formulation
and Self-Oscillation Prediction in
Multidimensional Nonlinear Closed-Loop
Autonomous Systems, Int. J. App. Math. And
Comp. Sci., Vol. 9, No. 2, 1999, pp. 327 -
346.
[33] HORI, Y. et al Slow resonance ratio control
for vibration suppression and disturbance
rejection in torsional system, IEEE Trans.
Ind. Electron., vol. 46, (1), 1999, pp.162-168.
[34] NORDIN, M. and Gutman, P. O Controlling
mechanical systems with backlash- a survey,
Automatica , vol. 38, (10), 2002, pp.1633-
1649.
[35] T. RAYMOND, T, et al Design of Feedback
Control Systems, Oxford University Press, 4th
edition, 2002, pp. 677-678.
[36] STANISLAW, H. ak Systems and Control’
Oxford University Press, 2003, pp. 77 – 83.
[37] CHIDAMBARAM, I.A, and VELUSAMI, S
Decentralized biased controllers for load-
frequency control of inter connected power
systems considering governor dead band non-
linearity, INDICON, Annual IEEE, 2005,
pp.521-525.
[38] EFTEKHARI, M and KATEBI, S. D
Evolutionary Search for Limit Cycle and
Controller Design in Multivariable non linear
systems, Asian Journal of Control, , Vol. 8,
No. 4, 2006, pp. 345 – 358.
[39] KATEBI, M, et al Limit Cycle Prediction
Based on Evolutionary Multi objective
Formulation, Hindawi Publishing
Corporation, , Mathematical Problems in
engineering Volume, Article ID 816707,
2009, 17pgs.
[40] GARRIDO, J, et al Centralized PID control
by Decoupling of a Boiler-Turbine Unit,
Proceedings of the European Control
Conference, Budapest, Hungary, Aug. 2009,
23-26.
[41] TSAY, T.S Load Frequency control of
interconnected power system with governor
backlash nonlinearities, Electrical Power and
Energy, vol. 33, 2011, pp.1542-1549.
[42] TSAY, T.S Limit Cycle prediction of
nonlinear multivariable feedback control
systems with large transportation lags,
Hindawi Publishing corporation journal of
control science and Engineering, Vol., article
id 169848, 2011.
[43] TSAY, T.S Stability Analysis of Non linear
Multivariable feedback Control systems,
WSEAS Transactions on systems, Volume 11,
Issue 4, 2012, pp. 140 – 151.
[44] SUJATHA, V., PANDA, R. C Relay
Feedback Based Time domain modelling of
Linear 3-by-3 MIMO System, American
Journal of System Science, Scientific &
Academic Publishing, 1(2) 2012, pp. 17-22.
[45] WANG, C, et al Vibration suppression with
shaft torque limitation using explicit MPC-PI
switching control in elastic drive systems,
IEEE Trans. Ind. Electron, vol. 62,(11),
2015, pp. 6855-6867.
[46] YANG, M, et al Suppression of mechanical
resonance using torque disturbance observer
for two inertia system with backlash Proc.
IEEE 9th Int. Conf. Power Electron., ECCE
Asia, 2015, pp. 1860 - 1866.
[47] SHI, Z, and ZUO, Z back stepping control for
gear transmission servo systems with
backlash nonlinearity IEEE Trans. Autumn.
Sci. Eng., vol. 12, (2), 2015, pp. 752-757.
[48] WANG, C, et al, Analysis and suppression of
limit cycle oscillation for Transmission
System with backlash Nonlinearity, IEEE
Transactions on Industrial Electronics, vol.
62, (12), 2017, pp. 9261-9270.
[49] PATRA, K. C, and DAKUA, B. K,
Investigation of limit cycles and signal
stabilisation of two dimensional systems with
memory type nonlinear elements, Archives of
Control Sciences, vol. 28, (2), 2018, pp. 285-
330.
[50] PATRA, K. C, KAR, N Signal Stabilization
of Limit cycling two Dimensional Memory
Type Nonlinear Systems by Gaussian
INTERNATIONAL JOURNAL OF APPLIED MATHEMATICS,
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Volume 5, 2023
Random Signal, International Journal of
Emerging Trends & Technology in Computer
Science, Vol. 9 Issue 1, 2020, pp. 10-17.
[51] PATRA, K. C, KAR, N Suppression Limit
cycles in 2 x 2 nonlinear systems with
memory type nonlinearities, International
Journal of Dynamics and Control, Springer
Nature’,34,95€, vol.10 Issue 3, 2022, pp 721-
733.
[52] LOPEZ, D.S, VEGA, A.P, Fuzzy Control of
a Toroidal Thermosyphon for Known Heat
Flux Heating Conditions, Proceeding of the
8th World Congress on Momentum, Heat and
Mass Transfer (MHMT’23), Lisbon Portugal-
March 26-28, 2023.
DOI:10.11159/enfht23.133
[53] CORRADO. C, et. al, Quantifying the impact
of shape uncertainty on predict arrhythmias,
Computers in Biology and Medicine, Elsevier
Ltd., 153, 2023, 106528.
[54] CHEN, W., et. al, Oscillation characteristics
and trajectory stability region analysis
method of hierarchical control microgrids,
Energy Reports, 9, 2023, pp 315-324.
[55] Kumar, U., et.al. The effect of sub diffusion
on the stability of autocatalytic systems,
Chemical Engineering Science, Elsevier Ltd.,
265, 2023, 118230.
[56] Marrone, J.I., et.al. A nested bistable module
within a negative feedback loop ensures
different types of oscillations in signalling
systems, Scientific reports| Nature portfolio,
2023, 13:529.
[57] MUNCH, S, B., et.al. Recent developments
in empirical dynamic modelling, Methods in
Ecology and Evolution, 2022, 14, pp 732-
745.
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
No source of funding excepting the computational
facilities used for preparation of the paper has been
extended by the C. V. Raman Global University,
where the authors are working.
Conflict of Interest
The authors have no conflict 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
https://creativecommons.org/licenses/by/4.0/deed.e
n_US
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