Abstract: Nonlinear system identification has been received more attention especially for control purposes. In this study, a new identification method is presented for nonlinear system model. Its basic idea is as follows. At first, the identification problem of nonlinear system is changed into a nonlinear function optimization problem over parameter space. Then, the estimates of the parameters of the nonlinear system are obtained using a proposed particle swarm optimization algorithm. Finally, in simulation, the method is applied to several different nonlinear systems and simulation results indicated that the presented method is feasible and reasonable.
INTRODUCTION
In recent years, system identification has been widely applied to the design and analysis of control system, such as biology, physiology and so on. Consequently, system identification is a very important in practical problems, attracting many scientific personal for their theoretical study to examine the practical issues in different application fields (Ljung, 1999; Sjoberg et al., 1995). In view of nonlinear systems widely exist in people’s production and life, one have plunged into researching them in the social sciences (Roll et al., 2005; Giri and Bai, 2010; Wills et al., 2013). Unfortunately, because many nonlinear systems are not well understood, moreover, the expressions of nonlinear models are more complex, there is not a uniform estimation method for all nonlinear models.
From the early 1990s, it produced a random global optimization technique that simulates natural biological communities’ behavior. Under the mechanism of biological evolution, Kennedy and Eberhart (1995) proposed particle swarm optimization algorithm by simulating birds and fish. These results can be called particle swarm optimization algorithm and its application has been extended to combinatorial optimization problems (Pobinson and Rahmat-Samii, 2004). In this work, a parameter estimation method for different nonlinear system model is proposed using an Improved Particle Swarm Optimization Algorithm (IPSOA). Finally, three kinds of nonlinear system models are taken as simulation examples to show the effectiveness of the proposed approach.
PROBLEM DESCRIPTION
In general, the model of the nonlinear system can be described as the following equation:
| (1) |
where, y(t) is the system output, u(t) is the system input, θ = (θ1, θ2,…, θk)T is the unknown parameter vector, the formulation of f is known and u(t) is provided.
In order to estimate the above nonlinear system, it must meet the following assumptions. (a) y(t) can be measurable. (b) Each parameter must be related with output y(t). (c) As long as the parameters are determined, the value of output y(t) can be obtained by simulation. (d) The system does not diverges in finite time. (e) The signal to noise ratio of the system is enough large.
PARTICLE SWARM OPTIMIZATION ALGORITHM
Particle swarm optimization algorithm was first introduced by Kennedy and Eberhart (1995). Particle swarm optimization algorithm is a population based heuristic searching algorithm guided by individuals’ fitness. In particle swarm optimization algorithm, candidate solutions of a specific optimization problem are called particle. Each particle in the D-dimensional search space is characterized by two factors, namely, its position Xi = (xi1, xi2,…, xiD) and velocity Vi = (vi1, vi2,…, viD), where i denotes the ith particle in the swarm. In the process of search optimal value, all particles in particle swarm optimization algorithm fly through the searching space and adjust its velocity and position to find a better position by its own experience and experience of neighboring particles iteratively. Let Pi(t) = (pi1, pi2,…, piD) expresses the best position found by particle i within t iteration steps, Pg(t) = (pg1, pg2,…, pgD) expresses the best position among all particles in the swarm so far.
In Standard Swarm Optimization Algorithm (SPSOA), particles update their positions and velocities by the following Eq. 2 and 3:
| (2) |
| (3) |
where, n expresses the number of particles in the swarm, Vi(t) and Xi(t) express the velocity and position of particle i in the solution space at tth iteration step, respectively; r1 and r2 are two random numbers uniformly distributed in the [0, 1]; c1 and c2 are acceleration constant, in general, c1 = c2 = 2.0; w is the inertia weight factor. Generally, the value of each component in Vi can be clamped to the interval [Vmin, Vmax]. Each particle flies toward a new position by Eq. 2 and 3. In this way, all particles in the swarm find their new position and use these new position to modify their individual best position Pi(t) and global best position Pg(t) of the swarm.
In application, the SPSOA algorithm shows some defects, mainly including premature convergence, poor local optimization capable, etc. Consequently, in this study, the w is selected to be a positive nonlinear function of time as follows:
| (4) |
where, tmax expresses maximal number of iterations, t expresses current iteration number, wstart and wend are initial and ultimate inertia weight factor, respectively.
To sum up, an Improved Particle Swarm Optimization Algorithm (IPSOA) is presented by Eq. 2-4 in this study.
PARAMETER ESTIMATION USING IPSOA
The specific steps of the estimation procedure are as follows:
| Step 1: | Parameter vector θ is taken as a particle of the swarm |
| Step 2: | Determine the fitness function: Using Eq. 1, system actual outputs y0(t) can be obtained in simulation experiment. The estimated output is y(t). Then, the following criterion function is taken as the fitness function: |
| (5) |
| Step 3: | Initialize the swarm: Let swarm size is n, initial position and velocity of particles are randomly set in the range allowed, the individual extreme (pbest) coordinates of each particle is set as the current position and compute the corresponding individual extreme while the global extreme value (gbest) is the best among the extreme of all the individuals, record the particle’s serial number of the best value and take the global extremum as the current best particle position |
| Step 4: | Compute the fitness value of each particle |
| Step 5: | For each particle, its fitness value is compared with the pbest, if it is excellent, then the current individual extreme is modified |
| Step 6: | For each particle, its fitness value is compared with the gbest, if it is excellent, then the current global extreme is modified |
| Step 7: | The position and speed of each particle are updated using Eq. 2-4 |
| Step 8: | If the maximum number of iterations or minimum error dose not reach, then, it returns to step 4. Otherwise, end, that is to say, the estimate of θ is obtained |
The pseudo code of the above presented algorithm is as follows:
SIMULATION STUDY
Now, in order to indicate the feasibility of the presented identification method, the following simulation examples are implemented, respectively.
Example 1: The transfer function model:
This system is a pure delay plus inertia segment model, where the parameters K, t and τ need to be estimated.
Example 2: The state space model:
Example 3: The Hammerstein model:
A(q-1)y(k) = B(q-1)x(k)+C(q-1)w(k)
where, A(q-1) = 1+a1q-1+a2q-2, B(q-1), = b1q-1+b2q-2, C(q-1) = 1+c1q-1, x(k) = u(k)+r1u2(k)+r3u4(k) and w(k) is a Gaussian white noise sequence with zero mean, standard variance τ = 0.01.
It can be seen that the above cases are not the same expression. However, the cases can all be rewritten to the form of Eq. 1 by a certain transformation. Namely, once the input is determined and the parameters are gotten, the output of the system is received by computing at any time.
In the estimation, the parameters of the algorithm are set as follows. The swarm size n = 15, the parameters c1 = c2 = 2, the maximum number of iterations tmax = 50, the parameter Xmax = 2, the value of the initial inertia weight wstart = 0.9, the value of the final inertia weight wend = 0.4. In Examples 1-3, the initialized search ranges of the parameters are [2, 9], [0.1, 1.9] and [-2, 2], respectively.
The estimates of parameters by the IPSOA and SPSOA are shown in Table 1-3, respectively. From the simulation results of Examples 1, 2 and 3, one can see that the estimates of the parameters obtained using the IPSOA are more approximate to the actual values than those obtained from the SPSOA. That is to say, the precision of identification using IPSOA is improved remarkably.
| Table 1: | True values and estimates of the transfer function model parameters K, t and τ in Example 1 using IPSOA and SPSOA, respectively |
| 1K, t and τ are the parameters of the transfer function model in Example 1, 2IPSOA expresses the estimates of model parameters using the Improved Particle Swarm Optimization Algorithm, 3SPSOA expresses the estimates of model parameters using Standard Particle Swarm Optimization Algorithm | |
| Table 2: | True values and estimates of the state space model parameters θ1, θ2, θ3 and θ4 in Example 2 using IPSOA and SPSOA, respectively |
| 2IPSOA expresses the estimates of model parameters using the Improved Particle Swarm Optimization Algorithm, 3SPSOA expresses the estimates of model parameters using Standard Particle Swarm Optimization Algorithm 4θ1, θ2, θ3 and θ4 are the parameters of the state space model in Example 2 | |
| Table 3: | True values and estimates of the Hammerstein model parameters a1, a2, b1, b2, c1, r1, r2 and r3 in Example 3 using IPSOA and SPSOA, respectively |
| 2IPSOA expresses the estimates of model parameters using the Improved Particle Swarm Optimization Algorithm, 3SPSOA expresses the estimates of model parameters using Standard Particle Swarm Optimization Algorithm5a1, a2, b1, b2, c1, r1, r2 and r3 are the parameters of the Hammerstein model in Example 3 | |
The above three cases are random selected based on the literature of nonlinear system identification; they have a certain representative in a variety of nonlinear system model. Accordingly, the presented estimation method is reasonable from the above simulation results for most nonlinear system models.
CONCLUSION
A parameter estimation method of nonlinear system model is proposed based on the IPSOA in this study and obtain satisfactory results. In simulation, IPSOA is shown that it has the advantages of multipoint optimization, simple, easy, etc. Moreover, due to it dose not depend on the model form in the optimization process, it is widely applied to various model parameter estimation. Accordingly, it is shown that the presented method is feasible and reasonable.
ACKNOWLEDGMENTS
This study is supported by the Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 2013JK0698), the Ph.D. Scientific Research Startup Funds of Teachers of Xi’an University of Technology of China (Grant No. 108-211006), the Industrialization cultivation project of Shaanxi’s Ministry of Education (Program No. 2012J C19), Technology Transfer Major Project of Xi’an Ministry of Science and Technology (Program No.CX12166) and Science and Technology Project of Xi’an City Science and Technology Bureau of China.