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A six-element Yagi-Uda array is optimally designed using Central Force Optimization (CFO) with a small amount of pseudo randomly injected negative gravity. CFO is a simple, deterministic metaheuristic analogizing gravitational kinematics (motion of masses under the influence of gravity). It has been very effective in addressing a wide range of antenna and other problems and normally employs only positive gravity. With positive gravity the six element CFO-designed Yagi array described here exhibits excellent performance with respect to the objectives of impedance bandwidth and forward gain. This paper addresses the question of what happens when a small amount of negative gravity is injected into the CFO algorithm. Does doing so have any effect, beneficial, negative or neutral? In this particular case negative gravity improves CFO’s exploration and creates a region of optimality containing many designs that perform about as well as or better than the array discovered with only positive gravity. Without some negative gravity these array configurations are overlooked. This Yagi-Uda array design example suggests that antennas optimized or designed using deterministic CFO may well benefit by including a small amount of negative gravity, and that the negative gravity approach merits further study.

The Yagi-Uda array (Yagi) has been around for nearly one hundred years [_{i}). A typical six-element geometry is shown in _{i} are shorter, but these characteristics do not by any means constitute absolute design requirements, as they simply are typical geometry that produces appropriate phase and current relationships between the array elements.

A complete Yagi design must include the lengths and diameters of each element and the inter-element spacings. A six-element array thus has seventeen design parameters (six element lengths, six diameters, and five spacings). If the feedpoint input impedance is considered a design variable as well (discussed below), then the design/optimization (D/O) problem comprises eighteen dimensions. Many Yagis have far more than six elements, and every additional one adds three more design parameters (variables) to the problem.

There is no accurate analytical approach to Yagi array design using models that assume purely sinusoidal element current distributions and that neglect their mutual couplings. These assumptions lead to inaccuracies, and as a result many design approaches are inherently approximate or based entirely on empirical data [

Over the past two decades or so, in response to the limitations described above, Yagi D/O largely has been done using metaheuristics, that is, algorithms that provide acceptable solutions in reasonable times without being exact or having a

mathematical proof that the optimal solution eventually will be found [_{0} [

This paper reports results for a typical 6-element Yagi design developed with a basic Central Force Optimization (CFO) implementation that includes a small amount of pseudo randomly injected negative gravity. CFO is a deterministic evolutionary metaheuristic that analogizes gravitational kinematics (motion of masses under the influence of gravity) [

CFO explores a D/O problem’s landscape by flying through its decision space (DS) “probes” whose trajectories are governed by equations analogous to those governing gravitational kinematics. DS comprises vectors X = ( x 1 , ⋯ , x N d ) whose components x_{i} are the N_{d} Yagi array design (decision) variables (element lengths, spacings, diameters, and Z_{0}) with their allowable ranges (min/max values for each). Each such vector represents a complete array design. A CFO run begins with an Initial Probe Distribution (IPD) whose configuration is determined by the algorithm designer. The possibilities are limitless, and some IPD’s work better than others. In many recent CFO implementations a “probe line” IPD has been used in which CFO’s probes are arranged uniformly on lines parallel to the DS axes and intersecting along DS’s principal diagonal. For the work reported here, however, a pseudo random π-fraction IPD was used because it is simpler. π-fractions also were used to determine the degree of injected negative gravity. As a result, and as expected, the target and actual levels of negative gravity are not precisely equal.

Even though CFO is deterministic, it is reasonable to speculate that CFO may benefit from the inclusion of a pseudo random component. A pseudo random variable (prv) is a number known precisely by enumeration or calculation, in contrast to a true random variable (rv) whose value is a priori unknowable because it must be calculated from a probability distribution. With respect to any D/O problem, the prv is “random” in the sense that it is uncorrelated with the problem’s topology (“landscape”). Additionally, the prv’s themselves must be uniformly distributed and uncorrelated. Pseudo randomness is included here using π-fractions generated by the Bailey-Borwein-Plouffe (BBP) algorithm. A detailed discussion of π-fractions and their use in a sample GSO algorithm, πGASR (Genetic Algorithm with Sibling Rivalry), appears in [

A detailed description of the basic CFO algorithm is in the Appendix Part 2. Run parameters and the decision space boundaries for the basic CFO implementation used in this paper appear in _{0}, the Yagi’s feedpoint impedance. Variable Z_{0} (VZ_{0}) is a patented “product-by-process” technology in which the antenna’s feedpoint impedance is treated as an optimization variable rather than as a fixed parameter as usually is the case. At this time VZ_{0} is publicly available for use by anyone who wishes to use it (U.S. Patent No. 8,776,002). CFO pseudo code appears in

CFO Parameter | Value |
---|---|

N_{d} (# dimensions) | 12 |

N_{p} (# probes) | 20 |

N_{t} (# steps) | 550 |

G (grav. const.) | +2 |

α (CFO exponent) | 2 |

β (CFO exponent) | 2 |

Δt (“time” increment) | 1 |

F_{rep} (repos. factor) | 0.5 init. |

ΔF_{rep} (F_{rep} increment) | 0.05 |

P_{neg} (target G < 0) | 0 init. |

Array Design Variable | Allowable Range |
---|---|

Element Spacing | 0.1 ≤ S ≤ 1 meters |

Element Length | 0.1 ≤ L ≤ 1 meters |

Element Radius (fixed @ 1/2” diam) | 0.00635 meter |

Feed Point Impedance | 30 ≤ Z_{0} ≤ 150 ohms |

Positive gravity causes CFO’s probes always to move toward greater fitnesses, never away, and consequently to some degree perhaps to impede CFO’s exploration. CFO often converges very quickly [

With respect to G’s sign, + or ‒, the specific question is, Does making it negative benefit CFO’s performance, and if so, why? Or does it impede it, and if so, why? To quote:

“‘Exploration and exploitation are the two cornerstones of problem solving by search.’ For more than a decade, Eiben and Schippers’ advocacy for balancing between these twoantagonistic cornerstones still greatly influences the research directions of evolutionary algorithms (EAs)...” [

At each step negative gravity was pseudo randomly injected into CFO using π-fractions whose values are input from an external file containing the BBP-computed fractions π_{k}, 1 ≤ k ≤ 215829 (k is the π-fraction index). In order to avoid correlations between the π-fractions (see [_{neg} ≤ 100 specifies the target amount of negative gravity in percent, for example, P_{neg} = 6.5 sets the target at 6.5%. At each step the then current value of π_{k} is tested against P_{neg} to determine whether or not gravity is negative at that step. Specifically, if π_{k} ≤ P_{neg}/100 then G = − | G | (negative gravity), otherwise G = | G | (positive gravity). Because prv’s are used, the nature of gravity at each step is a priori known precisely and is uncorrelated with the outcome at any other step. Also because prv’s are used, the desired (target) level of negative gravity and the actual level will not be precisely equal. As the CFO run includes more and more steps the difference between actual and target levels of G ≤ 0 grows smaller with a limiting value of zero because the π-fraction prv’s are uniformly distributed. For this Yagi D/O problem runs were made with non-uniformly spaced values of P_{neg} in [0, 20%]. The actual vs. target levels of G ≤ 0 is plotted in

Every CFO run comprised 550 steps, a large enough number to meet several

objectives: 1) allow fitness evolution essentially to saturate (more steps resulting only in incremental improvement); 2) create actual an actual degree of G ≤ 0 close to the target value; and 3) reasonable runtimes (~8 min on Win7/HP 64-bit SP1; Intel Core i7 4700MQ 2.4 GHZ). Because there is no accurate analytical model of the six-element Yagi-Uda array, the performance of each candidate design evolved by CFO was modeled using the NEC-4 Method of Moments code [

The performance of every candidate antenna design was measured by the following simple fitness function (of course, the choice of fitness function is entirely up to the algorithm designer):

F = c 1 ⋅ g L + c 3 ⋅ g M + c 5 ⋅ g U − c 2 ⋅ S L − c 4 ⋅ S M − c 6 ⋅ S U (1)

where subscripts L, M, and U denote lower, mid and upper frequencies at which the Yagi’s power gain, g, and feedpoint voltage standing wave ratio, S (not to be confused with spacing), are computed. The weights c 1 = c 2 = c 5 = c 6 = 1 and c 3 = c 4 = 3 were chosen for simplicity while intentionally favoring midband performance, slightly, with L = 294.8 MHz, M = 299.8 MHz, U = 304.8 MHz. VSWR (Voltage Standing Wave Ratio) is computed relative to the feed point impedance Z_{0} and denoted VSWR//Z_{0}. While Z_{0} usually is a fixed, user-supplied parameter, typically the industry standard value of 50 Ω, VZ_{0} treats it the same as any other decision space variable. This approach embraces array current distributions that otherwise would be excluded because they fail to adequately match a predetermined value of Z_{0}. Whether or not the CFO-returned value is feasible and desirable is an engineering and economic judgment, and more often than not it is worth impedance-matching the “non-standard” Z_{0}because the antenna’s performance is better, often much better [

The reference Yagi design for this study is, of course, the one that CFO generates with zero negative gravity, the usual CFO implementation.

How the fitness varies with the amount of negative gravity is plotted in

Returning to the fundamental question addressed in this paper, Has pseudo randomly adding a small amount of negative gravity improved CFO’s exploration of the six-element Yagi’s decision space? The fitness data clearly show that the answer is “yes.” However, adding too much G < 0 prevents CFO from exploiting the solutions it has found. How much negative gravity is appropriate no doubt is problem-specific, but this array design example strongly suggests that every CFO implementation should experiment with some measure of G < 0.

Step-by-step the normalized average distance from the probe with the best fitness to all other probes, denoted D_{avg}, is a good measure of CFO’s convergence (see Appendix Part 2 for details). In many cases it approaches zero meaning that all probes have very tightly coalesced around the probe with the best fitness (maximum value). In other cases, such as here, the probe distribution more or less stabilizes but on average at some distance from the best probe.

_{avg} for the cases of zero negative gravity, (a), and for 6% G < 0, (b). Both cases initially show an extreme oscillation that diminishes with increasing Step #. This oscillatory behavior in CFO’s D_{avg} appears to be correlated with local trapping and is not entirely unexpected in view of CFO’s metaphor of gravitational kinematics. In fact, in the real Universe similar oscillatory behavior is seen in the trajectories of Near Earth Objects (NEO’s) that become gravitationally trapped by the Earth’s gravity and eventually break loose without any energy loss [

In order to investigate D_{avg}’s behavior in very long runs, two were made, both with N_{t} = 10,000 steps, zero negative gravity and at a target value of G < 0 of 5%. As discussed in Section 2.3 the actual and target levels of negative gravity are not equal except in the limit of an infinite number of steps. While both the 550-step

and 10,000-step runs targeted G < 0 at 5%, in the first case the actual value was 6% and in the second a much closer 5.22%. The D_{avg} results, which are quite interesting, are plotted in _{avg} settles down to what appears to be a stable, uniform magnitude oscillation with much smaller amplitude in the 5.22% case. In fact, at 5.22% G < 0 D_{avg} does not oscillate at all, and from steps #676-10,000 it is equal to 0.2352928. CFO’s probe distribution is stable and no longer changes step-to-step. In contrast, for the zero gravity case D_{avg} oscillates in an erratic but repetitive pattern as seen in the expanded plot for steps 9900 - 10,000,

The variation of maximum Yagi gain with degree of negative gravity is plotted in

The array’s relative gain bandwidth

~ region of

optimality

~ region of

optimality

As a final measure of the optimized antennas’ performance

This paper reports the results of the D/O of a six-element Yagi-Uda array using Central Force Optimization with pseudo randomly injected negative gravity. Adding 6% G < 0 results in an array that out performs its 0% G < 0 counterpart and also discovers a range of designs with similar fitnesses. Negative gravity was inserted using π-fraction pseudo random variables thereby preserving CFO’s determinism, which is an important consideration in real-world problems that require the formulation of a suitable fitness function. G < 0 has the effect of spreading apart CFO’s probes because negative gravity is repulsive in nature instead of attractive. This wider dispersal improves CFO’s exploration of the decision space without sacrificing the algorithm’s demonstrated high level of exploitation. Injecting some level of negative gravity likely will improve CFO’s exploration across the board, not only in the Yagi array D/O problem but in other applications as well. This particular problem is only one example of the benefit provided by injecting some small measure of negative gravity, and it does not (necessarily) provide specific guidance as to how much G < 0 should be used in any other particular case. The effect of negative gravity in CFO is a research area that should be pursued because the work reported here strongly suggests that there may be considerable benefit in doing so.

The author declares no conflicts of interest regarding the publication of this paper.

Formato, R.A. (2021) Six-Element Yagi Array Designs Using Central Force Optimization with Pseudo Random Negative Gravity. Wireless Engineering and Technology, 12, 23-51. https://doi.org/10.4236/wet.2021.123003

CFO’s best fitness NEC input files at zero and 6% G < 0 fully define the array geometry and Z_{0}. NEC input files appear immediately below, 0% G < 0 first, followed by 6% G < 0. Dimensions are in meters, and since the wavelength, λ, at 299.8 MHz is 1 m the array dimensions also are in λ. The array can be scaled to any other frequency by scaling its dimensions by the wavelength ratio. Also in this Appendix are screenshots of the 4nec2 (ver. 5.8.17) output: 1) 3D radiation patterns @ 299.8 MHz; 2) power gain, VSWR//Z_{0}, and feedpoint impedance for 250 - 350 MHz and for 294.8 - 304.8 MHz, and 3) NEC Average Gain Test at 299.8/250/350 MHz.

In the following screenshots 0% G < 0 is on the left, 6% G < 0 on the right.

3D Radiation Patterns (299.8 MHz)

4nec2 Summary Data, 299.8 MHz (with Average Gain Test, AGT, results)

Total Gain (dBi), 294.8 - 304.8 MHz (note: 4nec2 does not compute F/B if Gain is selected)

VSWR//Z_{0}, 294.8 - 304.8 MHz (Z_{0} = 65.56Ω @ 0% G < 0; 59.8Ω @ 6% G < 0)

Feedpoint Impedance, 294.8 - 304.8 MHz

Total Gain (dBi), 250 - 350 MHz (note: 4nec2 does not compute F/B if Gain is selected)

VSWR//Z_{0}, 250 - 350 MHz (Z_{0} = 65.56Ω @ 0% G < 0; 59.8Ω @ 6% G < 0)

Feedpoint Impedance, 250 - 350 MHz

4nec2 Summary Data, 0% G < 0, 250 & 350 MHz (including AGT results)

4nec2 Summary Data, 6% G < 0, 250 & 350 MHz (including AGT results)

Central Force Optimization (CFO) analogizes gravitational kinematics, the motions of real bodies in the real Universe under the influence of gravity. The fundamental law of physics is Newton’s Universal Law of Gravitation, according to which the magnitude of the force between the two masses m 1 and m 2 is given by [

F = γ m 1 m 2 r 2 (A1)

where r is the distance between them and γ the “gravitational constant.” This force always is attractive, never repulsive, and mass in the real Universe always is positive, never negative. The force of gravity is a central force because it acts only along the line connecting the mass centers. Mass m 1 experiences a vector acceleration due to mass m 2 that is given by

a → 1 = − γ m 2 r ^ r 2 (A2)

where r ^ is a unit vector that points toward m 1 along the line joining the masses.

A2.2. Problem StatementThe CFO metaheuristic solves the following problem: In a decision space (DS) defined by x i min ≤ x i ≤ x i max , i = 1 , ⋯ , N d where the x i are decision variables, locate the global maxima of an objective function f ( x 1 , x 2 , ⋯ , x N d ) possibly subject to a set of constraints Ω among the decision variables. The value of f ( x 1 , x 2 , ⋯ , x N d ) is called the “fitness.” CFO explores DS by flying metaphorical “probes” whose trajectories are governed by equations of motion drawn from gravitational kinematics.

A2.3. Constant AccelerationThe vector location of a mass under constant acceleration is given by the position vector [

R → ( t + Δ t ) = R → 0 + V → 0 Δ t + 1 2 a → Δ t 2 (A3)

where R → ( t + Δ t ) is the position at time t + Δ t . R → 0 and V → 0 , respectively, are the position and velocity vectors at time t, and the acceleration a → is constant during the interval Δ t . In standard three dimensional Cartesian coordinates R → = x i ^ + y j ^ + z k ^ , where i ^ , j ^ , k ^ are the unit vectors along the x , y , z axes, respectively. The CFO metaphor analogizes Equations (A1)-(A3) by generalizing them to a decision space of N d dimensions.

A2.4. Probe TrajectoryCFO’s probes in a typical three-dimensional DS are shown schematically in FigureA1. The location of each probe at each time step is specified by its position vector R → j p , in which p and j are the probe number and time step index, respectively.

In an N d -dimensional DS the position vector is R → j p = ∑ k = 1 N d x k p , j e ^ k , where the x k p , j are probe p’s coordinates at time step j, and following standard notation e ^ k is the unit vector along the x k axis.

Consider a typical probe,p. It moves from position R → j − 1 p at time step j − 1 to position R → j p at time step j under the influence of the metaphorical “gravitational” forces that act on it. Those forces are created by the fitness at each of the other probes’ locations at time step j − 1 . The “time” interval between steps j − 1 and j is Δ t .

At time step j − 1 at probe p’s location the fitness is M j − 1 p = f ( x 1 p , j − 1 , x 2 p , j − 1 , ⋯ , x N d p , j − 1 ) . Each of the other probes also has associated with it a fitness of M j − 1 k , k = 1 , ⋯ , p − 1 , p + 1 , ⋯ , N p , N p being the total number of probes. In this illustration, the value of the fitness is represented by the size of the blackened circle at the tip of the position vector. In keeping with the gravity metaphor, the blackened circles may be thought of as “planets,” say, in our Solar System. Larger circles correspond to greater fitness values, that is, bigger planets with correspondingly greater gravitational attraction. In FigureA1 the fitnesses ordered from greatest to least occur at R → j − 1 s , R → j p , R → j − 1 n , and R → j − 1 p , respectively, as shown by the relative size of the circles.

Probe p’s trajectory in moving from location R → j − 1 p to R → j p is determined by its initial position and by the total acceleration produced by the “masses” that are created by the fitnesses (or some function defined on them) at each of the other probes’ locations. In the CFO implementation used in this paper the “acceleration,” analogous to Equation (A2), experienced by probe p due to the single probe n is given by

G ⋅ U ( M j − 1 n − M j − 1 p ) ⋅ ( M j − 1 n − M j − 1 p ) α ⋅ ( R → j − 1 n − R → j − 1 p ) | R → j − 1 n − R → j − 1 p | β (A4)

where G is CFO’s “gravitational constant” corresponding to γ in Equation (A1). Note that in the real Universe G > 0 , always. In CFO space, however, G can be positive (attractive force of gravity) or negative (repulsive force of gravity). Returning to the forces acting on probe p, in a similar fashion to probe n’s effect, the acceleration of probe p due to a different probe s is given by

G ⋅ U ( M j − 1 s − M j − 1 p ) ⋅ ( M j − 1 s − M j − 1 p ) α ⋅ ( R → j − 1 s − R → j − 1 p ) | R → j − 1 s − R → j − 1 p | β (A5)

Note that the minus sign in Equation (A2) has been included in the order in which the differences are taken in these acceleration expressions. “Mass” in Equation (A2) corresponds to the terms in the numerator involving the fitnesses. Importantly, it does not correspond to the fitness itself. In these equations U ( ⋅ )

is the unit step function U ( z ) = { 1 , z ≥ 0 0 , otherwise . And following standard notation the vertical bars denote vector magnitude, | X → | = ( ∑ i = 1 N d x i 2 ) 1 2 , where x i are the scalar components of X → .

There are no parameters in Equation (A2) corresponding to the “CFO exponents” α > 0 and β > 0 , nor to the unit step U ( ⋅ ) . In real physical space α and β would take on values of 1 and 3, respectively. Note, too, that the numerators in Equations (A4) and (A5) do not contain a unit vector like Equation (A2). The exponents are included to give the algorithm designer a measure of flexibility by assigning, if desired, a different variation of gravitational acceleration with mass and with distance.

A2.5. Mass in CFO SpaceTwo other important differences between real gravity and CFO’s version are: 1) the definition of “mass,” which above is the difference of fitnesses, for example, M j − 1 s − M j − 1 p , not the fitness value itself; and 2) inclusion of the unit step

U ( z ) = { 1 , z ≥ 0 0 , otherwise . The difference of fitnesses is used to avoid excessive

gravitational “pull” by other close by probes that presumably will have fitnesses with similar values. The unit step is included to avoid the possibility of “negative” mass. In the physical Universe, mass is positive, always, but in CFO-space the mass could be positive or negative depending on which fitness is greater. The unit step forces CFO to allow only positive masses, that is, attractive masses. If negative fitness differences were allowed, then some accelerations would be repulsive instead of attractive, thus forcing probes away from large fitness values instead of towards them. The algorithm designer is free to consider other definitions of mass as well. One possibility, for example, might be a ratio of fitnesses similar to the “reduced mass” concept in gravitational kinematics.

A2.6. Total Acceleration and Position Vector for a Single ProbeTaking into account the accelerations produced by each of the other probes on probe p, the total acceleration experienced by p as it “flies” from position R → j − 1 p to R → j p is given by the sum of the gravitational effects over all other probes, that is,

a → j − 1 p = G ∑ k = 1 k ≠ p N p U ( M j − 1 k − M j − 1 p ) ⋅ ( M j − 1 k − M j − 1 p ) α × ( R → j − 1 k − R → j − 1 p ) | R → j − 1 k − R → j − 1 p | β (A6)

Probe p’s new position vector at time step j is therefore given by

R → j p = R → j − 1 p + V → j − 1 p + 1 2 a → j − 1 p Δ t 2 , j ≥ 1 (A7)

which is the analog to Equation (A3), V → j − 1 p being the probe’s “velocity” at the end of time step j − 1 . In Equation (A7) the coefficient 1/2, the velocity term, and the time increment Δ t have been retained primarily as a formalism to highlight the analogy to gravitational kinematics, but they are not required. For the CFO implementation used here, as a matter of convenience, V → j − 1 p and Δ t were arbitrarily set to zero and unity, respectively. Of course, if desired, a constant value of Δ t and the factor 1/2 can be absorbed into the gravitational constant G.

A2.7. Errant ProbesAn important concern is how to handle an “errant” probe, that is, one that flies outside DS, because it is possible that the total acceleration experienced by a probe will fly it into regions of unfeasible solutions that are beyond the DS boundaries. There are many ways to deal with this contingency, and a simple one was implemented in the basic version of CFO used here, the use of a “repositioning factor,” 0 ≤ F r e p ≤ 1 . This factor is used to reposition an errant probe according to the formulas

If x i p , j < x i min ∴ x i p , j = x i min + F r e p ⋅ ( x i p , j − 1 − x i min ) (A8)

If x i p , j > x i max ∴ x i p , j = x i max − F r e p ⋅ ( x i max − x i p , j − 1 ) (A9)

F r e p is assigned an initial value and incremented at each step by a fixed amount Δ F r e p , and if it exceeds unity is reset to the initial value. This simple approach guarantees that all probes will remain inside the decision space. Note that this procedure is pseudo random in nature, but numerical experiments have shown that it is not as effective as pseudo randomly injecting a small amount of negative gravity.

A2.8. D_{avg}Convergence Metric

Perhaps the best measure of CFO’s convergence is the “Average Distance”

metric computed as D a v g = 1 L d i a g ⋅ ( N p − 1 ) ∑ p = 1 N p ∑ i = 1 N d ( x i p , j − x i p * , j ) 2 , where p * is the number of the probe with the best fitness; the superscripts p and j denote, respectively, the probe and step numbers as above; and L d i a g = ∑ i = 1 N d ( x i max − x i min ) 2

is the length of the decision space principal diagonal. If every one of CFO’s probes has coalesced onto a single point, then D a v g = 0 . How closely this metric approaches zero is a good indicator of how CFO’s probe distribution has evolved around a maxima. D a v g also is useful in identifying potential local trapping because oscillatory behavior in a D a v g plot appears to signal trapping at a local maxima.