Design for vegetable waste fermentation control systems based on semi-tensor product fuzzy controller

Abstract

A multivariable fuzzy controller was designed to solve the control problem of vegetable waste fermentation. The knowledge-based fuzzy reasoning was constructed according to the sample data from both the fermentation process and the rule of temperature change in the control process. It is represented using the mathematical method of semi-tensor product matrices. The structural matrix of the fuzzy logic was constructed and the complex fuzzy reasoning was converted into simple matrix operations. In addition, a new algorithm based on the least in-degree method is put forward, solving the problem of control rules being incomplete and inconsistent. It was used for the design of the fuzzy controller on vegetable waste fermentation. The result of experiment proved that this control method is effective; the control method improves the adjustment speed as well as the control precision.

Keywords

Fuzzy control semi-tensor product multi-input multi-output vegetable waste fermentation multi-sensor

Introduction

Vegetable waste fermentation requires a suitable temperature; from the perspective of biological enzyme dynamics, the best enzyme activity is closely related to temperature. Thus, temperature control during the vegetable waste fermentation process is a key environment parameter that should be under strict control. Various factors can affect the temperature during the vegetable waste fermentation. The vegetable waste fermentation temperature controlling system is a multivariable system. Due to the complexity of vegetable waste fermentation reactions, the controlled variables are with uncertainly and nonlinearity. The control problem of the nonlinear MIMO (multi-input multi-output) system is a key point in research difficulties.^1–7 During industrial production, the factors that affect the temperature change during the fermentation process include fermentation heat from the vegetable waste, mechanical heat caused by the motor-stirring, heat conduction from the heating devices, and the heat loss of the fermentation reactor shell. All these factors should be considered when adjusting the temperature of the fermentation reactor. Proper adjustment requires precisely identifying the reaction conditions and the fermentation heat from the vegetable waste in the environment. In order to adjust the inner temperature, certain approaches are applied to control the system actuators.

Fuzzy control is a new approach from which we do not need to build a precise mathematical model. This approach promotes the development of those nonlinear MIMO systems.^3–7 Although the T-S model of fuzzy control can design the MIMO systems, there is still a large gap between its theoretical research and practical applications. In practice, the design of a multivariable fuzzy controller involves converting the multivariable problem into a problem requiring the control of many single variables, respectively, which reduces precision.⁸ During the control process, the establishment of fuzzy reasoning and fuzzy rules relies mostly on experience-based knowledge. The knowledge library, as an identification form of language rules, will have a geometrically growing number of rules. Gradually, the design of the language rules will be more difficult and complicated. The fuzzy controller can be adopted to find the logic between the input and output. It is actually a mapping problem that relates the input multivariable to the output multivariable. The logical relations between the input and output multivariables can then be displayed through a matrix. The semi-tensor product theory in modern mathematics can turn their logical operations into algebraic matrix operations, which will greatly simplify the fuzzy reasoning process and solve the problems regarding incompleteness, inconformity and inaccuracy of the fuzzy rule.

Professor Daizhan Cheng proposed the matrix semi-tensor product theory. It is a new mathematical approach. Cheng et al.⁹ studied the effective combination of the matrix semi-tensor product calculation methods and the logical relations in a Boolean network. Cheng¹⁰ proposed the resolve scheme for a complex matrix and proved the simplification process of a complex matrix. Cheng et al.¹¹ proposed a solving method using the semi-tensor product for an interconnected polynomial matrix. Cheng and Zhao¹² proposed the method of using the semi-tensor product to realize arbitrary matrix’s multiply and divide. These methods have been successfully applied to many complex systems. Li and Wang¹³ present the application of a Boolean network semi-tensor product in a fault detection circuit, which improves the accuracy and efficiency of fault detection. Liu and Wang¹⁴ proposed the application of semi-tensor product matrix expressions to express a mixed valued logic network, which is a good solution to the decoupling problem that arises using two kinds of controllers. Li et al.¹⁵ proposed using the semi-tensor product matrix to represent Boolean functions. Lv et al.¹⁶ proposed to optimize the fuzzy relation using semi-tensor product matrices, which can take the fuzzy inference language into the matrix. Ge et al.¹⁷ proposed an approach that applies the semi-tensor product theory in order to design a multivariable controller, which is then used in the control field of hybrid cars. Duan et al.⁸ and Lv et al.¹⁸ studied the fuzzy control of air conditioning in an indoor thermal comfort environment, which builds the fuzzy relationship matrix and calculates a result based on the semi-tensor product theory.

The vegetable waste fermentation control process is an unknown, complicated MIMO system. The reaction temperature of the vegetable waste fermentation under the given environment is chosen as the input parameter, while the heat from motor-stirring and media conduction are chosen as the control output parameter. By analyzing the massive input and output data, we determined the inner logical relations. We use the semi-tensor product theory to complete the calculations, which can simplify the fuzzy reasoning process. The matrix operation based on logic is easier than the fuzzy relationship operation based on experience; it will reduce the time necessary to reach a fuzzy reasoning decision and improve the real-time performance of the control. Meanwhile, the data collection, data analysis, and data mining are the most difficult parts during the process involved in building the logic relations of input and output data, and the data collection, data analysis, and data mining are the difficult parts. For example, it is impossible or very difficult to assure the data completeness due to the system’s complexity.

The approach of the least in-degree is to input the unknown data into the structure matrix, which can simplify the structure matrix. This offers a new control method to solve the problem of an incomplete knowledge library caused by missing data. To a certain extent, the problem can be solved in this manner. In addition, data collection errors and analysis errors are inevitable, which will result in nonconformity and inaccuracy in the fuzzy rule library.

The method using the semi-tensor product theory to analyze the Boolean network is helpful in solving these kinds of problems. The semi-tensor product theory, when combined with the above calculation methods, results in a new approach that proposes designing a multivariable fuzzy controller for vegetable waste fermentation temperatures. Finally, experiments will be carried out on the constructed experiment platform. The results of which are analyzed to test the validity and effect of the approach.

The principle of the semi-tensor product

Expression of the semi-tensor product

Definition 1

$Col (A)$ is the set of columns of matrix A, and $Row (A)$ is the set of rows of matrix A. $Co l_{i} (A)$ is the column i of Matrix A, and $Ro w_{i} (A)$ is the row i of Matrix A.¹²

Definition 2

$n \times mn$ matrix A can divide into m blocks of $n \times n$ matrix. $Bl k_{i} (A)$ is the ith block $n \times n$ matrix, $i = 1, \dots, m$ . For example¹²

A = [Bl k_{1} (A) Bl k_{2} (A) \dots Bl k_{m} (A)]

Definition 3

$δ_{k}^{i} = Co l_{i} (I_{k})$ is the column i of unit matrix $I_{k}$ . Then $Δ_{k} : = {δ_{k}^{i} | i = 1, 2, \dots, k}$ .¹²

Definition 4

If $D_{k} : = {\frac{k - 2}{k - 1} | i = 0, 1, \dots, k - 1}$ ; k-value logic can be expressed in the form of vector¹¹

\frac{i}{k - 1} ~ δ_{k}^{k - i}, i = 0, 1, \dots, k - 1

then $D_{k} ~ Δ_{k}$ , $D_{k}$ is equal to $Δ_{k}$ .

Definition 5

If the set of columns of matrix L $Col (L) \subset Δ_{n}$ , then Matrix $L \in R^{m \times n}$ is a logical matrix. The set of $m \times n$ logic matrices is denoted by $ξ_{m \times n}$ . If L is a logical matrix, $L \in ξ_{n \times r}$ , then $L = [δ_{n}^{i_{1}} δ_{n}^{i_{2}} \dots δ_{n}^{i_{r}}]$ . It can be written as¹¹

L = δ_{n} [i_{1} i_{2} \dots i_{r}]

The semi-tensor product theorem

Theorem 1

If $A \in R^{m \times n}$ and $B \in R^{m \times n}$ , then the semi-tensor product of A and B is¹¹

A ⋉ B = (A \otimes I_{\frac{α}{n}}) (B \otimes I_{\frac{α}{p}})

(1)

Note 1

$α = lcm (n, p)$ is the lowest common multiple of n and p. ⊗ is the product of Kronecker. The matrix product in this article is semi-tensor product. For convenience, we omit the symbol ‘⋉’.

Theorem 2

Conversion matrix $W_{[m, n]}$ is an $mn \times mn$ matrix, its row and column are denoted by $(i, j)$ , column is according to the index $Id (i, j; m, n)$ , row is according to the index $Id (j, i; n, m)$ , and the value of the element $[(I, J), (i, j)]$ is¹¹

w_{(I, J) (i, j)} = δ_{i, j}^{I, J} = {\begin{matrix} 1, I = i and J = j \\ 0, others \end{matrix}

(2)

Suppose $X \in R^{n}$ and $Y \in R^{m}$ are column vector, then

W_{[n, m]} XY = YX

(3)

W_{[m, n]} YX = XY

(4)

Theorem 3

Suppose logic function is $f : \prod_{i = 1}^{n} Δ_{k_{i}} \to Δ_{k_{0}}$ . There is unique matrix $M_{F} \in ξ_{k_{0} \times k}$ , called the construction matrix, such that f is expressed as⁸

f (x_{1}, \dots, x_{m}) = M_{F} ⋉_{i = 1}^{n} x_{i}

(5)

Note 2

The dimension of matrix is $k = Π_{i = 1}^{n} k_{i}$ .

Theorem 4

Assume $x_{i} \in D_{k_{i}}, i = 1, \dots, n$ ; $z_{j} \in D_{s_{j}}$ , $j = 1, \dots, m$ . Assuming logical relationship is⁸

F : D_{k_{1}} \times \dots \times D_{k_{n}} \to D_{s_{1}} \times \dots \times D_{s_{m}}

It is expressed below

{\begin{matrix} z_{1} = f_{1} (x_{1}, x_{2}, \dots, x_{n}) \\ z_{2} = f_{2} (x_{1}, x_{2}, \dots, x_{n}) \\ ⋮ \\ z_{m} = f_{m} (x_{1}, x_{2}, \dots, x_{n}) \end{matrix}

(6)

Therein, $f_{j} : D_{k_{1}} \times \dots \times D_{k_{n}} \to D_{s_{j}}, j = 1, \dots, m$ .

Then logic function $z = F (x_{1}, x_{2}, \dots, x_{n})$ can be expressed as shown below

z = F (x_{1}, x_{2}, \dots, x_{n}) = M_{F} ⋉_{i = 1}^{n} x_{i}

(7)

Therein, $M_{F} \in ξ_{s \times k}$ is called as the construction matrix of F, $z \in Δ_{s}$ , $s = s_{1} s_{2} \dots s_{m}$ , $k_{1} k_{2} \dots k_{n}$ . Then

{\begin{matrix} z_{1} = M_{1} x \\ z_{2} = M_{2} x \\ ⋮ \\ z_{m} = M_{m} x \end{matrix}

(8)

Therein, $x = ⋉_{i = 1}^{n} x_{i} \in Δ_{k}$ , $k = Π_{i = 1}^{n} k_{i}$ , $s = Π_{j = 1}^{m} s_{j}$ . $M_{j} : M_{f_{j}} \in ξ_{s_{j} \times k}$ is the construction matrix of $f_{j}$ .

Equation (8) can be written in matrix form

z = M_{F} x

(9)

Therein, $z = ⋉_{j = 1}^{m} z_{i} \in Δ_{s}$ , $M_{F} = M_{1} * M_{2} * \dots * M_{m} \in ξ_{s \times k}$ .

Then, $C o l_{r} (M_{F}) = C o l_{r} (M_{1}) ⋉ \dots ⋉ C o l_{r} (M_{m}), r = 1, \dots, s$ .

Temperature control system of vegetable waste fermentation

In the reactor of vegetable waste fermentation, the environment temperature and the fermentation temperature for the vegetable waste are measured, and we control the fermentation temperature by changing the heated water’s temperature and the mechanical heat produced by the blender. Figure 1 is the fuzzy control system’s framework of vegetable waste fermentation reactor.

Figure 1.

Temperature fuzzy controlling system of vegetable waste fermentation.

The temperatures as a result of the vegetable waste fermentation process are collected by data sensors. The data will then be analyzed to obtain the comparative value, the D-value, the variation trend, and the change rate between the collected temperature and the suitable temperature. These four values will then be inputted into the fuzzy controller. Output control values will be found using fuzzy principle, fuzzy reasoning, and defuzzification. It will control the blender’s rotation rate and the power of the heating devices. Here, the comparative value is defined as C_in (°C). The D-value is D_in (°C). The temperature variation trend is Tr_in (°C). The temperature change rate is R_in (°C/s). The blender’s rotation rate is M_out (r/min). The power percentage of heating devices is $H_{out}$ . Thus, the inner temperature of the fermentation reactor is controlled. The structure matrix is established according to the relationship between input and output data, and the structure matrix completes the fuzzy reasoning.

Figure 2 is the experiment platform for the vegetable waste fermentation temperature control. The whole system includes the microfermentation reactor, the fermentation controller, and the human–computer interactive operation platform. The fermentation reactor has temperature sensors, which collect temperature data. The fermentation controller analyzes the temperature data and implements the output according to the control strategy. The human–computer interactive operation platform can monitor the temperature and any changes during the fermentation process and then send orders to the controller accordingly.

Figure 2.

The experiment platform for the vegetable waste fermentation control.

Outside the fermentation reactor, there is a heating water flume. The heating device adjusts the temperature through the water inside, the power of which is controlled by the pulse width modulation (PWM).

The analysis and design of an MIMO system fuzzy controller are based on the semi-tensor product theory. First, the logical vector is expressed as the fuzzy set of input and output linguistic variables; consequently, the new control rules of the multivariable fuzzy controller will be obtained. After eradicating the fuzzy rules of nonconformity and contradiction, the new structure matrix is constructed based on the new control rule. The complicated fuzzy reasoning is then transferred into an algebraic equation. Then, the approach of the least in-degree is used to simplify the data according to the constructed structure matrix, which can eradicate the redundant variable produced by fuzzy rules of incompleteness, nonconformity, and contradiction. Through this, a comparatively reasonable solution can be obtained as the expression of fuzzy control rules to perform the fuzzy reasoning operation. Finally, we merge these algebraic expressions and obtain the final structure matrix. Afterward, we apply the methods of JD (jointed defuzzification) and Boolean calculations based on semi-tensor product theory to solve ambiguity and find the final output to adjust and control. The structure matrix is updated according to the accumulated input and output data, the fuzzy rule is completed, and the control results are optimized. This approach solves three problems: first, the fuzzy rule’s incompleteness caused by incomplete data from complicated systems. Second, the fuzzy rule nonconformity and contradiction due to data mistakes caused by factors such as measuring interference. Third, the fuzzy rules are simplified.

Fuzzy controller design for vegetable waste fermentation temperature

Fuzzification of input and output variables

The fuzzy controller has four inputs and two outputs. The input variable $C_{in}$ indicates the comparison between the measured temperature and the suitable one, which has two linguistic values {low, high}, expressing the higher and lower temperatures. Its domain is $[- 0.3, 0.3]$ . The input variable $D_{in}$ refers to the D-value between the measured temperature and the suitable one, which has three linguistic values {small, normal, large} and expresses the small, average, and large D-values. Its domain is $[0, 1]$ .

The input variable $T r_{in}$ is the temperature variation trend, which has two linguistic values {up, down}. It expresses the rising and falling of temperature. Its domain is $[- 0.3, 0.3]$ .

The input variable $R_{in}$ is the temperature change rate, which has three linguistic values ${slow, normal, fast}$ , expressing the slow, average, and rapid temperature changes. Its domain is $[0, 1]$ . The output variable $M_{out}$ is the stirring speed, which has three linguistic values {stop, slow, fast} that indicate ceased stirring, slow stirring, and fast stirring. Its domain is $[0, 200]$ . The output variable $H_{out}$ is the heating power, which has four linguistic values {closed, small, big, open}. This indicates closed heating, small-power heating, big-power heating, and heating at full speed. Its domain is $[0, 100]$ .

Figures 3 and 4 are the membership functions of each input and output variables. Thus, the input and output data group can be expressed as

(C_{i n_{i}}^{*}, D_{i n_{i}}^{*}, {Tr}_{i n_{i}}^{*}, R_{i n_{i}}^{*}, M_{ou t_{i}}^{*}, H_{ou t_{i}}^{*}), i = 1, 2, \dots, n

(10)

Figure 3.

Membership functions of input variables: (a) the membership function of $C_{in}$ , (b) the membership function of $D_{in}$ , (c) the membership function of $T r_{in}$ , and (d) the membership function of $R_{in}$ .

Figure 4.

Membership functions of output variables: (a) the membership function of $M_{out}$ and (b) the membership function of $H_{out}$ .

Then, fuzzy variables can be transformed into column vector. The column vector of Group 1 should be as follows

C_{i n_{i}} = (μ_{low} (C_{i n_{i}}^{*}), μ_{high} (C_{i n_{i}}^{*}))^{T}

(11)

D_{i n_{i}} = (μ_{small} (D_{i n_{i}}^{*}), μ_{normal} (D_{i n_{i}}^{*}), μ_{large} (D_{i n_{i}}^{*}))^{T}

(12)

T r_{i n_{i}} = (μ_{up} ({Tr}_{i n_{i}}^{*}), μ_{down} ({Tr}_{i n_{i}}^{*}))^{T}

(13)

R_{i n_{i}} = (μ_{slow} (R_{i n_{i}}^{*}), μ_{normal} (R_{i n_{i}}^{*}), μ_{fast} (R_{i n_{i}}^{*}))^{T}

(14)

M_{ou t_{i}} = (μ_{stop} (M_{ou t_{i}}^{*}), μ_{slow} (M_{ou t_{i}}^{*}), μ_{fast} (M_{ou t_{i}}^{*}))^{T}

(15)

\begin{matrix} H_{ou t_{i}} = & (μ_{close} (H_{ou t_{i}}^{*}), μ_{small} (H_{ou t_{i}}^{*}), \\ μ_{big} (H_{ou t_{i}}^{*}), μ_{open} (H_{ou t_{i}}^{*}))^{T} \end{matrix}

(16)

So the relation between input and output is as follows

(M_{out}, H_{out}) = fuzzy (C_{in}, D_{in}, T r_{in}, R_{in})

(17)

From the above, the fuzzy control rule of the input and output is transformed into formula (17) from $IF \dots, THEN \dots$ sentences. Next is to build the construction matrix. Semi-tensor product matrices theory and fuzzy controller can be used to simplify the calculation and solve the problems of incompleteness, nonconformity, and inaccuracy of the fuzzy rule.

Building of the fuzzy reasoning based on structure matrix

The first is to confirm the input and output variables of fuzzy controller. For simplicity and convenience, the variables can be defined as follows

C_{in} ~ x_{1}, D_{in} ~ x_{2}, T r_{in} ~ x_{3}, R_{in} ~ x_{4}; M_{out} ~ y_{1}, H_{out} ~ y_{2}

According to every fuzzy set of variables, each variable belongs to the corresponding logic

x_{1} \in D_{2}, x_{2} \in D_{3}, x_{3} \in D_{2}, x_{4} \in D_{3}, y_{1} \in D_{3}, y_{2} \in D_{4}

Every fuzzy set of variables can be expressed as the form of semi-tensor product matrix

low ~ δ_{2}^{1}, high ~ δ_{2}^{2}

small ~ δ_{3}^{1}, normal ~ δ_{3}^{2}, large ~ δ_{3}^{3}

up ~ δ_{2}^{1}, down ~ δ_{2}^{2}

slow ~ δ_{3}^{1}, normal ~ δ_{3}^{2}, fast ~ δ_{3}^{3}

\begin{matrix} stop ~ δ_{3}^{1}, slow ~ δ_{3}^{2}, fast ~ δ_{3}^{3} \\ close ~ δ_{4}^{1}, small ~ δ_{4}^{2}, big ~ δ_{4}^{3}, open ~ δ_{4}^{4} \end{matrix}

The system’s input and output variables are mixed value. This can be thought of as a multiple variable mixed logic problem.

The second is to design fuzzy rule. The sample data in the fermentation process of some vegetable waste fungus is collected. A mass of input and output data were extracted. A total of 100 fuzzy rules were converted by fuzzy identify. The form of semi-tensor product is as follows

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{1}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{1}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{2}, 6 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{1}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{2}, THEN \\ y_{1} = δ_{3}^{3}, y_{2} = δ_{4}^{2}, 4 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{1}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{2}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{3}, 5 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{3}, THEN \\ y_{1} = δ_{3}^{3}, y_{2} = δ_{4}^{3}, 8 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{3}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{4}, 3 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{3}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{4}, 1 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{2}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{3}, 5 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{2}, THEN \\ y_{2} = δ_{3}^{3}, y_{3} = δ_{4}^{2}, 4 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{3}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{2}, THEN \\ y_{1} = δ_{3}^{1}, y_{2} = δ_{4}^{2}, 7 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{1}, x_{2} = δ_{3}^{3}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{2}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{1}, 2 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{1}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{2}, THEN \\ y_{1} = δ_{3}^{1}, y_{2} = δ_{4}^{1}, 9 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{1}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{2}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{1}, 1 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{1}, x_{3} = δ_{2}^{1}, x_{4} = δ_{3}^{2}, THEN \\ y_{1} = δ_{3}^{1}, y_{2} = δ_{4}^{2}, 2 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{1}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{3}, 8 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{1}, THEN \\ y_{1} = δ_{3}^{1}, y_{2} = δ_{4}^{4}, 2 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{1}, THEN \\ y_{1} = δ_{3}^{3}, y_{2} = δ_{4}^{2}, 3 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{3}, THEN \\ y_{1} = δ_{3}^{3}, y_{2} = δ_{4}^{3}, 4 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{2}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{3}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{4}, 5 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{3}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{3}, THEN \\ y_{1} = δ_{3}^{3}, y_{2} = δ_{4}^{4}, 10 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{3}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{1}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{4}, 4 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{3}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{3}, THEN \\ y_{1} = δ_{3}^{2}, y_{2} = δ_{4}^{4}, 1 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{3}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{3}, THEN \\ y_{1} = δ_{3}^{3}, y_{2} = δ_{4}^{3}, 2 times \end{matrix}

\begin{matrix} IF x_{1} = δ_{2}^{2}, x_{2} = δ_{3}^{3}, x_{3} = δ_{2}^{2}, x_{4} = δ_{3}^{1}, THEN \\ y_{1} = δ_{3}^{3}, y_{2} = δ_{4}^{3}, 4 times \end{matrix}

From the above fuzzy rules, a lot of repetitive fuzzy rules are extracted by the big sample data. A portion of rules violate consistency for fuzzy rules.

The third is to define the algebra expression of input and output by fuzzy rules. The algebraic expression of $y_{1}, y_{2}$ is as below

{\begin{matrix} y_{1} = M_{1} x_{1} x_{2} x_{3} x_{4}, M_{1} \in ξ_{3 \times 36} \\ y_{2} = M_{2} x_{1} x_{2} x_{3} x_{4}, M_{2} \in ξ_{4 \times 36} \end{matrix}

(18)

The system has four inputs. Their multiple value logic is 2,3,2,3. So there are 36 columns in construction matrix. The number of row is relevant with the corresponding output map k-value logic.

The fourth step is to build up construction matrix.

When there is the inconsistent case of fuzzy rules, we accept or reject by the number of appearing times. Through the rule table, the ninth column of construction matrix $M_{1}, M_{2}$ can be obtained as

Co l_{9} (M_{1}) = δ_{3}^{3}, Co l_{9} (M_{2}) = δ_{4}^{3} 8 times

Co l_{9} (M_{1}) = δ_{3}^{2}, Co l_{9} (M_{2}) = δ_{4}^{4} 3 times

Co l_{9} (M_{1}) = δ_{3}^{2}, Co l_{9} (M_{2}) = δ_{4}^{4} 1 times

The number of times the first fuzzy rule appears is greater than the other fuzzy rules; the first is selected, and the others are rejected.

When the times of fuzzy rules are approximate, this fuzzy rule is omitted. According to rule table, the second column of construction matrix $M_{1}, M_{2}$ can be obtained as shown below

Co l_{2} (M_{1}) = δ_{3}^{3}, Co l_{2} (M_{2}) = δ_{4}^{2} 4 times

Co l_{2} (M_{1}) = δ_{3}^{2}, Co l_{2} (M_{2}) = δ_{4}^{3} 5 times

Because these two rules are approximate, this rule is not consistent. It cannot be used. The elements of column express as “*.” It is as uncertain element. So

Co l_{2} (M_{1}) = *, Co l_{2} (M_{2}) = *

Using the similar way, the other column elements of construction matrix $M_{1}, M_{2}$ can be obtained by fuzzy rule table as below

Co l_{1} (M_{1}) = δ_{3}^{2}, Co l_{1} (M_{2}) = δ_{4}^{2}

Co l_{8} (M_{1}) = *, Co l_{8} (M_{2}) = *

Co l_{17} (M_{1}) = δ_{3}^{1}, Co l_{17} (M_{2}) = δ_{4}^{2}

Co l_{20} (M_{1}) = δ_{3}^{1}, Co l_{20} (M_{2}) = δ_{4}^{1}

Co l_{28} (M_{1}) = δ_{3}^{2}, Co l_{28} (M_{2}) = δ_{4}^{3}

Co l_{30} (M_{1}) = *, Co l_{30} (M_{2}) = *

Co l_{34} (M_{1}) = *, Co l_{34} (M_{2}) = *

Co l_{36} (M_{1}) = δ_{3}^{3}, Co l_{36} (M_{2}) = δ_{4}^{4}

2. Because of deficiency of fuzzy rule, the other rules cannot be determined. The columns of construction matrix $M_{1}, M_{2}$ without fuzzy rule table are unknown elements. They are expressed as “*.”

Then, the construction matrix $M_{1}, M_{2}$ can be obtained as

\begin{matrix} M_{1} = δ_{3} [2 * * * * * * * 3 * * * * * * * 1 * \\ * 1 * * * * * * * 2 * * * * * * * 3] \end{matrix}

(19)

\begin{matrix} M_{2} = δ_{4} [2 * * * * * * * 3 * * * * * * * 2 * \\ * 1 * * * * * * * 3 * * * * * * * 4] \end{matrix}

(20)

The fifth step is to simplify construction matrix.

1. In simplifying construction matrix $M_{1}$ , input variable $x_{1}$ is checked. Because of $M_{1} = [M_{1}^{1} M_{1}^{2}]$ and $M_{1}^{1} = M_{1}^{2}$ , the construction matrix is simplified as

M_{1}^{1} = M_{1}^{2} = δ_{3} [2 1 * * * * * * 3 2 * * * * * * 1 3]

(21)

Then, input variable $x_{2}$ is checked. Because of $M_{1}^{1} = [M_{1}^{11} M_{1}^{12} M_{1}^{13}]$ and $M_{1}^{11} = M_{1}^{12} = M_{1}^{13}$ , the construction matrix is simplified as

M_{1}^{11} = M_{1}^{12} = M_{1}^{13} = δ_{3} [2 1 3 2 1 3]

(22)

Then, input variable $x_{3}$ is checked. Because of $M_{1}^{11} = [M_{1}^{111} M_{1}^{112}]$ and $M_{1}^{111} = M_{1}^{112}$ , the construction matrix is simplified as

M_{1}^{111} = M_{1}^{112} = δ_{3} [2 1 3]

(23)

Then, input variable $x_{4}$ is checked. $M_{1}^{111}$ cannot be separated as three equal parts apparently. So, $x_{1}$ , $x_{2}$ , $x_{3}$ are redundancy variables. Then, output $y_{1}$ is

y_{1} = δ_{3} [2 1 3] x_{4}

(24)

2.Simplifying construction matrix $M_{2}$ . Input variable $x_{1}$ is checked. Because $M_{2} = [M_{2}^{1} M_{2}^{2}]$ and $M_{2}^{1} = M_{2}^{2}$ , the construction matrix is simplified as

M_{2}^{1} = M_{2}^{2} = δ_{4} [2 1 * * * * * * 3 3 * * * * * * 2 4]

(25)

Then, input variable $x_{2}$ is checked. Because $M_{2}^{1} = [M_{2}^{11} M_{2}^{12} M_{2}^{13}]$ and $M_{2}^{11} = M_{2}^{12} = M_{2}^{13}$ , the construction matrix is simplified as

M_{2}^{11} = M_{2}^{12} = M_{2}^{13} = δ_{4} [2 1 3 3 2 4]

(26)

Then, input variable $x_{3}$ is checked. Because of $M_{2}^{11} = [M_{2}^{111} M_{2}^{112}]$ and $M_{2}^{111} \neq M_{2}^{112}$ , $x_{3}$ is not redundancy variable.

Then, input variable $x_{4}$ is checked. Because variable $x_{4}$ is not adjacent to construction matrix $M_{2}^{11}$ , the matrix cannot be simplified without conversion matrix directly. The conversion matrix can be obtained by the dimension of variable $x_{3}$ and $x_{4}$ as shown below

W_{[2, 3]} = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

(27)

The construction matrix is transformed by the above conversion matrix as shown below

M_{2}^{11} W_{[2, 3]} = δ_{4} [2 3 2 1 3 4]

(28)

This matrix cannot be separated as three equal parts apparently. So $x_{1}$ and $x_{2}$ are redundancy variables. Then, output $y_{3}$ is

y_{2} = δ_{4} [2 1 3 3 2 4] x_{3} x_{4}

(29)

The input and output relevant expression after simplified can be obtained finally as shown below

{\begin{matrix} y_{1} = δ_{3} [2 1 3] x_{4} \\ y_{2} = δ_{4} [2 1 3 3 2 4] x_{3} x_{4} \end{matrix}

(30)

The above is the fuzzy reasoning construction matrix. If the input data increase, the fuzzy rule would be more complete. The related column elements will be updated in the structure matrix and redundant variables will change. When the input data are big enough to make the fuzzy rule complete, it will become a common structure matrix. This kind of structure matrix also applies when increasing the input and output. When the input is increased, columns in the structure matrix will change. Increases in output result in the increase of an additional structure matrix.

The control realization of fuzzy output

We often use the approach of SD (separated defuzzification) to defuzzify, and its formula is as follows

u_{α} = \sum_{j = 1}^{β α} (\frac{b_{j}^{α}}{\sum_{j = 1}^{β α} b_{j}^{α}}) μ_{U_{j}^{α}}^{- 1} (1), α = 1, \dots, m

(31)

This approach is not appropriate for multiple output variables’ existing coupling relations. Which system is appropriate by the approach of JD

\begin{matrix} (u_{1}, \dots, u_{m}) = \\ \sum_{j_{1} = 1}^{β_{1}} \dots \sum_{j_{m} = 1}^{β_{m}} (\frac{b_{j_{1} \dots j_{m}}}{\sum_{j_{1} = 1}^{β_{1}} \dots \sum_{j_{m} = 1}^{β_{m}} b_{j_{1} \dots j_{m}}}) μ_{U_{j_{1}}^{1} \times \dots U_{j_{m}}^{m}}^{- 1} (1) \end{matrix}

(32)

Note 3

$μ_{U_{j_{1}}^{1} \times \dots U_{j_{m}}^{m}}^{- 1} (1)$ is every output defuzzification, their membership is 1.

The above expressions need to be merged. Then the vector for defuzzification is as follows

V_{u} = [b_{1 \dots 11} \dots b_{1 \dots 1 β_{m}} \dots b_{1 β_{2} \dots β_{m}} \dots b_{β_{1} β_{2} \dots β_{m}}]^{T}

(33)

The expression $y_{1}$ need to be extended for merging by semi-tensor product. Because $x_{3} \in D_{2}$

y_{1} = δ_{3} [2 1 3 2 1 3] x_{3} x_{4}

(34)

So the input and output expressions are as follows

{\begin{matrix} y_{1} = δ_{3} [2 1 3 2 1 3] x_{3} x_{4} \\ y_{2} = δ_{4} [2 1 3 3 2 4] x_{3} x_{4} \end{matrix}

(35)

Then, the merging expression by semi-tensor product is as follows

y = δ_{12} [6 1 11 7 2 12] x

(36)

The expression after defuzzification by the approach of JD is as follows

(u_{M_{out}}, u_{H_{out}}) = \sum_{p = 1}^{3} \sum_{q = 1}^{4} (\frac{b_{p, q}}{\sum_{p = 1}^{3} \sum_{q = 1}^{4} b_{p, q}}) μ_{M_{out}^{p} \times H_{out}^{q}}^{- 1} (1)

(37)

In formula (37), the calculation process of $μ_{M_{out}^{p} \times H_{out}^{q}}^{- 1} (1)$ is as follows

μ_{M_{out}^{p} \times H_{out}^{q}} (u_{M_{out}^{p}}, u_{H_{out}^{q}}) = μ (u_{M_{out}^{p}}) \land μ (u_{H_{out}^{q}})

(38)

The known fuzzy set of output $M_{out}$ and $H_{out}$ can be expressed as

\begin{matrix} stop ~ δ_{3}^{1}, slow ~ δ_{3}^{2}, fast ~ δ_{3}^{3} \\ close ~ δ_{4}^{1}, small ~ δ_{4}^{2}, big ~ δ_{4}^{3}, open ~ δ_{4}^{4} \end{matrix}

According to the membership function and domain of discourse of $M_{out}$ and $H_{out}$

\begin{matrix} 20 r / \min ~ δ_{3}^{1}, 100 r / \min ~ δ_{3}^{2}, 180 r / \min ~ δ_{3}^{3} \\ 0 % ~ δ_{4}^{1}, 50 % ~ δ_{4}^{2}, 70 % ~ δ_{4}^{3}, 100 % ~ δ_{4}^{4} \end{matrix}

Because of formula (38), the parameter table of $μ_{M_{out}^{p} \times H_{out}^{q}}^{- 1} (1)$ is as shown below.

Finally, the vector formula (33) after fuzzy reference and the value of $μ_{M_{out}^{p} \times H_{out}^{q}}^{- 1} (1)$ are substituted into formula (37) to carry out the accurate value for output control.

Experiment and results

Process of experiment

To test the validity of the semi-tensor fuzzy control approach, the experiment of vegetable waste fermentation reaction is carried out on Figure 2 platform. First, some vegetable waste fermentation bacteria are put into the reactor and then some vegetable waste is added and the coil is heated to implement PWM control through the fermentation controller. The experiment room temperature is 23.4°. The starting temperature inside the fermentation controller as well as the water medium’s temperature is the same as the room temperature. It is already known that the suitable experiment temperature is 55°C. So set the temperature and carry out the experiment. Analyze and compare the experiment data obtained from the general fuzzy control algorithm and the fuzzy control algorithm based on the semi-tensor product.

The experimental data are extracted for calculating verification. The data are extracted by temperature sensor and handled as shown below

C_{in}^{*} = 0.3, D_{in}^{*} = 5, {Tr}_{in}^{*} = 0.3, R_{in}^{*} = 0.85

The vector forms after fuzzification according to formulas (11)–(14) are as shown below

C_{in} = {(0 1)}^{T}, D_{in} = {(0 1 0)}^{T}, T r_{in} = {(0 1)}^{T}, R_{in} = {(0 0.25 0.75)}^{T}

The vectors above are substituted into the relevant expression (36) of construction matrix

\begin{matrix} y = δ_{12} [6 1 11 7 2 12] \times_{B} [\begin{matrix} 0 \\ 1 \end{matrix}] \times_{B} [\begin{matrix} 0 \\ 0.25 \\ 0.75 \end{matrix}] \\ = δ_{12} [6 1 11 7 2 12] \times_{B} (0 0 0 0 0.25 0.75)^{T} \\ = (0 0.25 \underset{9}{\underset{︸}{0 0 \dots 0}} 0.75)^{T} \end{matrix}

The result above is the output vector as form of expression (33). Then, the parameter of $μ_{M_{out}^{p} \times H_{out}^{q}}^{- 1} (1)$ is found out by Table 1. Output vector is substituted into formula (37) for defuzzification. Then, the accurate output value is obtained as

\begin{matrix} (u_{M_{out}}, u_{H_{out}}) \\ = \frac{0.25 \times (20, 50) + 0.75 \times (180, 100)}{0.25 + 0.75} \\ = (140, 87.5) \end{matrix}

Table 1.

The parameter table of $μ_{M_{out}^{p} \times H_{out}^{q}}^{- 1} (1)$ .

$μ_{M_{out}^{p} \times H_{out}^{q}}^{- 1} (1)$	$H_{out}$
	$δ_{4}^{1}$	$δ_{4}^{2}$	$δ_{4}^{3}$	$δ_{4}^{4}$
$M_{out}$
$δ_{3}^{1}$	(20,0)	(20,50)	(20,70)	(20,100)
$δ_{3}^{2}$	(100,0)	(100,50)	(100,70)	(100,100)
$δ_{3}^{3}$	(180,0)	(180,50)	(180,70)	(180,100)

So the result of output control is that the stirring rate is 140 r/min and the duty ratio of heating coil’s PWM control is 12.5%.

Results of the experiment

We collected the data in the experiment, and the formed curve is shown in Figure 5.

Figure 5.

Fermentation experiment curves: (a) the temperature variation curve during the fermentation, (b) temperature variation curve during the fermentation initial phase, and (c) the temperature variation curve after the fermentation stabilization.

Figure 5(a) shows the temperature variation curve during the fermentation. Both algorithms can rapidly adjust the temperature to the set value. The fuzzy control algorithm based on the semi-tensor product can reach the set value in 20 min, while the general fuzzy control algorithm can reach the set value in 12 min. The latter’s overshoot is 4.55%, and the stability of general fuzzy control algorithm is lower than the semi-tensor product of the fuzzy control algorithm. The general fuzzy control algorithm reaches its stability to the set value at the 24th min, which indicates that the semi-tensor product of the fuzzy control algorithm is more time efficient. Figure 5(b) shows the temperature variation curve during the fermentation’s initial phase, which shows the slight changes between 23.3° and 23.5°C in the first 4 min. This stage reflects the lagging of temperature adjustment. When the temperature sensor detects the real temperature, the heating coils will be controlled to heat the water media. When the water is hot enough for the heat exchange, the heat will then be conducted to the fermentation controller’s experimental shell of the experiment to change the temperature inside the reactor. During this process, the experiment experimental reactor and the water media both are under room temperature; the rising temperature will cause certain heat loss. After the balance between the heat absorbed by the water media and the heat loss, the temperature begins to rise. It was proven that the response speeds of the two algorithms are the same. Figure 5(c) shows the temperature variation curve after the fermentation is stable. Because of the heat loss of the water media and the shell, which is under room temperature conditions, and the noise interference of the bacteria fermentation reaction, the set temperature value can never be absolutely realized. Tiny temperature shifts certainly exist. The fuzzy control algorithm based on the semi-tensor product theory shows advantages; it is clearly shown that the D-value of fuzzy control algorithm based on the semi-tensor product is ±0.3°C while the D-value of the general fuzzy control algorithm is 0.5°C. Thus, the fuzzy control algorithm based on the semi-tensor product theory can meet these requirement requirements with higher precision.

The results of the experiment show that the semi-tensor product fuzzy control approach rapidly adjusts the complicated vegetable waste fermentation temperature to a certain value. In addition, its precision can meet the requirements if vegetable waste fermentation reactions occur. The fuzzy controller design approach based on semi-tensor product theory is proved to be valid and effective in controlling complicated multivariable systems.

Conclusion

The semi-tensor product of the matrix solves the calculation problems in a fuzzy relationship matrix, offering an excellent solution to MIMO systems. This approach collects and analyzes the input and output data, turns them into logical variables, and builds the structure matrix according to the semi-tensor product of the matrix and the approach of the least in-degree as the basis of fuzzy reasoning. Thus, a multivariable temperature fuzzy controller was designed, which realized the efficient control of vegetable waste fermentation reaction temperatures. This method can implement fuzzy control to complicated MIMO systems. In addition, it can be easily realized in actual industrial production and has definite application value.

Footnotes

Handling Editor: Alexander Hosovsky

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by National Natural Science Foundation of China under Grant nos 61304133 and 61601256. This work was also supported in part by Science and Technology Development Project of Shandong Education Department under Grant no. J16LN84.

ORCID iD

Zuoxun Wang

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