Optimisation of PID Controllers in Active Suspension Systems: A Comparative Study of the Firefly Algorithm and the Particle Swarm Optimisation

Suspension systems provide a fundamental link between a vehicle's structure and its tyres. Among the main principles in vehicle engineering lies the need for passenger comfort, particularly minimizing disturbances caused by external vibrations. Such vibrations can emanate from multifarious causes, such as braking, engine operation, and unpaved roads. Referring to ISO2631-1 [1], Figure 1 indicates the frequency weightings that demonstrate human body susceptibility to vibrations. It is seen that the 0.5-2 Hz on the horizontal axis and 4-10 Hz on the vertical axis are the areas where the human body shows the most sensitivity. Therefore, to elevate ride comfort, efforts should concentrate on reducing the accelerations of the sprung mass, specifically within the 4-10 Hz range [2].

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Figure 1. Specified frequency weightings in ISO 2631-1 [2]

Vehicle suspensions can largely be categorised based on operational principles. The initial category is the passive suspension system, constructed predominantly from dampers and springs. When integrated with electric, hydraulic, or pneumatic devices, this system evolves into an active suspension. Challenges pertaining to active vehicle suspension systems predominantly surround the need for reliable control strategies. Such strategies must effectively modulate hydraulic actuators to deliver optimal suspension characteristics, whilst concurrently accounting for model uncertainties, parameter variations, and erratic feedback readings [3]. A distinctive capability of active suspensions, not seen in traditional passive systems, is the ability to introduce energy into the system [4]. Typically, in active suspension configurations, actuators are aligned in parallel with the spring and shock absorber components. The core philosophy of active suspension control rests in the deployment of requisite forces between the vehicle's body and the wheel axle through an active element, commonly recognised as the actuator.

In recent years, active suspension systems have been the subject of a wide variety of investigations. Many of these examinations have applied various approaches to the PID controller, owing to its cost-effectiveness, reliability and suitability for many usages. For example, attempts to nullify sprung mass acceleration, suspension deflection and tyre deflection led to the production of the H-infinity control system for actuator time-delayed active suspension systems [5]. A separate study [6] proposed the construction of a disturbance observer-based sliding mode controller in order to increase the efficacy of the vehicle's active suspension system when it comes to sprung mass displacement and acceleration. The aim of this exercise was to diminish body displacement and vibrational duration following exposure to exterior disruptions. Subsequently, a robust controller, a fractional order controller with PID, was formulated to mitigate the effects of such perturbations and optimised using Whale Optimisation Algorithm (WOA) and the PSO methodology [7]. On the opposite end of the spectrum, contrasting studies have explored fuzzy logic controllers (FLC) for vehicular suspensions, revealing a superior performance of FLC compared to other approaches [8]. In a groundbreaking study, an active suspension quarter was built using the first adaptive control method. This particular effort used an approximation dynamic programming (ADP) algorithm. This study highlighted the widespread use of linear control techniques in the field [9]. Subsequently, the introduction of the PID method was reported for the first time for a three-dimensional impact dynamics model [10]. indicating significant progress in the field. Advances in this field have not only incorporated sophisticated control techniques, but also for systems equipped with two hydraulic systems, it has been demonstrated that such devices this ability to independently operate two components provides high efficiency compared to other systems, as described by Nguyen [11]. Such innovations are essential in exploiting the infinite advantages of modern algorithms, which have implications for cost reduction and time efficiency in design and testing. The development of the field is evident, as exemplified by Rodriguez-Guerra et al. LPV) method used it is shown that the constantly changing parameters of this method are consistent with the selected model [12]. At the same time, stronger control mechanisms have been found to help reduce vehicle vibration [13]. Generally, artificial neural network (ANN)-based algorithms have been shown to enhance control process stability [14, 15]. In the realm of active suspension control, a unique method was developed that fuses a PID controller with a fuzzy neural network. The primary optimisation focus of this method lies in body acceleration, with real-time adjustments made to the parameters of the PID controller [16]. A specific method is introduced for active suspension systems using a fuzzy sliding mode control (SMC) method motivated by bionic nonlinear dynamics [17 that concludes by taking advantage of the nonlinear stiffness damping properties of the biomimetic reference model a are used efficiently Despite its energy efficiency performance, an adaptive neural network output feedback control case of a quarter car active suspension system was investigated [18].

Research efforts have expanded in the area of ​​control algorithms used for active suspension systems. One such study introduced a linear quadratic regulator (LQR) control algorithm, where an in-loop optimization procedure was used to optimize the parameters of the LQR controller Another work used a dynamic model with multiple parameters [19] to simulate vehicle oscillations in the four-way surface excitation scenarios [20] in this regard The study also laid the foundation for adaptive fuzzy sliding mode proportional (AFSP) control solutions for active suspension systems. Another notable approach focused on adaptive extended Kalman estimation filtering (AEKF-UI) without known inputs, with an explicit relationship between parking and road complexity as its cornerstone on the reference [21]. An active suspension test system was also established, in which stiffness and damping properties of front and rear suspensions were evaluated under various conditions [22]. The test used an A-level road power spectrum as an input excitation. Subsequently, Min et al. [23] introduced an adaptive fuzzy optimal controller with dynamic capabilities for active suspension systems. Lastly, Mustafa et al. [24] proposed a PSO-based model-free fuzzy intelligent PID controller, further advancing the field.

The present study focuses on the development of a PID controller optimized by FA and PSO. The PSO-PID, FA-PID, and uncontrollable schemes are compared. PID controllers, as designed, have fast and consistent power delivery, characterized by reduced error, steady-state error elimination, reduced overshoot and oscillations Key metrics for measuring suspension systems no efforts include spring mass acceleration, displacement, suspension deflection and tire acceleration Finally bound metrics are After this introduction, the analysis goes into detail on the numerical examples in Section 2. Section 3 covers the development of PID controller with optimization -communicated using FA and PSO capabilities. Finally, Section 4 presents simulation results and insights gained from the implemented controller.

3.1 PSO-based optimisation for PID control parameters

PSO emerged as a compelling intelligent swarm optimisation technique, inspired by the natural behaviours observed in flocks of birds. The inception of PSO is attributed to Kennedy, Eberhart, and Shi. Within the PSO algorithm, every particle in the population is orchestrated to gravitate towards the currently superior particle at a specific velocity. This gravitation seeks to discern the optimal solution among the available alternatives. Owing to its pronounced convergence properties and parallel global search capabilities, PSO frequently discovers the global optimum, adeptly addressing formidable optimisation challenges [26, 27]. The effectiveness of a control system is heavily influenced by the interaction among the particles, and their associated fitness values, as determined through the amalgamation of the PSO method and a pertinent simulation model [28].

Compared to other population-centric stochastic optimisation algorithms such as the genetic algorithm (GA) and Ant Colony Optimisation (ACO), PSO often exhibits comparable, if not superior, search performance for a myriad of optimisation quandaries. PSO's primary merits lie in its straightforward implementation, rapid convergence to satisfactory solutions, and efficacy. Nonetheless, PSO is not without limitations. For instance, a definitive convergence guarantee is absent. Given its stochastic nature, the algorithm can become ensnared in a local optimum, potentially missing the global pinnacle. Furthermore, the sensitivity of PSO to parameter selection poses difficulties. The inertia weight, which plays a crucial role in determining how much particles' ideal placements affect their movements, requires careful calibration. An excessive level of inertia might cause stagnation. However, a minimal level of it leads to unpredictable particle movements. Both of these situations limit the thorough exploration of the search domain. PSO's convergence speed has been found to be limited, especially in search spaces with several dimensions. The delayed convergence of the PSO method can be linked to the limited knowledge that each particle holds about the search space. It is a result of the system's fundamental architecture. To implement PSO, it is necessary to initialise a group of particles with random positions and velocities within the specified search area. The following steps consist of evaluating the fitness of each particle, updating the best positions for both individual particles and the entire swarm, modifying the velocity of each particle based on the optimal positions, and repeating this process until a specified termination condition is satisfied.

$v_{i+1}=w * v_i+c_1 r_1\left(\right.$ pbest $\left.-x_i\right)+c_2 r_2\left(\right.$ Gbest $\left.-x_1\right)$                    (12)

$x_{i+1}=x_i+v_{i+1}$                    (13)

The PSO algorithm's core variables include the inertia weight (w), which enhances search stability, alongside the cognitive (c₁) and social coefficients (c₂), with the former regulating the retention of a particle's initial velocity (v), and the latter two influencing the extent to which a particle is swayed by its own optimum position and that of its neighbours. The equilibrium between exploration and exploitation, pivotal for discerning the global optimum, is modulated by these variables. Random values, r₁ and r₂, fall within the [0-1] range. The terms pbest and Gbest correspond to the finest individual and collective particle positions in the swarm, while x_i represents the current position. Eq. (12) computes the new particle velocity given the prior velocity and the positional discrepancy with the optimum, whereas Eq. (13) facilitates the particle's trajectory towards a new position. The prowess of a particle is evaluated via its fitness function, enabling the pursuit of the finest global and local positions, reflected in diverse PSO performance metrics. Table 2 lists the settings for the PSO algorithm.

Table 2. Configuration parameters for the PSO algorithm

3.2 Description of the FA

The FA has been recognized in recent literature as a prominent method within evolutionary computation, primarily attributed to its transparent evolutionary mechanism, minimal parameter requirements, and notable efficacy in low-dimensional searches. A surge in interest in FA has led to its application in a plethora of domains. Notably, Chatterjee et al. demonstrated superior performance of the FA over the Artificial Bee Colony (ABC) algorithm in the realm of antenna design optimisation [29]. Elsewhere, the FA was successfully integrated for tuning the adaptive network-based fuzzy inference system (ANFIS) and PID controllers in managing water discharge levels in tanks [30]. Furthermore, the PID controller parameters within an Automatic Voltage Regulator (AVR) system were optimised using the FA [31]. Essentially, PID controller gains were adjusted during the FA's optimisation process to enhance control performance under typical conditions. Via this algorithm and subsequent simulations, attempts were made to refine the PID parameters: k_p, k_i and k_d.

Yang is credited with the development of FA, which was modeled on the basis of flame behavior and light-beam patterns, and is a population-based method [32] Each firefly generates a specific heat pattern that serves two purposes: attraction and communication. The absorbance is determined by the inherent brightness of the flame associated with the specified objective function. In this algorithm, each flame occupies the position x=(x1..., xd), the attraction value, the deviation (x) in the segment d-dimensional field, which corresponds to the function f(x) and then in form on the opposite light intensity I(x) is absorbed for the other fireflies ( . It is functional to evaluate f x) as depicted in Eq. (14) [33, 34].

$I=I_e e^{-\gamma r}$                   (14)

where, γ denotes the light absorption coefficient, and I₀ represents the initial light intensity. Given the inverse relationship between a firefly's allure and its perceived brightness by other fireflies, the attractiveness β is defined as:

$\beta(r)=\beta_0 e^{-\gamma r^2}$                    (15)

In this context, β symbolises the attractiveness at r=0, while Eq. (15) depicts the attractiveness's variation β₀ with distance r. Predicated upon these, an initial solution is proposed by Eq. (16):

$X_j=r$ and $\left(U_b-L_b\right)+L_b$                     (16)

where, upper and lower boundaries are denoted by U_b and L_b respectively. Eq. (17) provides an illustration of a firefly's trajectory towards another more luminous firefly.

$X_i^{r+1}=X_i^r+\beta_0 e^{-\gamma r_{i j}^2}\left(X_j-X_i\right)+\alpha \varepsilon_i$                    (17)

where, X_i and X_j delineate the distance between any two fireflies. The parameters implemented in this study associated with the firefly are exhibited in Table 3.

Table 3. Parameters of the FA

The parameter α was identified as being of paramount significance. Experimental sets were conducted, varying its value from 0 to 1 in increments of 0.25, culminating in an optimal value of 0.25. γ is the subsequent pivotal parameter, with its range typically spanning from 0.01 to 100 [35]. The final noteworthy parameter is the population size N, with experimental sizes extending from 100 to 1,000.

The suspension system stands as a pivotal component of a vehicle, instrumental in mitigating vibrations and impacts from uneven road terrains while transmitting force. For the quarter-car active suspension model under investigation, a PID controller was optimised using the FA and PSO. It was observed that with an increment in the number of iterations for both the FA and PSO algorithms, there was a significant reduction in the sprung mass displacement: 18.09% with PSO and 53.19% with FA. Similarly, car body acceleration witnessed a reduction of 44.2% with PSO and 55.75% with FA. These outcomes highlight enhanced system dynamic responsiveness, specifically in the displacement of the car body, an indicator of passenger comfort and vehicular stability.

Future avenues of study are proposed to investigate hybrid algorithms, encompassing combinations such as FA combined with grey wolf optimisation (FA+GWO), FA combined with PSO (FA+PSO), and FA combined with ABC (FA+ABC). Such hybrids promise to amalgamate the strengths of individual optimisation techniques, potentially circumventing their inherent limitations. Employing these multi-strategy approaches could facilitate a more expansive exploration of search spaces, thereby diminishing the risk of converging to local optima. Additionally, the synergistic employment of diverse optimisation strategies might streamline the search, cutting down computational time. This aspect is particularly critical, given that extended computation remains a significant impediment in such studies.