Swarm optimization of the drone s intelligent control system: comparative analysis of hybrid techniques

Abstract

Over recent years, bioinspired swarm techniques have gained significant popularity for addressing realworld engineering optimization challenges. One promising application of these methods is developing and optimizing of intelligent systems, specifically fuzzy control systems. This paper examines research issues and performs a comparative analysis of bioinspired swarm methods for parameter optimization in fuzzy control systems. It compares various hybrid modifications of particle swarm optimization and grey wolf optimization techniques, specifically adapted for fuzzy system parameter optimization, against traditional search methods. As a case study, the paper uses the parametric optimization of a TakagiSugeno fuzzy control system designed for a quadrotor-type unmanned aerial vehicle (UAV). The simulation results confirm the effectiveness of the presented swarm bioinspired optimization techniques, taking into account both the performance of the UAV's fuzzy control system and the computational costs involved.

Keywords

• Bio-inspired optimization
• hybrid swarm methods
• particle swarm optimization
• grey wolf optimization
• fuzzy control system
• unmanned aerial vehicle 1

1. Introduction

At designing intricate objects and systems in diverse fields like technology, agriculture, manufacturing, economics, and medicine, there's often a need to locate optimal solutions within a complex, multidimensional search space [ 1-4 ]. Standard optimization techniques frequently fail to meet these demands due to the unpredictable nature of the terrain and the existence of multiple local optima in the functions under consideration, which capture the complex relationships between solution effectiveness and unknown parameters [5-8]. Currently, approximate techniques of global optimization are gaining traction due to their capacity to find high-quality solutions efficiently in terms of both computational resources and time [9-12]. Bioinspired swarm and evolutionary algorithms stand out among these methods [13-15]. Unlike traditional methods of local search, bioinspired approaches are effective even with limited information about the nature and characteristics of the systems being optimized. They are proficient at avoiding local minima and are versatile enough to be applied to a wide array of real-world optimization problems. Additionally, these methods utilize straightforward computational procedures that mimic the behaviors of social animals, evolutionary concepts such as natural selection, and certain physical phenomena. In essence, bioinspired intelligent algorithms can be effectively combined with various These authors contributed equally. local search methods, significantly improving the optimization process by strategically merging global and local search principles [16, 17].

In recent years, several intelligent algorithms have proven particularly effective and widely adopted for tackling diverse optimization challenges. Notable among these are particle swarm optimization (PSO) [18], ant lion optimization (ALO) [19], cuckoo search (CS) algorithm [20], grey wolf optimization (GWO) [21], whale optimization algorithm (WOA) [22], firefly algorithm (FA) [23], and chaotic swarming of particles (CSP) algorithm [24]. Moreover, numerous hybrid techniques have been successfully applied in the development of complex systems [25].

The use of metaheuristic swarm and evolutionary techniques for designing and optimizing intelligent control and decision support systems is a promising field, particularly when applied to systems based on fuzzy logic [14, 26, 27]. Recent research shows that fuzzy systems (FS) optimized with bioinspired methods are highly effective in tackling complex control and decision support issues across various domains [13, 28]. For instance, studies [29-32] have adapted PSO methods for the parametric optimization of linguistic terms (LT) in fuzzy automatic control systems (ACS) for different technical applications. Specifically, [29] discusses the optimization of triangular LT vertices for a fuzzy power control system in an industrial wireless sensor network. Meanwhile, [31] focuses on optimizing 1st type Gaussian LT parameters for a fuzzy ACS in an autonomous mobile robot operating in uncertain environments. Additionally, the GWO method has shown highly competitive results in parametric optimization for various fuzzy system configurations, outperforming other well-known bioinspired techniques [17, 33, 34]. Therefore, ongoing research into developing, refining, and implementing bioinspired swarm methods for synthesizing and optimizing different types of fuzzy systems is essential and highly relevant.

This study centers on the research and comparison of bio-inspired swarm techniques for the parametric optimization of a Takagi-Sugeno fuzzy automatic control system, specifically designed for a quadcopter UAV. The quadcopter drone possesses the ability to navigate through various terrains and environments, serving as a versatile tool for conducting intricate inspection and monitoring tasks in various environments and conditions. To fully harness the potential of this UAV, an intelligent control system optimized using advanced methods is essential [35]. Selecting the most suitable method to achieve high efficiency while maintaining low computational costs is a challenging task. Hence, the main goal of this study is to carry out efficiency research and perform a comparative analysis of various adaptations (basic and hybrid) of PSO and GWO swarm techniques for the parametric optimization of fuzzy ACSs. 2. Hybrid fuzzy automatic control system for the unmanned aerial vehicle Quadcopter-type UAVs are among the most advanced and popular types of micro-class drones, which have recently been widely used for monitoring and inspection tasks in various civil and military sectors [ 36-38 ]. Also, such drones can be effectively used in critical infrastructure facilities along with other mobile robotic means to perform complex tasks of increased importance and danger [ 39 ]. These devices have several advantages over other types of unmanned aerial vehicles, including high maneuverability, the ability to take off and land vertically, hovering capability, high energy efficiency, and ease of maintenance. However, quadcopters are also quite complex to control, making it appropriate to use intelligent ACS, particularly fuzzy control systems, for their automation [ 35, 40 ]. One of the most important tasks in automating the spatial movement of such drones is the stabilization and automatic regulation of their flight altitude.

In this work, the fuzzy control system for quadcopter flight altitude is considered for researching and comparing different swarm optimization methods. To improve the quality of the altitude control and, accordingly, the overall efficiency of the processes of automating its vertical movement, it is advisable to use a hybrid fuzzy ACS based on a classical PID controller and a sliding mode controller, the structure of which is shown in Figure 1.

In turn, the following notations are used in Figure 1: SD is the setting device; CC is the classical PID controller; SMC is the sliding mode controller; FS is the fuzzy system for determining value of the aggregating weight coefficient KH; AS is the sensor for altitude measuring; LU is the limiting unit that passes only positive signals and limits them to a level of no more than one; zS and zR are the set and real values of the flight altitude of the UAV; uSD, uAS, uPID, uSC, and u1 are the corresponding output signals of the SD, AS, classical PID controller, sliding mode controller, and the entire hybrid controller z is the control error formed at the output of the adder in the main feedback channel; ε z and  εzdt are the control error derivative and integral, respectively; kP, kD, and kI are the PID controller gains; Fg and Fz are the gravity force and various disturbances acting on the UAV.

In this hybrid ACS, the classical PID controller is combined with the sliding mode controller using a specific fuzzy system that calculates the weighting factor KH. At the same time, the aggregation of the output control signals of the classical uPID and sliding mode uSC controllers is carried out using the KH coefficient, as a result of which the control signal of the hybrid controller u1 is calculated according to the expression: u 1 = K Hu PID + (1 −K H )u SC. (1)

At sufficiently small values of the control error z (in the vicinity of the set value of the controlled coordinate zS), the coefficient KH approaches unity, and the resulting control signal u1 for the UAV is largely determined by the output signal of the classical PID controller uPID. In turn, at significant values of the error z, the coefficient KH approaches zero, and the greatest contribution to the signal u1 is made by the control signal of the sliding mode controller uSC. Moreover, due to the additional use of the derivative of the flight altitude control error ε z , when calculating the KH coefficient, the dynamics of transitions from one control mode to another are better taken into turn, the limiting unit LU calculates the modulus of the output signal of the FS and limits it to a level of no more than one.

The output control signal of the sliding mode controller uSC is calculated based on the current values of the sliding surface SK according to relationship (2), and the sliding surface SK itself is calculated based on equation (3).

+u Cmax ,  at  S  К  0;  u SС =   0,         at  S  К = 0;

−u Cmax ,  at  S  К  0; S К = a1ε +a2ε + ... +an −1ε(n −2) + ε(n −1) ,

K H = f FS (K Pεz ,K Dεz ),

X = Ki ,PLT ,PC, where uCmax is the maximum possible value of the SMC control signal; a1, a2 an 1 are coefficients of the Hurwitz polynomial (a polynomial with real coefficients, all zeros of which are located in the left complex half-plane); n

In this case, a sliding surface of the first order is used, its only coefficient according to the Hurwitz polynomial a1 is equal to 2.4 (a1 = 2.4). The parameters of the traditional PID controller are as follows: kP = 2.54; kD = 1.43; kI = 0.513. In turn, these coefficients are calculated on the basis of a mathematical model of a quadcopter UAV [17, 37] for the conditions of constancy of all its coefficients.

In turn, the FS calculates the value of the KH weighting factor according to the dependence: where KP and KD are the normalizing factors of the FS.

The desired flight altitude of the UAV zS can be specified by the upper-level control system, which is recommended to be implemented using IoT (Internet of Things) technology [ 41-43 ].

In this study, to conduct a comparative analysis of the considered swarm techniques, a parametric optimization of the Takagi-Sugeno type FS was performed for the hybrid ACS of the X to be optimized is defined by the expression (5) (2) (3) (4) (5) where Ki is the vector of normalizing factors (KP and KD); PLT is the vector of adjustable parameters of the linguistic terms; PC is the vector of weighting gains for the consequences of the rule base rules.

For the input signals of the fuzzy system the following linguistic terms of the triangular type z 5 LTs (BN big negative; SN small negative; Z zero; SP small positive; BP big positive); ε z 3 LTs (N negative; Z zero; P positive). Therefore, the PLT vector contains 24 tunable parameters, each of which must be optimized. In turn, the rule base of this fuzzy system consists of 15 rules, each defined by expression (6)

IF “K Pεz = LT1 ” AND “K Dεz = LT2 ” THEN “K H = k 1r (K Pεz ) + k 2r (K Dεz ) + k 3r , (6) where LT1 and LT2 are certain linguistic terms; k1r, k2r, and k3r are the weighting gains of the r-th rule.

Thus, the PC vector of consequent weighting coefficients consists of 45 coefficients. Overall, the parameter vector X to be optimized in this case comprises 71 parameters.

Next, we move directly to optimizing the parameters of the proposed hybrid fuzzy control system. This step involves conducting efficiency research and a comparative analysis of various modifications (both basic and hybrid) of PSO and GWO swarm techniques. 3. Swarm-based parameter optimization of the hybrid fuzzy control system for a quadcopter UAV The a system using effective and well-established swarm methods like PSO and GWO [ 17, 18, 21, 44 ]. These methods, along with their various hybrid modifications integrated with local search techniques, aim to expedite convergence. Specifically, hybrid PSO modifications based on the elite strategy with gradient descent (GD) and extended Kalman filter (EKF) algorithms, as proposed in [ 44 ], will be applied. Furthermore, an enhanced GWO technique [ 45 ], along with its hybridization with GD and EKF techniques as suggested in [17], will be considered. Finally, for thorough stem using individual local search techniques as GD and EKF.

The fundamental principles of the PSO algorithm and its application for the synthesis and parametric optimization of FSs are thoroughly covered in [16, 18]. Additionally, the authors in [ 44 ] proposed enhancing the FS optimization processes by hybridizing PSO with GD and EKF, utilizing an elite strategy. The key idea behind these modifications is to enable an independent parallel search by the best particle in the swarm using GD or EKF, which can potentially speed up convergence and reduce the computational and time costs associated with these methods.

The basic and improved GWO methods are thoroughly detailed in [21] and [ 45 ]. The enhanced GWO method incorporates an additional dimension learning-based hunting (DLH) strategy to boost population diversity and prevent premature convergence to suboptimal solutions [ 45 ]. In [17], the authors suggested hybridizing the improved GWO with local search methods such as GD and EKF. To implement this, similar to hybrid PSO techniques, alpha, beta, and delta agents are assigned to perform local searches in their nearby areas using GD or EKF, alongside utilizing group hunting and DLH strategies.

In this research to conduct a comparative analysis during the optimization processes, the s actual transient response from the desired response was selected as the objective function J [17]. In turn, the desired response was calculated based on the reference model, which had the transfer function of the second-order dynamic object [17]. The objective function's optimal value was set to Jopt = 3100, which needed to be reached during the optimization process. To ensure comprehensive research, the maximum number of iterations was capped at Nmax = 200, serving as the termination criterion for the optimization. For the PSO algorithm and its hybrid modifications, the following adjustable parameters were used: the swarm size Zmax = 30, the maximum particle velocity Vmax = 10, and acceleration coefficients C1 = C2 = 0.1. Zmax = 30. The same constraints as in [ 44 ] and [17] were applied in this case.

The procedures for optimizing the parameter vector X were conducted sequentially using each of the investigated swarm methods, repeating the process five times and selecting the best outcomes. During each iteration, the objective function values were computed by simulating the operating modes, specifically addressing both large and small deviations from the specified values. In addition, all simulation calculations were performed using the mathematical model of the UAV flight detailed in papers [17, 37].

To evaluate the efficacy of the swarm optimization methods used in this study, it is suggested to compare the achieved minimum values of the objective function Jmin alongside the associated computational costs. Moreover, the computational resources required to reach the predefined optimal value Jopt of the objective function are considered for evaluation. In this regard, the computational costs of the analyzed techniques are largely determined by the total number of

J needed Jopt for the optimal value Jopt Jmin for the best value Jmin. This is explained by the fact that the calculation of the current value of the objective function using a complex simulation model of a fuzzy control system requires significantly greater computational and time costs compared to simpler computational operations of the considered optimization algorithms [17]. Also, based on a certain value of the total number of objective function evaluations, the total time of the calculations performed can be determined quite simply, depending on the power of the computing resources used.

The Figure 2 illustrates the progression of the best values of the objective function during the optimization of the vector X using the methods under study: 1 basic PSO; 2 hybrid PSO with GD; 3 hybrid PSO with EKF; 4 basic GWO; 5 IGWO (improved GWO); 6 hybrid IGWO with GD; 7 hybrid IGWO with EKF; 8 GD; 9 EKF.

Table 1 provides a summary of the best results from the computational experiments conducted for optimizing vector X with each of the investigated methods. In turn, when it comes to swarm methods and their diverse adaptations (including hybrid and

Jopt Jmin typically exceed the corresponding iteration counts NJopt and NJmin. This is because the objective function must be computed at each iteration for every agent within the swarm. In contrast, for standalone methods like the gradient method and the extended Jmin required to reach its minimum value Jmin equals the iteration count NJmin.

The presented outcomes in Figure 2 and Table 1 confirm that hybrid IGWO methods demonstrate higher effectiveness compared to hybrid PSO algorithms in the parametric optimization of the fuzzy system for computing the coefficient KH of the hybrid ACS for the UAV. Thus, to find the optimal value of the objective function J using hybrid IGWO methods with EKF and GD, in the best case scenario, it required 513 and 225 fewer evaluations of the objective function respectively compared to using hybrid PSO algorithms based on the elite strategy with EKF and GD. Furthermore, the implementation of hybrid IGWO methods on average ensured the achievement of a smaller minimum value of the objective function Jmin compared to hybrid PSO techniques.

For addressing this specific problem (optimization of the hybrid fuzzy ACS for the UAV), the most effective approach is the hybrid IGWO method with EKF. Implementing this method enabled the attainment of the optimal value of the objective function for the hybrid fuzzy ACS (J with the fewest number of evaluations of the objective function Jopt = 1797). Furthermore, when implementing this method on the 36th iteration (Figure 2, curve 7), the lowest value of the objective function was achieved (Jmin = 2712).

The separate application of the gradient method and the extended Kalman filter algorithm in this case did not allow achieving the optimal value of the objective function (J feature of this specific problem (parametric optimization of the hybrid fuzzy ACS for the UAV) is that the application of the hybrid PSO method with GD enabled faster attainment of the optimal technique with EKF. Additionally, the standalone gradient method showed better results than the standalone EKF algorithm.

Furthermore, the optimized parameters of the resulting vector Xbest exhibit the following specific values. Regarding the vector of normalization factors Ki, the components were determined to be KP = 0.0587; KD = 0.072. The representation of linguistic terms for the input variables of the FS with their optimized parameters is illustrated in Figure 3.

The fragment of the rule base within the optimized fuzzy system, achieved using the hybrid IGWO method incorporating EKF for minimizing the objective function, is detailed in Table 2. methods being investigated. 4. Simulation tests of the hybrid fuzzy control system for the UAV

To validate the effectiveness of the developed hybrid fuzzy ACS with optimized FS parameters based on the proposed hybrid IGWO method with EKF, transient flight processes of the UAV are depicted in Figure 4. Curves 1, 2, and 3 represent the system outputs (actual flight altitude values of the quadcopter at constant horizontal coordinates) with the hybrid controller (based on the optimized FS using hybrid GWO method with EKF), fuzzy PID controller, and optimally tuned traditional PID controller (developed in [17]). Line 4 represents the set altitude value of the UAV flight, while line 5 depicts the disturbance influence of wind Fz. Additionally, Table 3 provides a comparative analysis of the altitude control system performance metrics for the aforementioned transient processes.

In turn, the numerical data in Table 3 correspond to the change in time of the actual UAV flight altitude zR when varying the set altitude value zS from 0 to 40 m (Figure 4).

From the data presented in Figure 4 and Table 3, it is clear that the hybrid fuzzy control system implemented for the quadcopter UAV, utilizing optimized FS via the hybrid IGWO method with EKF, demonstrates significantly higher control quality metrics compared to configurations employing optimized fuzzy and conventional PID controllers. Specifically, this system exhibits faster response times and reduced overshoot, which are key indicators of dynamic control performance.

5. Conclusions

This research focuses on evaluating and comparing swarm bio-inspired techniques for parameter optimization in fuzzy control systems. In particular, it examines various hybrid adaptations of particle swarm optimization and grey wolf optimization methods, tailored for FS parameter optimization, comparing them both with each other and with traditional search methods.

The study involves research and comparative analysis using a specific case: the parametric optimization of a Takagi-Sugeno hybrid fuzzy automatic control system for an unmanned aerial vehicle of a quadcopter type. The simulation results indicate that hybrid IGWO methods generally outperform hybrid PSO methods in optimizing parameters of a particular fuzzy system that aggregates a classical PID controller with a sliding mode controller within the UAV's ACS. Among the evaluated methods, the hybrid IGWO method with EKF proves to be the most effective for this challenge. Its application achieves the optimal objective function value for the hybrid fuzzy ACS (J Jopt = 1797).

Additionally, the hybrid fuzzy ACS, by integrating the benefits of both a classical PID controller and a sliding mode controller via the FS, along with employing a highly efficient parameter optimization method, shows improved responsiveness and reduced overshoot compared to the ACS that utilizes a fuzzy PID controller. As a result, identifying the optimal vector for the FS of the hybrid ACS for the UAV using the hybrid IGWO technique with EKF did not demand substantial

Jmin = 2082). This overall confirms the high effectiveness of the proposed fuzzy hybrid ACS model and developed in [17] hybrid swarm parameter optimization approach.

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