Simulation-based Optimal Design of α-β-γ-δ Filter

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1 International Journal of Automation and Computing 7(2), May 2010, DOI: /s Simulation-based Optimal Design of α-β-γ-δ Filter Chun-Mu Wu 1 Paul P. Lin 2 Zhen-Yu Han 3 Shu-Rong Li 3 1 Department of Mechanical and Automation Engineering, Kao Yuan University, aiwan 821, aiwan, China 2 College of Engineering, Cleveland State University, Cleveland Ohio , USA 3 College of Information and Control Engineering, China University of Petroleum, Dongying , China Abstract: he existing third-order tracker known as α-β-γ filter has been used for target tracking and predicting for years. he filter can track the target s position and velocity, but not the acceleration. o extend its capability, a new fourth-order target tracker called α-β-γ-δ filter is proposed. he main objective of this study was to find the optimal set of filter parameters that leads to minimum position tracking errors. he tracking errors between using the α-β-γ filter and the α-β-γ-δ filter are compared. As a result, the new filter exhibits significant improvement in position tracking accuracy over the existing third-order filter, but at the expense of computational time in search of the optimal filter. o reduce the computational time, a simulation-based optimization technique via aguchi method is introduced. Keywords: Filter design, simulation-based optimization, aguchi methods, position tracking, velocity tracking, acceleration tracking. 1 Introduction Since the introduction of the α-β filter for target tracking used in radar systems [1], there have been many derivations of the optimal relationship between the tracking parameters for the second-order and third-order of the α-β filter [2 4]. Later, a new parameter called tracking index to characterize the behaviors of α-β-γ filters was developed [5]. In 2002, an optimal design of α-β-γ filters was studied [6] in which a constrained parameter optimization problem was formulated, where α, β, and γ were bounded to lie within the stability volume. More specifically, the study found the optimal ranges, but not the optimal values, for the α, β, and γ parameters. Unlike the α-β filter, the α-β-γ filter is capable of tracking an accelerating target without steady-state error [6]. More specifically, the third-order filter can predict the target s next position and velocity based on the current and past positions, velocities, and accelerations. o further predict the acceleration, a fourth-order filter called α-β-γδ filter will have to be developed. For the new filter, an additional state called jerk which is a time derivative of acceleration will need to be observed. he addition of jerk, however, greatly complicates the searching for the optimal filter parameters (i.e., α, β, γ, and δ). raditionally, the Monte Carlo method has been widely used for simulations. It is, however, very time consuming and therefore inefficient. In contrast, the aguchi method provides an efficient way to evaluate the contribution of each factor, and find the optimal set of factors that will optimize the output value. It is worth noting that the aguchi method requires only some pre-selected input-output data which can be the experimental data. he method is therefore best suited in a pure data-driven environment where the object or target s function is not known in advance. From that perspective, the traditional optimization techniques that require a well-defined objective function are not considered in this study. Conceptually speaking, the new filter acts like an observer that predicts the next value of Manuscript received February 23, 2009; revised July 27, 2009 each state (such as position, velocity, and acceleration). his paper is organized as follows. Section 2 introduces the derivations of the proposed α-β-γ-δ filter. Section 3 gives the performance comparisons between α-β-γ filter and α-β-γ-δ filter via numerical simulations of tracking a highly nonlinear function. Section 4 briefly introduces the aguchi method, followed by simulation-based optimal design by means of the aguchi method described in Section 5. Section 6 gives the conclusions that state the advantages of the proposed new fourth-order filter. Notations. X p, X s, X o: Predicted position, smoothed position, observed position. V p, V s, V o: Predicted velocity, smoothed velocity, observed velocity. a p, a s, a o: Predicted acceleration, smoothed acceleration, observed acceleration. J s, J o: Smoothed jerk, observed jerk. α, β, γ, δ: Filter parameters. : ime step. f: racking frequency. t: ime elapsed in simulation. t start: Starting time in simulation. t max: Maximum time allowed in simulation. L mean: Mean squared error. L max: Maximum absolute error. n: Number of steps. 2 Derivations of the α-β-γ-δ filter he α-β-γ-δ filter is a fourth-order tracker capable of predicting the object s next position, velocity, and acceleration by tracking the current and past positions, velocities, accelerations, and jerks. x p(k + 1) = x s(k) + v s(k) a s(k) J s(k) (1) v p(k + 1) = v s(k) + a s(k) J s(k) (2) a p(k + 1) = a s(k) + J s(k) (3)

2 248 International Journal of Automation and Computing 7(2), May 2010 where is time step or time increment, and subscripts p and s denote the predicted and smoothed state values, respectively. he smoothed parameters are very dependent on the discrepancy between the observed and predicted values, and the filter parameters (α, β, γ, and δ). Note that an additional state called jerk is used here for the smoothing purpose. x s(k) = x p(k) + α[x o(k) x p(k)] (4) v s(k) = v p(k) + β [xo(k) xp(k)] (5) a s(k) = a p(k) + γ [xo(k) xp(k)] 2 2 (6) J s(k) = J s(k 1) + δ [xo(k) xp(k)] (7) 6 3 where subscript o denotes the observed state values. In mathematics and signal processing, Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation. It is like a discrete equivalent of the Laplace transform. Just as analog filters are designed using the Laplace transform, recursive digital filters are developed with a parallel technique called the Z-transform. he overall strategy of these two transforms is the same: probe the impulse response with sinusoids and exponentials to find the system s poles and zeros. he Laplace transform deals with differential equations, the s-domain, and the s-plane. Correspondingly, the Z-transform deals with difference equations, the z-domain, and the z-plane. However, the two techniques are not a mirror image of each other; the s- plane is arranged in a rectangular coordinate system, while the z-plane uses a polar format. Applying the Z-transform to (1) through (7) and solving for the ratio x p/x o leads to the transfer function in z-domain as shown in (8). he roots of the characteristic polynomial (CP) (i.e., the denominator of the transfer function) are required to lie within the unit circle to guarantee stability). Jury s stability test [7] yields the constraints on the α, β, γ, and δ parameters for the α-β-γ-δ filter as follows. 0 < α < 2 (9) 0 < β < 13 (4 2α) (10) 6 0 < γ < 4αβ (2 α) (11) 0 < δ < 24(2 α). (12) For the comparison purpose, the constraints on the α-β-γ filter are 0 < α < 2 (13) 0 < β < 4 2α (14) 0 < γ < 4αβ (2 α). (15) In comparison with the α-β-γ filter, the α-β-γ-δ filter has a large constraint range for the additional parameter δ, and the range for β is more than doubled. his indeed, makes the search for the optimal parameter values much more time consuming. 3 Comparisons between α-β-γ and α-βγ-δ filters via numerical simulations 3.1 racking error calculation o compare the position tracking errors between using the α-β-γ filter and α-β-γ-δ filter, the following nonlinear function is chosen: x t = A t cos(2πft) (16) y t = B t sin(2πft) (17) z t = 1 lg(t 3 5t + 1) (18) where f is the tracking frequency, t is time, and A t and B t are defined by A t = 2.5(1 + t ) (19) t max B t = 1.5(1 + t t max ) (20) where t max is the preset maximum time for the simulation. he α-β-γ-δ tracker requires four time steps for initiation. j s(4) = a s(4) = x p(4) = x s(4) = x o(4) v s(4) = xo(4) xo(3) xo(4) + xo(2) 2xo(4) 2 xo(4) xo(1) 3(xo(3) xo(2)) 3. In comparison, the α-β-γ tracker requires only three time steps for initiation. a s(4) = x p(4) = x s(4) = x o(4) v s(4) = xo(4) xo(3) xo(4) + xo(2) 2xo(3) 2. he tracking error is quantified in terms of mean squared error (L mean) and maximum absolute error (L max) as shown below: n (x p(i) x o(i)) 2 L mean = i=1 n (21) L max = max x p(i) x o(i), i = 1, 2,, n (22) n = (tmax tstart). G(z) = xp x o = (α + β + γ + δ 4 36 )z3 + ( 3α 2β + δ 9 )z2 + (3α + β γ + δ )z α 4 36 z 4 + (α + β + γ + δ )z3 + ( 3α 2β + δ + 9 6)z2 + (3α + β γ + (8) δ 4)z + (1 α). 4 36

3 C. M. Wu et al. / Simulation-based Optimal Design of α-β-γ-δ Filter Numerical simulation A simulation was performed for each type of error. he ranges for α, β, γ, and δ parameters are [ ], [ ], [ ], and [0.1 45], respectively. he low and high ends of each range are determined by checking with the constraints listed in (9) (12). he increment for each of the four parameter values was chosen as 0.1. Each error was calculated with four different time steps (i.e., = 0.4, 0.2, 0.1, and 0.05), where t start = 2.5s. and t max = 100 s. he corresponding tracking parameters (i.e., α, β, γ, and δ) were recorded where the minimum of L mean and that of L max were found. A set of parameter combinations was discarded whenever it violates the constraints. he time step plays an important role in tracking error. Figs. 1 and 2 show the difference between using = 0.4 and = 0.05 with the α-β-γ-δ filter. he tracking error between the two filters in terms of L mean and L max are compared and shown in ables 1 and 2. he accuracy improvement shown in each table is the use of α-β-γ-δ filter without optimal search versus the use of α-β-γ filter without optimal search, based on L mean and L max, respectively. ables 1 and 2 clearly indicate that the α-β-γ-δ filter consistently outperforms the α-β-γ filter for every given time step. As expected, the tracking error monotonically decreases with decreased time step. In other words, the tracking accuracy improves as the time step decreases. When examining the accuracy improvement listed in able 1, it may appear that the trend is reversed between at = 0.4 s. and = 0.2 s. his is because the accuracy improvement percentage is the result of using the α-β-γ-δ filter in comparison with that of using the α-β-γ filter. Closely comparing the magnitudes of L mean between the two cases when using the α-β-γ-δ filter, one will find that the tracking errors greatly reduced from e-004 to e-005. In other words, the tracking accuracy improves by more than four times when the time step is reduced by half. In addition to tracking errors, the computational time has also been studied, and the result is shown in able 3. he computational time listed in this table is the time needed to calculate the two types of errors without optimal searches. Fig.1 racking plot at = 0.4 with the α-β-γ-δ filter able 3 Computational time comparison between the α-β-γ filter and α-β-γ-δ filter ime Number α-β-γ filter without α-β-γ-δ filter without step of steps optimal search optimal search s s s s s s s s Fig.2 racking plot at = 0.05 with the α-β-γ-δ filter able 3 clearly shows that the proposed α-β-γ-δ filter is much more computationally expensive than the α-βγ filter. his is mainly due to the additional parameter δ and its wide range. o reduce the optimum searching time, a simulation-based optimization technique by means of aguchi method is employed. able 1 racking error comparison on L mean between the α-β-γ filter and α-β-γ-δ filter ime α-β-γ filter without α-β-γ-δ filter without Accuracy 0.4 L mean = e-004 (α = 1.3, β = 1.0, γ = 4.3) L mean = e-004 (α = 0.4, β = 2.7, γ = 1.8, δ = 4.2) 15.4% 0.2 L mean = e-005 (α = 1.4, β = 0.9, γ = 5.9) L mean = e-005 (α = 0.3, β = 3.2, γ = 2.1, δ = 9.7) 11.3% 0.1 L mean = e-005 (α = 1.5, β = 0.8, γ = 7.8) L mean = e-005 (α = 0.2, β = 3.5, γ = 1.4, δ = 15.1) 19.7% 0.05 L mean = e-006 (α = 1.6, β = 0.7, γ = 9.0) L mean = e-006 (α = 0.2, β = 3.6, γ = 1.4, δ = 16.5) 31.8% able 2 racking error comparison on L max between the α-β-γ filter and α-β-γ-δ filter ime α-β-γ filter without α-β-γ-δ filter without Accuracy 0.4 L max = (α = 1.2, β = 0.8, γ = 4.7) L max = (α = 0.3, β = 2.6, γ = 1.3, δ = 8.4) 8.4% 0.2 L max = (α = 1.6, β = 0.6, γ = 9.4) L max = (α = 0.1, β = 3.6, γ = 0.7, δ = 22.9) 10.8% 0.1 L max = (α = 1.6, β = 0.7, γ = 10.7) L max = (α = 0.1, β = 3.8, γ = 1.3, δ = 8.4) 18.6% 0.05 L max = (α = 1.6, β = 0.7, γ = 11.0) L max = (α = 0.1, β = 3.8, γ = 0.7, δ = 22.6) 35.7%

4 250 International Journal of Automation and Computing 7(2), May Introduction to aguchi method aguchi methods are a statistical process that perturbs a parameter in order to study its influence on the overall output. aguchi methods strength lies in their ability to extract relatively large amounts of information from limited experiments (fractional factorial compared to full factorial). he basic tools used to obtain the information are orthogonal arrays and linear graphs. he former is more useful, and therefore will be elaborated more. An orthogonal array contains the number of experimental runs, the number of levels of each input factor or parameter (such as two levels: high and low), and the number of columns in the array. In an orthogonal array, every input factor is placed in one of the columns. A linear graph contains the relationship of input factor interactions. aguchi has created a transformation of the repetition data to another value, which is a measure of the variation present. his transformation is the signal-to-noise (S/N) ratio [8,9]. By examining the S/N ratios, the significant factors can be identified. he following four steps summarize the procedure for using the aguchi s orthogonal array. Step 1. Find the appropriate orthogonal array. For illustration purposes, able 4 shows a popular orthogonal array, L 27 (3 13 ), the subscript 27 stands for 27 experiment runs, the 3 stands for three levels (1 for low, 2 for medium, and 3 for high), and the superscript 13 stands for 13 columns which are for control factors and their interactions. More specifically, columns 1, 2, 4, and 7 are reserved for the control factors (such as α, β, γ, and δ) and the remaining columns are reserved for the factors interactions. able 4 aguchi s orthogonal array L 27(3 13 ) Run No. Column No Step 2. Construct the orthogonal array. Starting with the first control factor (column 1) in able 4, the values for the first nine experiments or cases are set at the low end (i.e., the lowest in the specified range), whereas the values for the next nine are set at the middle (i.e., the mean), and the remaining other nine are set at the high end (i.e., the highest). he process is repeated for each of the other parameters. Step 3. Find the corresponding output. For each experiment run, the corresponding output is recorded. hus, for the case of L 27(3 13 ), a total of 27 experiment runs are to be recorded. Step 4. Perform aguchi analysis. When performing aguchi analysis, the S/N ratio for each experiment run is computed and recorded. he S/N ratio is an indication of significance. here are three commonly-used types of S/N ratios: the-larger-thebetter, the-nominal-the-better, and the-smaller-the-better, depending on the type of design objective. For instance, the formula for calculating the S/N ratio for the-smallerthe-better type [8] is n η = 10lg[ yi 2 i=1 n ] (23) where n is the number of experiment runs, and y i is the corresponding output value for each run. 5 Simulation-based optimal design by means of aguchi method 5.1 Finding the optimal set of parameters In this study, the aguchi method was applied to optimal design of the α-β-γ-δ filter. Each of the four control factors (i.e., α, β, γ, and δ) has seven levels. hus, there are a total of 2401 experiment runs based on full-factorial design. Note that fractional-factorial design is often used in aguchi method in order to reduce the number of experiment runs. However, it was not considered in this study due to the fear of missing critical parameter combinations. Besides, computation of 2401 runs is not considered excessive. he seven levels used in the aguchi s orthogonal array are α : [0.05, 0.1, 0.2, 0.4, 0.8, 1.6, 1.9] β : [0.1, 0.6, 0.9, 1.35, 2.0, 3.5, 3.9] γ : [0.1, 0.4, 0.8, 1.3, 3.2, 6.4, 15.9] δ : [4, 6, 9,14, 20, 30, 45]. As can be seen, the levels do not need to be equally spaced. he selected level values, however, should satisfy all the constraints in order for the S/N ratios to be calculated. he optimal design objective is to minimize the target s position tracking errors (L mean and L max). he S/N ratio for the-smaller-the-better is calculated by [ ] 1 n η = 10 lg (x p(i) x o(i)) 2. (24) n i=0 All the S/N ratios are compared, and the experiment run is recorded for the highest η value. It should be noted that

5 C. M. Wu et al. / Simulation-based Optimal Design of α-β-γ-δ Filter 251 the η value is the-higher-the-better even though the formula is for the-smaller-the-better. his is simply because a negative sign has been added in front of the formula. he parameter design of aguchi method works well only with dimensional constraints. he constraints derived by satisfying the Jury s stability test are, however, a mixture of dimensional and functional constraints. his implies that some experiment runs may violate the constraints. When this occurs, a very large value of (x p(i) x o(i)) is assigned, which leads to a large negative number that will be automatically discarded. It is worthwhile to note that the parameter design in the aguchi method only finds the set of control factors (i.e., the experiment run) that achieves the optimum design objective. However, the optimal design solution does not necessarily reside at one of the 2401 runs. o address this concern, the selected best experiment run (or parameter combination) is further examined. 5.2 Parameter fine-tuning he parameters of the selected set of run can be finetuned. If, for instance, the best level for parameter α was found at level 4, the best parameter level could actually lie between levels 3 and 4 or between levels 4 and 5. o find the exact parameter level, the η values at levels 3, 4, and 5 are fitted to a second-order curve from which the parameter level corresponding to the highest η value can be found. Note that each parameter level has its corresponding parameter value. he process is repeated for every parameter. his parameter fine-tuning technique is, of course, based on the assumption that the number of levels for each parameter is appropriate. A detailed description of simulation-based design optimization can be found in [10]. he tracking errors are compared for each type of error with and without optimal searches. he results are listed in ables 5 and 6. ables 5 and 6 indicate that the optimized α-β-γ-δ filter sacrifices little in L mean error, but much more in L max error except when the time step is 0.2 or 0.4. he fine-tuning seems to affect only the β and γ values. In most cases, β was the only parameter needed to be fine-tuned. his is because β was found to be the most sensitive or significant factor in the orthogonal array. his finding also suggests that it is more desirable to increase the number of levels for β. he ability to identify the most significant parameter provides a direct means to further improve the output performance without having to increase the number of levels for each parameter. his leads to less experimental runs or less computational time. he computational time for the α-β-γ-δ filter is greatly reduced, as shown in able 7. able 7 Computational time for the α-β-γ-δ filter with and without optimal search ime α-β-γ-δ filter without α-β-γ-δ filter with step optimal search optimal search s s s s s s s s It is interesting to note that even though the α-β-γ-δ filter with optimal search requires additional vigorous search for the δ parameter, its computational time is comparable to that for the α-β-γ filter without optimal search. able 8 shows the comparison. able 8 Comparison of computational time between using the α-β-γ filter without optimal search and the α-β-γ-δ filter with optimal search ime α-β-γ filter without α-β-γ-δ filter with step optimal search optimal search s s s s s s s s able 5 racking error comparison on L mean with and without optimal searches ime α-β-γ-δ filter without α-β-γ-δ filter with Accuracy 0.4 L mean = e-004 L mean = e-004 (α = 0.4, β = 2.7, γ = 1.8, δ = 4.2) (α = 0.4, β = , γ = 1.3, δ = 4) 1.68% 0.2 L mean = e-005 L mean = e-005 (α = 0.3, β = 3.2, γ = 2.1, δ = 9.7) (α = 0.2, β = , γ = 1.3, δ = 9) 2.56% L mean = e-005 L mean = e-005 (α = 0.2, β = 3.5, γ = 1.4, δ = 15.1) (α = 0.2, β = , γ = 1.36, δ = 14) L mean = e-006 L mean = e-006 (α = 0.2, β = 3.6, γ = 1.4, δ = 16.5) (α = 0.2, β = , γ = 1.3, δ = 14) able 6 racking error comparison on L max with and without optimal searches 4.41% 3.13% ime α-β-γ-δ filter without α-β-γ-δ filter with Accuracy 0.4 L max = (α = 0.3, β = 2.6, γ = 1.3, δ = 8.4) L max = (α = 0.1, β = , γ = 0.4, δ = 9) 1.34 % 0.2 L max = (α = 0.1, β = 3.6, γ = 0.7, δ = 22.9) L max = (α = 0.2, β = , γ = 1.3, δ = 14) 8.07 % 0.1 L max = (α = 0.1, β = 3.8, γ = 1.3, δ = 8.4) L max = (α = 0.2, β = , γ = 1.3, δ = 14) 21.69% 0.05 L max = (α = 0.1, β = 3.8, γ = 0.7, δ = 22.6) L max = (α = 0.2, β = 3.772, γ = 1.3, δ = 14) 33.33%

6 252 International Journal of Automation and Computing 7(2), May 2010 able 9 racking error comparison on L mean between α-β-γ filter without optimal search and α-β-γ-δ filter with optimal search ime α-β-γ filter without α-β-γ-δ filter with Accuracy L mean = e-004 L mean = e-004 (α = 1.3, β = 1.0, γ = 4.3) (α = 0.4, β = , γ = 1.3, δ = 4) L mean = e-005 L mean = e-005 (α = 1.4, β = 0.9, γ = 5.9) (α = 0.2, β = , γ = 1.3, δ = 9) L mean = e-005 L mean = e-005 (α = 1.5, β = 0.8, γ = 7.8) (α = 0.2, β = , γ = 1.36, δ = 14) L mean = e-006 L mean = e-006 (α = 1.6, β = 0.7, γ = 9.0) (α = 0.2, β = , γ = 1.3, δ = 14) % 9.05% % % able 10 racking error comparison on L max between α-β-γ filter without optimal search and α-β-γ-δ filter with optimal search ime step α-β-γ filter without optimal search α-β-γ-δ filter with optimal search Accuracy improvement 0.4 L max = (α = 1.2, β = 0.8, γ = 4.7) L max = (α = 0.1, β = , γ = 0.4, δ = 9) 7.17% 0.2 L max = (α = 1.6, β = 0.6, γ = 9.4) L max = (α = 0.2, β = , γ = 1.3, δ = 14) 3.60% 0.1 L max = (α = 1.6, β = 0.7, γ = 10.7) L max = (α = 0.2, β = , γ = 1.3, δ = 14) 0.98% 0.05 L max = (α = 1.6, β = 0.7, γ = 11.0) L max = (α = 0.2, β = 3.772, γ = 1.3, δ = 14) 14.29% In addition to the comparable computational time, the tracking accuracy between the two filters are compared and shown in ables 9 and 10. he results tabulated in ables 9 and 10 clearly demonstrate the superiority of the new α-β-γ-δ filter with optimal search over the existing α-β-γ filter without optimal search in terms of tracking accuracy, and yet the new filter only takes a little more computational time. 6 Conclusions he proposed new α-β-γ-δ filter has been described and compared with the existing α-β-γ filter. he biggest advantage of using the new filter is its ability to quickly track and predict the targets in a pure data-driven environment, such as missile tracking and interception where the function of the target s moving path is unknown. he simulation results indicated that the new filter offers significant improvement over the existing one in terms of position tracking accuracy. he only drawback of the filter is the expensive computational time, which has been addressed with the presented simulation-based optimization technique by means of aguchi method. he optimal solution found from the traditional aguchi method is limited to pre-selected parameter combinations (i.e., preset parameter levels). he technique presented in this paper is capable of fine-tuning the solution between two identified levels. hus, it is essentially an extension of the existing aguchi method. Evolution algorithms (EAs) [11] are a wide class of randomized problem solvers based on principles of biological evolution. hey have been used in many computational areas such as optimization, learning, and others [12,13]. However, they were not considered in this study because they generally take more time to find the optimal solution than the presented simulation-based optimization technique that is based on the extended aguchi method. Overall, this study shows that the proposed new α-β-γ-δ filter in conjunction with the simulation-based optimization technique is superior to the existing α-β-γ filter in terms of target tracking accuracy. With the technique, the time needed to search for the optimal parameter values was greatly reduced. It is worthwhile to note that the new filter not only results in better position tracking accuracy, but also provides the ability to track the object s acceleration which the existing α-β-γ filter is incapable of. he new filter guarantees tracking stability since all the parameter constraints were yielded from the Jury s stability test. his means that the constraints as described in (9) (12) remain the same regardless of the application (i.e., the moving path). In other words, the presented formulation for the new filter and the optimal design methodology by means of the aguchi method remain valid even if the tracking data changes. References [1] J. Sklansky. Optimizing the Dynamic Parameter of a rackwhile-scan System, New Jersey, USA: RCA Laboratories, Princeton, [2]. R. Benedict, G. W. Bordner. Synthesis of an optimal set of radar track-while-scan smoothing equations. IRE ransactions on Automatic Control, vol. 7, no. 4, pp , [3] H. R. Simpson. Performance measure and optimization condition for a third-order sample-data tracker. IEEE ransactions on Automatic Control, vol. 8, no. 2, pp , [4] S. R. Neal. Discussions on parametric relations for the α-β-γ filter predictor. IEEE ransactions on Automatic Control, vol. 12, no. 3, pp , [5] P. R. Kalata. he tracking index: A generalized parameter for α-β and the α-β-γ trackers. IEEE ransactions on Aerospace and Electronic Systems, vol. AES-20, no. 2, pp , [6] D. enne,. Singh. Characterizing performance of α-β-γ filters. IEEE ransactions on Aerospace and Electronic Systems, vol. 38, no. 3, pp , 2002.

7 C. M. Wu et al. / Simulation-based Optimal Design of α-β-γ-δ Filter 253 [7] K. Ogata. Discrete-ime Control Systems, Engelwood Cliffs, New Jersey, USA: Prentice-Hall Inc., [8] M. S. Phadke. Quality Engineering Using Robust Design, Prentice-Hall Inc., pp , [9] P. J. Ross. aguchi echniques for Quality Engineering, 2nd Edition, New York, USA: McGraw-Hill Inc., [10] P. P. Lin, K. Jules. Fast multidisciplinary design optimization via aguchi method and soft computing. AIAA Journal of Aerospace Computing, Information and Communication, vol. 3, no. 6, pp , [11] A. E. Eiben, G. Rundolph. heory of evolutionary algorithms: A bird s eye view. heoretical Computer Science, vol. 229, no. 1 2, pp. 3 9, [12] S. C. Chian, K. C. an, A. Al Mamum. Evolutionary multiobjective portfolio optimization in practical context. International Journal of Automation and Computing, vol. 5, no. 1, pp , [13] P. S. Oliveto, J. He, X. Yao. ime complexity of evolutionary algorithms for combinatorial optimization: A decade of results. International Journal of Automation and Computing, vol. 4, no. 3, pp , Chun-Mu Wu received the B. Sc. and M. Sc. degrees from amkang University, aiwan, China, and Ph. D. degree from National Cheng Kung University, aiwan, China. He is currently an associate professor in Department of Mechanical and Automation Engineering at Kao Yuan University, aiwan, China. His research interests include numerical analysis, heat transfer analysis, automation engineering, and system monitoring. wtm@cc.kyu.edu.tw Paul P. Lin received the B. Sc. degree from atung University, aiwan, the M. Sc. degree from University of Florida, USA, and Ph. D. degree from University of Rhode Island, USA. He is currently an associate dean of College of Engineering and a professor of mechanical engineering at Cleveland State University, USA. He is a fellow of American Society of Mechanical Engineers (ASME) and the recipient of many awards including 2009 Cleveland Engineering Society Leadership Award. His research interests include intelligent systems, fault diagnosis, robotics, optimization, and optical inspection. p.lin@csuohio.edu (Corresponding author) Zheng-Yu Han received the B. Sc. degree from China University of Petroleum, China in He is currently a Ph. D. candidate at the College of Information and Control Engineering, China University of Petroleum. His research interests include optimal control and robotics, especially the control of robots and trajectory planning. hanzhenyu0315@gmail.com Shu-Rong Li received the B. Sc. degree from Shandong University, China in 1987, and M.Sc. and Ph.D. degrees from Chinese Academy of Sciences, China in 1990 and He is currently a professor at the College of Information and Control Engineering, China University of Petroleum, China. His research interests include nonlinear system, optimal control, and robotics. lishuron@hdpu.edu.cn