## 3 Proposed Identification Methods
### Particle Swarm Optimization  
Particle swarm optimization (PSO) is a meta-heuristic optimization algorithm that was developed by [PERSON] and [PERSON] in 1995 [PERSON] and [PERSON] (1995). It is a population-based algorithm that simulates the social behavior observed in swarms of biological individuals, such as flocking birds and insects searching for food sources [PERSON] and [PERSON] (1995). The PSO algorithm is easy to implement and has been applied to a wide range of scientific and engineering problems [PERSON] and [PERSON] (1995); [PERSON] and [PERSON] (1995).  
In the PSO algorithm, a swarm of particles (\(N_{PSO}\)) is initialized randomly within a search space defined by upper (\(x_{max}\)) and lower (\(x_{min}\)) bound limits. Each particle is evaluated against a pre-defined cost function (\(Fcost\)) to search for an optimal solution. The particles update their velocity vectors, which control their movement in the next iteration [PERSON] and [PERSON] (1995); [PERSON] and [PERSON] (1995); [PERSON] and [PERSON] (1995); [PERSON] and [PERSON] (1995). The velocity updates are influenced by two key factors: personal best (\(pBest\)) and global best (\(gBest\)). The \(pBest\) represents the best solution found by an individual particle, indicating its optimal position within the search space. The \(gBest\) refers to the best solution discovered by any particle in the entire swarm, representing the overall optimal position identified by the collective behavior of the swarm. The velocity and position of each particle are updated as follows [PERSON] and [PERSON] (1995); [PERSON] and [PERSON] (1995):  
\[v_{k+1}(i)=v_{k}(i)+c_{1}r_{1}(pBest_{k}(i)-x_{k}(i))+c_{2}r_{2}(gBest_{k}-x_{ k}(i)) \tag{9}\]  
\[x_{k+1}(i)=x_{k}(i)+v_{k+1}(i) \tag{10}\]where \(i\) denotes the current particle; \(c_{1}\) and \(c_{2}\) are the acceleration coefficients for \(pBest\) and \(gBest\) terms; and \(r_{1}\) and \(r_{2}\) are random numbers generated by uniform distribution in the interval \([0,\ 1]\)[45; 46]. The PSO offers several advantages that make it suitable for various optimization problems. It excels at quickly finding solutions, even in high-dimensional search spaces, and is capable of handling complex and noisy optimization problems [45]. Additionally, the PSO has the ability to adapt to changes in the optimization landscape, allowing it to continue searching for better solutions, even when the search space becomes more challenging [47]. However, the PSO also has some limitations. One of the main drawbacks of the PSO is its tendency to get attracted to sub-optimal solutions that are not globally optimal [48]. This can result in premature convergence, where the particles get trapped in a locally optimal solution and continue to search within reduced and limited regions of the search space. Moreover, the performance of the PSO algorithm is sensitive to the selection of control parameters, such as the inertia weight and acceleration constants. Inappropriate parameter choices can lead to poor performance and potential convergence to sub-optimal solutions [49].  
To enhance the performance of PSO and address its limitations, various PSO variants have been developed, such as constriction PSO, adaptive inertia weight PSO, and hybrid PSO [50; 51]. These variants incorporate additional strategies like dynamic control of the inertia weight, self-adaptive acceleration constants, and hybridization with other optimization methods. Some prior studies have explored variants such as constricted PSO with linearly decreasing inertia weight, adaptive weight factor PSO, and chaos initialized PSO for the identification of 2 MM systems [2; 3]. Among these, constricted PSO with a linearly decreasing inertia weight showed particularly promising performance, making it our selected base PSO variant for the hybrid framework. For more detailed information on these PSO variants and their application in 2 MM system identification, readers are encouraged to refer to [2; 3].