System identification in the delta domain: a unified approach using FAGWO algorithm

AbstractThe identification of linear dynamic systems with static nonlinearities in the delta domain has been presented in this paper applying a firefly based hybrid meta-heuristic algorithm integrating Firefly algorithm (FA) and Gray wolf optimizer (GWO). FA diversifies the search space globally while GWO intensifies the solutions through its local search abilities. A test system with continuous polynomial nonlinearities has been considered for hammerstein and wiener system identification in continuous, discrete and delta domain. Delta operator modelling unifies system identification of continuous-time systems with discrete domain at higher sampling frequency. Pseudo random binary sequence, contaminated with white noise, has been taken up as the input signal to estimate the unknown model parameters as well as static nonlinear coefficients. The hybrid algorithm not only outperforms the parent heuristics of which they are constituted but also proves better as compared to some standard and latest heuristic approaches reported in the literature.


Introduction
System identification is an approximate modeling for a specific application on the basis of observed data and prior system knowledge. The literature on the system identification problem is extensive [1]. Meta-heuristic algorithms and their hybridizations have also taken active participation in the literature of system identification and control [2][3]. Linear systems with static nonlinearities at the input termed as the Hammerstein model, and linear systems with static nonlinearities at the output known as the Wiener model are two widely prevailing models used for system identification [4]. Parameter estimation of these models has traditionally been carried out in discrete-time using either shift operator in the time domain and z-transformation in the complex domain via soft computing approaches [5][6][7][8]. Likewise, a huge volume of literature also exists in the continuous-time system [9].
In the literature of identification and control, there have been several methods developed over the last five decades on discrete time systems utilizing the potential of digital computers.
Concurrently, there has been a similar attempt in developing methods in continuous time identification and control in system theory due to the very fact that the physical signals are continuous time in nature. Modelling, identification and control with the help of delta operator is a holistic approach in which the signals and systems are modelled in discrete domain and leads to converge to its corresponding continuous time signals and systems at a high sampling frequency thus unifying both discrete and continuous time signals and systems [10].
Though hammerstein and wiener model identification with meta-heuristic approaches are quite popular in the discrete-time domain, similar analyses are rarely investigated for continuous-time systems. Hence system identification with hybrid meta-heuristic techniques can be thought of to unify both continuous and discrete time systems leveraging the properties of delta operator. A hybrid algorithm namely FAGWO developed by Ganguli et al. [11] has been utilized to identify the unknown hammerstein and wiener model parameters in a unified delta operator framework. Continuous and discrete time analyses are carried out simultaneously to highlight upon the usefulness of delta operator modelling.
The rest of the paper is developed as follows. Section 2 discusses the problem of wiener and hammerstein models in the delta domain. Section 3 gives a brief overview of the parent algorithms FA and GWO algorithms. Section 4 discusses the hybrid FAGWO algorithm.
Section 5 presents the results while Section 6 concludes the paper.

Delta operator modelling
The  -operator, an alternative formulation of discrete-time system [10] is defined in the time domain as: where  denotes the sampling period while q is the forward shift operator. Operating  on a differential signal () xt gives It is straightforward to see that which indicate the close relationship between the discrete-time  -operator and the continuous-time differential operator d dt at high sampling rate.
Similarly relation exists in the complex domain as well. The delta transform operator  is defined as

Wiener system identification in delta domain
The wiener model is characterized by a linear dynamic part followed by a static nonlinearity shown in the Fig. 1.
B and A are two polynomials of unknown orders and coefficients, u and y represents system input and output respectively. The non-measured intermediate variable x(k) is the input to the static nonlinearity given by- f () is any nonlinear function and is a set of parameters describing the nonlinearity. Thus, the problem of the wiener model identification is to estimate the unknown parameters 01 , , , , , mn b b a a from the input-output data. Further, 'w' represents the white gaussian noise of fixed signal-to-noise ratio (SNR). In case the structure of the nonlinear function f () is not known, a polynomial of degree L can be used to approximate the nonlinearity as-

Hammerstein model identification in the delta domain
Hammerstein model is a good example of nonlinear dynamic systems in which nonlinear static system and linear dynamic systems are separated in different order. The block diagram of Hammerstein model is shown below in Fig. 2.

Fig. 2. Hammerstein model
The input/output relation of discrete-delta system is represented as: The objective is to estimate the parameters 10 , , , , , , Here 'N' denotes the number of input-output data points used in the identification and the parameter estimates 10, , , , , ,

Brief overview of FA and GWO algorithms
In this section, the two parent algorithms viz. FA and GWO are introduced to set up an appropriate background for the hybrid method. The hybrid technique is then utilized to solve hammerstein and wiener model identification in the delta domain.

Firefly algorithm (FA)
Xin-She Yang developed the firefly algorithm [12] considering the following assumptions:  All fireflies are unisex so that one firefly is attracted to other fireflies regardless of their sex  Attractiveness is proportional to their brightness, thus for any two flashing fireflies, the less bright one will move towards the brighter one. The attractiveness is proportional to the brightness and they both decrease as their distance increases. If no one is brighter than a particular firefly, it moves randomly  The brightness or light intensity of a firefly is affected or determined by the landscape of the objective function to be optimized.
In the firefly algorithm, there are two salient aspects: the variation of light intensity and formulation of attractiveness. For the sake of simplicity, it is assumed that the attractiveness of a firefly is determined by its brightness or light intensity which in turn is correlated with the encoded objective function. For maximum optimization problems, the brightness   it will be seen in the eyes of the beholder or to be judged by the other fireflies. So it will vary with the distance ij r between firefly i and firefly j. As light intensity decreases with the distance from its source, the light is also absorbed in the media, hence it is concluded that the attractiveness should vary with the degree of absorption. The light intensity   r I varies with the distance r monotonically and exponentially as: (10) where 0 I is the original light intensity and  is the light absorption coefficient. As a firefly's attractiveness is proportional to the light intensity seen by adjacent fireflies, the attractiveness  of a firefly is defined as (11) where 0  is the attractiveness at r=0. It is worth mentioning that the exponent 2 r  can be replaced by other functions such as m r  when 0  m . The distance between any two fireflies i and j at i x and j x respectively, is the Cartesian distance is calculated as where d i x , is the dth component of the spatial coordinate i x of ith firefly. The movement of a firefly i is attracted to another more attractive (brighter) firefly j is determined by

Gray wolf optimizer (GWO)
Gray wolf optimizer (GWO) is a population based meta-heuristic algorithm 7ehaviour the leadership hierarchy and hunting mechanism of gray wolves found in nature [13]. Gray wolves are considered as apex predators, belonging at the top of the food chain. They live in groups (packs), each group containing 5-12 members on average. All the members in the group maintain a strict social hierarchy as shown in Fig. 3.

Fig. 3. Social hierarchy of gray wolves
As seen from Fig. 3, four types of gray wolves such as alpha, beta, delta, and omega are employed for simulating the leadership hierarchy. In the hierarchy, alpha    is considered the most dominating member among the group. The rest of the subordinates to α are beta    and delta ( ), which help to control the majority of wolves in the hierarchy that are considered as omega    .The ω wolves are of the lowest ranking in the social hierarchy.
In GWO algorithm, the hunting is guided by ,  and  . The  solutions follow these three wolves. During hunting, the gray wolves encircle the prey. The mathematical model of the encircling 8ehaviour is presented as: where t indicates the current iteration, A and C are the coefficient vectors, p X denotes the position vector of the prey while X represents the position vector of a gray wolf. The vectors A and C are computed using the following equations: where a is linearly decreased from 2 to 0 over the course of iterations while 1 rand and 2 rand denote random numbers lying in the range (0,1). The hunting operation of the gray wolves is usually guided by the alpha wolves. The beta and delta wolves occasionally participate in the hunting process. Thus, in the mathematical model for the hunting 8ehaviour of gray wolves, it is assumed that the alpha, beta and delta type gray wolves have better knowledge about the potential location of prey. Hence, the first three best solutions acquired are saved and the other search agents are obliged to update their positions according to the location of the best search agents. The following mathematical equations are thus framed as: and finally The gray wolves complete the hunt by attacking the prey when it stops moving. In this phase, the value of a is decreased and thereby the fluctuation range of A is reduced. When A has random values in the range   1,1 ,  the search agent's next location will be in anywhere between its current position and the position of the prey.

FAGWO algorithm
Ganguli et al. [11] integrated FA with GWO as a low level relay type heterogenous hybrid topology, coined as FAGWO algorithm. From the literature it has been found that FA can subdivide the whole population into subgroups automatically in terms of the attraction mechanism with the variation of light intensity. Further, FA can also escape from the local minima by virtue of long-distance mobility via Lévy flight. Such advantages clearly indicate that FA has good exploration capabilities. Thus FA is used to explore the solution vector globally whereas GWO is employed to exploit the solutions through its local search abilities.
The search operation begins with FA with the help of initialization through a group of random agents. The computation continues with FA for a certain number of iterations to search for the global best position in the specified search domain. The search process then shifts to GWO to speed up the convergence for global optimum. Thus the hybrid algorithm finds an optimum more accurately and precisely. The pseudocode of the hybrid algorithm to solve identification problem is provided in Fig. 4 given below. Step1: Excite the static nonlinear system by PRBS sequence.
Step2: Generate random initial solutions for zeros and poles of the linear part, and the parameters of the nonlinearity in the appropriate search space.
Step3: Evaluate the fitness function defined in equation (9)

Results and discussions
The input-output relation of the continuous-time plant model to be identified [9] is given by- The delta operator form of the transfer function at the same sampling time is represented as-    The hybrid method employed outperforms the parent algorithms as well as some standard heuristics in all the three domains. In addition, the results are better than a popular hyrbrid algorithm PSOGSA. Further, the continuous-time and discrete-delta parameters show close resemblance. The statistical measures of the test system in respective domains are narrated in Table 7. The statistical assessments viz. best, worst, average and standard deviation of the fitness function for each algorithm are provided in this table. In addition to this, the best results obtained with respect to the best and worst values, average, standard deviations are highlighted with bold letters. The hybrid method has the least value in all the statistical measures in all the domains. The continuous-time and discrete-delta results are in close match. Since standard deviation turns out to be least with the hybrid method, it can be concluded the algorithm is more stable than those considered for comparison.

Conclusions
A hybrid technique referred to as FAGWO is applied to identify hammerstein and wiener systems in the delta domain. Parameter estimation has been carried out in continuous, discrete and delta domain respectively. Delta operator modelling provides unification of continuous and discrete-delta results. The unknown model parameters are estimated through the minimization of mean square error (MSE). The fitness value of the hybrid technique not only surpasses that obtained by some popular metaheuristic algorithms but also the parent heuristics of which they are constituted for the test system under consideration. Wilcoxon test also validates the significance of the results obtained by the hybrid approach. The hybrid method also exhibits better convergence in the delta domain as compared to other algorithms.