**3. Employed optimization methods**

In infinite impulse response (IIR) systems, the error surface is generally nonquadratic and multimodal with respect to the system parameters. Minimization of such error fitness function using derivative-based search algorithm is difficult. This is due to the fact that the derivative-based search algorithm may not converge to the global minima and get stuck in local minima. Moreover, IIR systems are associated

with the stability issues as the poles of the systems may lie outside the unit circle. Such techniques are found unfit to solve multi-objective, multi-modal complex problems and a fine tuning of algorithm parameters is required. To conquer these disadvantages, several practitioners rely on meta-heuristic algorithms, which are based on natural evolution. The meta-heuristic algorithms are nature inspired population-based search techniques which have the ability to serve a global optimal solution with high convergence by accumulating random search and selection principle. Therefore, intelligent search paradigms and optimization methods are adopted in this work for an optimal differentiator design (a multi-modal problem) in short computation time and with high accuracy. Three population-based heuristic search algorithms PSO, CSA, and GSA are employed in this section to diminish the cost function in order to find optimum values of characteristic impedances for designed wideband SOMIs of line elements. **Table 1** displays the optimum set of algorithm control parameters for designed SOMIs. A brief description of all three algorithm is discussed below.

*v* ð Þ *k*þ1 *<sup>i</sup>* ¼ *w* ∗ *v*

**Table 1**.

**137**

where, *v<sup>k</sup>*

iteration and *pbest*ð Þ*<sup>k</sup>*

ð Þ*k*

*DOI: http://dx.doi.org/10.5772/intechopen.94843*

ters has range from [0, 1]. *gbest<sup>k</sup>*

**3.2 Cuckoo search algorithm**

an infinite variance and infinite mean.

**3.3 Gravitational search algorithm**

*Fd t*ð Þ

*ij* <sup>¼</sup> *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>M</sup>*ð Þ*<sup>t</sup>*

*<sup>i</sup>* <sup>þ</sup> *<sup>C</sup>*<sup>1</sup> <sup>∗</sup> *rand*<sup>1</sup> <sup>∗</sup> *gbest*ð Þ*<sup>k</sup>*

*Analysis of Wideband Second-Order Microwave Integrators*

*p*ð Þ *<sup>k</sup>*þ<sup>1</sup> *<sup>i</sup>* <sup>¼</sup> *<sup>p</sup>*ð Þ*<sup>k</sup>*

*<sup>i</sup>* � *<sup>p</sup>*ð Þ*<sup>k</sup> i*

*<sup>i</sup>* þ *v*

*<sup>i</sup>* is the particular best value of *i th* particle. *pbest*ð Þ*<sup>k</sup>*

are positive cognitive parameters while *rand*1 and *rand*2 are two random parame-

position vector of *i th* particle at *k th* iteration. Some parameters are selected to obtain the characteristic impedances of line elements, which are provided in

The cuckoo search algorithm is developed by Yang and Deb in 2009 which is inspired by the concept of unique breeding behavior of cuckoo bird in combination with Lèvy (λ) fights. The theoretical concepts of CSA are well developed and tested in [15–17]. The single parameter setting in CSA is proven to be a crucial superiority factor as compared to other nature-based algorithms. As a result, the optimized results are executed in very less time. CSA is employed in this section for determining the optimum characteristic values of impedances of transmission line ele-

The species of cuckoo birds lay eggs in other bird nests, where the host birds either throw off the detected strange eggs or leave their nests and move into a new spot. The algorithm symbolizes each host nest to a capable solution for this design problem and assigns it a fitness value, as defined in Eq. (24). Furthermore, CSA starts to exchange the current fitness value with a better solution iteratively. The concept of Lèvy flights is then introduced in the process for exploration of new solutions, mathematically modelled using the Lèvy distribution, 1 < λ ≤ 3 [17] with

This algorithm is based on Newton's theory of gravitational force and was introduced by E Rashedi et al. in 2009 [19]. Candidates in GSA are supposed to be objects with a given mass. In accordance with Newton's Law, every candidate in a space has an attraction force with every other candidate. This force correlates inversely to the square of the distance between the candidates and is directly proportional to the product of their masses [20, 21]. Each candidate in the GSA has certain parameters: position of candidate's mass, inertial mass of candidate, active gravitational mass and passive gravitational mass between candidates. The position of the mass of candidates resembles the solution of the problem devised [21]. The gravitational mass and inertial mass are determined using a cost function of designed problem. The mathematical representation of GSA is considering *N* candidates within a system and all candidates are randomly positioned in search space. The gravitational force on candidate *i* from candidate *j* at time *t* in dimension *d* is defined as [19].

> *pi* � *<sup>M</sup>*ð Þ*<sup>t</sup> aj*

*ij* <sup>þ</sup> *<sup>ε</sup> <sup>x</sup>*

*d t*ð Þ *<sup>j</sup>* � *x*

*d t*ð Þ *i*

(27)

*R*ð Þ*<sup>t</sup>*

ments on account of minimizing the magnitude error of designed SOMI.

<sup>þ</sup> *<sup>C</sup>*<sup>2</sup> <sup>∗</sup> *rand*<sup>2</sup> <sup>∗</sup> *pbest*ð Þ*<sup>k</sup>*

is the global best position component at *k th*

ð Þ *k*þ1

*<sup>i</sup>* is the velocity of the *i th* particle and *ω* is the weight factor. *C*<sup>1</sup> and *C*<sup>2</sup>

*<sup>i</sup>* � *<sup>p</sup>*ð Þ*<sup>k</sup> i*

(25)

*<sup>i</sup>* is the

*<sup>i</sup>* (26)
