INTRODUCTION
The Particle Swarm Optimization (PSO) Algorithm was initially introduced by
Kennedy and Eberhart (1995). PSO is an algorithm to solve
optimization problem based on the study on a swarm of birds’ looking for
food. The algorithm is developed rapidly because of its simple concept, easy
to implement suitable for solving nonlinear, multipeak problem (Eberhart
and Shi, 2001). However, the basic PSO is very easily to stagnate, or fall
into optima, many scholars have made a study and proposed different solutions.
For example, change the particle’ position with a certain probability,
change the particle’ velocity with a certain probability, the strategy
of adaptive and dynamic adjustment of the inertia weight (Shi
and Eberhart, 1998) imitation on adjacent (Dautenhahn,
2002) particles combined with other optimization algorithm and so on (Allahverdi
and AlAnzi, 2006). This study studies the identification process of PSO
and analysis some method which can solve the stagnation of PSO algorithm. Improved
PSO algorithm is been proposed.
Finally, the improved algorithm is applied to the identification of a model
of electric steering gear.
METHODS
This section’s main idea is about the basic PSO algorithm. Suppose S is
a subspace in Ddimensional space nonempty. Define a real function f on S and
N is the number of particles. For the ith particle, x_{i } = (x_{i1},
x_{i2}, ..., x_{iD})^{T} ε S is the location of
particle i. v_{i} = (v_{il}, v_{i2}, ..., v_{iD})^{T}
is the velocity of particle i. Also define eval_{i} = f(x_{i})
as the function value of particle i. The value of i is 1 to N.
Define pbest_{i} as the minimum function value, X_{i}^{pbest}
= (X_{il}^{pbest}, X_{i2}^{pbest},..., X_{iD}^{pbest})^{T}
as the corresponding position of particle i, nbest_{i }as the minimum
function value and X_{i}^{pbest} = (X_{il}^{pbest},
X_{i2}^{pbest},..., X_{iD}^{pbest})^{T}
as the corresponding position of the adjacent particles. Define P = {p = (e_{1},
e_{2},...,e_{D})^{T} ε, e_{d}^{min}≤e_{d}≤e_{d}^{max},
d = 1, 2,..., D} as a nonempty search space (Eberhart and
Shi, 2001).
Above all, the basic PSO process can be summarized as follows:
Step 1: 
Randomly generate the initialization of location and velocity
of each particle in the particle swarm based on the search space 
Step 2: 
Evaluate the function value of each particle. eval_{i }i = 1,
2, ..., N 
Step 3: 
For i = 1, 2,..., N, compare eval_{i} and pbest_{i}, if
eval_{i} is less than pbest_{i}, let pbest_{i} equals
eval_{i}, X_{i}^{pbest} equals x_{i} 
Step 4: 
For i = 1, 2, ..., N, calculate the minimum function value n min_{i}
and the corresponding location X_{i}^{nmf} of adjacent particle
i, if n min_{i} is less than nbest_{i}, let nbest_{i},
equals n min_{i}, X_{i}^{nbest} equals X_{i}^{nmf} 
Step 5: 
Update the velocity and location of particles as follows: 
Check whether the value of the x_{id} and v_{id} are out of
given range or not, the range of v_{id} is as follows:
where,i = 1, 2,.., N, d = 1, 2, ..., D. In the above equations, K, w, C_{1},
C_{2} and V^{max }= (V_{1}^{max}, V_{2}^{max},
..., V_{D}^{max})^{T }is the design parameters and rand_{1},
rand_{2} is independent random number range in [0, 1].
Step 6: 
If the termination condition has been satisfied, as min (pbest_{1},
pbest_{2}, ..., pbest_{N}) is less than a threshold value
or has reached the maximum number of iteration, the calculation is terminated.
Otherwise, algorithmic process jumps to step 2 
The basic PSO has five design parameters, K, w, C_{1}, C_{2}
and V^{max}.
STAGNATION PROCESS
In this section, basic particle Swarm identification algorithm is analyzed
in order to find the reasons that may affect the stagnation.
Generally while using particle swarm algorithm for identifying a mathematical
model, the initial position and velocity of the particles were assigned by the
random number, the initial state of the particle is random.
Since, the algorithm is convergence, therefore, the position and speed of the
particles will be convergent with the identify process. In the whole search
process, the particles which have excellent location and search speed are always
updated to the current value.
Whether the particle’s position and particle’s earch speed update
or not, depending on calculating for each particle's current fitness. Each particle’s
fitness, needs to be compared with the best fitness’ history records. If
the calculated fitness is smaller, it was recorded as the best fitness. If the
calculated fitness is greater, remains the best historical position of fitness.
Therefore, with the increase of the iterative process, particles’ fitness
may no longer be reduced which will result in stagnation.
Algorithm is no longer going on. Based on the above analysis, the stagnation
in the identification process of the basic PSO algorithm is due to the way of
updating the optimal fitness. For this, the study improves the basic PSO.
In essence, the concept of fitness and variance are approximate, in practice
solving process, the algorithm can determine whether the current location is
the optical location by taking advantage of both fitness value and variance.
MODIFICATION ON PSO
Discussion in the previous section has set out the possible improvements to
improve the PSO. The stagnation and optimization inaccurate can be improved
in several respects. This section focuses on the major improvements of improved
particle swarm. A flowchart of improved particle swarm algorithm is drawn.
Application of particle swarm algorithm for model identification, first, the
need to solve the problem is that how to determine the particle solver has stalled.
Based on the above analysis, some definitions are as follows:
• 
Index: Used to describe the optimal position of the
particle in the population in the process of solving the result. Optimal
location of the particle at each identification process needs to be recorded 
• 
Width_indx: Used to describe the number of the recording position.
Particle’s position which is been recorded should be the latest. Similarly,
identification by the improved PSO algorithm should record the latest real
recognition system output and model output, to figure out the variance 
• 
K_Variance: Used to describe the relationship between current calculated
variance and the minimum variance given in multiples. The minimum variance
can be given by experience, or rely on statistics 
• 
K_Pbest: Used to describe the relationship between fitness of the
current calculated and history optimal fitness given in multiples. Figure
1 shows the flowchart of Improved Particle Swarm Optimization 
Improved Particle Swarm Optimization first determines whether the optimal particles
positions in record are the same or not.
If all the optimal particles positions in record are consistent with the current
value, judgment is that the algorithm is in stagnation, enter restart judgment.
Otherwise, enter the next optimization judge.
Judging whether the algorithm is in the restart status or not, if it is, the
current variance should be figure out. A minimum variance K_Variance times is
given to make sure the current variance is bigger enough. While the current
variance value is appropriate, the current fitness is used to compare with the
optimal fitness in history.
K_Pbest is a value for the current fitness and the optimal fitness in history
like K_Variance. Based on the above analysis, in this study, the improvement
of particle swarm optimization algorithm for identification, focused on particle
swarm identify whether in stagnation or not a judgment on whether restart or
not.
After the improvement, the first to fifth steps are the same between the improved
particle swarm identification algorithm and the particle swarm algorithm. After
calculating the particle swarm first five steps, there are width_indx historic
optimal particle’s positions being recorded. Algorithm enters the improved
algorithm. Step 6:
Step 6: 
Determine the best locations in history records are the same
or not. If yes, go to step 7, otherwise, to step 9 
Step 7: 
Judge the current variance is greater than the given minimum variance
K_Variance times or not. If so, rerandom position and velocity of a given
particle, skip to step 9, otherwise go to step 8 
Step 8: 
Determine if the current fitness is greater than the historical optimal
fitness K_Pbest times, if so, rerandom position and velocity of a given
particle, skip to step 2. Otherwise, skip to the step 9 
Step 9: 
Determine whether the algorithm needs to be terminated, if so, end of
the algorithm, otherwise skip to step 2 
IDENFICATION OF ELECTRIC STEERING GEAR BASED ON PSO AND IMPROVED PSO
In the previous sections, analysis of the improved PSO is been done. In this
section, for a certain type of electric steering gear, an experiment was done
to verify the improved reliability of the algorithm. First, the approximate
mathematical model of electric steering gear is as follows:
In practical applications, during operation, the model there may be greater
changes in parameter values. The former 5 sec, the system’ parameters are
constant while in the latter 5 sec, the gain of the system becomes large. This
model is used to verify the performance of improved particle swarm optimization
algorithm for identification process. By comparing the identification results
of simple PSO algorithm and improved PSO algorithm, demonstrate the better properties
of improved algorithm in stagnation during the identification progress.
During the identification, the system needs to be discrete; identification
algorithm is based on the differential equation. The above formula discrete
model is given below:
b_{0}, b_{1}, b_{2}, a_{1}, a_{2},
are the parameters need to identify, b_{0 }= b_{2} so this identification
parameters is 4, v is white noise the variance is 0.01, u is the input signal
for the system, y is the system’s output signal.
In order to verify the superiority of the improved algorithm and make the results
more apparent, identification results are obtained for comparison in the case
of the same input, the electric steering gear model is same.
The input signal of model shown in Fig. 2, is a sinusoidal
signal, frequency is 1.6 Hz, amplitude is 6 V.
Identification results figure out by basic particle swarm optimization algorithm
are shown in Fig. 3.
Obviously, the identification results are constant. Stable value of each parameters
are:
Using basic particle swarm optimization algorithm, the electric steering gear
model’s output and the real output is shown in Fig. 4.
As can be seen from the Fig. 4, when the system’s parameters
have changed after 5 sec, the system will change the output while the estimating
of the output from identification using particle swarm algorithm cannot follow
the system output tightly.
The results of identification using improved algorithm is shown in Fig.
5.
Before the model’ parameters are changing, the constant parameters are:

Fig. 2: 
Input sinusoidal signal of electric steering gear model 
During the identify process, there is an obvious algorithm restart operation.
After a period of Identify, the constant parameters are:
The improved PSO algorithm can make sure restart the identify process while
model parameters have been constant.
The electric steering gear’ model output and the real output using improved
algorithm is shown in Fig. 6.
Comparing the identification results in Fig. 3 and 5,
it is easily to find that the improved PSO has a restart process while parameters
in the system are changed. Also, Fig. 4 and 6
show that while Particle swarm optimization algorithm has an obvious error between
the model output and the real output, the improved PSO can make sure a precise
follow.

Fig. 4: 
Electric steering gear model (identified by basic PSO) output
and the real output of electric steering gear 

Fig. 6: 
Electric steering gear model (identified by improved PSO)
output and the real output of electric steering gear 
The improved PSO can achieve better identification results, model output followed
the real output very tight. Also, the improved algorithm achieves to identify
timevarying parameters.
CONCLUSION
This study implements an improved particle swarm algorithm which is better
than the basic particle swarm algorithm for identification. In this study, two
main strategies adapted for improvement is given out. The first is a method
to determine whether identify process using particle swarm algorithm is entering
the stagnation or not. Which can be achieved by means of recorded multiple times
of the optimal position of the optimal particle. The second is a rule that strictly
determine whether to restart or not. By programming algorithm, a model identification
experiment is done, the result proves that the improved strategy is effective,
can be used for Identify timevarying systems. The improved particle swarm algorithm
in this study has a better characteristics compared with the original algorithm
for electric steering gear identification experiments.