81. Takagi-Sugeno Fuzzy Inference - Parametric Fuzzy System -
Takagi and Sugeno introduced a new inference structure based on fuzzy sets theory. Such structure is either called a Takagi-Sugeno fuzzy inference system or a parametric fuzzy system. It has been demonstrated to function as a efficient model for systems that can be fully represented by their input / output relationships
Like Mamdani’s Rule-Based Fuzzy Systems, parametric fuzzy systems are also based on a rule base approach. But the rules consequents, instead of being formed by fuzzy relations, are linear parametric equations in terms of the inputs of the system.

82. Theoretically the consequents could be any function, even non-linear
Theoretically the consequents could be any function, even non-linear. However, linear functions have been employed most of the time. Adaptive training have been used for further non-linear capabilities
Combination of a global rule-base description with local linear approximations by means of a linear regression model corresponding to a linear input-output model that one would use for describing the system locally.
The idea behind the approach is to combine a global rule-base description with local linear approximations by means of a linear regression model corresponding to a linear input-output model that one would use for describing the system locally. In other words, it is a hybrid approach that combines a fuzzy rule-based and a mathematical method: the antecedents (IF part) define the overlapping patches related to the input-output transfer, and the consequents (THEN part) define linear approximations for the patches. Therefore, the premise portion of the fuzzy rules is identical with the rule base approach, but the consequents are described by mathematical equations

83. Parametric Fuzzy Controllers
The parametric form of fuzzy rules has the following structure:
IF s1 = S1i AND s2 = S2 i THEN vouti = a0i + a1i s1+a2i s2+…+ api spi
where si is an input variable; vout is an output variable, Sji is a linguistic fuzzy membership function, and the coefficient set {aji} is the parameter set to be identified.
The idea behind the approach is to combine a global rule-base description with local linear approximations by means of a linear regression model corresponding to a linear input-output model that one would use for describing the system locally. In other words, it is a hybrid approach that combines a fuzzy rule-based and a mathematical method: the antecedents (IF part) define the overlapping patches related to the input-output transfer, and the consequents (THEN part) define linear approximations for the patches. Therefore, the premise portion of the fuzzy rules is identical with the rule base approach, but the consequents are described by mathematical equations.

84. In the parametric method, the linear equation coefficients Aij are trained by the example data. This is comparable with the learning phase of a neural network !
The linear equation outputs are then defuzzified, i.e., the weighted average of the consequents is evaluated by the respective membership values to determine the crisp output.

85. Example, consider two rules :
R1 : IF x1 is BIG AND x2 is MEDIUM THEN u1 = x1-3x2
R2 : IF x1 is SMALL AND x2 is BIG THEN u2 = 4 +2x1
mBIG(x1) = 0.3 mBIG(x2) =0.35
mSMALL(x1) = 0.7 mMED(x2) = 0.75
Thus the weighted normalized sum one gets
U = [0.3(-176) (12)]/( ) =

86. Power Electronics Problem:
Look at the current waveform for a three-phase rectifier and by measuring the width and height of the current pulses try to figure out the RMS total current value

87. Assume that a table of values, gained either from measurements or simulations, is available for a two-input (W and H) single-output (Is) system. The task is to find linear segments of the output function by fitting a straight line on those of its values which correspond to the fuzzy inputs defined by linguistic membership functions.

88. The parametric fuzzy approach has been applied to the estimation of diode bridge rectifier input current waveform distortion. The input diode bridge current in a three-phase systems is distorted, leading to poor power factor and harmonic pollution. Such a current can be viewed as a pulse having a height (H) and a width (W), from which the RMS values for such waves can be calculated. The proposed fuzzy algorithm uses input variables W and H, converting to per unit value and fuzzifying by means of 8 membership functions. The values of W and H were fed as inputs to the linear equations. The estimation algorithm was developed by using a numerical relationship between input and output, while the coefficients of the linear equations were determined by multi-regression linear analysis and subsequently fine-tuned by a number of simulation runs. The output consists of: the RMS value of the total current (Is), the RMS value of the fundamental component current (If), and the displacement power factor (DPF) which measures the phase lag between the fundamental current and the fundamental voltage. In turn, the power factor was calculated from these three estimated waveform parameters. There were only 8 rules while a previous work with a rule based approach required 66 rules.

89. The fuzzy parametric estimation algorithm shown can be summarized as follows:
Read the parameters W and H for an operating condition.
Convert W to a normalized value after dividing by the Wmax value to get W(pu).
Identify the interval in which W(pu) lies.
Fuzzify by calculating the degree of membership 1 and 2, as shown
Fire the two relevant rules and calculate Is1 - Is2, If1 - If2 and DPF1 - DPF2 from the linear equations using the known W and H.
Defuzzify the crisp output by the weighted average (C-o-A) method.

90. Linear Filtering with Parametric Fuzzy Systems
The beauty of Parametric Fuzzy Systems is the ability of representing very complex systems and to embedded linear controllers.
Linear controllers are linear filters as :
This equation describes a filter of n-th order whose coefficients dependent on the state vector

91. The parametric fuzzy controller permits a smooth transition between the individual controllers with the following compact representation :
which expressed linguistically says that
IF state vector x has the property Li
THEN apply controller fi(x)

92. Training Parametric Fuzzy Systems
In order to train the parametric fuzzy system it is necessary to have input-output data sets for the system to be modeled.
Fuzzy partitions are then created for the input variables of the data sets, i. e., fuzzy membership functions are defined to fill in the universe of discourse of the input variables.
Then, the combination of these membership functions for each input variable form the antecedent part of the fuzzy rule base.

93. Equations below show the input-output expressions for a general parametric fuzzy system that uses a set of n fuzzy rules and k inputs.
where
Using the existing data sets, the consequent parameters are estimated to minimize the quadratic error between the parametric fuzzy system output and the available output data. A recursive least-squares (RLS) algorithm is often used to perform this estimation task. Once the fuzzy partitions and the parameters are optimized, the system is ready to replace the original one. Equations above show the input-output expressions for a general parametric fuzzy system that uses a set of n fuzzy rules and k inputs.
The principles for training parametric fuzzy rules are described in:
[1] Takagi, T. & Sugeno, M. “Fuzzy Identification of Systems and Its Applications to Modelling and Control”, IEEE Trans. On Systems, Man and Cybernetics, Vol.SMC-15, N.1, pp , January/February, 1985.
[2] Ying, Hao. “General SISO Takagi–Sugeno Fuzzy Systems with Linear Rule Consequent Are Universal Approximators”, IEEE Transactions On Fuzzy Systems, V.6, N.4, pp , November, 1998.
Given a set of input-output data represented by x1j, x2j,…, xkj, yj, (j=1,2,…,m), the consequent parameters can be estimated by RLS

94. Let X (m x n.(k+1) matrix), Y (m vector) and
P (n.(k+1) vector) be :
Then the parameter vector P can be calculated by

95. The parameter vector can be recursively estimated by a stable-state Kalman filter like:
where xi are lines of the matrix X. The initial values of P0 e S0 are set as follows, with  being a large initial value and I the identity matrix. :

96. The following Matlab programs were developed to implement Parametric Fuzzy Models:
Init_TSModel.m
Adjust_TSModel.m
Run_TSModel.m
A test was made with a straight line data set (Adj1Reta.m) as well as a two straight lines data set (Adj2Reta.m).
The TS system is configured with rectangular membership functions covering the entire input space, one at a time, only two rules and 4 parameters. This will allow the system to learn the exact parameters of the straight lines used to generate the data sets.

97. Estimation of a system with two straight lines

98. Estimated Parameters :
Actual parameters :
a1 = 0.2
b1 = -5.0
a2 = -0.7
b2 = 1.5
Estimated Parameters :
a1=
b1 =
a2 =
b2 =
=> Discontinuity is very hard to capture
=> Convex membership functions will help to smooth out the discontinuity
=> System has good convergence !

99. Approximation of a Nonlinear Curve
Function approximated:
Y = x3 + 1
Fuzzy partition with 6 terms, yielding 12 parameters to be estimated.
Average Error of 1 %.

100. Approximation of a Nonlinear Surface
Function approximated:
Z = 1 – (x2 + y2)
Fuzzy partition with 10 terms, yielding 300 parameters to be estimated.
Average error of 2 %.

101. Performance Criteria
Accuracy: The accuracy of the parametric approach is generally superior to the rule based approach for the same number of rules. Of course, accuracy can be improved by a larger number of membership functions and a correspondingly larger number of rules.
Response time: One outstanding advantage of fuzzy estimation is a very fast that the response time is very fast when compared to conventional hardware or software estimation. This is because the fuzzy method tends to estimate the output instantaneously from the input pattern. The reason is that fuzzy systems are memoryless input-output mapping systems and pattern recognizers like a neural network.
Robustness: Robustness means the system’s relative insensitivity to both external and internal disturbances. The fact that a system can be updated regularly on line in real time assures that the system is rapidly adapted to the latest changes that occurred.

102. Comparison between rule based and parametric fuzzy approaches
Rule based fuzzy approach is more suitable for acquiring and implementing expert human operator knowledge, while the parametric fuzzy approach is best used when input/output numerical data are available.
Parametric fuzzy approach yields a better estimation accuracy because it is a hybrid of rule based fuzzy and numerical components. The rule based fuzzy approach requires no training, while the parametric fuzzy approach requires linear coefficient adjustment performed by statistical multi-linear procedures.
The rule based fuzzy approach is more suitable for acquiring and implementing expert human operator knowledge, while the parametric fuzzy approach is best used with input/output numerical data, if available. The parametric fuzzy approach yields a better estimation accuracy because it is a hybrid of rule based fuzzy and mathematical components. Whenever algorithms are developed in conventional languages, the development time of parametric fuzzy systems is shorter than that of the rule based approach. However, modern commercially available rule based fuzzy development software speeds up the development of rule based systems and it can also improve the accuracy considerably. Such software unfortunately does not cater to the design of parametric fuzzy controllers. The reason is that such software is geared to applications in industrial systems where the emphasis is on capturing the linguistic know-how of human operators rather than resorting to training by means of measurements. The rule based fuzzy approach requires no training, while the parametric fuzzy approach requires linear coefficient adjustment performed by statistical multi-linear procedures.
The parametric fuzzy algorithm is inherently adaptive, because the coefficients Aij can be altered for system tuning. Thus a real-time adaptive implementation of the parametric approach is feasible by dynamically changing the linear coefficients by means of a recursive least-square algorithm repeatedly on a recurrent basis, i.e. repeatedly either with a certain fixed frequency or whenever determined by observed factors. In adaptive versions of the rule-based approach , changing the rule weights (Degree of Support) or the membership functions recurrently is theoretically possible. Commercially available neurofuzzy development software (i.e. a combination of neural and fuzzy techniques) does make non-recurrent adaptation practically achievable. However, the recurrent updating of the appropriate parameters is presently not catered to by commercial software packages. With both the rule based and the parametric approach, tuning requires extensive iteration with the input-output training data generated by simulation or experiment.
disadvantage of the parametric fuzzy approach is the loss of the linguistic formulation of output consequents. In a heavy-industrial plant/process control environment the rule-based approach is still more attractive. However, the parametric fuzzy approach is not restricted to industrial control.

103. Parametric fuzzy algorithm is inherently adaptive, because the coefficients Aij can be altered for system tuning. Thus a real-time adaptive implementation of the parametric approach is feasible by dynamically changing the linear coefficients by means of a recursive least-square algorithm repeatedly on a recurrent basis
Adaptive versions of the rule-based approach , changing the rule weights (Degree of Support) or the membership functions recurrently is possible.
Disadvantage of the parametric fuzzy approach is the loss of the linguistic formulation of output consequents, sometimes important for industrial plant/process control environment