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1 Introduction
Fuzzy system technology has the ability of word calculation and processing inaccuracy, uncertainty and fuzzy information. In recent years, it has been proved to be an effective method to solve many practical complex modeling and control problems. However, many black systems still use the black box approach because their structural complexity has become a major obstacle to traditional mathematical analysis. This black box method is very different from the widely used analytical method-based design method in classical control theory. It can neither provide analytical insights for fuzzy systems nor perform effective mathematical analysis of system characteristics and performance. This kind of defect will bring unreality and insecurity to its application of control problems in many fields.
In order to develop fuzzy control theory and allow it to have a solid theoretical foundation, analytical methods have attracted the attention of many scholars. Based on analytical methods, many mature methods in classical system theory can be used for some analysis and design of fuzzy systems. The analysis of the analytical structure of the fuzzy controller and the stability analysis of the fuzzy control system are two of the main directions. This article will systematically review the latest research results in both directions and look forward to future research work.
2 Fuzzy control system composition
The fuzzy controller generally consists of five parts: 1) Fuzzification interface: Convert the real determinant to the fuzzy quantity through the membership function; 2) Database: The membership function for storing all fuzzy subsets of the input and output variables; 3) Fuzzy rules Set: The information given in the form of IF-THEN control rules, according to the form of fuzzy rules, the fuzzy controller can be mainly divided into Mamdani and Takagi-Sugeno (TS); 4) Fuzzy reasoning mechanism: based on fuzzy rules, using fuzzy logic Operation and reasoning methods to obtain fuzzy output; 5) Defuzzing interface: used to convert fuzzy output into the system's numerical output.
According to the number of input and output variables, the fuzzy control system can be divided into univariate and multivariable fuzzy control systems. Most of the fuzzy systems are complex nonlinear systems, and the nonlinearity between the input and output is caused by the above-mentioned various components of the fuzzy controller.
3 Structure Analysis of Fuzzy Controllers
Analyzing and analyzing the structure of fuzzy controllers based on conventional control theory is an important way to develop fuzzy control technology. This has a certain guiding significance for the practical application of the fuzzy controller, because many important and difficult aspects (such as analysis, design, stability and robustness) of the fuzzy control technology can be used in (non-linear) control theory. Some mathematical techniques are effectively studied. At present, many scholars have analyzed the Mamdani fuzzy controller and TS fuzzy controller well, including the analytical structures of some more complex fuzzy controllers. We discuss the following aspects.
3.1 fuzzy controller is a nonlinear PID controller <br> linear discrete expression for PID
(1)
Many fuzzy controllers can be expressed in the form of (1), except that the gain of the controller varies with the input. Therefore, the fuzzy controller is a nonlinear PID controller. Ying [1] first proposed the analytical structure of the fuzzy PID controller and proved the simplest Mamdani fuzzy controller using two linear input fuzzy sets, four fuzzy rules, Zadeh fuzzy logic AND and OR operations and barycentric defuzzifier. It is a non-linear PI controller; then it is further extended to various mamdani fuzzy controllers [2] using other inference methods (such as Mamdani minimum, Larsen product, drastic product, and bounded product). The more complex case is the use of two input variables, multiple symmetric or asymmetric triangular input fuzzy sets, linear control rules, uniformly distributed single output fuzzy sets, different reasoning methods, and a Mamdani fuzzy controller of the center-of-mass defuzzifier. It has been proved to be the sum of a global two-dimensional multi-valued relay controller and a local nonlinear PI controller [3, 4]. These results are generalized to single-input single-output [5] and two-input two-output fuzzy controller [6] using nonlinear control rules. Other similar results can be found in [7-10].
Various extended designs and structural analysis of basic Mamdani fuzzy controllers have been studied, and fuzzy PIDs[11-13], fuzzy PI+D[14], fuzzy PD+I[15], serial fuzzy PI+PD[16], and parallel have been proved. The fuzzy PI+PD[17] and fuzzy(PI+D)2[18] controllers are all nonlinear PID controllers and a clear expression of their nonlinear gain is derived. In addition, a structure of a time-varying fuzzy controller based on the open-close control technique is analytically linked to a nonlinear PD controller and proved to be a nonlinear PD controller with a nonlinear control bias [19 ].
Recently, we started to discuss the analytical structure of the TS fuzzy controller and used a simple 2×2 fuzzy rule set structure to analyze the nonlinearity of a class of TS fuzzy PI (or PD) controllers [20]. A clear expression for the gain of the TS fuzzy PI (or PD) controller is deduced, and the range and geometry of the gain change are studied. The TS fuzzy PI (or PD) controller is actually a nonlinear PI (or PD) controller. The analytical results of the above simple TS fuzzy PI (or PD) controller structure have also been generalized to more typical and complex TS fuzzy controllers [21-23]. These TS fuzzy controllers consist of three or more trapezoidal (or arbitrary) input fuzzy sets, TS fuzzy rules with linear back entries, Zadeh fuzzy logical AND operations, and barycentric defuzzifiers.
The analytical structure associated with the fuzzy controller and the linear PID controller on the one hand reveals that the fuzzy controller is superior to the linear PID controller in the applications of nonlinear, time-varying and pure-delay systems, and also provides The relationship between the gains comes from a way of parsing a fuzzy control system and ensuring its stability.
3.2 Fuzzy controller as <br> Sliding Mode Controller for a large class of nonlinear systems, fuzzy controller is of the state x (n), and 
(n) Corresponding deviation e(n) and deviation change rate 
(n) Designed by determining the phase plane. For a two-dimensional fuzzy controller, the general design method is to divide the phase plane into two half planes through a switch line. Its switching function is defined as
(2)
The control input on the zero-diagonal line of the two-dimensional control rule set is zero. In terms of working principle, the fuzzy controller is similar to the sliding mode variable structure controller [10,24-28]. Wu and Liu represent the fuzzy control as a type of variable structure control, and the sliding mode is used to determine the best parameter value in the fuzzy control rule [29]. If a rule set of variable structure type is used, the fuzzy controller has variable structure characteristics in both semantic and quantitative aspects. For two-dimensional and three-dimensional fuzzy controllers, a specific mathematical expression has been derived [30]. Compared with the common sliding mode control, the fuzzy control has stronger robustness, and the variable structure of the fuzzy controller helps people design a robust and stable fuzzy controller.
3.3 fuzzy controller is a non-linear gain control <br> typical planning and complex types of TS fuzzy controller, from the structure that has been proven to be non-linear gain planner [20, 21]. These TS fuzzy controllers consist of multiple trapezoid (or triangle) input fuzzy sets, TS fuzzy rules with linear back entries, Zadeh fuzzy logical AND operations, and barycentre defuzzifiers. Unlike a conventional linear controller with different constant gains at different operating points, the gain of the nonlinear gain planner varies with the output of the controlled system. These proofs not only make up for a simple explanation of the relationship between the fuzzy controller and the gain controller by some scholars in the past, but also explain the effectiveness of the fuzzy controller in dealing with nonlinear problems.
3.4 Relationship between fuzzy controllers and multi-valued relay controllers Kickert and Mamdani revealed the relationship between fuzzy controllers and multi-valued relay controllers. A class of simple fuzzy controllers, whose input-output characteristics have multi-valued relay characteristics, can be considered as multi-valued relay controllers [31]. Ying [3] proved that the Mamdani fuzzy controller using two input variables, multiple triangular input fuzzy sets, linear control rules, uniformly distributed single output fuzzy sets, different inference methods and barycentre defuzzifiers is a global one. The sum of two-dimensional multi-valued relay controllers and a local nonlinear PI controller [3]. These results are generalized to SISO using a non-uniform distribution of multiple triangular input fuzzy sets, SISO and MIMO Mamdani fuzzy controllers using nonlinear control rules [4-6]. According to the relationship between the fuzzy controller and the multi-valued relay controller, the method described in the classical control theory can be used to analyze and design the fuzzy control system and ensure its stability.
Limit structure theory <br> 3.5 Fuzzy Controller Some scholars have noted that when the number of fuzzy rules increases large enough to affect the controlled process little or no effect, resulting in limit structure theory of fuzzy controller [32-34]. For a general fuzzy controller using linear control rules, as the number of control rules increases, its output becomes a linear function of the input [32]. Especially when the number of control rules is large, for a two-input fuzzy controller, the output is approximately equal to the output of a linear PI controller; for a three-input fuzzy controller, the output is approximately equal to the output of a linear PID controller. These structures are generalized to fuzzy controllers using multi-state variables and multiple output fuzzy sets [33]. If a nonlinear control rule expressed by an arbitrary function f is used, the analytical structure of the fuzzy controller is the sum of a global nonlinear controller that depends on f and a local nonlinear controller, with the number of control rules increasing to ∞, the local nonlinear controller will also disappear [34]. If linear control rules are used, then as the number of control rules increases to ∞, the global controller will become a multidimensional multi-valued relay controller with a global approximation to a linear controller [34]. These results apply to any fuzzy logic operation, inference method, and defuzzifier. The theory of the limit structure shows that the number of fuzzy sets and fuzzy rules in a fuzzy controller is not as effective as the number of fuzzy rules. Therefore, in the actual design, the number of fuzzy sets and rules should be properly selected according to specific problems.
3.6 MIMO structure of fuzzy controller <br> decomposition of many industrial processes controlled objects are complex, often require the use of MIMO fuzzy controller. The general MIMO Mamdani fuzzy controller can always be decomposed into a sum of a global nonlinear controller determined only by fuzzy rules and a local nonlinear controller determined by all components of the fuzzy controller [35]. In addition, fuzzy systems based on N-variables of product-and-barycentric reasoning can be decomposed into additions or multiplications of N univariate fuzzy subsystems [36].
4 Stability Analysis of Fuzzy Control System
By analyzing the stability of the fuzzy control system, the designer can understand all the steps of the design method. Because the fuzzy control system is a complex nonlinear system and has a variety of different forms, its stability analysis is difficult. At present, the stability analysis methods of fuzzy control systems based on classical control theory mainly include the following:
4.1 <br> Lyapunov method based on Lyapunov direct method, many scholars discussed the discrete-time and continuous-time fuzzy stability analysis and design of control systems [37-44]. Among them, Tanaka and Sano generalized the basic stability condition in [43] to the (non)robust stability condition of the SISO system, and the stability criterion became a common Lyapunov from a set of Lyapunov inequalities. In the function problem [44], since there is no general effective way to analytically find a public Lyapunov function, neither Tanaka et al. [43, 44] provide a method for finding the public matrix P of Lyapunov stability conditions. In order to solve this problem, the paper [45-47] proposes to describe the stability conditions using linear matrix inequalities, and some scholars use a set of P matrices instead of a common matrix P of the Lyapunov function in the literature [43, 44]. A piecewise approximately smooth quadratic Lyapunov function for stability analysis [37]. Each matrix P corresponds to only one subsystem and shows that the fuzzy control system is globally stable if and only if a set of suitable Riccati equations have a positive definite symmetric solution and these solutions can be obtained.
Using Lyapunov linearization method, Ying establishes the necessary and sufficient conditions for the local stability of TS fuzzy control systems including nonlinear objects [23]. In addition, a vector Lyapunov direct method used in large systems is used to derive the stability conditions of multivariable fuzzy systems [48]; Lyapunov's second method is used to determine the stability of quantized factor selection for fuzzy systems. [49]; Popov-Lyapunov method was used to study the robust stability of fuzzy control systems [50].
However, some stability conditions of Lyapunov are usually conservative, that is, when the stability conditions are not satisfied, the control system is still stable.
4.2 System-based sliding mode fuzzy controller because <br> method is the use of semantic expressions, easy system design to ensure the stability and robustness of the fuzzy control system. Sliding mode control has a distinct feature that can handle the nonlinearity of the control system and is robust. Therefore, some scholars proposed to design a fuzzy controller with a fuzzy sliding surface, so that the stability of the closed-loop control system can be obtained by using Lyapunov theory [25,27,51-54]. Palm and Driankov used the concept of sliding mode control to analyze the stability and robustness of closed-loop fuzzy control systems for gain planning [55]. Some scholars use fuzzy inference to deal with the nonlinearity of the control system and reduce the control of tremor, making the stability of the control system based on Lyapunov method [26].
Based on the variable structure system theory, the relationship between the tracking accuracy of the control system and the I/O fuzzy set map shape of the fuzzy controller can be obtained, so that the robustness and control performance of the fuzzy controller can be explained. Literature [24,56,57] studied the stability of fuzzy control system based on variable structure control framework, and adopted fuzzy variable structure control of output feedback, and proved that the closed-loop control system is a global bounded input bounded output by Lyapunov method. Stable [58]. If a fuzzy rule set of variable structure control type is used, the fuzzy controller can show variable structure characteristics semantically and quantitatively. In order to facilitate Lyapunov's stability criteria to guide the design and adjustment of fuzzy controllers, the literature [30] derives the specific mathematical expressions of fuzzy controllers.
4.3 small gain theoretical approaches <br> small gain nonlinear theory is a theory in very general tool for continuous and discrete control systems. Based on the analytical structure of the fuzzy controller and combining the nonlinear nature of the object and the fuzzy controller, some scholars adopted the small gain theory to establish the Mamdani fuzzy PI[59], PD[9], PID[14] and a class of simple and typical The sufficient conditions for the stability of the bounded input bounded output (BIBO) of the TS [20,21] fuzzy control system; and proved that replacing the conventional PI controller with a nonlinear fuzzy PI controller does not affect the stability at the equilibrium point. . Because these stability results are based on the structure of the controller, stability results that are unknown to those of the fuzzy controller's analytical structure are less conservative.
4.4 <br> phase plane analysis technique using a phase plane analysis and understanding of the low-order to facilitate describing the dynamic behavior of the fuzzy control system, it is used for plane analysis method Stability Analysis of some fuzzy systems [60-62], but This technique is limited to the fuzzy system of the two-dimensional regular structure.
4.5 <br> function method described function method described can be used to predict the presence of limit cycle frequency, and amplitude stability. By establishing the relationship between the fuzzy controller and the multi-value relay controller, the description of the function method can be used to analyze the stability of the fuzzy control system [31]. In addition, the exponent-input description function technique can also be used to investigate the transient response of a fuzzy control system [63]. Although the described functional methods can be used for SISO and MISO fuzzy controllers as well as some nonlinear object models, they cannot be used for three-input and above fuzzy controllers. Since this method is generally used to determine the existence of periodic oscillations in a nonlinear system, it is only an approximate method.
4.6 Method <br> circle round criterion stability criterion for the analysis and re-design of a fuzzy control system. Using the concept of sector bounded nonlinearity, the generalized Nyquist (circular) stability criterion can be used to analyze the stability of SISO and MIMO fuzzy systems [62], and the extended circle criterion can be used to derive a class of simple Sufficient conditions for the stability of fuzzy PI control systems [63]. Due to the strictness of the circle criteria, Furutani proposed a moving Popov criterion for analyzing the stability of the fuzzy control system. When the parameter θ is set to zero in this criterion, the criterion is consistent with the circular criterion [64].
Other methods TS 4.7 <br> robust control techniques (such as a vector stabilized, H∞ control theory and LMI) with uncertainty is used to derive fuzzy control system [46]. In addition, the generalized energy concept [65], the cell mapping concept [66], the geometric state space method [67], the Hurwitz stability condition method [68], the absolute stability criterion method [69], and the Kudrewicz theory method [70 ] and extended Haddad method [71], have been used to analyze the stability of the fuzzy control system.
From the results of the stability analysis of the fuzzy control system, we can see that the most general method is the Lyapunov method, but it is more conservative and the circular criterion is more conservative. For some other typical fuzzy control system stability analysis methods, the object model is required to determine and should meet some continuity constraints. As described in the functional analysis limit cycle, a linear time-invariant object or an object with a certain mathematical form is essentially required, so that the nonlinearity is bounded by a non-linear element in the cycle.
5 Conclusions and Outlook
Through the analysis of the fuzzy controller structure, the nature and working mechanism of the fuzzy controller can be revealed, and the relationship between the fuzzy controller and the classic controller is established. The stability analysis result can be used to guide the analysis and design of the fuzzy control system. Although many analytical results have been obtained so far, compared with classical control theory, analytical fuzzy control theory is still not mature. Many classical control theories and concepts need to be further extended to the analysis and design of fuzzy control systems:
1) In terms of analytical structure analysis, the structural analysis of TS fuzzy controllers needs to be generalized to more general situations and MIMO controllers; the limit structure theory of general Mamdani and TS fuzzy controllers needs further discussion; at the same time, the analysis of fuzzy control of complex systems The theory needs further in-depth. For example, a hierarchical hierarchical fuzzy control system can solve the dimensional disaster problem of multivariable fuzzy controllers and has been proved to be a universal controller [72]. Therefore, it is necessary to analyze and analyze the structure of such controllers.
2) The fuzzy control stability analysis results are still lacking. The current methods are somewhat limited, and more generalized stability criteria, especially those based on the fuzzy controller's analytical structure, should be studied.
3) Other robustness, observability, and controllability, such as fuzzy control systems, need to be further studied and further developed.
4) It is necessary to develop computer-aided design software for fuzzy systems based on analytic theory. Given that many current fuzzy system development software lacks analytical capabilities, we are developing a software package that can be used to analyze, design, and develop fuzzy systems.
In short, the combination of mature and rich classical control theory and fuzzy control will help to better analyze and design the fuzzy control system, thereby laying the foundation for fuzzy control theory and making the core technology of human intelligence more vital in the 21st century. .
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