Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
8
Athenea Journal
Vol.4, Issue 14, (pp. 8-21)
ISSN-e: 2737-6419
https://doi.org/10.47460/athenea.v4i14.64
Algorithm for Diffuse TSK Modeling of SNL MIMO with
Undefined Operation Points
Received (22/09/2023), Accepted (10/11/2023)
Abstract. - This paper presents an algorithm for constructing fuzzy models in linear state subspaces from the nonlinear
MIMO dynamic model of plants whose operating points are not defined within the permissible physical range for the
system. It is based on the fuzzy Takagi-Sugeno-Kang model and Kawamoto's ideas of non-linearity sectors. The
relevant functions in the antecedent are modeled with linear functions, while functions model the consequent in
Discrete State Space. The application of the algorithm to the model of a thermoelectric plant widely studied in the
specialized literature is discussed.
Keywords: Fuzzy modeling algorithm, undefined points of operation, Kawamoto non-linearity sectors, nonlinear MIMO
systems, Takagi-Sugeno-Kang.
Algoritmo para el modelado TSK difuso de SNL MIMO con puntos de operación indefinidos
Resumen: En este artículo se presenta un algoritmo para la construcción de modelos difusos en subespacios de estado
lineales a partir del modelo dinámico MIMO no lineal de plantas cuyos puntos de operación – dentro del rango físico
permisible para el sistema – no se encuentran definidos. Se toma como base el modelo difuso Takagi-Sugeno-Kang y
las ideas de sectores de no linealidad de Kawamoto. Las funciones de pertinencia en el antecedente se modelan con
funciones lineales, en tanto que el consecuente se modela mediante funciones en Espacio de Estado Discreto. Se
discute la aplicación del algoritmo al modelo de una planta termoeléctrica ampliamente estudiada en la literatura
especializada.
Palabras clave: Algoritmo modelado difuso, puntos de operación indefinidos, sectores de no-linealidad Kawamoto,
sistemas MIMO no-lineales, Takagi-Sugeno-Kang.
Luis J. Gonzalez Lugo
https://orcid.org/0000-0001-9933-4147
lgonzalez@ingenieriala.com
LA Engineering
Frutillar-Chile
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Vol.4, Issue 14, (pp. 8-21)
ISSN-e: 2737-6419
Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
I. INTRODUCTION
To control means to exert the actions necessary to produce a desired result, but to do so, the system to be
commanded must exhibit "reasonably predictable" behavior. In the specialized literature [1], it is proposed that the
systems to be controlled can be classified into two large groups: deterministic and non-deterministic. Classical control
algorithms are based on the hypothesis that the system to be controlled is deterministic, for which a series of restrictions
are applied to guarantee the functionality of the algorithm, ranging from restricting the work area to a small region
around a point of operation, to "disregarding" the probabilistic nature of the present and future state of the system.
There is also a presumption that the mathematical model of the system is time-invariant. However, these classical
control techniques work very well for various physical systems and have been successfully employed for over a century,
as stated in [2]: "Most physical systems contain complex non-linear relationships, which are difficult to model with
conventional techniques." That is why, in advanced process control, non-linear control techniques are used.
One of the ways to mathematically model the non-linear nature of systems is by using models based on fuzzy logic
systems. This theory is supported by fuzzy logic systems being universal approximators [2]. In particular, the fuzzy
system model developed by Takagi and Sugeno [3] and Sugeno and Kang [4], called the TSK fuzzy model in the
literature, is suitable for a broad class of non-linear systems because the consequent is a linear function or even a state-
space system. Interestingly, the TSK model allows the use of equations in State Spaces in the consequent, thus being
able to obtain a fuzzy model for a Non-linear System (SNL) of multiple inputs and outputs (MIMO), which allows the
application of modern control algorithms based on models in State Space such as Optimal Control, H∞, Genetic
Algorithms, Predictive Control.
The fuzzy model of an SNL MIMO is robust in applications where the plant has more than one operating point.
However, following conventional techniques, as the number of operation points increases, the fuzzy model increases
significantly in complexity since, in general, a linear subspace is generated for each operation point, that is, a rule in the
consequent. It has also been sufficiently studied that as the number of rules in the consequent increases, the fuzzy TSK
model exhibits a behavior closer and closer to the non-linear model of the system. Thus, there is a dilemma between
keeping the complexity of the fuzzy model low – few rules – and, at the same time, ensuring that it represents the
dynamics of the SNL as accurately as possible.
Now, imagine for a moment that a TSK model is required for an SNL MIMO whose operating points – within the
permissible physical range for the system – are not defined, as might be the case with the design of a fuzzy servo
controller for such a system. Undoubtedly, this last proposition introduces an additional level of complexity to the
previously posed dilemma between keeping the number of rules of the fuzzy model to a minimum and, at the same
time, representing the system as accurately as possible, the complexity of not having defined the points of operation.
Next, an algorithm is presented to solve the problem: synthesize the fuzzy TSK model of a MIMO SNL with undefined
operation points.
II. DEVELOPMENT
In the first instance, a synthesis of the theoretical foundations of the developed algorithm is presented, and then a
detailed description is given.
A. Takagi-Sugeno-Kang Fuzzy Model (TSK)
Takagi and Sugeno [3], and later Sugeno and Kang [4], developed the structure of a Fuzzy Model that has been
widely studied. They denoted the relevance function of a fuzzy set A as A(x), with x
X, and defined that all fuzzy sets
are associated with linear relevance functions, such that a relevance function is characterized by having two limit values:
1 for the highest degree of relevance and 0 for the lowest degree of significance. Thus, the Truth Value of a linguistic
proposition of the type "(x is A) Y (y is B)" is expressed as:
(1)
In addition, these researchers defined the format of a fuzzy R implication as:
(2)
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ISSN-e: 2737-6419
Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
Where:
y : a variable of the consequent whose value is inferred.
x
1
– x
k
: premise variables that also appear in the consequent.
A
1
– A
k
: fuzzy sets with linear relevance functions representing a subspace in which the implication R can be
applied by reasoning.
f : logical function that relates the propositions in the antecedent.
G : a function that involves the value of “y” in the consequent when x
1
– x
k
satisfies the antecedent.
If only logical functions are used in the antecedent, and a linear function is adopted in the consequent, implication
(2) is written as:
(3)
In addition, if i have implications R
i
(i =1,
, n) according to the format indicated in (3). When given:
1 10 0
,,
kk
x x x x
(4)
where x
10
- x
k0
are singletons, the value of y is inferred using the following algorithm:
1) For each implication Ri, y
i
is computed by the function g
i
of the consequent:
00
0 1 1
i i i i
kk
y p p x p x
(5)
2) The Truth Value of the proposition y = y
i
is calculated by the equation:
1 10 0
i i i
kk
y y A x A x
(6)
Where denotes the Truth Value of the proposition, represents the minimum operation, and A(x
0
) is the degree
of pertinence of x
0
to the fuzzy set A.
3) Finally, the value of y is inferred from the n implications as its weighted average such that:
1
n
ii
i
i
y y y
y
yy
(7)
B. Kawamoto's Non-Linearity Sectors
According to Mehran [5], the idea of using Non-Linearity Sectors in the construction of fuzzy models was first
proposed by Kawamoto [6]. Consider an SNL such that:
x f x t
(8)
Where
00f
. The aim is to define the global sector in such a way that:
12
,x f x t a a x t
(9)
Figure 1 illustrates the nonlinearity sector approach.
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Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
Figure 1. Non-Linearity Sectors.
C. Algorithm for Constructing the TSK Model of a MIMO SNL with Undefined Points of Operation.
Mehran [5] argues that, although the consequent variables can be continuous or discrete in theory, they must be
discrete because virtually all fuzzy systems are implemented and modeled using digital systems. Based on the ideas of
Takagi, Sugeno, Kawamoto, and Mehran, the following algorithm was designed for the TSK modeling of a MIMO SNL
whose operation points are undefined.
Algorithm:
1) From the differential equations of the MIMO SNL, algebraically determine the state-space model for the SNL by
considering a generic operation point (X
0
, U
0
). Such that:
(10)
2) Analyze the Jacobian matrices A, B, C, and D to determine which variables they depend on. These will be the k
premise variables of the TSK model.
3) Analyze the maximum and minimum allowable physical values for the premise variables according to the
characteristics of the SNL, or what the maximum and minimum values are for which the SNL model in differential
equations is valid. From these, some steady-state pseudo-points of operation of the SNL MIMO are defined and
used to obtain the relevance functions of the antecedent and the Jacobian matrices of the consequent. For
pseudo-points of operation, the following must be met:
0 0 0 0
0 , ,A X U X B X U U
(11)
4) Define the linear relevance functions based on the previous step's results.
5) Define the n=2
k
implications of the TSK model according to the following structure:
(12)
6) Infer the value of the output vector from (6) and (7).
III. ALGORITHM AND APPLICATION STUDY
To illustrate the application of the presented algorithm and to verify its effectiveness, the comparison of the results
obtained in the modeling of a Thermoelectric Plant widely worked in the specialized literature [7], [8], [9], [10], [11],
using both the traditional algorithm based on operating points and the algorithm developed in the research carried
out will be presented as a case study.
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Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
A. SNL MIMO: Thermoelectric Plant
The model is based on the P16/G16 Thermoelectric Plant at the Sydvenska Kraft AB Plant in Malmö, Switzerland.
The power of the Plant is 160 MW, with a Boiler-Turbine-Alternator structure. The 3rd-order model of this SNL MIMO
was developed by Bell and Aström [12]. The differential equations that define the dynamics of this MIMO SNL are
presented in (13) , (14) and (15):
98
2 1 3
0,0018 0,9 0,15p u p u u
(13)
(14)
32
141 1,1 0,19
85
f
u u p
(15)
Where p is the Pressure in the boiler [kg/cm
2
], P
o
is the electrical power generated [MW],
f
is the density of the
fluid [kg/m
3
], u
1
is the position of the fuel flow valve, u
2
is the position of the steam control valve, and u
3
is the position
of the water flow valve. All u
i
inputs are normalized in the range [0,1]. Without losing generality, the system outputs are
considered the state variables for this study case.
Dimeo and Lee [7] presented the seven points of operation of the Thermoelectric Plant under study, reproduced
in Table 1.
Table 1. Thermoelectric Plant Operation Points.
B. Discrete TSK Model Based on Points of Operation
To construct the Discrete TSK model Based on the Operating Points of the thermoelectric plant, in the first instance,
seven linear subspaces are defined – one for each point of operation indicated in Table 1, then the triangular relevance
functions for each of the state variables are described in the antecedent as shown in Figures 2, 3 and 4. Thus, for a
given linear subspace, the Truth Value of the proposition, according to (6), is provided by the expression:
0 0 0
1 1 2 2 3 3
i i i i
X X A x A x A x
(16)
In the case of the consequent, we work with discrete state space models around each point of operation presented
in Table 1. Linearized models are obtained from the truncated Taylor series expansion of the SNL represented by the
equations(13), (14) and (15). To do this, it is necessary to calculate the following Jacobian matrices:
00
,XU
F
A
X
(17)
00
,XU
F
B
U
(18)
So, the linear approximation of the system will be:
X AX BU
(19)
where:
0
1 2 3
,
T
T
of
X X X X p P x x x
(20)
0
1 2 3
,
T
U U U U u u u
(21)
1
2
3
4
5
6
7
p
0
75,60
86,40
97,20
108,0
118,8
129,6
140,4
P
o0
15,27
36,65
50,52
66,65
85,06
105,8
128,9
f0
299,6
342,4
385,2
428,0
470,8
513,6
556,4
u
10
0,156
0,209
0,271
0,340
0,418
0,505
0,600
u
20
0,483
0,552
0,621
0,690
0,759
0,828
0,897
u
30
0,183
0,256
0,340
0,435
0,543
0,663
0,793
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Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
Figure 2. Relevance Functions of the Traditional Discrete TSK Model for Boiler Pressure.
Figure 3. Relevance Functions of the Traditional Discrete TSK Model for Generated Electrical Power.
Figure 4. Relevance Functions of the Traditional Discrete TSK Model for Fluid Density.
It was subsequently, using a zero-order retainer-type conversion algorithm, with a sampling time of 60 seconds,
the seven pairs of matrices presented from (22) to (28).
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Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
11
0,9355 0 0 73,1364 14,6469 12,1894
0,1416 0,0025 0 , 9,1908 100,2430 1,5318
0,1172 0 1,0000 4,4779 64,3485 140,0875
dd
AB
(22)
22
0,9237 0 0 73,6787 18,8617 12,1131
0,2121 0,0025 0 , 13,8405 128,6948 2,3067
0,1549 0 1,0000 5,9344 80,6090 140,3303
dd
AB
(23)
33
0,8895 0 0 71,3414 21,5197 11,8902
0,4317 0,0025 0 , 28,6071 145,1289 4,7678
0,2779 0 1,0000 10,7098 90,6696 141,1262
dd
AB
(24)
44
0,8749 0 0 70,7640 24,3791 11,7940
0,5217 0,0025 0 , 34,8055 163,6185 5,8009
0,3258 0 1,0000 12,5898 101,3357 141,4395
dd
AB
(25)
55
0,8603 0 0 70,1853 27,2193 11,6975
0,6108 0,0025 0 , 41,0290 181,7735 6,8382
0,3729 0 1,0000 14,4483 111,7990 141,7493
dd
AB
(26)
66
0,8457 0 0 69,6057 30,0499 11,6009
0,6988 0,0025 0 , 47,2645 199,6554 7,8774
0,4191 0 1,0000 16,2853 122,1120 142,0554
dd
AB
(27)
77
0,8313 0 0 69,0260 32,8642 11,5043
0,7854 0,0025 0 , 53,4991 217,2193 8,9165
0,4646 0 1,0000 18,1007 132,2649 142,3580
dd
AB
(28)
Finally, the defusification is performed using the equations (6) and (7).
C. TSK Discrete Model for Undefined Points of Operation
The following is the development of the proposed algorithm to obtain the TSK Discrete model of the thermoelectric
plant.
1) The differential equations of the SNL MIMO presented in (13), (14) and (15), are linearized using (17) and (18)
around a generic point of operation (X
0
, U
0
), yielding:
0 0 0 0
, ( , )X A X U X B X U U
(29)
00
00
0
18
21
18
0 0 2 1
2
0,002025 0 0
, 0,082125 0,018 0,1 0
0,01294118 0,002235294 0 0
ux
A X U u x
u
(30)
0
0
0
98
1
98
0 0 1
1
0,9 0,0018 0,15
, 0 0,073 0
0 0,01294118 1,658824
x
B X U x
x
(31)
2) Analyzing (30) and (31), it is concluded that the Jacobian matrices of the linearized system only depend on the
variables x
1
and U
2
, so these become the premise variables of the TSK model (k=2).
3) By analyzing the maximum and minimum physical values of the premise variables x
1
and U
2
and considering
that the condition is met (11), the pseudo-points of operation presented in Table 2 are obtained.
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Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
Table 2. Pseudo-Operation Points of the Thermoelectric plant.
T
o
B
c
d
x
10
60,02
60,02
165,00
165,00
x
20
8,03
51,32
25,02
159,90
x
30
275,00
165,80
462,30
474,20
u
10
0,0780
0,2427
0,2388
0,7358
u
20
0,3287
0,9208
0,3287
0,9208
u
30
0,0730
0,3502
0,2008
0,9630
From Table 2, note that for x
1
(Pressure), 60.02 [kgf/cm
2
] and 165.00 [kgf/cm
2
] have been defined as the minimum
and maximum values, respectively, for this premise variable, while for the position of the steam control valve, U
2
0.3287
and 0.9208 have been defined as their maximum and minimum values; from these and from (11), (13), (14) and (15), the
rest of the data contained in Table 2 were obtained.
4) Based on the results presented in Table 2, the relevance functions shown in Figures 5 and 6 are defined.
Figure 5. Relevance Functions of the Proposed Discrete TSK Model for Boiler Pressure.
Figure 6. Relevance Functions of the Proposed Discrete TSK Model for Steam Control Valve Position.
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Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
Figure 6. Relevance Functions of the Proposed Discrete TSK Model for Steam Control Valve Position.
5) The n=22 implications of the Discrete TSK model is defined:
low"
closed"
(32)
Where:
11
0,9355 0 0 73,1364 14,6469 12,1894
0,1416 0,0025 0 , 9,1908 100,2430 1,5318
0,1172 0 1,0000 4,4779 64,3485 140,0875
AB
(33)
low"
opened"
(34)
Where:
22
0,8297 0 0 68,9638 13,8113 11,4940
0,8208 0,0025 0 , 55,9540 90,8778 9,3257
0,5299 0 1,0000 20,6518 61,1094 142,7832
AB
(35)
high"
closed"
(36)
Where:
33
0,9272 0 0 72,8127 45,4890 12,1355
0,1595 0,0025 0 , 10,3892 311,9603 1,7315
0,1166 0 1,0000 4,4647 176,5755 140,0853
AB
(37)
high"
opened"
(38)
Where:
44
0,8091 0 0 68,1305 42,5638 11,3551
0,9121 0,0025 0 , 62,8271 279,2003 10,4712
0,5235 0 1,0000 20,4850 166,5670 142,7554
AB
(39)
In general:
00
( ) ( ) , ( ) ( )
i i i i
X n X n X U n U n U
(40)
7) The value of the output is inferred from (6) and (7).
IV. RESULTS
Two comparative studies were conducted to show the performance of the Discrete TSK model developed with the
proposed algorithm. The first study consisted of subjecting the three models (the non-linear, the Fuzzy model with
seven rules, and the Fuzzy model formulated with the proposed algorithm) to a sequence of ascending and descending
inputs according to the seven points of operation presented in Table 1. The second comparative study involved
subjecting the three models to a pseudo-random sequence of the three input variables, respecting the established
limits (minimum and maximum values) for each input in Table 1.
For both comparative studies, Variation Accounting (VAF) and Mean Square Error (RMSE) were considered as
performance indices, indices used by Castillo, Sarmiento, and Sanz [13] when performing the comparative evaluation
of a similar discrete TSK model. Recall that as two signals in a time series are almost identical, the RMSE tends to zero,
while the VAF tends to 100%.
Figures 7 to 10 present the most relevant results of the first comparative study, while Figures 11 to 14 present the results
of the second study.
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Gonzalez L. Algorithm for Diffuse TSK Modeling of SNL MIMO with Undefined
Operation Points
Figure 7. Comparison of the estimation of the Pressure in the Boiler of the Thermoelectric Plant for case 1.
Figure 8. Comparison of the estimation of the Power Output of the Thermoelectric Plant for case 1.
Figure 9. Comparison of the Fluid Density estimation in the Thermoelectric Plant for Case 1.
0 2 4 6 8 10 12
x 10
4
-20
0
20
40
60
80
100
120
140
Time (seconds)
Potencia [MW]
Time Series Plot:Potencia [MW]
SNL
TS Discreto Puntos Operacion (7 Subespacios Lineales)
TS Discreto Sin Puntos Operacion (4 Subespacios Lineales)
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Figure 10. Comparison of the Truth Value of the TSK Models for Case 1.
Figure 11. Comparison of the estimation of the Pressure in the Boiler of the Thermoelectric Plant for case 2.
Figure 12. Comparison of the Output Power estimation of the Thermoelectric Plant for case 2.
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Figure 13. Comparison of the Fluid Density estimation in the Thermoelectric Plant for Case 2.
Figure 14. Comparison of the Truth Value of the TSK Models for Case 2.
Table 3 shows the results of the measurement of performance indices for each study.
Table 3. TSK Model Performance Indices.
TSK Model – 7 Rules
TSK Model – 4 Rules
Case 1
Case 2
Case 1
Case 2
VAF%
RMSE
VAF
RMSE
VAF%
RMSE
VAF%
RMSE
x
1
Nan
Nan
Nan
Nan
99,96
0,51
99,93
5,08
x
2
Nan
Nan
Nan
Nan
99,05
3,63
97,58
7,19
x
3
Nan
Nan
Nan
Nan
99,94
1,40
100,0
0
2,80
Table 4 presents the percentage error of the 4-rule TSK model implemented with the proposed algorithm with
respect to the nonlinear model, both for case study 1 and case 2.
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Table 4. TSK Model Error – 4 Rules Regarding the Non-linear Model.
TSK Model – 4 Rules
Case 1
Case 2
%Error
%Error
x
1
0.04%
0.07%
x
2
0.95%
2.42%
x
3
0.06%
0.00%
A. Discussion of the Results
For case study 1 (sequence of ascending and descending inputs), in Figures 7, 8, and 9, it is evident that the Discrete
TSK model obtained from the points of operation (7 Rules) is not able to follow the sequence of inputs continuously,
this is corroborated when analyzing Figure 10, where it is shown that the Global Truth Value of the model becomes
zero at multiple points in the trajectory. This cancels the output of the TSK model at those intervals; as a consequence
of this discontinuity, it is impossible to calculate the VAF and RMSE performance indices for this TSK Model in case
study 1, as shown in Table 3. In contrast, the Discrete TSK Model obtained from the proposed algorithm, if it was able
to follow the trajectory imposed by the sequence of inputs, is corroborated in Figures 7, 8, and 9 (red line) and even in
Figure 10, in which the Global Truth Value of the model is never canceled. This makes it possible to calculate the VAF
and RMSE performance indices, which were above 99.00% and below 3.7 respectively, values that are very close to the
ideal case (VAF=100% and RMSE=0).
For case study 2 (sequence of pseudo-random inputs), in Figures 11, 12, and 13, it is again evident that the Discrete
TSK model obtained from the points of operation is not able to follow the sequence of inputs, even the performance is
still lower than case 1, which is corroborated when analyzing Figure 14 and comparing it with Figure 11; again as a
consequence of such discontinuity, it is not possible to calculate the VAF and RMSE performance indices for this TSK
Model in case study 2, as shown in Table 3. In contrast, the Discrete TSK Model obtained from the proposed algorithm
was once again able to follow the trajectory imposed by the sequence of inputs, which are even more demanding
(because they are random), which is corroborated in Figures 11, 12 and 13 (red line), and even in Figure 14, in which
the Global Truth Value for this model is clearly never cancelled; As for the VAF and RMSE performance indices, which
were above 97.50% and below 7.20 respectively, results that, although they show lower performance than that obtained
in case 1, turn out to be extremely interesting given the rigorous sequences of inputs to which the model was subjected.
For both cases, the discrete TSK model obtained from the proposed algorithm was able to follow the SNL
satisfactorily.
CONCLUSIONS
In this paper, we have presented an algorithm to synthesize the fuzzy discrete TSK model in linear state subspaces
for a MIMO SNL, from the dynamical model in differential equations, without the SNL operating points having been
previously defined. The relevance functions in the antecedent are modeled with linear functions. The application of the
algorithm to the model of a Thermoelectric Power Plant has been discussed, widely studied in the specialized literature,
obtaining satisfactory values in the chosen performance indices (VAF and RMSE). It is expected that this methodology
will serve to promote the application of modern trajectory tracking control algorithms based on models in State Space
such as: Optimal Control, H∞, Genetic Algorithms, Predictive Control.
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Operation Points
REFERENCES
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