A study on automated ribbon bridge installation strategy and control system design

Introduction

Background and motivation

As the global demand for water crossings increases due to the prevalence of rivers, lakes, and channels, various efficient methods for overcoming these water obstacles have emerged.

The bridging system remains the most widely used structure for water crossings, designed to connect land across rivers, lakes, and oceans Various bridge designs cater to specific objectives and conditions, but they primarily fall into two categories: permanent (fixed) bridges, which are constructed in a specific location, and temporary bridges, which are portable and can be relocated as needed This article focuses on the ribbon floating bridge, a notable example of a temporary bridge system.

The second type of water obstacle crossing is the tunnel, which is distinct from other methods due to its underground structure, created by excavating through the surrounding soil, earth, or rock.

Ferries are a popular type of water obstacle crossing, operating similarly to surface vessels but over shorter distances Their primary functions include transporting passengers, vehicles, and cargo across bodies of water efficiently.

The ribbon floating bridge, also known as a foldable float bridge, plays a crucial role in transportation, particularly for emergency restoration in military and civil applications It offers several advantages over traditional land-based bridges, including the capacity to transport heavy loads, rapid installation, a simple structure, and minimal environmental impact Its most notable feature is its portability, allowing for easy relocation when needed.

Fig 1.1 The actual ribbon bridge system

The ribbon floating bridge, developed in the 1970s in Germany by EWK and in the US, was designed to be fully interoperable under a NATO agreement Since its inception, these bridges have become essential for military forces in various countries, including Germany, Canada, and Australia.

The ribbon floating bridge, as illustrated in Fig 1.1, utilizes buoyancy to support its own weight and vehicular loads, effectively reducing costs and extending its lifespan Constructed from lightweight materials such as concrete, steel, aluminum alloys, and composites, these bridges demonstrate significant success in various applications across the US.

Fig 1.2 Conventional methods for ribbon bridge installation

The success of constructing and operating ribbon floating bridges hinges on two key factors: safety and speed Traditional installation relies on manual labor, utilizing erection boats and cables to position each bay, which is then floated into place and connected by operators using support boats This method can be time-consuming due to challenges faced by workers in navigating rivers or lakes While ribbon floating bridges are often used for rapid deployment, particularly in military contexts, manual installation can pose significant risks in hazardous situations, such as combat Consequently, there is a pressing need for an advanced installation strategy that incorporates autonomous, self-operated systems to mitigate these risks Various approaches to the dynamic analysis of ribbon floating bridges have been explored, including studies on their behavior under moving loads and the analysis of dynamic displacement and connection forces.

Research on the dynamic response of floating bridges with transverse pontoons is essential Most existing studies focus on analyzing the dynamic responses and structures of these bridges However, understanding the dynamic characteristics and thoroughly analyzing the bridge system is crucial for developing appropriate mathematical models for effective control design.

Floating bridges share dynamic characteristics with water surface vehicles, making their control a natural extension of managing specialized floating units Numerous strategies have been documented for both surface vehicles and dynamic positioning, highlighting the evolution of this technology.

Dynamic positioning (DP) has its origins in the 1970s, with significant advancements in control techniques introduced by Balchen, who expanded his work in the 1980s Additionally, Grimble utilized stochastic optimal control theory to enhance DP systems, while Strand explored nonlinear control theory for surface vessel positioning Fossen contributed general solutions for vessel control system designs focused on active propulsion for course tracking and station keeping Furthermore, Sorensen proposed a model-based control technique that effectively manages both course tracking and station keeping, and Katebi and colleagues developed a robust controller for a DP system.

Numerous researchers focus on the dynamics of propulsion systems and their applications in marine system control John provided an overview of marine propellers and propulsion systems, while ỉyvind and Damir explored various topics related to marine electrical power systems and propeller control Sứrensen et al introduced torque and power control for electrically driven marine propellers Additionally, Timothy reviewed trends in ship electric propulsion, highlighting the future of electric use in marine vehicle control.

The use of ribbon floating bridges for crossing water bodies is gaining popularity, yet research and information on the subject remain limited A study by Yasuhiro and colleagues introduced a measurement system for tracking the positional displacement of floating units in pontoon bridges Additionally, Kim et al proposed an installation strategy for ribbon bridges, facilitating the design of control systems for these innovative structures.

Problem Statements

Based on aforementioned analyses, the problem statements to in- troduce a new autonomous installation strategy and control system for the ribbon floating bridge are described as follows:

This article introduces an innovative automated installation strategy for ribbon floating bridges, aimed at significantly reducing the risks associated with manual labor It discusses various ideas that can be implemented to enhance the floating bridge system's efficiency and safety.

This article focuses on the mathematical modeling of a ribbon floating bridge that utilizes active electrical propulsion systems The aim is to drive the entire bridge and derive the system dynamics, which are expressed through state equations.

• To identify characteristic parameters and define the external water current disturbance The state estimator is designed based on observability estimation.

This article presents the design of a yaw motion controller utilizing the Linear Quadratic Regulator (LQR) control technique, integrated with a pre-defined state observer The effectiveness and robustness of the controller have been validated through numerical investigations and experimental tests.

A robust sliding mode control (SMC) strategy has been developed to effectively address a range of disturbances and uncertainties in under-actuated systems This SMC controller has been validated through various experimental conditions, demonstrating its effectiveness and reliability.

Objective and researching method

This dissertation aims to create a control system utilizing propulsion systems for the automated installation and self-operation of ribbon floating bridges, thereby reducing the challenges and risks associated with manual labor Two observer-based controllers are proposed to achieve these objectives effectively.

• The whole bridge will be rotated from the initial position by a parallel control actions: the yaw motion and yaw displacement among floating units.

Once the bridge is in its intended position over the water barrier, the control system's primary role is to maintain the linearity of the entire structure by adjusting the yaw displacement of the floating units in response to the current water flow and wave forces.

This article focuses on the mathematical modeling of a ribbon floating bridge by analyzing its mechanical and electrical behaviors Key parameters in the model will be identified through experiments and estimated using the Matlab Identification Toolbox.

To create an effective controller using the LQR technique, a linear observer is established to estimate unmeasured states Various simulations are conducted, exploring different desired outputs and initial yaw angles under diverse conditions To validate the proposed controller, an experimental model is developed, applying the controller to replicate the actual processes involved in bridge installation and operation.

An advanced controller utilizing Sliding Mode Control (SMC) has been developed to effectively manage external disturbances, including current flow and wave attacks The proposed controller's robustness and effectiveness are validated through both simulation and experimental studies.

The ribbon floating bridge system consists of five interconnected floating units, with three units powered by electrical propulsion and two driven passively To measure yaw displacements, five incremental encoders are utilized The control system is managed through a National Instruments PXIe-8115 embedded controller, paired with acquisition cards NI PXIe-6363 and PXI 6221 Programming is conducted using NI LabVIEW 2016, with a sample time of 0.03 seconds Experimental results demonstrate the proposed strategy and control system's feasibility, robustness, and effectiveness.

Organization of dissertation

This chapter outlines the background and motivation behind the dissertation, detailing the problem statements, objectives, and research methods employed Additionally, it provides an overview of the dissertation's organization.

Chapter 2: Introduction of the Ribbon Bridge, Modeling, and Identification

This chapter outlines the structure of a widely implemented ribbon floating bridge and describes the model used for both modeling and experimentation It details the scale model of the ribbon floating bridge system, which is crucial for control system design and utilized in this dissertation Due to the unknown system parameters mentioned in the previous chapter, essential parameters are identified through necessary experiments, aided by computational software.

Chapter 3:Observer-Based Optimal Control Design with Lin- ear Quadratic Regulator Technique

This chapter presents a yaw motion controller designed using optimal control and a state estimator, focusing on the critical need to derive unmeasured state variables since only yaw angles are directly measurable A linear observer is developed based on observability estimation, with its effectiveness confirmed through simulation investigations Additionally, a servo-system is introduced for precise control of yaw displacement among floating units and the overall yaw motion of the bridge Simulation and experimental results demonstrate the efficacy of the proposed approach.

Chapter 4: Motion Control Performance with Sliding Mode Control Design

Chapter 4 proposes the sliding mode controller design for under- actuated ribbon bridge system The sliding mode control law guar- antees both fast response and robustness of desirable performance of position keeping under environmental disturbances In combination with the pre-defined observer, the sliding mode controller shows the outstanding control performance verified by both numerical investi- gation and experimental studies.

Chapter 5: Conclusion and Future Study

The major research results of this study are summarized and sug- gestions for further research are presented.

Induction of the Ribbon Bridge and Modeling 10

System description

2.1.1 Overview of the ribbon floating bridge

The Ribbon Floating Bridge (RFB), also known as a foldable float bridge, is a modular structure designed for single or double-lane traffic, featuring a floating superstructure made of aluminum that functions as a pontoon This innovative bridge system includes interior and ram bays that can be easily assembled and disassembled, allowing for convenient transportation by trucks, helicopters, aircraft, and railway cars when folded Upon reaching the water, the ribbon bays automatically open to create a bridge or ferry-like structure Traditionally, the segments are positioned using erection boats or manual labor, with the aid of cables, while workers connect the individual bays using mechanical locks and couplings.

RFBs are engineered to transport heavy combat vehicles and trucks as floating bridges or ferries, offering significant advantages over traditional bridge systems such as rapid installation, straightforward structures, and minimal environmental impact Their exceptional portability allows for easy relocation, enhancing their versatility However, Chapter 1 outlines several limitations related to the installation and operation of RFBs To address these challenges, an automated solution is proposed in the following section.

2.1.2 An automated installation and operation strategy for RFBs

The manual operation of RFBs (Radio Frequency Beacons) poses significant safety risks and inefficiencies, particularly in hazardous situations or combat scenarios To address these challenges, it is essential to implement an autonomous driving system for the installation and operation of RFBs This study introduces a novel strategy for automated control through yaw motion control, enhancing both safety and operational speed.

The RFB will be assembled on one side of the water body by linking the ram bays and interior bays Once fully constructed, it will be maneuvered into position across the water body to connect both sides of the obstacle During this phase, the controller must ensure the bridge remains linear by adjusting the displacement of the floating units, while also ensuring a smooth transition to the target location.

In the next stage, the control system must maintain the bridge in its target position while ensuring its linearity To achieve this, the controller must demonstrate stability and quick adaptability to external disturbances such as current flow, wave attacks, and moving loads.

The ribbon floating bridge model description

This study examines the installation and operation of redox flow batteries (RFBs) using a fully automated control system that incorporates propulsion systems To address this topic, the article presents the mechanical and electrical designs associated with the RFBs.

For modeling, a ribbon floating bridge system consisting five ribbon bays is designed as shown in Fig 2.2 There are three active bays

Fig 2.1 A proposed installation strategy for the ribbon bridge

The five-bay ribbon bridge model structure features propulsion systems driving the active bay, while the remaining bays are passively driven through a coupling relationship Detailed illustrations of both the active bay and passive bay can be found in Figures 2.3 and 2.4, respectively.

The five floating units share a similar structure and dimensions in terms of length, width, height, and weight Each unit features two adjustable slots on the left and right sides for connector insertion, along with four additional axes equipped with integrated holes for spring installation All bays are constructed from acrylic, a type of polymer material, while the active unit includes extra bases for housing electrical control components such as the DC motor, motor drive, and propeller Detailed specifications can be found in Table 2.1.

In contrary, the passive unit does not contain additional parts as the active one Therefore, to maintain the balance in term of weight,

Fig 2.3 Structure of the active bay

Table 2.1 Parameters of floating unit

Type of connection rotate connector the additional weight must be inserted.

To effectively connect five floating units, incorporating connectors is essential Figure 2.2 illustrates the various designs of these connectors, showcasing two distinct structures utilized on the left and right sides.

The passive bay side structure features a connector on the left that aligns with the axis of the incremental encoder, while the right side includes a protective component to shield the encoder from water To ensure precise measurements, it is essential that the connectors allow for unrestricted movement.

The control system structure utilized in this study is illustrated in Fig 2.5 This proposed control system for the RFB is designed as a closed-loop system, which includes essential components such as sensors, controllers, and actuators.

Fig 2.5 The configuration diagram of the control system

The sensor system features five incremental encoders positioned at the connectors between each pair of floating units, enabling the measurement of yaw motions for controller feedback Detailed specifications of these encoders are illustrated in Fig 2.6.

The control configuration for the RFB system, illustrated in Fig 2.5, utilizes a National Instruments PXIe-8815 embedded controller housed in a PXIe-1078 chassis to execute control algorithms This main controller is complemented by data acquisition cards, specifically the PXIe-6363 and PXI-6221, which facilitate signal acquisition from encoders and the transmission of control signals, including both digital and voltage outputs, to the actuators Detailed specifications of the embedded controller are provided in Table 2.2.

Table 2.2 Detailed specification of the PXIe-8115 embedded controller

Hard disk 2.5 inch SATA SSD 128 GB

Power supply 150 VAC to 260 VAC

The specifications and structures of the DAQ cards PXIe-6363 and PXI-6221 are illustrated in Figures 2.7 and 2.8 The actuator system consists of motor drivers, DC motors, and rotating propellers, with each DC motor managed by a KDC248H 12-24V motor driver The main controller sends an analog signal to the motor driver through the DAQ card Detailed specifications and a photograph of the motor drive can be found in Figure 2.9.

In this study, the Graupner Speed 700BB Turbo 12V Brushed Mo-

Fig 2.7 The photo and specification of NI PXIe-6363

The NI PXI-6221 DC motor is utilized to power the azimuth propeller, generating the necessary thrust for the bridge system Detailed specifications and structural designs of both the DC motor and the Propulsion Schottel Graupner are illustrated in Figures 2.10 and 2.11.

Motor drivers convert digital signals and analog control inputs from the main controller into output voltages for DC motors, generating the necessary forces to operate the bridge system effectively.

Fig 2.9 The photo and specification of NI PXI-6221 the measured yaw position, the suitable control inputs are utilized for each propulsion system placed on active floating unit.

Fig 2.10 The photo and specification of DC motor

Fig 2.11 The photo and specification of the propeller

The RFBs Modeling

2.3.1 General Modeling for Control of the RFBs

For simplification of the mechanical system modeling, the follows are assumed:

• The RFB model is considered as a rigid body.

• The joint connecting floating units is fully constrained in the translation direction and completely free in the direction of rotation.

• Rigid floating units have entirely symmetrical structures in left- right and up-down direction.

• The spring parameters are the same for all springs except the ones in starting and ending positions.

• The movement in z-axis (water surface and vertical direction) is negligible.

The following notations are defined for developing the mathematical modeling of the RFB:

The moment of inertia for the i-th floating unit is denoted as J_i, while the absolute damping coefficient for the same unit is represented by ˆc_i The stiffness coefficient of the upper translation spring linking the i-th and (i+1)-th floating units is indicated as k_u_i, whereas the stiffness coefficient of the lower translation spring connecting these units is k_d_i Additionally, the damping coefficient for the joint between the i-th and (i+1)-th floating units is referred to as c_i.

In this study, we analyze various parameters related to the floating units, including T_i, which represents the external torque acting on the i-th floating unit, and θ_i, the yaw displacement, where counterclockwise is defined as the forward rotation direction Additionally, we examine θ˙_i, the yaw velocity, and θ¨_i, the yaw acceleration of the i-th unit The upper and lower linear relative displacements between the i-th and (i+1)-th floating units are denoted as x_u_i and x_d_i, respectively Furthermore, we define w as the distance from the left and right sides of the floating unit to the center of the joint, and h as the distance from the top and bottom sides of the floating unit to the center of the joint.

In this case, as previous mentioned, we assumed k u i =k u i ≡k(i 1,2,⋯,n−1) The exception of the first and the last springs have different stiffness coefficientsk 0 u ,k d 0 ,k u n ,k n d

In this study, we assume that the damping coefficients for all joints are uniform, denoted as \( c_i \) for \( i = 1, 2, \ldots, n-1 \), with the exception of the first and last joints, \( c_0 \) and \( c_n \) Additionally, the absolute damping coefficients for the floating units are represented as \( \hat{c}_i \) for \( i = 2, 3, \ldots, n-1 \), while \( \hat{c}_1 \) and \( \hat{c}_n \) differ from \( \hat{c} \).

Therefore, the motion equation of the first floating unit and the n t hone will be expressed as follows:

(2.2) The equation that describes the motion of thei th is expressed as:

2.3.2 The Pilot Model of the RFB Modeling for Control Design

This section presents a comprehensive mathematical model of the pilot system for the Redox Flow Battery (RFB), which is utilized for control design and experimental analysis As previously discussed in the mechanical and electrical design sections, the RFB model incorporates five floating units.

Generally, the dynamics of floating units can be described using kinematic and kinetic model [13]:

The equation Jν˙ + Cν = τ (2.4) illustrates the relationship between control input forces and the dynamics of a floating unit Here, τ denotes the forces and moments generated by propulsion systems and spring forces, while the vector ν = [u, v, r] T ∈ R 3 captures the body-fixed velocities, including surge, sway, and yaw rate Additionally, J ∈ R 3 represents the mass/inertia matrix of the system.

C v is the hydrodynamic damping matrix J andC v are expressed as follows:

(2.6) wheremis the floating unit’s mass andI z is the inertia moment of the floating unit on the fixed z-axis.

Fig 2.12 The structure of five-floating unit bridge system

The five-bay RFB structure, depicted in Fig 2.12, demonstrates that one end of the system is fixed, reflecting its real operational dynamics Consequently, the control challenge is framed as managing planar motion in relation to yaw angles.

The yaw motion of each individual floating unit is defined by equations (2.1) to (2.3) Utilizing these definitions, the mathematical modeling of the RFB system, which comprises five floating units, can be detailed through the following equations.

C i ,θ i (i=0∼5)are the damping coefficient and the yaw rate of the corresponding floating unit, respectively;

F i lu and F i ld are the spring forces at the left-hand-side on upper and lower positions of the floating unit;

F i ru andF i rd are the spring forces at the right-hand-side on upper and lower positions of the floating unit;

F i p (i=1,3,5) is the force generated by the propulsion systems installed in three active floating units;

F i w (i=0∼5) is the water current flow force attacking to each floating unit. hand w v are the corresponding arms of spring force, propulsion and water current forces.

System Identification

To effectively design control systems, it is crucial to acquire the dynamic parameters of the system Experiments conducted in the absence of water flow currents or external disturbances, while assuming a homogeneous mass distribution in the floating unit and symmetrical geometry for xz and yzh, indicate that the center of gravity aligns with the center of geometry Consequently, the floating unit model can be represented as specified in equation (2.4), leading to the determination of the inertia moment and damping matrices.

This dissertation focuses on yaw motion control, emphasizing the identification of key parameters, specifically (I z −N r ˙ ) and N r in the yaw direction The mathematical equation for system identification is outlined in detail.

Due to the lightweight nature of the floating unit, utilizing a propulsion system for generating pushing force is more effective By analyzing the correlation between input voltage and output force, along with the connection between input voltage and resulting yaw motion, it becomes feasible to determine the required parameters The necessary processes to achieve these objectives are then implemented.

Firstly, the relation between input voltage and output force is identified by experiment The experiment setup is shown in Fig 2.13.

Fig 2.13 The experiment setup for propulsion system identification

Fig 2.14 The input step voltage and the obtained output force

Fig 2.15 The fitting result of identified model for propulsion system

The propulsion system receives a step input voltage, and the resulting pushing force is measured using a load cell, as illustrated in Fig 2.14 The collected data is then processed using the Matlab Identification Toolbox, leading to the formulation of the transfer function for the propulsion system, represented by equation (2.15), with the corresponding matching displayed in the analysis.

Fig 2.16 The experiment setup for inertia and damping coefficient identification

Fig 2.17 The least square data fitting result

The floating unit is rotated using a pushing force generated by applying the appropriate voltage, allowing for the estimation of the equivalent moment based on this input voltage The yaw angle is determined through an incremental sensor attached to the coupling joint, along with data recorded by a camera The experimental setup is illustrated in Fig 2.16.

The two unknown parameters(I z −N r ˙ )andN r were estimated by using least square fitting technique The fitting result for the obtained data from experiment is presented in Fig 2.17.

To obtain the reliable and accurate data, at least 10 times of experiments were conducted Then, the unknown data is estimated as follows:

Summary

This chapter outlines the structure of the ribbon floating bridge system utilized in both military and civil applications globally It provides a detailed description of the bridge model system used for simulation and control design, highlighting the mechanical and electrical design aspects A mathematical model for the ribbon bridge, comprising five floating units, was developed for controller design purposes Furthermore, the characteristic parameters of the floating unit and the propulsion system transfer function were effectively identified through experimental execution and computational software analysis.

Observer-Based Optimal Control Design

Introduction

In the previous chapter, we successfully established the mathematical modeling of the RFB and identified the necessary parameters for controller design However, a challenge arises as only yaw motion angles can be directly measured by sensors Therefore, it is essential to develop a controller that incorporates a state estimator.

This chapter discusses the design of yaw motion control for the RFB system, focusing on installation and station keeping through an observer-based optimal control strategy Optimal control, particularly for MIMO systems in state-space form, is favored for its robustness and effectiveness The Linear Quadratic Regulator (LQR) technique is highlighted for its excellent disturbance response and system stability To implement full-state feedback control, a state estimator is essential for acquiring unmeasurable data The proposed controller's effectiveness and performance are validated through experiments conducted under various working conditions.

Control System Framework

The dynamic equation of the five-floating unit RFB can be expressed in a state space model, represented by the equation x˙=Ax+Bu+dw, where the output is given by y=Cx The state matrix A, input matrix B, and output matrix C are defined accordingly.

The variables in above matrices representing for the bridge system are determined and stated as follows: a 21 =a 65 =a 87 =a 1210 = −4kih 2 /J= −2.10; a 22 =a 66 =a 88 =a 1212 =a 1414 = −(C v +2C0)/J= −4.40; a 23 =a 89 =a 1416 =w/J'.30; a 25 =a 61 =a 67 =a 85 =a 810 =2kih 2 /J=1.05; a 127 =a 1213 =a 1411 = −a 1412 =2k i h 2 /J=1.05; a 26 =a 62 =a 68 =a 812 =a 128 =a 1214 =a 1412 =C 0 /J=1.81; a 43 =a 109 =a 1615 = −55.74; a 44 =a 1010 =a 1616 = −6.95;

−d 1 = −d 2 = −d 3 = −d 4 = −d 5 =w/J'.30; b 1 =b 2 =b 3 50 and x= [ θ 1 θ˙ 1 x f 1 x˙ f 1 θ 2 θ˙ 2 θ 3 θ˙ 3 x f3 x˙ f3 θ 4 θ˙ 4 θ 5 θ˙ 5 x f 5 x˙ f 5 ] T is the state variable u= [u 1 u 3 u 5 ] T is the control input voltage provided to the propulsion systems.w d = [F 1 w F 2 w F 3 w F 4 w F 5 w ] T is the vector that represents the current flow disturbance forces on each floating unit.

The position control servosystem for the RFB, utilizing an optimal control strategy based on an observer, is illustrated in Fig 3.1 It is essential to establish the controller and observer gains The subsequent section will focus on designing the state estimator and the Linear Quadratic Regulator (LQR) controller.

Fig 3.1 The servosystem for positional control of the RFB system

Observer-based Control Design

To achieve effective feedback performance, it is essential to have access to all state variables for feedback, leading to the formulation of the control input as u = K_f w x, where K_f is the feedback gain matrix However, implementing full-state feedback control poses challenges, as the requirement for all state variables can result in complex feedback loops, causing economic inefficiencies and practical difficulties Moreover, this study identifies several states that cannot be directly measured for feedback Therefore, it becomes crucial to utilize an observer to continuously estimate and monitor these state variables for effective feedback.

The full-state observer for the system, as described by Luenberger, is represented by the equation ˙ˆ x = Aˆx + Bu + L(y − Cˆx), where ˆx is the estimated state of x and L is the observer gain matrix that must be determined The observer's structure is illustrated in Fig 3.2 With the outputs of the system available at all times for feedback, this information can be utilized to construct an estimator that accurately estimates the system's state variables.

Fig 3.2 The diagram of a full-state observer structure

The final goal is that the observer provides an estimated ˆxso that ˆx→xwhent→ ∞ Let’s define the error of the estimator as e x (t) =x(t)−ˆx(t) (3.3)

To ensure the accuracy of proposed estimator, it is required thatex(t) →

0 ast→ ∞ If the system is observable, there will always existL to satisfy that the tracking error is asymptotically stable.

The derivative of the estimation error yields ˙e x =x˙−x By substi-˙ˆ tuting the system model of the RFB and the observer model in (3.2), we obtain ˙ e x =Ax+Bu−Aˆx−Bu−L(y−Cˆx) (3.4) or ˙ e x = (A−LC)e x +dw d (3.5)

The corresponding observability matrix of the given system in (3.1) is

By checking the observability, the Phas full rank, therefore, the system is completely observable.

The next task is finding the observer gainLthat ensures the roots of the following characteristic equation lie in the left half-plane. det(λI−(A−LC)) =0 (3.7)

The computational process is carried out by employing Matlab, the observer gain matrixLis obtained as

The dynamics of the RFB are modeled as a linear system in state-space form, with the objective function being a quadratic functional of the system states and control inputs This model features three control inputs and five desired outputs, classifying it as an under-actuated MIMO system.

The mathematical modeling of the RFB system is represented by a state-space model, with detailed system matrices available in Appendix A For linear state-space MIMO systems, the LQR controller is a recognized design method that ensures robust control performance This approach aims to minimize a quadratic cost function, reflecting the weighted sum of the energy of both the state and the control inputs.

The integral expression \(2\int_0^T (x(t)^T Q x(t) + u(t)^T R u(t)) dt\) represents a cost function where \(Q \geq 0\) and \(R \geq 0\) are symmetric, positive (semi-) definite weighting matrices that assign different weights to states and control inputs In this context, \(x(t)\) denotes the yaw angles, command forces, and their derivatives, while \(u(t)\) signifies the control inputs to the propulsion systems, specifically represented as voltage signals in this study.

Selecting the elements of Q is directly linked to the control effort required for the relevant state, with a greater number of elements leading to increased control effort Additionally, the selection of R is essential for achieving an optimal solution.

Finding the feedback control inputu= −kx(t)that minimizes the

Jfor the system (3.1) means that the optimal control problem of the initial objectives is successfully resolved.

The problem of minimizing the cost function leads to solving the algebraic Riccacti equation

The state feedback control input obtained from the above Riccacti equation is u(t) = −R −1 B ′ Px(t) (3.11)

For practical and effective purpose, the choice ofQandRmust be carefully considered through multi-times of designing and trial before implementing on actual system.QandRare chosen as follows

Based on the servosystem structure shown in Fig 3.1, we define w e =r−y=r−Cx (3.12) whereris the desired output.

As previous section shown, the state estimator provides the estimate state ˆx Therefore, the full state feedback control law can be barely expressed as u= −K f x xˆ (3.13)

Substituting the state feedback law in (3.13) into the system model in

Thus, the (3.5), (3.12), and (3.14) can be written in matrix form as follows

For accurate computation, the linear optimal control gain matri- cesK f , K w and the observer gain matrixL (3.8) are determined by support of Matlab as follows

Simulation Results

In this section, the simulation results of the proposed control system design presented in aforementioned section will be shown and discussed.

Yaw Angle 1 [deg] reference yaw angle estimated angle

Fig 3.3 The yaw angle deviation of floating unit no 1

Control Input [V] zero voltage control input 1 control input 3 control input 5

Fig 3.8 The control input voltage for propulsion systems in ideal condition

To assess the effectiveness of the proposed control system under ideal conditions, a simulation study was conducted using the designed controller The initial yaw angle positions of the five floating units were set at 90 degrees each The goal of the simulation is to successfully transition the entire bridge system from its original position to the desired position of 0 degrees for all units.

Figures 3.3 to 3.7 illustrate the yaw motion of each controlled floating unit, demonstrating that they rapidly reach their target positions within approximately 40 seconds Additionally, the effective performance of the designed observer is validated Figure 3.8 displays the estimated control input voltages for the three propulsion systems.

For further eliminate the effectiveness and robustness of the designed control system, an additional assumption of considering external disturbance effects is investigated.

Fig 3.9 The yaw motion of floating unit no 1 under disturbance

Control Input [V] zero voltage control input 1 control input 3 control input 5

Fig 3.14 The control input for propulsion systems under disturbance

Figures 3.9 to 3.13 illustrate the controller's effectiveness in managing external disturbances while ensuring each floating unit maintains its desired position Furthermore, the state estimator demonstrated satisfactory tracking performance with minimal discrepancies.

In next section, the experimental study is carried out to verify the feasibility and stability of the proposed controller in actual model.

Experimental Results

To assess the effectiveness and reliability of the proposed control system, several experimental studies were conducted, as depicted in Fig 3.15 The experimental setup, detailed in Chapter 2, features five interconnected floating units that simulate a ribbon bridge system Among these, three units are equipped with electrical propulsion systems to generate driving forces, while the other two units remain passive Yaw motions are monitored using five incremental encoders.

Fig 3.15 The experiment setup for RFB installation and position keeping control

The experiment demonstrates the installation and yaw motion control for maintaining the position of the bridge system Initially, the bridge system is aligned parallel to the riverside, and it is then maneuvered to achieve the desired target yaw angle.

The bridge will be positioned over the river, and the subsequent objective is to maintain the stability of the bridge system in the crossing position This stability will be tested against external disturbances, which will be simulated by continuous waves produced by a wave generator.

To demonstrate the effectiveness of the proposed installation strategy and the robustness of the designed control system, experiments were conducted under two distinct conditions: one in calm water and the other in the presence of continuous wave disturbances The following section presents the experimental results, followed by a detailed discussion.

Yaw Angle 1 [deg] reference actual angle estimated angle 24.6 24.64 24.68

Fig 3.16 The yaw motion of floating unit no 1 in calm water

Yaw Angle 2 [deg] reference actual angle estimated angle

Control Input [V] control input 1 control input 3 control input 5

Fig 3.21 The control input for propulsion systems in calm water

The bridge will be maneuvered from its starting position at a yaw angle of 0 degrees to a crossing position at a yaw angle of 90 degrees To ensure the linearity of the entire bridge system, all floating units must remain parallel with an approximate relative angle of 0 degrees throughout the process.

The yaw motion of the bridge system is illustrated in Figures 3.16 to 3.20, demonstrating that each floating unit smoothly approaches the target position with minimal fluctuations during the initial approach Consequently, the bridge maintains its predefined position Additionally, the control input voltages for the three propulsion systems are depicted in Figure 3.21.

To maintain the linearity of the bridge system, the angular displacements among the five floating units should remain near 0 degrees Figures 3.22 to 3.25 effectively demonstrate the performance of the proposed controller in achieving linearity control by illustrating the yaw angle displacement between two continuous floating units.

Fig 3.22 The yaw angle displacement between #1 unit and #2 unit

Figures 3.22 to 3.25 illustrate that the maximum displacement between two floating units is around 0.2 degrees, while during the position-keeping stage, the displacements approach the ideal value of 0 degrees This indicates that the linearity of the bridge system is effectively preserved.

For further estimation of the proposed control system, the control performance of the ribbon floating bridge under continuous wave disturbance is presented.

Fig 3.26 The yaw motion of unit #1 with external disturbance

Control Input [V] control input 1 control input 3 control input 5

Fig 3.31 The control input for propulsion systems with external disturbance

The experiment revealed the yaw motion of individual floating units under continuous wave disturbances, illustrated in Figures 3.26 to 3.30 The control inputs for the propulsion systems are depicted in Figure 3.31 The data indicates that the addition of continuous wave disturbances caused fluctuations in the yaw angles of the floating units, deviating from their target positions However, through appropriate controller adjustments, the bridge system successfully maintained its intended crossing position despite the disturbances created by the wave generator.

When analyzing yaw displacement among floating units, it is crucial to maintain a focus on their interactions Figures 3.32 to 3.35 illustrate that the displacement between adjacent floating units remains notably linear, supported by acceptable recorded data indicating a maximum yaw displacement of 0.5 degrees.

Additionally, in two different cases, the designed observer showed its dramatical tracking ability with the average tracking discrepancy of less than 0.6%.

Summary

This chapter presents a control system utilizing an observer-based optimal strategy for bridge installation and position maintenance A linear state observer was designed to estimate unmeasured states, complemented by a linear quadratic regulator for optimal control Both simulated and experimental results demonstrated the effectiveness of the proposed control system in achieving target positions and maintaining stability under disturbances Additionally, the displacement data confirmed the controller's capability to uphold the linearity of the bridge system across various operational conditions.

Motion Control Performance with Sliding

Introduction

The LQR technique effectively guides the bridge system to its crossing position and maintains stability amid disturbances; however, it faces challenges such as unstable approaches and slow responses to external disturbances To enhance controller performance, a sliding mode control technique is introduced, which excels in positional tracking, minimizes control effort, and significantly reduces noise compared to traditional methods The proposed sliding mode controller's effectiveness is validated through both simulation and experimental results.

Sliding Mode Control of MIMO Underactuated System 59

The integration of state-space models with sliding mode control presents significant potential for designing MIMO systems However, this approach is not suitable for under-actuated multi-body systems, where the number of inputs falls short of the required outputs, making direct inversion solutions impractical in these applications.

Schkoda et al [29] developed a technique to 'square up' the influence input matrix, addressing challenges in sliding mode control equations This section outlines the initial derivation of this method and its application to the ribbon floating bridge system.

The system is given as follow ˙ x=A s x+B s u (4.1) whereA s ∈R 10×10 andB s ∈R 10×3 indicates thatB s is non-square and the system is under-actuated.

The sliding surface can be defined as s=x−x d +γ∫ 0 t (x−x d )dr (4.2) The derivative of the above sliding surface is ˙ s=x˙−x˙ d +γx˜ (4.3)

Let ˙s=0 and substitute the initial equation of model in Eq (4.1), we obtain:

Therefore, the control input can be obtained as u=B s −1 [−A s x+x˙ d −γx˜] (4.6)

To solve the control command, the influence matrix B must be inverted Schokoda [28] previously introduced a coordinate transformation that ensures the resulting matrix is invertible This transformation is defined by the equation y = Tx, where T is a 3x10 matrix designed to make TB invertible.

By differentiating the original state-space model, we obtain the equation ˙y = T˙x, where ˙x = Ax + Bu Substituting this into the model gives us ˙y = T[Asx + Bsu] Consequently, a new sliding surface is defined as s = y - yd + γ∫0t (y - yd) dr Taking the differential form of this equation and applying Leibniz's rule results in ˙s = y˙ - y˙d + γỹ = 0.

Finally, substituting the state space model into new sliding surface yields:

TB s u= −TA s x+y˙ d −γy˜ (4.13) at last, we obtain: u= (TB s ) −1 [−TA s x+y˙ d −γy˜] (4.14)

The challenge lies in determining T, which has dimensions that are the transpose of B's, while ensuring selected states are tracked and control efforts are minimized As discussed in the previous chapter, a common approach to address this optimization problem is to utilize a Linear Quadratic Regulator (LQR) cost function to identify the optimal values for the necessary variables.

The chosen cost function for solving the optimal problem of choos- ingTis:

Q s andR s are selected to weight the desired states and allocate th control effort, respectively Generally, the feedback control signal will be formed as: u= −K s x (4.16)

Substituting Eq (4.14) into Eq (4.16) and solving forT: u= −(TB s ) −1 T[A s +γ˜I]x (4.17)

Ks= (TBs) −1 T[As+γI] (4.18) finally, it can be obtained:

SubstitutingT back into the control law, the final control law is defined as: u= (T ⋆ Bs) −1 T[x˙ d −Asx−γx˜] (4.20)

The obtained controller has complete properties of a sliding mode controller including noise rejection, stability assurance with the additional control allocation and control effort minimization.

Simulation results

To demonstrate the effectiveness of the derived control technique presented in previous section, the design process and simulation are studied.

The weight matrices are chosen as follows:

The computational process are carried out by employing Matlab and the necessary variables and gain matrices are obtained as follows:

To select specific feedback states that cannot be measured directly, it is necessary to incorporate an additional state estimation The observer's design has been detailed in Chapter 3.

The observer gain matrixL s is obtained as

To assess the effectiveness of the proposed sliding mode controller for RFB motion control, numerical simulations were conducted The tracking reference is modeled as a sine function, with the control input representing the command force produced by propulsion systems.

Yaw Angle 1 [deg] reference simulation

Fig 4.1 Yaw angle deviation of floating unit #1

Control Force [N] control force 1 control force 3 control force 5

Fig 4.6 Force command inputs for propulsion systems

The yaw motions of individual floating units, illustrated in Figures 4.1 to 4.5, demonstrate nearly perfect control performance, especially in floating units #1 and #3.

#5 in which active driven by propulsion systems For the floating unit

#2 and #4, the overall tracking performance is good.

The application of identical reference outputs to five floating units equipped with the same amplifier and period ensures that the displacement among them remains nearly zero, thereby preserving the linearity of the bridge system.

Figure 4.6 illustrates the command forces produced by the propulsion systems that drive the bridge system, highlighting the allocation property of the sliding mode controller through the designated control forces for each propulsion system in the three active floating units.

Experimental results

To assess the performance of the modified sliding mode controller, numerous experiments were conducted using the setup described in the previous chapter (refer to Fig 3.15) The implementation of the controller was carried out using LabVIEW 2016.

NI DAQ and embedded controller The yaw motion is obtained by incremental encoder Thanks to efficiency of the state estimator, the essential force command can be obtained.

Similar to the previous experiment execution, there are two cases of real plant being considered: calm water condition and under continuous water disturbance attack condition.

Fig 4.7 The yaw motion of unit #1 with SMC in calm water

Figures 4.7 to 4.11 demonstrate the yaw motions of five floating units under ideal conditions, free from external disturbances With a target yaw angle of 90 degrees, the bridge system transitions smoothly from an initial yaw angle of 0 degrees to the desired position, exhibiting no overshoot or fluctuations These findings highlight the enhanced performance of the sliding mode controller over the linear optimal controller discussed in Chapter 3.

Fig 4.12 The yaw displacement between unit #1 and unit #2 with

Fig 4.16 The control forces generated by propulsion system in calm water condition

Figures 4.12 to 4.14 illustrate the yaw angle displacements along the bridge, with the maximum recorded displacement being approximately 0.1 degrees This finding confirms the exceptional linearity of the bridge system.

The command forces generated by three propulsion systems are shown in Fig 4.16.

To assess the stability and effectiveness of the proposed sliding mode controller, experiments were conducted using a continuous wave generated by a wave generator, as illustrated in Fig ??.

The generated wave exhibits characteristics of a 20 mm amplifier and a 2-second period, continuously impacting the bridge system This ongoing wave action will influence the yaw position of each floating unit and necessitate adjustments in the displacements between the floating units.

Fig 4.17 The yaw motion of unit #1 with SMC under disturbance

To validate the effectiveness of the proposed sliding mode controller under challenging conditions, experiments were conducted involving significant external wave disturbances The yaw motion responses of the floating units are illustrated in Figures 4.17 to 4.21.

The installation phase under disturbance effects proceeded steadily without fluctuations However, at 120 seconds, a sudden and strong wave was introduced, causing the entire bridge system to deviate from its target position Consequently, the controller's effective control actions swiftly realigned the floating unit to the desired position.

Fig 4.22 The yaw displacement between unit #1 and unit #2 under disturbance

Fig 4.22 ∼ Fig 4.25 illustrate the displacements between two continuous floating units Before the extreme wave attack, the displacements among these floating units were significantly kept under

During periods of strong wave activity, displacements reached a peak of 0.8 degrees However, the SMC controller effectively managed this disturbance, reducing displacements to 0.2 degrees The data on yaw displacements indicates that the designed controller successfully mitigates external disturbances, maintaining the linearity of the entire bridge system by minimizing displacements among the floating units.

The control command forces generated by three propulsion systems are shown in Fig 4.26 The control allocation property of the designed controller is illustrated by the obtained data.

Fig 4.26 The force commands generated by propulsion systems under disturbance condition

Summary

This chapter presents a modified sliding mode control approach for MIMO state space systems, leveraging the controller's benefits such as noise rejection, low control effort, and stability The controller, designed in conjunction with a state estimator, effectively manages the installation and operation of the ribbon floating bridge The proposed controller's performance was validated through simulation and experimental studies, particularly under extreme conditions with sudden strong wave disturbances Results demonstrate precise yaw motion control and the ability to maintain displacement in a multi-floating unit bridge system, confirming the robustness and stability of the designed controller amidst strong wave effects.

Conclusions and Future Works

Conclusions

This dissertation presents an innovative automatic installation strategy and control system design for Ribbon Bridge Systems, utilizing multiple floating units that implement yaw motion control and minimize yaw displacement The key findings of this research highlight the effectiveness of the proposed methods in enhancing the stability and performance of floating structures.

Chapter 2 introduces a modern ribbon bridge structure and proposes an automated installation method to replace traditional manual operations Detailed mechanical and electrical designs of the bridge model are presented for modeling and experimentation Additionally, a mathematical model for the bridge system's control design is proposed, and key characteristics of the floating unit, such as the inertia matrix and damping coefficient, are identified through experimental results.

In Chapter 3, an observer-based optimal controller was developed to facilitate an automated installation strategy for a bridge system Due to the inability to directly measure several states, a state estimator was implemented for full-state feedback This led to the design of an optimal controller using a linear quadratic regulator, specifically for managing the yaw motion of the bridge and maintaining position stability amid disturbances The installation of the bridge system was successfully completed with minimal fluctuations, and the system demonstrated the capability to maintain the desired position even under continuous wave disturbances, with displacements between floating units kept below 0.5 degrees to ensure linearity Experimental results indicate that the proposed control system effectively accomplished automated installation and maintained the desired position through yaw motion control, achieving an impressive state tracking accuracy with an average discrepancy of less than 0.8%.

In Chapter 4, a sliding mode controller is implemented for an under-actuated ribbon bridge system, showcasing its advantages such as effective positional tracking, reduced control effort, and noise rejection, which enhance installation and position maintenance A state observer is designed to gather unmeasured data for the controller, resulting in smooth yaw motion that approaches the target position without fluctuations or overshoot The control system's stability is validated against strong external wave disturbances, demonstrating the sliding mode controller's ability to quickly manage disruptions Data on yaw displacement among floating units confirm the bridge system's linearity is preserved Overall, the sliding mode controller outperforms the optimal controller from the previous chapter, indicating that the proposed installation strategy and control design are stable and feasible for real-world application.

Future works

As for this dissertation, the future works will be made as follows:

• Developing the less complicated model for the general bridge system containingn-floating unit bridge for reduction of computational time consuming.

• The combination of active propulsion system and rope system for station keeping.

• Developing the installation and positional keeping using rope tension control.

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1 Van Trong Nguyen, Myong-Soo Choi, and Young-Bok Kim, “A

Study on Automated Ribbon Bridge Installation Strategy,” Jour- nal of Institute of Control, Robotics and Systems, Vol 23 No.

2 Van Trong Nguyen, Yong-Woon Choi, Jung-In Yoon, Kang-Hwan

Choi, and Young-Bok Kim, “Modeling, Identification, and Simu- lation of Positional Displacement Control for Ribbon Bridges,”

MATEC Web of Conferences, Vol 159 No 02026, 2018 SCO-

3 Van Trong Nguyen, Young-Bok Kim, and Sang-Won Ji, “Installa- tion Strategy and Control System Design for Floating Bridges,”International Journal of Engineering and Innovative Technology, Vol 7, issue 12,

1 Van Trong Nguyenand Young-Bok Kim, “A Study on Positional Displacement Control for the Pontoon Ribbon Bridges for River- crossing Purpose,” 2017 17th International Conference on Con- trol, Automation and Systems (ICCAS 2017), Jeju, Korea, pp 93–

2 Van Trong Nguyen, Yong-Woon Choi, Jung-In Yoon, Kwang- Hwan Choi, and Young-Bok Kim, “Modeling, Identification, and Simulation of Positional Displacement Control for Ribbon Bridges” The 2nd International Joint Conference on Advanced Engineering and Technology (IJCAET 2017) and International Symposium on Advanced Mechanical and Power Engineering (ISAMPE 2017), Bali, Indonesia, 2017.

3 Van Trong Nguyen, C Kang, J Jeong, C Son, K Choi, S Jung,

J Yoon, J Yang, and Young-Bok Kim, “A Control Strategy for Multiply Connected Floating Units: Experimental Study,”The 20th The Korea Society for Power System Engineering (KSPSE 2017), pp 11–2, Busan, Korea, 2017.

4 D.H Lee, T W Kim,Van Trong Nguyen, C H Son, J I Yoon,

K H Choi, and Young-Bok Kim, “Installation Strategy and Con- trol System Design for Floating Bridges,”International Congress on Engineering and Information (ICEAI 2018), ISBN: 978-986- 88450-4-6, Hokkaido, Japan, 2018.

5 T W Kim, D.H Lee,Van Trong Nguyen, C H Son, J I Yoon,

K H Choi, and Young-Bok Kim, “A Study on Motion Control of

Tiêu đề	A Study on Automated Ribbon Bridge Installation Strategy and Control System Design
Tác giả	Van Trong Nguyen
Người hướng dẫn	Prof. Young-Bok Kim
Trường học	Pukyong National University
Chuyên ngành	Mechanical System Engineering
Thể loại	thesis
Năm xuất bản	2018
Thành phố	Busan

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Số trang	113
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