# A Study of Friction Nonlinearity and Compensation for Turntable Servo Systems
## 2 Friction Modeling and Identification
### Friction Parameter Identification  
#### 2.2.1 Introduction to the Identification Algorithm  
In this study, the particle swarm optimization (PSO) algorithm is employed for parameter identification. The PSO algorithm is a branch of evolutionary computational methods that seek optimal solutions through iterative processes in a swarm-based approach. As the PSO algorithm converges quickly, occupies less computational space, and can adapt to the needs of complex nonlinear dual-inertia system parameter identification, it was selected to replace the traditional single-inertia parameter identification and optimization algorithm. The principles of PSO are as follows [16]:  
\(M\) particles exist in a \(D\)-dimensional space, and each particle has only two vector attributes: position and velocity. An evaluation function \(f_{m}\) is defined to calculate the distance of each particle from the optimal solution in the space, which is referred to as fitness. By sharing and comparing the fitness of each particle within the current group, the global best solution \(g_{best}\) can be determined. Each particle compares its own fitness and flight experience to determine its current best position \(p_{best}\). The positions and velocities of all particles are then updated, causing each particle to move toward positions with better individual fitness and better group fitness until the optimal solution in the space is reached. The iterative formulas are given in (2).  
\[\left\{\begin{array}{l}x_{i}(t+1)=x_{i}(t)+v_{i}(t+1)\\ v_{i}(t+1)=\gamma v_{i}(t)+c_{1}r_{1}[p_{best}-x(t)]+c_{2}r_{2}[g_{best}-x_{i} (t)]\end{array}\right. \tag{2}\]  
In Formula (2), the learning factors \(c_{1}\) and \(c_{2}\) are both set at 1.7, representing the particle's self-learning capability and social sharing ability, respectively. Random numbers \(r_{1}\) and \(r_{2}\) range from 0 to 1. The variables \(x_{i}(t)\) and \(v_{i}(t)\) denote the current position and velocity of the \(i\)th particle, respectively, while \(x_{i}(t+1)\) and \(v_{i}(t+1)\) represent the position and velocity of the particle at the next time step. The inertia weight \(\gamma\) characterizes the confidence level of the particle; a higher value indicates a stronger global search capability and a weaker local search capability. To better balance between the global and local search abilities, [PERSON] et al. proposed the linear decreasing inertia weight (LDIW) [17].  
Figure 1: Static friction + Coulomb friction + viscous friction model.  
In the friction model, the parameters to be identified include \(T_{c}\), \(T_{s}\), and \(B_{v}\); hence, the dimension of the search space is \(D=3\). By substituting a particle's current position and the time series, the identified speed sequence is obtained. The reciprocal of the sum of squared differences between this sequence and the actual speed sequence in the deceleration phase is defined as the particle's fitness function. Smaller errors indicate stronger particle fitness and a more optimal particle position. The steps for parameter identification using PSO are as follows:  
1. Randomly initialize the particle population's initial positions and velocities within the range of dynamic parameters, with a population size of \(M=100\) and a maximum identification number of 500.
2. Evaluate the fitness of each particle, update the personal best and global best values, and retain relatively better parameter identification results.
3. Update the particle positions and velocities, forming new identification parameters.
4. Repeat iterations until reaching the maximum number of identifications, and select the particle corresponding to the global extremum as the best identification value.