Modeling,stability analysis and computational aspects of nonlinear fuzzy PID controllers

Abstract

This paper presents mathematical models of the simplest fuzzy PID controller of Mamdani type. This controller is called the “simplest” as it employs minimum number of fuzzy sets (two on each input variable and four on output variable) while satisfying the control rule base that contains four rules relating all six input fuzzy sets to all four output fuzzy sets. L - type, Γ - type and Π - type membership functions are considered in fuzzification process of input and output variables. Controller modeling is done via algebraic product AND operator-maximum OR operator-Larsen product inference method-Centre of Sums (CoS) defuzzification process combination. The new model obtained in this manner turns out to be nonlinear, and its properties are studied. Since digital controllers are implemented on the digital processors, the computational and memory requirements of the fuzzy controller and conventional (nonfuzzy) controller are compared. Stability analysis of closed loop systems containing the fuzzy controller models is done using the small gain theorem.

Keywords

Mathematical model fuzzy control PID controller Mamdani type controller center of sums defuzzification

1 Introduction

Conventional (linear) PID controllers are the most widely used controllers in industry (Kumar et al. [28]) due to their ease of implementation and well known tuning methods. However, they generally do not ensure satisfactory performance for nonlinear systems, higher order linear systems and time - delay systems. To overcome this limitation, various kinds of nonlinear PID controllers such as auto tuned and adaptive PID controllers have been developed. In these cases of nonlinear PID controllers usually its analytical structure u = f (x), where x and u are the inputs and outputs of the controller, is known. So their design involves selection of suitable f and tuning the parameters to meet the performance specifications. However, these methods require accurate mathematical model of the plant. Alternatively, controllers employing fuzzy logic were also used. Fuzzy logic controllers perform well for complex plants without accurate mathematical models because, to a large extent, the system’s behavior is captured and represented in the forms of fuzzy sets and fuzzy control rules provided by the human control expert. The drawbacks of this method are

Lot of trials are required to choose fuzzy sets and control rules and are subject to experience of human control expert.

Analysis and precise understanding of the controller is not possible.

In this context, finding analytical structure f with various components of fuzzy logic controllers which gave better results previously will have the following advantages:

Analysis and design can be done with the helpof various time-tested nonlinear control methods, thereby minimizing trial and error effort.

The structure can be easily implemented on various digital platforms like Field Programmable Gate Arrays (FPGAs), digital computers, microcontrollers etc. at lower price.

They can be applied to safety critical industries such as nuclear power plants as stability of the overall control system is guaranteed.

Let us now take a look at the historical developments in fuzzy control technology. The superiority of fuzzy PID controller over conventional PID controller was validated with experimental studies by Lim [3]. Fuzzy rules for tuning of PID controllers for better performance without system identification was presented in Baras and Patel [4]. It has been shown by Mizumoto [5] that PID controllers can be realized by product-sum-gravity method and simplified fuzzy reasoning method. Analytical structure for a fuzzy PID controller and its Bounded Input Bounded Output (BIBO) stability analysis have been studied in Misir et al. [6]. A systematic and hierarchical approach to the design of a hybrid fuzzy-PID controller through the application of a learning - based algorithm was described in Homaifar et al. [7]. The stability of fuzzy controllers is analyzed with the help of passivity theorem by Sio and Lee [8]. Mann et al. [9] investigated different fuzzy PID controller structures. By expressing the fuzzy rules in different forms, each PID structure was distinctly identified. It was shown by Hayashi et al. [10] that nonlinear PID controller, based on conventional PID controller, could be easily realized by applying the simplified indirect fuzzy inference method. Fuzzy PI and fuzzy PD controllers have been combined in parallel to get a fuzzy PID controller by Kim and Oh [11]. A fuzzy gain scheduling scheme that allows for the online replacement and subsequent improvement of existing conventional PID control performance has been developed by Blanchett et al. [12]. Some practical recommendations for replacing control by a human operator with a fuzzy controller, and for choosing the structure and parameters of this fuzzy controller were given in Reznik et al. [13]. Tao and Taur [14] proposed a flexible complexity reduced design approach for PID-like fuzzy controllers. Analytical structures of various fuzzy controllers were presented in the book Ying [15]. Stability analysis, design and real time implementation of some of the developed analytical structures were nicely discussed in this book.

Several forms of decomposed PID fuzzy logic controllers have been tested and compared in Golob [16]. A function-based evaluation approach has been proposed in Hu et al. [17] for a systematic study of fuzzy PID controllers. Analytical structures of fuzzy PID controllers were obtained by Du et al. [19] using triangular membership functions for the input variables, singletons for the output variable, linear control rules and center of area defuzzification. Using the analytical structures it is proved that the fuzzy PID controller is the sum of a global three-dimensional multi-level relay and a local nonlinear controller. When the number of fuzzy sets tends to infinity, the local nonlinear controller will disappear, and the degree of nonlinearity of the fuzzy PID controller becomes zero. Structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear PID control with variable gains was done by Haj - Ali and Ying [20]. Mann and Gosine [21] proposed a novel algorithm to produce an analytical solution for three-input fuzzy PID systems. They provided an efficient solution algorithm to produce the general fuzzy output solution using a minimum number of nonlinear expressions. Mohan and Sinha [22] have shown that algebraic product triangular norm - bounded sum triangular co - norm - Larsen product inference method - CoS defuzzification method combination leads to a linear fuzzy PID controller. Mohan and Sinha [23] have shown that the analytical structures of fuzzy PID controllers derived via minimum triangular norm are not suitable for control. Mathematical models of fuzzy controllers with arbitrary number of inputs were presented by Du et al. [24]. They investigated BIBO global stability as well as local stability and presented a systematic design procedure for regulating nonlinear plants. Mohan and Sinha [25] introduced an analytical structure for fuzzy PID controller by employing algebraic product triangular norm, bounded sum triangular co - norm, minimum inference method and CoS defuzzification method. They also have derived conditions for stability using the Small Gain theorem. Analytical structures of fuzzy PID controllers and their applications were presented by Edara et al. [26]. An error based on-line rule weight adjustment method for fuzzy PID controller was proposed by Karasakal et al. [27]. Mohan and Neethu [29] have presented a new nonlinear fuzzy PID controller model by using algebraic product triangular norm, bounded sum triangular co - norm, Larsen product inference method and CoS defuzzification. Design procedure for function based fuzzy PD and fuzzy I control in parallel was outlined by Dadam and Eldukhri [30]. Das et al. [31] proposed a novel fractional order fuzzy PID controller and its optimal time-domain tuning. Stabililty analysis of parallel fuzzy P plus fuzzy I plus fuzzy D control systems was done with the help of mathematical models of the controllers (Vineet et al. [32]). Karasakal et al. [33] proposed a method of online tuning of fuzzy PID controllers via rule weighting based on normalized acceleration. Gharghory and Kamal [34] proposed modified PSO for optimal tuning of fuzzy PID controller. Zhang et al. [35] proposed a nonlinear fuzzy PID control algorithm, whose membership function is adjustable. Appling this function to fuzzy control will increase system’s tunability. Effectiveness and feasibility of this function are verified in the simulation and experimental results. Savran and Kahraman [36] proposed a fuzzy model based adaptive PID controller design for nonlinear and uncertain processes. Fereidouni et al. [37] proposed a new adaptive configuration of PID type fuzzy logic controller. Gil et al. [38] proposed a performance driven approach for tuning of fuzzy PID controllers for multi - input multi - output systems.

In the literature very few models of three input fuzzy PID controllers of Mamdani type are available. As the mathematical models of fuzzy logic controller are useful in the analysis, design and implementation, in this paper an attempt is made to derive a nonlinear three - input fuzzy PID controller using L-type, Γ-type and Π-type membership functions for input and output variables, algebraic product AND operator, maximum OR operator, Larsen product inference and CoS defuzzification. We study and compare the computational aspects of the proposed fuzzy controller and the fuzzy PID controller derived via bounded sum OR operator (Mohan and Neethu [29]). We also establish a sufficient condition for BIBO stability of a closed loop control system containing one of these fuzzy PID controller models. The paper is organized as follows: The next section deals with fundamental components of a typical fuzzy PID controller. Section 3 presents mathematical models of two classes of fuzzy PID controllers. Properties of the models are discussed in Section 4. In Section 5 computational aspects of fuzzy PID controller models are discussed. In Section 6 BIBO stability analysis of feedback systems involving the simplest fuzzy PID controller is presented. In Section 7 illustrative examples showing computation of gains of nonlinear plants are provided. Section 8 concludes the paper.

2 Principal components of a typical fuzzy PID controller

The block diagram of a typical computer controlled system is shown in Fig. 1. For brevity we drop T in kT in the following expressions.

The incremental control effort generated by a discrete-time PID controller is given by: $Δ u (k) = K_{P}^{d} Δ e (k) + K_{I}^{d} e (k) + K_{D}^{d} Δ^{2} e (k)$ (1) $Δ u (k) = Δ e_{s} (k) + e_{s} (k) + Δ^{2} e_{s} (k)$ (2) where Δu(k) = u(k) − u(k − 1), Δe(k) = e(k) − e(k − 1), Δ²e(k) = Δe(k) − Δe(k − 1) and $K_{P}^{d}, K_{I}^{d}$ and $K_{D}^{d}$ are respectively the proportional, integral and derivative constants of discrete-time PID controller. The inputs to fuzzy PID controller are shown in Fig. 2. The block diagram of a typical fuzzy PID controller is shown in Fig. 3 in which S_e, S_Δe, S_Δ²e and $S_{Δ u}^{- 1}$ represent the scaling factors of the fuzzy controller. The scaled inputs are given by e_s(k) = S_e · e(k), Δe_s(k) = S_Δe · Δe(k) and Δ²e_s(k) = S_Δ²e · Δ²e(k). The principal components of fuzzy PID controller are fuzzification and defuzzification modules, control rule base, and inference engine which are described in the following.

2.1 Fuzzification module

The inputs are fuzzified by L - type and Γ - type membership functions, shown in Fig. 4, whose mathematical description is given by $μ_{N_{e}} (e_{s} (k)) = {\begin{matrix} 1 & if - H_{e} \leq e_{s} (k) \leq - h_{e} \\ \frac{- e_{s} (k) + h_{e}}{2 h_{e}} & if - h_{e} \leq e_{s} (k) \leq h_{e} \\ 0 & if h_{e} \leq e_{s} (k) \leq H_{e} \end{matrix}$ (3) $μ_{P_{e}} (e_{s} (k)) = {\begin{matrix} 0 & if - H_{e} \leq e_{s} (k) \leq - h_{e} \\ \frac{e_{s} (k) + h_{e}}{2 h_{e}} & if - h_{e} \leq e_{s} (k) \leq h_{e} \\ 1 & if h_{e} \leq e_{s} (k) \leq H_{e} \end{matrix}$ (4)

Similarly, the mathematical descriptions of the other membership functions on two scaled inputs Δe_s(k) and Δ²e_s(k) can be defined. Notice that μ_{N
_e}(e_s(k)) + μ_{P
_e}(e_s(k)) =1, ∀ e_s(k). This is true for the other two inputs also. The membership functions (ΔU₋₂, ΔU₋₁, ΔU₊₁ and ΔU₊₂) of the scaled output fuzzy sets are shown in Fig. 5, where Δu_s(k) = S_Δu · Δu(k).The interval [-B B] represents the universe of discourse of the scaled output. The value of parameter A should lie in $[0 \frac{B}{3})$ . When the value of A = 0 the trapezoidal membership functions ΔU₋₁ and ΔU₊₁ become triangular membership functions.

2.2 Control rule base

Total eight (2³) rules are required as there are two fuzzy sets defined on each of the three input variables. The following is the standard rule base followed in Mohan and Sinha [22]

R₁ : IF e_s(k) is N_e AND Δe_s(k) is N_Δe AND Δ²e_s(k) is N_Δ²e THEN Δu_s(k) is ΔU₋₂

R₂ : IF e_s(k) is N_e AND Δe_s(k) is N_Δe AND Δ²e_s(k) is P_Δ²e THEN Δu_s(k) is ΔU₋₁

R₃ : IF e_s(k) is N_e AND Δe_s(k) is P_Δe AND Δ²e_s(k) is N_Δ²e THEN Δu_s(k) is ΔU₋₁

R₄ : IF e_s(k) is P_e AND Δe_s(k) is N_Δe AND Δ²e_s(k) is N_Δ²e THEN Δu_s(k) is ΔU₋₁

R₅ : IF e_s(k) is N_e AND Δe_s(k) is P_Δe AND Δ²e_s(k) is P_Δ²e THEN Δu_s(k) is ΔU₊₁

R₆ : IF e_s(k) is P_e AND Δe_s(k) is N_Δe AND Δ²e_s(k) is P_Δ²e THEN Δu_s(k) is ΔU₊₁

R₇ : IF e_s(k) is P_e AND Δe_s(k) is P_Δe AND Δ²e_s(k) is N_Δ²e THEN Δu_s(k) is ΔU₊₁

R₈ : IF e_s(k) is P_e AND Δe_s(k) is P_Δe AND Δ²e_s(k) is P_Δ²e THEN Δu_s(k) is ΔU₊₂

Notice that rules R₂, R₃ and R₄ fire the same output fuzzy set ΔU₋₁. Therefore, the above researchers used the fuzzy OR operator (s - norm) on the THEN (consequent) parts of rules R₂, R₃ and R₄. Similarly fuzzy OR operator (s - norm) is used on the THEN (consequent) parts of rules R₅, R₆ and R₇. Absolutely nothing wrong in this approach! In practice it is always desirable to have less number of rules. This is possible by clubbing the rules R₂, R₃ and R₄ together into a single rule and R₅, R₆ and R₇ together into a single rule. So, the modified rule base containing only four rules is as follows:

R₁ : IF e_s(k) is N_e AND Δe_s(k) is N_Δe AND Δ²e_s(k) is N_Δ²e THEN Δu_s(k) is ΔU₋₂

R₂ : IF (e_s(k) is N_e AND Δe_s(k) is N_Δe AND Δ²e_s(k) is P_Δ²e) OR (e_s(k) is N_e AND Δe_s(k) is P_Δe AND Δ²e_s(k) is N_Δ²e) OR (e_s(k) is P_e AND Δe_s(k) is N_Δe AND Δ²e_s(k) is N_Δ²e) THEN Δu_s(k) is ΔU₋₁

R₃ : IF (e_s(k) is N_e AND Δe_s(k) is P_Δe AND Δ²e_s(k) is P_Δ²e) OR (e_s(k) is P_e AND Δe_s(k) is N_Δe AND Δ²e_s(k) is P_Δ²e) OR (e_s(k) is P_e AND Δe_s(k) is P_Δe AND Δ²e_s(k) is N_Δ²e) THEN Δu_s(k) is ΔU₊₁

R₄ : IF e_s(k) is P_e AND Δe_s(k) is P_Δe AND Δ²e_s(k) is P_Δ²e THEN Δu_s(k) is ΔU₊₂

Notice that the IF (premise) part of rules R₂ and R₃ contain three parts and are combined by the fuzzy OR operator.

Algebraic product t - norm is considered to perform the AND operation in the rule base, and is defined as $Algebraic product : t (μ_{A} (x), μ_{B} (y)) = μ_{A} (x) \cdot μ_{B} (y)$

Maximum t-co-norm is considered to perform the OR operation in the rule base, and is defined as: $Maximum : s (μ_{A} (x), μ_{B} (y)) = \max {μ_{A} (x), μ_{B} (y)}$

We consider all possible combinations of three input variables in a 3D space. A point, say (x_l, y_l, z_l), in a 3D space can always be distinctly shown by taking its projections on the xy, yz and zx planes. So, as shown in Fig. 6, twenty input combinations are considered in each input plane (e_s(k) Δe_s(k), Δe_s(k) Δ²e_s(k) and Δ²e_s(k) e_s(k)) so that the input point (e_s(k), Δe_s(k), Δ²e_s(k)) can be uniquely located in the 3D cell (subspace) represented by the triplet (n_a, n_b, n_c) where n_a, n_b, n_c = 1, 2, 3,. . . , 20. For example, the triplet (9, 11, 20) represents the 3D cell with 9 from (a), 11 from (b), and 20 from (c) of Fig. 6. A cell is said to be valid if and only if the relations between e_s(k) and Δe_s(k), and Δe_s(k) and Δ²e_s(k) produce the relation between Δ²e_s(k) and e_s(k). For example, the cell (13, 14, 16) is a valid cell because the relations |Δe_s(k) | ≥ e_s(k) (valid in 13 of Fig. 6(a)) and |Δ²e_s(k) | ≥ |Δe_s(k) | (valid in 14 of Fig. 6(b)) will produce the relation |Δ²e_s(k) | ≥ e_s(k) which is valid in 16 of Fig. 6(c). The control rules R₁ to R₄ are used to evaluate appropriate control law in each valid cell (n_a, n_b, n_c). Let the outcomes of premise part of rules R₁, R₂, R₃ and R₄ be μ₋₂, μ₋₁, μ₊₁ and μ₊₂ respectively. By using the algebraic product triangular norm and the maximum triangular co - norm, the values of μ₋₂, μ₋₁, μ₊₁ and μ₊₂ are found in each valid cell and are shown in Table 1.

2.3 Inference engine

Larsen product inference method is used here. The expression of μ₋₂ is used to modify the reference output fuzzy set ΔU₋₂ in rule R₁. Similarly, the expressions of μ₋₁, μ₊₁, μ₊₂ are used to modify ΔU₋₁, ΔU₊₁ and ΔU₊₂, respectively. The reference membership functions and inferred membership functions ( $Δ {\tilde{U}}_{- 2}$ , $Δ {\tilde{U}}_{- 1}$ , $Δ {\tilde{U}}_{+ 1}$ and $Δ {\tilde{U}}_{+ 2}$ ) obtained after employing Larsen product inference are shown in Fig. 5 with hatching.

2.4 Defuzzification module

Defuzzification is done here using the well known Centre of Sums (CoS) method (Drainkov et al., [2]). According to this method, the crisp value of scaled control output is given by

$\begin{matrix} Δ u_{s}^{*} (k) \\ = \frac{\begin{matrix} [\bar{A} (Δ {\tilde{U}}_{- 2}) \bar{C} (Δ {\tilde{U}}_{- 2}) + \bar{A} (Δ {\tilde{U}}_{- 1}) \bar{C} (Δ {\tilde{U}}_{- 1}) \\ + \bar{A} (Δ {\tilde{U}}_{+ 1}) \bar{C} (Δ {\tilde{U}}_{+ 1}) + \bar{A} (Δ {\tilde{U}}_{+ 2}) \bar{C} (Δ {\tilde{U}}_{+ 2})] \end{matrix}}{\begin{matrix} [\bar{A} (Δ {\tilde{U}}_{- 2}) + \bar{A} (Δ {\tilde{U}}_{- 1}) \\ + \bar{A} (Δ {\tilde{U}}_{+ 1}) + \bar{A} (Δ {\tilde{U}}_{+ 2})] \end{matrix}} \end{matrix}$ (5) where $\bar{A} (.)$ and $\bar{C} (.)$ are respectively, the area and centroid corresponding to the inferred output membership function. It is easy to compute the following from Fig. 5:

$\bar{A} (Δ {\tilde{U}}_{- 2})$ = $\frac{1}{3} μ_{- 2} B$

$\bar{C} (Δ {\tilde{U}}_{- 2})$ = $\frac{- (14 B^{2} + 6 AB - 9 A^{2})}{18 B}$

$\bar{A} (Δ {\tilde{U}}_{- 1})$ = $\frac{2}{3} μ_{- 1} B$

$\bar{C} (Δ {\tilde{U}}_{- 1})$ = $\frac{- B}{3}$

$\bar{A} (Δ {\tilde{U}}_{+ 1})$ = $\frac{2}{3} μ_{+ 1} B$

$\bar{C} (Δ {\tilde{U}}_{+ 1})$ = $\frac{B}{3}$

$\bar{A} (Δ {\tilde{U}}_{+ 2})$ = $\frac{1}{3} μ_{+ 2} B$

$\bar{C} (Δ {\tilde{U}}_{+ 2})$ = $\frac{14 B^{2} + 6 AB - 9 A^{2}}{18 B}$

3 Mathematical models of the simplest fuzzy PID controllers

We have the following two classes of controllers:

Class 1. (Using maximum OR): Δu_s(k) = $\frac{N_{1}}{D_{1}}$

Class 2. (Using bounded sum OR): Δu_s(k) = $\frac{N_{2}}{D_{2}}$

The models obtained in Mohan and Neethu [29] are presented here with the notation followed in this paper for the sake of completeness.

For simplicity let h₁ = h_e, h₂ = h_Δe, h₃ = h_Δ²e, x₁ = e_s(k), x₂ = Δe_s(k), and x₃ = Δ²e_s(k). Then upon substituting the values of μ₋₂, μ₋₁, μ₊₁ and μ₊₂, defined in Table 1, in $\bar{A} (.) s$ and $\bar{C} (.) s$ in Equation (5), we have the following expressions for class 1 controller:

Case 1: In cells (1, 2, 3), (1, 3, 4), (2, 3, 1), (3, 1, 2), (3, 4, 1), (4, 1, 3) $\begin{matrix} N_{1} = N_{2} = S_{1} B \\ D_{1} = D_{2} = 3 \end{matrix}$ and S₁ is as defined in Table 1. Here S₁ is the sign term (either + or -). Such a term is introduced here to present the mathematical models in all the possible cells in a concise manner.

Case 2: In cells (2, 2, 2), (4, 4, 4) $\begin{matrix} N_{1} = N_{2} = S_{1} (14 B^{2} + 6 AB - 9 A^{2}) \\ D_{1} = D_{2} = 18 B \end{matrix}$

Case 3: In cells (1, 7, 9), (1, 8, 10), (3, 11, 5), (3, 12, 6) $\begin{matrix} N_{1} = N_{2} = S_{1} {Bx}_{3} \\ D_{1} = D_{2} = 3 h_{3} \end{matrix}$

Case 4: In cells (2, 7, 6), (2, 8, 5), (4, 11, 10), (4, 12, 9) $\begin{matrix} N_{1} & = & N_{2} = (6 AB - 9 A^{2}) (x_{3} + S_{1} h_{3}) \\ + 2 B^{2} (x_{3} + S_{1} 13 h_{3}) \\ D_{1} & = & D_{2} = 18 B [3 h_{3} - S_{1} x_{3}] \end{matrix}$

Case 5: In cells (5, 2, 8), (6, 2, 7), (9, 4, 12), (10, 4, 11) $\begin{matrix} N_{1} & = & N_{2} = (6 AB - 9 A^{2}) (x_{1} + S_{1} h_{1}) \\ + 2 B^{2} (x_{1} + S_{1} 13 h_{1}) \\ D_{1} & = & D_{2} = 18 B [3 h_{1} - S_{1} x_{1}] \end{matrix}$

Case 6: In cells (5, 3, 11), (6, 3, 12), (9, 1, 7), (10, 1, 8) $\begin{matrix} N_{1} = N_{2} = S_{1} {Bx}_{1} \\ D_{1} = D_{2} = 3 h_{1} \end{matrix}$

Case 7: In cells (7, 6, 2), (8, 5, 2), (11, 10, 4), (12, 9, 4) $\begin{matrix} N_{1} & = & N_{2} = (6 AB - 9 A^{2}) (x_{2} + S_{1} h_{2}) \\ + 2 B^{2} (x_{2} + S_{1} 13 h_{2}) \\ D_{1} & = & D_{2} = 18 B [3 h_{2} - S_{1} x_{2}] \end{matrix}$

Case 8: In cells (7, 9, 1), (8, 10, 1), (11, 5, 3), (12, 6, 3) $\begin{matrix} N_{1} = N_{2} = S_{1} {Bx}_{2} \\ D_{1} = D_{2} = 3 h_{2} \end{matrix}$

Case 9: In cells (5, 7, 16), (5, 7, 17), (5, 8, 18), (6, 7, 15), (9, 11, 13), (9, 11, 20), (9, 12, 14), (10, 11, 19) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) {h_{3} x_{1} + h_{1} x_{3} + S_{1} (x_{1} x_{3} + h_{1} h_{3})} \\ + B^{2} {38 h_{1} x_{3} - S_{1} 10 x_{1} x_{3} + 14 h_{3} (x_{1} + S_{1} h_{1})} \\ D_{1} & = & 18 B [5 h_{1} h_{3} + x_{1} x_{3} - S_{1} {3 h_{3} x_{1} - h_{1} x_{3}}] \end{matrix}$

$\begin{matrix} N_{2} & = & (26 B^{2} + 6 AB - 9 A^{2}) (h_{3} x_{1} + h_{1} x_{3} + S_{1} h_{1} h_{3}) \\ + S_{1} (- 22 B^{2} + 6 AB - 9 A^{2}) x_{1} x_{3} \\ D_{2} & = & 18 B [7 h_{1} h_{3} - x_{1} x_{3} - S_{1} (h_{3} x_{1} + h_{1} x_{3})] \end{matrix}$

Case 10: In cells (5, 8, 19), (6, 7, 14), (6, 8, 13), (6, 8, 20), (9, 12, 15), (10, 11, 18), (10, 12, 16), (10, 12, 17) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) {h_{3} x_{1} + h_{1} x_{3} + S_{1} (x_{1} x_{3} + h_{1} h_{3})} \\ + B^{2} {38 h_{3} x_{1} - S_{1} 10 x_{1} x_{3} + 14 h_{1} (x_{3} + S_{1} h_{3})} \\ D_{1} & = & 18 B [5 h_{1} h_{3} + x_{1} x_{3} - S_{1} {3 h_{1} x_{3} - h_{3} x_{1}}] \end{matrix}$ N₂ and D₂ are same as in case 9.

Case 11: In cells (7, 14, 6), (8, 13, 6), (8, 19, 5), (8, 20, 6), (11, 18, 10), (12, 15, 9), (12, 16, 10), (12, 17, 10) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) {h_{3} x_{2} + h_{2} x_{3} + S_{1} (x_{2} x_{3} \\ + h_{2} h_{3})} + B^{2} {38 h_{2} x_{3} - S_{1} 10 x_{2} x_{3} \\ + 14 h_{3} (x_{2} + S_{1} h_{2})} \\ D_{1} & = & 18 B [5 h_{2} h_{3} + x_{2} x_{3} - S_{1} {3 h_{3} x_{2} - h_{2} x_{3}}] \\ N_{2} & = & (26 B^{2} + 6 AB - 9 A^{2}) (h_{3} x_{2} + h_{2} x_{3} + S_{1} h_{2} h_{3}) \\ + S_{1} (- 22 B^{2} + 6 AB - 9 A^{2}) x_{2} x_{3} \\ D_{2} & = & 18 B [7 h_{2} h_{3} - x_{2} x_{3} - S_{1} (h_{3} x_{2} + h_{2} x_{3})] \end{matrix}$

Case 12: In cells (7, 15, 6), (7, 16, 5), (7, 17, 5), (8, 18, 5), (11, 13, 9), (11, 19, 10), (11, 20, 9), (12, 14, 9) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) {h_{3} x_{2} + h_{2} x_{3} \\ + S_{1} (x_{2} x_{3} + h_{2} h_{3})} \\ + B^{2} {38 h_{3} x_{2} - S_{1} 10 x_{2} x_{3} \\ + 14 h_{2} (x_{3} + S_{1} h_{3})} \\ D_{1} & = & 18 B [5 h_{2} h_{3} + x_{2} x_{3} - S_{1} {3 h_{2} x_{3} - h_{3} x_{2}}] \end{matrix}$ N₂ and D₂ are same as in case 11.

Case 13: In cells (13, 6, 8), (14, 6, 7), (15, 9, 12), (16, 10, 12), (17, 10, 12), (18, 10, 11), (19, 5, 8), (20, 6, 8) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) {h_{2} x_{1} + h_{1} x_{2} \\ + S_{1} (x_{1} x_{2} + h_{1} h_{2})} \\ + B^{2} {38 h_{1} x_{2} - S_{1} 10 x_{1} x_{2} \\ + 14 h_{2} (x_{1} + S_{1} h_{1})} \\ D_{1} & = & 18 B [5 h_{1} h_{2} + x_{1} x_{2} \\ - S_{1} {3 h_{2} x_{1} - h_{1} x_{2}}] \\ N_{2} & = & (26 B^{2} + 6 AB - 9 A^{2}) (h_{1} x_{2} + h_{2} x_{1} \\ + S_{1} h_{1} h_{2}) + S_{1} (- 22 B^{2} \\ + 6 AB - 9 A^{2}) x_{1} x_{2} \\ D_{2} & = & 18 B [7 h_{1} h_{2} - x_{1} x_{2} - S_{1} (h_{1} x_{2} + h_{2} x_{1})] \end{matrix}$

Case 14: In cells (13, 9, 11), (14, 9, 12), (15, 6, 7), (16, 5, 7), (17, 5, 7), (18, 5, 8), (19, 10, 11), (20, 9, 11) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) {h_{2} x_{1} + h_{1} x_{2} + S_{1} (x_{1} x_{2} + h_{1} h_{2})} \\ + B^{2} {38 h_{2} x_{1} - S_{1} 10 x_{1} x_{2} + 14 h_{1} (x_{2} + S_{1} h_{2})} \\ D_{1} & = & 18 B [5 h_{1} h_{2} + x_{1} x_{2} - S_{1} {3 h_{1} x_{2} - h_{2} x_{1}}] \end{matrix}$ N₂ and D₂ are same as in case 13.

Case 15: In cells (13, 14, 16), (14, 14, 15), (15, 15, 14), (15, 16, 13), (15, 17, 13), (15, 17, 20), (16, 18, 13), (16, 18, 20), (17, 18, 20), (18, 18, 19), (19, 13, 16), (19, 13, 17), (19, 19, 18), (19, 20, 17), (20, 14, 16), (20, 14, 17) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) (h_{2} h_{3} x_{1} + h_{1} h_{3} x_{2} + h_{1} h_{2} x_{3} \\ + x_{1} x_{2} x_{3}) + B^{2} {26 h_{1} h_{3} x_{2} \\ + 2 x_{1} x_{2} x_{3} + 14 h_{2} (h_{1} x_{3} + h_{3} x_{1}) \\ + S_{1} 12 x_{2} (h_{3} x_{1} - h_{1} x_{3})} \\ D_{1} & = & 18 B [3 h_{1} h_{2} h_{3} + x_{2} (h_{3} x_{1} \\ + h_{1} x_{3}) - h_{2} x_{1} x_{3} \\ + S_{1} 2 h_{2} (h_{3} x_{1} - h_{1} x_{3})] \\ N_{2} & = & (26 B^{2} + 6 AB - 9 A^{2}) (h_{2} h_{3} x_{1} \\ + h_{1} h_{3} x_{2} + h_{1} h_{2} x_{3}) \\ + (- 22 B^{2} + 6 AB - 9 A^{2}) x_{1} x_{2} x_{3} \\ D_{2} & = & 18 B [7 h_{1} h_{2} h_{3} - h_{3} x_{1} x_{2} - h_{2} x_{1} x_{3} - h_{1} x_{2} x_{3}] \end{matrix}$

Case 16: In cells (13, 15, 16), (13, 15, 17), (13, 16, 18), (14, 15, 15), (15, 14, 14), (16, 13, 14), (16, 19, 13), (16, 20, 14), (17, 19, 13), (17, 19, 20), (17, 20, 14), (18, 19, 19), (19, 18, 18), (20, 15, 17), (20, 16, 18), (20, 17, 18) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) (h_{2} h_{3} x_{1} + h_{1} h_{3} x_{2} \\ + h_{1} h_{2} x_{3} + x_{1} x_{2} x_{3}) + B^{2} {26 h_{1} h_{2} x_{3} \\ + 2 x_{1} x_{2} x_{3} + 14 h_{3} (h_{1} x_{2} + h_{2} x_{1}) \\ + S_{1} 12 x_{3} (h_{2} x_{1} - h_{1} x_{2})} \\ D_{1} & = & 18 B [3 h_{1} h_{2} h_{3} + x_{3} (h_{1} x_{2} \\ + h_{2} x_{1}) - h_{3} x_{1} x_{2} + S_{1} 2 h_{3} (h_{2} x_{1} - h_{1} x_{2})] \end{matrix}$ N₂ and D₂ are same as in case 15

Case 17: In cells (13, 16, 19), (13, 17, 19), (14, 15, 14), (14, 16, 13), (14, 16, 20), (14, 17, 20), (15, 14, 15), (16, 13, 15), (17, 13, 15), (17, 20, 15), (18, 13, 16), (18, 19, 18), (18, 20, 16), (18, 20, 17), (19, 18, 19), (20, 17, 19) $\begin{matrix} N_{1} & = & (6 AB - 9 A^{2}) (h_{2} h_{3} x_{1} + h_{1} h_{3} x_{2} \\ + h_{1} h_{2} x_{3} + x_{1} x_{2} x_{3}) + B^{2} {26 h_{2} h_{3} x_{1} \\ + 2 x_{1} x_{2} x_{3} + 14 h_{1} (h_{3} x_{2} + h_{2} x_{3}) \\ + S_{1} 12 x_{1} (h_{2} x_{3} - h_{3} x_{2})} \\ D_{1} & = & 18 B [3 h_{1} h_{2} h_{3} + x_{1} (h_{3} x_{2} + h_{2} x_{3}) \\ - h_{1} x_{2} x_{3} + S_{1} 2 h_{1} (h_{2} x_{3} - h_{3} x_{2})] \end{matrix}$ N₂ and D₂ are same as in case 15.

4 Properties of the simplest fuzzy PID controllers

Fuzzy controllers presented in the previous section possess certain interesting properties which are discussed here.

The control surface generated by Δu_s(k) is continuous at any point in the 3D input space. This can be verified by calculating the data of Δu_s(k) at various points in the 3 - dimensional input space.

The magnitude of incremental control effort increases monotonically as the distance of the input point increases from the origin of 3D input space.

Fuzzy PID controllers are different nonlinear controllers. This is evident from the expressions of Δu_s(k) in Section 3.

Fuzzy PID controllers are variable structure controllers as their structures are different in different input cells.

The minimum incremental control effort,given by $\frac{- (14 B^{2} + 6 AB - 9 A^{2})}{18 B}$ , occurs at (e_s(k), Δe_s(k), Δ²e_s(k)) = (−h_e, −h_Δe, −h_Δ²e). See Case 2 of Section 3.

The incremental control effort is zero at the origin of 3D input space.

The maximum incremental control effort, given by $\frac{(14 B^{2} + 6 AB - 9 A^{2})}{18 B}$ , occurs at (e_s(k), Δe_s(k), Δ²e_s(k)) = (h_e, h_Δe, h_Δ²e). See Case 2 of Section 3.

5 Computational aspects of the simplest fuzzy PID controllers

Conventional PID controllers are still widely used in industries because they provide control output quickly. The computational delay introduced by them in the loop is almost insignificant. In fact, it is the smallest when compared with any other control scheme. It can be seen from the mathematical model of PID controllers (Equation 2) that they require only two mathematical operations (additions). But conventional PID controllers do not work satisfactorily for nonlinear, higher order and time - delay systems. Nonlinear controllers like fuzzy controllers will provide better performance for such systems. As the mathematical models of different classes of the simplest fuzzy PID controllers are available, we can find the number of mathematical operations and memory locations required during their implementation. The number of such mathematical operations and memory locations required for class 1 fuzzy controller is explained here first.

The expression for class 1 fuzzy controller in cell (13, 14, 16) is $\begin{matrix} Δ u_{s} (k) \\ = \frac{\begin{matrix} [N_{c 10} \cdot x_{1} + N_{c 11} \cdot x_{2} + N_{c 12} \cdot x_{3} \\ + N_{c 5} \cdot x_{1} \cdot x_{2} \cdot x_{3} + N_{c 6} \cdot x_{2} \cdot (h_{3} x_{1} - h_{2} x_{1})] \end{matrix}}{\begin{matrix} {D_{c 1} \cdot [D_{c 2} + x_{2} \cdot (h_{3} x_{1} + h_{1} x_{3}) \\ - h_{2} \cdot x_{1} \cdot x_{3} + D_{c 3} \cdot x_{1} - D_{c 4} \cdot x_{3}]} \end{matrix}} \end{matrix}$ which requires 29 operations (18 multiplications, 7 additions, 3 subtractions and 1 division). We call these operations on - line operations because these twenty nine operations are required to be carried out for every k^th sampling instant. In the above equation $N_{c 1} = 6 AB - 9 A^{2}$ which requires 5 operations (4 multiplications and 1 subtraction) and 1 memory location, $N_{c 2} = B^{2}$ which requires 1 operation (multiplication) and 1 memory location, $N_{c 3} = N_{c 1} + 14 N_{c 2}$ which requires 2 operations (1 multiplication and 1 addition) and 1 memory location, $N_{c 4} = N_{c 1} + 26 N_{c 2}$ which requires 2 operations (1 multiplication and 1 addition) and 1 memory location, $N_{c 5} = N_{c 1} + 2 N_{c 2}$ which requires 2 operations (1 multiplication and 1 addition) and 1 memory location, $N_{c 6} = N_{c 4} - N_{c 3}$ which requires 1 operation (subtraction) and 1 memory location, $N_{c 7} = h_{2} h_{3}$ which requires 1 operation (multiplication) and 1 memory location, $N_{c 8} = h_{1} h_{3}$ which requires 1 operation (multiplication) and 1 memory location, $N_{c 9} = h_{1} h_{2}$ which requires 1 operation (multiplication) and 1 memory location, $N_{c 10} = N_{c 3} N_{c 7}$ which requires 1 operation (multiplication) and 1 memory location, $N_{c 11} = N_{c 4} N_{c 8}$ which requires 1 operation (multiplication) and 1 memory location, $N_{c 12} = N_{c 3} N_{c 9}$ which requires 1 operation (multiplication) and 1 memory location, $D_{c 1} = 18 B$ which requires 1 operation (multiplication) and 1 memory location, $D_{c 2} = 3 h_{1} N_{c 7}$ which requires 2 operations (2 multiplications) and 1 memory location, $D_{c 3} = 2 N_{c 7}$ which requires 1 operation (multiplication) and 1 memory location, and $D_{c 4} = 2 N_{c 9}$ which requires 1 operation (multiplication) and 1 memory location.

We call the operations required for computation of N_c1, N_c2,....N_c12, D_c1, D_c2, D_c3 and D_c4 as off - line operations because, once they are computed they can be stored in memory locations so that these stored values can be used at every kth instant. Hence the total number of mathematical operations and memory locations required during the implementation of class 1 fuzzy controller to compute Δu_s(k) at kth instant are 53 and 16, respectively. Similarly the mathematical operations and memory locations required during the implementation of class 2 controller in cell (13, 14, 16) are 39 and 10, respectively. Thus, as far as computational aspects are concerned, class 2 controller is better than class 1 controller.

From the mathematical models of class 1 fuzzy controller in different cells, it can be observed that the computational and memory burden on digital controller is the same when all the three inputs lie in the inner cuboid (Case d in Table 1) and is the highest when compared to that in Cases a, b, and c. This is true even for class 2 fuzzy controller. As we are comparing the computational and memory burdens on digital computer during the implementation of different classes of fuzzy controllers it is good enough to consider the cell (13, 14, 16) where the computational effort is maximum. The number of mathematical operations required to compute e_s(k) and Δe_s(k), and Δ²e_s(k) using e(k), e(k − 1), e(k − 2), S_e, S_Δe and S_Δ²e is the same for both the classes of simplest fuzzy PID controllers, and hence they are not counted here in the relative assessment.

6 BIBO stability analysis of feedback systems containing the simplest fuzzy PID controller

Using the Small Gain theorem [1] we establish the sufficient condition for BIBO stability of feedback systems that contain the class 1 PID controller. Consider the feedback system shown in Fig. 7. According to the small gain theorem, if γ₁ (G₁), the gain of G₁, and γ₂ (G₂), the gain of G₂, have a product smaller than unity, then any bounded - input pair (u₁, u₂) produces a bounded - output pair (y₁, y₂). We consider the general case wherein the process G₂ under control is nonlinear, denoted by N. If the plant is continuous - time plant, then we discretize it and use in digital computer simulation. Hence, by defining r(k) = u₁(k), u(k − 1) = u₂(k), Δu(k) = y₁(k) = G₁e₁(k), y(k) = y₂(k) = Ne₂(k), e(k) = e₁(k) and u(k) = e₂(k) in Fig. 7, we obtain an equivalent closed - loop system as shown in Fig. 1. Let

M_e = $sup_{k \geq 0} | e (k) |$ = $sup_{k \geq 0} | e_{1} (k) |$ ; S_e · M_e = h_e = h₁

M_Δe = $sup_{k \geq 0} | Δ e (k) |$ = $sup_{k \geq 0} | e (k) - e (k - 1) |$ ≤ 2M_e; S_Δe · M_Δe = h_Δe = h₂

M_Δ²e = $sup_{k \geq 0} | Δ^{2} e (k) |$ = $sup_{k \geq 0} | Δ e (k) - Δ e (k - 1) |$ ≤ 2M_Δe or 4M_e; S_Δ²e · M_Δ²e = h_Δ²e = h₃

From the mathematical model of class 1 fuzzy controller in the cell (13, 14, 16) we get

∥Δu(k)∥ = ∥y₁(k)∥ = ∥G₁e₁(k)∥ = $∥ \frac{Q}{R} ∥$ ≤ $\frac{14 B^{2} + 6 AB - 9 A^{2}}{54 {BS}_{Δ u} h_{1} h_{2} h_{3}} [(h_{2} h_{3} S_{e} + h_{1} h_{3} S_{Δ e} + h_{1} h_{2} S_{Δ^{2} e}) | e_{1} (k) | + (h_{1} h_{3} S_{Δ e} + 3 h_{1} h_{2} S_{Δ^{2} e}) M_{e}]$

where Q = $[(14 B^{2} + 6 AB - 9 A^{2}) h_{2} h_{3} + \frac{14 B^{2} + 6 AB - 9 A^{2}}{3}$ $S_{Δ e} S_{Δ^{2} e} {e_{1} (k) - e_{1} (k - 1)} {e_{1} (k) - 2 e_{1} (k - 1) + e_{1} (k - 2)}] S_{e} e_{1} (k) + [(26 B^{2} + 6 AB - 9 A^{2}) h_{1} h_{3} + \frac{- 22 B^{2} + 6 AB - 9 A^{2}}{3} S_{e} S_{Δ^{2} e} {e_{1} (k)} {e_{1} (k) - 2 e_{1} (k - 1) + e_{1} (k - 2)} + 12 B^{2} (h_{3} S_{e} e_{1} (k) - h_{1} S_{Δ^{2} e} {e_{1} (k) - 2 e_{1} (k - 1) + e_{1} (k - 2)})] S_{Δ e} {e_{1} (k) - e_{1} (k - 1)} + [(14 B^{2} + 6 AB - 9 A^{2}) h_{1} h_{2} + \frac{14 B^{2} + 6 AB - 9 A^{2}}{3}$ S_eS_Δee₁(k) {e₁(k) − e₁(k − 1)}] S_Δ²e {e₁(k) −2e₁(k − 1) + e₁(k − 2)}

and R = 18B [3h₁h₂h₃ +(h₃S_ee₁(k) + h₁S_Δ²e {e₁(k) −e₁(k − 1) + e₁(k − 2)}) S_Δe {e₁(k) − e₁(k − 1)} − h₂S_eS_Δ²ee₁(k) {e₁(k) −2e₁(k − 1) + e₁(k − 2)} +2h₂(h₃S_ee₁(k) − h₁S_Δ²e {e₁(k) −2e₁(k − 1) + e₁(k − 2)})]

$\begin{matrix} so that γ_{1} \\ = \frac{\begin{matrix} (14 B^{2} + 6 AB - 9 A^{2}) (h_{2} h_{3} S_{e} + h_{1} h_{3} S_{Δ e} \\ + h_{1} h_{2} S_{Δ^{2} e}) \end{matrix}}{54 {BS}_{Δ u} h_{1} h_{2} h_{3}} \end{matrix}$ (6)

Considering the other class of controller as a subsystem instead of class 1 controller, the expression obtained for γ₁ is the same as in Equation (6). Next, we have

$\begin{matrix} ∥ y_{2} (k) ∥ = ∥ G_{2} e_{2} (k) ∥ = ∥ {Ne}_{2} (k) ∥ \leq ∥ N ∥ | e_{2} (k) | \\ so that γ_{2} = ∥ N ∥ \end{matrix}$ (7)

At origin, ∥Δu(k)∥ = 0. This implies that γ₁ = 0. Since γ₁ = 0 and γ₂ = ∥N∥, to ensure γ₁γ₂ < 1, ∥N∥ < ∞ follows. So, the sufficient condition for the nonlinear fuzzy PID control system to be BIBO stable can be stated as follows:

Theorem 1.A sufficient condition for the closed - loop system shown inFig. 1with the fuzzy PID controller (class 1, class 2) to be BIBO stable is:

the given nonlinear process N has a bounded norm, i.e., ‖N‖ < ∞, and

the parameters S_e, S_Δe, S_Δ²e, h₁, h₂, h₃, A,B and S_Δuof fuzzy PID controller should satisfy the inequality $\frac{\begin{matrix} (14 B^{2} + 6 AB - 9 A^{2}) (h_{2} h_{3} S_{e} \\ + h_{1} h_{3} S_{Δ e} + h_{1} h_{2} S_{Δ^{2} e}) \end{matrix}}{54 {BS}_{Δ u} h_{1} h_{2} h_{3}} ‖ N ‖ < 1$ .

7 Illustrative examples showing computation of γ ₂

Here we consider couple of examples and show how to compute γ₂ = ∥N∥ using Theorem 5.1 in [18].

Example 1. Consider the single - input single - output first order system $\begin{matrix} \dot{x} = f (x, u) = - x - x^{3} + u \\ y = h (x, u) = x \end{matrix}$

By choosing the Lyapunov function $V (x) = \frac{x^{2}}{2}$ , we have $c_{1} = c_{2} = \frac{1}{2}$ where c₁ and c₂ are positive constants in Equation (5.6) of [18].

Since $\frac{\partial V (x)}{\partial x} \cdot f (x, 0) = x \cdot (- x - x^{3}) = - x^{2} - x^{4}$ , c₃ = 1 where c₃ is a positive constant in Equation (5.7) of [18].

Moreover, since $| \frac{\partial V (x)}{\partial x} | = | x |$ , c₄ = 1 where c₄ is a positive constant in Equation (5.8) of [18].

Now |f(x, u) − f(x, 0) | = | − x − x³ + u + x + x³| = |u| and |h(x, u) | = |x|. Hence, from Equations (5.9) and (5.10) of [18] we have L = 1, η_₁ = 1 and η_₂ = 0 where L, η_₁ and η_₂ are nonnegative constants. Therefore $γ_{2} = η_{2} + \frac{η_{1} c_{2} c_{4} L}{c_{1} c_{3}} = 1.$ .

Example 2. Consider the single - input single - output second - order system $\begin{matrix} {\dot{x}}_{1} = f_{1} (\underline{x}, u) = x_{2}; \\ {\dot{x}}_{2} = f_{2} (\underline{x}, u) = - x_{1} - x_{2} - 0.5 \tanh x_{1} + u \\ y = h (\underline{x}, u) = x_{1} \end{matrix}$

Using V $(\underline{x})$ $= 1.5 x_{1}^{2} + x_{1} x_{2} + x_{2}^{2}$ as a Lyapunov function candidate for the unforced system, we have

$\dot{V} (\underline{x})$ = $- x_{1}^{2} - x_{2}^{2} - 0.5 x_{1} \tanh x_{1} - x_{2} {tanhx}_{1} \leq$ $t a n h x_{1} \leq {||\underline{x}||}_{2}^{2} + |x_{1}| |x_{2}| \leq - 0.5 {||\underline{x}||}_{2}^{2}$ and positive definite V $(\underline{x})$ $\underline{x}$ . Thus, V $(\underline{x})$ satisfies Equations (5.6)–(5.8) in [18] with c₁ = 0.6909, c₂ = 1.809, c₃ = 0.5 and c₄ = 3.618. The functions f₁, f₂, and h satisfy Equations (5.9) and (5.10) in [18] globally with L = η₁ = 1 and η₂ = 0.

Hence, $γ_{2} \frac{1.809 \times 3.618}{0.6909 \times 0.5}$ = 18.9461

Since finding V $(\underline{x})$ to a nonlinear system is not easy, finding γ₂ = ∥ N ∥ is also equally difficult, in general.

8 Conclusions

In this paper, a new mathematical model for a fuzzy PID controller has been derived using L-type, Γ-type and Π-type fuzzy membership functions, algebraic product triangular norm, maximum triangularco - norm, Larsen product inference method, and CoS defuzzification method. The model obtained is shown to be nonlinear. Sufficient conditions for a closed loop system with fuzzy PID controller in the loop to be BIBO stable are established. We believe that this work helps in replacing the conventional PID controllers by fuzzy controllers for better performance.

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