**3.1 Particle swarm optimization**

Particle swarm optimization was proposed by Kennedy and Eberhart in nineties based on swarm behavior. PSO has simplicity, high solution quality and superior convergence characteristics as compared to other algorithms. It is more efficient, easy to implement and flexible to control between global and local exploration of the search space [13, 14]. In PSO, every particle has a candidate solution and each candidate solution has its position and velocity. The position vector gives the required solution and the velocity vector gives the current position by which it reaches at the new position. After every iteration, the position and the velocity vectors are updated unless final coefficients are obtained. The velocity and position vector are updated according to the following equations:


#### **Table 1.**

*Control parameters of PSO, CSA and GSA for optimization.*

*Analysis of Wideband Second-Order Microwave Integrators DOI: http://dx.doi.org/10.5772/intechopen.94843*

$$v\_i^{(k+1)} = w \* v\_i^{(k)} + C\_1 \* rand \mathbf{1} \* \left(gbest\_i^{(k)} - p\_i^{(k)}\right) + C\_2 \* rand \mathbf{2} \* \left(gbest\_i^{(k)} - p\_i^{(k)}\right) \tag{25}$$

$$p\_i^{(k+1)} = p\_i^{(k)} + v\_i^{(k+1)} \tag{26}$$

where, *v<sup>k</sup> <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> are positive cognitive parameters while *rand*1 and *rand*2 are two random parameters has range from [0, 1]. *gbest<sup>k</sup>* is the global best position component at *k th* iteration and *pbest*ð Þ*<sup>k</sup> <sup>i</sup>* is the particular best value of *i th* particle. *pbest*ð Þ*<sup>k</sup> <sup>i</sup>* is the 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 **Table 1**.

#### **3.2 Cuckoo search algorithm**

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 prin-

ciple. 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

Particle swarm optimization was proposed by Kennedy and Eberhart in nineties based on swarm behavior. PSO has simplicity, high solution quality and superior convergence characteristics as compared to other algorithms. It is more efficient, easy to implement and flexible to control between global and local exploration of the search space [13, 14]. In PSO, every particle has a candidate solution and each candidate solution has its position and velocity. The position vector gives the required solution and the velocity vector gives the current position by which it reaches at the new position. After every iteration, the position and the velocity vectors are updated unless final coefficients are obtained. The velocity and position

**Parameters PSO CSA GSA** Population size *np* ¼25 *nc* ¼25 *ng* ¼25 Maximum Iteration 400 400 400 *C*<sup>1</sup> 2 — — *C*<sup>2</sup> 2 — —

*<sup>i</sup>* 0.01 — —

*<sup>i</sup>* 1.0 — *wmin* 0.1 — *wmax* 1.0 — — Discovering rate of alien egg, Pa — 0.25 — *G*<sup>0</sup> — — 100 *α* — — 20 *ϵ* — — 0.001 *rNorm* — — 2 *rPower* — — 1

algorithm is discussed below.

*vmin*

*vmax*

**Table 1.**

**136**

**3.1 Particle swarm optimization**

*Innovations in Ultra-WideBand Technologies*

vector are updated according to the following equations:

*Control parameters of PSO, CSA and GSA for optimization.*

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

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 an infinite variance and infinite mean.

#### **3.3 Gravitational search algorithm**

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].

$$\boldsymbol{F}\_{ij}^{d(t)} = \mathbf{G}^{(t)} \frac{\mathbf{M}\_{pi}^{(t)} \times \mathbf{M}\_{aj}^{(t)}}{R\_{ij}^{(t)} + \boldsymbol{\varepsilon}} \left(\boldsymbol{\varkappa}\_{j}^{d(t)} - \boldsymbol{\varkappa}\_{i}^{d(t)}\right) \tag{27}$$

where, *G*ð Þ*<sup>t</sup>* represents gravitational constant at time *t*, *M*ð Þ*<sup>t</sup> pi* represents the passive gravitational mass related to candidate *i*, *M*ð Þ*<sup>t</sup> aj* represents the active gravitational mass related to candidate *j*, *R*ð Þ*<sup>t</sup> ij* represents the Euclidian distance between two candidates *i* and *j* and *ε* is a small constant. Then *G*ð Þ*<sup>t</sup>* is calculated as [19].

$$\mathbf{G}^{(t)} = \mathbf{G}\_0 \times \exp\left(-a \times \text{itr}/m\omega \text{air}\right) \tag{28}$$

**Algorithm 1**

Pseudo code for GSA for the design of SOMI-GSA Define *H* ð Þ *ω* and fitness function, Eq. (24)

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

*Analysis of Wideband Second-Order Microwave Integrators*

While iteration, l increases, *N* <400 do

Evaluate gravitational constant, *G*<sup>0</sup> and *gbest*

Compute fitness using Eq. (24)

For minimzation, If *El*þ<sup>1</sup> <*E<sup>l</sup>*

EndIf EndWhile

**Table 2.**

**139**

*Optimized characteristic impedances of the design-1 SOMI.*

New solutions are updated

Record the best solution

**4. Simulation results**

Characteristic impedances optimized

Initialize population size of candidates, ng and other control parameters Set upper and lower bounds, maximum iterations, generate population

Updae the velocity and position of each candidate using Eqs. (35), (36)

Compute masses, gravitational forces, and acceleration using Eqs. (27)–(32). for each candidate

In this section, simulation results are discussed and analyzed. All the simulation results are carried out in MATLAB environment. The same control parameters for PSO, CSA and GSA have been selected for appropriate comparison of optimization algorithms. The lower and upper limits of the optimized coefficients are set to be 10 and 150 for functional realizability. Absolute magnitude error (AME), phase response, pole-zero plot, convergence rate and improvement rate are taken into account in assessing the performance of the proposed SOMI magnitude response.

**Figure 3** presents the configuration of the design-1 SOMI, and **Table 2** displays the characteristic impedances of design-1 SOMI achieved with PSO, CSA and GSA. The algorithm steps have shown in section-3 by which these characteristic impedances are attained. The magnitude response, the AME, the phase response, the polezero plot, the fitness rate and the improvement graph are the parameters selected in **Figure 4(a)**–**(f)** for the frequency response analysis of the design-1 SOMI. **Table 3**

**Characteristic impedance PSO CSA GSA** *Zoc*<sup>1</sup> 12 112.2 148.8 *ZTL*<sup>1</sup> 49 17 10 *ZTL*<sup>2</sup> 120 114 149 *ZTL*<sup>3</sup> 14 11 43 *Zoc*<sup>2</sup> 115 14 21

**4.1 Design-1 [SOMI using two open stub and three serial line sections]**

where, *G*<sup>0</sup> represents the initial value of gravitation constant, *α* is descending coefficient, the current iteration is represented by *itr*, and *maxitr* represents the maximum number of iterations.

Then the total force that acts on candidate *i* in dimension *d* is calculated as

$$F\_i^{d(t)} = \sum\_{\substack{j=1,\ j \neq 1}}^N rand\_j F\_{ij}^{d(t)} \tag{29}$$

where *rand <sup>j</sup>* is a random number. As reported to the motion's law, the acceleration of candidate *i* is given as

$$acc\_i^{d(t)} = \frac{F\_i^{d(t)}}{\mathbf{M}a\_{ii}^{(t)}} \tag{30}$$

where *Maii* represents the mass of the candidate *i*. The inertial mass and the gravitational mass are updated as follows [19].

$$m\_i^{(t)} = \frac{CF\_i^{(t)} - worst^{(t)}}{best^{(t)} - worst^{(t)}} \tag{31}$$

where *CF* represents the cost function of the candidate *i*.

$$\mathcal{M}\_{pi}^{(t)} = \frac{m\_i^{(t)}}{\sum\_{j=1}^{J} m\_j^{(t)}} \tag{32}$$

The best and worst value are given by

$$best^{(t)} = \min\_{j \in (I.J)} CF\_j^{(t)} \tag{33}$$

$$
\omega\_{\text{worst}^{(t)}} = \max \, \mathbf{x}\_{\text{ }j \in (l..J)} \mathbf{C} \mathbf{F}\_{\text{ }j}^{(t)} \tag{34}
$$

Then the position and the velocity of candidates are given by [19].

$$position: \mathcal{x}\_i^{d(t+1)} = \mathcal{x}\_i^{d(t)} + \boldsymbol{\upsilon}\_i^{d(t+1)} \tag{35}$$

$$\boldsymbol{\nu}\_{\text{velocity}} : \boldsymbol{\nu}\_{i}^{d(t+1)} = \boldsymbol{r} \boldsymbol{\nu} \boldsymbol{d}\_{i} \times \boldsymbol{\nu}\_{i}^{d(t)} + \boldsymbol{a} \boldsymbol{c}\_{i}^{d(t)} \tag{36}$$

where *randi* represents a random number. Finally, the position and velocity of the candidate are obtained.

