Dynamic particle swarm optimisation for the design of loaded wire ...

www.ietdl.org Published in IET Microwaves, Antennas & Propagation Received on 2nd August 2010 Revised on 23rd December 2010 doi: 10.1049/iet-map.2010.0375

ISSN 1751-8725

Dynamic particle swarm optimisation for the design of loaded wire antennas C. Leone1 A. Rogovich2 C. Marasini3,4 S. Genovesi2 A. Monorchio2 1

Institute for Telecommunications and Electronics of Italian Navy ‘G. Vallauri’, Viale Italia 72, 57127 Livorno, Italy Department of Information Engineering, University of Pisa, Via G. Caruso 16, I-56122 Pisa, Italy 3 TNO Defence, Security & Safety, The Hague, The Netherlands 4 ISS Payload & Operation Division, European Space Research and Technology Centre, ESTEC, 2201 AG Noordwijk, The Netherlands E-mail: [email protected] 2

Abstract: A modified particle swarm optimisation (PSO) algorithm, the dynamic PSO, is proposed to obtain a better convergence rate, thus increasing the overall efficiency of the algorithm. To prove the effectiveness of the method, wire antennas operating over a wide frequency range have been optimised and tested.

1

Introduction

The particle swarm optimisation (PSO) is a stochastic algorithm, originally proposed by Kennedy and Eberhart [1], which mimics the behaviour of animals organised in groups, such as a flock of birds or a swarm of bees. Following this metaphor, each member of the swarm, referred as agent or particle, is free to fly through the multidimensional search space whose dimensions represent the parameters of the addressed problem. Hence, the position of the agent is an array containing the optimised values of a candidate solution. The particle is constantly attracted both towards the best position visited by itself and the best position found by the swarm as a whole. The intensity of these two distinct poles of attraction determines a different pattern for the search, encouraging an independent pursuing of the optimum instead of using the shared information. The PSO has proven to be effective in managing a large number of parameters and to cope with local minima owing to its inherent global-search nature. Moreover, much effort has been devoted to improve its performance in terms of convergence speedup [2 – 4]. In fact, the PSO ability to reach the global optimum and the velocity of convergence depend on a critical trade-off between the exploration attitude, which tends to encourage at each step the search in a wide area of the solution space, and the exploitation of the already gained information, which refines at each step the search in the neighbourhoods of best-known locations. These two components can be more effective during some stages of the evolutionary process, the former being more useful at the first stage of the search process and the latter more determinant in the following. Owing to the importance of this concept, we present an evolved search paradigm, the dynamic PSO (DPSO), which dynamically activate and tune the level of the two attractions. As IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 5, pp. 611– 615 doi: 10.1049/iet-map.2010.0375

introduced in [5], this procedure takes advantage of both exploration and exploitation at different stages of the searching process thus guaranteeing an improvement of the convergence rate. Accounting for the flexibility of PSO algorithm in dealing with multi-parameter optimisation problems, we have proposed a combined method of moments (MoM)/DPSO procedure for broadband and miniaturised antenna design in ultra high frequency (UHF) or/and very high frequency (VHF) frequency bands. The proposed methodology reveals reliable in designing a monopole operating in a wide frequency range both with a high-gain level and a good voltage standing wave ratio (VSWR) [6]. In Section 2, the standard PSO algorithm is briefly described and then the DPSO mechanism is introduced. A design example with a large number of parameters is reported in Section 3, where it is highlighted the improvement in terms of convergence with respect to the standard PSO. Finally, in Section 4 the effectiveness of the proposed methodology is proven by realising and measuring a wide-band simplified monopole prototype.

2

PSO algorithm improvement

A problem that requires the simultaneous optimisation of N parameters varying in a reasonable range of values results in an N-dimensional bounded space where the optimal solution has to be found. In a standard PSO algorithm we define a collection of M agents that change their positions iteratively within this N-dimensional space. In particular, for each agent the updated step in each dimension n [ [1, N ] is specified accordingly to xn (t + 1) = xn (t) + vn (t + 1)

(1)

where the initial position xn(0) is randomly defined while 611

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www.ietdl.org vn(t + 1) represents the nth component of the agent velocity at the instant t + 1. The conventional stochastic iterative equation vn (t + 1) = wvn (t) + c1 rn,1 (t)(pbest,n − xn (t)) + c2 rn,2 (t)(gbest,n − xn (t))

(2)

describes the PSO mechanism 6. The first right-hand-side term in (2) represents the inertia of the agent to maintain the actual direction, being w referred as ‘inertial weight’. The other two terms express the attraction towards the best position ( pbest) ever found by the agent and the impulse towards the direction of the best position ( gbest) found by any particle. The constant coefficients c1 and c2 are also referred as the ‘cognitive’ and the ‘social’ rate, respectively, since the former biases the search around the personal best location while the latter encourages to chase the solution as a result of a collective process. The random numbers rn,1 and rn,2 are uniformly distributed within the range [0, 1] and introduce a chaotic component in the velocity update process. The PSO capability to converge to the global optimum and its velocity of convergence are strongly dependent on the weighting factors w, c1 and c2 in (2). In particular, large w values give impulse for an adequate exploration of the solution domain and force the particles to expand their search space area, since heavy particles are less affected by the attraction of pbest and gbest . This attitude is particularly important in the first stages of the search to prevent an early stagnation in the local maxima, and therefore we have defined w as a descendant function in the first Nw iteration steps, starting from a high w value. Moreover, noticing that the balance between global and local search through the course of run is critical to the convergence of the algorithm, we also decided to vary both the cognitive and social rate. To be more specific, at certain instants, we set c1 and c2 , respectively to a high and a low value recovering their initial values after a pre-defined number of iterations (Nc 2 Nw). This procedure is applied to encourage a new phase of exploration with respect to exploitation, both at instant Nw and every time a stagnation in the fitness occurs.

a numerical code such as the MoM represents the most suitable full-wave method for an accurate and efficient analysis. By combining the reliability of the MoM with the optimisation capability of the DPSO, we propose a methodology that reveals to be effective in designing a monopole operating in a wide frequency range both with a high-gain level and a good VSWR. The adopted procedure is based on a parametrical description of an antenna model where possible parameters are size and/or positions and values of lumped embedded loadings. The DPSO algorithm is therefore used to find the optimal configuration which meets the specifications and each agent is the set of codified parameters which represents a particular antenna model. The performance of each particle is measured by using a proper fitness function defined according to the imposed antenna requirements. As preliminary design, a monopole of a fixed length L ¼ 1.75 m, loaded with four parallel RLC lumped circuits and fed through a matching network, is considered (see Fig. 1). In order to guarantee broadband characteristics, positions and values of the embedded loads as well as matching network elements are the parameters to be optimised [10]. The wire radius is a ¼ 0.5 cm and the lumped elements vary in the ranges 0 , Rpi , 1550 V, 0 , Lpi , 2 mH and 0 , Cpi , 200 pF. The constraints for the matching network components are 0 , Ls , 1 mH, 0 , Lp , 2 mH and 0 , Cp , 10 pF whereas the transformer ratio is in the interval 0 , n2/n1 , 3. Reasonable requirements are a system gain G S at the horizon greater than G0 ¼ 25 dBi and a VSWR , 3.5 within 30 and 450 MHz [11]. To satisfy the design goals, the following fitness function has been minimised F = k1 Fg + k2 FS + k3 FVSWR

(3)

where k1 , k2 and k3 are weighting constants. The first two terms Fg and Fs control at each frequency fi (i [ [1, Nf ]) within the frequency band the value at the horizon [10] of both the

3 Application to wide-band wire antenna design Wire antennas are over the years widely used in many communication systems and their various fields of application requires the design of wide-band radiating elements. Conventional antennas operating at low frequencies results in very large dimensions therefore in naval applications, where antennas operate in UHF or/and VHF frequency bands, particular miniaturising techniques should be implemented to realise small and efficient devices. Different methods relevant for the miniaturisation as well as for broadband antennas have been proposed [7–9]. Loading wire antennas with resonant traps, as, for instance, resistor, inductor, capacitor (RLC) parallel circuits, is one possible way to obtain broadband performance. In particular, inductors and capacitors guarantee a multi-resonant frequency system, whereas resistors permit achieving wide-band characteristics [7]. When designing such kind of antennas, the different loading configurations (positions, RLC values) play a crucial role in achieving the required performance in terms of gain and VSWR. An accurate electromagnetic model of the radiating element is therefore fundamental in the design of the loaded system. Since antennas under analysis are realised by wire structures, 612 & The Institution of Engineering and Technology 2011

Fig. 1 Loaded monopole antenna with matching network structure IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 5, pp. 611 –615 doi: 10.1049/iet-map.2010.0375

www.ietdl.org system gain G S and the smoothness of the antenna gain G a Fg =

Nf 

(G0 − GS (fi , u = 908))3

(4)

|Ga (fi+1 , u = 908) − Ga (fi , u = 908)|2

(5)

i=1 Nf −1

Fs =

 i=1

The third term FVSWR in (3) FVSWR =

Nf 

x(|Gs (fi )| − |G|)

(6)

i=1

has been added to emphasise solutions having VSWR , 3.5. Indeed, the function x is defined according to 

x(|Gs (fi )| − |G|) =

log ||Gs (fi )| − |G||,

|Gs (fi )| , |G|

s

log [e||G (fi )|−|G|| − 1],

|Gs (fi )| . |G| (7)

where Gs( fi) is the reflection coefficient at the matching network input port and G corresponds to a goal (VSWR ¼ 3.5). The PSO combined with NEC4.1 EM solver [12] converges to a design whose load values (Rpi , Lpi and Cpi) and positions respect to ground (hi) (see Fig. 1) are reported in Table 1. The matching network results to be defined by Ls ¼ 0.1 mH, Lp ¼ 1.31 mH, Cp ¼ 1.19 pF and n2/n1 ¼ 2.1. By the numerical analysis of the solution, both gain system and VSWR meet the requirements in the frequency range. The DPSO algorithm has proved to guarantee a faster convergence and to illustrate the effectiveness of our algorithm, we report in Figs. 2 and 3 a comparison between the fitness behaviour for the dynamic (continuous line) and Table 1

Optimised positions and values of the RLC traps

Load

hi , cm

Rpi , V

Lpi , mH

Cpi , pF

1 2 3 4

30.5 46.3 112 128.1

695 1431 625 473.3

0.82 0.3 1.5 1.9

198 0.21 181 3.2

Fig. 3 Fitness comparison between DPSO (continuous line) and conventional PSO (dotted line) Effectiveness of the DPSO algorithm is apparent

conventional (dotted line) PSO (these curves have been obtained by using N ¼ 32 agents). As apparent from Fig. 2, in the interval [N1 ¼ 100; N2 ¼ 130] the algorithm stagnates in a local minimum, indeed resulting in a smooth behaviour of the fitness function. The dynamic change in the update of the velocity, as described in the previous section, takes place starting from N3 (141 iterations, as shown in Fig. 3) and allows the DPSO to converge to an optimal solution faster than the conventional PSO.

4

Wire antenna prototype

To prove the reliability of our approach, an antenna operating in the 150 – 1000 MHz frequency band has been optimised and tested. In order to produce a structure easy to fabricate, a monopole with a length L ¼ 30 cm and 1 cm of diameter has been considered, with only one embedded lumped parallel RLC circuit and without a transformer. For the same reason the lumped elements have been chosen by the algorithm within a dataset of commercially available components [13]. By requiring a system gain G S at the horizon greater than G0 ¼ 25 dBi and a VSWR , 3.5, the solution converges to Rp ¼ 255 V, Lp ¼ 270 nH and Cp ¼ 0.2 pF. The corresponding components were embedded on a printed circuit board (PCB), as shown in Fig. 4, and the PCB has been placed by cutting the monopole at a 20 cm height. Finally, the monopole is fed by a sub miniature version A (SMA) connector through a ground plane of copper of 90 cm × 90 cm. The antenna has

Fig. 2 Fitness behaviour of DPSO (continuous line) and conventional PSO (dotted line) at the initial iteration steps In the interval [N1 ¼ 100; N2 ¼ 130] a smooth behaviour of the DPSO fitness function can be noticed IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 5, pp. 611– 615 doi: 10.1049/iet-map.2010.0375

Fig. 4 Prototype with PCB and SMA feeding connector 613

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www.ietdl.org therefore been tested at an open site with conducting ground to avoid the influence of the finite copper plane. By observing the measured VSWR drawn in Fig. 5, it results to be below 3.5 in all the frequency band of our interest. Moreover, the measured results relevant to the input impedance of the prototype have been compared against simulated data: very good agreement has been obtained as shown in Figs. 6 and 7, confirming the effectiveness of the DPSO/MoM algorithm. The radiation efficiency of the optimised monopole reveals a typical behaviour that, starting from a minimum value of 26% at 150 MHz, grows up to the maximum (86% at 1100 MHz).

Fig. 8 Normalised radiation pattern on E plane at 100 MHz (continuous line), 600 MHz (dotted line) and 1100 MHz (dashed line)

Fig. 5 Measured VSWR

Fig. 9 Estimated gain in the operative frequency band

Fig. 6 Simulated (continuous line) and measured (dashed line) monopole input resistance

In Fig. 8, the numerical radiation patterns on the vertical plane are shown at 100, 600 and 1100 MHz. Each pattern is normalised to the gain, whose values for the three frequencies are 21.19, 3.42 and 6.38 dBi, respectively, as inferable from Fig. 9, where the overall gain of the antenna within the operative bandwidth is shown. The higher-frequency gain pattern is consistent with the case of the unloaded ideal monopole. In fact, as the antenna length increases beyond half a wavelength, side lobes start appearing and the direction of the maximum gain begins tilting towards the vertical off the horizontal plane.

5

Fig. 7 Simulated (continuous line) and measured (dotted line) monopole input reactance 614 & The Institution of Engineering and Technology 2011

Conclusions

An enhanced version of the PSO exploiting a dynamic tuning of the velocity update has been proposed for improving the rate of convergence and therefore reducing the simulation time. The proposed DPSO algorithm has proven to be effective when a large number of parameters are involved in the optimisation process as in the case of the design of a matching network and RLC loads for wide-band wire IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 5, pp. 611 –615 doi: 10.1049/iet-map.2010.0375

www.ietdl.org antennas. The reliability of this approach has been demonstrated in the realisation of an optimised monopole prototype and the level of accuracy of the method has been illustrated by measurements.

6

Acknowledgments

The measurements have been performed at facilities of Institute for Telecommunications and Electronics of Italian Navy ‘G. Vallauri’; the help of its personnel is kindly acknowledged.

7

References

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