Most of the models considered in the previous section consist of many parameters to build the shape of the hysteresis curve. In the first place, a suitable model has to be assigned to describe a nonlinear behavior of the system properly, and then the param-eters of the proposed model have to be estimated. This matter can be considered from two different points of views. In one hand, an identifier can be designed and substi-tuted into the model of the system for imitating behavior of the real system as nearly as possible with a minimum error. This kind of identification is known as non-parametric
identification. On the other hand, parameters of the proposed model can be estimated through an optimization tool. This type of identification is known as parametric iden-tification in which the parameters of the system are estimated using several methods such as least mean square, recursive least square, genetic algorithm, particle swarm optimization, etc.
2.3.1 Least Square Identification
In this method, the unknown parameters in a certain model are estimated by find-ing numerical values for the parameters that minimize the sum of the squared de-viations. Normally, the model is expressed in a regression form such as autoregres-sive with exogenous model (ARX), autoregresautoregres-sive-moving-average model (ARMA), autoregressive-moving-average with exogenous model (ARMAX), and so on.
For an illustration, consider a second-order discrete model of the ARX form given as
y(k) +a1y(k−1) +a2y(k−2) =a3u(k) +a4u(k−1) (2.28) The objective is to estimate the parameter vectorθT = [a1, a2, a3, a4]using the vector of input and output measurements. Define,
φT(k) = [−y(k−1), −y(k−2), u(k), u(k−1)] (2.29)
Then, we can write (2.28) as follows
y(k) =φT(k)·θ (2.30)
In the least-square (LS) estimation, the following cost function is used
J(ˆθ) =
N
X
k=1
[y(k)−φT(k)ˆθ]2 (2.31)
whereφT(k)ˆθis the predicted output andy(k)is the real output which measured in the experiment. To determine the parameter vectorθ, the cost function (2.31) is minimized,ˆ
i.e., dJ(ˆθ) dθˆ = 0.
dJ(ˆθ) dθˆ =−2
N
X
k=1
φ(k)(y(k)−φT(k)ˆθ) = 0 (2.32) in which we obtainθˆN as
θˆN = N
X
k=1
φ(k)φT(k) −1 N
X
k=1
φ(k)y(k) (2.33)
Results on modeling and identification of hysteresis behaviour pertaining to the LS method can be found in Iyer and Shirley, M. E.,2004; Stakvik et al.,2015; Tan et al.,2001.
The LS identification is also known as an off-line parameter estimation method. For on-line parameter estimation, it is extended to recursive least square (RLS) identification and is discussed in the following subsection.
2.3.2 Recursive Least Square Identification
In the recursive least square (RLS) technique, the evolution or estimation of parameters is updated at every time when a new set of observation data is obtained. Compared to LS approach, RLS algorithm has a faster convergence speed and do not exhibit the eigenvalue spread problem. However, it entailed more complicated mathematical op-erations and require more computational resources than LS method. The standard RLS algorithm is described as follows (Goodwin and Sin, K. S.,2009):
θ(k) = ˆˆ θ(k−1) + P(k−2)φ(k−1)
1 +φT(k−1)P(k−2)φ(k−1)(y(k)−φT(k−1)ˆθ(k−1)) (2.34)
P(k−1) =P(k−2)− P(k−2)φ(k−1)φT(k−1)P(k−2)
1 +φT(k−1)P(k−2)φ(k−1) (2.35) withθ(0)ˆ given andP(k−1)is any positive definite matrixP0.
There are many variants of RLS algorithm that have been developed to solve the identification problems in the linear as well as nonlinear systems. For example, Zhang et al.,2013apply an extended RLS method to adaptively identify the parameters of the controlled autoregressive moving average (CARMA) model which is used to describe the hysteresis phenomenon in a smart beam. In this study, the CARMA model is used
to characterize the relationship between the output strain near the fixed end of the can-tilever beam and the input voltage applied on the piezoelectric actuator. It is shown that the hysteresis effects in the smart beam could be well identified by the above extended RLS algorithm.
Meanwhile, Zhou et al., 2013proposed a variable step-size RLS estimation algo-rithm in order to reduce the computation overhead in the identification process. It is then used to identify the weighting parameters of the Krasnosel’skii-Pokrovskii model in modeling hysteresis nonlinearity of a magnetic shape memory alloy (MSMA) ac-tuator. For a benchmarking purpose, an improved gradient correction identification method is used. Through simulation and experimental studies, it is verified that the variable step-size RLS has a better performance over the gradient approach.
2.3.3 Particle Swarm Optimization Method
The particle swarm optimization (PSO) method is inspired by the flocking and school-ing patterns of birds and fish. Its establishment is relatively new (in the 1990s) in com-parison to Genetic Algorithm (GA) and Fuzzy Logic (FL) but has become one of the most powerful methods for solving unconstrained and constrained global optimiza-tion problems (Bergh and Engelbrecht, A.P.,2006). Essentially, it consists of a number of individuals that denote particles to simulate social behavior that ‘flying’ around in a multidimensional search space. The individuals thus have a position and a velocity.
The particles evaluate and update their positions with a fitness value at each iteration.
By attracting the particles to better positions with good solutions, each particle remem-bers its own previously best-found position, and particles in the group (a.k.a swarm) share memories of their “best” positions, and then use those memories to adjust their own velocities, and thus subsequent positions.
The algorithm of the original PSO is described as follows (Hassani et al.,2014):
1. Initialize the time to zero and set a number for initial position xi,d(0) and initial velocityv(0)i,d.
2. Evaluate the fitness of each particleF(xi,dk ).
3. Set theP bi,dk to the better performance as follows
P bi,dk =
( P bi,dk−1, F(xi,dk )≥F(P bi,dk−1);
xi,dk , F(xi,dk )< F(P bi,dk−1).
(2.36)
4. Set theGbi,dk to the position of particle with the best fitness within the swarm as Gbi,dk ∈P b1,dk , P b2,dk ,· · · , P bNks,d/F(Gbi,dk )
=min{F(P b1,dk ),· · ·, F(P bNks,d)}
5. Update the velocity vector for each particle according to the following rule:
vk+1i,d =Vmax, if vi,dk ≥Vmax (2.37a)
vk+1i,d =−Vmax, if vvki,d<−Vmax (2.37b) vk+1i,d =Iw·vki,d+ρ1·r1·(P bi,dk −xi,dk ) +ρ2·r2·(Gbi,dk −xi,dk ), otherwise
(2.37c)
6. Update the position of each particle according to
xi,dk+1 =xi,dk +vi,dk+1 (2.38)
7. Letk=k+ 1.
8. Compute the newF(xi,dk )until the iteration to be terminated or the least value for F to be achieved.
whereIw is inertia weight; ρ1 is cognitive learning gain; ρ2 is social learning gain; r1
andr2are random numbers,uniformly distributed in the range of [0,1];P bi,dk is the best known position along thedth dimension of particleiin iterationk; Gbi,dk is the global best known position among all particles along thedth dimension in iterationk; and k = 1,2,· · · , N,denotes the iteration number,N is the maximum allowable iteration number.Nsis the population size. In addition,Vmaxis the maximum velocity evolution which is usually selected to be half of the length of the search space.
Since its establishment, it has been applied in many areas, such as function opti-mization, artificial neural network training, pattern classification and so forth (Pant et al.,2007). Among the advantages of PSO are including rapid convergence, less compu-tation overhead, and ease of implemencompu-tation. However, the standard PSO does exhibit some disadvantages: it is sometimes easy to be trapped in local minima, and the con-vergence rate decreased considerably in the later period of evolution; when reaching a near optimal solution, the algorithm stops optimizing, and thus the accuracy that the algorithm can achieve is limited ( Yang et al.,2007).
To attend the aforementioned problems, the standard PSO has received various modifications and upgrades. For example, Evers and Ghalia, M. B., 2009introduce a mechanism for overcoming the stagnation problem of PSO. This mechanism triggers the swarm regrouping whenever premature convergence is detected and helps liber-ate the swarm from the stliber-ate of premature convergence in order to enable continued progress toward the true global minimum. In Alrasheed et al., 2007, a chaotic accel-eration function is introduced into the PSO algorithm. The modified version of PSO is then empirically tested with the well-known benchmark functions include sphere, rosenbrock and rastrigin functions. A real case application is also considered to further evaluate the modified PSO. From the simulation and experimental results, it is proven that the modified version outperforms the standard PSO with better enhancement of convergence rate and accuracy. Other results related to the improvement of PSO tech-nique are include Fan,2002; Pant et al.,2007; Yang et al.,2007.