A Novel Sliding Mode Control with Low-Pass Filter for Nonlinear Handling Chain System in Container Ports

Nonlinearities in a container port handling chain include mainly nonnegative arrive rate of container cargoes, limited container handling completion rate, and nonnegative unsatisfied freight requirement constraints. -e nonlinearity influences the operation resources availability and consequently the planned container port handling strategies. Developments presented in this work are devoted to a novel design of sliding mode control with low-pass filter (SMC-LPF) to nonlinear handling chain system (HCS) in container ports. -e SMC-LPF can effectively reduce unsatisfied freight requirement of the HCS and make chattering decrease significantly. To illustrate the effectiveness and accuracy of the proposed SMC-LPF, an application to a real container port in China is outlined. -e performances of the SMC-LPF for the nonlinear HCS in container ports outperform those of the traditional method, particle swarm optimization algorithm, and slide mode control under simulations with a unit step signal and a sinusoidal signal with offset as the freight requirements. -e contributions herein demonstrate the proposed control strategy in weakening chattering, reducing the unsatisfied freight requirements to 0 as close as possible in the HCS, maximizing the operation resilience and robustness of port and shipping supply chain against parametric perturbation, external disturbances, and fluctuant handling abilities.


Introduction
Over the last decades, there has been a significant growth of global container transportation due to the fast development of global trade. Container ports play a significant role in commercial trade, serving as the crucial nodes in the global port and shipping supply chain. China has become the world's largest container gathering area [1]. Seven Chinese ports have ranked in the top ten container ports in the world since 2012. e throughput of China's ports increased from a mere 2.7 million TEUs in 1993 to 251 million TEUs in 2018 [2]. e increasing amount of container volumes undoubtedly brings much pressure to port operations. Hence, the improved operational efficiency in container ports needs to accommodate this growing demand for container freight service [3].
Container port handling is a complex and dynamic system where different types of machines (such as truck, yard crane, and gantry crane) cooperate closely to complete the transport of inbound and outbound containers between quayside and stack [4]. e container port handling chain system (HCS) may be divided into five main individual operations [5]. More specifically, they are the delivery/ receipt of containers, yard crane loading and unloading operations, yard truck/AGV horizontal transportation between the berth and yard, gantry crane loading and unloading operations, and container vessel berthing [6,7]. A close container port handling chain of efficient landside and seaside operations are inevitable to reduce dwell times and improve the service level as far as possible. As it is usually not easy to expand the infrastructure of a container port, how to effectively establish an efficient container port HCS is of great importance for container port operations management.

Literature Review.
During the past decades, academies, society, and port industry have been striving for an advanced container port handling chain system (HCS) based on analytical, simulation, and control approaches in order to stay competitive [8][9][10]. e analytical approaches search for the optimal solution of the handling operations by addressing the mathematical formulation. For example, Crane Choe et al. aimed at scheduling for opportunistic remarshaling of containers in an automated stacking yard to minimize the delay of the current main jobs as well as the makespan of all the jobs in the horizon [11]. Li and Jia modeled the traffic scheduling problem as a mixed integer linear program (MILP) to minimize the berthing and departure delays of vessels and the number of vessels that could not berth or depart successfully [12]. Kasm and Diabat considered the Quay Crane Scheduling Problem (QCSP) with noncrossing and safety clearance constraints and proposed a two-step approach initiated by a partitioning heuristic and terminated by a Branch and Price algorithm to acquire operationally practical solutions by minimizing crane repositioning movements [13]. Hu et al. addressed the joint vehicle dispatching and storage allocation problem in automated container terminals and proposed a PSO-based greedy search method to solve it, with the goal of minimizing the vehicle operating costs [4]. However, the vast majority disregard the operation resilience problem inherent in the container port HCS when freight requirement changes. e simulation approaches evaluate different management policies by modeling the dynamical behavior of the container port HCS. For example, in terms of agent-oriented programming, Yin et al. described a distributed agent system for dynamic port planning and scheduling [14]. Cubillos et al. presented a decision support system based on the agent technology that helped in solving the problem of berth allocation for ships within a port [15]. With respect to objectoriented programming, Lee and Cho discussed a simulationbased dynamic planning system based on real-time tracking of yard tractors, with two scheduling rules: shortest operation time and equal load distribution among yard blocks [16]. Ursavas and Zhu studied the optimal policies for the berth allocation problem, taking into consideration the uncertain nature of the vessel arrival and handling times [17]. e simulation approaches provide port authorities with enhanced information to decide on the quality and robustness of the proposed schedules, resulting in better solutions for the HCS. e control approaches improve the performance of the container port handling chain by optimizing control decisions on the system behaviors. For example, optimal control approach [18], model predictive scheduling [19], eventdriven receding horizon control [3], robust control [7], and parameter control strategies (such as deterministic, adaptive, self-adaptive, and augmented self-adaptive) [20][21][22]). Despite the diversity of control approaches used to enhance port productivity and efficiency, nonlinearities of the container port HCS has been largely ignored, which does not match the actual situation of container ports and limits the practical applications. However, nonlinearities play an important role in the container port HCS, which is not only directly related to the production efficiency of terminals but also restricts and influences the operation of the other partners in the port and shipping supply chain. erefore, it is essential to consider the nonlinearities in the container port HCS.

Motivation for Work.
Port authorities need to deal with a wide variety of interrelated handling problems, and the productivity and effectiveness of the port depends on their control and scheduling solution. Management strategies are therefore necessary to increase their productivity and effectiveness, and thereby enhancing the resilience of the container port HCS. Up to present, most of the literatures about the container port HCS with a focus on its linearity from the perspective of container ports, few research efforts have been devoted to the nonlinearity of the whole HCS from the standpoint of the port and shipping supply chain. Fortunately, the nonlinearities in the container port HCS gradually begin to receive increasing attention among academies, society, and port industry. Xu et al. modeled the nonlinear handling chain in the container port with control approach. In addition, the authors demonstrated a starting point in the development of a quantitative resilience decision-making framework and mitigating the negative impacts for port authorities and other players in the port and shipping supply chain [5]. However, the exiting models and solution approaches have the following limitations that might hinder practical applications: (i) their model relies on highly simplified assumptions that reducing the complexity of the problem; as a result, the accuracy of their models may vary a lot under different dynamical conditions, and (ii) since their solution method is not a practically applicable exact approach, the method may incur unsatisfied freight service requests even if the handling plans proposed by the port operators are feasible; as a result, the port operators may need to revise their plans frequently, which is undesirable in practice. erefore, there is a need for operational control decisions to appropriately adapt the dynamically operating environments in container ports.
Consequently, our study adds to the literature by emphasizing three important nonlinearities of the HCS in container ports, which includes nonnegative arrive rate of container cargoes, limited container handling completion rate, and nonnegative unsatisfied freight requirement constraints. e main contributions of this work are as follows: (i) a new nonlinear HCS in container ports is developed, which can be used for container handling with different freight requirements; thus, it helps to improve port service levels. (ii) A sliding mode control with the low-pass filter (SMC-LPF) for the nonlinear HCS is formulated for efficiently obtaining enhanced solutions, where strategies for coping with nonlinear constraints are proposed, a sliding mode controller with LPF is designed, and exponential reaching law is employed to improve the dynamic quality of the reaching process. (iii) e performance of the SMC-LPF for the nonlinear HCS in container ports is measured and validated by comparing with those of the traditional method, particle swarm optimization algorithm, and slide mode control under simulations on the certain HCS (the values of freight requirements are a unit step signal and a sinusoidal signal with offset, respectively) and uncertain HCS (the uncertainties are parametric perturbation, external disturbances, and fluctuant handling abilities, respectively). e main improvements of the sliding mode control method are (i) a low-pass filter is added to weaken chattering of sliding mode control; (ii) a stable sliding mode motion with the largest sliding mode area is designed; (iii) the upper and lower limits of control signal are well constructed to deal with the nonlinear segments; (iv) the original arate, which reflects the handling requirements, is employed in the design of control signal.

Linear
Model. e linear model of the handling chain system in container ports is shown in Figure 1. For detailed description of the handling chain system in container ports, we refer the readers to Xu et al. [5], where quay cranes, trucks, and yard cranes etc., collaborate to handle containers from the berth to yard and vice versa. However, tuning the control parameters in the linear model for implementation to real container ports is not an easy task due to coping with the dynamics of uncertainties and disturbances.
As described in Figure 1, the UFR transfer function in relation to the input FR can be calculated as follows: If fr(t) is a unit step signal, ufr(∞) (final value of ufr(t) ) can be obtained by final value theorem: It can be seen that ufr(∞) is equal to 0 only when T G � T Q . does not exist in the actual operation [5]. Under real environments of container ports, the value of arate(t) cannot be negative, although mathematically it holds; the upper value of comrate(t) is limited; fr(t) is not constant; ufr(t) should not be negative even in the case of a sharp drop in fr(t). However, the linear model in Figure 1 cannot guarantee these three nonlinear natures. To reflect them, a novel nonlinear handling chain system (HCS) model consisting of three nonlinear segments is given in Figure 2. ere are three nonlinear segments in this model: the value of arate(t) cannot be less than 0, there is an upper limit value COM-RATE m for comrate(t) because of the actual limited handling ability, and the value of ufr(t) cannot be less than 0, where, COMRATE m should not be less than fr(t) all the time, otherwise ufr(t) will continuously increase, it would congest the HCS.

Nonlinear
Compared to the linear model (Figure 1), the nonlinear system has three more nonlinear segments. It is more complex and difficult to control ufr(t) in the nonlinear system for enhancing the service level and operation performance.

Sliding Mode Control with Low-Pass Filter for Nonlinear Handling Chain Model
A motivation to use the sliding mode control (SMC) is that it has many essential advantages such as fast response, insensitivity to parameters and disturbance changes, and simple implementation. It has been widely applied to many kinds of the nonlinear control system [23,24]. However, there is a drawback of chattering in traditional SMC, which directly influences the performance of the controlled system. To reduce chattering in SMC and improve the control efficiency, a novel SMC-LPF in Figure 3 is designed for the nonlinear HCS in container ports. e dashed box contains two parts: the SMC and the LPF, which together form the SMC-LPF system. Similar to the nonlinear system, in Figure 2, the input signals of the sliding mode controller include avfr(t), ufr(t)/T U , and efrip(t)/T I . arate(t), the output control signal of the SMC-LPF is set as the new input to the lead policy of handling operation 1/(1 + T G s).

Strategies for Coping with Nonlinear Constraints.
Considering the nonlinear segments, the nonnegative arate(t) and the limited comrate(t), it is set that e value of u(t) is exponentially smoothed by a firstorder lag to be arate(t), and the value of arate(t) is exponentially smoothed by a first-order lag to be comrate(t). When the initial values of arate(t) and comrate(t) are both in [0, COMRATE m ], it can be ensured that arate(t) ≥ 0 and comrate(t) ≤ COMRATE m . And the fixed upper limit of u(t) is also more in line with the actual handling volume because the shoreline resources of container terminals are finite and the increase of comrate(t) is limited.
In addition, ufr 1 (t) instead of ufr(t) is managed by the SMC-LPF. e control objective is to make ufr 1 (t) reach 0 as Complexity closely as possible. en, ufr(t) would also be as close as possible to 0.
In summary, it can be seen that all these three constraints have been well coped with the above strategies. Figure 3,

Switching Function of Sliding Mode Motion. From
en, It is set that X 1 (s) � UFR 1 (s) and x 1 (t) � ufr 1 (t). To describe simply, "(t)" is omitted in the following text. en, It is set that x 2 � dx 1 /dt and x 3 � dx 2 /dt, then the state equations are e switching function is employed: It can be known that formulas (10) and (11) will definitely hold according to the generalized sliding mode condition: To make the sliding mode area as large as possible, it is set that Due to T 1 > 0 and T G > 0, the whole line of s � 0 is the sliding mode area if formulas (14) and (15) are satisfied: where u s � fr

Stability Analysis of Sliding Mode Motion. From equation (8), it is known that the sliding mode equation is
en, e characteristic equation of equation (17) is en, Due to T 1 > 0 and T G > 0, the roots of the characteristic equation are both negative. It can be seen that the stability condition of sliding mode motion is satisfied, and the SMC-LPF for the nonlinear HCS in container ports is stable.

Exponential Reaching
Law-Based u. u is the output control signal of the sliding mode controller and the input control signal of the low-pass filter. Exponential reaching law is employed here to improve the dynamic quality of the reaching process, and saturation function is used to replace sign function to further reduce chattering. en, , Δ is a small positive real number, and k Δ � 1/Δ. From equations (9), (12), (13), and (20), It can be derived that It is obvious that formulas (14) and (15) are satisfied if u is always equal to equation (22). However, there are limits for u in the handling chain system of container ports: u min ≤ u ≤ u max (u min and u max are the minimum and maximum values of u, which will be given in Section 3.4; then, To satisfy formulas (14) and (15) in every situation, u min , u s , and u max are required to meet the following formula: And from Section 3.1, it is concluded that 0 < u s < COMRATE m .

Output of Sliding Mode Controller with LPF.
e output of the SMC-LPF is arate. From Figure 3, For the convenience of analysis, let u 1 � arate, then It can be calculated that where u 1 (0 + ) is the initial value of u 1 .

e Impact of Original Arate on u.
In the abovementioned SMC-LPF method, the original arate has not been considered. However, the original arate, which reflects the handling requirements, is very valuable information that should not be ignored. In this work, it is assumed that where μ(μ ≥ 1) is the coefficient of the arrive rate. From Figure 3, oarate � avfr + (ufr/T U ) + (efrip/T I ). To enhance the handling efficiency of container ports, we make some modifications in the SMC-LPF. If frip is lower than dfrip, it is expected that oarate is larger; else, if frip is more than dfrip, it is set that efrip has no influence on oarate. en, where efrip � dfrip − frip and dfrip � T Q · avfr. In equation (29), frip and avfr should be calculated. From Figure 2, en, frip(t) can be calculated through inverse Laplace transform of FRIP(s): e calculation of avfr depends on fr. In Section 4, avfr are given in the cases of specific fr.

Minimum and Maximum Values of u.
e minimum value of u is only limited by formula (3); therefore, it is set that u min � 0. e maximum value of u is subject to formulas (3), (24), and (28). Among them, formula (24) maintains the basic performance of the SMC-LPF. To avoid conflicts among these formulas while maintaining the basic performance of the SMC-LPF, it is set that formula (28) e pseudocode of the maximum value of u is as follows:

Simulation Analysis
e handling chain system that motivated this study is typically a real container port in China. It has four container docks. Under normal operations, its annual container throughput would be around 4 million TEU. In the system under consideration, its resilience can be defined as its ability to preserve the specifications facing variable freight requirements. is work interprets the resilience into ITAE (the integral of time multiplied by the absolute error) of ufr. To validate the efficiency of the SMC-LPF for the nonlinear handling chain system in container ports, five cases are simulated and analyzed. In the first two cases, the handling chain system is certain and the values of fr are a unit step signal and a sinusoidal signal with offset, respectively. In the subsequent three cases, the handling chain system is uncertain. And the robustness of the proposed method is fully shown by case studies on parametric (T G ) perturbation, external disturbances, and fluctuant COMRATE m . e performance of the SMC-LPF is verified by comparing simulation results with those of three other methods: 6 Complexity traditional method (TM), particle swarm optimization (PSO) algorithm, and slide mode control (SMC). In the TM and PSO, the nonlinear model of the handling chain system in container ports in Figure 2 is directly employed. To highlight the effect of the low-pass filter, the application of SMC is the same as that of the SMC-LPF in Figure 3 except that the low-pass filter is not used.
In all four methods, T G � 2, T Q � 1.5, and COMRATE m � 1.3. T G is determined by the handling ability in container ports, and T Q is different with T G to compare ufr(∞) among different methods. In TM, SMC, and SMC-LPF, T F � 3, T U � 0.5, and T I � 0.5. In the PSO method, T G and T Q are fixed values, while T F , T U , and T I are optimized with ITAE of ufr as the fitness function. e ranges of T F , T U , and T I are [3,30], [0. 1,7], and (0, 1], respectively. In the SMC-LPF, it is assumed that the handling chain system has reached stability at the initial moment, and u 1 (0 + ) � u(0 + ).
In all cases, the initial values of ufr, arate, and comrate are 1, 0, and 0, respectively. It is expected that unsatisfied freight requirement would reduce to 0 or as small as possible. And then the expected ufr in the calculation of ITAE is set to 0.

Simulations on Certain HCS.
In this section, the handling chain system is certain, and there are no uncertain factors such as parametric perturbation, external disturbances, and other contingencies.

fr Is a Unit
Step Signal. In Case 1, fr is a unit step signal, and it can be calculated that And it can be proved that formula (24) is met. e best T F , T U , and T I obtained by the PSO method are T F � 6.8120, T U � 0.1, and T I � 0.4313. e comparisons of ufr, arate, and ITAE are plotted in Figure 4. To facilitate the visualization of the trend of each curve, the curves are not drawn in the same figure. ufr(max) (the maximum value of ufr), asymptotic value, settling time, ufr(25) (the last value of ufr in 25 days), and ITAE(25) (the last value of ITAE in 25 days) of four methods are compared in Table 1. Based on the asymptotic value of ufr and the error value 0.01, the settling time is calculated.
From Figure 4(a), the values of ufr obtained by TM and PSO both increase first and then decrease. ufr(max) obtained by TM is much larger than that obtained by PSO. e values of ufr obtained by SMC and SMC-LPF, which are almost the same, both decrease from the beginning. ufr obtained by four methods gradually reaches their respective asymptotic value. e asymptotic value obtained by PSO is smaller than that obtained by TM, while it is still larger than those obtained by SMC and SMC-LPF.
From Figure 4(b), arate obtained by TM declines from 2 and eventually reaches 1. arate obtained by PSO decreases from 10 to 0, then gradually increases to 1, and finally stabilizes at 1. e values of arate obtained by SMC and SMC-LPF are both 1.3 at the beginning, then decreases to 1, and eventually stabilizes at 1. ere is a significant chattering between 2.5 and 5 days in arate obtained by SMC. e chattering has been greatly weakened in arate obtained by the SMC-LPF.
From Figure 4(c), the values of ITAE obtained by TM and PSO have been gradually increasing within 25 days. e growth rate of ITAE obtained by TM is significantly higher than that obtained by PSO. e values of ITAE obtained by SMC and SMC-LPF, which are almost the same, both reach a value less than 10.
From Table 1, ufr(max), asymptotic value, ufr(25), and ITAE(25) obtained by TM are significantly larger than those obtained by other methods. Settling time obtained by TM is shorter than that obtained by PSO, while it is obviously longer than those obtained by SMC and SMC-LPF. All results obtained by PSO are larger than those obtained by SMC and SMC-LPF. ufr(max), asymptotic value, ufr(25), and ITAE(25) obtained by SMC are same or similar to those obtained by the SMC-LPF.
It can be seen that TM and PSO cannot make ufr reach 0 because T Q ≠ T G . It easily results in congestion in container ports in the long term. e asymptotic value obtained by TM is consistent with equation (2), while the asymptotic value obtained by PSO is less than equation (2) because of the nonlinear segments. In SMC and SMC-LPF, ufr can be controlled to reach 0 regardless of whether T Q and T G are equal.
In addition, although most results obtained by PSO are better than those of TM, the arate obtained by PSO is too large at the beginning, which would bring great pressure on the handling chain system in container ports. And all results obtained by the SMC-LPF are better than those obtained by TM and PSO, whereas the control effect of SMC is similar to that of the SMC-LPF. e SMC-LPF overcomes the chattering problem of the sliding mode control system for the nonlinear handling chain model in container ports better.

fr Is a Sinusoidal Signal with Offset.
In Case 2, fr is a sinusoidal signal with offset. When fr fluctuates as fr � (A 1 + A 2 sin ωt) · 1(t) (to avoid misunderstanding, "(t)" in "1(t)" has not been omitted), it can be calculated that en, avfr, the inverse Laplace transform of AVFR(s), is It can be proved that formula (24) is met. T F , T U , and T I in the PSO method are the same as those in Section 4.1. e comparisons of ufr, arate, and ITAE are plotted in Figure 5. In order to fully reflect the trends of ufr, arate, and ITAE, each curve is drawn from 0 to 50 days. In the 2 nd figure of Figure 5(b), the ordinate of the curve for the first 10 days is Complexity on the left, while the ordinate of the curve for the last 40 days is on the right. e thick dashed line is the dividing line. e ordinate on the left can completely present the initial curve, and the ordinate on the right is consistent with other figures to promote convenient comparison among these figures: ufr(max), asymptotic value, settling time, ufr(50) (the last value of ufr in 50 days), and ITAE(50) (the last value of ITAE in 50 days) of four methods are compared in Table 2.
e values of ufr obtained by TM and PSO are always in flux, so there is no data of asymptotic value and settling time in the rows of TM and PSO. From Figure 5(a), the values of ufr obtained by TM and PSO both initially increase, then decrease, and finally remain in the flux. e maximum value and the fluctuation of ufr obtained by PSO are much smaller than those of TM. e values of ufr obtained by SMC and SMC-LPF, which are almost the same, both decrease from the beginning and then gradually reach 0.  From Figure 5(b), arate obtained by TM begins with a gradual decline from 2 and then has been in volatility. arate obtained by PSO first falls fast from 10 and then continues to fluctuate. arate obtained by SMC and SMC-LPF are both 1.3 at the beginning and then remains in flux. ere is little difference among four curves in the fluctuation state between 10 and 50 days. e main difference lies in the variation between 0 and 10 days. Similar to Section 4.1, there is a significant chattering between 5 and 10 days in arate obtained by SMC. e chattering has been greatly weakened in arate obtained by the SMC-LPF.
From Figure 5(c), the values of ITAE obtained by TM and PSO continue to increase within 50 days, although these increases are fluctuating. ITAE obtained by TM is significantly larger than that obtained by PSO. ITAE obtained by SMC and SMC-LPF, which are almost the same, both reach a value less than 30.  From Table 2, ufr(max), ufr(50), and ITAE(50) obtained by SMC are the first largest among four methods. ufr(max), ufr(50), and ITAE(50) obtained by PSO are the second largest among four methods. ufr(max), asymptotic value, and settling time obtained by SMC and SMC-LPF are the same. ere is no overshoot in ufr obtained by SMC and SMC-LPF. e differences among ufr(50) and ITAE(50) obtained by SMC and SMC-LPF are very small.
From result comparisons among four methods, PSO can reduce the numerical size and fluctuation of ufr, but it cannot make it close to 0. SMC and SMC-LPF can make ufr reach 0 although T Q ≠ T G and fr is fluctuating. Similar to those described in Section 4.1, arate obtained by PSO is too large at the beginning, which would bring great pressure on the handling chain system in container ports. arate obtained by SMC and SMC-LPF is between 0 and 1.3, which will not cause this problem brought by PSO. e control effect of the SMC-LPF, which is close to that of SMC, is better than those of TM and PSO. And the chattering problem that exists in SMC is better solved by the SMC-LPF.

Parametric Perturbation.
In Case 3, parametric T G is not constant, as shown in formula (36), when t is less than 27 days, T G is equal to 2. When t is in the interval [27, 34), the port's handling ability decreases significantly for some reasons such as equipment failures and operation conflicts, and T G becomes 14. After active respond, the port's handling ability gradually recoveries. From the 34 th day, T G is 1.9: 14, 27 ≤ t < 34, 1.9, t ≥ 34.
e comparisons of ufr, arate, and ITAE are plotted in Figure 6. In order to fully display the changes of ufr, arate, and ITAE from the 27 th day when T G begins to vary, each curve is drawn from the 25 th to 50 th days. To facilitate the identification of chattering in arate, the dotted grid is not drawn in Figure 6(b), and ufr(max), asymptotic value, settling time, ufr(50), and ITAE of four methods are compared in Table 3. Here, we only care about the impact of changes in T G , the data in the column of "ufr(max)" are the maximum value of ufr from the 25 th to 50 th days, and the data in the column of "ITAE" only calculate the integral of time multiplied by the absolute error for the 27 th to 50 th days. Similar to Table 2, there is no data of asymptotic value and settling time in the rows of TM and PSO because the values of ufr obtained by TM and PSO cannot converge within 50 days.
From Figure 6(a), the values of ufr obtained by TM and PSO still fluctuate during the 27 th to 50 th days. Compared with those in Figure 5(a), they have gradually increased with the increase of T G . ey cannot converge to some asymptotic values in 50 days. e values of ufr obtained by SMC and SMC-LPF begin to increase gradually owing to the change in T G , reach the peaks around the 36 th day, and then decrease gradually. On the 45 th day or so, they approach 0.
From Figure 6(b), arate obtained by TM has not changed much compared to that in Figure 5(b). is means that TM is not sensitive to the change in T G . arate obtained by PSO increases significantly from the 28 th day, reaches the maximum value on the 34th day, then gradually decreases, and lastly returns to the original fluctuation state. e values of arate obtained by SMC and SMC-LPF have changed since the 27 th day. ey continue to increase after small fluctuations, reach 1.3 on the 31 st day, keep them until the 39 th day, then rapidly decrease, and lastly return to the original fluctuation state. arate obtained by SMC has significant chatters around the 27 th day and 39 th to 41 st days while arate obtained by the SMC-LPF, and these chatters have been significantly weakened.
From Figure 6(c), the values of ITAE obtained by TM and PSO continue to increase especially from the 30 th day, and they are significantly larger than those in Figure 5(c). And ITAE obtained by TM is still much larger than that obtained by PSO. e values of ITAE obtained by SMC and SMC-LPF gradually increase from the original stable value, especially from the 32 nd day; since, about the 44 th day, they have generally reached stability and rarely increased again.
ere is not much difference between them, which are much smaller than those obtained by TM and PSO.
From Table 3, ufr(max), ufr(50), and ITAE obtained by TM are the first largest among four methods. ufr(max), ufr(50), and ITAE obtained by PSO are the second largest among four methods. Asymptotic values obtained by SMC and SMC-LPF are both 0. ufr(max), settling time, ufr(50), and ITAE obtained by the SMC-LPF are slightly larger than those obtained by SMC.
It can be seen that perturbation in T G causes significant deterioration in ufr obtained by TM and PSO. However, the parametric perturbation has limited impacts on ufr obtained by SMC and SMC-LPF. SMC and SMC-LPF respond to T G perturbation more effectively and timely. After limited increases for a short time, the values of ufr obtained by SMC and SMC-LPF are able to converge to 0 again. ere is not much difference between results obtained by SMC and SMC-LPF. SMC has the disadvantage of chattering, which has been significantly improved in the SMC-LPF.

External Disturbances.
In Case 4, there are external disturbances in this uncertain HCS of container ports, such as bad weather, equipment failure, and traffic congestion. ese external disturbances cause fr and comrate' change. After the external disturbances, the freight requirement and container handling completion rate become fr + f 1 and comrate'+f 2 . f 1 and f 2 are 10 Complexity where f 10 � 0.12, A 1 � 0.01, ω 1 � 0.5, φ 1 � 7π/12, t 1 � 30, f 20 � 0.15, A 2 � 0.02, ω 2 � 0.6, φ 2 � − 5π/12, and t 2 � 40. e comparisons of ufr, arate, and ITAE are plotted in Figure 7 ufr(max), asymptotic value, settling time, ufr(60), and ITAE of four methods are compared in Table 4. Due to the occurrence of external disturbances on the 30 th day, each curve in Figure 7 is drawn from the 25 th to 60 th days, data in the column of "ufr(max)" are the maximum value of ufr from the 30 th to 60 th days, and data in the column of "ITAE" only calculate the integral of time multiplied by the absolute error for the 30 th to 60 th days.
From Figure 7(a), the values of ufr obtained by TM and PSO both increase significantly from the 30 th day, and the values of ufr obtained by SMC and SMC-LPF both increase from the 32 nd day. ey all peak around the 37 th day and then gradually decrease. e peak of ufr obtained by TM is the first largest among four methods. e peak of ufr      Figure 8. ufr(max), asymptotic value, settling time, ufr(50), and ITAE of four methods are compared in Table 5. Similar to Section 4.2.1, each curve in Figure 8 is drawn from the 15 th to 50 th days, data in the column of "ufr(max)" are the maximum value of ufr from the 20 th to 50 th days, and data in the column of "ITAE" only calculate the integral of time multiplied by the absolute error for the 20 th to 50 th days. And there is no data of the asymptotic value and settling time in the rows of TM and PSO because the values of ufr obtained by TM and PSO cannot converge within 50 days.
From Figure 8(a), the values of ufr obtained by TM and PSO continue to fluctuate from the 20 th day, and they change little compared with Figure 5(a). After the 25 th day, they quickly recover to their original fluctuation state. It can be seen that the fluctuation of COMRATE m in TM and PSO has a small impact on ufr. e values of ufr obtained by SMC and SMC-LPF are always kept at 0. ey are completely unaffected by the fluctuation of COMRATE m in SMC and SMC-LPF.
From Figure 8(b), arate obtained by TM maintains its original fluctuation and does not respond to the change of COMRATE m in real time. From the 20 th to 25 th day, arate obtained by PSO experiences a large fluctuation. Around the 20 th day, arate obtained by SMC shows a severe chatter and then returns to the original fluctuation state, while arate obtained by the SMC-LPF only changes slightly; at other times, arate obtained by SMC and SMC-LPF are roughly the same.
From Figure 8(c), the values of ITAE obtained by TM and PSO both continue to increase, but obviously the former is much larger than the latter. e values of ITAE obtained by SMC and SMC-LPF are an order of magnitude smaller than that obtained by PSO. After the 25 th day, ITAE obtained by SMC increases slightly while that obtained by the SMC-LPF hardly increases.
From Table 5, ufr(max), ufr(50), and ITAE obtained by TM are the largest in those obtained by four methods. ufr(max) and ufr(50) obtained by SMC and SMC-LPF are very close to 0. ITAE obtained by SMC and SMC-LPF are also much smaller than that obtained by PSO. Asymptotic value and settling time obtained by SMC and SMC-LPF are the same with each other. e values of ufr(max) obtained by SMC and SMC-LPF are almost equal. ufr(50) and ITAE obtained by the SMC-LPF are still obviously smaller than those obtained by SMC.
It can be seen that the impact of fluctuant COMRATE m on results obtained by four methods are not as strong as those in Sections 4.2.1 and 4.2.2. e reason may be COMRATE m only decreases in a short time and the decrease is small. TM responds less to it. Although the response of PSO is very violent, the control effect is still not as good as SMC and SMC-LPF. To cope with the fluctuation of COMRATE m , arate obtained by SMC generates significant chatter, which is greatly smoothed in arate obtained by the SMC-LPF.
In general, sliding mode control can significantly improve the control effect of the nonlinear handling chain system in container ports and make ufr reach 0 although T Q ≠ T G and fr is fluctuating. In the cases of parametric (T G ) perturbation, external disturbances, and fluctuant COM-RATE m , sliding mode control which can also reduce the impact of these uncertainties on ufr has strong robustness.
ere exists significant chattering problem in arate obtained by SMC, while the SMC-LPF can effectively overcome the chattering problem. e results of the two cases discussed above show the usefulness of the SMC-LPF, which is highlighted in the analysis. e presented control strategy with little chattering tries to reduce ufr to 0 as close as possible in the nonlinear handling chain system and to maximize the operation resilience. ese can be recommended for the partners of port and shipping supply chain to make decisions.

Conclusion
is work extends the existing nonlinear model of Xu et al. and explicitly considers the nonnegative unsatisfied freight requirement constraint. Correspondingly, a novel SMC-LPF is designed for the nonlinear handling chain system in container ports. e upper and lower limits of the control signal are well designed to deal with the nonlinear segments. A low-pass filter is added to weaken the chattering problem of sliding mode control. Switching function of sliding mode motion is constructed and the stability of sliding mode motion is analyzed. Exponential reaching law is employed to improve the dynamic quality of the reaching process. e simulation results of fr for a unit step signal and a sinusoidal signal with offset show that the proposed SMC-LPF method can significantly improve the control effect of nonlinear handling chain system in container ports, effectively weaken the chattering problem, and has strong robustness for  uncertainties. ough this work concentrates on container port handling chain system, the proposed framework and method can be applied to the other areas of the port and shipping supply chain.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this article.