Artificial bee colony with predator effect algorithm for proportional integral derivative controller tuning

Abstract

This paper proposes a novel artificial bee colony with predation effect (ABCPE) algorithm for tuning a proportional integral derivative (PID) controller. The mathematical model of ABCPE algorithm to introduce predator effect in the foraging behavior of artificial bees colony algorithm has been formulated. The proposed algorithm has been tested on tuning problems of different process models. The simulation results reveal that the closed-loop responses are relatively fast and non-oscillatory as compared to the frequency response analysis method for reference tracking. Further, the results obtained using ABCPE are also compared with other evolutionary algorithms. The exhaustive analysis shows that the ABCPE-based solution approach leads to a set of tuning parameters having smaller overshoot, less setting time, and rise time compared to other solution approaches. The stability analysis using Nichols plot reveals that the phase margin of proposed algorithm is higher as compared to other tuning methods. Finally, the convergence behavior and robustness analysis reveals the effectiveness of the proposed approach to solve engineering design problems.

Keywords

artificial bee colony ball and hoop system frequency response method proportional integral derivative controller tuning predator effect

1. Introduction

Proportional integral derivative controllers are widely accepted due to their simple structure, cost effectiveness, and optimal performance. These controllers have the advantage of lesser computational complexity and smaller hardware requirements in comparison to artificial intellect based systems which require more computational effort and memory. These advantages of PID controllers are of utmost importance, when user is opting for a real-time control solution. PID parameter tuning is a core part of the controller design. A properly tuned controller helps to achieve the desired performance of a system. Although the proportional, integral, and derivative gains are only three adjustable parameters of a PID controller, however they are difficult to be tuned properly. One of the complexity is that tedious plant tests are required to achieve controller setting for desired performance Cao (2020); O’Dwyer (2009).

The conventional approaches for tuning a PID controller include the Ziegler–Nichols (ZN) method, Cohen Coon method, and the relay auto-tuning method Bequette (2003). The other conventional methods include internal model control, direct synthesis approach, and pole placement-based methods Ogata and Yang (2002). Although these methods are easy to implement, the performance of the controller may deteriorate with changing environment/operating points Gontijo et al. (2023). In such conditions, artificial intelligence based algorithms have been explored to tune a controller Jood et al. (2019). These algorithms can be classified as artificial neural network Kumar et al. (2014), adaptive neuro-fuzzy inference system Chopra et al. (2014), and fuzzy logic Blanchett et al. (2000). These controllers give quite satisfactory performance in the presence of changing environment or model uncertainties Jood et al. (2019). However, the training of these controllers is a complex task, requires good human knowledge, as well as additional computational resources.

In recent literature, a lot of meta-heuristic techniques have been used as a solution procedure to solve the PID controller tuning problems. These methods include application of genetic algorithm Cao (2020); Pereira and Pinto (2005); Ganapathy et al. (2016), differential evolution Saad et al. (2015); Gün (2023), and particle swarm optimization Nagaraj and Murugananth (2010). Further, the other natural phenomenon based algorithms such as ant colony optimization Dhieb et al. (2019), binary-coded extremal optimization Zeng et al. (2014), chaotic PSO algorithm Pluhacek et al. (2013), artificial bee colony Martins et al. (2022), etc. have been proposed to solve engineering problems.

To obtain the controller parameters, these methods require the minimization of certain performance criteria such as integral square error (ISE) or integral of time multiplied by absolute error (ITAE). As per literature, a balanced exploration and exploitation in a swarm-based meta-heuristic algorithms is a well challenging issue Hussain et al. (2019); Singh et al. (2018a). The predator-based concepts have been introduced as a meta-heuristics to improve the performance of population-based search algorithms Silva et al. (2002). Over the past three decades, the study of behavioral interactions between predator and prey has exposed mystery of prey behavior. These studies lead to a conclusion that the predating risks influence every aspect of prey decision-making such as foraging behavior Lima (2002). In the past decade, one of the most important concepts of nature, predator-prey behavior, has been mimicked as search algorithms and presented in the literature viz. adaptive predator-prey optimization Singh et al. (2018b), integrated predator optimization Narang et al. (2012), NSGA Amiri et al. (2023), spatial predator-prey approach Laumanns et al. (1998), real coded predator-prey genetic algorithm Li (2003), synergic predator-prey optimization Singh et al. (2016), etc. These algorithms have been highly recommended as solution procedure for the diverse engineering problems.

In this paper, a novel artificial bee colony with predation effect (ABCPE) technique has been proposed, where the artificial bee colony (ABC) algorithm has been modified with predator effect. ABCPE differs itself from ABC as it emphasis on dynamic spatial structure of the bee population by considering predation effect. ABCPE introduces dynamic changes of the bee position by employing Euclidean distance based random operator to encourage self-adaptive search in the available space. The predators chase the center of bees’ swarm, thereby the bees while changing the position takes care of predation effect. This effect contributes to local optimal avoidance and strengthens global optimal solution search capability. The proposed tuning algorithm is applied to standard test systems. The main objectives of the work are outlined as follows:

• Development of ABCPE algorithm, wherein artificial bee colony algorithm is modified with introduction of predator effect.

• Development of solution approach, using ABCPE, for the solution of PID controller tuning.

• Analyze the performance of ABCPE algorithm using simulation results of PID controller tuning problem.

The paper is organized as follows: Firstly, the PID controller problem formulation is described and the standard test system used in this research work are elaborated. Thereafter, the working principle of ABCPE algorithm is presented followed by mathematical model and algorithmic solution approach to solve PID controller tuning problem. The simulation results and observations of the proposed algorithm are presented finally, followed by the conclusions of the research work.

2. Problem formulation

Mathematically, a PID controller is represented as:

g_{c} (s) = k_{p} + k_{i} / s + k_{d} s

(1)

Controller tuning problem is stated as to search a suitable set of proportional, integral, and derivative gains, that is, k_p, k_i and k_d in the available problem hyperspace such that a certain objective function is to be minimized or maximized. These objective functions are based upon error performance criteria, which reflect the expectation from the system under unity feedback condition. The performance index (PI) is calculated over t normally in the region from 0 < t < T. The performance indices used are explained as follows:

2.1. Performance index

The commonly used performance index functions are Joseph et al. (2022):

2.1.1. Integral of absolute error (IAE)

\int_{0}^{T} | e (t) | d t

(2)

In this performance index, the negative error component is removed and gives the absolute value of overall error of the system.

2.1.2. Integral of time absolute error (ITAE)

\int_{0}^{T} t | e (t) | d t

(3)

This performance index employs time-based scaling of error and thereby emphasizes on the error values during the latter part of the response. It ensures faster damping of oscillations and hence improves the steady state response of system.

2.1.3. Integral of square error (ISE)

\int_{0}^{T} e {(t)}^{2} d t

(4)

In ISE, the negative error is rectified using square operation and provides the overall area under the error curve through the integral operation. This performance index accomplishes the transient part of response as compared to steady state.

2.1.4. Mean square error (MSE)

\frac{1}{n} \sum_{i = 1}^{n} e {(t)}^{2}

(5)

MSE reflects all variation and deviation from the target value.

In order to observe the effectiveness of the proposed solution approach to solve the problem of controller’s tuning parameters, three different types of test system are considered. These systems are 3rd order process, a ball and hoop system, and a DC motor position control system. The explanation of these systems is as follows:

2.2. Test systems

2.2.1. 3rd order system

A majority of practical systems are usually higher order systems with oscillatory transient response. Therefore, the first system considered for performance evaluation is a 3rd order system, mathematically represented as follows:

G (s) = \frac{0.1}{0.24 s^{3} + 3.8 s^{2} + s}

(6)

The transient response of this system is sluggish in nature Ogata and Yang (2002).

2.2.2. Ball and hoop system

The ball and hoop system is a bench mark control problem and its transient response exhibits a complex dynamics which is modeled as a fourth order system. The transfer function of system is represented by equation (7) as below:

G (s) = \frac{Y (s)}{θ (s)} = \frac{1.0}{s^{4} + 6 s^{3} + 11 s^{2} + 6 s}

(7)

Here Y(s) is the position of a ball in the inner periphery of hoop measured with respect to reference point; θ(s) represents angular position of hoop with respect to reference point Sreekanth and Hari (2016).

2.2.3. DC motor position control system

The transfer function of DC motor position control system is given by following equation:

G (s) = \frac{θ (s)}{V (s)} = \frac{0.9}{0.0005 s^{3} + 0.2104 s^{2} + 0.8193 s}

(8)

Here θ(s) is the angular position of shaft and V(s) is the voltage applied to the motor field winding Kumar et al. (2014).

3. Artificial bee colony with predator effect algorithm

Artificial bees colony with predator effect (ABCPE) algorithm mimics the idea of honeybee swarm’s collective intelligence for foraging activity. ABCPE owes employed bees to search for food source, onlooker bees for selection of the best quality food source, and the scout bees identify new food source Karaboga and Basturk (2007). The food source position in the feasible search domain represents the possible solution of the problem, whose quality is indicated by the nectar amount. The predation psychological feature imposes an extremely high fitness cost Bednekoff (2007). Hence, selection process has to prefer foraging activity such that forage gains balance-out the predation risk Brown et al. (1999). The predator-prey foraging game analysis indicates that the predator opt prey hunting in the high quality areas Hugie and Dill (1994). Therefore, the prey has to negotiate between forage advantage and predation risk Sih (1998).

In ABCPE, the employer bees and onlooker bees visit the most promising locations to maximize their foraging success. As the predators opt high quality foraging arena for prey bee, the bees often prefer to visit less rewarding locations to trade-off foraging benefits and predation risk Jones and Dornhaus (2011). An experimental study on the bumble bees’ foraging behavior, associated with predation risk, shows that the bees experiencing a simulated attack on a most rewarded foraging locations were more likely to visit little degraded arena on consequent forage trips. The predator effect has been successfully mathematically modeled and analyzed for multi-behavioral strategies Mitchell (2009), swarm-based optimization Silva et al. (2002) and behavioral strategies in predators and prey Haynes and Sen (1995). Silva et al. (2002) suggested a predator’s mathematical model, wherein Px_j represents the predator’s position in the jth dimension of N_D dimensional search space, is randomly initialized as follows:

\begin{array}{lc} P x_{j} = x_{j}^{\min} + r a n d () & (x_{j}^{\max} - x_{j}^{\min}) (j = 1, 2 \dots, N_{D}) \end{array}

(9)

where

x_{j}^{\min}

and

x_{j}^{\max}

represents the decision boundary of bee’s position in jth dimension of N_D dimensional search space, respectively.

The predator’s physiology is to visit high rewarded areas identified and this behavior is reflected by a mathematical expression as:

\begin{array}{l} {P x}_{j}^{t + 1} = P x_{j}^{t} + c_{p} r a n d () & (g_{j}^{b e s t} - P x_{j}^{t}) \\ (j = 1,2, \dots, N_{D}) \end{array}

(10)

where

g_{j}^{b e s t}

is the bee’s global best position, c_p is the predator position scaling factor.

The ABC algorithm mimics the honey bee activity specific to the foraging interests only represented by mathematical equation (11). More detail of ABC algorithm is presented in Appendix A. The updated position of ith honey bee among population of N_P honey bees has been represented by xb_ij. It is related to present position x_ij as follows:

x b_{i j} = x_{i j} + r a n d [- 1,1] (x_{i j} - x_{l j})

(11)

where l ∈ [1, N_P] is a randomly generated number.

In the proposed ABCPE, in order to trade-off the foraging gains and predation risk, the employer bees and onlooker bees have been modeled to change their foraging preferences in the presence of predators, as the predators are attracted towards high quality foraging arena to locate prey bee. Therefore, the mathematical equation to represent new bee position xp_ij has been updated considering predation effect as follows:

\begin{array}{l} x p_{i j} = x_{i j} + r a n d [- 1, 1] (x_{i j} - x_{l j}) + r a n d [- 1, 1] a γ e^{- b β} \end{array}

(12)

where x_i is the ith bee’s present location, a represents the maximum amplitude of the predator’s effect over prey, b represents the effective proximity of predator’s influence, β is the Euclidean distance between ith bee’s position vector X_i and predator position vector Px.

The bees maximize their foraging success by visiting the safer foraging locations. A greedy search-based decision approach has been applied to replace the foraging position vector $X_{i}^{t}$ with a better one out of position vectors Xb_i and Xp_i at each iteration as follows:

X_{i}^{t + 1} = {\begin{cases} X_{i}^{t} & F (X_{i}^{t}) > \max (F (X b_{i}^{t}), F (X p_{i}^{t})) \\ X b_{i}^{t} & F (X b_{i}^{t}) > \max (F (X_{i}^{t}), F (X p_{i}^{t})) \\ X p_{i}^{t} & F (X p_{i}^{t}) > \max (F (X_{i}^{t}), F (X b_{i}^{t})) \end{cases}

(13)

The process explained above has been employed in both employer bee phase and onlooker bee phase, as shown in Algorithm 1 pseudo code. The ABCPE solution approach starts with a uniformly distributed, randomly generated population of bees and a predator, both having N_D dimension. The population size has been represented by N_P. The iterative process to improve the quality of solutions, indicated by a maximization process using objective function equations (2)–(5), proceeds until termination criteria has been satisfied. The ABCPE algorithm to solve problem in-hand is as follows:

4. Results and discussions

The artificial bee colony with predator effect algorithm based solution approach to solve the PID controller design problem has been explained in the previous section. The solution approach described in pseudo code has been implemented in MatLab 2019, i5 processor 8 GB RAM. The solution approach has been applied to solve test systems namely 3^rd order system, ball and hoop system and DC motor position control system considering the performance index listed by equations (2)–(5), respectively. The boundary range for decision variables is [0, 20]. In order to obtain the solution of problem, the population size N_B has been fixed at 10. The number of iterations considered are 200. The detailed discussion of key observations related to solution obtained for three different systems has been presented. The observations are depicted using a comparison of tuning parameters, graphical representation of transient response, examination of transient behavior specifications, stability analysis and convergence behavior in the following section.

4.1. 3rd order system

Firstly, a 3rd order system represented by equation (6) has been considered to obtain solution using ABCPE. The tuning parameters, namely, k_p, k_i, and k_d of PID controller have been obtained. The values of tuning parameters obtained using ABCPE along with DE-CHAOS and SOMA algorithms (listed in literature) are tabulated in Table 1. It can be clearly observed from the listed values that the parameter values obtained using ABCPE are smaller in magnitude as well as in the prescribed decision boundary range, for all the performance indexes.

Table 1.

Comparison of controller parameters of 3^rd order system.

	DECHAOS Davendra et al. (2010)			SOMA Davendra et al. (2010)			ABCPE
	k _p	k _i	k _d	k _p	k _i	k _d	k _p	k _i	k _d
IAE	5.856	4.34 × 10⁻³	11.835	33.477	0.079	148.74	6.2384	5.51 × 10⁻⁴	20.0
ISE	5.204	1.56 × 10⁻¹	20.804	5.248	1.135	150.0	8.3396	0.0	20.0
ITAE	4.843	2.50 × 10⁻⁴	7.023	5.248	1.135	150.0	6.3086	1.94 × 10⁻⁴	20.0
MSE	5.240	1.56 × 10⁻¹	20.801	5.248	1.135	150.0	6.2384	5.50 × 10⁻⁴	20.0

The transient response for all the performance indices corresponding to tuning parameters obtained using ABCPE algorithm listed in Table 1, has been graphically depicted in Figure 1. For the comparison purpose a standard method namely frequency response analysis approach for reference tracking has been applied using PID tuner application tool available in Matlab 2019. The comparison of transient response plots show that the parameters obtained by ABCPE algorithm offers advantage of less oscillatory and faster response with smaller steady state error.

Further, the tabular comparison of transient response specifications in Table 2 reveals that for the performance index IAE, ITAE and MSE the tuning parameters obtained by ABCPE have advantage of smaller peak overshoot (M_P) and faster transient response in terms of rise time (t_r) except for SOMA. However, the settling time (t_s) is quite smaller than its counterparts. Further, the ABC algorithm designed controller has reflected higher t_r and t_s than the proposed ABCPE algorithm.

Table 2.

Comparison of transient response specifications of 3rd order system.

Algorithm	P.I.	M_P	t_r (sec)	t_s(sec)
Frequency response method		9.69	11.02	25.92
DECHAOS Davendra et al. (2010)	IAE	10.44	4.33	20.31
	ISE	2.56	3.33	26.82
	ITAE	14.36	5.58	13.08
	MSE	2.56	3.33	26.82
SOMA Davendra et al. (2010)	IAE	38.98	0.52	5.65
	ISE	29.60	0.97	41.87
	ITAE	29.60	0.97	41.87
	MSE	29.60	0.97	41.87
ABC	IAE	2.72	17.71	41.88
	ISE	2.93	10.66	28.81
	ITAE	2.77	17.01	49.52
	MSE	2.89	10.92	37.55
ABCPE	IAE	1.01	3.72	4.56
	ISE	4.62	3.03	3.49
	ITAE	1.01	3.68	4.50
	MSE	1.01	3.72	4.55

The stability analysis of 3rd order system along with the proposed ABCPE control method and with existing methods has been carried out here in this section. The Nicholas stability plot of the given system with various controllers is shown in Figure 2. To obtain the stability plot of the system, the open loop transfer function given in equation (6) and the tuned parameter values of PID controller given in Table 1 have been used. In the case of DE-CHAOS and SOMA, the system is showing a phase margin (PM) of 30.8° and 79.5°, respectively. However, in the case of proposed ABCPE method, the PM has been improved considerably to a value of 84.4°.

Further, in order to shift the focus of experimentation towards practical systems, the ball and hoop system has been considered for analysis. The unforced response of ball and hoop system is highly oscillatory in nature as discussed in Section 2. Therefore, the aim in this problem is to obtain such a set of decision variables that the system response can be as smooth as possible without steady state error. The ball and hoop system is represented by equation (7) and the decision variables are tuning parameters, namely, k_p, k_i, and k_d. The values of tuning parameters obtained using ABCPE along with DE-CHAOS, SOMA, and ABC algorithm are tabulated in Table 3. It can be clearly observed from the listed values that the parameters values obtained using ABCPE are comparable with other algorithms and are within the prescribed decision boundary range for all the performance indexes.

Table 3.

Comparison of controller parameters of ball and hoop system.

	DECHAOS Davendra et al. (2010)			SOMA Davendra et al. (2010)			ABCPE
	k _p	k _i	k _d	k _p	k _i	k _d	k _p	k _i	k _d
IAE	5.856	4.30 × 10⁻³	11.835	5.856	4.3 × 10⁻³	11.835	6.637	0.0	12.578
ISE	5.204	1.56 × 10⁻¹	20.804	5.204	1.56 × 10⁻¹	20.804	5.522	7.77 × 10−2	20.0
ITAE	4.843	2.50 × 10⁻⁴	7.023	4.963	3.0 × 10⁻⁴	7.308	4.587	0.0	6.296
MSE	5.204	156 × 10⁻¹	20.804	5.204	1.56 × 10⁻¹⁰	20.804	6.623	3.29 × 10−4	12.571

Ball and Hoop System.

The transient response for all the performance indices’ corresponding to tuning parameters obtained using ABCPE algorithm, listed Table 3, has been graphical depicted in Figure 3. The comparison of transient response plots show that the parameters obtained by ABCPE algorithm offers advantage of smaller overshoots and acceptable transient behavior specifications.

The tabular comparison of transient response in Table 4 exhibits that for the performance index IAE, the decision variables obtained using ABCPE have higher overshoot but results in lesser settling time. The comparison of ISE performance index results reveals that peak overshoot is little bit higher, but both rise time and settling time are highly reduced. Further, the numerical comparison of time response for ITAE and MSE performance index indicates that ABCPE has obtained better quality solution for all the three performance specifications. In brief, in case of ball and hoop system, the parameters obtained by ABCPE solution approach are better in terms of improved rise time and settling time with lesser control effort than other algorithms. In case of ball and hoop system, both DE-CHAOS and SOMA control methods possess the same PM of 38.7°. In the case of ABCPE, it improves almost 20% in PM and shown in the Figure 4 the 46.4° for the overall system.

Table 4.

Comparison of transient response specifications of ball and hoop System.

Algorithm	PI	M _P	t_r (sec)	t_s(sec)
Frequency response method		25.8	1.96	9.93
GA Griffin and Bruton (2003)	IAE	44.97	1.2	9.3
	ISE	28.80	1.2	20.4
	ITAE	57.91	1.3	8.2
	MSE	28.55	1.2	20.4
DECHAOS Davendra et al. (2010)	IAE	14.50	1.66	5.19
	ISE	24.52	1.31	9.22
	ITAE	6.71	2.15	6.09
	MSE	24.52	1.31	9.22
SOMA-CHOS Davendra et al. (2010)	IAE	14.50	1.66	5.19
	ISE	24.52	1.31	9.22
	ITAE	6.71	2.15	6.09
	MSE	24.52	1.31	9.22
GWO Jain et al. (2018)	IAE	14.20	1.28	4.86
	ISE	20.57	1.35	5.18
	ITAE	5.02	2.63	4.15
	MSE	16.11	1.63	4.07
ABC	IAE	56.88	1.09	5.63
	ISE	28.12	1.13	4.86
	ITAE	22.37	1.16	6.82
	MSE	73.37	0.9	5.84
ABCPE	IAE	19.31	1.36	4.68
	ISE	25.29	1.08	6.54
	ITAE	5.51	2.11	3.46
	MSE	19.23	1.36	4.69

4.2. DC motor position control system

In another effort for analysis of proposed solution approach using physical system, an industrial application of position control using a DC motor has been considered. The DC motor under no load conditions has a fast transient behavior; its position control is a tedious task. The expectation from the system is a smoother transient behavior with minimum peak overshoot and no steady state error Chopra et al. (2022). The DC motor system is represented by equation (8). The values of tuning parameters obtained for this system using ABCPE along with DE-CHAOS and SOMA algorithms are tabulated in Table 5. It can be clearly observed from the listed values that the parameters obtained using ABCPE are smaller in magnitude as well as in the prescribed decision boundary range, for all the performance indexes.

Table 5.

Comparison of controller parameters of DC motor position control system.

	DECHAOS Davendra et al. (2010)			SOMA Davendra et al. (2010)			ABCPE
	k _p	k _i	k _d	k _p	k _i	k _d	k _p	k _i	k _d
IAE	158.05	3.33 × 10⁻¹⁰	36.50	117.60	5.22 × 10⁻¹⁰	27.16	20.0	0.0	4.617
ISE	158.05	2.57 × 10⁻⁰⁷	36.50	117.60	1.42 × 10⁻⁰⁷	34.60	20.0	0.0	4.617
ITAE	158.05	8.28 × 10⁻¹⁰	36.50	149.98	0.13 × 10⁻⁰¹	34.64	19.97	0.0	4.612
MSE	158.05	2.57 × 10⁻⁰⁷	36.50	117.60	1.38 × 10⁻⁰⁷	27.16	20.0	0.0	4.617

Figure 5 graphically depicts the transient response for all the performance indices corresponding to tuning parameters obtained using ABCPE algorithm, listed in Table 5. The comparison of transient response plots show that the parameters obtained by ABCPE algorithm offers advantage of negligible overshoots as shown in Table 6.

Table 6.

Comparison of transient response specifications of DC motor position control system.

Algorithm	PI	M _P	t_r (sec)	t_s(sec)
Frequency response method		8.90	0.54	1.30
EP Nagaraj et al. (2008)		8.81	0.01	0.20
GA Nagaraj et al. (2008)		13.0	0.03	0.32
PSO Nagaraj et al. (2008)		12.9	0.03	1.15
DECHAOS Davendra et al. (2010)	IAE	11.97	0.015	0.035
	ISE	11.97	0.015	0.035
	ITAE	11.97	0.015	0.035
	MSE	11.97	0.015	0.035
SOMACHAOS Davendra et al. (2010)	IAE	6.835	0.02	0.04
	ISE	6.835	0.02	0.04
	ITAE	10.475	0.02	0.035
	MSE	6.835	0.02	0.04
ABC	IAE	0.09	0.17	2.63
	ISE	0.05	0.17	2.07
	ITAE	0.16	0.21	7.74
	MSE	0.06	0.20	2.17
ABCPE	IAE	0.0	0.11	0.14
	ISE	0.0	0.11	0.14
	ITAE	0.0	0.11	0.15
	MSE	0.0	0.11	0.14

The Nicholas chart for stability analysis is plotted in Figure 6. In the case of DE-CHAOS and SOMA, the overall system shows the PM of 70.2° and 74.6°, respectively, while with ABCPE PM is 86.2° which proves the dominance of ABCPE control method.

4.3. Convergence behavior and robustness analysis

The convergence behavior analysis of the proposed ABCPE algorithm for different objective functions is presented from Figures 7 –10, for all the three test systems. It has been observed from the convergence plots that ABCPE swiftly moves toward the optimal solutions at a uniform pace under the influence of predator effect.

In order to verify the robustness of the proposed ABCPE solution approach to obtain a suitable solution of problem, 30 independent trial runs are performed for all the three systems. The statistical observations for the three considered systems are tabulated in Tables 7 –9, which shows that the solution obtained by proposed technique shows very small standard deviation. For all the three systems, the proposed approach is able to generate acceptable solution at each trial. Hence, it can be concluded that the proposed ABCPE technique has ability to generate a highly precised solution in small number of iterations (Table 10).

Table 7.

Statistical observations of 3rd order system.

PI	Mean	Maxima	S.D.
IAE	0.200	0.200	1.81 × 10⁻⁰⁵
ISE	0.264	0.264	5.70 × 10⁻¹⁰
ITAE	0.174	0.176	1.13 × 10⁻⁰²
MSE	0.938	0.938	7.12 × 10⁻⁰⁶

Table 8.

Statistical observations of ball and hoop system.

PI	Mean	Maxima	S.D.
IAE	0.069	0.069	3.06 × 10⁻⁰⁴
ISE	0.118	0.118	6.27 × 10⁻¹⁰
ITAE	0.059	0.061	7.05 × 10⁻⁰⁴
MSE	0.957	0.957	3.28 × 10⁻⁰⁵

Table 9.

Statistical observations of DC motor system.

PI	Mean	Maxima	S.D.
IAE	0.162	0.162	1.23 × 10⁻⁰⁵
ISE	0.414	0.414	1.17 × 10⁻⁰⁹
ITAE	0.772	0.820	1.67 × 10⁻⁰¹
MSE	0.989	0.989	9.68 × 10⁻⁰⁹

Table 10.

Convergence behaviour of ABCPE for various performance index.

PI	3rd order system	Ball and hoop system	DC motor system
IAE	15	20	30
ISE	20	11	6
ITAE	3	31	35
MSE	3	3	5

5. Conclusion

An ABCPE algorithm has been proposed to design PID controller to improve the transient and steady state response of the systems. The performance of the proposed solution approach has been verified using three different test problems namely, 3^rd order system, ball and hoop system and DC motor position control system. The error based performance index such as IAE, ISE, IATE and MSE have been used in the ABCPE algorithm’s performance analysis. The controller designed using proposed method has been compared with the other design methods presented in the literature. The experimental simulations confirm that the ABCPE-based solution approach has been able to search control parameters smaller in magnitude thereby facilitating controller's practical implementation possible. The transient behavior comparison shows the supremacy of ABCPE over other methods. The ABCPE approach has successfully determined the tuning parameters that results in smaller overshoots, lesser rise time and settling time for all the considered test systems. The stability analysis of the proposed algorithm has been carried out using the Nicholas plot which reveals an improvement in the PM of systems under consideration using the proposed ABCPE algorithm. It clearly shows the improved PM of the systems under consideration when the proposed ABCPE has been applied. The exhaustive analysis shows that the ABCPE has the ability to obtain feasible solutions with lesser computational efforts, which is a desirable feature for online tuning of controllers. Finally, the convergence behavior and robustness analysis proves the effectiveness of proposed solution approach to solve the controller design problem.

Footnotes

Author contributions

The first author has specialization in the field of hybrid optimization algorithms and he has contributed for the development of ABCPE algorithm. The second author has expertise in the field of control engineering and they are responsible for PID controller tuning part. The last author has contributed in the compilation of the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Ethical statement

ORCID iD

Nirbhow Jap Singh

Appendix

References

Amiri

Sohrabi

Eftekharian

, et al. (2023) Optimization and control of an energy-efficient vibration-driven robot. Journal of Vibration and Control 29. doi: 10.1177/10775463231175543.

Bednekoff

(2007) Foraging in the Face of Danger. IL: The University of Chicago Press Chicago.

Bequette

(2003) Process Control: Modeling, Design, and Simulation. Upper Saddle River, NJ: Prentice Hall Professional.

Blanchett

Kember

Dubay

(2000) PID gain scheduling using fuzzy logic. ISA Transactions 39(3): 317–325.

Brown

Laundré

Gurung

(1999) The ecology of fear: optimal foraging, game theory, and trophic interactions. Journal of Mammalogy 80(2): 385–399.

Cao

(2020) PID controller optimized by genetic algorithm for direct-drive servo system. Neural Computing & Applications 32(1): 23–30.

Chopra

Singla

Dewan

(2014) Comparative analysis of tuning a PID controller using intelligent methods. Acta Polytechnica Hungarica 11(8): 235–249.

Chopra

Singh

Kumar

, et al. (2022) Position control of pmbldc motor using SVR-and ANFIS-based controllers. National Academy Science Letters 45(1): 57–60.

Davendra

Zelinka

Senkerik

(2010) Chaos driven evolutionary algorithms for the task of PID control. Computers & Mathematics with Applications 60(4): 1088–1104.

10.

Dhieb

Yaich

Guermazi

, et al. (2019) PID controller tuning using ant colony optimization for induction motor. Journal of Electrical Systems 15(1): 133–141.

11.

Ganapathy

Kumar C

Jerome

(2016) Optimal tuning of PID controllers for dead-time systems using stud genetic algorithms. Journal of Vibration and Control 22(10): 2503–2518.

12.

Gontijo

Araújo

Santos

, et al. (2023) Proportional-integral-derivative-acceleration robust controllers for vibrating systems. Journal of Vibration and Control 29(5-6): 1243–1253.

13.

Griffin

Bruton

(2003) On-Line PID Controller Tuning Using Genetic Algorithms. Dublin, Ireland: Dublin City University.

14.

Gün

(2023) Attitude control of a quadrotor using PID controller based on differential evolution algorithm. Expert Systems with Applications 229: 120518.

15.

Haynes

Sen

(1995) Evolving behavioral strategies in predators and prey. In: International joint conference on artificial intelligence. Quebec, Canada, 20–25 August 1995, 113–126. Springer.

16.

Hugie

Dill

(1994) Fish and game: a game theoretic approach to habitat selection by predators and prey. Journal of Fish Biology 45: 151–169.

17.

Hussain

Salleh

MNM

Cheng

, et al. (2019) On the exploration and exploitation in popular swarm-based metaheuristic algorithms. Neural Computing & Applications 31(11): 7665–7683.

18.

Jain

Parmar

Gupta

, et al. (2018) Performance evaluation of GWO/PID approach in control of ball hoop system with different objective functions and perturbation. Cogent Engineering 5(1): 1–18.

19.

Jones

Dornhaus

(2011) Predation risk makes bees reject rewarding flowers and reduce foraging activity. Behavioral Ecology and Sociobiology 65(8): 1505–1511.

20.

Jood

Aggarwal

Chopra

(2019) Performance assessment of a neuro-fuzzy load frequency controller in the presence of system non-linearities and renewable penetration. Computers & Electrical Engineering 74: 362–378.

21.

Joseph

Dada

Abidemi

, et al. (2022) Metaheuristic Algorithms for PID Controller Parameters Tuning: Review, Approaches and Open Problems. Cambridge, MA: Heliyon.

22.

Karaboga

Basturk

(2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm. Journal of Global Optimization 39(3): 459–471.

23.

Kumar

Gaur

Mittal

(2014) ANN based self-tuned PID like adaptive controller design for high performance pmsm position control. Expert Systems with Applications 41(17): 7995–8002.

24.

Laumanns

Rudolph

Schwefel

(1998) A spatial predator-prey approach to multi-objective optimization: a preliminary study. In: International Conference on Parallel Problem Solving from Nature. Berlin, Heidelberg, 27–30 September 1998, 241–249. Springer.

25.

(2003) A real-coded predator-prey genetic algorithm for multiobjective optimization. In: International conference on evolutionary multi-criterion optimization, Faro, Portugal, 8–11 April 2003, 207–221. Springer.

26.

Lima

(2002) Putting predators back into behavioral predator–prey interactions. Trends in Ecology & Evolution 17(2): 70–75.

27.

Martins

Adekunle

Arowolo

, et al. (2022) The effect of an evolutionary algorithm’s rapid convergence on improving dc motor response using a PID controller. Scientific African 17: e01327.

28.

Mitchell

(2009) Multi-behavioral strategies in a predator–prey game: an evolutionary algorithm analysis. Oikos 118(7): 1073–1083.

29.

Nagaraj

Murugananth

(2010) A comparative study of PID controller tuning using GA, EP, PSO and ACO. In: 2010 international conference on communication control and computing technologies, Nagercoil, Tamilnadu, 7–9 October 2010, 305–313. IEEE.

30.

Nagaraj

Subha

Rampriya

(2008) Tuning algorithms for PID controller using soft computing techniques. International Journal of Computer Science and Network Security 8(4): 278–281.

31.

Narang

Dhillon

Kothari

(2012) Multiobjective fixed head hydrothermal scheduling using integrated predator-prey optimization and powell search method. Energy 47(1): 237–252.

32.

Ogata

Yang

(2002) Modern Control Engineering. Hoboken, NJ: Prentice hall India, Vol. 4.

33.

O’Dwyer

(2009) Handbook of PI and PID Controller Tuning Rules. London, UK: Imperial College Press.

34.

Pereira

Pinto

(2005) Genetic algorithm based system identification and PID tuning for optimum adaptive control. In: Proceedings, 2005 IEEE/ASME international conference on advanced intelligent mechatronics, Monterey, USA, 24-28 July, 2005, 801–806. IEEE.

35.

Pluhacek

Senkerik

Davendra

, et al. (2013) Designing PID controller for dc motor by means of enhanced pso algorithm with dissipative chaotic map. In: Soft Computing Models in Industrial and Environmental Applications. Springer, 475–483.

36.

Saad

Jamaluddin

Darus

IZM

(2015) Active vibration control of a flexible beam using system identification and controller tuning by evolutionary algorithm. Journal of Vibration and Control 21(10): 2027–2042.

37.

Sih

(1998) Game theory and predator–prey response races. Game Theory and Animal Behavior: 221–238.

38.

Silva

Neves

Costa

(2002) An empirical comparison of particle swarm and predator prey optimisation. In: Irish conference on artificial intelligence and cognitive science, Limerick, Ireland, 12–13 September 2002, 103–110. Springer.

39.

Singh

Dhillon

Kothari

(2016) Synergic predator-prey optimization for economic thermal power dispatch problem. Applied Soft Computing 43: 298–311.

40.

Singh

Dhillon

Kothari

(2018a) Multi-objective thermal power load dispatch using chaotic differential evolutionary algorithm and Powell’s method. Soft Computing 22(7): 2159–2174.

41.

Singh

Dhillon

Kothari

(2018b) Multiobjective thermal power load dispatch using adaptive predator–prey optimization. Applied Soft Computing 66: 370–383.

42.

Sreekanth

Hari

(2016) Genetic algorithm based self-tuning regulator for ball and hoop system. In: 2016 Conference on emerging devices and smart systems (ICEDSS), Namakkal, India, 4–5 March 2016, 147–152. IEEE.

43.

Zeng

Dai

, et al. (2014) Binary-coded extremal optimization for the design of PID controllers. Neurocomputing 138: 180–188.