Particle Swarm Optimization (PSO) PSO is a robust stochastic optimization technique based on the movement and intelligence of swarms. PSO applies the concept of social interaction to problem solving. It was developed in 1995 by James Kennedy (social-psychologist) and Russell Eberhart (electrical engineer). It uses a number of agents (particles) that constitute a swarm moving around in the search space looking for the best solution. Each particle is treated as a point in a N-dimensional space which adjusts its “flying” according to its own flying experience as well as the flying experience of other particles.

Particle Swarm Optimization (PSO) Each particle keeps track of its coordinates in the solution space which are associated with the best solution (fitness) that has achieved so far by that particle. This value is called personal best , pbest. Another best value that is tracked by the PSO is the best value obtained so far by any particle in the neighborhood of that particle. This value is called gbest. The basic concept of PSO lies in accelerating each particle toward its pbest and the gbest locations, with a random weighted accelaration at each time step as shown in Fig.1

Particle Swarm Optimization (PSO) y x Fig.1 Concept of modification of a searching point by PSO sk : current searching point. sk+1: modified searching point. vk: current velocity. vk+1: modified velocity. vpbest : velocity based on pbest. vgbest : velocity based on gbest

Particle Swarm Optimization (PSO) Each particle tries to modify its position using the following information: the current positions, the current velocities, the distance between the current position and pbest, the distance between the current position and the gbest. The modification of the particle’s position can be mathematically modeled according the following equation : Vik+1 = wVik +c1 rand1(…) x (pbesti-sik) + c2 rand2(…) x (gbest-sik) ….. (1) where, vik : velocity of agent i at iteration k, w: weighting function, cj : weighting factor, rand : uniformly distributed random number between 0 and 1, sik : current position of agent i at iteration k, pbesti : pbest of agent i, gbest: gbest of the group.

Particle Swarm Optimization (PSO) The following weighting function is usually utilized in (1) w = wMax-[(wMax-wMin) x iter]/maxIter (2) where wMax= initial weight, wMin = final weight, maxIter = maximum iteration number, iter = current iteration number. sik+1 = sik + Vik+1 (3)

Particle Swarm Optimization (PSO) Comments on the Inertial weight factor: A large inertia weight (w) facilitates a global search while a small inertia weight facilitates a local search. By linearly decreasing the inertia weight from a relatively large value to a small value through the course of the PSO run gives the best PSO performance compared with fixed inertia weight settings. Larger w ----------- greater global search ability Smaller w ------------ greater local search ability.

Particle Swarm Optimization (PSO) Flow chart depicting the General PSO Algorithm: Start Initialize particles with random position and velocity vectors. For each particle’s position (p) evaluate fitness Loop until all particles exhaust If fitness(p) better than fitness(pbest) then pbest= p Loop until max iter Set best of pBests as gBest Update particles velocity (eq. 1) and position (eq. 3) Stop: giving gBest, optimal solution.

Comparison with other evolutionary computation techniques. Unlike in genetic algorithms, evolutionary programming and evolutionary strategies, in PSO, there is no selection operation. All particles in PSO are kept as members of the population through the course of the run PSO is the only algorithm that does not implement the survival of the fittest. No crossover operation in PSO. eq 1(b) resembles mutation in EP. In EP balance between the global and local search can be adjusted through the strategy parameter while in PSO the balance is achieved through the inertial weight factor (w) of eq. 1(a)

Variants of PSO Discrete PSO ……………… can handle discrete binary variables MINLP PSO………… can handle both discrete binary and continuous variables. Hybrid PSO…………. Utilizes basic mechanism of PSO and the natural selection mechanism, which is usually utilized by EC methods such as GAs.

Intialization parameters used for PSO: Application of PSO ALGORITHM to Optimize a Meander-line Polarizer for LI→CP conversion Intialization parameters used for PSO: wMax=0.41 wMin=0.4 (Note:The inertial weight ,w is linearly decreased from wMax to wMin according the Eq. (2), w is chosen virtually constant in this case for better local search near the Sun’s Optimized parameters. ) c1=c2=1.49 maxIter=2000 The above parameters are used in conjuction with eqs.(1) & (2) Swarm size/Population size used for solution search : 25

Application of PSO ALGORITHM to Optimize a Meander-line Polarizer for LI→CP conversion Frequency band of interest: 3.5 to 6.5 (GHz) (evaluated at 12 frequency points) Desired VSWR <= 1.2 Desired AR <= 0.5 (dB) Total number of fitness evaluations: 100025 Note: For my implementation of the PSO the number of fiteness evaluations are calculated as follows: (2 x swarmsize x maxIter)+ swarmsize = (2 x 25 x 2000)+ 25 The following slides include the results for the broadband case.

Mean best & Best fitness over 50 runs

VSWR

Axial Ratio (dB)

Optimized dimensions for 4-layer Meander Line Polarizer Spacer (inches) loi dielectric (inches) lei Line Width w1 w2 Height h Period b Pitch a Layer ---- 8.4705018E-03 1.5984001E-02 2.2501351E-02 0.2520565 0.7283382 0.3449360 4 0.5428128 3.4676325E-03 9.4296653E-03 5.3999661E-03 0.4707420 0.8704398 0.3798469 3 0.4211845 2 1 Dielectric constants: Dielectric Sheet Metal Layer Spacer 4 Layers for CP

Application of PSO ALGORITHM to Optimize a Meander-line Polarizer for LI→CP conversion Frequency bands of interest: Band1: 3.7 to 4.2 (GHz) Band2: 5.9 to 6.4 (GHz) (evaluated at 2 frequency points: 3.95 (GHz), 6.15 (GHz)) Desired VSWR <= 1.2 Desired AR <= 0.5 (dB) Total number of fitness evaluations: 100025 The following slides include the results for the dualband case.

Mean best & Best fitness over 50 runs

VSWR

Axial Ratio (dB)

Optimized dimensions for 4-layer Meander Line Polarizer Spacer (inches) loi dielectric (inches) lei Line Width w1 w2 Height h Period b Pitch a Layer ---- 4.1147252E-03 2.7704202E-02 4.0456183E-02 0.2480143 0.8081669 0.2528913 4 0.4533823 3.9593712E-03 4.4981677E-02 5.0358579E-02 0.3433722 0.9529658 0.5425724 3 0.4482195 2 1 Dielectric constants: Dielectric Sheet Metal Layer Spacer 4 Layers for CP

Intialization parameters used for PSO: Application of PSO ALGORITHM to Optimize a Meander-line Polarizer for LP rotation Intialization parameters used for PSO: wMax=0.41 wMin=0.4 (Note:The inertial weight ,w is linearly decreased from wMax to wMin according the Eq. (2), w is chosen virtually constant in this case for better local search near the Sun’s Optimized parameters.) c1=c2=1.3 maxIter=1000 The above parameters are used in conjuction with eqs.(1) & (2) Swarm size/Population size used for solution search : 25

Application of PSO ALGORITHM to Optimize a Meander-line Polarizer for LP rotation Frequency band of interest: 3.5 to 6.5 (GHz) (evaluated at 12 frequency points) Desired VSWR <= 1.2 Phase Difference around 180° Total number of fitness evaluations: 50025 The following slides include the results for the broadband case.

Mean best & Best fitness over 15 runs

VSWR

Axial Ratio (dB)

Phase Difference

Optimized dimensions for 8-layer Meander Line Polarizer Spacer (inches) loi dielectric (inches) lei Line Width w1 w2 Height h Period b Pitch a Layer ---- 2.1623570E-02 3.0709708E-02 2.8606838E-02 0.2148045 1.020600 0.3631878 4, 8 0.4550797 4.0660784E-02 3.4942929E-02 4.1542474E-02 0.4374999 0.8225765 0.3848170 3, 7 0.3959468 2, 6 1, 5 Dielectric constants: Dielectric Sheet Metal Layer Spacer 8 Layers for LP

Application of PSO ALGORITHM to Optimize a Meander-line Polarizer for LP rotation Frequency bands of interest: Band1: 3.7 to 4.2 (GHz) Band2: 5.9 to 6.4 (GHz) (evaluated at 2 frequency points: 3.95 (GHz), 6.15 (GHz)) Desired VSWR <= 1.2 Phase Difference around 180° Total number of fitness evaluations: 50025 The following slides include the results for the dualband case.

Mean best & Best fitness over 15 runs

VSWR

Axial Ratio (dB)

Phase Difference

Optimized dimensions for 8-layer Meander Line Polarizer Spacer (inches) loi dielectric (inches) lei Line Width w1 w2 Height h Period b Pitch a Layer ---- 2.2299249E-02 3.2010745E-02 3.3202391E-02 0.2747569 1.055596 0.3150869 4, 8 0.3544725 3.1127717E-02 0.1126298 6.0811251E-02 0.3565608 1.164576 0.4085801 3, 7 0.3159389 2, 6 1, 5 Dielectric constants: Dielectric Sheet Metal Layer Spacer 8 Layers for LP

END