Data-Driven Fractional-Order PID Controller Tuning for Liquid Slosh Suppression Using Marine Predators Algorithm

Recently, the studies on the liquid slosh problem have become popular topics among researchers due to the urgency to suppress the liquid slosh in various applications. For example, in industrial fluid packaging, the residual vibration induced by the intermittent motion profile of the liquid can limit the machine’s operating speed, thus, reducing productivity [1, 2]. Meanwhile, an unsteady internal liquid movement in the large aerospace structures can modify the behaviour of the system by producing an extra excitation load at the internal wall [3]. This problem may influence the system stability margins. Furthermore, the internal sloshing forces produced by the partially filled liquid tank can damage the tank walls structure and also influence the seakeeping behaviour of the on-board ship [4]. Additionally, in the metal industries, the pouring work of the molten metal is usually conducted by the human operator which tends to cause the metal sloshing and consequently can cause the molten metal spill out from the ladle [5]. Moreover, the stability of the tank trucks during the overturning has significantly decreased due to the sloshing force produced by the movement of the liquid in the tank. This liquid slosh can increase the roll moment and can cause the rollover of the tank trucks [6]. Therefore, the liquid slosh suppression in aforementioned industries is vital to ensure the safety during the system’s operation because the effect from the partially-filled liquid slosh can generate extra forces and moments in which the overall system’s performance can be degraded significantly. Unfortunately, suppressing the liquid slosh in the container is challenging due to chaotic motion and turbulent behavior of the liquid slosh. Therefore, it is essential to develop an effective control method in ensuring the minimal liquid slosh effect and improving the performance of the system.

There is a growing body of literature that developed an effective method for solving the liquid slosh problem. The most popular methods in suppressing the liquid slosh can be divided into two main methods, which are passive method and active method. For passive methods, there is no requirement for actuator and measurement of liquid slosh. It solely involves the modification of mechanical parts to reduce the liquid slosh in the container. For instance, the porous media has been used inside the rectangular tank as a tuned liquid damper (TLD) to suppress the fluid oscillation [7]. They have shown that the porous media has successfully reduced the liquid sloshing with the presence of impulsive and harmonic excitations. Another passive method was proposed in the study [8] which introduced the vertical baffles in suppressing parametric sloshing in the clean tank. The vertical baffles with uniformly distributed holes are placed inside the tank and create vortexes that change the natural frequency of the fluid system, thus, helping in dissipating the fluid energy. Regrettably, these passive methods require a lot of time to construct because of the complex design, bulky and heavy structure. Meanwhile, an active method seems to be a good choice since it does not involve complex mechanical modification directly. An active method involved the actuation of the container to suppress the liquid slosh during the system operation. The control strategies for the active method can be divided into two types, which are feed-forward control and feed-back control. For the first type, there is no involvement of the liquid slosh measurement by the sensor since this technique provides the prescribed motion to the actuator in reducing the residual slosh of the liquid. This includes the command shaping techniques which generates a command from the convolution between the reference and a multi-sine wave function [9]. Another feed-forward control was introduced by Alshaya and Alshayji [10] which proposed an equidistant multistep input command for eliminating the liquid slosh. They have modeled the liquid sloshing based on finite element method (FEM) to numerically predict the elevation of the free-surface liquid level. The last example of the feed-forward control is combined command smoothing and input shaping architecture which was introduced by Zang et al. [11]. They have proposed the command smoothing which consists of low-pass and multi-notch filters to eliminate the slosh for the third and higher mode frequency, while the input shaping is responsible to reduce the slosh for the first mode frequency. Unfortunately, there is a drawback of these feed-forward control methods which is lack of robustness when dealing with external disturbances that happen in the system. Alternatively, the feed-back control which has more robustness in dealing with external disturbances, has gained more attention among researchers in suppressing the liquid slosh inside the container. Various attempts have been made in controlling the liquid slosh by using the feed-back control method. These studies include, H-infinity controller [12], adaptive robust wavelet control [13], and sliding mode controller [14].

In aforementioned feed-back control literature, they only focused on the model-based approach for controller design which requires an accurate mathematical model of liquid slosh problem. Unfortunately, the modeling of the liquid slosh involved a very complicated mathematical model due to turbulent behavior of the liquid inside the container. As an alternative, a data-driven control method can also be applied in solving the liquid slosh problem since this data-driven control method does not require an exact mathematical model of the system. The only procedure to implement the data-driven control method is by fully utilizing the measurement of input and output data of the actual system [15]. Over the past few decades, the proportional-integral-derivative (PID) controller has been widely used as a data-driven control method thanks to its ability to provide a robust performance with simple structure and low execution effort [16]. However, solely using the standard structure of PID controllers may degrade the system performance especially for highly nonlinear systems. Therefore, many researchers have proposed a variety of PID controllers to provide a more precise and robust control system. One of the most popular PID variant is a fractional order PID (FOPID) controller which introduces two additional controller gains i.e., fractional exponential terms of integral λ and derivative μ [17]. Despite the FOPID controller being able to enhance the system’s overall performance in various applications [18], the tuning process of FOPID parameters is more difficult and requires a lot of time. Thus, there is a high demand to tune the FOPID parameters by implementing variety optimization algorithms. For example, Mok and Ahmad [19] have proposed a modified smoothed function algorithm (MSFA) to tune the FOPID controller of the automatic voltage regulator (AVR) system. The MSFA has successfully produced superior results compared to other methods. Moreover, the fractional order particle swarm optimization (FOPSO) has been used as a tuning tool for the FOPID controller of magnetic levitation plants [20]. The proposed FOPSO-FOPID controller has outperformed other tuning tools such as genetic algorithm (GA) and PSO in terms of time response specification and bode plot. Lastly, the study [21] has suggested an artificial bee colony (ABC) as a tuning tool for the FOPID controller of brushless DC (BLDC) motor. The ABC optimization algorithm has obtained better results as compared to GA and modified GA in terms of time domain characteristics, performance indices, and control effort. Meanwhile, a marine predators algorithm (MPA) [22] which inspired by the ocean predators in finding the food, has been widely used as a data-driven tuning tool for various control applications such as tuning the controller of the wind plant [23], fine-tuned PID controller of static synchronous compensator [24] and tuning the PID controller of IEEE-39 bus system [25]. MPA has also been implemented to tune FOPID controllers such as in smart grid power systems [26]. They have shown that the MPA is an effective tuning method for the FOPID controller by outperforming other variants of the PID controller. Interestingly, research to date has not yet implemented the MPA to fine-tune the FOPID controller for the liquid slosh suppression system. Thus, it is worth choosing the MPA for this study.

The purpose of this paper is to introduce the data-driven tuning tool for the FOPID controller of liquid slosh control system based on marine predators algorithm (MPA). MPA is responsible to tune ten parameters of two dedicated FOPID controllers of liquid slosh suppression system in order to track the input of the cart position and in reducing the slosh level of the partially-filled liquid. In this study, a small motor-driven liquid container moving in a horizontal direction which is taken from the study [27] is considered as a basis for the nonlinear liquid slosh model. Here, the effectiveness of the MPA-FOPID controller is assessed by evaluating the performance criteria in regards to the convergence curve of the average fitness function, the statistical analysis of the fitness function, the total norm of tracking error, total norm of slosh angle, and the total norm of control input, as well as Wilcoxon’s rank test. Then, the comparative findings between the MPA-FOPID controller and other metaheuristic optimization based controllers viz. grey wolf optimizer (GWO)-FOPID, sine cosine algorithm (SCA)-FOPID, slime mould algorithm (SMA)-FOPID, and multi-verse optimizer (MVO)-FOPID controllers are conducted to show superiority of the proposed method. In addition, the MPA-FOPID controller performance criteria are also benchmarked with the previously published simultaneous perturbation stochastic approximation (SPSA)-PID controller [27].

This paper is organized as follows. The explanation of the problem formulation of data-driven FOPID controller for liquid slosh control is specified in Section 2. Section 3 is then reviewed on MPA and the application of MPA based method for data-driven FOPID controller design. The results and analysis are presented in Section 4. Lastly, the conclusion regarding the research is concluded in Section 5.

In this section, the discussion on a data-driven FOPID controller by using the proposed MPA is explained. Firstly, the review on MPA is presented. Then, the procedure in applying the MPA as a tool in tuning the data-driven FOPID controller of the liquid slosh problem is discussed in detail.

3.1 A review on MPA

The marine predators algorithm (MPA) [22], is principally motivated by the foraging strategy of the ocean predators. The main motivation of MPA is to be well-balanced between Brownian walk and Levy walk.

The MPA procedure starts with the randomly distribute the initial solution of each agents within given search space in solving the optimization problem as follows

$\arg \min _{X_i(1), X_i(2) \ldots} f_i\left(X_i(k)\right)$            (14)

for iterations $k=1,2, \ldots, k_{\max }$. In Eq. (14), $X_i$ represents the position vector of agent $i, f_i$ represents the fitness function of agent $i$, and the maximum number of iterations is being denoted by $k_{\max }$. The prey and predators are the agents which can defined by two matrices which are the prey matrix $P$ in Eq. (15) and elite matrix $E$ consisting the best predator in Eq. (16), respectively, as follows

$P=\left[\begin{array}{cccc}X_{1,1} & X_{1,2} & \cdots & X_{1, d} \\ X_{2,1} & X_{2,2} & \cdots & X_{2, d} \\ \vdots & \vdots & \vdots & \vdots \\ X_{n, 1} & X_{n, 2} & \cdots & X_{n, d}\end{array}\right]$,             (15)

$E=\left[\begin{array}{cccc}X_{1,1}^{\prime} & X_{1,2}^{\prime} & \cdots & X_{1, d}^{\prime} \\ X_{2,1}^{\prime} & X_{2,2}^{\prime} & \cdots & X_{2, d}^{\prime} \\ \vdots & \vdots & \vdots & \vdots \\ X_{n, 1}^{\prime} & X_{n, 2}^{\prime} & \cdots & X_{n, d}^{\prime}\end{array}\right]$.                   (16)

In Eq. (15) and Eq. (16), n represents the total number of agents, and the number of dimensions is being denoted by d. In (15), $X_{i, j}$ represents the j-th dimension of i-th prey or agent $X_i$ in Eq. (14). In Eq. (16), the matrix E consists of $X_{i, j}^{\prime}$ which is the j-th element of the best predator vector $X_i^{\prime}$. Then, $X_i^{\prime}$ is replicated n times to form an elite matrix. Next, the prey and predators are updated by using the MPA structure which consists of three main phases, fish aggregating devices’ effect (FADs) and marine memory.

3.1.1 Phase 1

The prey moves faster than the predators in Phase 1 which shows that the prey is the most active agent. Meanwhile, the predators are static, only waiting for the prey. This situation is known as full exploration phase which occurred during the first tierce of the maximum iterations ($k_{\max }$). Thus, the updated equation for the prey in Phase 1 is given as

$P_i=P_i+Q \cdot r_1 \otimes\left[R_B \otimes\left(E_i-R_B \otimes P_i\right)\right], \quad$ if $k<\frac{1}{3} k_{\max }$,               (17)

for $i=1,2, \ldots, n$, where $P_i$ is a $i$-th row of matrix $P, E_i$ represent the $i$-th row of matrix $E$, and $R_B$ is a vector of random numbers based on Brownian distribution. Also, the element-wise multiplications is being denoted by the notation $\otimes$. The symbol $Q$ represents the constant number which is set to 0.5 . The random number that is uniformly distributed across the range of $[0,1]$ is represented by symbol $r_1$.

3.1.2 Phase 2

In this phase, both prey and predators are moving at the same pace. The MPA structure in Phase 2 is ready to transit from exploration to exploitation phase by dividing the prey into two groups. The first group of prey perturb its current position towards the exploitation phase by moving in Levy walk. The second group of prey updated its position based on the best predator position towards the exploration phase in which the prey are moving in Brownian walk. This phase occurred after one third of k_max until two third of k_max. Consequently, the first group of prey for i=1,2,…,n⁄2 are updated its location by following equation:

$P_i=P_i+Q . r_1 \otimes\left[R_L \otimes\left(E_i-R_L \otimes P_i\right)\right]$,

if $\frac{1}{3} k_{\max }<k<\frac{2}{3} k_{\max }$,            (18)

while, for second group of prey for i=n⁄2, n⁄2+1,…,n, the updated equation is derived as follows:

$P_i=E_i+Q . C F \otimes\left[R_B \otimes\left(R_B \otimes E_i-P_i\right)\right]$,

if $\frac{1}{3} k_{\max }<k<\frac{2}{3} k_{\max }$.           (19)

In Eq. (18), the Levy walk is represented by R_L which consists of a random number based on Levy distribution. Meanwhile, CF in Eq. (19) represents an adaptive coefficient that controls the predator motion step size which is derived by following equation:

$C F=\left(1-\frac{k}{k_{\max }}\right)^{\left(2 \times \frac{k}{k_{\max }}\right)}$.              (20)

3.1.3 Phase 3

The Phase 3 of the MPA structure is performed on the final tierce of the maximum iteration ($k_{\max }$). This phase is also known as the exploitation phase in which the predators follow the Levy walk and move faster than the prey. Hence, the equation to update each prey (i=1, 2,…,n) is given as follows:

$P_i=E_i+Q \cdot C F \otimes\left[R_L \otimes\left(R_L \otimes E_i-P_i\right)\right]$,

if $k>\frac{2}{3} k_{\max }$.               (21)

3.1.4 Fish aggregating devices’ effect

Next, the MPA behavior is also considered a fish aggregating devices (FADs) by performing longer vertical jumps into the sea in hopes to locate the location with ample prey. This behavior also can assist the predators to jump out from the local optima region and continue a new search region. Thus, the updated equation for prey by considering FADs effect is given as follows:

$P_i=\left\{\begin{array}{cc}P_i+C F\left[L B+r_1 \otimes(U B-L B)\right] \otimes U, & \text { if } r_2 \leq F A D s, \\ P_i+\left[F A D s\left(1-r_2\right)+r_2\right]\left(P_{r_3}-P_{r_4}\right), & \text { if } r_2>F A D s .\end{array}\right.$              (22)

In Eq. (22), the upper bound is being denoted by UB, the lower bound is being denoted by LB, and FADs is a coefficient that has been set to 0.2. Two random numbers which are $r_1$ and $r_2$ are uniformly produced in range of [0, 1]. Meanwhile, the binary vector which consists of value 0 or 1 is being denoted by symbol U. In particular, U produces array to 1 if $r_2>0.2$. Or else, U generates its array to 0. Additionally, the random indexes of the column of the P matrix are represented by $r_3$ and $r_4$.

3.1.5 Marine memory

The last feature of the MPA structure is marine memory in which the location of high production foraging is saved into memory to avoid local optima solutions. The marine memory can enhance the quality of the produced solutions by comparing the fitness of each solution of the current iteration with the previous iteration. If the current fitness is more fitted, it will replace the previous one. If not, it will maintain the previous solution. This method imitates the predator’s behavior in returning to the most abundant foraging location.

For more detailed explanation on the MPA structure, please refer to the MPA pioneer in the study [22].

3.2 Application of MPA for tuning the data-driven FOPID controller

Here, the procedure in utilizing the MPA as a data-driven tuning tool for the FOPID controller of liquid slosh problem is described in detail. The operational of MPA is focused in minimizing the fitness function in Eq. (10) by tuning ten design parameters of two FOPID controllers of liquid slosh suppression system. The step-by-step procedure for tuning the FOPID controller is stated as follows:

Step 1: Map the fitness function of $f_i=J\left(\boldsymbol{K}_{\boldsymbol{P}}, \boldsymbol{K}_{\boldsymbol{I}}, \boldsymbol{K}_{\boldsymbol{D}}, \boldsymbol{\lambda}, \boldsymbol{\mu}\right)$ and design parameter of $X_{i, j}=\log \left(\boldsymbol{K}_{\boldsymbol{P}}, \boldsymbol{K}_{\boldsymbol{I}}, \boldsymbol{K}_{\boldsymbol{D}}, \boldsymbol{\lambda}, \boldsymbol{\mu}\right)$. Then, determine UB, LB, $k_{\max }, n, d, Q,$ and FADs.

Step 2: Execute the MPA structure in Section 3.

Step 3: Record the optimal design parameters $X_i^{\prime}$ from the best predators in the matrix E as soon as achieving the maximum iteration $k_{\max }$. Then, apply this optimal design parameters to FOPID controllers $K_1(s)$ and $K_2(s)$ in Figure 3.

Remarks 3.1. In this paper, it is highlighted that the tuning of the FOPID controller is based on a data-driven method which solely uses the input and output data with no knowledge towards the model of the plant.

Figure 4 illustrates a detailed flow diagram of the FOPID controller tuning approach using the MPA based method. There are two main blocks: the implementation of the FOPID controller for the AVR system block and the MPA based method block. In the first block, the aim is to obtain the minimal objective function J in Eq. (10). By mapping $f_i$ to $J\left(\boldsymbol{K}_{\boldsymbol{P}}, \boldsymbol{K_I}, \boldsymbol{K}_{\boldsymbol{D}}, \boldsymbol{\lambda}, \boldsymbol{\mu}\right)$, the MPA structure can be executed in the second block to obtain the updated design variable of each prey $X_i$, which is obtained from $P_i$. This updated design variable is then applied to the first block by defining $X_{i, j}=\log \left(\boldsymbol{K}_{\boldsymbol{P}}, \boldsymbol{K}_{\boldsymbol{I}}, \boldsymbol{K}_{\boldsymbol{D}}, \boldsymbol{\lambda}, \boldsymbol{\mu}\right)$. This bidirectional flow between these two main blocks is repeated until reaching the maximum iterations to obtain the optimal design parameter $X_i^{\prime}$.

4.png

Figure 4. Flow diagram of MPA implementation for FOPID controller of liquid slosh suppression

In this paper, we have proposed a data-driven tuning tool for the FOPID controller of liquid slosh suppression system based on marine predators algorithm (MPA). From the obtained results, the MPA-FOPID controller has outperformed other recent metaheuristic optimization tuned FOPID controller as well as SPSA-PID controller in terms of average convergence curve, statistical results of fitness function, total norm of tracking error, total norm of slosh angle and total norm of control input. Additionally, the MPA based method is significantly different with other metaheuristic optimization algorithms as has been verified by the Wilcoxon’s rank test. Hence, it has been justified that the proposed MPA based method is a good choice in tuning the FOPID controller for solving the liquid slosh problem. Lastly, due to feasibility and applicability of the algorithm, it can be a good candidate as a data-driven tuning tool for other controllers of practical applications in improving the system performance.