This article presents a comprehensive approach to the structural-parametric optimization of Mamdani-type fuzzy control systems (FCS) using a set of bioinspired methods. The proposed methodology integrates optimization of the number of linguistic terms (LT), rule base (RB) synthesis, selection of membership function (MF) types, parametric tuning, as well as the adjustment of fuzzy inference engine (FIE) operations and defuzzification methods in the most rational sequence. The approach is validated through its application to a FCS for an unmanned aerial vehicle (UAV). Experimental results demonstrate that the greatest improvements in control accuracy are achieved through the parametric optimization, with reductions in the objective function of up to 27.5%. The final optimization stage yielded only minor gains (5.6%), indicating its lower impact. The proposed optimization strategy enables the development of high-performance fuzzy systems with simplified implementations and reduced computational costs, making it suitable for embedded control applications in robotics and autonomous systems.
The design and development of complex technical systems across diverse domains of human
activity, ranging from robotics and industrial automation to medicine and agriculture, are
invariably associated with multifaceted optimization problems [
Conventional optimization These authors contributed equally. techniques, though effective in narrowly constrained contexts, frequently fall short when addressing the intricate interrelations and uncertainties inherent in large-scale, real-world systems [8-10]. These limitations have catalyzed the exploration of more flexible, adaptive, and computationally intelligent approaches to system optimization.
One of the most promising directions in contemporary optimization research is the
development of novel algorithms inspired by natural phenomena and biological systems
[1113]. Bioinspired optimization approaches have gained increasing attention due to their ability
to efficiently explore complex and high-dimensional search spaces, often avoiding local optima
and adapting well to non-linear and multi-objective formulations [14-17]. Among the most
widely adopted and well-tested methods are genetic algorithms (GA) [18], particle swarm
optimization (PSO) [19], ant colony optimization (ACO) [
Moreover, recently, several newer bioinspired methods have emerged, aiming to improve
convergence speed, adaptability, and robustness. These include the Tianji s horse racing
optimization (THRO) [
Fuzzy control systems, in turn, are increasingly adopted in domains requiring robust
operation under uncertainty and imprecision [
In this context, bioinspired algorithms offer a particularly effective toolkit for the
structuralparametric optimization of fuzzy systems [
Despite the substantial progress achieved in recent years, most existing studies primarily address isolated aspects of fuzzy system optimization, either focusing on parameter tuning or structural adjustment. However, the problem of holistic or complex optimization, which entails the systematic execution of all key synthesis procedures in a coherent and rational sequence using appropriate methods and technologies, remains largely unresolved. A fragmented optimization approach often limits the attainable performance and adaptability of fuzzy systems, particularly in complex, real-world applications.
Accordingly, the principal objective of this paper is to develop and validate a comprehensive approach to structural-parametric optimization, which integrates all critical stages of fuzzy system design within a unified, logically structured and the most rational sequence, which will allow to create fuzzy control systems for nonlinear dynamic objects with high quality indicators and robust properties with the shortest duration of the synthesis and implementation processes. The core contributions of this work are threefold: (1) a detailed analysis of the influence of each individual optimization procedure on the efficiency of the fuzzy system, as well as the necessary conditions for its direct execution; (2) the formulation of a stepwise approach to complex structural-parametric optimization, ensuring the most rational sequencing at integration of the core synthesis stages; and (3) a study of the effectiveness of the proposed approach using the example of a UAV's fuzzy control system.
In the design of real-world fuzzy systems under conditions of limited expert knowledge and absence of a priori information, complex structural-parametric optimization tasks may arise. Successful resolution of such tasks requires not only a suite of high-performance structural and parametric optimization methods but also their application in an appropriate sequence. An improper order of application can lead to suboptimal use of these methods, reducing their effectiveness and significantly increasing the overall computational cost of the design and optimization process. Therefore, determining an optimal sequence for structural-parametric optimization procedures based on the available methods and technologies is essential for enhancing the efficiency and success of FCS design.
To determine the optimal sequence of structural-parametric optimization procedures in the development of Mamdani-type FCSs, it is advisable to analyze the impact of each individual procedure on the overall effectiveness of the system, as well as the necessary conditions for its direct implementation. The main procedures involved in the structural-parametric optimization of Mamdani-type FCSs typically include: (1) optimization of the rule base; (2) optimization of the number of linguistic terms for input and output variables; (3) optimization of the membership function types for linguistic terms; (4) optimization of the FCS parameters; and (5) optimization of the types of core operations in the fuzzy inference engine (aggregation, activation, accumulation), as well as the defuzzification method.
The rule base optimization procedure involves determining the optimal consequents vector
and the optimal number of rules (i.e., rule base reduction in terms of rules count or rules
components). This procedure is effectively implemented using the multi-agent method based
on ACO developed in [
The procedure for optimizing the types of fuzzy membership functions enables the selection
of the most suitable membership function for each linguistic term of all input and output
variables in a FCS. This optimization aims to enhance system accuracy and performance while
also reducing the complexity of subsequent parametric optimization by minimizing the total
number of tunable parameters. The procedure can be effectively implemented using the method
proposed in [
Parametric optimization of Mamdani-type FCSs involves the tuning of adjustable parameters
of LTMFs as well as normalization coefficients associated with input and output variables. This
process is essential for enhancing system accuracy and improving the effectiveness of solving
specific tasks. Since setting specific LTMFs parameter values is only feasible after their types
have been determined, the LTMFs parametric optimization must follow their types
optimization. Otherwise, any later change in LTMFs types would require re-performing the
parameter optimization procedure, resulting in unnecessary computational overhead. In turn,
the procedures for optimizing fuzzy system parameters, namely the parameters of LTMFs and
normalization coefficients, can be effectively implemented using hybrid multi-agent methods,
such as the hybrid improved GWO developed and investigated in [
The optimization procedures for selecting the types of core operations in the fuzzy inference
engine (aggregation, activation, and accumulation) as well as the defuzzification method, can
be applied to improve the accuracy of a FCS and enhance its effectiveness in solving the target
tasks. However, the implementation of these procedures is advisable only after the prior
timization of the
number of LTs, which must precede the RB optimization. In the process of optimizing the fuzzy
inference operations, the following core operators can be considered: for aggregation "min"
or "prod"; for activation "min", "prod", or "average"; for accumulation "max", "sum",
-sum operation [
Considering the above-mentioned requirements for the rational sequence of structuralparametric optimization procedures in Mamdani-type fuzzy systems, Table 1 outlines, for each optimization procedure, all the prerequisite procedures that must precede it. Based on the analysis of the requirements presented in Table 1 and the aforementioned considerations, a step-by-step approach to comprehensive structural-parametric optimization of Mamdani-type FCS can be formulated. 3. Approach to complex structural-parametric optimization of Mamdani-type fuzzy control systems
The developed structural-parametric optimization approach is presented as a block diagram
in Figure 1. In the proposed approach to comprehensive structural-parametric optimization of
Mamdani-type fuzzy systems (Figure 1), the initial procedures involve the optimization of the
number of linguistic terms and the synthesis and optimization of the rule base. These
procedures are executed concurrently using the structural optimization method described in
[
Upon completion of these procedures, a verification step is conducted to determine whether the design objectives of the fuzzy system, such as the required accuracy and other performance indicators, have been achieved. If the verification yields a positive result, the process of structural-parametric optimization is considered complete, and the developed and optimized FCS may be implemented using appropriate hardware-software platforms and subsequently applied to solve the targeted tasks. Otherwise, the process proceeds to the next optimization stages according to the proposed approach (Figure 1). As outlined in the approach, the subsequent procedures are performed sequentially: optimization of the LTMFs types, system parameters, fuzzy inference engine operations, and the defuzzification method.
The optimization of LTMFs types is carried out using a bioinspired method developed in
[
After the execution of each of these procedures, a verification step is performed to assess whether the system has achieved the desired performance objectives. If the required level of effectiveness is reached, subsequent optimization steps may be skipped to avoid unnecessary computational overhead.
The developed approach to the comprehensive structural-parametric optimization of
Mamdani-type fuzzy systems (Figure 1) can also be applied to the optimization of Takagi
Sugeno fuzzy systems. In this case, instead of optimizing the rule base using ACO-based
methods, the optimization of the rule consequent weight coefficients is performed using the
improved hybrid GWO method proposed in [
To validate the effectiveness and rationality of the proposed approach, this study performs a comprehensive structural-parametric optimization of a Mamdani-type fuzzy system for UAV flight control. In turn, this study does not present a comparison of the employed bioinspired methods with their counterparts in terms of efficiency, time, and computational costs during the optimization procedures, as such comparisons have already been conducted extensively in previous research, including benchmarks against classical optimization methods. 4. Study of the effectiveness of the proposed approach using the example of a UAV's fuzzy control system
The structural-parametric optimization procedures were carried out in accordance with the
proposed approach for the altitude automatic control system of the UAV, whose detailed
description, mathematical model, and key technical specifications are provided in [
The comprehensive structural-parametric optimization procedures of the described FCS
were conducted for the case of UAV flight at a fixed altitude over mountainous terrain with
complex topography, which is thoroughly examined in [
J 1 (x ,v x ,S) = 1 x max
(z D (x ) −z R (x ,v x ,S))2 dx , x max 0 where x is the horizontal coordinate; xmax is the length of the terrain section for which the calculations were made; vx is the flight speed along the coordinate x; S is the vector of optimized parameters or components of the structure for each specific optimization procedure of the approach; zD is the specified value of the flight altitude over mountainous terrain; zD is the real value of the flight altitude.
Thus, for the purpose of detailed analysis, no fixed target value of the objective function (2) was predefined at the beginning of the fuzzy system design, and no verification of goal achievement was performed after each optimization procedure during the process. All major optimization procedures of the proposed approach (Figure 1) were executed sequentially, with a specified number of iterations allocated to each procedure.
Since the first two procedures, namely, the optimization of the number of linguistic terms
and the synthesis and optimization of the RB, were successfully carried out for the given UAV
altitude control system in [
Following this, in accordance with the proposed approach (Figure 1), the third procedure
optimization of the types of LTMFs was carried out to further enhance the accuracy of altitude
control. This procedure was performed using a bioinspired method for selecting optimal
membership functions based on evolutionary algorithms, as developed in [
As a composite objective function JC for executing the LTMFs type optimization, expression
(3) was selected [
J C = J 1 + k J 2J 2 , where J2 is the objective function that determines the complexity of further parametric optimization of a fuzzy system; kJ2 is the weighting factor.
In turn, the component of the objective function J1 was calculated according to expression
(2), while J2, which evaluates the complexity of subsequent parameter optimization, was
computed as the total number of adjustable parameters of the LTMFs set based on expression
(4) [
n τi m τj J 2 = k iin (q ) + kojut (k ),
i =1 q =1 j =1 k =1 where k iin (q ) and k ojut (k ) are the numbers of optimized parameters of the q-th linguistic term for the i-th input variable and the k-th term for the j-th output variable depending on their membership functions type; n and m are the total numbers of input and output variables; i and j are the total numbers of LTs for the i-th input variable and j-th output variables.
The weighting coefficient for J2 was set kJ2 = 0.002. Prior to the execution of the LTMFs optimization procedure for the fuzzy altitude controller with triangular MFs, the initial values of the objective functions were as follows: JC = 0.227, J1 = 0.119, and J2 = 54.
Since the previous optimization procedures (i.e., optimization of the number of LTs and rule
base) had already enabled the fuzzy altitude control system for the UAV to achieve sufficiently
high control accuracy (J1 = 0.119), the iterative search for the optimal vector of membership
functions was conducted using only a single global optimization evolutionary algorithm,
namely, the biogeography-based optimization algorithm. This choice was made to reduce
computational and time costs, as BBO demonstrated the best performance for membership
functions types optimization in the studies reported in [
During the execution of the LTMFS type optimization using the BBO algorithm, its core parameters were experimentally tuned for this specific task. In particular, an ecosystem was initialized with Zmax = 100 habitats (islands). The species migration rates as functions of the number of species per island, NS) and NS), were assumed to be linear, with maximum values max = max = 1. The mutation operator coefficient was set to r = 0.1, and the maximum allowable number of species per island (corresponding to the optimal habitat suitability index fopt) was set at NSmax = 10. The habitat suitability index f was calculated as the inverse of the composite objective function JC. The stopping criterion for the optimization process was defined as reaching the maximum number of iterations Nmax = 100.
Upon completion of the LTMFs type optimization using the BBO algorithm for the FCS, the composite objective function JC was successfully reduced to 0.184 (an 18.9% decrease), the performance objective J1 was lowered to 0.098 (a 17.6% improvement), and the complexity criterion J2 decreased to 43 (a reduction of 11 parameters). These results indicate that the optimization procedure was effectively carried out in accordance with the proposed comprehensive algorithm. As a result, both the altitude control accuracy and the overall simplicity of the LTMFs structure were improved, significantly facilitating the subsequent parametric optimization of the designed fuzzy control system.
In turn, the optimal membership function vector S obtained using the BBO algorithm has the form:
S = {Gs 1FN ,TrpFN ,TrFN ,Gs 1FN ,SgFN ,ZFN ,Gs 1FN ,SFN , Gs 1FN ,Gs 1FN ,TrFN ,ZFN ,GbFN ,Gs 1FN ,TrFN ,TrFN ,Gs 1FN ,SFN }, (5) where Gs1FN, TrpFN, TrFN, SgFN, ZFN, SFN, and GbFN are the Gaussian 1st type, trapezoidal, triangular, sigmoid, Z-shaped, S-shaped, and bell-shaped functions.
Subsequently, in accordance with the approach presented in Figure 1, the accuracy and efficiency of the UAV altitude control system were further improved by performing the next procedure, namely, the parametric optimization. Since the preceding optimization of the number of LTs, the RB, and the types of LTMFs had already yielded sufficiently high accuracy and performance for the developed system (J1 = 0.098), only the tunable parameters of the membership functions were optimized in this stage. To significantly reduce computational costs, this procedure was carried out using a single local search algorithm, specifically the Extended Kalman Filter algorithm. In the application of the EKF algorithm, the initial values of the a posteriori error covariance matrix were selected as P0 = 40900I43, where I43 is the identity matrix of size 43. The process noise covariance matrix was set to Q = 3900·I43. The measurement noise covariance matrix R was considered scalar in this case, with R = 500, as the fuzzy altitude controller has only one output (the control signal u). The stopping criterion for the optimization process was defined as the maximum number of iterations Nmax = 100. The objective function for this stage of optimization was J1, computed according to (2).
The performed parametric optimization of the LTMFs resulted in a significant improvement in the accuracy of UAV flight control. Specifically, the value of the objective function J1 was reduced to 0.071 (a 27.5% decrease), which confirms the effectiveness and appropriateness of with the optimized parameters is presented in Figure 2.
The final optimization procedure within the proposed approach involves the optimization
of the types of FIE operations and the defuzzification method. This procedure was carried out
after the parametric optimization of the LTMFs to further improve the accuracy of UAV altitude
control, using a sequential search method proposed in [
As a result of the optimization, the following configuration was identified as optimal: aggregation "min", activation "prod", accumulation "max", and defuzzification center of gravity. As seen from the results, only the activation operation changed (from "min" to "prod"), while the remaining FIE operations and the defuzzification method remained unchanged. This modification led to a further reduction in the value of the objective function from 0.071 to 0.067 (a 5.6% improvement), thereby slightly increasing the accuracy of the UAV's altitude control. Conversely, changes to other types of FIE operations and defuzzification methods only resulted in an increase in the objective function value.
Table 2 presents the overall results of the comprehensive structural-parametric optimization procedures for the UAV's fuzzy altitude control system, implemented using the proposed approach, where N denotes the number of iterations for each corresponding procedure.
As a result of the optimization process, the objective function value (2) was reduced to J1 = 0.067 through the execution of 8 iterations in the first two procedures (optimization of the LTs number and the rule base), 100 iterations in the third procedure (optimization of LTMF types), 100 iterations in the fourth procedure (parametric optimization), and 24 iterations in the final procedure (optimization of FIE operations and the defuzzification method).
As shown in Table 2, the UAV flight control system optimized using the proposed integrated
approach demonstrates significantly higher control accuracy and improved performance
compared to the system with only the number of LTs and the RB optimized, as described in
[
This study proposes an advanced approach to comprehensive structural-parametric optimization of fuzzy control systems. The main novelty of the developed approach is that it combines all critical stages of fuzzy system design within a unified, logically structured and the most rational sequence, which makes it possible to create fuzzy control systems for nonlinear dynamic objects with high quality indicators, robust properties and simplified software and hardware implementation while maintaining shortest duration of the synthesis process and minimal computational costs. This approach includes the optimization of the number of linguistic terms for input and output variables, synthesis and optimization of the rule base, selection of optimal types of membership functions, parametric optimization, as well as the identification of optimal fuzzy inference engine operations and defuzzification method. Moreover, the core optimization procedures of this approach are performed using a specially selected set of highly efficient and well-established bioinspired algorithms that have repeatedly proven their superiority in a number of previous studies.
The effectiveness of the proposed approach has been validated through its application to the comprehensive structural-parametric optimization of a fuzzy altitude control system for the UAV. Analysis of the experimental results indicates that the first two optimization procedures (LTs number and RB optimization) are foundational within the proposed methodology. They allow for the identification and implementation of highly flexible and effective fuzzy control and decision-making strategies by forming optimized rule bases in number and consequents. As a result, this ensures high efficiency, interpretability, and logical transparency of the fuzzy systems while preserving implementation simplicity. Subsequently, the optimization of LTMF types significantly improves the accuracy and performance of the FCS following the foundational procedures. It also simplifies the next phase (parametric optimization) by reducing the number of LTMF parameters subject to optimization. Parametric optimization, in turn, provides the most substantial improvement in system performance at the penultimate stage of design, requiring relatively low computational effort, which makes it one of the most important stages of comprehensive optimization. For instance, this procedure reduced the objective function J1 for the UAV's control system by 27.5%. The final optimization stage, involving the tuning of FIE operations and the defuzzification method, was found to have the least impact on overall system performance. In the case of UAV's FCS, it resulted in only a 5.6% improvement in performance. Therefore, this procedure can be omitted in a number of cases, which will further reduce the computational and time costs of the entire design and optimization process.
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