## 2 Hybrid Model Updating Method  
### Gmyo  
GMPSO is a stochastic global search algorithm based on adjustment of standard PSO. Supposing there are an amount of \(N\) random initialization particles with a \(d\)-dimensional searching space in which the position and velocity of particle \(i\) are \(\mathbf{X}_{i}=[x_{i1},x_{i2},\cdots,x_{id}]\) and \(\mathbf{V}_{i}=[v_{i1},v_{i2},\cdots,v_{id}]\), respectively. In every iteration process, particle \(i\) can yield two best solutions of the objective function, one is the personal best (_pbest_), \(\mathbf{P}_{i}=[p_{i1},p_{i2},\cdots,p_{id}]\) and the other is the global best (_gbest_), \(\mathbf{P}_{g}=[p_{g1},p_{g2},\cdots,p_{gd}]\). For a standard PSO, the particle \(i\) adjusts its velocity and position according to the following equations:  
\[v_{ij}(t+1)=\omega v_{ij}(t)+c_{1}r_{1}\big{[}p_{ij}-x_{ij}(t)\big{]}+c_{2}r_{ 2}\big{[}p_{gj}-x_{ij}(t)\big{]} \tag{1}\]  
\[x_{ij}(t+1)=\ x_{ij}(t)+\ v_{ij}(t+1)\times\Delta t \tag{2}\]  
where \(i\) = 1, 2,..., \(N\), \(j\) = 1, 2,..., \(d\), \(\Delta t\) = 1, \(t\) is the \(t\)-th iteration, \(w\) is inertia weight, \(c_{1}\) and \(c_{2}\) are positive learning coefficients, namely acceleration coefficients, \(r_{1}\) and \(r_{2}\) are random numbers uniformly distributed between 0 and 1. In the Equation (1) of subsequent velocity, the first part represents the inheritance of current state, the second part means self-cognition namely experiences and memories of itself which enables global search capability and the last part reflects group-cognition based on experiences of entire particles which improves the global search ability. As aforementioned, the drawback of standard PSO is the lack of swarm diversity at the later stage which influences the global convergence as the algorithm may end prematurely. Through introducing mutation of Gaussian white noise (GWN) disturbance for position of every particle at a probability, GMPSO apparently increases the diversity of particle swarm [23]. For a GMPSO algorithm, the position of particle \(i\), \(\mathbf{X}_{i}=[x_{i1},x_{i2},\cdots,x_{id}]\), is mutated by disturbance of GWN at a probability of \(P_{\text{m}}\) around the global best position before the adjustment of Equation (2). The value of \(P_{\text{m}}\) is generally between 0.05 and 0.15 and the mutation formula is as follows  
\[x_{ij}=p_{gj}\times(0.5+\sigma) \tag{3}\]  
where \(\sigma\) is a random number of GWN and the other meanings of items are like before.  
The detail steps shown in Figure 1 are as follows:  
1. Randomly initializing positions and velocities of \(N\) particles and setting the values of serials of parameters such as \(w\), \(c_{1}\), \(c_{2}\), \(d\), \(P_{\text{m}}\), maximum iteration number, and the upper and lower bounds of search space;
2. Establishing the objective function, called fitness function and calculating its value of each particle. Recording each particle's current position and function value in _pbest_ in which the best position and best value are recorded in _gbest_;
3. Renewing velocity of particles based on Equation (1);
4. Mutating position of particles at a probability of Pm based on Equation (3);Renewing position of particles based on Equation (2);
* Calculating function value again and replacing _pbest_ if the current value is better than the former one.
* Replacing _gbest_ in the current _pbest_ and _gbest_ according to the best value.
* Quitting searching if the iteration number reaches the maximum one or accuracy meets the limitation, otherwise, returning to Step 3.