Reformative artificial bee colony algorithm based PID controller for radar servo system

1.   Introduction

Radar servo systems (RSS) are widely used in various fields, such as meteorology, satellites, rockets, etc. ^[1]. As the diversity of tasks and the complexity of the environment in these fields increase, the requirements for the stability and tracking accuracy of the RSS are further increased ^[2]. However, compared with other servo systems, the RSS is a complex system with uncertainties. First, parameters in the RSS, such as drag, and viscosity, change with the environment. Second, the radar antenna orientation will be deflected under interference. In addition, the position change of the tracking target is random ^[3]. Since the tracking performance of the RSS is affected by many uncertainties, it is necessary to design a complex robust controller for it ^[4].

Current controllers for servo systems are sliding mode controllers (SMC), the neural network controller (NNC), and the proportional-integral-derivative controller (PID) ^[5]. SMC is widely used for servo systems. Integrations of SMC with intelligent methods, such as SMC with fuzzy ^[6], SMC with neural network ^[7], SMC with genetic algorithm (GA) ^[8], and SMC with adaptive backstepping ^[9,10] were proposed to reduce the jitter phenomenon and ensure stability. NNC is another promising controller. Cao et al. ^[11] designed a composite neural network adaptive controller for RSS. Lu et al. ^[12] designed an adaptive neural network controller to achieve accurate control of an RSS with time-varying state constraints. Huang et al. ^[13] developed an NNC based on a sliding mode variable structure for improving the stability of the missile servo system. He et al. ^[14] proposed a neural network parallel self-learning tracking controller that was used in large-scale radar control. Gong et al. ^[15] proposed a monitoring NNC in which feedback is used in the initial stage to further improve the tracking capability of the radar.

The structure of the SMC is not fixed but is constantly changing as the system state changes, resulting in a range of parameter variations and disturbances. The disadvantages of NCC are slow parameter updates and poor control performance. Therefore, designing a good controller for RSS is still a challenge.

The PID controller is still receiving attention in various fields due to its simplicity and ease of implementation ^[16]. Chen et al. ^[17] designed a neuronal adaptive PID controller to improve the response speed of the RSS. However, optimization of the PID controller poses a challenging problem for selecting the right parameters. Swarm intelligence (SI) algorithms have been applied in the real world ^{[18,19,20,21]}. SI algorithms have attracted a lot of attention for the optimization of PID controllers in recent years. Özdemir et al. ^[22] proposed the optical-inspired optimization (OIO) algorithm to optimize the PID controller. The proposed controller has better performance in terms of maximum overshoots and stabilization time. Chen et al. ^[23] proposed a fuzzy PID controller optimized by an improved ant colony (IACO) algorithm. Nonlinear incremental evaporation rates and pheromone incremental updates were employed in the IACO algorithm to improve the quality of the solution. Hekimoğlu et al. ^[24]. proposed the atomic search optimization (ASO) algorithm to determine the optimal parameters of fractional-order proportional-integral-derivative (FOPID) controller. Although many SI algorithms have been applied to PID controller optimization, some SI algorithms exhibit local convergence or achieve global solutions at the cost of a high computational burden. Therefore, other SI algorithms are further tried to achieve faster convergence and globally optimal solutions.

The ABC algorithm is a SI algorithm proposed by Karaboga by simulating the characteristics of honey bee colony honey harvesting behavior ^[25], which is easy to implement with few parameters. The motivation for this paper was to develop an ABC-PID controller for improving the tracking accuracy of the RSS. This paper substantially extends the previous work on PID controllers based on SI algorithms, and can be distinguished from it from three aspects as follows:

● The application of PID controller based on ABC algorithm in RSS has been rarely reported. Here is an addition to the PID-SI controller application. The PID controller based on the RABC algorithm (RABC-PID) is proposed to apply to the RSS.

● A RABC optimization algorithm is conducted in this work. In the RABC algorithm, the solution search equation and select probability are modified to balance its explorative and exploitative capabilities.

● The superiority of the RABC-PID controller is confirmed against various input tracking signals and the RSS parameters uncertainties.

The rest of this paper is organized as follows. Section 2 introduces the position loop model of the RSS, and Section 3 introduces the basic ABC algorithm, the RABC algorithm, and the superiority of the RABC algorithm verified by six benchmark functions. The principle of optimized PID controller parameters and the whole optimization process of the RABC algorithm is introduced. Experimental simulations are performed in Section 4 to verify the superiority of the RABC algorithm by comparing it with three other SI algorithms. Section 5 concludes the full paper.

2.   RSS modeling

The RSS is a multi-loop feedback controls system, which is composed of three closed loops: current loop, speed loop, and position loop ^[26]. The current, speed and position feedback are completed by corresponding sensors ^[27]. The structure of the RSS is shown in Figure 1.

2.1.   Current loop model of RSS

The structure of the current loop is shown in Figure 2. Low-pass filters are added to the feedback loop and input to improve the quality of the feedback signal.

In Figure 2, U₁(s) is the current loop input; ACR is the current controller; T_oi is the current filtering time constant; $\alpha$ is the current feedback coefficient.

Since the inverter timed constant T_PWM and the filter time constant T_oi is much smaller than T₁. Thus multiple small inertia links are approximated as one inertia link, and the following expression is given by Eq (1).

When Eq (2) is satisfied, the transfer function of the current loop can be given by Eq (3)

where K_PWM is the amplification factor; T₁ is the electromagnetic time constant, R is the average resistance of armature winding, ${\omega _{{\text{oi}}}}$ and is the cut-off frequency for the current loop.

The PI controller is used for the current loop, and its transfer function can be given by the following equation (4)

where K_i is the proportional coefficient of the PI controller; ${\tau _{\text{i}}}$ is the overrun time constant of the PI controller.

If ${\tau _{\text{i}}}$ = T₁, then the open-loop transfer function of the current loop is expressed as Eq (5)

where K_I is a constant and ${K_{\text{I}}} = {{{K_{\text{i}}}{K_{{\text{PWM}}}}\alpha } / {{\tau _{\text{i}}}R}}$.

The transfer function of the closed-loop of the current loop can be given by equation(6)

2.2.   Speed loop model of RSS

When Eq (7) is satisfied, The transfer function of the closed-loop of the current loop can be simplified as Eq (8)

where ${\omega _{{\text{cn}}}}$ is the cut-off frequency for the speed loop.

If Eq (8) is considered as a link in the speed loop, then the structure of the speed loop is shown in Figure 3.

In Figure 3, U_n(s) is the speed loop input; ASR is the speed controller; T_on is the speed filter time constant; β is the speed feedback coefficient.

Multiple small inertia links are approximated as one inertia link, and the following expression is given by Eq (9)

The PI controller is used for the speed loop, and its transfer function can be given by the following Eq (10)

where K_n is the proportional coefficient of the PI controller; τ_n is the overrun time constant of the PI controller. The open-loop transfer function of the speed loop can be given by

Equation (11) can be simplified as Eq (12)

where K_N is a constant and ${K_{\text{N}}} = {{{K_{\text{n}}}\beta R} / {{\tau _{\text{n}}}{C_{\text{e}}}{T_{\text{m}}}}}$.

If higher-order terms are ignored, according to Eq (12) the transfer function of the closed-loop of the speed loop can be given by Eq (13)

2.3.   Position loop model of RSS

If Eq (13) is considered as a link in the position loop, then the structure of the position loop is shown in Figure 4.

In Figure 4, U_P(s) is the position loop input; ART stands for the position controller; T_op is the position filter time constant; $\gamma$ is the angle feedback coefficient.

The open-loop transfer function of the position loop can be given by Eq (14).

When Eq (15) is satisfied, the transfer function of the closed-loop of the position loop can be simplified as Eq (16)

where ${T_{{\rm{\Sigma p}}}} = {T_{{\rm{\Sigma n}}}} + {T_{{\text{op}}}} + {\tau _{\text{n}}}$

In this paper, the PID controller is optimized by the RABC algorithm (RABC-PID) as the ART controller. The RABC-PID controller is described in detail in the next section.

3.   Proposed controller

3.1.   Basic ABC algorithm

In the basic ABC algorithm, the bees are divided into three types: employed bees, follower bees, and scout bees. A search phrase is set for one type of bees, namely: employed bees phase, follow bees phase, and scout bees phase ^[28]. In the beginning, the algorithm uses a random initialization method to generate the initial population, and its generation method is shown in the following Eq (17)

where i = 1, ..., SN, n = 1, ..., D. SN represents the population size, D represents the dimension of the problem, rand(0, 1) is a random number between 0-1, $X_{{\text{max}}}^{\text{j}}$ and $X_{{\text{min}}}^{\text{j}}$ represents the upper and lower bounds of the individual jth dimension, respectively.

In the stage of the employed bees, the employed bees search for a food source according to Eq (18)

where k∈(1, ..., SN) and k ≠i; j∈(1, ..., D). $\varphi _{\text{i}}^{\text{j}}$ is a uniformly distributed random number on [−1, 1].

In the following bees stage, the follower bees use the information that the employed bees share, to select the food source according to the probability calculated by the following equation (19). The search process for following bees still proceeds according to Eq (18)

where fit_i represents the fitness value of the ith food source.

In the scout bee stage, a food source, the quality that has not been getting better, will be discarded. The following bee searching for this food source will turn into the scout bee. In the ABC algorithm, the food source corresponds to the candidate solution of the optimization problem, and the quality of the food source represents the good or bad candidate solution.

3.2.   RABC algorithm

The exploration and the exploitation conflict with each other. Enhancement of the exploration will inevitably affect the exploitation reducing the convergence speed of the algorithm and vice versa. ABC algorithm has a strong exploration ability and weak exploitation ability ^[29]. This paper proposes an RABC algorithm to balance exploration and exploitation. The search equation in the RABC algorithm is added to the best-positioned food source so that the bees search around better quality food sources. Using the optimal food source as a reference improves the searchability of the algorithm to some extent. The search equation in the RABC algorithm can be given by Eq (20)

where $X_{{\text{i, best}}}^{\text{j}}$ is the best-positioned food source.

The selection probability of bees is determined by the proportion of fitness of the current food source among all food sources. Little difference between the better food source and the optimal food source leads to a lower probability of selection of the optimal food source. The chance to search for some bad food source results in slowing down the speed of finding the optimal food source due to the random selectivity in the ABC algorithm. In this paper, we propose to use the current optimal fitness as a reference to improve the speed of the colony in finding the optimal food source. Thus the selection probability can be given by Eq (21)

where fit_max is the fitness value for the optimal solution.

Using Eq (20) to update the food source location ensures that the bees are not disturbed by the locally optimal bees, but also that the bees move to the best-positioned food source. Equation (21) enhances the probability of exploitation of high-quality food sources.

3.3.   Simulation verification of benchmark functions

To investigate the solution performance of the RABC algorithm, comparisons with ABC, GABC ^[30], and GBABC ^[31] algorithms were performed using six benchmark functions that are shown in Table 1. The parameters of the four algorithms were set as follows: D = 50, SN = 100, limit = 50, MaxCycle = 5000. Each algorithm runs independently 30 times. The results are presented in Table 2, Figures 5 and 6.

Table 2 gives the accuracy results of four algorithms for the six benchmark functions. In Table 2, the mean represents the mean of the error and std represents the standard deviation of the error. The results of the Wilcoxon statistical test at a 5% confidence level are also given, with "+, = , -" indicating that the RABC algorithm is better, the same, or worse than the other algorithms, respectively. From Table 2, it can be seen that the RABC outperforms the ABC, the GABC, and the GBABC in terms of solution accuracy.

Figure 5 shows the convergence curve of four algorithms on the benchmark function. From Figure 5, it can be seen that the RABC is superior to the ABC, the GABC, and the GBABC in terms of convergence speed. Figure 6 shows a boxplot of iterative data for four algorithms. The top and bottom edges of the boxplot indicate the 75th percentile and 25th percentile, respectively. From Figure 6, it can be further seen that the top and bottom edges of the boxplot from the RABC almost overlap, which further illustrates the fast convergence of the RABC.

3.4.   RABC-PID controller

The schematic of the RABC-PID controller is shown in Figure 7. The essence of the RABC-PID controller is to find the optimal values of K_p, K_i, and K_d using the RABC with the objective function as the criterion, which results in the best control performance of RSS. In Figure 7, r(t) is the RSS input, y(t) is the RSS output, and e(t) is the error between input and output.

It is also important for the system to choose an objective function for calculating the performance index (PI). The objective function used in this paper is the following equation (22)

where J₁, J_2, and J₃ are weighting factors, which are 0.999, 0.001, and 100, respectively, in this paper. t is the time.

To better clarify our controller, referred to as RABC-PID, its pseudocode is described in the following Table 3.

4.   Simulation verification

In this section RABC-PID controller is applied to the position loop of RSS for simulation verification. In the simulation experiments, the parameters of the RSS are set as follows: $\gamma$ = 100, ${T_{{\rm{\Sigma p}}}}$ = 0.25. With the above parameters, the transfer function of the closed-loop of the position loop can be given by Eq (23)

To fully evaluate the performance of the RABC-PID controller, three SI-PID controllers were used for comparison. The basic SI algorithms are particle swarm optimization (PSO) ^[32], differential evolution (DE) ^[33], and genetic algorithm (GA) ^[34], respectively. In the simulation experiments, the specific parameters of four controllers are given below:

RABC-PID settings: population number SN is set 50, the limit is set 50, and the maximum number of cycles MaxCycle is set 50.

PSO-PID settings: acceleration constant c₁ and c₂ are set 2, inertia weight W is set 0.9, maximum particle flight speed V_max is set 1, minimum particle flight speed V_min is set −1, population number SN is set 50, the maximum number of cycles MaxCycle is set 50.

DE-PID settings: crossover probability P_cr, scaling factor F, population number SN, and the maximum number of cycles MaxCycle are set at 0.8, 0.85, 50, and 50, respectively.

GA-PID settings: crossover probability P_c, mutation probability P_m, population number SN and the maximum number of cycles MaxCycle are set at 0.7, 0.3, 50, and 50, respectively.

To simulate different tracking targets, the inputs of the RSS are set as a square wave, triangle wave, and sine wave, respectively. The experimental results, including PI, optimum value, mean, and std are presented in Tables 4–7, respectively. Figures 8–13 show the convergence speed of the four controllers for different inputs.

For square wave input, in Table 4, it is clear that our controller achieves the minimum PI value among four controllers, which indicates that the performance of our controller is better than that of the other three controllers. From Figure 8, the GA-PID controller has the fastest convergence, but its PI is the largest. Although our controller converges slightly slower than PSO-PID and DE-PID, its PI is nearly 15% smaller than theirs. Figure 11 shows a tracking performance comparison for square wave input. From Figure 11(a), the tracking curve obtained by our controller is closer to the input curve than that obtained by the other three controllers. The tracking errors of the four controllers are indistinguishable from each other as seen in Figure 11(b), but the mean and std of the tracking error of our controllers are smaller than those of the other three controllers as seen in Table 7.

For triangular wave input, in Table 5, it can be seen that the PI value of RABC-PID is still the smallest, but all PI values do not differ much. Figure 9 shows convergence that is similar to that shown in Figure 8. From Figure 12, the tracking curves of all controllers are almost coincident in the simulation time except for the first 0.2 s. In Table 7, the tracking performance of our controllers is still ahead of the other three controllers, but that leading trend is decreasing.

For sine wave input, in Table 6, all PI values are similar. The convergence curves in Figure 10 almost overlap except for the curve of GA-PID, resulting in the inability to identify the convergence speed of RABC-PID, PSO-PID, and DE-PID. From Figure 13, the tracking curves of all controllers overlap, but our controller maintains a slight lead given mean and std as shown in Table 7. It is important to note that our controller performs well regardless of the input.

For RSS, the controller is required to have not only good tracking performance but also good robustness to cope with sudden changes in parameters of RSS. In the simulation experiments, the parameter ${T_{{\rm{\Sigma p}}}}$ of RSS varied to 0.5, 0.75, and 1, respectively. The simulation results are presented in Figures 14–22 and Tables 8–10. From Figures 14–22, as the parameter ${T_{{\rm{\Sigma p}}}}$ of RSS increases, the tracking performance of all controllers becomes worse, especially under square wave input. From Tables 8–10, it can be seen that a similar conclusion can be drawn as in Figures 14–22. It can be observed that, although the uncertainty of RSS increases with its parameter, the performance of our controller is not always affected. In summary, our controller has good robustness.

5.   Conclusions

The purpose of the current study was to design an RABC-PID controller to improve the tracking performance of RSS. In the RABC-PID controller, a novel RABC algorithm is proposed to enhance the basic ABC algorithm by introducing the best-positioned food source and modifying the food source probability. With a set of six benchmark functions, RABC is proven to have better performance than ABC, GABC, and GBABC. The simulation studies, which conduct on RSS, show that the RABC-PID controller obtains statistical better than the PSO-PID, DE-PID, and GA-PID controllers in terms of tracking performance and robustness. The significance of our work lies in the application of the RABC-PID controller to improve the tracking performance of RSS and enriches the application of SI in RSS.

Acknowledgments

This study was fully supported by Liaoning Provincial Department of Education Basic Research Projects for Higher Education Institutions, China (No. LJKZ0301); The Scientific Research Foundation of the Education Department of Liaoning Province, China (No. 2017LNQN22); Young Teachers Foundation of University of Science and Technology Liaoning, China (No. 2017QN04).

Conflict of interest

The authors declare no conflict of interest.