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

Introduction

Background and motivation

The growing importance of water crossing is driven by the extensive network of rivers, lakes, and channels worldwide Various efficient methods exist for overcoming these water obstacles, catering to the increasing demand for effective water transportation solutions.

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

The tunnel represents a distinct category of water obstacle crossing, characterized by its underground structure that is excavated through the surrounding soil, earth, or rock.

A ferry is a popular type of water obstacle crossing that operates over short distances, primarily transporting passengers, vehicles, and cargo across bodies of water.

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 This type of bridge offers several advantages over traditional land-based bridges, including the capacity to carry heavy loads like vehicles, quick installation, a simple structure, and minimal environmental impact Its most notable feature is its portability, allowing for easy relocation as needed.

Fig 1.1 The actual ribbon bridge system

The ribbon floating bridge, developed in the 1970s by Germany's former EWK and a U.S company, was designed for full interoperability under a NATO agreement Since its inception, these bridges have been widely utilized by military forces in various countries, including Germany, Canada, and Australia.

The ribbon floating bridge, as illustrated in Fig 1.1, is a successful design primarily composed of lightweight materials such as concrete, steel, aluminum alloys, and composites This innovative structure utilizes buoyancy to support its own weight and vehicular loads, leading to reduced costs and an extended lifespan.

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 methods rely heavily on manual labor, utilizing erection boats and cables to position individual bays on the water Operators then connect these bays using supporting boats, a process that can be time-consuming and challenging, particularly in hazardous environments like rivers or lakes While these bridges are often used for rapid deployment, especially in military contexts, manual installation poses significant risks during combat or dangerous situations To address these challenges, it is essential to develop an advanced installation strategy that incorporates autonomous self-operated systems Various studies have explored the dynamic behavior of ribbon floating bridges, including analyses of displacement and connection forces, highlighting the need for innovative approaches in their design and operation.

Recent studies, including those by [8], have focused on the dynamic response of floating bridges equipped with transverse pontoons While many researchers have analyzed the dynamic behavior and structural integrity of these bridges, it is essential to further investigate their dynamic characteristics This analysis is crucial for developing appropriate mathematical models that facilitate effective control design for floating bridge systems.

Floating bridges share dynamic characteristics with water surface vehicles, making their control akin to that of specialized floating units The control of ribbon floating bridges can be viewed as an advanced application of these principles Numerous strategies have been documented for managing surface vehicles and their dynamic positioning.

Dynamic positioning (DP) technology has evolved since its inception in the 1970s, with significant advancements in control techniques introduced by Balchen, who expanded his work in the 1980s Grimble further contributed to the field by developing a dynamic positioning control system grounded in stochastic optimal control theory Meanwhile, Strand explored nonlinear control theory for surface vessel positioning, while Fossen provided general solutions for vessel control system designs, focusing on active propulsion systems for effective course tracking and station keeping Additionally, Sorensen proposed a model-based control technique to enhance both course tracking and station keeping, and Katebi and colleagues designed a robust controller for DP systems.

Numerous researchers are focused on the dynamics of propulsion systems and their applications in marine system control John provided an overview of marine propellers and propulsion systems, while Iyvind and Damir explored various aspects of marine electrical power systems and propeller control Additionally, Sørensen et al discussed torque and power control for electrically driven marine propellers Timothy's review on trends in ship electric propulsion highlights the growing role of electric technology in marine vehicle control.

The growing popularity of ribbon floating bridges for crossing water bodies is not matched by extensive research on the topic A recent 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, which facilitates the design of control systems specifically 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 outlines various ideas and methods that can be effectively implemented within the floating bridge system to enhance safety and efficiency.

This article focuses on the mathematical modeling of a ribbon floating bridge that utilizes active electrical propulsion systems for its movement The goal is to derive the system dynamics, which are articulated through state equations, to better understand the bridge's operational mechanics.

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

This article discusses the design of a yaw motion controller utilizing the Linear Quadratic Regulator (LQR) control technique integrated with a predefined state observer The effectiveness and robustness of this controller are validated through numerical simulations and experimental testing.

An advanced Sliding Mode Control (SMC) strategy has been proposed to effectively manage a range of disturbances and uncertainties in under-actuated systems The effectiveness of the SMC controller has been validated through various experimental conditions.

Objective and researching method

This dissertation aims to create an advanced control system that utilizes propulsion systems for the automated installation and self-operation of ribbon floating bridges, addressing the challenges and risks associated with manual labor To achieve this, two innovative observer-based controllers have been proposed to enhance the system's efficiency and safety.

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

The control system's primary role is to maintain the linearity of the bridge as it crosses the water barrier, adjusting the yaw displacement of the floating units in response to 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 Essential parameters within the model will be identified through experimental methods and estimated using the Matlab Identification Toolbox.

To develop an effective controller using the LQR technique, a linear observer is utilized to estimate unmeasured states Various simulations are conducted, exploring different desired outputs and initial yaw angles under multiple conditions To validate the proposed controller, an experimental model is established, 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 like current flow and wave attacks The controller's robustness and effectiveness have been 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 functioning passively To measure yaw displacements, five incremental encoders are utilized The control system is based on the National Instruments PXIe-8115 embedded controller, which is paired with the NI PXIe-6363 and PXI 6221 acquisition cards The programming is done using NI Labview 2016, operating at a sample time of 0.03 seconds Experimental results demonstrate the proposed strategy's feasibility, robustness, and overall 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 details the structure of a widely used ribbon floating bridge and describes the model employed for both modeling and experimentation It outlines the scale model of the ribbon floating bridge system, which is essential for control system design in this dissertation Due to the unknown system parameters discussed in the previous chapter, necessary experiments are conducted to identify these critical parameters with the help of computational software.

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

This chapter presents a yaw motion controller that utilizes optimal control and a state estimator, focusing on the necessity of estimating unmeasured state variables since only yaw angles are directly measurable A linear observer is designed based on observability estimation, with its effectiveness confirmed through simulation studies Additionally, a servo-system is introduced for precise control of yaw displacement in floating units and the overall bridge motion Simulation and experimental results demonstrate the proposed approach's effectiveness in controlling yaw motion.

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 guarantees 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 lanes, supported by a lightweight aluminum superstructure that functions as a pontoon This innovative bridge system features interior and ram bays that can be easily assembled and disassembled, allowing for convenient transportation via trucks, helicopters, aircraft, and railway cars when folded Upon reaching the water, the ribbon bay automatically expands to create either a bridge or ferry configuration Traditionally, erection boats or manual labor, aided by cables, are used to position the ribbon segments, after which 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 portability allows for easy relocation, enhancing their versatility However, Chapter 1 highlights several limitations related to the installation and operation of RFBs To address these challenges, an automated solution will be proposed in the following section.

2.1.2 An automated installation and operation strategy for RFBs

The manual operation of RFBs poses significant safety risks and inefficiencies, particularly in hazardous situations or combat scenarios To address these challenges, an autonomous driving system is essential for the installation and operation of RFBs This study introduces a novel strategy for automated control of RFBs through yaw motion control, streamlining the entire process for enhanced safety and efficiency.

The RFB will be assembled on one side of the water body by linking the ram bays and interior bays Once the entire bridge is constructed, it will be maneuvered into position to span the obstacle, connecting both sides During this process, the controller is tasked with ensuring the bridge maintains its linearity by adjusting any displacement among the floating units, while also ensuring a smooth transition to the target location.

In the next phase, the control system must maintain the bridge in its designated position while ensuring its linearity It is essential for the controller to demonstrate stability and rapid adaptability in response to external disturbances such as current flow, wave impact, 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 integrated with propulsion systems, addressing the relevant mechanical and electrical designs.

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, depicted in Fig 2.2, features propulsion systems that actively drive one bay, while the remaining bays are passively driven through a coupling relation Detailed illustrations of both the active and passive bays can be found in Fig 2.3 and Fig 2.4, respectively.

The five floating units share a similar structure and dimensions in length, width, height, and weight Each bay features two adjustable slots on the left and right sides for connector insertion, along with four additional axles equipped with integrated holes for spring placement Constructed from acrylic, a type of polymer material, the active unit includes extra bases for housing electrical control components such as the DC motor, motor drive, and propeller Detailed parameters are available 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.

Connecting five floating units requires the inclusion of connectors, which are illustrated in Fig 2.2 The figure showcases two distinct structures utilized on the left and right sides.

The passive bay side structure includes a connector on the left that aligns with the incremental encoder's axis, while the right side features an additional component designed to protect the encoder from water damage To ensure precise measurements, it is essential that the connectors allow for unrestricted motion.

The control system structure utilized in this study is illustrated in Fig 2.5 It features a closed-loop design for the Redox Flow Battery (RFB), incorporating essential components such as sensors, a controller, and actuators.

Fig 2.5 The configuration diagram of the control system

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

The control configuration for the RFB system is illustrated in Fig 2.5, featuring a National Instrument PXIe-8815 embedded controller housed in a PXIe-1078 chassis, which serves as the main controller for executing control algorithms This setup 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 can be found 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 being controlled 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 are provided 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 motor is utilized to operate the azimuth propeller, generating the necessary thrust to propel the bridge system Detailed specifications and structural information about the DC motor and the Propulsion Schottel Graupner can be found 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 is represented as ˆc_i The stiffness coefficient of the upper translation spring linking the i-th and (i+1)-th floating units is indicated by k_u_i, and the stiffness coefficient for the lower translation spring between these units is k_d_i Additionally, the damping coefficient for the joint connecting the i-th and (i+1)-th floating units is referred to as c_i.

In this study, the external torque acting on the i-th floating unit is denoted as T_i, while θ_i represents the yaw displacement, with counterclockwise rotation considered as the forward direction The yaw velocity is indicated by θ˙_i, and the yaw acceleration is represented as θ¨_i Additionally, x_u_i refers to the upper linear relative displacement between the i-th and (i+1)-th floating units, whereas x_d_i denotes the lower linear relative displacement between the same units The variable w indicates the distance from the left and right sides of the floating unit to the center of the joint, and h represents 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

All joints in the system have equal damping coefficients, represented as \( c_i \equiv c \) 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 floating units are defined as \( \hat{c}_i \equiv \hat{c} \) 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 crucial for control design and experimental analysis As mentioned earlier, the RFB model comprises five floating units, integrating both mechanical and electrical design elements.

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

The equation Jν˙ + Cν = τ describes the dynamics of a floating unit, where τ denotes the control input forces and moments from propulsion systems and spring forces The vector ν, consisting of body-fixed velocities—surge, sway, and yaw rate—is represented as ν = [u, v, r] T ∈ R³ Additionally, J represents the mass/inertia matrix in R³, which is crucial for understanding the unit's motion.

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 is fixed to accurately represent the system's operation Consequently, the control challenge is framed as planar motion control concerning yaw angles.

The yaw motion of each individual floating unit is defined by equations (2.1) to (2.3) Using 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

For effective control design, acquiring the dynamic parameters of the system is crucial Experiments conducted without water flow currents or external disturbances, while assuming a homogeneous mass distribution in the floating unit, reveal that the center of gravity aligns with the center of geometry due to symmetrical geometry in the xz and yz planes Consequently, the model of the floating unit can be represented as (2.4), leading to the determination of the inertia moment and damping matrices.

This dissertation focuses on yaw motion control, identifying essential parameters such as (I z − N r ˙ ) and N r in the yaw direction The mathematical equation for system identification is outlined as follows.

The lightweight nature of the floating unit makes it more effective to utilize a propulsion system for generating thrust By analyzing the correlation between input voltage and output force, along with the connection between input voltage and output yaw motion, we can determine the essential parameters needed for optimal performance This process involves several key steps to achieve the desired outcomes.

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, with the resulting pushing force measured by a load cell (refer to 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, as indicated in equation (2.15), along with the corresponding matching results.

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 a specific voltage, allowing for the estimation of the equivalent moment based on the input voltage The yaw angle position is measured through an incremental sensor attached to the coupling joint, with additional 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 ribbon floating bridge systems used in military and civil applications globally It presents a detailed description of the bridge model system utilized for simulation and control design, focusing on both mechanical and electrical aspects The mathematical modeling of the ribbon bridge, consisting of five floating units, was developed for controller design purposes Furthermore, the characteristic parameters of the floating unit and the transfer function of the propulsion system 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 Redox Flow Battery (RFB) and identified the necessary parameters for controller design However, a significant 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 to address this limitation.

This chapter discusses a yaw motion control design for the RFB system's installation and station keeping, utilizing an observer-based optimal control strategy The optimal control method, 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 exceptional response to disturbances while ensuring system stability To implement full-state feedback control, a state estimator is necessary to gather 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, defined by the equations x˙=Ax+Bu+dw and y=Cx The state matrix A, input matrix B, and output matrix C are specified 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 utilizes an optimal control strategy based on an observer, as illustrated in Fig 3.1 It is crucial to establish the controller and observer gains The following section will detail the design of the state estimator and the LQR controller.

Fig 3.1 The servosystem for positional control of the RFB system

Observer-based Control Design

To achieve optimal feedback performance, it is essential to have access to all state variables for feedback, which is typically represented by the control feedback input \( u = K_f w x \), where \( K_f w \) is the feedback gain matrix However, implementing full-state feedback control presents challenges, as the requirement for all state variables can lead to excessive feedback loops, resulting in economic inefficiencies and impractical solutions Moreover, in this study, several states cannot be directly measured for feedback Therefore, utilizing an observer to estimate and monitor state variables continuously is necessary for effective feedback implementation.

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 and L is the observer gain matrix that must be determined The observer's structure is illustrated in Fig 3.2, highlighting that continuous access to the system's outputs enables the construction of an estimator to accurately estimate 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 represented as a quadratic functional of the system states and control inputs This system comprises three control inputs and five desired outputs, categorizing it as an under-actuated MIMO model.

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

In this study, we analyze the control inputs to propulsion systems represented as voltage signals, focusing on the optimization of yaw angles and command forces The objective function is defined by the integral \(2\int_{0}^{T} \left( x(t)^T Q x(t) + u(t)^T R u(t) \right) dt\), where \(Q\) and \(R\) are symmetric, positive (semi-) definite weighting matrices that assign different weights to states and control inputs, respectively, ensuring a balanced approach to system performance.

Selecting the elements of Q is directly linked to the control effort required for the relevant state, with a greater number of elements resulting in increased control effort Additionally, the choice 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 angles of the five floating units were set at 90 degrees each The primary goal of this simulation is to transition the entire bridge system from its original position to the target position of 0 degrees for all units.

Figures 3.3 to 3.7 illustrate the yaw motion of each controlled floating unit, demonstrating that each unit swiftly reaches its target position within approximately 40 seconds Additionally, the effective performance of the designed observer validates its effectiveness The estimated control input voltages for the three propulsion systems are shown in Figure 3.8.

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

The analysis in Figures 3.9 to 3.13 demonstrates the controller's effectiveness in managing external disturbances while successfully maintaining the desired position of each floating unit Furthermore, the state estimator exhibited commendable 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 validate the performance and robustness of the proposed control system, several experimental studies were conducted The experimental setup, depicted in Fig 3.15, includes five individual floating units connected in series to demonstrate the ribbon bridge system Three of these units are equipped with electrical propulsion systems to generate driving forces, while the other two units remain passive Yaw motions are measured 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 adjusted to achieve the desired target yaw angle.

The bridge will be positioned over the river crossing, and the subsequent task involves ensuring the stability of the bridge system in the crossing position This stability must be maintained despite external disturbances, which are 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 continuous wave disturbance The subsequent section will present the experimental results, followed by a 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 It is essential to maintain all floating units in parallel with an approximate relative angle of 0 degrees to ensure the linearity of the entire bridge system.

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 supplied to the three propulsion systems are depicted in Figure 3.21.

To maintain the linearity of the bridge system, it is crucial to keep the angle displacements among the five floating units near zero Figures 3.22 to 3.25 effectively demonstrate the proposed controller's ability to control linearity by showcasing 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 the two floating units is around 0.2 0, while during the position-keeping stage, the displacements approach the ideal value of 0 0 This indicates that the linearity of the bridge system is effectively maintained.

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 demonstrated the impact of continuous wave disturbances on the yaw motion of individual floating units, as illustrated in Figures 3.26 to 3.30 The propulsion system's control inputs 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, displacing them from their target positions However, with appropriate controller adjustments, the bridge system successfully maintained its intended crossing position despite the disturbances generated by the wave generator.

Attention should be given to the yaw displacement of floating units, as illustrated in Figures 3.32 to 3.35 The data indicates that the displacement between adjacent floating units maintains a notable linearity, with an acceptable maximum yaw displacement recorded at 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 developed 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, showcasing its capability to approach and maintain the target position despite disturbances Furthermore, the displacement data confirmed that the controller successfully preserves the linearity of the bridge system across various operating conditions.

Motion Control Performance with Sliding

Introduction

The LQR technique effectively guides the bridge system to its crossing position and maintains stability under disturbances; however, issues such as unstable approaching and slow response to external disturbances persist To enhance controller performance, a sliding mode control technique is introduced, known for its superior capabilities in positional tracking, reduced control effort, and improved noise rejection compared to traditional methods The proposed sliding mode controller for the under-actuated system is validated through both simulation and experimental results, demonstrating its effectiveness.

Sliding Mode Control of MIMO Underactuated System 59

The integration of state-space models with sliding mode control presents significant potential for MIMO system control design However, in the case of under-actuated multi-body systems, where the number of inputs falls short of the required outputs, the direct inversion solution cannot be effectively applied.

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 the 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, it is essential to invert the influence matrix B 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 3×10 matrix that guarantees the invertibility of TB.

By differentiating the original state-space model, we obtain the equation ˙y = T˙x, where ˙x is represented as Ax + Bu This leads to the expression ˙y = T[Asx + Bsu] Consequently, a new sliding surface is introduced, defined by s = y - yd + γ∫0^t (y - yd) dr Taking the differential form of this equation and applying Leibniz's rule results in ˙s = ˙y - ˙yd + γȳ = 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, while ensuring the tracking of selected states and minimizing control efforts As discussed in the previous chapter, a common approach to address this optimal problem is to utilize a Linear Quadratic Regulator (LQR) cost function to identify the optimal values of 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 effectively select specific feedback states that cannot be directly measured, it is essential to incorporate an additional state estimation The design of the observer necessary for this process has been detailed in Chapter 3.

The observer gain matrixL s is obtained as

Numerical simulations were conducted to validate the effectiveness of the proposed sliding mode controller for the motion control of the RFB The tracking reference is modeled as a sine function, while the control input consists of the command force generated 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 the individual floating units, illustrated in Figures 4.1 to 4.5, demonstrate nearly perfect control performance, especially for 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, all utilizing the same amplifier and period, ensures that the displacement among them is 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 assigned control force for each propulsion system in the three active floating units.

Experimental results

To assess the performance of the modified sliding mode controller, several 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 smoothly reached the desired position from an initial yaw angle of 0 degrees, exhibiting no overshoot or fluctuations These findings highlight the enhanced performance of the sliding mode controller compared to 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 a maximum recorded displacement of approximately 0.1 degrees, confirming 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 amplitude and a 2-second period, continuously impacting the bridge system As a result, the yaw position of each floating unit will be influenced, leading to 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.

During the installation phase, the system remained stable despite disturbances 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 implemented effective control actions that swiftly repositioned the floating unit to the desired location.

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 significant wave activity, displacements reached a peak of 0.8 degrees However, the SMC controller effectively managed the disturbance, reducing displacements to 0.2 degrees The data on yaw displacements indicate that the designed controller successfully mitigates external disturbances, ensuring 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 advantages such as noise rejection, low control effort, and stability The controller, combined with a state estimator, was implemented to manage the installation and operation of the ribbon floating bridge The proposed controller's effectiveness was assessed through simulation and experimental studies, including tests under extreme conditions with sudden strong wave disturbances Experimental results demonstrated precise yaw motion control and maintained displacement within the multiple floating unit bridge system, confirming the robustness and stability of the designed controller amidst challenging 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 to enhance yaw motion control and minimize yaw displacement The key findings are summarized in the conclusion.

Chapter 2 introduces a recent ribbon bridge structure and proposes an automated installation approach to replace traditional manual methods The article details the mechanical and electrical designs of the bridge model used for modeling and experimentation Additionally, it presents a mathematical model for the control design of the bridge system, while experimental results identify key characteristics of the floating unit, including the inertia matrix and damping coefficient.

In Chapter 3, an observer-based optimal controller was developed for an automated installation strategy, utilizing a state estimator for full-state feedback due to the inability to directly measure several states This led to the design of an optimal controller using a linear quadratic regulator to manage the yaw motion of the bridge system and maintain position stability amid disturbances The installation of the bridge system was successfully completed with minimal fluctuations, and the desired position can now be sustained under continuous wave disturbances, with displacements between floating units remaining below 0.5 degrees, ensuring linearity along the bridge Experimental results demonstrate that the proposed control system effectively accomplishes automated installation and maintains the desired position through yaw motion control, with the observer achieving an impressive state tracking capability, showing less than 0.8% average discrepancy.

In Chapter 4, a sliding mode controller is applied to an under-actuated ribbon bridge system, showcasing its advantages such as effective positional tracking, reduced control effort, and enhanced noise rejection, which are crucial for maintaining the bridge's position A state observer is implemented 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 adapt to disturbances Additionally, data on yaw displacement among floating units confirm the bridge system's linearity is preserved Overall, the sliding mode controller outperforms the optimal controller discussed in the previous chapter, indicating that the proposed installation strategy and control design are both stable and feasible for real-world applications.

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	Luận án
Năm xuất bản	2018
Thành phố	Tai

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Số trang	113
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