應用最佳化方法於自組式電磁散射體及天線效能之提升

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(1)國立臺灣大學電機資訊學院電信工程學研究所博士論文 Graduate Institute of Communication Engineering College of Electrical Engineering and Computer Science. National Taiwan University Doctoral Thesis. 應用最佳化方法於自組式電磁散射體及天線效能之提升 Applications of Optimization Techniques to Self-Structuring Electromagnetic Scatterers and Antenna Performance Enhancements 陳晏笙 Yen-Sheng Chen 指導教授：陳士元博士 Advisor: Shih-Yuan Chen, Ph.D. 指導教授：李學智博士 Advisor: Hsueh-Jyh Li, Ph.D. 中華民國 101 年 6 月 June, 2012.

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(3) 國臺灣大國立臺大學博士士學位位論文文. 口委員會口試委會審定書書應用用最佳化化方法於於自組式式電磁散散射體及及天線效效能之提提升 Appplicationns of Opptimizatiion Tech hniquess to Selff-Structu uring E Electrom magnetic Scatteerers and d Antennna Perfo formancee Enhhancemeents. 本論論文係陳陳晏笙君君（D989942005）在國立臺臺灣大學學電信工工程學研研究所完成成之博士士學位論論文，於民民國一百百零一年年六月四日日承下列列考試委委員審查通通過及口口試及格格，特此證證明.

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(5) 致謝首先感謝李學智教授的教育與提攜；李教授除了教導學生應有嚴謹之治學態度之外，也提點學生成為肩負社會責任的知識份子。謝謝指導教授陳士元教授的啟蒙與栽培；陳教授淬練學生具備嚴謹的推論能力，並要求學生於大格局上由本質著手求新求變。兩位指導教授是我人生路上的貴人。如果沒有遇到兩位老師，我想我將一味地隨波逐流而最終淪於平庸。除此之外，也謝謝教導過我的師長們。特別感謝工業工程所所長陳正剛教授、電機系教授鄭士康教授以及于天立教授給予我研究上的寶貴建議。謝謝我的女朋友忻恩，謝謝妳總是如此體諒我、支持我。我是個無能享樂的工作狂。我明知人生在各階段的付出最終都是徒勞無功也沒有意義，卻還是盡全部的努力。在這不斷循環的過程裡，只有妳是我人生唯一的溫暖與美好，挽救了我使我不致成為冰冷孤獨的機器人。有妳作為我的靈魂知己是我這輩子最幸運的事。最後，謝謝台灣大學這九年來給我的完整教育。特別感謝中文系教授歐麗娟教授，拓展了我電機系教育之外的廣度以及人格的深度。在這學校有那麼多珍貴的回憶，我還有很多想做的事、想修的課，這些遺憾都無法彌補了。九年好像是很長的時間，是我人生至此的三分之一，可是除了一張文憑證明我在這學校就讀了九年，九年前與九年後的台大有什麼不一樣？誰又能證明我在這裡發生的事或那些所做的努力與掙扎過的痕跡？只是一切都來不及了，也沒辦法了，我無從補償也無從挽救。各階段的回憶就像過眼雲煙，在遺忘的基礎上我無法再現，而人活著就是不斷地失去我們失去不起的東西。這是我在完成學位論文後最大的感慨，其成也毀，其毀也成。.

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(7) 摘要本論文提出三種創新的電磁應用以改善傳統架構之限制並提升工作效率。藉由最佳化方法的智能輔助，吾人所提出之精密結構得以實現，而複雜的可重組元件之合成問題亦得以更有效率地解決。本論文所使用之最佳化方法包含實驗計畫法與演化式演算法，分別應用於下列創新元件中。吾人首先提出一款適用於無線射頻標籤系統之新型雙天線標籤架構。此新型架構使用兩支獨立工作之天線，一支專職接收來自讀取機之訊號與功率，另一支專職將所載資料後散射回讀取機。若妥善將接收天線之輸入阻抗與後級整流電路之晶片阻抗設計為共軛匹配，接收天線便能連續接收來自讀取機之功率，使標籤電路之供電更為穩定；此外，若將後散射天線於開路與短路間切換，並將其輸入阻抗設計為純實數，則後散射天線能回傳最大之散射訊號差給讀取機，可大幅提升系統之讀取距離及讀取可靠性。由於雙天線架構須整體考慮所有設計目標，並降低兩支天線之互耦合量，因此吾人利用實驗計畫法來掌握多目標與天線幾何參數間的函數關係，於 0.1λ0 × 0.1λ0 的面積下成功實做出此雙天線架構。此新型標籤之效能經實驗佐證，可大幅改善傳統結構下之接收及後散射限制。其次，吾人發展出網格化天線自動設計程式。此設計工具整合全波電磁模擬軟體以及數種單目標及多目標演化式演算法，當天線設計情境為給定設計面積並須考量周圍環境時，此工具僅需將設計面積切割為若干網格，就能找出工作目標下最適合的天線幾何形狀。吾人以一多輸入多輸出天線系統來驗證多目標最佳化方法的效能，並針對實際應用中的頻寬考量設計一更有效率的演算法，藉以改善傳統方法的限制。此方法以手機天線設計為例，設定工作目標為同時涵蓋 698–960 兆赫以及 1710–2170 兆赫，其最佳化結果經模擬及量測佐證後，證實所提出之策略確實比傳統方式更勝任多頻且寬頻的工作目標。最後，吾人提出一款創新之自組式電磁散射體。此自組式電磁散射體為首創之智能散射平面；它能根據下達指令自行調整其電氣形狀，完成雷達截面積最小化或最大化等工作目標。此自組式電磁散射體利用自組式元件之原理，使用多枚繼電器連接細長金屬片；當繼電器各自於開、關兩狀態間切換，數十億種散射組態因應產生。藉由適當之搜尋演算法來尋找各工作目標下之最佳開關組態，雷達截面積特性得以隨心所欲地控制。吾人提出一創新之搜尋演算法，利用部分因子實驗設計來估計各開關之作用以及開關間的交乘影響，能極有效率地解決此合成求解問題。吾人亦實做出此自組式電磁散射體之量測系統，以實驗佐證雷達截面積之最小化與最大化效能，證實它能自行重組為多角度之吸收體或增強反射面。 i.

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(9) ABSTRACT In this dissertation, three innovative electromagnetic (EM) applications are proposed to improve the efficiency and limitation of conventional employments. By using the intelligence of optimization methodologies, including design of experiments (DOE) and evolutionary algorithms, complex design processes and arduous synthesis problems are simplified and solved, and the proposed applications exhibit powerful and sophisticated capabilities which satisfy the original need. The first application is a novel dual-antenna structure for passive radio-frequency identification (RFID) tags. It is formed by two linearly tapered meander dipole antennas that are perpendicular to each other and connected to the slightly modified tag chip. One of the antennas is for receiving, while the other is for backscattering. The input impedance of the receiving antenna is designed to be conjugate matched to the highly capacitive chip impedance for the maximum power transfer. Meanwhile, the backscattering antenna is alternatively terminated by an open or a short circuit to modulate the backscattered field. By making the input impedance of the backscattering antenna real-valued, the maximum differential radar cross section (RCS) may be achieved leading to a longer read range and better reliability. With the aid of DOE, the proposed dual-antenna structure is designed to fit within a compact area of 32.8 × 32.8 mm2 while keeping relatively low mutual coupling between the two antennas. The impedance, receiving, and backscattering performances of the proposed dual-antenna structure are measured and simulated, and they agree very well. Also, it is demonstrated that the proposed dual-antenna structure outperforms the conventional single-antenna tag design in every respect. The second application is a competent antenna design tool based on the pixelized design technique. Merely with only a roughly-formed solution domain, this pixelized design tool is capable of automatically finding an antenna layout with performance satisfying the design needs. The pixelized design tool integrates a full-wave simulator and. external. optimization. schemes,. including. various. single-objective. and. multiobjective evolutionary algorithms. The capability of multiobjective operations is demonstrated by a multiple-input-multiple-output (MIMO) antenna system for handset applications, where the impedance matching of each antenna should be optimized and the mutual coupling between them should be minimized. In addition, an innovative iii.

(10) approach for designing wide- and multi-band antennas within a small area is proposed and incorporated into this tool. The proposed method is verified through a handset antenna design, covering 698–960 MHz and 1710–2170 MHz. The simulated and measured results confirm that the proposed method can find an antenna configuration with satisfactorily wide bandwidth and outperforms the conventional approaches. The third application is a self-structuring electromagnetic scatterer (SSES). The SSES is the first intelligent reflective surface that can alter its electrical shape to fulfill various operational objectives, such as RCS reduction or RCS enhancement. The SSES template. comprises. segments. of. metallic. thin. strips. interconnected. via. voltage-controlled switches. By opening or closing the switches, the phase of the field scattered by the strips changes, resulting in destructive or constructive interference in the total scattered field. The RCS of the SSES can thus be controlled. An efficient search algorithm based on the fractional factorial design of experiments (FFD) is adopted to find a suitable switch configuration for the SSES system. A SSES prototype was built and a series of RCS measurements were performed to demonstrate its capability to adaptively control the RCS. It is shown that the bistatic RCS can be significantly reduced in any specified direction and that the main beam maximum of the RCS pattern can be enhanced and steered within an angular range of 30 degrees.. iv.

(11) CONTENTS 摘要. i. Abstract. iii. Contents. v. List of Figures. ix. List of Tables. xv. Chapter 1. Introduction. 1. 1.1 Research Motivation..........................................................................................1 1.2 General Concepts of Evolutionary Algorithms and Design of Experiments.....2 1.3 Chapter Outlines................................................................................................4. Chapter 2. Overview of Optimization Methods in Electromagnetic Problems. 7. 2.1 Introduction.......................................................................................................7 2.2 Genetic Algorithms............................................................................................8 2.2.1. Simple GA Mechanisms....................................................................9. 2.2.2. Multiobjective GA...........................................................................12. 2.3 Particle Swarm Optimization...........................................................................13 2.3.1. PSO Mechanisms.............................................................................14. 2.3.2. Binary Particle Swarm Optimization...............................................16. 2.3.3. Multiobjective PSO.........................................................................17. 2.4 Design of Experiments....................................................................................19 2.4.1. Pre-Experimental Planning..............................................................19. 2.4.2. Phase 1: Design of Experiments......................................................20. 2.4.3. Phase 2: Analysis of Experiments...................................................25. 2.4.4. An Example: Design and Optimization of an Inductive-Loop-Fed Meander Dipole for Passive RFID Tags..........................................32. Chapter 3. A Novel Dual-Antenna Structure for Passive UHF RFID Tags. 47. 3.1 Introduction.....................................................................................................47 3.2 Proposed Tag Structure....................................................................................48 3.2.1. Reception and Backscattering in RFID Tags...................................48 v.

(12) 3.2.2. Dual-Antenna Tag Structure............................................................50. 3.3 Design Methodology........................................................................................52 3.3.1. Statistical Model Building...............................................................55. 3.3.2. Optimization of Multiple Responses...............................................70. 3.4 Experimental Verification................................................................................70 3.4.1. Isolation and Antenna Impedances..................................................72. 3.4.2. Receiving Performance....................................................................75. 3.4.3. Backscattering Performance............................................................78. 3.5 Summary and Discussion................................................................................83. Chapter 4. A Competent Pixelized Design Tool and Wide- and Multi-Band Antenna Designs. 85. 4.1 Introduction.....................................................................................................85 4.2 An Overview of the Pixelized Design Technique............................................86 4.2.1. Problem Formulation.......................................................................87. 4.2.2. Continuous Optimization Approach................................................87. 4.2.3. Discrete Optimization Approach.....................................................89. 4.3 Implementation of the Pixelized Design Tool.................................................90 4.3.1. Integration of Evolutionary Algorithms with HFSS........................90. 4.3.2. Capability of the Pixelized Design Tool Developed at NTU..........91. 4.4 Investigation of Single-Objective Competence...............................................93 4.4.1. Initialization Condition....................................................................94. 4.4.2. Searching Efficacy of Simple GAs..................................................98. 4.4.3. Searching Efficacy of BPSO.........................................................101. 4.4.4. Remark...........................................................................................105. 4.5 Demonstration of Multiobjective Competence..............................................105 4.5.1. Design Considerations...................................................................105. 4.5.2. Results and Discussion..................................................................106. 4.6 Wide- and Multi-Band Pixelized Antenna Designs.......................................110 4.6.1. Design Considerations...................................................................110. 4.6.2. Conventional Objective Functions.................................................111. 4.6.3. General Rules for Objective Functions..........................................114. 4.6.4. A Novel Multiobjective-Based Method for Wide- and Multi-Band vi.

(13) Antenna Designs............................................................................117 4.6.5. Second-Stage Design.....................................................................119. 4.7 Summary........................................................................................................120. Chapter 5. A Self-Structuring Electromagnetic Scatterer. 123. 5.1 Introduction...................................................................................................123 5.2 SSES Design..................................................................................................124 5.2.1. System Setup.................................................................................124. 5.2.2. SSES Template..............................................................................125. 5.3 Implementation of Search Algorithms...........................................................128 5.3.1. Phase 1: Design of Experiments....................................................130. 5.3.2. Phase 2: Analysis of Experiments.................................................130. 5.4 Prototype and Experimental Results..............................................................132 5.4.1. RCS Reduction..............................................................................136. 5.4.2. RCS Enhancement.........................................................................142. 5.5 Evaluation of Search Algorithms...................................................................147 5.6 Summary........................................................................................................147. Chapter 6. Conclusions. 149. 6.1 Summary........................................................................................................149 6.1.1. A Novel Dual-Antenna Structure for passive RFID Tags.............149. 6.1.2. A Competent Pixelized Design Tool..............................................150. 6.1.3. A Self-Structuring Electromagnetic Scatterer................................150. 6.2 Future Work...................................................................................................151 6.2.1. Efficacy Enhancement of the Pixelized Design Tool....................151. 6.2.2. Enhancement of Tunable Phase Range in SSES...........................152. References. 153. List of Publications. 169. vii.

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(15) LIST OF FIGURES Chapter 2. Overview of Optimization Methods in Electromagnetic Problems. 7. Fig. 2-1 Flow chart describing a simple GA. ...............................................................10 Fig. 2-2 Flow chart describing PSO. ............................................................................15 Fig. 2-3 Flow chart describing BPSO. .........................................................................18 Fig. 2-4 General model of a system. ............................................................................20 Fig. 2-5 Individual desirability function for the objective is (a) a maximization, (b) a minimization, and (c) to hit the target T. ........................................................32 Fig. 2-6 Geometry of the benchmarking antenna structure. .........................................33 Fig. 2-7 Normal probability plot of residuals for (a) Ra and (b) Xa. .............................38 Fig. 2-8 Plot of residuals versus predicted values for (a) Ra and (b) Xa. ......................39 Fig. 2-9 Comparison of the three experimental designs. ..............................................45. Chapter 3. A Novel Dual-Antenna Structure for Passive UHF RFID Tags. 47. Fig. 3-1 Configuration and equivalent circuit of a passive RFID tag. .........................49 Fig. 3-2 Block diagram of proposed dual-antenna tag. ................................................50 Fig. 3-3 Schematics of (a) conventional tag and (b) proposed dual-antenna tag. ........51 Fig. 3-4 Geometry of the proposed dual-antenna structure. .........................................53 Fig. 3-5 Photograph of the prototype of the proposed dual-antenna structure. ............72 Fig. 3-6 Simulated and measured isolation between the receiving and backscattering antennas. .........................................................................................................73 Fig. 3-7 Simulated and measured input impedances of the receiving antenna with the backscattering antenna being open-circuited (ZL = ∞). ...................................73 Fig. 3-8 Simulated and measured input impedances of the receiving antenna with the backscattering antenna being short-circuited. (ZL = 0). ..................................74 Fig. 3-9 Simulated and measured input impedances of the backscattering antenna with the receiving antenna being conjugate matched. ............................................74 Fig. 3-10 (a) E- and (b) H-plane patterns measured at 915 MHz for the conventional tag antenna and the proposed dual-antenna structure with the backscattering antenna being open- and short-circuited. ........................................................76 Fig. 3-11 Experimental setup for measuring the receiving performance of the proposed dual-antenna tag structure. ..............................................................................77 ix.

(16) Fig. 3-12 Measured receiving performance of the proposed dual-antenna tag for various antenna spacing d as the backscattering antenna is alternatively switched between open and short circuits. (Frequency: 915 MHz). ..............................77 Fig. 3-13 Experimental setup for measuring the backscattering performance. ..............79 Fig. 3-14 Simulated, measured, and theoretically calculated backscattering RCS of the proposed dual-antenna structure when the receiving antenna is conjugate matched. ..........................................................................................................79 Fig. 3-15 Simulated, measured, and theoretically calculated backscattering RCS of the conventional tag structure. ..............................................................................81 Fig. 3-16 Calculated detection ranges of the proposed dual-antenna tag and the conventional tag. .............................................................................................81 Fig. 3-17 Difference between backscattered fields of the proposed dual-antenna structure normalized to that of the conventional design. ................................83. Chapter 4. A Competent Pixelized Design Tool and Wide- and Multi-Band Antenna Designs. 85. Fig. 4-1 Design problems based on (a) a shape-optimization approach and (b) a pixelized design approach. .............................................................................86 Fig. 4-2 Example of mapping topology into a bitstream of 1’s and 0’s. ......................88 Fig. 4-3 Flowchart of the pixelized design tool. ..........................................................92 Fig. 4-4 Different optimization schemes implemented in the pixelized design tool. ...92 Fig. 4-5 The physical environment and geometry of a handset antenna. .....................93 Fig. 4-6 The candidate pixels of the antenna (a) discretizing in elaborate pixels and (b) incorporating with prior knowledge. ..............................................................94 Fig. 4-7 Investigation of initialization condition via the iteration histories of (a) best fitness value and (b) average fitness value (two-point crossover pc = 0.5, pm = 0.01, s = 2, Nite = 40, Npop = 64, and Nelit = 4). ...............................................96 Fig. 4-8 Investigation of initialization condition via the iteration histories of (a) best fitness value and (b) average fitness value (uniform crossover pc = 0.8, pm = 0.01, s = 2, Nite = 40, Npop = 64, and Nelit = 4). ...............................................97 Fig. 4-9 Investigation of initialization condition via the iteration histories of (a) best fitness value and (b) average fitness value (two-point crossover pc = 0.5, pm = 0.1, s = 2, Nite = 40, Npop = 64, and Nelit = 4). .................................................97 x.

(17) Fig. 4-10 Investigation of initialization condition via the iteration histories of (a) best fitness value and (b) average fitness value (c1 = c2 = 2, c0 = 1, Nite = 40, and Npop = 64). .......................................................................................................99 Fig. 4-11 Investigation of initialization condition via the iteration histories of (a) best fitness value and (b) average fitness value (c1 = c2 = 1, c0 = 1, Nite = 40, and Npop = 64). .......................................................................................................99 Fig. 4-12 Investigation of mutation operator via the iteration histories of (a) best fitness value and (b) average fitness value (two-point crossover pc = 0.5, s = 2, Nite = 40, Npop = 64, and Nelit = 4). ..........................................................................100 Fig. 4-13 Investigation of crossover operator via the iteration histories of (a) best fitness value and (b) average fitness value (two-point crossover, pm = 0.01, s = 2, Nite = 40, Npop = 64, and Nelit = 4). .......................................................................100 Fig. 4-14 Investigation of mutation operator via the iteration histories of (a) best fitness value and (b) average fitness value (pc = 0.5, pm = 0.01, s = 2, Nite = 40, Npop = 64, and Nelit = 4). ...........................................................................................102 Fig. 4-15 Investigation of population sizing via the iteration histories of (a) best fitness value and (b) average fitness value (two-point crossover pc = 0.5, pm = 0.01, s = 2, Nite = 40, and Nelit = 4). ..........................................................................102 Fig. 4-16 Investigation of personal best’s attraction via the iteration histories of (a) best fitness value and (b) average fitness value (c2 = 2, c0 = 1, Nite = 40, and Npop = 64). ................................................................................................................103 Fig. 4-17 Investigation of global best’s attraction via the iteration histories of (a) best fitness value and (b) average fitness value (c1 = 2, c0 = 1, Nite = 40, and Npop = 64). ................................................................................................................104 Fig. 4-18 Investigation of population sizing via the iteration histories of (a) best fitness value and (b) average fitness value (c1 = c2 = 2, c0 = 1, and Nite = 40). .......104 Fig. 4-19 The candidate pixels for the multiobjective optimization problem of multiple-antenna system. ..............................................................................106 Fig. 4-20 Pareto fronts found by the four multiobjective optimization algorithms. ....108 Fig. 4-21 Simulated scattering parameters of the posteriori-selected optimum solution by (a) NSGA-II, (b) SPEA2, (c) NSPSO, and (d) c-MOPSO. Operational frequency: 900 MHz. ....................................................................................109 Fig. 4-22 Optimal performances obtained by minimizing the maximum of |S11| j. xi.

(18) Operational bands: 698–960 MHz and 1710–2170 MHz. ............................112 Fig. 4-23 Optimal performances obtained by minimizing the sum of |S11|j,dB. Operational bands: 698–960 MHz and 1710–2170 MHz. ................................................114 Fig. 4-24 Functional behaviors of (a) |S11|k and (b) logk|S11| with respect to |S11|. ........115 Fig. 4-25 Optimal performances obtained by minimizing the sum of |S11|jk. Operational band: 700–960 MHz. ....................................................................................116 Fig. 4-26 Optimal performances obtained by minimizing the sum of (logk|S11|)j. Operational band: 700–960 MHz. ................................................................116 Fig. 4-27 Topology of the optimum antenna structure for wide- and dual-band operation. ......................................................................................................119 Fig. 4-28 Comparison of the optimal performances obtained by the proposed method and the minimization of the maximum of |S11|j. Operational bands: 698–960 MHz and 1710–2170 MHz. ..........................................................................119 Fig. 4-29 The candidate pixels for second-stage optimization. ....................................120 Fig. 4-30 Photograph of the optimum antenna structure for wide- and dual-band operation. ......................................................................................................121 Fig. 4-31 Simulated and measured reflection coefficients of the optimum antenna structure. Operational bands: 698–960 MHz and 1710–2170 MHz. ............121. Chapter 5. A Self-Structuring Electromagnetic Scatterer. 123. Fig. 5-1 Block diagram of the SSES system. .............................................................125 Fig. 5-2 Simulated scattered phase in the observation direction θobs = 0° versus strip lengths for normal (θin = 0°) and oblique (θin = 30°) incidence. ...................126 Fig. 5-3 (a) Top view of the proposed SSES template and (b) enlarged side view of switch connections. .......................................................................................127 Fig. 5-4 Simulated reflected phase versus strip length for various operational frequency. ......................................................................................................128 Fig. 5-5 Photographs of the SSES prototype. (a) Top and (b) bottom views. ............133 Fig. 5-6 System setup for measuring the bistatic RCS (pattern) of SSES. ................134 Fig. 5-7 Bistatic RCS patterns measured in x–z plane for (a) normal (θin = 0°) and (b) oblique (θin = 30°) incidence, respectively. ...................................................135 Fig. 5-8 Bistatic RCS patterns of the SSES with its RCS minimized in various objective angles for normal incidence (θin = 0°). ..........................................137 xii.

(19) Fig. 5-9 Bistatic RCS patterns of the SSES with its RCS minimized in various objective angles for oblique incidence (θin = 30°). .......................................138 Fig. 5-10 Bistatic RCS patterns of the SSES with its RCS minimized in various objective angles for normal incidence (θin = 0°). (a) θopt = 90°, (b) θopt = 70°, (c) θopt = 50°, (d) θopt = 30°, (e) θopt = 10°, and (f) θopt = 0°. ...............................140 Fig. 5-11 Bistatic RCS patterns of the SSES with its RCS minimized in various objective angles for oblique incidence (θin = 30°). (a) θopt = 90°, (b) θopt = 60°, (c) θopt = 30°, (d) θopt = 0°, (e) θopt = –20°, (f) θopt = –30°, (g) θopt = –40°, (h) θopt = –60°, and (i) θopt = –90°. ......................................................................142 Fig. 5-12 Bistatic RCS patterns of the SSES with its RCS maximized in various objective angles for normal incidence (θin = 0°). (a) θopt = 0°, (b) θopt = 5°, (c) θopt = 10°, (d) θopt = 15°, and (e) θopt = 20°. ...................................................145 Fig. 5-13 Bistatic RCS patterns of the SSES with its RCS maximized in various objective angles for oblique incidence (θin = 30°). (a) θopt = –30°, (b) θopt = –15°, (c) θopt = 0°, (d) θopt= 15°, and (e) θopt = 30°. ...................................146. xiii.

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(21) LIST OF TABLES Chapter 1 Table 1.1. Introduction. 1. Characteristics comparison between evolutionary algorithms and design of experiments. ...................................................................................................4. Chapter 2. Overview of Optimization Methods in Electromagnetic Problems. 7. Table 2.1. Ranges for the input geometric parameters (mm). ......................................34. Table 2.2. 25 full factorial experiment (at 915 MHz). ..................................................35. Table 2.3. ANOVA table for the 25 experiment of Ra. ..................................................36. Table 2.4. ANOVA table for the 25 experiment of Xa. ..................................................36. Table 2.5. Optimization results for the 25 experiment. .................................................40. Table 2.6. Resolution-V 25–1 fractional factorial experiment (at 915 MHz). ................41. Table 2.7. ANOVA table for the 25–1 experiment of Ra. ...............................................42. Table 2.8. ANOVA table for the 25–1 experiment of Xa. ...............................................42. Table 2.9. Resolution-III 25–2 fractional factorial experiment (at 915 MHz). ..............44. Table 2.10 ANOVA table for the 25–2 experiment of Ra. ...............................................44 Table 2.11 ANOVA table for the 25–2 experiment of Xa. ...............................................44 Table 2.12 Performance comparisons between the three experimental designs (at 915 MHz). ...........................................................................................................45. Chapter 3. A Novel Dual-Antenna Structure for Passive UHF RFID Tags. 47. Table 3.1. Ranges for the input geometric parameters (mm). ......................................56. Table 3.2. 26 full factorial experiment (at 915 MHz). ..................................................57. Table 3.3. ANOVA table for the 26 experiment of Rre. .................................................59. Table 3.4. ANOVA table for the 26 experiment of Xre. .................................................59. Table 3.5. ANOVA table for the 26 experiment of Xsc. .................................................60. Table 3.6. ANOVA table for the 26 experiment of |S21|. ...............................................60. Table 3.7. Resolution-VI 26–1 fractional factorial experiment (at 915 MHz). ..............62. Table 3.8. ANOVA table for the resolution-VI 26–1 experiment of Rre. ........................63. Table 3.9. ANOVA table for the resolution-VI 26–1 experiment of Xre. ........................63. Table 3.10 ANOVA table for the resolution-VI 26–1 experiment of Xsc. ........................63 Table 3.11 ANOVA table for the resolution-VI 26–1 experiment of |S21|. .......................64 xv.

(22) Table 3.12 Resolution-IV 26–2 fractional factorial experiment (at 915 MHz). ..............66 Table 3.13 ANOVA table for the resolution-IV 26–2 experiment of Rre. ........................67 Table 3.14 ANOVA table for the resolution-IV 26–2 experiment of Xre. ........................67 Table 3.15 ANOVA table for the resolution-IV 26–2 experiment of Xsc. ........................67 Table 3.16 ANOVA table for the resolution-IV 26–2 experiment of |S21|. .......................68 Table 3.17 Performance comparisons between the three experimental designs (at 915 MHz). ...........................................................................................................71 Table 3.18 Simulated and measured backscattering RCS at 915 MHz. ........................82. Chapter 4. A Competent Pixelized Design Tool and Wide- and Multi-Band Antenna Designs. 85. Table 4.1. Parameters employed in NSGA-II. ............................................................107. Table 4.2. Parameters employed in SPEA2. ...............................................................107. Table 4.3. Parameters employed in NSPSO. ..............................................................107. Table 4.4. Parameters employed in c-MOPSO. ..........................................................108. Table 4.5. Parameters employed in the simple GA. ...................................................112. Table 4.6. Sample frequencies of each objective (MHz). ...........................................118. Table 4.7. Parameters employed in the multiobjective-based GA for wide- and multi-band antenna designs. ......................................................................118. Chapter 5. A Self-Structuring Electromagnetic Scatterer. 123. Table 5.1. Results of the SSES prototype for RCS reduction when θin = 0°. .............137. Table 5.2. Results of the SSES prototype for RCS reduction when θin = 30°. ...........139. Table 5.3. Results of the SSES prototype for RCS enhancement when θin = 0°. .......144. Table 5.4. Results of the SSES prototype for RCS enhancement when θin = 30°. .....144. xvi.

(23) CHAPTER 1. Introduction 1.1. RESEARCH MOTIVATION Many practical electromagnetic (EM) problems, such as antenna design, mutual coupling reduction, or synthesis of reconfigurable microwave devices, can be generalized into a relation between response variables and input factors. Mostly, engineers study how the response variables are affected by the input factors and then determine the values for input factors so that the response variables can provide the desired performance. To obtain a satisfactory result, two strategies have been extensively used in the EM fields. The fist one is the trial-and-error approach. That is, engineers select an arbitrary combination of input factors and examine if the resultant performance meets the specifications. If not, the combination is changed based on the engineers’ intuition or experience, and the process is repeated until the performance criterion is met. However, such an approach may take a long time and without any guarantee of success; moreover, when the process is terminated, there is no guarantee that the best levels of input factors are found. The other strategy is the one-factor-at-a-time tuning, also known as the parameter study. That is, a baseline set of values for each parameter is first assigned, and then engineers repeatedly scan each parameter over its range with all other parameters held constant at the baseline value. After all the parameter tunings are respectively conducted, a series of figures are constructed showing how they help achieve desirable performance or how the responses are affected by these parameters. However, such an approach merely represents a subjective opinion because it offers no validity that the responses would hold unchanged as the other parameters are set at different levels. In particular, as the EM problems encountered become more and more complicated, such as multiple design goals to be considered simultaneously, the conventional approaches, namely the trail-and-error approach and the parameter study, usually provide very poor results. To overcome these difficulties, optimization methodologies are developed. By definition, an optimization technique is a step-by-step procedure for obtaining the best 1.

(24) result under the given constraints [1]. It shows three significant gains over the conventional approaches. First, an optimization methodology is a systematic procedure. It models a problem as mathematical expressions and gives unambiguous instructions to solve them, enabling the solving process in quick and inexpensive manner. Second, an optimization methodology ensures that the best solution, which has been found, is a global optimum, or at least a local optimum. In contrast, with the conventional approaches, engineers stop tuning merely at an acceptable solution. Lastly, and most important of all, the intelligence of optimization techniques helps engineers develop sophisticated and powerful applications. With the conventional approaches, engineers are only capable of handling easy design tasks with a limited number of objectives; however, with the rapid development in EM applications, the requirements of compact size, low cost, high robustness, reconfigurable operation, and diverse functionality become more and more indispensable. Only by intellectually dealing with every detail in system level, innovative applications with more sophisticated competence can be realized. And this is what this dissertation is devoted to. With the help of two optimization techniques, including design of experiments (DOE) and evolutionary algorithms, this dissertation proposes three innovative and feasible applications showing significantly improved efficiency than conventional methods.. 1.2. GENERAL CONCEPTS OF EVOLUTIONARY ALGORITHMS AND DESIGN OF EXPERIMENTS Since all the applications proposed in this dissertation employ either evolutionary algorithms or DOE, their general concepts are provided and compared in this section. Evolutionary algorithms, also known as evolutionary computation methods, are a field of problem solving techniques inspired from the process of natural evolution. The original definition of evolutionary algorithms is a stochastic adaption procedure based on the mechanism of natural genetics and natural selection [2]. That is, in order to solve a problem, a population of candidate solutions is formed and their performances are tested; after evaluating their performance measures, called fitness, the candidate solutions with larger fitness have more odds to reproduce themselves in this problem. At early stages, evolutionary algorithms only consider evolutionary programming (EP), evolutionary strategy (ES), genetic algorithm (GA), and genetic programming (GP) as 2.

(25) subareas belonging to the associated algorithm variants. Recently, the definition of evolutionary algorithms has been relaxed as a stochastic procedure that launches a population of candidate solutions and recollects each other’s information to generate a new promising one [3]. Different individual and collective processes produce different techniques, such as particle swarm optimization (PSO), ant-colony optimization (ACO), and learning classifier system (LCS), etc. Among them, GAs and PSO are the most commonly-used methods in the EM fields due to their robustness and efficiency. These two methods are adopted in this dissertation to develop an automatic antenna design tool and to synthesize an innovative reconfigurable device. On the other hand, DOE possesses a completely different framework from evolutionary algorithms. Table 1.1 compares the characteristics of evolutionary algorithms and DOE. Evolutionary algorithms treat the problem to be solved as a black box; no matter what the problem it is, they just launch a population of solutions, searching the solution domain by executing specific operators until a termination criterion is met. In contrast, DOE attempts to uncover the black box by approximating the problem by mathematical models. By performing the predesigned treatment combinations of input factors, called the experiment, DOE builds response surface models by statistically analyzing the associated experimental results. If the models are inferred as proper expressions, numerical optimization techniques can be applied to solve them effortlessly. In other words, the main computational effort of DOE is not focused on iteratively “searching” the entire solution domain, but in contrast on sequentially “building” the interested solution sub-domain. The other discrepancy between DOE and evolutionary algorithms is that the former investigates the problem specifics. During the process of estimating the “effects” of input factors on the response variables, additional knowledge of EM phenomenon can be obtained; engineers may identify the influential factors and screen out those insignificant ones to the next-stage analysis. In contrast, not until good-enough solutions have been found, evolutionary algorithms perform a blind search and treat every input factor as equally important. However, blind search is not necessarily a disadvantage in the optimization process. Although investigating problem specifics helps engineers determine operational policy at the next stage, less human intervention results in less human bias. In sum, DOE helps gain a thorough understanding of all the factors involved in the problem. It is applied in this dissertation to design a complex antenna 3.

(26) Table 1.1 Characteristics comparison between evolutionary algorithms and design of experiments Characteristics. Evolutionary algorithms. Design of experiments (DOE). Optimization approach. Treat a problem as black box and blind search for solutions. Investigate a problem structure and uncover the black box. Computational effort. Search the solution space. Build the solution sub-space. Operational policy. Every decision variable is treated as equally important. Differentiate the significance between decision variables. Human intervention. Less human bias. More human interpretation. structure for passive radio-frequency identification (RFID) tags and to synthesize an innovative reconfigurable device.. 1.3. CHAPTER OUTLINES The focus of this dissertation is on the development of innovative and feasible EM applications by means of the intelligence of optimization methods. The associated applications and the corresponding methodologies are presented in the remaining chapters, and they are organized as follows: In Chapter 2, we provide a conceptual overview of different optimization schemes employed in this dissertation. The optimization methods cover DOE and evolutionary algorithms, including GAs and PSO as well as their multi-objective versions. Since evolutionary algorithms have been widely discussed in the literatures, we describe DOE in more detail, and a design example of RFID tag antenna is selected to demonstrate its capability. In Chapter 3, a novel dual-antenna structure for passive RFID tags is presented. Such a dual-antenna structure has several design considerations, which aim to simultaneously optimize the power reception and the level difference of signal backscattering. DOE is adopted to efficiently achieve the multiple design goals. Simulated and measured results are also provided to validate the effectiveness of the proposed antenna structure. In Chapter 4, a competent pixelized design tool for practical design situations is 4.

(27) presented. Evolutionary algorithms are implemented as the kernel of this pixelized design tool, which are capable of handling single- and multiple-objective design tasks. Also, an innovative approach for wide- and multi-band antenna designs is presented, and the experimental results are also provided to verify the associated effectiveness. In Chapter 5, a novel self-structuring electromagnetic scatterer (SSES) is presented. The SSES is able to adapt its electrical configurations to new environment/objective for reconfigurable radar cross section (RCS) control. To search for an associated optimum configuration, a novel method based on the fractional factorial design of experiments (FFD) is adopted as the searching algorithm. A prototype SSES was implemented and a series of bistatic RCS measurements were performed to validate the concepts of adaptive RCS control. The experimental results for RCS reduction and RCS enhancement are also provided. Finally, a summary and several future research directions are presented in Chapter 6.. 5.

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(29) CHAPTER 2. Overview of Optimization Methods in Electromagnetic Problems 2.1 INTRODUCTION Instead of continually finding solutions from physical domain, an optimization method is a well-organized procedure, which solves a physical problem by modeling it into mathematical expressions:. Find : x∗ = arg min Fm ( x ) x. ⎧⎪ gi ( x ) ≤ 0, i = 1, 2,..., p Subject to : ⎨ ⎪⎩ l j ( x ) = 0, j = 1, 2,..., q. (2.1). where x is the N-vector of decision variables, Fm(x) are objective functions, gi(x) are inequality constraints, and lj(x) are equality constraints. The decision variables are the input factors being investigated, which by nature can be continuous or discrete. The objective functions are the response variables of the problem being solved, and they are functions of decision variables. The constraints are restrictions that must be satisfied in order to obtain a solution as good as we can. Before elaborating on various optimization schemes for solving (2.1) modeled from EM problems, it is important that the problem nature must be investigated first. This is because various distinct optimization methods have been developed to tackle problems with extremely diverse natures. Only after the problem specifics are properly addressed, one could declare that a particular algorithm is more efficient than a random search method [4]. There are two important features for general EM problems. First, the objective functions, for instance, the input impedance of antennas, isolation between multiple antennas, and/or the scattering properties of a scatterer, are usually non-convex functions. It makes conventional mathematical programming methods, which evolve a solution by collecting neighbor information around one single candidate solution with deterministic rules, very easy to be trapped into a local optimality. The other feature is 7.

(30) that it is difficult to formulate objective functions into closed-form expressions in terms of decision variables. This raises difficulty to derive the derivative or second-order differentiation with respect to the decision variables, which is necessity for conventional gradient methods and second-order methods [5]. Consequently, the mathematical programming methods are short of correctness and robustness when handling EM problems. Considering the nature of EM problem, the evolutionary algorithms such as GAs and PSO are well suited due to their population-based heuristics and non-deterministic searching. In addition, DOE has also been found to have high capability for handling EM problems. In this chapter, details of algorithm implementation and their main features are provided. Since the GA and PSO routines have been well-described in literatures [6–10], less description is presented for them than for DOE, which will be thoroughly reported in Section 2.4. Also, an antenna design problem for passive RFID tags is used as an example to demonstrate the full operation and capability of DOE.. 2.2 GENETIC ALGORITHMS Being the most famous optimization method for engineering problems, GAs have arisen great concern for decades. GAs were first introduced by Holland in 1975 [11]. In 1983, Goldberg proposed the first GA application, where he optimized the distribution problem of gas by pipeline [12]. Later on, not only the applications of GAs but also their fundamental theorems are extensively published. While other evolutionary algorithms, such as PSO and ACO, are still employed based on empirical rules, the underlying theorems of GAs have been established and mathematically shown that the quality of solutions in subsequent iterations will indeed improve [2]. Also, GAs are the most widely-applied evolutionary algorithm. They have been extensively applied to various fields, such as optimal designs, control engineering, decision analysis, image registrations, even the design of Shinkansen in Japan, and so forth. The concept of GAs is based on Darwin’s principle of survival of the fittest. Just as creatures evolve themselves to fit an unknown environment, GAs attempt to evolve a solution that yields the optimum value with an unknown function. That is, GAs handle a problem in a black-box manner, despite the type of the problem is unaware. By naturally selecting the solutions fittest in the problem and exchanging the information between them, it forms a canonical search procedure that is capable of addressing a 8.

(31) broad spectrum of problems. In contrast to deterministic local-search techniques, which iteratively move from one current solution to a neighbor solution until the solution improve no more, GAs belong to the class of stochastic population-based techniques. That is, GAs search from a population of solutions, instead of a single one. In addition to the individual process, where each solution individually evolved according to certain operators, a more important feature of population-based techniques is the collective process: the solutions of the current population are compared and combined to generate new ones. Also, GAs use randomized operators so that they are more capable of detecting the local optimum solutions for non-convex problems. In addition, GAs are zeroth order method and hence they do not require gradient computations. The problems, in which first-order or second-order gradient computations are difficult to implement, can be easily casted into the GA framework. These features render GAs attractive for many EM problems [13]. In the last decade, many branches have emerged in the GA area that vary from the Holland’s early idea of genetic plans, and they are denoted as “competent GAs” [3]. Although competent GAs, such as dependency structure matrix genetic algorithm (DSMGA) [14] and hierarchical Bayesian optimization algorithm (hBOA) [15], perform satisfactorily because they identify and preserve the interacting bits. However, in the EM fields, simple GAs are still predominant over all GA applications. Here the term “simple GAs” refers to the class of GAs, which involve no problem structure learning [14]. Throughout this dissertation, it is the simple GAs that we used in all the proposed applications. The implementation details are provided in the following.. 2.2.1. Simple GA Mechanisms. Fig. 2-1 depicts the flow chart of a simple GA. The process begins by coding the decision variables into a binary bit stream, called chromosome, and determining the initial population of chromosomes. Each chromosome of the population owns a fitness value, which represents the performance index for the given decision variables. Fitness is evaluated from a specified objective function, which is the link between the optimization algorithm and the physical problem. The evaluated fitness is the figure of merit for performing selection. This is the survival-of-the-fittest step, where we generate more copies to the chromosomes having better fitness within a fixed population size. By emphasizing the influence of the better 9.

(32) Initialize the process (Assign parameters). Evaluate fitness. Perform selection. Perform crossover. Perform mutation. Evaluate fitness No Replace population. Iteration > Nite? Yes Stop. Fig. 2-1 Flow chart describing a simple GA.. individuals and searching in their neighborhood via the succeeding operators, GAs are capable of exploiting the existing promising sub-domain. The degree of exploitation is controlled by selection pressure s, which means the ratio of the number of the best individual’s copy to the average number of all individuals’ copies. The larger the selection pressure, the more thoroughly the current information is used, but the less the effort of exploring the global solution domain is spent. There are diverse categories of the selection operators, such as Roulette-Wheel selection [11], truncation selection, stochastic remainder selection (SRS), and stochastic universal selection (SUS). We implemented the tournament selection without replacement in all of the proposed applications because this operator is less noisy and time saving. Such a procedure 10.

(33) randomly picks s chromosomes into a tournament, selecting the best one and put it into the mating pool. The other losers are dropped, and the process repeats until the mating pool is filled. Next, the promising chromosomes in the mating pool are recombined. Such a process is called crossover, which intends to generate new chromosomes by combing potential bits. The m-point crossover operator is executed by first randomly selecting two individuals as parents, then, m cross sites are randomly selected and cut, resulting in m + 1 pieces for each parent. Afterward, the new offspring are generated by a probability pc through swapping the odd pieces while retaining the even pieces from each parent. In other words, it is of probability 1 – pc that the chromosomes of the offspring are identical to those of the parents. Another category of crossover operator is uniform crossover, which is implemented by means of swapping every bit of parents independently by a probability pc. Once the offspring are generated, each bit is flipped by a low probability pm. Such a mutation operator aims to explore the solution sub-domains outside the span of the current population. The resultant set of the chromosomes becomes a new population. However, before repeating the procedure until the maximum iteration number Nite is reached, note that the best individual in the new population might be worse than that in the preceding iteration due to the stochastic nature of the genetic operators. Therefore, before performing selection again, an elitist strategy is usually added when replacing the population. By saving and inserting the top Nelit individuals from the previous iteration, it is for sure that there is a monotonic improvement in the best fitness for every succeeding iteration. It has been shown that using the simple GA procedure, the quality of solutions in subsequent iteration will improve by Holland’s theory of schemata [2]. Also, Goldberg has, via analysis of the mixing boundary, the drift boundary, the schema theorem, and the cross competition, expressed the sweet spot for s and pc rendering simple GAs to converge successfully [16]. However, such a sweet spot vanishes as problem difficulty grows, and the population size must expand exponentially to retain convergence. This is because simple GAs treat every bit in every chromosome as equal weight. It recognizes no information about which group of interacting bits, called building blocks, should be kept intact [14]. Instead, simple GAs monotonously perform the genetic operators in stochastic manners, so building blocks may be disrupted over a long period of iteration, 11.

(34) leading to the waste of computational resources. Therefore, the expense of robustness is usually a large number of functional evaluations, making simple GAs computationally intensive and time-consuming.. 2.2.2. Multiobjective GA. In practical engineering problems, it is often desired to find a combination of decision variables that compromise between more than one response variables, which results in multiobjective optimization problems. However, selection operators can only compare one single fitness, so simple GAs can only handle one objective function. Intuitively, multiple fitness components can be weighted and summed up, leading back to a single-objective problem. Although it is simple, however, the determination of the weighting coefficients is troublesome. Also, the validity of the linear-combination operation is difficult to be justified. Another multiobjective approach is Pareto optimization [17], which was proposed by a French engineer (V. Pareto, 1848–1923). He suggested that instead of obtaining one single best solution, the optimum solutions for a multiobjective optimization problem should be a set of solutions whose fitness components are not all worse than other solutions’ components. In other words, if a solution cannot be dominated in all fitness components by other solutions, the solution is regarded as a Pareto optimum. Accordingly, the set of all nondominated solutions is called the Pareto front, and the approach that finds the Pareto front is called the Pareto optimization. Two multiobjective GAs based on Pareto optimization are adopted in this dissertation. The first one is nondominated sorting genetic algorithm-II (NSGA-II) [18]. NSGA-II possesses a similar infrastructure to simple GAs. It evolves better solutions by those genetic operators, but the operation of selection operator and elitist replacement operator are modified. Since each solution has multiple fitness components, the selection operator in NSGA-II is applied by the overall rank information rather than the individual fitness value. More specifically, the set of solutions in the objective space which are not dominated by any individual is identified as the first rank. After discarding the rank-1 solutions, the subsequent nondominated set is thus identified as the second rank, and the procedure continues until all the solutions are classified. Such rank information is utilized to determine which solution survives the tournament selection. Also, to break the tie for two solutions being in the same rank, the NSGA-II 12.

(35) stipulates the secondary measure, namely crowding distance, for each solution. The crowding distance of a solution is assigned as a distance value equal to the absolute normalized difference of its two adjacent solutions. The solution with larger crowding distance will win the selection competition since it is more representative. Therefore, the quality of each solution is expressed by its primary rank information and secondary crowding distance, and selection can be performed accordingly. On the other hand, the elitist replacement operator in NSGA-II is performed by retaining Nelit = Npop; that is, after evaluating the fitness’s of the offspring, the offspring population are combined with the parent population, and the top Npop solutions according to the rank information and crowding distance are selected subsequently. The other multiobjective GA being implemented is the strength Pareto evolutionary algorithm 2 (SPEA2) [19]. SPEA2 also possesses the framework of simple GAs, merely re-evaluating the overall fitness of a solution. The overall fitness of the i-th solution F(i) is defined by first calculating the number of solutions it dominates, denoted as S(i); afterward, the raw fitness of the i-th solution R(i) is assigned by summing up S(j), where j is the solution that dominates i. Clearly, the smaller R(i), the better the i-th solution is. Note that for nondominated solutions their R(i) should be zero. Besides R(i), another index D(i) for density estimation is also applied. D(i) is defined by 1⁄. 2 , where d(i) is the distance to the kth nearest solution. Finally, the overall. fitness F(i) is computed by F(i) = D(i) + R(i). Therefore, multiple fitness components are summarized into one, and the simple GA routine can be thus applied.. 2.3 PARTICLE SWARM OPTIMIZATION Recently, another popular evolutionary algorithm is the PSO algorithm. PSO also possesses a stochastic population-based framework but has far fewer categories of operators and parameters than simple GAs, making them relatively simple in implementation. The idea of PSO is derived from a swarm-intelligence concept. The swarm intelligence means that despite how complex an individual is, when there are plenty of individuals gathering together into an organization, they will abide by some simple patterns and alter its behavior to adapt to the environment, making the organization as well as each individual become more accommodating to the situation. For example, it is generally believed that every human has his/her own free will, and often we will assume 13.

(36) that all the economical activities are created by reasonable thought of every individual participant. However, during the middle of 1980s, it is observed that people may not always take each act by their own logical reasoning; instead, they take a little step by customs, seeing what consequences occur and what accomplishment other people achieve, then regulating themselves toward the next step. Interactions among people produce self reasoning and social learning; thus, people adapt themselves into a better situation by the course of evolution [20–22]. Such a concept is adopted and transformed into an optimization algorithm by Kennedy in 1995 [23]. A population of individual, called particle, is modeled in the PSO algorithm, and each explores the solution domain by incorporating the information of other particles with self experiences, step-by-step regulating their searching directions to approach the optimum solution. The movement of a particle is realized by a Newtonian mechanics model, where the position of a particle represents a candidate solution, and the imposed forces which result in acceleration represent the attraction of self organization and social learning. The population-based and extensively-search characteristics make PSO suitable for handling non-convex EM problems, and it has been successfully applied to various EM problems, such as pixelized antenna designs, wide-band antenna designs [24], pattern synthesis of linear array [25], reflector antenna shaping [26], and EM absorber designs [27]. In this section, we start from the description of the PSO implementation, and then provide its extensions that are used in this dissertation, including a binary version and multiobjective versions.. 2.3.1. PSO Mechanisms. Fig. 2-2 depicts the flow chart of the PSO algorithm. The algorithm starts from initialization of the process, including establishing a feasible solution space, defining an objective function, and assigning the PSO parameters. Conventional PSO is only suitable for real-number problems; therefore, the N decision variables are expressed into an N-digit real-number stream. A population of Npop real-number streams, which represent the positions of particles, are randomly generated and expressed as matrix-form X(0). In order to search the solution space, we assign another randomly-generated parameter, namely velocity V(0), for the population of particles to vary their positions. Next, each particle possesses a fitness, which is evaluated from the objective 14.

(37) Initialize the process (Assign parameters). Evaluate fitness. Stop Yes. Update locations of personal best and global best. Iteration > Nite? No. Update velocities by (2.1). Update positions by (2.2). Fig. 2-2 Flow chart describing PSO.. function. To analogize the self reasoning of humans, the location with the best fitness discovered by an individual particle is recorded as the personal best position. Also, to analogize the social learning, the location with the best fitness discovered by the entire population is recoded as the global best position. Each particle is attracted by the forces of self reasoning and social learning, thereby intending to move toward a direction which comprises the weighted sum of these two positions. Consequently, the particle’s velocity is updated by. (. ). (. V (i +1) = c0 V (i ) + c1r1 P (i ) − X(i ) + c2 r2 G ( i ) − X( i ). ). (2.1). where P(i) and G(i) are the matrices collected by the personal best locations of each particle and the global best location of all particles found at the ith iteration, respectively. At the first iteration, or i = 1, P(1) = X(1). Also, G(i) is always a rank-one matrix where the global best location is replicated for Npop times. Note that P(i) and G(i) will be updated iteration-by-iteration, analogizing that a human takes his/her act step-by-step. Parameters c0, c1, and c2 are the weighting constants of old velocity V(i), personal best locations P(i), and global best location G(i), respectively. In order to introduce stochastic 15.

(38) characteristics to the algorithm, c1 and c2 are multiplied by two random numbers uniformly distributed in [0, 1], namely r1 and r2. In addition, every element of V(i) is restricted within a given limit [–vmax, vmax] so that the velocities are acceptable. Once the velocity has been determined, the position of particle is updated by. X(. i +1). = X( ) + V( i. i +1). (2.2). After the position is updated, the fitness of each particle can be evaluated, and the personal best locations as well as the global best location can be reassigned. Such a procedure is repeated until a termination criterion is met. Throughout this dissertation, the termination criterion of PSO is that the maximum iteration number Nite is reached. Much work has been proposed for the selection of parameter values [28–30]. The general idea behind them is to have PSO explore the entire solution space as one could at the beginning and exploit the promising sub-region as thoroughly as possible at the end, so PSO may has more odds to converge to a global optimum efficiently. In particular, linearly varying c0 from 0.9 to 0.4 is suggested in [28]. It is proposed in [29] that once velocities are updated, they should be multiplied by a constriction factor K. Linearly decreasing c1 from 3.5 to 0.5 while linearly increasing c2 from 0.5 to 3.5 is suggested in [30]. However, the selection of parameter values is problem-dependent. It is not practical to determine the best values for all problems; instead, it is critical to understand the problem nature so that engineers know how to coordinate the degree of exploration and exploitation by adjusting these parameters. We can see that the number of parameters and the categories of operators in PSO are indeed fewer than those of GAs. Some literatures claim that this is the predominance of PSO; however, the larger number of useful parameters may provide much diversity to modulate the searching capabilities for wide-range problems, while the smaller number of parameters makes it difficult to look after both the convergence rate and the quality of attained solutions for every problem. For example, PSO usually suffers from premature convergence to a local optimum as it handles a complex problem. Therefore, it is meaningless to argue which algorithm is better; instead, it is important to investigate the problem complexity and choose the most suitable algorithm.. 2.3.2. Binary Particle Swarm Optimization 16.

(39) PSO was first proposed for solving continuous problems. In 1997, Kennedy and Eberhart proposed a binary version using discrete or binary decision variables [31]. Fig. 2-3 depicts the flow chart of binary particle swarm optimization (BPSO). BPSO process starts from randomly generated X(0) and V(0), formed by binary bit streams and real numbers, respectively. While the equation (2.1) remains unchanged, the updated velocities are readily applied with the Sigmund transformation by. (. ). S V(i +1) =. 1. (2.3). ( i+1). 1 + e− V. Now the value of S(V(i+1)) is restricted to the range [0, 1] and can be regarded as a probability that each bit takes a value of 1 or 0. Then, Npop × N random numbers that uniformly distributed in [0, 1] are generated and respectively compared to each element in S(V(i+1)). The position updating rule is expressed by. X. ( i +1). ⎧1 if rand ( ⎪ =⎨ ⎪0 if rand ( ⎩. ( )) ) ≥ S ( V( ) ). ). < S V(. i +1. i +1. (2.4). where rand( ) is a random number with a uniform distribution in [0, 1]. Therefore, the position of particle is confined to either 0 or 1. Also, it is shown that the probability interpretation provides validity for determining which value the bits should be [10].. 2.3.3. Multiobjective PSO. PSO has also been applied to multiobjective domain. Two Pareto-based multiobjective PSO algorithms are implemented in this dissertation. The first algorithm is nondominated sorting PSO (NSPSO) [32]. NSPSO only modified the definition of the global best location G(i) and the personal best locations P(i), while all other operations remain the same as the PSO routine. For all the solutions in the global objective space, their fitness components are reassigned by primarily the rank information and secondarily the crowding distance, just as the nondominated sorting procedure in NSGA-II. Once we obtain the sorted list where all the solutions are sorted in descending order, the global best position for particle i is randomly assigned from the position on the top M of the list. Note that the particles may have distinct global “bests”, so the 17.

(40) Initialization (Assign parameters). Evaluate fitness. Stop Yes. Update locations of personal best and global best. Iteration > Nite? No. Update velocities by (2.1). Transform the velocities by (2.3). Update positions by (2.4). Fig. 2-3 Flow chart describing BPSO.. global best locations G(i) is no longer always a rank-one matrix. As for the personal best locations, they are randomly assigned from one of the nondominated solutions in each individual objective space. With such modified G(i) and P(i), the velocities and positions of particles can be updated by (2.1) and (2.2), respectively, and the PSO routine is thus performed. The other multiobjective PSO is composite point based multiobjective PSO (c-MOPSO) [33]. c-MOPSO also utilizes (2.1) and (2.2) to update the velocity and position of particles, respectively, but both the definitions of global best position and personal best position are changed. c-MOPSO divides the global objective space into several sub-regions formed by composite points. The composite point ci is constructed by taking the maximum value of the ith objective of each nondominated solution. If a solution has at least one of its fitness components larger than the corresponding component of ci, it is assigned as in the sub-region i. Therefore, each solution pj belongs to a particular sub-region, and the global best of pj is assigned as the nondominated 18.

(41) solution within the same sub-region. On the other hand, the personal best location of each particle is randomly selected from the nondominated solution set in its individual objective space. As a result, the customary PSO routine can be thus performed.. 2.4 DESIGN OF EXPERIMENTS DOE is a statistical optimization method with a rich history [34]. In the 1920s, Fisher introduced the concept of data analysis from systematically designing the treatment combinations of input factors. He developed the two fundamental techniques of DOE, including the full factorial design of experiments and the analysis of variance (ANOVA). At that time, the applications of DOE are primarily focused on agricultural systems. In the 1950s, DOE spread to the chemical and the process industries, where Box developed a very efficient experimental strategy that can learn crucial information from a small fraction of experiments, called the fractional factorial design of experiments. After that, DOE has been successfully applied to quality improvement in industry, medical treatment in therapy, risk assessment of chemical mixtures, and many other areas due to its effectiveness and efficiency. Yet, despite DOE has been extensively used, only a few applications have been done by the DOE optimization in the antenna community. In this section, we describe DOE in detail and cast its implementation into two phases: the design of experiments and the analysis of experiments. Before elaborating on the implementation of DOE, its specific terms and some pre-experimental planning must be introduced.. 2.4.1. Pre-Experimental Planning. In general, an EM problem can be casted into a system as represented in Fig. 2-4 [35]. The output responses y1, y2,…, yn are usually the questions to be answered, such as quantitative performance measures of the design process. They are either the goals being optimized or the important information being investigated. The input factors are classified into two categories. The first one is the explanatory factors x1, x2,…, xk, which are the decision variables actually selected for study. The other type is the noise factors ε1, ε2,…, εm. They are not necessary to represent only unobservable random errors. Any factor, whose effect on the response variables should be kept as small as possible, can be regarded as the noise factor. 19.

(42) x1 x2. …. System. …. Explanatory factors. y1 y2. xk. Response variables. yn. … ε1. ε2. εm. Noise factors. Fig. 2-4 General model of a system.. To quantify how influential the explanatory factors are, engineers purposefully vary them at distinct levels within specified ranges. The degree of a factor’s individual influence on a response variable is called the main effect Ei of the factor xi. Usually, the main effect Ei of xi does not hold constant when other factors alter their levels. The influence of xj (j ≠ i) on Ei is called the two-factor interaction Eij between them. Likewise, two-factor interaction Eij between xi and xj may change when other factor xl (l ≠ j ≠ i) is involved, and the resultant influence is called the three-factor Eijl between them. Similarly, other higher-order interactions between multiple factors can thus be defined. It is the major computational effort in DOE that estimating these effects as precise as possible. The above pre-experimental planning facilitates the following procedures. Since experiments are performed sequentially, these pre-experimental planning should be done with every detail before the DOE process is initiated.. 2.4.2. Phase 1: Design of Experiments. The specified level of each factor results in a treatment combination, and all of the treatment combinations form an experimental unit. Phase 1 focuses on how to allocate these treatment combinations. In general, an ideal experimental design provides information about the effects as accurate as possible based on a minimum number of experimental runs [36]. To obtain a good experimental design, several issues must be considered, such as the number of factors, complexity of the response surface, size of the experimental unit, computational resources at hand, and intentions [37]. Here, we provide three fundamental experimental designs and detail their potential applications. 20.