Satellite Reconfigurable Contour Beam Reflector Antennas by Multi-objective Evolutionary Optimization

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1 101 Satellite Reconfigurable Contour Beam Reflector Antennas by Multi-objective Evolutionary Optimization Sérgio L. Avila Fundação CERTI / UFSC, Campus Universitário, CP 5053, Florianópolis Walter P. Carpes Jr. GRUCAD / CTC / UFSC, Campus Universitário, CP 476, Florianópolis Jose R. Bergmann CETUC / PUC-Rio, Rua Marques de São Vicente, 225, Rio de Janeiro Abstract Due to the growing demand for traffic communication, the requirement for reconfigurable antennas for future generation of satellites is growing steadily. This article presents designs examples of reconfigurable satellite contour beam reflector antennas via multi-objective evolutionary optimization. One configuration is composed of shaped reflector illuminated by a single feed horn. To explore the concept, we consider the case of a satellite that can be placed in three orbital positions to provide different coverage: Europe, North America and South America. A second example considers a single shaped reflector illuminated by two feed horns to simultaneously provide dual coverage with frequency reuse. Index Terms Evolutionary Optimization, Multi-objective Problem, Reconfigurable System, Reflector Antennas, Satellite. I. INTRODUCTION Fixed satellite services have to deal with many requirements such as high gain at a specific coverage area, wide frequency operation band, isolation for the main and cross-polarization, and gain slope. However, during the satellite operational life, the geographic characteristic of the traffic communication can change, due to sporadic events like sport competitions, unpredicted political or military conflicts, or due to fast transformations in economic activities, like the fast economic growth of some Asian countries, making certain fixed satellite services less attractive from the commercial point of view. Besides the limitations of space in the geosynchronous orbit, changes in the traffic communications bring the requirement for reconfigurable antennas for future generation of satellites [1]. Antenna configurations composed of shaped reflector fed by a single feed have been employed to produce contour beam for satellite applications, where the antenna radiation characteristics are obtained by optimizing the feed horn parameters and a set of coefficients that define the reflector shape. For the optimization, deterministic methods are usually employed [2,3,4,5], leading to single optimum solution, and leaving the antenna designer with no freedom to accommodate further adjustments. These deterministic algorithms optimize an objective function that requires the

2 102 specification of the minimum gain in the coverage area. However, in a reconfigurable scenario [5], the previous definition of the minimum coverage gain in each region poses as limitation for application of deterministic optimization methods to adjust the parameters of the shaped reflector configuration. The limitation in building an efficient objective function increases when the shapes of the coverage areas diverge and the antenna has to operate over wide frequency band. Alternatively, Multi-Objective Evolutionary Optimization with Genetic Algorithms has been extensively studied, becoming increasingly attractive to engineers [6,7,8,9]. In comparison with the classic deterministic methods, one of the advantages of Genetic Algorithms is the possibility of identifying a group of feasible solutions, given the opportunity of a post-synthesis selection with the intervention of the customer. The Genetic Algorithm (GA) is founded on the concepts of natural selection and genetics. They indeed bypass the main drawbacks of simple deterministic optimizers: they are particularly effective when the goal is to find an approximate global optimum (maximum or minimum) for multimodal functions, in a near-optimal manner. Among the advantages of GAs, we can quote that it can optimize with continuous or discrete parameters and it do not require gradient information; the possible discontinuities present on the fitness function have little effect on overall optimization performance; they are resistant to becoming trapped in local optima; they provide a list of semioptimum parameters instead of a single solution; they are well suited for parallel computers; and they can be employed for a wide variety of optimization problems. Additionally, for shaped reflector applications, they do not require any previous gain specification for the coverage areas. Here, we employ Multi-Objective Evolutionary Optimization with Genetic Algorithms (MGA), to design reconfigurable shaped reflector antenna for satellite applications. We present two design examples of antenna configurations for multiple beam applications. In the first case, the configuration is composed of rigid shaped reflector fed by a single feed horn where the radiation pattern reconfigurability is obtained by modified the feed-reflector relative position in order to emphasize different portions of the reflector surface. In the second case, two feed horns are employed to obtain multiple beams each one illuminates different portions of the shaped reflector. However, before present the design examples, the next section presents a brief about GA and MGA optimization tools. II. OPTIMIZATION PROCEDURE GA starts with a set of solutions called population where a solution is defined by a set of coefficients and parameters associated with a specific reflector antenna. Solutions from a population are used to form a new population. This is motivated by the hope that the new population (by genetic operators) will be better than the old one. Solutions that will form new solutions are selected according to their fitness (evaluation): the more suitable they are, more chances they have to reproduce. This is repeated until some condition (for example, number of generations or improvement of the best solution) is satisfied (different set of coefficients can give almost the same evaluation). The

3 103 niching technique is a genetic operator that helps to explore distinct areas simultaneously (different set of coefficients can give almost the same evaluation), by discovering local and/or global optima. Figure 1 illustrates a niche genetic algorithm [7,9] solving a multimodal function. Fig. 1. Genetic Algorithm in a multimodal function. As mentioned before, in the reconfigurable antennas design several goals, or multiple coverages, must be satisfied simultaneously in order to obtain an optimal solution. In the reflector shaping, as the optimization objectives conflict, no single solution may exist that is best regarding all considered criteria. Multi-objective optimization seeks to optimize the components of a vector-valued cost function. Unlike single objective optimization, the solution to this problem is not a single one, but a family of efficient solutions. Each one is optimal in the sense that no improvement can be achieved in a cost vector component that does not lead to degradation in at least one of the remaining objectives. Each element of this family constitutes a non-dominated (non-inferior or non-superior) solution to the multi-objective problem. The main action of the multi-objective optimization is to determine the efficient front. With this set of solutions, it is possible to understand the dependence between each objective, which allows making efficient choices for the final solution decision displays in Fig. 2. Fig. 2. Multi-objective optimization. The analysis of the Pareto-front behavior permits to understand the tradeoff between the different objectives. Compared with the deterministic optimization methods, which lead to unique solution, MGA offers the possibility to the designer to make the final choice among the set of solutions by

4 104 considering additional constraints not included in the initial steps. Like a GA procedure, the MGA starts with a set of solutions randomly created where the number of elements depends on the complexity of the problem. For reflector antennas cases shown here, we used an initial population with 200 individuals. These solutions are evaluated (fitness functions are presented in the problems description) and the Pareto-optimal condition is tested, giving two groups of solutions: one formed by efficient solutions, called non-dominated population (NDOM); and another by non-efficient solutions, called dominated population (DOM). After the Pareto s check, it is time to apply the Clearing technique, whose purpose is to obtain a sparse and well-established Pareto front. If similarities among individuals are detected (in parameters or/and objectives spaces [10]), one or some of them are converted to dominated solutions. The others genetic operators employed are the classical ones: selection with sampling and tournament, crossover and mutation (with probabilities of 80% and 5%, respectively), variable reflection and global elitism [9]. Figure 3 presents the flowchart of the MGA procedure employed in our studies. Fig. 3. Multi-objective Genetic Algorithm flowchart. The evolutionary process stops when the convergence criteria is achieved, which happens when goals became stable. Nevertheless, in both design cases presented here, number of generations never exceed to 100 generations. This stochastic optimization procedure is repeated several times to guarantee that found solutions are the optimal ones, or at least the best solutions we can found.

5 105 III. CASE I: OPTIMIZATION AND RESULTS To explore the concept of antenna reconfigurability, we consider the case of a satellite that (depending on the demand) can be placed in three orbital positions to provide different coverage: Europe, North America and South America, as shown in Fig. 4. The differences in the coverage shapes indicate three independent objectives. Similar antenna design challenge was reported in [5]. There, the antenna performance was obtained by employing several actuators to modify the shape of a malleable reflector surface and, consequently, the antenna radiation pattern. These actuators are activated by land command to reconfigure the antenna surface in order to attend the desired gain coverage. (a) Europe (b) North America (c) South America Fig. 4. Satellite on geostationary orbit up each coverage area. Longitude: Europe = 15 ; South America = -60 ; North America = -100 ; The satellite just moves on geostationary orbit, thus the latitudes are 0 always. For this exercise, the antenna configuration is composed of a fixed feed horn and a rigid reflector, where the change of the antenna radiation pattern is made by land command set acting on devices (actuators) to modify the position of reflector with respect to the feed horn and to reflect the energy radiated from the feed in different directions. With the help of Fig. 5, the action of the actuators can be described by the modifying the distance OO =(Δx,Δy,Δz) between the origins. The antenna operates at the Ku Band (12GHz), has an aperture projected diameter D=2a=3m and the feed radiation pattern is represented by a raised cosine model cos N (θ) [6]. The optical reference geometry has equivalent focal distance F = 2.74m, offset displacement feeder H = 1.75m, and offset angle θ 0 =35.3, as illustrated in Fig. 5.

6 106 Fig. 5. Offset Antenna Parameters optimization. The antenna optimization procedure adjusts the reflector shape, the feed radiation pattern (N) and the reflector-feed relative position (Δx,Δy,Δz,θ 0 ) defined by the actuators. To control the reflector dimension during the optimization procedure, the coordinate system xyz is supposed fixed and the z axis points to the center of the coverage, while the modification in imposed to the x y z system of coordinate. The reflector surface is described by using a Jacobi s polynomial expansion [3] where the paraboloid is one of the terms. For the optimization, 200 expansion coefficients are used to describe the reflector (n=20 and m=10 the first coefficient is not adjusted [3]). Ten additional optimization variables are used to define the three set of coordinates of the reflector position associated with each coverage area and the feed radiation pattern (N), resulting in a total of 209 optimization parameters. The limits for each variable are specified to allow a compromise between the numerical efficiency and freedom to ensure the inclusion of the optimum solution. There are three objectives to be pursued, corresponding to the maximization of the minimum directive copolar gain G i in each one of the three continents. Some simulations have been carried out to acquire confidence in the achieved optimized solutions, as every single optimization procedure demands. Figure 6 illustrates the performance of set of the possible solutions given by the MGA. The best solution for each goal is graphically represented by a triangle with vertices at the axis associated with the gain coverage achieved for each independent beam G i. It shows the performance of examples of antennas selected by maximizing the gain G i in one of the three coverage areas (Europe, North America, and South America) when consider independently. In addition, it shows the performance of a solution obtained from the maximization of the average of the objectives [W 1 G 1 +W 2 G 2 +W 3 G 3 ], where the weight W i is proportional to the respective coverage area.

7 107 Fig. 6. Best solutions for each objective [dbi]. Figure 7 illustrates the antenna radiation patterns at the three continents obtained for the four examples of antennas where the dashed line indicates the coverage contours. Table I and II list the gain values G i and the reflector displacement (Δx,Δy,Δz,θ 0 ) required for each objective. The radiation patterns are depictured in the uv plane where u and v are the cosine director in degrees. Best Solutions TABLE I. REFLECTOR SPACE POSITION Europe ΔY(cm) ΔZ(cm) θ( 0 ) Objective Objective Objective 3 0, Average Best Solutions North America ΔY(cm) ΔZ(cm) θ( 0 ) Objective Objective Objective Average

8 108 Best Solutions South America ΔY(cm) ΔZ(cm) θ( 0 ) Objective Objective Objective Average As observed in Fig. 7, the high level contours of the radiation pattern are responsible for accommodating the European and North-American coverages, while the lower level contours are shaped to accommodate the South-American coverage. As observed in Table I, the action of actuators produce small displacement of the reflector with respect to the feed phase center, affecting particularly the higher levels counters of radiation pattern. The displacement Δx is negligible and has not being listed. The critical objective is the European coverage as the high energy concentration required for this region implies on lower gain in the other continents, especially for the South-American coverage. In other words, as the reflector is shaped to produce uniform illumination of the center of the antenna beam, outside the European coverage it shows higher gain slope and, as consequence, brings lower gain coverage for the large South-American coverage. On the other hand, when searching for a solution that maximizes the gain at the largest area, the South American Coverage, lower the gain level are found at the center of the beam. Alternatively, a solution obtained from the average of the objectives shows a compromise between these limitations, as observed from the gain values listed in Table II. TABLE II. DIRECTIVE GAIN MAIN POLARIZATION [dbi] Best Solutions Found Coverage Area Europe North America South America Objective Objective Objective Average solution From the Table II, small differences in the values can be seen, when compared with the limits for the optimization variables, reassuring the initial assumption that the optimum solution would comprise an overlapping of the coverage areas. Also, it is noticeable that the best solutions are similar themselves. It suggests that the obtained solutions belong to the same niche - particularly the parameters used to locate the reflector.

9 109 Figure 7.a Radiation Pattern for the best solution for the Objective 1 Figure 7.b Radiation Pattern for the best solution for the Objective 2 Figure 7.c Radiation Pattern for the best solution for the Objective 3 Figure 7.d Radiation Pattern for the best solution for the Average Solution

10 110 To better illustrate the effect of the optimization on the reflector shape, we chose the solution provided by the average of the objectives. Figure 8 shows the difference in the z coordinate between the synthesized reflector and the initial parabolic reflector. As observed, the deviations from initial reflector are lower than 5 cm and they grow towards the reflector rim. Fig. 8. Difference (m) between the shapes of parabolic reflector and the optimized one. IV. CASE II: OPTIMIZATION AND RESULTS Fig. 9. European coverage scenario for satellite TV distribution The design of large Ku-band antenna system for multiple regional beams tailored to specific geographical areas is presented in [4]. It considers a coverage scenario for satellite TV distribution composed of several beams illustrated in Fig. 9. The polygons some distance apart form separate beams, allowing the reuse the same frequency band by sidelobe isolation. The beams A, B and F would share one set of frequency channels, beams C and D a second set, E and I a third set, and beams G and H a fourth. The design example employs one 4m reflector for the most closely spaced

11 111 beams A-D and two 3m reflectors for the five remaining beams E-I to allow extensive frequency reuse by imposing a sidelobe isolation of 27 db or more. The reconfigurability is obtained by using a large linearly polarized array-fed to illuminate the single offset reflector. Here, to explore the concept of reconfigurability in the European multiple beam coverage described above, we employ an alternative antenna configuration to provide the simultaneous coverage of the areas E and I with the same channel frequency. The antenna configuration is composed of a single offset reflector fed by two corrugated horns, as illustrated in Fig. 10. When compared with the configuration employed in [4], it may result lighter antenna design by avoiding the use of an array feed with 90 elements and a Beam Form Network. For this exercise, the antenna operates at the Ku Band (12GHz), and the feed radiation patterns are represented by a raised cosine model cos N (θ). The reflector has an aperture projected diameter D=2a=4m to allow frequency reuse with spacing between coverages down to The optical reference geometry has equivalent focal distance F = 4.4m, and offset displacement feeder H = 2.375m, as illustrated in Fig. 4. The antenna optimization procedure adjusts the reflector shape, the feed radiation pattern (N), the offset angle (θ 1, 2 ) and the feed phase center positions (y 1,2,z 1,2 ). The reflector surface is described by using a Jacobi s polynomial expansion [3] where the paraboloid as one of the terms. For the optimization, 120 expansion coefficients are used to describe the reflector (n=15 and m=8 the first coefficient is not adjusted [3]), an amount that satisfies the antenna gain specifications. The optimization procedure also adjusts the feed radiation pattern (N) to ensure the required 27 db isolation between the beams. It results a total of 127 optimization parameters and the limits for each variable are specified to allow a compromise between the numerical efficiency and freedom to ensure the inclusion of the optimum solution. Fig. 10. Offset Antenna Parameters optimization.

12 112 There are four objectives to be pursued, corresponding to the maximization of the minimum directive copolar gain G E,I in each one of the areas E and I, respectively, and simultaneously, minimize the peak sidelobe level SL E,I in the corresponding isolation area E and I, respectively. Some simulations have been carried by imposing the 27 db isolation as restriction to define the set of possible solutions. As observed in Case I, the choice of a solution that leads to the best performance for one of the objectives is associated to poor performance for the other three objectives. As a compromise, we choose in the set the solution obtained from the maximization of the average of the minimum directive copolar gain (G E +G I )/2 for each one of the beams while keeping the isolation 27dB. Figure 11 illustrates the antenna radiation patterns for the two beams. E-beam coverage I-beam coverage Fig.11. Radiation pattern of average solution, contours in dbi. Coverage contour (--0--) For the solution, Table III list the radiation pattern performance for both beams and Table IV lists the feed phase center position (y 1,2,z 1,2 ), the offset θ 1,2 angle and feed radiation pattern N 1,2. TABLE III. RADIATION PATTERN PERFORMANCE E-beam I-beam G E [dbi] SL E [dbi] G I [dbi] SL I [dbi] Solution 41,41 14, TABLE IV. FEED POSITION Feed parameters for E-beam Feed parameters for I-beam y 1 [cm] z 1 [cm] θ 1 [ 0 ] N 1 y 2 [cm] z 2 [cm] θ 2 [ 0 ] N 2 Solution

13 113 To illustrate the effect of the optimization on the reflector shape, Fig. 12 shows the difference in the z coordinate between the synthesized reflector and the initial parabolic reflector. As observed, the deviations from initial reflector are lower than 2cm and they grow towards the reflector rim. Fig. 12. Difference (m) between the shapes of parabolic reflector and the optimized one. V. FINAL COMMENTS In this work, Multi-Objective Evolutionary Optimization with Genetic Algorithms was employed for the design of a reconfigurable satellite antenna. It shows two designs of satellite antenna for multiple beam applications at the Ku band. In contrast to the single solution yielded by the usual deterministic numerical algorithms, the MGA employed here yields a set of possible solutions for the shaped reflector antenna. It permits the choice of a final solution in a post-synthesis analysis that considers the whole set of possible solutions and the variable elements, leaving the designer with freedom in the negotiations with the customer. For example, if a solution with a better European coverage is essential, the required antenna is available in the set of possibilities generated by the numerical scheme, as illustrated by the examples shown in Fig. 6. This work was partially supported by the CAPES and CNPq Brazilian agencies. REFERENCES [1] INTELSAT, Commercial Global Satellite Communications System, [Online] [2] J.R.Bergmann, R.C.Brown,P.J.B.Clarricoats and Z. Hai,:"Synthesis of Shaped-Beam Reflector Antennas Patterns", Proc. IEE Microwaves Antenas and Propagation, Vol.135, February, 1988,pg [3] D.W. Duan and Y. Rahmat-Samii, A Generalized Diffraction Synthesis Technique for High Performance Reflector Antennas, IEEE Antennas and Propagation Magazine, v. 43, n. 1, pp , Jan [4] P. Balling, M. van der Vorst, H. Wolf, Design and analysis of large linearly polarized array-fed offset reflector antennas with frequency reuse, Proceeding of the 26 th ESA Antenna Technology Workshop on Satellite Antenna Modelling and Design Tools, V.1, pp , Noordwijk, The Netherlands, November [5] W. H. Theunissen et al., Reconfigurable contour beam reflector antennas using adjustable subreflector and adjustable single feed, Microwave and Optical Technology Letters, v. 21, n. 6, pp , Jun

14 114 [6] S.L.Avila,. L. Krähenbühl, W. P. Carpes Jr., and J. R. Bergmann, Conception and comprehension of the contour beam antenna in a satellite system by multi-objective genetic algorithm with sensitive analysis, 6th WCSMO World Congress on Structural Multidisciplinary Optimization, Rio de Janeiro, Brazil, May-Jun,2005. [7] K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Apr [8] S. L. Avila, A. C. Lisboa, R. H. Takahashi, et al., Sensitivity Analysis in the Parameters Space Applied to Decision Making in Multi-objective Evolutionary Optimization, IEEE Transactions on Magnetics, v. 42, n. 4, pp , Apr [9] M. Schoenauer et al., Evolutionary Computation Journal, MIT Press, [Online] [10] S. L. Avila et al., A Multi-Niching Multi-Objective Genetic Algorithm for Solving Complex Multimodal Problems, In: OIPE 2006, The 9th Workshop on Optimization and Inverse Problems in Electromagnetics, September 13-15, 2006, Sorrento (Italy).