# Ship-Borne Phased Array Radar Using GA Based Adaptiveα-β-γ Filter for Beamforming Compensation and Air Target Tracking

Authors: J. Mar; Chen-Chih Liu; M. B. Basnet
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

Beam pointing error caused by ship motion over the ocean affects the tracking performance of the ship-borne phased array radar. Due to the dynamic nature of the sea environments, the ship-borne phased array radar must be able to compensate for the ship’s motion adaptively. In this paper, the adaptiveα-β-γ filter is proposed for the ship-borne phased array radar to compensate for the beam pointing error and to track the air target. The genetic algorithm (GA) and the particle swarm optimization (PSO) methods are applied to estimate the gain parameters of adaptive α-β-γ filters, while achieving the optimum objective of minimum root mean square error (RMSE). The roll and pitch data measured from a gyroscope of the sea vehicle and generated from ship motion mathematical model are used in the experiments. The tracking accuracy of adaptive α-β-γ filter using the GA method is compared with PSO method under different ship motion conditions. The convergent time and tracking accuracy of ship-borne phased array radar using the proposed GA based adaptive α-β-γ filter are also compared with the adaptive extended Kalman filter (AEKF). Finally, it is proved that the proposed GA based adaptive α-β-γ filter is a real time applicable algorithm for ship-borne phased array radar.


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## Body

## 1. Introduction

Sea wave causes the effect of roll and pitch motions on ships. These ship rotational motions result in measurement error in phased array radar aboard the ship. The antenna stabilization to achieve the beam pointing accuracy over the long dwell time is an important issue for ship-borne phased array radar [1]. There are two ship motion compensations: compensation for rotational motion (i.e., pitch, roll, and heading angle) and compensation for translational motion (i.e., radial speed relative to the earth). Gyroscope provides pitch, roll, heading angles of the ship, speed, course, and vertical velocity of antenna installed on the ship at a data rate of 10 Hz. To compensate for the translational motion, the speed, course, and vertical velocity acquired from the gyroscope are averaged for the duration of radar-dwell time. Then the Doppler shift introduced by the translational motion is estimated in the digital signal processor (DSP) of the radar to compensate the radial velocity. The radar control computer (RCC) provides the target locations relative to earth, schedules beam directions, and predicts beam pointing error to the beam steering controller (BSC), which compensates for the beam pointing error and controls the phased array antenna to point the beam at the target direction relative to the ship coordinates.The motion compensation method based on coordinate conversion has been described in [2, 3], which requires the measurement of a device’s coordinates, the ship’s coordinates, and the earth’s coordinates systems to compensate the device errors due to motion disturbances. In [1], Kalman filtering along with coordinates conversion is applied to reduce the beam pointing error and to stabilize a tracking beam. The α-β-γ  filter, which is more easily implemented than a generalized Kalman filter, is used for motion compensation under different sea states in [1]. The sea environments are very dynamic; hence, there is need of an adaptive system for controlling and compensating devices regardless of ship motion. Most recent works have used Kalman filtering (KF) [4] and extended Kalman filtering (EKF) [5] to estimate the ship’s attitude. Estimation accuracy of KF and EKF depends on the values of different parameters, such as error covariance matrices; thus, the KF and EKF require knowledge of covariance. In [6], the automatic beam pointing error compensation mechanism employs the parallel fuzzy basis function network (FBFN) architecture to estimate the beam pointing error caused by roll and pitch of the ship. The effect of automatic beam pointing error compensation mechanism on the tracking performance of adaptive extended Kalman filter (AEKF) implemented in ship-borne phased array radar is investigated. It shows that the tracking error of AEKF converges to less than about 20 m at 550 iterations (sec/iteration) and the estimation error will remain within the range of about 20 m when beam pointing error is compensated by FBFN controller.Theα-β-γ filter has a similar structure with the KF but depends on the gain value of α, β, and γ which are limited and interdependent [7–9]. The uses of various methods to adjust parameter values for Kalman filter, EKF, and α-β-γfilter have been proposed and implemented over the years. Fuzzy membership function is used in [10] to estimate and correct the target position for the signal being tracked. Genetic algorithm (GA) is used in [9] to search the suitable parameter values for the α-β-γfilter, which can provide the approximate position, velocity, and acceleration signal and simultaneously decrease the measurement noise. Particle swarm optimization (PSO) is used in [11] to tune the noise covariance of a Kalman filter. In [12], the PSO is used to find the optimal gain parameter values for an α-β-γ filter. PSO and GA methods are suitable for the real time application because they do not require differentiation and are less complicated than other methods.In this paper, GA based adaptiveα-β-γ filter is proposed for automatic beam pointing error correction and air target tracking in three-dimensional space. Continuous monitoring of the environment and adapting filter gain parameter with less computational burden is needed for real time application. The adaptive α-β-γfilter requires knowledge of the coefficients for which GA algorithm is used only in certain time intervals. GA is used to find the gain values by minimizing the objective function, that is, root mean square error (RMSE). The proposed adaptive α-β-γ  filter algorithm is implemented and compared with using PSO method. The roll and pitch data were recorded by a gyroscope of the sea vehicle to simulate the tracking performance of ship-borne phased array radar using the proposed adaptive α-β-γ  filter. In addition, the roll and pitch data generated from ship motion mathematical model at sea states 2 and 3 are also applied for the proposed adaptive α-β-γfilter to verify the correctness of the test results.The rest of this paper is organized as follows: the model of ship rotational motion, coordinates transform, and planar array antenna of ship-borne phased array radar are described in detail in Section2. The algorithm of the proposed adaptive α-β-γfilter based on the GA is presented in Section 3, where the α-β-γ  filter, GA gain estimator, PSO gain estimator, and optimization problem formulation are described. In Section 4, six different experimental cases are performed; the variations of roll and pitch angles in sea states 2 and 3 are analyzed; the GA and PSO methods are used to determine the optimal filter gain of adaptive α-β-γ filter that minimizes RMSE; the tracking performance of ship-borne phased array radar using the proposed adaptive α-β-γ filter is simulated. Finally, conclusions are made in Section 5.


## 2. Ship Rotational Motion Compensation

The ship-borne phased array radar must be able to compensate the ship’s motion and track the maneuvering targets automatically. The adaptiveα-β-γfiltering algorithm is designed to real time compensate the errors caused by the ship’s motion. The block diagram of rotational motion compensation system for ship-borne phased array radar is shown in Figure 1, which consists of adaptive α-β-γfilter for beam pointing error prediction, ship coordinates/earth coordinates conversion, earth coordinates/ship coordinates conversion, and adaptive α-β-γfilter for target tracking. The roll angle ϕR(t) and the pitch angle θP(t) of the ship angular motion are measured by the gyroscope. (θA,ϕA) is the antenna beam pointing angle relative to the ship body, where θA is the angle off antenna boresight, and ϕA is the azimuth angle counterclockwise from the bow of the ship. Assume that the beam steering angle at a certain time instant is (θ0,ϕ0), the antenna point angle offset caused by the ship motion (pitch, roll) is (θe,ϕe), and then the current antenna pointing angle is (θE,ϕE) with respect to the earth coordinates:
(1)θE=θ0+θe,ϕE=ϕ0+ϕe.
If the beam pointing error is predicted as (ΔθE,ΔϕE), then the beam pointing angle is corrected as
(2)θEC=θ0+θe-ΔθE,ϕEC=ϕ0+ϕe-ΔϕE.
The value of (θEC,ϕEC) is approximate to (θ0,ϕ0).Figure 1
High level structure of ship rotational motion compensation for phased array radar.


### 2.1. Ship Rotational Motion Model

Ship is affected by the waves in the ocean, resulting in six degrees of freedom of movement. In this paper, assuming zero yaw angle, the simplified roll and pitch model [1, 13] is adopted to describe the ship rotational motion in the earth coordinates. Ship’s rotational motion is modeled with sinusoidal signal. The roll angle is
(3)ϕRt=ARsinωRt+nRt.
The pitch angle is
(4)θPt=APsinωPt+nPt,
where AR and AP are the amplitude of ship’s roll and pitch angles, nR(t) and nP(t) are assumed to be zero mean Gaussian noise, TR and TP are the roll and pitch periods, and ωR=2π/TR and ωP=2π/TP are the roll and pitch angular frequencies.The random roll and pitch angle errors caused by other unknown factors, including the change of ship’s traveling direction and weather, in the actual ship navigation environment will be considered into the standard deviation of the noise terms. The amplitude and period parameters of sea states 2 and 3 are listed in Table1 [1].Table 1
Sea state parameters.
Roll
Pitch
A
R (°)
T
R (sec)
A
P (°)
T
P (sec)
Sea state 2
1
2.5
0.2
1.2
Sea state 3
1
2.0
0.2
0.6


### 2.2. Coordinates Transform

Since the phased array antenna is installed on the ship, the beam steering control employs the ship body coordinates. But the target tracking of adaptiveα-β-γ filter employs the Earth coordinates. The Euler coordinates transform formula is used to convert antenna beam pointing angle relative to the earth (θE,ϕE), where θE is the angle from vertical and ϕE is the angle clockwise from true north, into antenna beam pointing angle relative to the ship body (θA,ϕA) [13]:
(5)sinθAcos⁡ϕAsinθAsinϕAcos⁡θA=TϕTθTψsinθEcos⁡ϕE-sinθEsinϕEcos⁡θE,
where the coordinates transform matrices are defined as
(6)Tϕ=1000cos⁡ϕsinϕ0-sinϕcos⁡ϕ,Tθ=cos⁡θ0sinθ010-sinθ0cos⁡θ,Tψ=cos⁡ψ-sinψ0sinψcos⁡ψ0001,
where the ship roll angle ϕ is positive with downward roll of starboard, the ship pitch angle θ is positive with bow pitch upward, and the ship yaw angle ψ is positive clockwise from the north.


### 2.3. Planar Array Antenna

AnM×N element planar array [6] is designed for the ship-borne phased array radar system, which includes the beamforming (BF) mode and direction of arrival (DOA) mode. The planar array can produce multibeams in the azimuth and elevation by using the beamformer network (BFN), which consists of a set of power dividers and phase shifters. The planar array using amplitude comparison method [14] generates the difference signal patterns for the DOA estimation. The difference signals obtained from two neighboring beams can measure the DOA of the target signal of the ship-borne phased array radar system. The beam pattern is expressed as [14, 15]
(7)AFθ,ϕ=∑n=1NI1n∑m=1MIm1ej(m-1)(kdxsinθcos⁡ϕ+βx)llllliiillllllll×ejn-1kdysinθsinϕ+βy=SxmSyn,
where
(8)Sxm=∑m=1MIm1ej(m-1)(kdxsinθcos⁡ϕ+βx),Sym=∑n=1NI1nej(n-1)(kdysinθsinϕ+βy),βx=-kdxsinθtcos⁡ϕt,βy=-kdysinθtsinϕt,
where Im1 = I1n = Chebyshev weighting [13], M = the number of array elements in the x-axis, N = the number of array elements in y-axis, ϕt = the azimuth steering angle of the main beam, θt = the elevation steering angle of the main beam, dx = the interelement spacing in x-axis, dy = the interelement spacing in y-axis, k=2π/λ= the phase propagation constant, and λ is the wavelength of the carrier.The phase of the RF signal at each array element is adjusted to steer the beam to the coordinates(θt,ϕt).


## 2.1. Ship Rotational Motion Model

Ship is affected by the waves in the ocean, resulting in six degrees of freedom of movement. In this paper, assuming zero yaw angle, the simplified roll and pitch model [1, 13] is adopted to describe the ship rotational motion in the earth coordinates. Ship’s rotational motion is modeled with sinusoidal signal. The roll angle is
(3)ϕRt=ARsinωRt+nRt.
The pitch angle is
(4)θPt=APsinωPt+nPt,
where AR and AP are the amplitude of ship’s roll and pitch angles, nR(t) and nP(t) are assumed to be zero mean Gaussian noise, TR and TP are the roll and pitch periods, and ωR=2π/TR and ωP=2π/TP are the roll and pitch angular frequencies.The random roll and pitch angle errors caused by other unknown factors, including the change of ship’s traveling direction and weather, in the actual ship navigation environment will be considered into the standard deviation of the noise terms. The amplitude and period parameters of sea states 2 and 3 are listed in Table1 [1].Table 1
Sea state parameters.
Roll
Pitch
A
R (°)
T
R (sec)
A
P (°)
T
P (sec)
Sea state 2
1
2.5
0.2
1.2
Sea state 3
1
2.0
0.2
0.6


## 2.2. Coordinates Transform

Since the phased array antenna is installed on the ship, the beam steering control employs the ship body coordinates. But the target tracking of adaptiveα-β-γ filter employs the Earth coordinates. The Euler coordinates transform formula is used to convert antenna beam pointing angle relative to the earth (θE,ϕE), where θE is the angle from vertical and ϕE is the angle clockwise from true north, into antenna beam pointing angle relative to the ship body (θA,ϕA) [13]:
(5)sinθAcos⁡ϕAsinθAsinϕAcos⁡θA=TϕTθTψsinθEcos⁡ϕE-sinθEsinϕEcos⁡θE,
where the coordinates transform matrices are defined as
(6)Tϕ=1000cos⁡ϕsinϕ0-sinϕcos⁡ϕ,Tθ=cos⁡θ0sinθ010-sinθ0cos⁡θ,Tψ=cos⁡ψ-sinψ0sinψcos⁡ψ0001,
where the ship roll angle ϕ is positive with downward roll of starboard, the ship pitch angle θ is positive with bow pitch upward, and the ship yaw angle ψ is positive clockwise from the north.


## 2.3. Planar Array Antenna

AnM×N element planar array [6] is designed for the ship-borne phased array radar system, which includes the beamforming (BF) mode and direction of arrival (DOA) mode. The planar array can produce multibeams in the azimuth and elevation by using the beamformer network (BFN), which consists of a set of power dividers and phase shifters. The planar array using amplitude comparison method [14] generates the difference signal patterns for the DOA estimation. The difference signals obtained from two neighboring beams can measure the DOA of the target signal of the ship-borne phased array radar system. The beam pattern is expressed as [14, 15]
(7)AFθ,ϕ=∑n=1NI1n∑m=1MIm1ej(m-1)(kdxsinθcos⁡ϕ+βx)llllliiillllllll×ejn-1kdysinθsinϕ+βy=SxmSyn,
where
(8)Sxm=∑m=1MIm1ej(m-1)(kdxsinθcos⁡ϕ+βx),Sym=∑n=1NI1nej(n-1)(kdysinθsinϕ+βy),βx=-kdxsinθtcos⁡ϕt,βy=-kdysinθtsinϕt,
where Im1 = I1n = Chebyshev weighting [13], M = the number of array elements in the x-axis, N = the number of array elements in y-axis, ϕt = the azimuth steering angle of the main beam, θt = the elevation steering angle of the main beam, dx = the interelement spacing in x-axis, dy = the interelement spacing in y-axis, k=2π/λ= the phase propagation constant, and λ is the wavelength of the carrier.The phase of the RF signal at each array element is adjusted to steer the beam to the coordinates(θt,ϕt).


## 3. Adaptiveα-β-γ Filter

As shown in Figure1, the adaptive α-β-γ filters are applied to beam pointing error prediction and target tracking, respectively.


### 3.1. Parallel Adaptiveα-β-γ Filter for Beam Pointing Error Prediction

Figure2 shows the beam pointing error prediction system, which consists of separate adaptive α1-β1-γ1 filters to predict the roll and pitch angles of ship motion and their outputs are fed into the beam steering controller (BSC). The BSC compensates the beam pointing errors of the phased array antenna onboard the ship to reduce the impact of the roll and pitch motion on the phased array radar. Roll and pitch data are random but they are similar in nature and they can be characterized with different amplitudes and frequencies [1]. The α1-β1-γ1 filter predicts the next positions using current error also known as innovation [8]. This innovation process has two steps: prediction and estimation.Figure 2
Parallel adaptiveα-β-γ filter for beam pointing error prediction.The pitch prediction equations are expressed as(9)θ^pk+1=θe,pk+T1ωe,pk+T122ae,pk,ω^pk+1=ωe,pk+T1ae,pk,a^pk+1=ae,pk.The pitch estimation equations are expressed as(10)θe,pk=θ^pk+α1,pθm,pk-θ^pk+n1,pk,ωe,pk=ω^pk+β1,pT1θm,pk-θ^pk,ae,pk=a^pk+γ1,p2T12θm,pk-θ^pk,
where θ^p(k+1), w^p(k+1), and a^p(k+1) are pitch angular position, pitch angular velocity, and pitch angular acceleration values predicted at (k+1)th interval, respectively. The sampling time T1=0.1sec, θe,p(k), we,p(k), and ae,p(k) are pitch angular position, pitch angular velocity, and pitch angular acceleration estimated at kth interval. α1,p, β1,p, and γ1,p are the smoothing parameters whose region of stability and parameter constraint have been studied in [8]. The relationship among α1,p, β1,p, and γ1,p parameters can be related to pitch gain g1,p, where 0<g1,p<1. The formula is described as follows:
(11)α1,p=1-g1,p3,β1,p=1.5·1+g1,p1-g1,p2,γ1,p=1-g1,p3.
The roll prediction equations, roll estimation equations, and roll smoothing parameters have similar forms as the pitch prediction equations, pitch estimation equations, and pitch smoothing parameters, respectively.


### 3.2. Adaptiveα-β-γ Filter for Target Tracking in Three-Dimensional Space

The GA basedα2-β2-γ2 filter algorithm for target tracking is used by ship-borne phased array radar to predict the next target positions using current error and further improves its tracking accuracy. The tracking process of α2-β2-γ2 filter is shown in Figure 3, where the prediction equations are
(12)X^k+1=Xek+T2Vek+T222Aek,V^k+1=Vek+T2Aek,A^k+1=Aek.
The estimation equations are
(13)Xek=X^k+α2Xmk-X^k+n2k,Vek=V^k+β2T2Xmk-X^k,Aek=A^k+γ22T22Xmk-X^k,
where Xk=[xk,yk,zk]T, Vk=[vxk,vyk,vzk]T, and Ak=[axk,ayk,azk]T are the position vector, velocity vector, and acceleration vector, respectively, at time instant k; n2(k) is the zero mean white Gaussian noise and the sampling time T2=1sec. The relationship among α2, β2, and γ2 parameters can be related to gain factor vector g2=[g2,g2,g2]T. The formula is described as follows:(14)α2=α2,x000α2,y000α2,z=1-g230001-g230001-g23,β2=β2,x000β2,y000β2,z=1.51+g21-g220001.51+g21-g220001.51+g21-g22,γ2=γ2,x000γ2,y000γ2,z=1-g230001-g230001-g23.Figure 3
Adaptiveα-β-γ filter for target tracking in three-dimensional space.


### 3.3. GA Gain Estimator

The flow chart of GA roll/pitch gain estimators of adaptiveα-β-γ filter is shown in Figure 4. The GA roll/pitch gain estimators are used to estimate the roll and pitch angles gain (g1,r/g1,p) of adaptive α1-β1-γ1 filters. The GA tracking gain estimator is used to estimate the optimal gain g2 of adaptive α2-β2-γ2 filter. The flow chart of GA tracking gain estimator is similar to Figure 4. The GA method uses selection, crossover, and mutation [16] techniques to find the solutions. At first, the initial population which contains randomly generated chromosomes is made. Chromosomes are chains of 0 and 1 bits whose length is defined as string length. Chromosomes number is defined as the population size. These chromosomes from the mating pool are shuffled to create more randomness and then applied to the problem to find their outputs and fitness scores accordingly. Fitness value defines how well the chromosome solves the problem. Two chromosomes are selected from the mating pool depending on selection method. Depending on the crossover rate, all bits between two randomly chosen points selected from two different chromosomes are interchanged, which is known as two-point crossover. Chromosomes may undergo mutation where random bits of chromosomes are flipped depending on the mutation rate. Termination criteria are defined by the number of generations.Figure 4
Flow chart of GA roll/pitch gain estimator.Selection is done usually by using two methods as follows: roulette wheel selection and tournament selection. In roulette wheel selection, the chromosomes selected from population are put into a mating pool. The chromosome selection probability is proportion to the fitness value. In tournament selection, chromosomes have to go through several tournaments. The chromosome, which has the fittest value, is selected and put into the mating pool. Here the roulette wheel selection method is chosen to determine the solution.Two-point crossover calls for two random points selected from two different parent chromosome strings; all bits between two points are swapped for two parent chromosomes. Crossover rate is commonly chosen in a range of 0.60 to 0.80 [16]. Mutation rate describes the probability that a bit in a chromosome will be flipped (0 flips to 1 and 1 flips to 0). Mutation brings diversity to the population as mutation of a chromosome is a random process which will bring randomness to present chromosome group. Generation defines iteration, the GA method runs for a defined number of generations, and the final best solution is considered as the result.


### 3.4. PSO Gain Estimator [9, 12, 17]

PSO method was first proposed in [18]. PSO is based on the number of particles flying around the solution space to find the best solution that minimizes the cost function. Particles move around the solution space as soon as a particle detects a better solution; the information is passed to other particles and then particles change their position with respect to their best position observed among all the particles. Figure 5 describes the PSO method in detail. The PSO changes the velocity and position of the particle according to 15 and 17, respectively, to achieve the fitness [16]:
(15)vidn+1=K*pgdn-xidnvidn+c1·rand·pidn-xidnlllllllllll+c2·Rand·pgdn-xidn,(16)K=22-φ-φ2-4φ,where,φ=c1+c2,φ>4,(17)xid(n+1)=xid(n)+vid(n),
where xid(n) and vid(n) are position and velocity of (id)th particle at (n)th iteration, respectively. pid(n) and pgd(n) are the best position and global best position of (id)th particle at (n)th iteration, respectively. Rand() and rand() are random number between 0 and 1, respectively. Equation 15 updates the velocity of the particles and 17 updates the position of particles. K means inertia, c1 and c2 are correction factors, and swarm size means number of particles for an iteration, as described in [16].Figure 5
Flow chart of PSO method.Particles in the PSO method sometimes go out of constraints while changing their position. Reinitialization of the particle’s position randomly to fall within the constraints is done as soon as a particle moves out of constraints. This ensures full use of particles and also increases the chance of discovering a better solution. To reduce the chance of getting trapped in a local minimum, the advantages of using PSO method are that it is derivative-free, easy to implement, and easy to comprehend and the solution is independent of the initial point [17]. One of the drawbacks of the PSO method is that it does not guarantee success; that is, the solution found may not be the best solution.


### 3.5. GA/PSO Based Adaptiveα-β-γ Filter

The flow chart of the adaptiveα-β-γ filter algorithm is shown in Figure 6, where N is the number of sampling data to determine the filter gain. In this algorithm, the parameters α, β, and γ, which are related to the filter gain g, are optimized by minimizing the RMSE using GA or PSO method. The proposed adaptive α-β-γ filter processes the current sampled data xi which predicts the next sample data xi+1 as iteration continues. The adaptive GA or PSO process only occurs at certain time interval. The previous N data, that is, xp(i-N)⋯xp(i-1), is sent to the adaptive GA or PSO algorithm where the optimum value of filter gain (g) is determined for the α-β-γ filter. Then xi+1 is predicted and the process continues. The value of N determines running time and accuracy of the algorithm, so N must have small enough value to support real time implementation and large enough value to provide acceptable accuracy.Figure 6
Block diagram of adaptiveα-β-γ filter.Since the roll and pitch angles of ship motion change over time randomly, in order to speed up the processing time and to reach the objectives of minimum estimation error, the filter gain values are real time updated by using the pipelined architecture. The recorded roll and pitch signals are segmented into block of 1000 sampled data (100 sec) for linear trajectory target and 420 sampled data (42 sec) for circular trajectory target, respectively. The gain value ofα2-β2-γ2 filter in 101–200 sec is estimated by GA and PSO methods using the collected data during 1–100 sec, the gain value of α2-β2-γ2 filter in 201–300 sec is estimated by the GA and PSO methods using the collected data in 101–200 sec, and so on. Similarly, the measured flight target signals are segmented into block of 100 sampled data (100 sec) for linear trajectory target and 42 sampled data (42 sec) for circular trajectory target, respectively. The gain value of α2-β2-γ2 filter in 101–200 sec is estimated by the GA and PSO methods using the collected data during 1–100 sec, the gain value of α2-β2-γ2 filter in the interval of 201–300 sec is estimated by the GA and PSO methods using the collected data in 101–200 sec, and so on.For the PSO process, it was found that gain parameter converged within 100 iterations, so the number of iterations for an adaptiveα-β-γ filter was kept at constant 100 iterations per processing cycle, swarm size = 30, inertia = 0.7298, and correction factors c1=2.1, c2=2. For the GA process, we have considered population size = 8, string length = 12, crossover rate = 0.8, and mutation rate = 0.05.Ifθi,p,i=1,2,…,N and θ^i,p,i=1,2,…,N are real measurement pitch’s values and estimated pitch’s values of adaptive α1-β1-γ1 filter, respectively, then θi,r,i=1,2,…,N and θ^i,r,i=1,2,…,N are real measurement roll’s values and estimated roll’s values of adaptive α1-β1-γ1 filter, respectively, where i is the current iteration and N is the total number of sampling data. The RMSEs of pitch and roll estimations are defined as follows:
(18)RMSEp=1N∑i=1Nθ^i,p-θi,p2,(19)RMSEr=1N∑i=1Nθ^i,r-θi,r2,
where the main purpose of GA and PSO methods is to find the optimum values of g1,p and g1,r which minimize the RMSEp and RMSEr, respectively.IfXi and X^i, i=1,2,…,N are real measurement target position vector values and estimated target position vector values of adaptive α2-β2-γ2 filter, the RMSE of target tracking is defined as
(20)RMSE=1N∑i=1Nxi-x^i2+yi-y^i2+zi-z^i2,
where xi, yi, and zi are the components of target position vector Xi in x, y, and z axes. The main purpose of GA and PSO algorithms is to find the optimum gain value of g2 which minimizes the RMSE.


## 3.1. Parallel Adaptiveα-β-γ Filter for Beam Pointing Error Prediction

Figure2 shows the beam pointing error prediction system, which consists of separate adaptive α1-β1-γ1 filters to predict the roll and pitch angles of ship motion and their outputs are fed into the beam steering controller (BSC). The BSC compensates the beam pointing errors of the phased array antenna onboard the ship to reduce the impact of the roll and pitch motion on the phased array radar. Roll and pitch data are random but they are similar in nature and they can be characterized with different amplitudes and frequencies [1]. The α1-β1-γ1 filter predicts the next positions using current error also known as innovation [8]. This innovation process has two steps: prediction and estimation.Figure 2
Parallel adaptiveα-β-γ filter for beam pointing error prediction.The pitch prediction equations are expressed as(9)θ^pk+1=θe,pk+T1ωe,pk+T122ae,pk,ω^pk+1=ωe,pk+T1ae,pk,a^pk+1=ae,pk.The pitch estimation equations are expressed as(10)θe,pk=θ^pk+α1,pθm,pk-θ^pk+n1,pk,ωe,pk=ω^pk+β1,pT1θm,pk-θ^pk,ae,pk=a^pk+γ1,p2T12θm,pk-θ^pk,
where θ^p(k+1), w^p(k+1), and a^p(k+1) are pitch angular position, pitch angular velocity, and pitch angular acceleration values predicted at (k+1)th interval, respectively. The sampling time T1=0.1sec, θe,p(k), we,p(k), and ae,p(k) are pitch angular position, pitch angular velocity, and pitch angular acceleration estimated at kth interval. α1,p, β1,p, and γ1,p are the smoothing parameters whose region of stability and parameter constraint have been studied in [8]. The relationship among α1,p, β1,p, and γ1,p parameters can be related to pitch gain g1,p, where 0<g1,p<1. The formula is described as follows:
(11)α1,p=1-g1,p3,β1,p=1.5·1+g1,p1-g1,p2,γ1,p=1-g1,p3.
The roll prediction equations, roll estimation equations, and roll smoothing parameters have similar forms as the pitch prediction equations, pitch estimation equations, and pitch smoothing parameters, respectively.


## 3.2. Adaptiveα-β-γ Filter for Target Tracking in Three-Dimensional Space

The GA basedα2-β2-γ2 filter algorithm for target tracking is used by ship-borne phased array radar to predict the next target positions using current error and further improves its tracking accuracy. The tracking process of α2-β2-γ2 filter is shown in Figure 3, where the prediction equations are
(12)X^k+1=Xek+T2Vek+T222Aek,V^k+1=Vek+T2Aek,A^k+1=Aek.
The estimation equations are
(13)Xek=X^k+α2Xmk-X^k+n2k,Vek=V^k+β2T2Xmk-X^k,Aek=A^k+γ22T22Xmk-X^k,
where Xk=[xk,yk,zk]T, Vk=[vxk,vyk,vzk]T, and Ak=[axk,ayk,azk]T are the position vector, velocity vector, and acceleration vector, respectively, at time instant k; n2(k) is the zero mean white Gaussian noise and the sampling time T2=1sec. The relationship among α2, β2, and γ2 parameters can be related to gain factor vector g2=[g2,g2,g2]T. The formula is described as follows:(14)α2=α2,x000α2,y000α2,z=1-g230001-g230001-g23,β2=β2,x000β2,y000β2,z=1.51+g21-g220001.51+g21-g220001.51+g21-g22,γ2=γ2,x000γ2,y000γ2,z=1-g230001-g230001-g23.Figure 3
Adaptiveα-β-γ filter for target tracking in three-dimensional space.


## 3.3. GA Gain Estimator

The flow chart of GA roll/pitch gain estimators of adaptiveα-β-γ filter is shown in Figure 4. The GA roll/pitch gain estimators are used to estimate the roll and pitch angles gain (g1,r/g1,p) of adaptive α1-β1-γ1 filters. The GA tracking gain estimator is used to estimate the optimal gain g2 of adaptive α2-β2-γ2 filter. The flow chart of GA tracking gain estimator is similar to Figure 4. The GA method uses selection, crossover, and mutation [16] techniques to find the solutions. At first, the initial population which contains randomly generated chromosomes is made. Chromosomes are chains of 0 and 1 bits whose length is defined as string length. Chromosomes number is defined as the population size. These chromosomes from the mating pool are shuffled to create more randomness and then applied to the problem to find their outputs and fitness scores accordingly. Fitness value defines how well the chromosome solves the problem. Two chromosomes are selected from the mating pool depending on selection method. Depending on the crossover rate, all bits between two randomly chosen points selected from two different chromosomes are interchanged, which is known as two-point crossover. Chromosomes may undergo mutation where random bits of chromosomes are flipped depending on the mutation rate. Termination criteria are defined by the number of generations.Figure 4
Flow chart of GA roll/pitch gain estimator.Selection is done usually by using two methods as follows: roulette wheel selection and tournament selection. In roulette wheel selection, the chromosomes selected from population are put into a mating pool. The chromosome selection probability is proportion to the fitness value. In tournament selection, chromosomes have to go through several tournaments. The chromosome, which has the fittest value, is selected and put into the mating pool. Here the roulette wheel selection method is chosen to determine the solution.Two-point crossover calls for two random points selected from two different parent chromosome strings; all bits between two points are swapped for two parent chromosomes. Crossover rate is commonly chosen in a range of 0.60 to 0.80 [16]. Mutation rate describes the probability that a bit in a chromosome will be flipped (0 flips to 1 and 1 flips to 0). Mutation brings diversity to the population as mutation of a chromosome is a random process which will bring randomness to present chromosome group. Generation defines iteration, the GA method runs for a defined number of generations, and the final best solution is considered as the result.


## 3.4. PSO Gain Estimator [9, 12, 17]

PSO method was first proposed in [18]. PSO is based on the number of particles flying around the solution space to find the best solution that minimizes the cost function. Particles move around the solution space as soon as a particle detects a better solution; the information is passed to other particles and then particles change their position with respect to their best position observed among all the particles. Figure 5 describes the PSO method in detail. The PSO changes the velocity and position of the particle according to 15 and 17, respectively, to achieve the fitness [16]:
(15)vidn+1=K*pgdn-xidnvidn+c1·rand·pidn-xidnlllllllllll+c2·Rand·pgdn-xidn,(16)K=22-φ-φ2-4φ,where,φ=c1+c2,φ>4,(17)xid(n+1)=xid(n)+vid(n),
where xid(n) and vid(n) are position and velocity of (id)th particle at (n)th iteration, respectively. pid(n) and pgd(n) are the best position and global best position of (id)th particle at (n)th iteration, respectively. Rand() and rand() are random number between 0 and 1, respectively. Equation 15 updates the velocity of the particles and 17 updates the position of particles. K means inertia, c1 and c2 are correction factors, and swarm size means number of particles for an iteration, as described in [16].Figure 5
Flow chart of PSO method.Particles in the PSO method sometimes go out of constraints while changing their position. Reinitialization of the particle’s position randomly to fall within the constraints is done as soon as a particle moves out of constraints. This ensures full use of particles and also increases the chance of discovering a better solution. To reduce the chance of getting trapped in a local minimum, the advantages of using PSO method are that it is derivative-free, easy to implement, and easy to comprehend and the solution is independent of the initial point [17]. One of the drawbacks of the PSO method is that it does not guarantee success; that is, the solution found may not be the best solution.


## 3.5. GA/PSO Based Adaptiveα-β-γ Filter

The flow chart of the adaptiveα-β-γ filter algorithm is shown in Figure 6, where N is the number of sampling data to determine the filter gain. In this algorithm, the parameters α, β, and γ, which are related to the filter gain g, are optimized by minimizing the RMSE using GA or PSO method. The proposed adaptive α-β-γ filter processes the current sampled data xi which predicts the next sample data xi+1 as iteration continues. The adaptive GA or PSO process only occurs at certain time interval. The previous N data, that is, xp(i-N)⋯xp(i-1), is sent to the adaptive GA or PSO algorithm where the optimum value of filter gain (g) is determined for the α-β-γ filter. Then xi+1 is predicted and the process continues. The value of N determines running time and accuracy of the algorithm, so N must have small enough value to support real time implementation and large enough value to provide acceptable accuracy.Figure 6
Block diagram of adaptiveα-β-γ filter.Since the roll and pitch angles of ship motion change over time randomly, in order to speed up the processing time and to reach the objectives of minimum estimation error, the filter gain values are real time updated by using the pipelined architecture. The recorded roll and pitch signals are segmented into block of 1000 sampled data (100 sec) for linear trajectory target and 420 sampled data (42 sec) for circular trajectory target, respectively. The gain value ofα2-β2-γ2 filter in 101–200 sec is estimated by GA and PSO methods using the collected data during 1–100 sec, the gain value of α2-β2-γ2 filter in 201–300 sec is estimated by the GA and PSO methods using the collected data in 101–200 sec, and so on. Similarly, the measured flight target signals are segmented into block of 100 sampled data (100 sec) for linear trajectory target and 42 sampled data (42 sec) for circular trajectory target, respectively. The gain value of α2-β2-γ2 filter in 101–200 sec is estimated by the GA and PSO methods using the collected data during 1–100 sec, the gain value of α2-β2-γ2 filter in the interval of 201–300 sec is estimated by the GA and PSO methods using the collected data in 101–200 sec, and so on.For the PSO process, it was found that gain parameter converged within 100 iterations, so the number of iterations for an adaptiveα-β-γ filter was kept at constant 100 iterations per processing cycle, swarm size = 30, inertia = 0.7298, and correction factors c1=2.1, c2=2. For the GA process, we have considered population size = 8, string length = 12, crossover rate = 0.8, and mutation rate = 0.05.Ifθi,p,i=1,2,…,N and θ^i,p,i=1,2,…,N are real measurement pitch’s values and estimated pitch’s values of adaptive α1-β1-γ1 filter, respectively, then θi,r,i=1,2,…,N and θ^i,r,i=1,2,…,N are real measurement roll’s values and estimated roll’s values of adaptive α1-β1-γ1 filter, respectively, where i is the current iteration and N is the total number of sampling data. The RMSEs of pitch and roll estimations are defined as follows:
(18)RMSEp=1N∑i=1Nθ^i,p-θi,p2,(19)RMSEr=1N∑i=1Nθ^i,r-θi,r2,
where the main purpose of GA and PSO methods is to find the optimum values of g1,p and g1,r which minimize the RMSEp and RMSEr, respectively.IfXi and X^i, i=1,2,…,N are real measurement target position vector values and estimated target position vector values of adaptive α2-β2-γ2 filter, the RMSE of target tracking is defined as
(20)RMSE=1N∑i=1Nxi-x^i2+yi-y^i2+zi-z^i2,
where xi, yi, and zi are the components of target position vector Xi in x, y, and z axes. The main purpose of GA and PSO algorithms is to find the optimum gain value of g2 which minimizes the RMSE.


## 4. Experimental Results

Six scenarios are used to simulate the tracking performance of ship-borne phased array radar using the proposed adaptiveα-β-γ filter for beam pointing error compensation and target tracking in three-dimensional space.Case 1 (roll and pitch estimations using GA based adaptiveα-β-γ filter and measured data).
The roll and pitch data recorded by a gyroscope of the ship were used in the experiments. The total number of data is 1000 and the sampling time is 0.1 sec. The angle variation of roll and pitch signals measured by ship gyroscope are shown in Figure7, which will result in the beam pointing errors of phased array antenna. The GA based adaptive α-β-γ filter is used to estimate the roll and pitch angles. Figures 8(a) and 8(b) show that the convergent time of roll and pitch angles are 3 sec and 5 sec, respectively, under simulated sea states 2 and 3 and measured data. The ship’s roll and pitch signals are simulated according to 3 and 4 with roll and pitch angle parameters of sea states 2 and 3 listed in Table 1 and standard deviation of 0.01. It concludes that the proposed GA based adaptive α-β-γ filter is suitable to compensate the ship motion. As shown in Table 1, the period of sea state 2 is longer than sea state 3. Therefore, the convergent RMSE (about 0.19° for roll, about 0.1° for pitch) of sea state 2 is less than sea state 3 (about 0.29° for roll, about 0.22° for pitch) because the roll and pitch signals with shorter period in sea state 3 have rapid oscillations, which are more difficult to be predicted precisely.Figure 7
Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope.
(a)
(b)Figure 8
RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data.
(a)
(b)Case 2 (tracking linear moving target in three-dimensional space using GA based adaptiveα-β-γ filter (without beam pointing error)).
Assuming the ship is not affected by the sea waves (antenna beam pointing error is zero), the tracking performance of GA based adaptiveα-β-γ filter for a straight flight target trajectory is simulated. The initial position of radar is (0,0,0). The ship moves with a speed of 10 m/sec in the Y axial direction. The linear path equation of the ship is given by
(21)Xsxt,yt,zt=X0s+Vst,
where t=1,2,3…,1000 sec, Vs is ship velocity vector and X0s is initial ship position vector in three-dimensional space. The radar position is updated every second. The initial position of flying target is (-74840,-129620,9100) m, which is about 150 km from the radar. The flight speed of target is 300 m/sec (X axial velocity of 150 m/sec and Y axial velocity of 260 m/sec). The target location is updated every second. The linear target trajectory equation is
(22)Xmxt,yt,zt=X0m+Vmt,
where Vm is the target velocity vector and X0m is the initial target position vector in three-dimensional space. The tracking accuracy of GA based adaptive α-β-γ filter for a straight flight target is shown in Figure 9, where the trajectory estimation error is calculated by
(23)Et=xt-x^t2+yt-y^t2+zt-z^t2,
where (xt,yt,zt) and (x^t,y^t,z^t) denote the real target location and estimated target location, respectively. It shows that the GA based adaptive α-β-γ filter converges to less than about 2 m after 5 sec.Figure 9
Estimation error for linear target trajectory without beam pointing error.Case 3 (tracking circular moving target in three-dimensional space using GA based adaptiveα-β-γ filter (without beam pointing error)).
Figures10(a) and 10(b) show the simulation scenario of beam pointing error compensation for circular maneuvering air target and linear moving ship. The ship has the same linear moving speed and path equation as Case 2. The flying target is parallel to the XY plane, the initial position of flying target is (8885, 4590, 9100) m, the velocity is 300 m/sec, and the angle (θ0) between the initial position and the X-axis is 30°. The trajectory of the flight vehicle is a circle, which has a radius of 10 km. The center of circular trajectory is Y-axis. The rotational cycle time of the flight target is 209 seconds and the total observation time is 419 sec.
The circular trajectory equation is(24)Xmxt,yt,zt=rcos⁡θ0-ωt,rsinθ0-ωt,9100,
where t=1,2,3,…,418,419sec, r is the circular flight radius in meter, θ0 is the initial angle between the flight vehicle, and the X-axis, ω is the angular speed of flight vehicle. The accuracy of GA based adaptive α-β-γ filter for tracking a circular flight target is shown in Figure 11, where the GA based adaptive α-β-γ filter converges to less than 3 m after 8 sec. Therefore, using the GA based adaptive α-β-γ filter to track the circular trajectory target has the larger average estimation error and longer convergent time than linear trajectory target.Figure 10
Simulation scenarios of Case3 for (a) 3D and (b) top view diagrams.
(a)
(b)Figure 11
Estimation error for circular target trajectory without beam pointing error.Case 4 (estimating roll and pitch angles and tracking linear trajectory target using GA/PSO based adaptiveα-β-γ filter and measured data (with beam pointing error)).
The measured roll and pitch signals shown in Figure7 are used to evaluate the performance of GA/PSO based adaptive α-β-γ filter for estimating the roll and pitch angles and tracking the linear trajectory target. The beam of 6×6 planar array antenna is steered to track the linear trajectory target. The sampling frequency is set as 10 Hz. Therefore, the ship’s roll and pitch signals are sampled 10000 points within 1000 seconds. As the scenario described in Case 2, the ship is along the Y axis of XY plane in the earth coordinates, the ship speed is 10 m/sec, and the flying target speed is 300 m/sec. The flying target is parallel to the XY plane, 9.1 km above the ground, and the angle between the flight direction and the Y-axis is 120°. The largest reconnaissance distance of ship-borne radar is 150 km. When the air target is flying within the radar detection range, the beam pointing error estimation and linear trajectory target tracking of ship-borne phased array radar are simulated.
Figure12 is a simulation flow chart used to verify the effect of beam pointing error compensation on the tracking accuracy of adaptive α-β-γ filter. Tables 2 and 3 demonstrate the optimal gain parameters of adaptive α-β-γ filter for roll gr,1, pitch gp,1, and target tracking g2 obtained by the GA and PSO methods, respectively, that minimizes the RMSE. Figures 13–17 show the estimation errors for five different gain values, which are summarized in Table 4. It demonstrates that the greater gain value (g=0.1, 0.75 and 0.9) will generate the greater error and need the longer convergence time. The estimation errors presented in Figures 16 and 17 are too large to be applicable when the gain values are equal to 0.75 and 0.9. The GA and PSO have smaller errors compared with the fixed gain values, and the GA obtains the best estimation accuracy. The estimation errors of GA and PSO methods were found to be close. From Figures 13, 14, and 15, we can observe that there is a peak value that occurred at the time duration about 510 to 515 seconds, because the horizontal beam direction of ship-borne phased array radar is steered according to the linear trajectory target, which is flying through sky just above the ship at the time instant of 510 second. As shown in Figure 18, the horizontal beam steering angle of ship-borne phased array radar at about the time of 510 second is changing from descending into ascending. When the GA based adaptive α-β-γ filter tracks the linear trajectory target, the estimation error values are less than 30 meters or less if the peak estimation errors during these 5 seconds are ignored.Table 2
Estimated gain parameters for GA based adaptiveα-β-γ filter.
Time (sec)
1~100
101~200
201~300
301~400
401~500
501~600
601~700
701~800
801~900
901~1000
g
r
,
1
0.574
0.6399
0.652
0.6904
0.6537
0.674
0.6286
0.6313
0.6044
0.5392
g
p
,
1
0.629
0.6664
0.622
0.7756
0.664
0.7399
0.663
0.6652
0.6816
0.6498
g
2
0.285
0.5827
0.3282
0.3851
0.0054
0.3480
0.2916
0.3736
0.547
0.4864Table 3
Estimated gain parameters for PSO based adaptiveα-β-γ filter.
Time (sec)
1~100
101~200
201~300
301~400
401~500
501~600
601~700
701~800
801~900
901~1000
g
r
,
1
0.557
0.5986
0.357
0.5345
0.6142
0.5774
0.5544
0.529
0.4592
0.5626
g
p
,
1
0.612
0.5898
0.4129
0.5675
0.5286
0.3953
0.5565
0.6199
0.5214
0.6285
g
2
0.326
0.07
0.1136
0.146
0.098
0.2365
0.21
0.2547
0.2474
0.1245Table 4
Estimation errors for different gains.
Calculated by GA
Calculated by PSO
Fixed gain
g
r
,
1
0.5392~0.6904
0.357~0.6142
0.1
0.75
0.9
g
p
,
1
0.622~0.7756
0.3953~0.6285
0.1
0.75
0.9
g
2
0.054~0.5827
0.07~0.3259
0.1
0.75
0.9
AverageEt (m)
14.781
16.3224
16.9663
53.4804
399.1188Figure 12
Flow chart of tracking performance simulation for Case4 experiment.Figure 13
Estimation error of GA based adaptiveα-β-γ filter.Figure 14
Estimation error of PSO based adaptiveα-β-γ filter.Figure 15
Estimation error ofα-β-γ filter with gr,1=gp,1=g2=0.1.Figure 16
Estimation error ofα-β-γ filter with gr,1=gp,1=g2=0.75.Figure 17
Estimation error ofα-β-γ filter with gr,1=gp,1=g2=0.9.Figure 18
Azimuth beam steering angle of ship-borne phased array radar.Case 5 (estimating roll and pitch angles and tracking linear trajectory target using GA/PSO based adaptiveα-β-γ filter and simulated data (with beam pointing error)).
The ship’s roll and pitch signals are simulated according to3 and 4 with parameters of sea state 2 listed in Table 1 and standard deviation of 0.01. The simulated ship’s roll and pitch signals are applied for five different methods to estimate the gain of GA/PSO based adaptive α-β-γ filter. The total observation time is 1000 sec and sampling time is 0.1 sec. The simulation replicates 10000 times. The input samples are updated by one new sample for the next iteration. The other simulation scenario and parameters are the same as Case 4. Tables 5 and 6 demonstrate the optimal gain parameters of adaptive α-β-γ filter for roll gain gr,1, pitch gain gp,1, and target tracking gain vector g2 obtained by the GA and PSO methods, respectively, that minimizes RMSE. Figures 19, 20, 21, 22, and 23 show the estimation errors for five different gain values, which are summarized in Table 7. The estimation error of GA based adaptive α-β-γ filter for tracking a linear trajectory target is shown in Figure 19, where the convergence time of GA based adaptive α-β-γ filter is about 3 sec and the average estimation error is about 16.0756 m. It concludes that the experimental results in Case 5 have the same trend as in Case 4; therefore, the simulated data can be used to observe the effect of ship motion on the tracking performance of ship-borne phased array radar under different sea state conditions. The convergence time and average estimation error of GA based adaptive α-β-γ filter are better than the method used in [6], where the tracking error of AEKF converges to less than about 20 m at 550 iterations (sec/iteration) and the estimation error will remain within the range of about 20 m when the beam pointing error is compensated by FBFN controller.Table 5
Estimated gain parameters for GA based adaptiveα-β-γ filter.
Time (sec)
1~100
101~200
201~300
301~400
401~500
501~600
601~700
701~800
801~900
901~1000
g
r
,
1
0.543
0.5653
0.5424
0.5455
0.5665
0.56
0.58
0.532
0.5414
0.5385
g
p
,
1
0.464
0.484
0.4459
0.422
0.4667
0.4654
0.4509
0.4459
0.4308
0.4651
g
2
0.161
0.0929
0.1937
0.0865
0.0241
0.0609
0.1174
0.1897
0.1713
0.1727Table 6
Estimated gain parameters for PSO based adaptiveα-β-γ filter.
Time (sec)
1~100
101~200
201~300
301~400
401~500
501~600
601~700
701~800
801~900
901~1000
g
r
,
1
0.366
0.3645
0.532
0.4519
0.5151
0.5002
0.4797
0.5254
0.4146
0.5351
g
p
,
1
0.366
0.3647
0.3634
0.3644
0.3643
0.363
0.3644
0.3684
0.3643
0.3642
g
2
0.125
0.1271
0.2286
0.2166
0.1169
0.1799
0.1735
0.1564
0.0767
0.1346Table 7
Estimation errors for different gains.
Calculated by GA
Calculated by PSO
Fixed gain
g
r
,
1
0.459~0.583
0.3645~0.5352
0.1
0.75
0.9
g
p
,
1
0.498~0.667
0.363~0.3684
0.1
0.75
0.9
g
2
0.0586~0.1697
0.0367~0.2291
0.1
0.75
0.9
AverageEt (m)
16.0756
18.7684
19.9594
108.0515
427.9449Figure 19
Estimation error of GA based adaptiveα-β-γ filter.Figure 20
Estimation error of PSO based adaptiveα-β-γ filter.Figure 21
Estimation error ofα-β-γ filter with gr,1=gp,1=g2=0.1.Figure 22
Estimation error ofα-β-γ filter with gr,1=gp,1=g2=0.75.Figure 23
Estimation error ofα-β-γ filter with gr,1=gp,1=g2=0.9.Case 6 (estimating roll and pitch angles and tracking circular moving target using GA based adaptiveα-β-γ filter and measured data (with beam pointing error)).
As the scenario described in Case3, the measured roll and pitch signals shown in Figure 7 are used to evaluate the performance of GA based adaptive α-β-γ filter for estimating the roll and pitch angles and tracking the circular trajectory target. The estimation errors of α-β-γ filter with fixed gains of gr,1=gp,1=g2=0.1 and gr,1=gp,1=g2=0.75 are shown in Figures 24 and 25, respectively. The average estimation errors for fixed filter gains of 0.1 and 0.75 are about 4.1638 m and 57.2386 m, respectively. The optimal roll gain gr,1, pitch gain gp,1, and target tracking gain g2 of adaptive α-β-γ filter estimated by the GA method during the observation time of 419 sec are listed in Table 8. The estimation error of GA based adaptive α-β-γ filter for tracking a circular flight target is shown in Figure 26, where the convergence time of GA based adaptive α-β-γ filter is about 5 sec and the average estimation error is about 3.6667 m. It concludes that the GA based adaptive α-β-γ filter can greatly improve the tracking accuracy and convergence time of the conventional α-β-γ filter for tracking the circular trajectory target.Table 8
Estimated gain parameters for GA based adaptiveα-β-γ filter.
Time (sec)
1~42
43~84
85~126
127~168
169~210
211~252
253~294
295~336
337~378
379~419
g
r
,
1
0.619
0.4996
0.5316
0.644
0.5052
0.5568
0.5316
0.4459
0.5092
0.5924
g
p
,
1
0.582
0.6727
0.7201
0.4913
0.4168
0.7169
0.5983
0.2036
0.5858
0.621
g
2
0.214
0.2215
0.1612
0.2474
0.1822
0.157
0.2073
0.1211
0.2278
0.1074Figure 24
Estimation error ofα-β-γ filter with gr,1=gp,1=g2=0.1.Figure 25
Estimation error ofα-β-γ filter with gr,1=gp,1=g2=0.75.Figure 26
Estimation error of GA based adaptiveα-β-γ filter.


## 5. Conclusions

This paper proposed an intelligent beam pointing error compensation mechanism for ship-borne phased array radar. The GA based adaptiveα-β-γ filter estimates the roll and pitch angles of the ship moving in the sea and thus compensates for the antenna beam pointing error in order to enhance the tracking accuracy of phased array radar system.Six experimental cases are conducted to verify the performance of ship-borne phased array radar using the proposed GA based adaptiveα-β-γ filter. In Case 1, the GA based adaptive α-β-γ filter is used to estimate the roll and pitch angles with the simulated and measured roll and pitch signals, respectively. The simulation results show that the measurement data has the minimum estimation error, the estimation error in the sea state 2 is the second, and the estimation error in the sea state 3 is the worst. Cases 2 and 3 show that when the GA based adaptive α-β-γ filter is applied to estimate the beam pointing error and to track the target in a ship-borne phased array radar, the circular trajectory target will be tracked with the larger average estimation error and longer convergence time than linear trajectory target. The experimental results of Cases 4 and 5 demonstrate that when a linear trajectory target is tracked by ship-borne phased array radar, the average estimation error and convergence time of ship-borne phased array radar using the GA based adaptive α-β-γ filter are better than PSO based adaptive α-β-γ filter and fixed filter gain α-β-γ filter. The convergence time and tracking accuracy of ship-borne phased array radar using the proposed GA based adaptive α-β-γ filter are superior to the AEKF significantly. In conclusions, the proposed GA based adaptive α-β-γ filter is a real time applicable algorithm whose accuracy is good with relatively less complexity. In addition, the roll and pitch signals measured in real operation environments can be replaced with the simulated roll and pitch signals to test all the relevant properties of practical ship motion compensation and air target tracking for ship-borne phased array radar.



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*Source: 563726-2015-05-20.xml*