#### Abstract

Lifting systems are in great demand since more and more massive buildings or bridges tend to be shifted or lifted integrally. Hydraulic cylinders in traditional lifting systems are usually supplied by a common pump with an oil tank, which brings long distance hydraulic pipes and signal lines. This paper designs a new architecture of an intelligent lifting system, with self-contained hydraulic power supply system, wireless communication modules, and distributed controller. Based on the designed architecture of the intelligent lifting system, an advanced iterative learning control strategy is proposed to enhance its synchronization performance. With the proposed advanced control strategy, synchronization is achieved in a finite time interval even under the effect of communication time delays and saturations. The distributed controller of a lifting subsystem only uses the delayed information received from subsystems around. This is distinguished from traditional lifting systems, in which all of the lifting subsystems are normally controlled in a centralized way.

#### 1. Introduction

In recent years, there have been many applications of hydraulic lifting systems for buildings or bridges’ maintenance or movement while keeping their structural integrity [1]. A hydraulic lifting system usually consists of several lifting subsystems. In order to guarantee the safety of lifting process and avoid tilting or even capsizing, the most important indicator of the lifting performance is synchronization of the lifting subsystems.

To achieve high-precision synchronization, many researchers have done significant work, and there have been some relative research results. Zhang et al. [2] propose a scheme of multicylinder synchronous lifting system, which included 32 hydraulic subsystems. In this system, every 8 hydraulic subsystems are supplied by a common pump station. In Ref. [3], the authors design a master-slave mechanism solution for the synchronous hydraulic system from the hardware aspect. The communication between the master hydraulic control system and the slave systems is realized by PROFIBUS-DP. All of the hydraulic systems directly receive control signals from the server station through long wires. At the same time, the hydraulic oil for all the lifting systems is supplied through pipe lines from a centralized pump station. This is reasonable when lifting a small building, and all of the hydraulic subsystems are deployed in a compact area. However, for the applications of lifting massive bridges/buildings, the lifting system usually contains numerous hydraulic lifting subsystems and is deployed in a large-scale area. It is apparent that for these applications, the traditional centralized controlled structure is no longer applicable. This requires a distributive control scheme and self-contained hydraulic lifting subsystems, which communicate through wireless data transfer units (DTUs).

At present, the wireless communication technique and networked control technique have developed with high speed. This makes it realizable to replace the long wires with wireless data transfer units in large-scale hydraulic synchronization lifting systems. In addition, the advanced manufacturing technique facilitates production of integrated intelligent hydraulic lifting systems. Therefore, intelligent lifting system architecture composed of self-contained hydraulic lifting subsystems which are equipped with DTUs and distributed controllers is the trend of development and is realizable by modern techniques [4, 5].

The above practical applications and development trends motivate the research work in this paper. Therefore, in this paper, the authors design an intelligent hydraulic synchronization lifting system architecture, where hydraulic lifting subsystems are equipped with wireless DTUs and deployed in an extensive region. The lifting subsystems could access information of other subsystems within the communication range. Each lifting subsystem calculates its control output using received information. Through the coupling of neighbouring information, subsystems gradually reach synchronization based on the condition that the communication topology is connected. Then, the mathematical model of a lifting subsystem is established based on existing literature studies about valve-controlled cylinders [6]. The mathematical model of the entire lifting system is established based on the complex system theory [7–9]. In this model, the coupling of lifting subsystems’ information interacted through DTUs is also considered.

In terms of a separate hydraulic cylinder system, it is able to achieve satisfactory control performance even with a simple PID controller. In fact, many experiments are conducted with the PID controller [10]. Nevertheless, for large-scale synchronization lifting systems, the controller is designed with local acquired information from neighbouring subsystems in a distributive way [11]. Specifically, the local information is acquired through DTUs instead of receiving reference signals directly from a centralized control station. Moreover, time delays are not avoidable in a practical system [12–14]. To deal with the teleoperated hydraulic lifting systems working with delayed communication networks, Maddahiet al. [15] proposed a master-slave control structure. The simple PID controller cannot guarantee synchronization of the lifting subsystems with time delays in the whole lifting process. Thus, we propose an iterative learning distributed controller for each hydraulic subsystem in the large-scale lifting system. The iterative controller is an advanced control technique for improving the synchronization response in systems that repeatedly track a given reference signal or operation [16–18].

To the best of our knowledge, for a hydraulic lifting system, the iterative learning distributed controller designed with local information transferred by DTUs from around intelligent hydraulic lifting subsystems has not been presented in the literature. The rest of this paper is presented as follows: Section 2 addresses the designed intelligent lifting system and analyzes its working principle, based on which the mathematical model is established. The advanced distributed iterative controller for designed large-scale intelligent lifting system is presented in Section 3. Then, the synchronization analysis of the overall lifting system with designed iterative controller is explored in Section 4. Finally, Section 5 carries out simulations and comparisons to illustrate the effectiveness and feasibility of the designed intelligent lifting system and proposed advanced iterative controller.

#### 2. Scheme and Mathematical Model of Intelligent Hydraulic Lifting Systems

##### 2.1. Scheme of Designed Intelligent Hydraulic Lifting System

In this paper, the authors design a scheme of intelligent hydraulic lifting systems. As described in Figure 1, several intelligent hydraulic lifting subsystems hold a common heavy load. All of the hydraulic subsystems are expected to lift the heavy load synchronously in order to avoid damage or even capsizing caused by tilting torque. Except general hydraulic components, each intelligent lifting subsystem is also equipped with a displacement sensor for obtaining current lifting status, a wireless data transfer unit for transferring its status to neighbouring subsystems, and a distributed controller for computing the control value using received local information. Furthermore, the hydraulic subsystem is self-contained with an independent pump rather than pumped by a centralized oil station.

Since the intelligent lifting subsystems are equipped with DTUs, they can interact with neighbouring subsystems; afterwards, the communication topology is formed. In the proposed intelligent hydraulic lifting system, the one nearest to the work station is defined as a leading cylinder and receives the reference signal directly from the server station. On the contrary, all the other lifting subsystems are defined as followers and are not connected to the station.

If the wireless DTU of the intelligent hydraulic subsystem can exchange information with another , then they are set as neighbours. The notation represents that subsystems and are connected, and its value stands for the connection weight, while denotes that subsystems and are not connected and cannot exchange information. Another notation represents that subsystem can obtain information from the leading subsystem. Otherwise, . Assume that the lifting system consists of hydraulic lifting subsystems. Then, the Laplacian matrix of communication topology is , in which and . Denote . Then, the Laplacian matrix reflecting the communication topology with the leading subsystem is extended as .

The notations used throughout this paper are listed in Table 1.

##### 2.2. Mathematical Model of Designed Intelligent Lifting System

This section establishes the mathematical model of the designed lifting system. From the scheme of the intelligent lifting system as shown in Figure 1, it is easy to conclude that the lifting subsystem is mainly a valve-controlled cylinder. The dynamics of the cylinder is

Furthermore, we denote

Then, we reorganize the mathematical model in a compact form:where denotes the nonlinear part of the system, which could be caused by friction or external disturbances. In addition, nonlinear function satisfies the following assumptions.

*Assumption 1 [19]. *The nonlinear function satisfies the Lipschitz condition with respect to over the time interval .where is the Lipschitz constant.

In practical applications, the physical control variable is a voltage or current signal, which linearly correlates with the equivalent area . Therefore, a proportional coefficient is introduced to construct the relationship between the actual control variable and the equivalent area , i.e., . Then, dynamics of the leading subsystem is reconstructed aswhere represents the state of the leading subsystem, denotes the controller, and stands for the lifting height of the leading subsystem’s hydraulic cylinder.

Similarly, the dynamics of the th lifting subsystem iswhere represent the state and lifting height of the th lifting subsystem, respectively, and denotes the distributed controller of th lifting subsystem, which will be designed in the following section.

The objective of a lifting system is that all of the lifting subsystems move synchronously considering the nonlinear effect caused by load disturbance or other issues, i.e., in this paper, all of the following hydraulic lifting subsystems follow the movement of the leading subsystem. To achieve this objective, an advanced control strategy is proposed.

#### 3. Advanced Iterative Controller Design for Lifting Systems with Communication Time Delays

In this section, the synchronization problem of hydraulic lifting systems with communication delays is explored. A distributed iterative learning controller (ILC) with delayed information received from around intelligent hydraulic subsystems is presented, with which synchronization is achieved within finite time interval even under unsatisfactory communication circumstances. Let be the communication time delay between the th hydraulic lifting subsystem and the th subsystem. stands for the around lifting subsystems within the th lifting subsystem’s communication range. Denote as the maximum possible communication time delay between intelligent lifting subsystems at time . In addition, is assumed to be varying between , in which , , and are constants.

As illustrated in Figure 2, each lifting hydraulic subsystem is controlled by an independent controller with information detected from subsystems around. An iterative controller (7) for the th intelligent lifting subsystem is designed for the synchronization objective. For a valve-controlled hydraulic cylinder, input saturation exists since the spool displacement suffers saturation. Thereafter, the ILC algorithm is designed considering the input saturation. As described in (7), the controller not only evolves in time domain but also iterates with respect to the iteration domain .where describes the input saturation function with saturation value , which is defined as

As presented in designed controller (7), the information from around lifting subsystems at time and is adopted, which makes the synchronization performance depend on the estimation of the time delay bound. Then, why not ? This is because at time , the information received by the lifting subsystem is actually . Since , it is easy to know that is not available at time . However, is available by searching the history of received data.

For the sake of analysing the performance of the lifting subsystems as an entire system, we denote . The lifting synchronization error with respect to iteration is . Then, according to (7), the overall controller at iteration *k* is deduced to a compact form:where .

*Remark 1. *In terms of the case where the bottom of the weight is uneven, we can assume that the difference between the length of the rod that each agent extends and the length of the rod that leader extends is . The matrix of all agents’ rod extension lengths is expressed as . Then, the lifting synchronization error with respect to iteration can be described as . Because the difference *e* is constant in the whole process of weightlifting, the whole operation process still satisfies the controllable strategy proposed in this paper.

The overall error of control inputs between lifting subsystems and the designated control input isUsing the fact thatequation (10) can be expressed asThe state error is . Considering the mathematical model of lifting subsystems at time , the lifting error system can be formulated in a compact form:where , , and .

Before analysing the lifting synchronization condition, it is necessary to introduce an important Lemma 1.

Lemma 1 [20] (Bellman–Gronwall). * Over time interval [0, T], if some constants , and some nonnegative piecewise-continuous function , satisfyingthen,*

#### 4. Synchronization Analysis of Lifting Systems with Communication Time Delays

Theorem 1. *Consider the intelligent lifting subsystems (6) and (5) with distributed iterative controller (7), if the following condition satisfies**The lifting synchronization is achieved in finite time .*

*Proof. *The initial cylinder positions of cylinders in all of the lifting subsystems are assumed to be identical, implying . Afterwards, the state error equation isIn light of Assumption 1 and taking norm on both sides of equation (17), we havewhere .

In order to get the relationship between and through eliminating at the right hand side of (18), Lemma 1 is adopted.Through multiplying , it is able to convert the norm into norm:Next, we will get the constraint on . Substitution of the overall dynamic equation of the lifting synchronization error (13) into (12) gives rise toSince the objective is to ensure the convergence of the synchronization error of the large-scale lifting system, some simple manipulations on are implemented, after which it hasFrom the expression of controller (9), it is obvious that . Then, imposing the norm on equation (22) and in light of the Lipschitz condition on external load disturbance, we obtainConsidering the fact that the valve-controlled hydraulic cylinder suffers saturation effect, i.e., , the above inequality (23) can be rearranged:Substituting (19) into (24) and converting the norm into norm by multiplying , then the following inequality holds:Straightforwardly, mathematical derivations on the supremum lead toSince , it has , which leads towhere

The bound of is determined by the control gain and the Laplacian matrix of the communication topology. From the above derivations, it is straightforward to conclude that there exists a sufficient large to make sure is bounded. Therefore, if , we haveThen, the upper bound of the lifting synchronization error is derived asObviously, if is large enough, the synchronization error converges to zero when within the time interval . This completes the proof.

#### 5. Numerical Examples

In order to illustrate the performance of the designed iterative learning controller, this section firstly conducts a simulation where the lifting system is controlled by the PID controller , which is easy to be designed with LMI tools. Then, we carried out simulation with designed controller (7) and compared the synchronization performance to show the advantage of our proposed advanced distributed controller.

The simulation is carried out in a powerful and reliable hydraulic system simulation tool: AMESim. The hydraulic lifting system constructed in AMESim is composed of 1 leading subsystem and 5 followers. As proposed in the scheme of the intelligent lifting system, independent hydraulic pumps are configured for the lifting subsystems, as shown in Figure 3. From the connection status as depicted in Figure 3, we can obtain that . Therefore, it is straightforward to obtain the Laplacian matrix of the communication topology of the intelligent lifting system:

A back pressure valve is added to the oil return circuit to maintain a constant pressure at the pump outlet and to reduce the fluctuation of flow and pressure caused by the siphon. The accumulator in the simulation can improve the stability of the hydraulic cylinder by stabilizing the flow and pressure of the hydraulic pump. In this article, the synchronous hydraulic cylinder is assumed to be fixed. However, the application scenario of the synchronous hydraulic cylinder is quite extensive. This paper mainly discusses the hydraulic cylinder in the vertical direction of the force. When multiple hydraulic cylinders can offset the external influence in the horizontal direction, the same excellent control effect can be obtained. Therefore, for mobile application scenarios, it is necessary to ensure that multiple hydraulic cylinders always have the same motion state, which puts forward higher requirements for automobiles, which is beyond the scope of this article.

The lifting heights of the subsystems are presented in Figure 4. Correspondingly, the lifting synchronization errors are described in Figure 5. From Figures 4 and 5, it is easy to see that the lifting subsystems generally appear to be synchronous under the effect of this distributed controller. However, the synchronization error peaks at the rising stage with maximum value 0.04 m, which could not meet the high requirements. (Table 3).

In order to verify the effect of load size on synchronization accuracy errors, we conduct more simulations to get the table above. According to the simulation data shown in the table above, it can be concluded that the load size affects the synchronization accuracy slightly.

*Remark 2. *It is reasonable to only present the following hydraulic subsystems’ lifting heights in the simulation results. This is because the leading hydraulic lifting system is directly controlled by the server station. It is independent from the following subsystems. In practical applications, the leading hydraulic subsystem is recommended to be deployed separately outside the heavy load for the sake of achieving synchronization performance.

In order to deal with this issue and enhance the synchronization precision in a finite time interval, this paper designs an advanced distributed controller. Next, a simulation is implemented with the iterative controller, which is based on mathematical model (3). Through analysing the AMESim configuration parameters, the state matrix and control matrix are specified asThe nonlinear part of the lifting system is specified asthus, .

The desired trajectory of the lifting system over time interval is configured as a sine curve with mean value 0.1. The communication time delay is . The estimation of communication time delay’s lower bound is , while the estimation of the upper bound is . The saturation of the control input is designed as , and the control parameter is designed asThe lifting heights and the synchronization errors of the lifting system with designed iterative controller at iteration 20 are presented in Figures 6 and 7, respectively. With the proposed advanced control algorithm in this paper, the maximum synchronization error reduces to 0.004 m, which clearly proves the effectiveness of the proposed advanced iterative controller.

#### 6. Conclusions and Future Work

This paper presents an intelligent hydraulic synchronization lifting system scheme, in which wireless DTUs and distributed controllers are equipped in each self-contained lifting subsystem. The intelligent hydraulic lifting subsystems interact with subsystems within their communication range through DTUs. Based on the communication interaction and coupling information, the mathematical model of the overall lifting system is established, and the controller is designed. The proposed advanced iterative controller effectively improves the synchronization performance when lifting a heavy load. The results are based on the assumption that all hydraulic subsystems are working in order. However, malfunctioning of hydraulic systems is possible. It is worth to further design the iterative controller under node failures in the future work. In this paper, the effectiveness of the proposed distributed controller is proved by simulations. However, the proposed controller needs further verification in experiments with the test bench.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 2018RC011).