Estrategias de control robusto para procesos de soldadura po arco eléctrico

Estrategias de control robusto para procesos de soldadura por arco eléctrico Autor: Manuel Masenlle Núñez Directores: J. Xabier Ostolaza Zamora Jorge Elso Torralva 2025 (cc) 2025 Manuel Masenlle Núñez (cc by 4.0) iii EUSKAL HERRIKO UNIBERTSITATEA Abstract Faculty of Engineering, Gipuzkoa Systems Engineering and Control Department Doctor of Philosophy Estrategias de control robusto para procesos de soldadura por arco eléctrico by Manuel Masenlle Núñez iv Welding, a fundamental process in modern manufacturing, joins metal components by melting them and allowing them to solidify, often with a filler material. Its widespread use is evident in the fact that over half of manufactured products are estimated to contain welded joints. However, achieving precise control over the welding process to ensure desired mechanical properties, such as strong, distortion-free joints with minimal residual stress, remains a significant challenge. These properties are heavily dependent on the heating and cooling cycles of the molten zone, making their precise manipulation essential. Despite welding’s importance, the application of sophisticated control techniques has not kept pace with other manufacturing domains. This research aims to bridge this gap by developing models that link controllable parameters (power and travel speed) to temperature, and by designing control systems to maintain the process within specified parameters. This study investigates two distinct welding applications. First, a long-duration welding process, common in large structure fabrication, is examined. A simplified 1D model serves as the basis for a decentralized control system that manipulates power and travel speed to achieve a target cooling curve. Second, the complexities of additive manufacturing are addressed. The layer-by-layer nature of this process and the associated heat accumulation demand a more advanced control strategy. A multivariable controller, designed using sequential Quantitative Feedback Theory (QFT), regulates the surface temperature profile by v adjusting power and travel speed, with particular emphasis on the challenges posed by direction changes during layer creation. A critical aspect of heat transfer modeling in welding is the representation of moving heat sources. Building on the foundational work of Rosenthal, which established the theory of heat flow from a moving source under quasi-stationary conditions, this research aims to derive equations that describe the dynamic relationship between temperature and the control parameters. Specifically, transfer functions are derived that relate temperature variations to changes in both heat source power (for 1D, 2D, and 3D cases) and travel speed (for 1D and 2D cases). These transfer functions are then used to create a 1D welding control model. For the more complex 3D case, a purely analytical approach proves difficult, especially in deriving the transfer function for travel speed. Therefore, a numerical approach is employed, linearizing an FEM model to create a state-space system. While sacrificing some analytical precision, this method offers greater geometric flexibility. This 3D FEM-based model is then used to design a control system aimed at improving performance during direction changes, a critical aspect of additive manufacturing. This model is further enhanced by incorporating a disturbance input that captures the dynamics of direction flips. A modified 2-Degree-of-Freedom (2DOF) MIMO QFT methodology is introduced for disturbance rejection, where specifications vi are defined as deviations from a desired response. The mathematical functions needed to generate QFT bounds for this type of problem are derived, expanding the applicability of the QFT technique. An algorithm for automatically optimizing the coefficients used to balance the tolerance of the bounds is also developed, further enhancing the MIMO QFT toolbox. The enhanced QFT method is applied to the defined benchmark problem, yielding satisfactory results in both the time and frequency domains. Comparison with the uncontrolled process demonstrates significant improvements, particularly in maintaining lower temperatures at the points representing the isothermal zone. This is expected to improve the quality of deposited layers in additive manufacturing by minimizing the need for height corrections or cooling periods between layers. In summary, this work provides valuable contributions to both the modeling and control of welding processes, including the development of novel modeling techniques, the extension of the QFT control methodology, and the demonstration of its effectiveness in challenging welding scenarios. vii Contents Abstract iv 1 Introduction 1 1.1 Automatization of a welding process ........ 5 1.2 Importance of welding in the local industry .... 7 1.3 Challenges ....................... 8 1.4 Objectives ....................... 12 1.5 Structure ........................ 14 2 State of the Art 17 2.1 Moving heat source model .............. 17 2.2 Monitoring of welding ................ 19 2.3 Modelling and control of welding .......... 23 2.4 Summary ........................ 28 3 Modelling of arc welding 31 3.1 Green’s functions ................... 34 3.2 Obtaining transfer functions ............. 38 3.2.1 Transfer function of welding for the 1-D case 39 3.2.2 Transfer function of welding for the 2-D case 41 3.2.3 Transfer function of welding for the 3-D case 42 viii 3.3 Steady-state model validation ............ 43 3.4 Modelling the welding travel speed as the input of the system ...................... 44 3.4.1 Transfer function for the 1-D case ...... 45 3.4.2 Transfer function for the 2-D case ...... 47 3.5 FEM welding model and simulation ........ 48 3.5.1 Simulation Model ............... 51 3.5.2 Linear Model ................. 55 3.5.3 Validation of 1-D analytical model in the frequency domain .............. 58 4 Decentralized MIMO control of the 1-D case 63 4.1 Control properties ................... 64 4.1.1 Outputs selection ............... 64 4.1.2 RGA analysis ................. 66 4.2 Decentralised multivariable control ......... 68 4.2.1 Benchmark problem statement ....... 68 4.2.2 Controller design ............... 69 4.2.3 Results ..................... 71 5 Multivariable QFT control of the 3D case 75 5.1 The flip of direction problem ............ 78 5.1.1 Direction flip modelled as an impulse disturbance .................... 79 5.2 Control approach ................... 81 5.2.1 Benchmark system .............. 82 5.2.2 Analysis .................... 83 ix 5.2.3 Control strategy ................ 83 5.3 The QFT method ................... 84 5.3.1 Disturbance rejection in QFT MIMO . . . . 89 5.3.2 Measured disturbance rejection ....... 94 5.3.3 Measured disturbance rejection with model matching .................... 96 5.3.4 Obtaining an optimal division of tolerances 99 5.4 Design .........................100 5.4.1 Specifications .................101 5.4.2 Controller design ...............104 5.5 Results .........................108 5.5.1 Specification compliance ...........108 5.5.2 Time response .................108 5.5.3 Nonlinear simulation .............110 5.5.4 Temperature profiles .............112 6 Conclusion 115 6.1 Contributions .....................120 6.2 Future work ......................121 7 Resumen 123 7.1 Introducción ......................123 7.2 Estado del arte .....................127 7.3 Modelado .......................131 7.3.1 Obtención de funciones de transferencia . 132 7.3.2 Obtención de modelo numérico ......135 7.4 Control MIMO descentralizado para el caso 1-D . 138 xvii List of Abbreviations AM Additive Manufacturing BJ Binder Jetting DED Directed Energy Deposition DOF Degree OfFreedom FEM Finite Elements Method HAZ Heat Afected Zone MAG Metal Active Gas MIG Metal Inert Gas MIMO Multiple Input Multiple Output PBF Powder Bed Fusion QFT Quantitative Feedback Theory SISO Single Input Single Output TIG Tungsten Inert Gas WAAM Wire Arc Additive Manufacturing xix List of Symbols ACross-sectional area m2 CpSpecific heat capacity J/(KgK) hConvection coefficient W/(m2K) KThermal conductivity J/(msK) PCross-sectional perimeter m QHeat input power W TTemperature ◦C vTravel speed mm/s XDimension in X axis mm YDimension in Y axis mm ZDimension in Z axis mm αThermal diffusivity m2/s ρMaterial density kg/m3 ωangular frequency rad 1 Chapter 1 Introduction Welding is a precise, safe and cost-effective method of joining materials in manufacturing industries. It is estimated that more than fifty percent of manufactured products contain welded joints, this technology being the predominant method of joining metallic materials. To melt the metals to be welded, usually steel, a considerably powerful heat source must be employed. There are three main types of heat sources that can be applied: 1. Gas Welding: It is typically used in manual welding and uses the combination of oxygen and acetylene gases to create a high temperature flame. It can also be used for cutting and shaping. 2. Electric Arc: It is the most used method. It consists in creating an electric arc by applying a voltage between an electrode and the materials. The electrode can be fungible or non-fungible, with the fungible method being the most widespread because it is faster. The non-fungible 2Chapter 1. Introduction method, known as TIG (Tungsten Inert Gas), uses a tungsten electrode and is predominantly used when precision or quality is more important than speed. In both cases, an inert gas is used to stabilise the arc. Depending on the gas used, further distinctions arise between arc methods such as MIG (Metal Inert Gas) and MAG (Metal Active Arc). This technology can be applied manually or through a robotic or automatic method. 3. Laser: Through a laser energy source, higher precision and smoother heat input can be achieved. This method can only be used in automated welding and requires more sophisticated equipment. Apart from the main methods above, there are specific niche technologies that are used in certain cases: 1. Friction Welding: It is normally used to weld cylindrical parts without filler material. It consists of rotating one of the pieces over the other part until the pieces reach the melting temperature through friction. Then the rotating part is stopped and pressure is applied. It can be used to join pieces of different metals. 2. Resistance Welding: This method consists of applying a current through the parts that are in contact where there is an electrical resistance that will produce heat that will melt the area. It can be used to join parts using attachment points. Chapter 1. Introduction 3 In the First World War, welding was used mainly to make repairs, but later, the manufacturing methods of many parts made by casting using moulds, for example tank rollers, were replaced by manufacturing methods using welded steel parts. [1]. The evolution of manufacturing methods continues today, where cast aluminium parts are continually being replaced by welded parts, which are somewhat lighter and much more economical [2]. Additive manufacturing (AM), also known as 3D printing, is the natural path of this evolution trend of manufacturing methods. It consists in the process of creating three-dimensional objects by building them up layer by layer. This is opposite to subtractive manufacturing techniques, like milling, which remove material from a solid block to create the desired shape. Metal 3D printing is an evolving manufacturing process, with new methods continually emerging. The technologies can be classified as Powder Bed Fusion (PBF), Directed Energy Deposition (DED) and Binder Jetting (BJ), mainly. The technology that shares the most with traditional welding is DED. Within DED technology, finer classifications can be made depending on the heat source and how the material is provided. The heat source can be an electric arc or a laser, while the added material can come in the form of powder or wire. Each technology has its advantages and disadvantages in terms of cost of equipment, speed of operations, quality of the results, limitations in the shapes of the parts and limitations in the size of the parts. When the heat source is an electric arc and the filler material 4Chapter 1. Introduction is wire, the advantages are reduced equipment cost, rapid processing, and the ability to produce large parts. Disadvantages, on the other hand, are limitations in part shapes and final quality, which typically require some additional machining. Wire Arc Additive Manufacturing (WAAM) has a surprisingly long history, even though it’s receiving renewed attention recently. The roots of WAAM can be traced back to the 1920s. This is when the core concept of using an electric arc to melt and deposit material layer-by-layer emerged [3]. Despite the longevity of the fundamental idea, WAAM is becoming increasingly attractive today due to its advantages. These include cost-effectiveness for large parts, faster build times compared to other AM techniques, and efficient use of materials [4]. Overall, WAAM represents an interesting bridge between traditional techniques like welding and modern advancements in additive manufacturing. Its long history and ongoing development position it as a potentially valuable tool for various industrial applications. At the same time, parts manufacturers are increasingly seeing AM as a viable alternative to traditional machining. However, while the technology’s potential to minimize material waste has long been acknowledged, defects such as deformation, porosity, and cracking have impeded its broader industrial adoption. Fortunately, existing manufacturing workshop technologies and control systems can repurpose arc welding robots into versatile 3D metal printers. This transformation, enabled by Wire 1.1. Automatization of a welding process 5 Arc Additive Manufacturing (WAAM), offers the benefits of additive manufacturing, including reduced production costs, without requiring significant investments in new machinery. WAAM democratises additive manufacturing, making it accessible to medium-sized welding workshops. 1.1 Automatization of a welding process There are different levels of development in relation to the process of robotization and digitalization of electric arc welding processes. Starting from the traditional situation in which the welder performs the work manually, this development goes through the following stages: 1. The incorporation of robots to the manual task. In addition to the safety and economical reasons, the repeatability and accuracy of the process is improved. 2. The evolution towards the application of sensors and process monitoring. With the gathering and registering of data, knowledge of the process is acquired. 3. The adjustments of the process inputs through feedback control to improve quality and reduce defects. 4. Incorporation of artificial intelligence to make decisions and act. 12 Chapter 1. Introduction to estimate penetration and cooling rate. 4. Maximum temperature: It is the maximum temperature that the material reaches at each point. All in all, welding control aims to achieve consistent, highquality welds by regulating the heating and cooling processes. However, welding presents significant control challenges: nonlinearity, multivariable nature, and the difficulty of measuring key variables. A control system must cope with these challenges to deliver reliable and high-quality results. 1.4 Objectives With the previous challenges in mind, the objective of this work is to develop models that relate controllable parameters to temperature, and to design control systems to maintain the process in the desired state. Two use cases are considered. First, a long welded joint process (case of a ship, bridge, large container, etc.), in which the objective is to adjust the power and advance speed to maintain a desired cooling curve. Using a one-dimensional model, the objective is to design a decentralised multivariable control system. Figure 1.1 shows a large gantry used to weld ship parts that match the use case considered for the one-dimensional model. The second use case focuses on an additive manufacturing process using electric arc welding. This case has been selected 1.4. Objectives 13 FIGURE 1.1: One of the largest laser welding installation in the world, operating at Meyer Werft, Papenburg since 2010, Source: Meyer Werft because it is a longer process and can benefit more than a simple welding bead of a few centimetres. In additive manufacturing, parts are built in layers and heat accumulates, especially when changing the direction of travel of the welding torch. The objective of this case is, using robust multivariable control (more specifically, Quantitative Feedback Theory, or QFT), to adjust the power and forward speed to regulate the geometric temperature profile. Figure 1.2 shows a picture of a WAAM process where the torch needs to perform constant flips of direction to form the part. In summary, the objectives of this work are: 1. Develop control models to aid in designing control systems that improve weld quality in manufacturing. 14 Chapter 1. Introduction FIGURE 1.2: Industrial gas turbine blade built by WAAM technology, Source: WAAM3D Limited 2. Design a multivariable decentralised control system to regulate the cooling curve of a long welding process. 3. Design a robust multivariable control system using an advanced technique to compensate for heat accumulation disturbances in an additive manufacturing welding process. 1.5 Structure Chapter 2 presents an overview of the current advancements and existing research on welding process control. It provides a comprehensive overview of the field, outlining established techniques and highlighting areas for potential improvement. Chapter 3 delves into the relationship between Green’s functions and transfer functions in the context of heat conduction during welding processes. The chapter presents analytical solutions for calculating transfer functions in both one-dimensional 1.5. Structure 15 (1-D) and two-dimensional (2-D) welding configurations. Furthermore, it develops a 3-D control model based on a numerical model derived using the Finite Element Method. Additionally, a non-linear simulation model based on the Finite Element Method to validate control models and control designs is presented. In chapter 4 a decentralised control strategy is designed upon the 1-D model developed in the previous chapter with the purpose of regulating the cooling rate of the welded part. A benchmark problem for the 1-D welding control scenario is presented, followed by the proposition of a practical decentralised control design approach. Chapter 5 aims to design a MIMO controller using the QFT sequential technique. The focus here is on maintaining the desired shape of the Heat Affected Zone (HAZ) during wire arc additive manufacturing. The control strategy is designed to address disturbances introduced by layer transitions, where the welding direction flips between layers. These transitions disrupt the thermal equilibrium of the weld pool. To counteract this, the control strategy utilises feedforward control to expedite the system’s return to a stable state. Chapter 6 summarises the key findings and contributions of the research work. 17 Chapter 2 State of the Art There is an abundance of published research on arc welding. However, most of it is focused on the search for optimum parameters for specific processes, the search for new materials or additives to facilitate the joints or improve their mechanical properties. Many works also seek to improve the final quality in terms of reduction of pores, cracks or projections with parameter adjustments, systems related to the protective atmosphere or additives in the filler materials. The bibliography used in this thesis is focused on the control of thermal aspects in electric arc welding, although in some cases works using lasers are also presented due to their relevance to the modelling of this work. 2.1 Moving heat source model In heat transfer, a critical challenge for engineers, particularly in welding, is modelling moving heat sources. The early 20th 18 Chapter 2. State of the Art century saw the dawn of research on this topic by welding engineers, who employed both empirical and theoretical approaches to understand how heat behaves during welding [12]. Theoretical solutions to these categories can be used to predict temperature distribution and cooling rates within the weld. This knowledge empowers engineers to grasp the impact of heat sources on a weld’s quality and the final product’s performance. The solution to this problem depends on welding parameters, material properties and geometry of the part. Until the mid-1930s, the study of the theory of heat transfer from a moving source was neglected, and temperature distribution due to moving heat sources could only be calculated approximately [13]. But, in 1935, Rosenthal [14] published the theory of heat flow from a moving source to arc welding, where the heat distribution of welding in the quasi-stationary conditions is given by T(x,y,z)=Q 2πKe−λvx e−λv√x2+y2+z2 px2+y2+z2(1) where Tis the temperature, Qis the heat input, vthe speed of welding, Kthe thermal conductivity and 1 2λthe thermal diffusivity. This model assumes that the material properties are constant throughout the work piece and that the heat source is a point source. The assumption that the heat source is a point leads to inaccuracy in the vicinity of the fusion zone, and it is not good for predicting the temperature within the weld pool, but it can be considered accurate enough for temperatures within 2.2. Monitoring of welding 19 about 20% of the melting temperature [15]. As it is considered that the quality of a weld is influenced by the dynamics of the temperature between the ranges below the melt point — from 800 to 1200ºC in the case of steel — due to the cooling rate of the heat affected zones [16], the same assumptions are considered in this work. For the same reason, the dynamics of the state change of the material and radiation are not considered as they have a negligible effect at lower temperatures compared to heat conduction [14]. 2.2 Monitoring of welding The first attempts to apply feedback control in TIG welding took place in the 1980s [17]. From the beginning, it was clear that the key factor for the development of this technology lay in the development of suitable sensors for the process [18,19]. Thus, the type of signal fed back largely determined the control technique to be employed. The two most widely used measurement methods in welding have been the measurement of arc parameters and the use of imaging. In the first case, the non-linear relationship between arc length, arc current and arc voltage is exploited. Since these last two parameters are easily measurable, it is possible to design actuation strategies that maintain a constant relationship between them, achieving a higher quality weld than in the nonfeedback case [20–22]. Other more sophisticated schemes use 20 Chapter 2. State of the Art arc measurements to estimate the weld penetration depth, trying to control it by adaptive strategies [23,24], or even for weld joint tracking [25]. However, the most widespread option in the literature is the use of images captured by cameras. Since this is an environment characterised by the high brightness caused by the arc and the high reflectivity of many of the materials to be welded, a robust imaging system is required. The most widespread option in the literature is the one that uses the laser triangulation principle to extract the profile of the part to be welded, [26–35]. In general, these techniques include algorithms for joint type recognition and joint tracking, determining, among other variables, the amount of material to be deposited. In this regard, several welding product manufacturers offer "welding sensors" (see Figure 2.1) whose triangulation principle provides data of the process. These welding sensors are ready to connect with industrial robots through drivers for the main manufacturing firms and are mainly used to follow the weld seam and to precompute the welding gap. Laser-scanning technology has also been used to control weld penetration based on a multivariable model that relates welding current and arc length to two geometrical parameters that serve to estimate weld penetration [36,37]. In recent years, the high cost of laser technology and improvements in machine vision techniques have led some researchers to opt for direct image acquisition of the welding process as a 2.2. Monitoring of welding 21 FIGURE 2.1: Welding Sensor sensing method [6,38–48]. Evidently, these images are subjected to a filtering process by both physical devices and software. A third group of control methods are based on the use of infrared thermography. Schemes have been implemented that adjust the penetration and width of the weld, as well as its position, based on the temperature profiles detected by this type of camera [49]. Other work focuses on weld depth, but reducing the need for post-processing by using strategically placed point sensors [50]. Infrared technology has also been frequently used in the field of GMAW welding [51,52]. Finally, the use of ultrasonic sensors has been proposed as a low-cost alternative to vision systems. Their use allows weld joint localization even on highly reflective surfaces [53–55]. In any case it must be considered several factors to determine the possibility of being used in industrial applications by part manufacturing companies. The following list gathers some factors that may determine the suitability of their use: 1. Availability of commercial products of the technology. 28 Chapter 2. State of the Art approaches for thermal management of thin-walled structures. They conclude that similar geometries can be achieved using either active or passive cooling techniques, but passive cooling results in longer deposition times. A monovariable PID controller is proposed in [98] to adjust the heat power using the melt pool width instead of using the temperature of the surface. [99] implements also a PID controller to address the resulting geometry, using the height of the deposited layer as input and the wire feed speed of the WAAM system as control output. The issue of layer direction flip in WAAM has been explored in [100] and [101]. These studies propose open-loop variations in speed at the beginning and end of thin parts to mitigate heat accumulation. To date, no research has addressed the control of surface temperature during layer direction flips in WAAM using a multivariable, closed-loop approach based on a robust control technique. 2.4 Summary In the case of monitoring, practically all the work carried out opts for a specific technology, which allows them to deal with a particular problem within the complex welding process. There is still a lot of work to be done in the integration of different types of sensors and the creation of control algorithms to manage all this information in a robust way, generating an optimal 2.4. Summary 29 welding process in all its aspects. Despite its importance as a manufacturing process, the use of control loops to improve welding results is not generalised in industry. There seems to be a need for a framework where control engineers can work comfortably without depending on low-order identified models or black boxes. The lack of control-oriented models is therefore perceived as one of the limiting factors for the generalisation of feedback loops in the welding industry. In particular, transfer function models, which are still the preferred choice for the practising control engineer, are shockingly out of the picture. This work aims to be part of the path to fill that void, presenting transfer function models for 1D and 2D welding processes, and a preliminary control design to serve as a benchmark for the control community. Inspired by Rosenthal’s quasistationary model, dynamical models to determine how the temperature profile changes when the heat input and travel speed changes will be obtained. The transfer functions are derived using Green’s functions, as they provide a very convenient method in this regard. For the 3D case, the approach followed is a numerical model based on the Finite Element Method, where a state-space control model is derived. In this way, by being able to create control loops for industrial systems, this work bridges the gap between pure theoretical models and the practical applications. The model can be applied 30 Chapter 2. State of the Art to various heat treatments or processes in which there is movement like welding where heat is applied through laser, arc or plasma [102]. Also to cutting processes with heat [103], curing materials [104] or controlled cooling of metals [105] for example. 31 Chapter 3 Modelling of arc welding The quality of a weld depends on many factors. Some can be directly controlled in real time, such as the heat power and its positioning and the amount of filler material. Others, however, are factors that cannot be controlled, such as impurities in the materials or geometric imperfections. During the welding process, each spatial point of the materials involved experiences a temperature variation over time. A quality weld requires each point to reach a sufficiently high temperature and a sufficiently slow cooling rate. Therefore, it is considered that regulating the temperature by adjusting the controllable parameters can improve the quality of the welds. This chapter presents models of temperature evolution of a weld, suitable for use in the design of control systems. The weldment can be modelled in one-dimensional, two-dimensional or three-dimensional space, depending on the shape of the part. For the one-dimensional and two-dimensional cases, a purely analytical model is presented. For the three-dimensional case, 32 Chapter 3. Modelling of arc welding the approach followed is a linearisation of a numerical model based on the finite element method. In both cases the starting point is the heat transfer equation. Heat transfer means the energy transport between material bodies due to a temperature difference. There are three modes of heat transfer: conduction, convection and radiation. For heat conduction, the rate equation is known as Fourier’s law, which is expressed as q=−k∇T(2) where qis the local heat flux density (W/m2), kis the thermal conductivity of the material (W/mK) and ∇Tis the temperature gradient (K/m). For convective heat transfer, the rate equation is given by Newton’s law of cooling as q=h(Tw−Ta)(3) where qis the convective heat flux (W/m2), (Tw−Ta)is the temperature difference between the wall and the ambient and his the convection heat transfer coefficient (W/m2K). The flux emitted by radiation from a surface is given by the Stefan-Boltzmann Law q=ϵσT4 w(4) Chapter 3. Modelling of arc welding 33 where qis the radiative heat flux (W/m2), ϵis the radiative property of the surface referred to as the emissivity, σis the StefanBoltzmann constant (5.669x10−8) in W/m2K4and Twis the surface temperature (K). For the purpose of developing a control model of welding of a part, for the 3-D case, only the mode of conduction is taken into consideration as its contribution to heat transfer is much greater than the transfer in the surface. For the 1-D and 2-D models, the convection mode is also considered but not the radiation mode. When considering also an external heat input source, the variation of the temperature in a welding part in the 3-D space, is given by ρCp ∂T ∂t=k∇2T+Q(5) where ρis the density of the material (kg/m3), Cpis the specific heat capacity of the material (J/(KgK)) and Qis the heat input from an external source. In a welding or similar heat input treatment process, the heat input comes from a torch device that moves along some surface of the part. If it is considered a point input heat source at the origin of coordinates, and a movement of speed vof the coordinates, the transient heat transfer equation for this case becomes ρCp ∂T ∂t=k∇2T+δQ−v∇T. (6) 34 Chapter 3. Modelling of arc welding It should be noted that although the temperature T(x,y,z,t) is a function in R4, in practice, the temperature that can be measured is only the surface temperature, and that in a continuous process it would not make sense to measure each point over time. But if continuous movement is considered, and the system is in a quasi-stationary state, a geometric profile of surface temperature takes into account the temporal effect. 3.1 Green’s functions A Green’s function is an integral kernel used to solve various differential equations, ranging from simpler ordinary differential equations with initial or boundary value conditions to more complex inhomogeneous partial differential equations (PDEs) with boundary conditions. They provide a general way to describe the response of a differential-equation solution to an arbitrary source term. They represent the impulse response at point x′of an inhomogeneous linear differential operator (L=L(x)) defined on a domain (x∈X⊂R3), LGx,x′=δ(x−x′), (7) where G(x,x′)is the Green’s function and δthe delta of Dirac’s function. By multiplying the above identity by a function f(x′) 3.1. Green’s functions 35 and integrating with respect to x′yields ZLGx,x′f(x′)dx′=Zδ(x−x′)f(x′)dx′. (8) The right-hand side reduces merely to f(x)due to properties of the delta function, and because Lis a linear operator acting only on xand not on x′, the left-hand side can be rewritten as LZGx,x′f(x′)dx′. (9) This reduction is particularly useful when solving for u=u(x) in differential equations of the form Lu(x) = f(x), (10) because they hold that Lu(x) = LZGx,x′f(x′)dx′, (11) and thus, u(x)has the specific integral form u(x) = ZGx,x′f(x′)dx′. (12) The functions originated in the 1820s with the work of British mathematician George Green. He was particularly interested in solving boundary value problems, specifically for electricity 36 Chapter 3. Modelling of arc welding and magnetism. Its applicability for solving ODEs is very convenient when the function for the specific differential operator and the boundary and initial conditions is known, since through the superposition principle, given a linear ordinary differential equation, one can first solve (7) for each point and then sum the results. There are several methods for finding Green’s functions, including eigenvalue expansions, separation of variables, and Laplace transforms. Due to the research in different fields, given the differential operator and the domain, the specific Green’s function for some problems are known and are gathered in the literature. They have been extensively analysed for the problem of heat conduction [106], and they have also been used for the development of analytical models in welding [58]. For the case of the heat equation (3) in steady state, the distribution of temperature along the space x∈X⊂R3is given by k∇2T(x) = Q(x), (13) in which the linear differential operator Lis the Laplacian, ∇2, and in the absence of boundary conditions the value of its Green’s function is known, Gx,x′=1 4π|x−x′|. (14) 3.1. Green’s functions 37 Using (7), T(x)can be calculated by adding all the impulse contributions of Qas follows. If Qis expressed as Q(x) = Zδ(x−x′)Q(x′)dx′, (15) the distribution of temperature T(x)can be obtained by T(x) = ZV Q(x′) 4πk|x−x′|d3x′. (16) When transient values are considered, Green’s function G(r,t|r′,τ)represent the temperature at the location r, at time t, due to an instantaneous point source of unit strength, located at a point r′in a region R, releasing its energy spontaneously at time t=τ. The main advantage of Green’s functions is that once G(r,t|r′,τ)is known for a specific problem [107], the temperature distribution T(r,t)in the medium is obtained [108] as T(r,t)= Initial Conditions, T0(r) z }| { ZRGr,t|r′,0T0r′dv + Boundary conditions, fi(r,t) z }| { αZt 0dτ N ∑ i=1ZSiGr,t|r′,τ1 ki fir′,τdsi + Heat Inputs, g(r,t) z }| { α kZt 0dτZRGr,t|r′,τgr′,τdv . (17) where α(m2/s) is the thermal diffusivity and is calculated from the thermal conductivity and capacity by the relation α=k ρCp. 44 Chapter 3. Modelling of arc welding be rapidly evaluated by means of Laplace’s final-value theorem. Thus, for the 1-D case, the steady-state value is T0(x)=lim s→0sT (x,s)=αe−1 2αhvx+√x2(4αb+v2)i k√4αb+v2·q0, (32) which fully agrees with the 1-D Rosenthal solution [109]. In a similar manner, for the 2-D and 3-D problems the final value theorem leads to T0(x,y)=e−vx 2α 2πkK01 2αq(x2+y2) (4αb+v2)·q0, (33) and T0(x,y,z)=e−v(x+√x2+y2+z2) 2α 2πkpx2+y2+z2·q0(34) which are in full agreement with the expressions given by Rosenthal [14]. 3.4 Modelling the welding travel speed as the input of the system As stated before, equations (28), (31) and (25) represent the mathematical models of the welding process for the 1-D, 2-D and 3-D respectively, where the obtained temperature profile is the output, the heat source power is the input and the torch travelling speed acts as a parameter. 3.4. Modelling the welding travel speed as the input of the system 45 But, inasmuch as torch speed is an operational parameter that is easily modifiable, it seems reasonable to include this parameter as a direct control action in the welding control model. This can be achieved by means of linearization techniques. 3.4.1 Transfer function for the 1-D case When differential equation (23) is expressed about an equilibrium point, assuming there are small perturbations among parameters and variables —[∆Q(t),∆v(t),∆T(x,t)]— we get ∂∆T ∂t−v0 ∆T ∂x−∆v∂T0 ∂x=α∂2∆T ∂x2+δ(x)∆Q ρCp−b∆T. (35) As T0=T0(x)is known from (32), the differential equation (35) is the same as (23), but considering that inputs terms come from two sources, i.e. g∆(x,t)=δ(x) ρCp ∆Q(t)+∂T0(x) ∂x∆v(t), (36) where the first one is located at the origin and is due to increments in the heat input, and welding torch speed increments act in a distributed manner. As the effect of ∆Q(t)is exactly the same as in eq. (23), the first transfer function is straightforward from eq. (25): ∆Tq(x,s) ∆Q(s)=T(x,s) Q(s)v=v0 =1 ρCp e− v0x+rx2(4αs+4αb+v2 0) 2α q4αs+4αb+v2 0 . (37) 46 Chapter 3. Modelling of arc welding On the other hand, the effect of travelling speed increments on temperature profiles can be obtained following (17), but the Green’s function of the problem (24) is modified [107] to consider distributed heat inputs as follows: Gx,t|x′=e−tb+v2 4α 2√παtexp "−v(x−x′) 2α−(x−x′)2 4αt#. (38) Then, it is possible to write ∆Tv(x,t)=α kZt 0dτZ∞ −∞Gx,t−τ|x′∂T0(x′) ∂x′∆v(τ)dx′, (39) and, applying Laplace transform properties, ∆Tv(x,s)=α kZ∞ −∞Gx,s|x′∂T0(x′) ∂x′dx′∆V(s). (40) Once the space integral is analytically calculated, we obtain the transfer function that describes the effect of changes in torch travelling speeds on the temperature profile: ∆Tv(x,s) ∆V(s)=q0e−v0x 2α 2sk hχ(x,s)−χ(x,0)i, (41) where χ(x,s)= sgn(x)+v0 q4αs+4αb+v2 0 e−rx2(4αs+4αb+v2 0) 2α. (42) 3.4. Modelling the welding travel speed as the input of the system 47 3.4.2 Transfer function for the 2-D case For the 2-D case, the same process is followed, by which the transfer function of the temperature profile with respect to the travel speed is obtained: ∆Tv(x,y,s) ∆V(s)=q0e−vx 2a 4πaks [χ(x,y,s)−χ(x,y,0)], (43) where χ(x,y,s) = vK0 px2+y2√4ab +4as +v2 2a! +x px2+y22|v|− v2 √4ab +4as +v2 K1 px2+y2√4ab +4as +v2 2a!, (44) where K0denotes the Bessel function of second kind and zero order and K1the Bessel function of second kind and first order. For 3-D case, the following integral function must be solved to obtain the transfer function: T(x,y,z) = Z∞ −∞Z∞ −∞Z∞ 0 1 (x2+y2+z2)3 2 e−(z+z′)2 b+e−(−z+z′)2 b v0x2−v0xqx2+y2+z2+v0y2+v0z2−x ev0(−x+x′)−(−x+x′)2+(−y+y′)2 bdz′dy′dx′(45) 48 Chapter 3. Modelling of arc welding but for the time being it has not been achieved due to its complexity. In subsection 3.5.2, a numerical approach is employed to obtain the transfer function of the temperature change with respect to travel speed for the 3-D case. The numerical model will also enable simulation and validation of controllers. 3.5 FEM welding model and simulation Due to the difficulty of solving (45) and obtaining an analytical transfer function as a control model for the 3-D case, a numerical model is used to derive a state space system from a Finite Element Model. The Finite Element Method (FEM) follows a general structure, but there are variations within that framework. The variational approaches are the most popular variation, being the Galerkin method the most widely used. The Galerkin method aims to convert a continuous PDE into a system of algebraic equations that can be solved numerically on a computer. It achieves this by: 1. Dividing the domain: The physical domain of the problem (e.g., welding part) is first discretized into smaller subdomains called elements. These elements can be triangles, squares, tetrahedrons, or other shapes depending on the problem’s dimensionality. 3.5. FEM welding model and simulation 49 2. Choosing basis functions: Simple functions with specific properties (often polynomials) are chosen within each element. These functions are called basis functions and are used to approximate the unknown solution of the PDE within that element. 3. Projecting the residual: The governing PDE is reformulated into a weak form, which often involves integrating the equation over the entire domain. The Galerkin method then enforces this weak form by requiring the residual (the difference between the left and right sides of the equation) to be orthogonal (zero average) to a set of weighting functions. 4. Solving the system: By applying the orthogonality condition over each element and using the chosen basis functions, a system of algebraic equations is obtained. This system relates the unknown coefficients of the basis functions, essentially representing the approximate solution at specific points (nodes) within the elements. The system under study is the three-dimensional version of the system presented in the previous section. It consists in a dynamic model that simulates processes in which there is a moving heat input. The inputs to the system are the power of the heat source and the travel speed. Its outputs are the temperature at two given points on the surface of the piece. 50 Chapter 3. Modelling of arc welding To develop the control system, a model of the welding process (the plant) is created. For this purpose, the system is asumed to be in steady state, where the base metal has infinite length, the energy input and velocity are constant, and the temperature profile over the entire part with respect to the point of the heat source is in equilibrium. Considering the geometric coordinates relative to the heat source, this allows working with a bounded coordinate system even if the torch is in continuous motion. In this coordinate system, under steady state conditions, temperatures are constant in space, whose state is said to be quasistationary. Figure 3.4 shows this condition. Considering Rosenthal equations [14], the control challenge of obtaining a good temperature profile could be translated to the temperature control of two representative points: P1behind the heat input, and P2on its side; both belonging to the same isotherm. Once again, inputs are the torch speed vand the input power Q. The next section develops in detail a FEM simulation model that will allow for the testing and verification of the control systems to be developed. 3.5. FEM welding model and simulation 51 FIGURE 3.4: Control inputs and positioning of points for temperature measurement. 3.5.1 Simulation Model Recall that the system can be modelled with the heat equation, given by ∂T(x,y,z,t) ∂t−k ρCp∇2T(x,y,z,t) +v(t)∂T(x,y,z,t) ∂x=δ(x,y,z)Q(t) ρCp , (46) where T(x,y,z,t)is the temperature at any point in the space of the domain, tis the time, v(t)is the travel speed, kis the thermal conductivity constant of the material, ρis the mass density of the material, Cpis the specific heat capacity and Q(t) the heat input. Convection and radiation are not considered because heat exchanges (losses) through the surface to surrounding atmosphere can be neglected in regard to the heat flow in the piece itself. This assumption is experimentally supported and is explained by the fact that the heat conductivity of metals is much greater than their heat transmission through the surface [14]. 52 Chapter 3. Modelling of arc welding The method of the Finite Elements uses the weak or variational formulation to establish the numerical approximation in the domain of some space discretization. To convert a partial derivative equation into a variational problem the equation is multiplied by a function wand integrate the resulting equation over the domain Ω, ZΩ ∂T(x,y,z,t) ∂twdΩ−k ρCpZΩ∇2T(x,y,z,t)wdΩ +v(t)ZΩ ∂T(x,y,z,t) ∂xwdΩ=ZΩ δ(x,y,z)Q(t) ρCp wdΩ, (47) where the function wwhich multiplies the PDE is called a test function and the unknown function Tto be approximated is referred to as the trial function. Then integration by parts of terms with second-order derivatives is then performed, ZΩ∇2TwdΩ=−ZΩ(∇T·∇w)dΩ+Z∂Ω ∂T ∂nwds (48) where ∂T ∂n=∇T·nis the derivative of Tin the outward normal direction non the boundary. In the FEM method, the test function is made to be zero in the boundary of the problem resulting in a variational form for the equation (46), of ρCpZΩ δT δtwdΩ+kZΩ(∇T·∇w)dΩ +ρCpZΩ(v(t)·∇T)wdΩ=δ(xq,yq,zq)Q(t). (49) where wis the test function and Ωthe domain of the problem. 3.5. FEM welding model and simulation 53 Using one of the several FEM methods (see for example the Galerkin method in [110]) a system of equations is built that can be expressed in matrix form as L˙ T(t) + K(v(t))T(t) = f Q(t), (50) where Land K(v(t)) are known as the elasticity and stiffness matrices respectively, and fas the load vector. Obtaining the values of the matrices in (50) gives the dynamic FEM model of the heat equation for the domain. To perform simulations and to derive the problem matrices, the FEniCS software [111] is used. The FEniCS software facilitates the assembly of matrices of differential equations expressed in the weak formulation. The specific parameters used in the spatial discretization are shown in Table 3.1. To improve the precision of the numerical model, an irregular mesh (A.2) is used where greater granularity is applied in the vicinity of the heat source point, where greater temperature gradients are expected. TABLE 3.1: Discretization parameters. Parameter Value Description X120 Dimension of the X axis Y40 Dimension of the Y axis Z10 Dimension of the Z axis N1484 Number of points M5899 Number of cells DOF 1 Degree of freedom FS Lagrange Function space 60 Chapter 3. Modelling of arc welding FIGURE 3.7: FEM Simulation for dynamic validation and the simulation is run from 0 to 1000 seconds with dt =0.01 seconds. Once temperature data are obtained, the Fast Fourier transform of the inputs and outputs are obtained as U(ω)and Y(ω) and from those the transfer function is calculated as H(ω) = U(ω)Y(ω) U(ω)2. (58) Then, gains and phases are computed to represent and compare with the analytical functions in the Bode plot diagram. This way, Figure 3.8 represents the comparison between the obtained simulation results and the frequency response ∆Tq(xi,jω)/∆Q(jω)for the same set of points {xi}with very good agreement among them. The model has also been validated considering variable frequency torch travelling speeds and, as Figure 3.9 shows, obtained behaviour matches that of ∆Tv(xi,jω)/∆V(jω)with great accuracy. 3.5. FEM welding model and simulation 61 a) b) xi=−0.04 xi=−0.03 xi=−0.02 xi=−0.01 Magnitude (dB) Frequency (rad/s) Phase (deg) 10-3 10-2 10-1 100101 -400 -300 -200 -100 0 10-3 10-2 10-1 100101 -2000 -1500 -1000 -500 0 FIGURE 3.8: Comparison of frequency responses of FEM and ∆Tq(xi,jω) ∆Q(jω): a) Magnitude, b) Phase a) b) xi=−0.04 xi=−0.03 xi=−0.02 xi=−0.01 Magnitude (dB) Frequency (rad/s) Phase (deg) 10-3 10-2 10-1 100101 60 70 80 90 100 110 10-3 10-2 10-1 100101 -135 -90 -45 0 45 90 135 180 FIGURE 3.9: Comparison of frequency responses of FEM and ∆Tv(xi,jω) ∆V(jω): a) Magnitude, b) Phase 63 Chapter 4 Decentralized MIMO control of the 1-D case Now that we have models for the welding process we can face control problems. In this chapter, a control system is designed for the use case of a long welded joint process (a ship, bridge, large container, etc.) in which, using a one-dimensional model, the power and advance speed are adjusted to maintain a desired cooling curve, using a decentralised control. The model predicts the temperature distribution within a material based on its power input and travel speed. In the onedimensional case, temperatures along the material’s x-axis, and in the two and three-dimensional cases, the temperatures of some superficial points can be measured using a non-contact sensor like an infrared pyrometer. The following section discusses the specific temperature measurement points required for implementing a decentralised multivariable control strategy for the 1-D case. 64 Chapter 4. Decentralized MIMO control of the 1-D case 4.1 Control properties 4.1.1 Outputs selection A key objective of thermal systems is to control the cooling curve of the material [115–117]. In this work, this is achieved by governing the temperature of two points along the slab —see x1,x2 on Figure 3.6—. The choice of two points has to do with the fact that there are two manipulated variables: Heat source power and torch travelling speed. This way, a 2 ×2 multivariable system is configured. However, its properties depend very much upon the particular two points whose temperature is measured. Based on classical multivariable control tools, this section defines a procedure to make such choice. To begin with, solid lines in Figures 3.8 and 3.9 show the Bode plots defined by the transfer functions (37) and (41) at points xi=−0.01, −0.02, −0.03, −0.04m. Similarly, Figure 4.1 shows the responses of the system, obtained with FEniCS, when each of the two control inputs suffers a step deviation from its equilibrium value. a) b) xi=−0.04 xi=−0.03 xi=−0.02 xi=−0.01 Time (s) ∆Tq(xi,t) Time (s) ∆Tv(xi,t) 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 -2.4 -1.8 -1.2 -0.6 0 0.6 1.2 FIGURE 4.1: Normalised step responses at points xi=−0.01, −0.02, −0.03, −0.04: a) Heat source power, b) Torch travelling speeds 4.1. Control properties 65 From a control point of view, Figure 3.8 shows that low frequency gain between power and temperature is almost the same for all xipoints. However, as the distance between the source and the sensor increases, the Bode plot of the irrational transfer function reduces its bandwidth and accumulates phase lag, which justify the slower response observed in the time domain —Figure 4.1a—. To minimise this harmful delay, the first point should be placed in the closest position to the source that still provides a good reading of the temperature. In the example, this point is assumed to be x1=−0.01 m. The plots defining the relation between torch travelling speeds and temperature —Figure 3.9— are more complex. Speeding up the advance reduces the final temperature on points near the source, but increases it as the distance grows. In other words, the gain goes from negative values to positive ones. At the same time, step responses —Figure 4.1b— show that the plant is clearly non-minimum phase on points near the source. With all this in mind, it is hard to decide where to place the second sensor: too close to the first one implies negative gain and undershoot, and too far from it ensures limited bandwidth and a slow response. To get out of this conundrum, and also to define the best input-output pairing, a Relative Gain Array (RGA) analysis of the plant is presented in the next section. 66 Chapter 4. Decentralized MIMO control of the 1-D case 4.1.2 RGA analysis In multivariable process control systems, RGA helps determine the best pairings between manipulated variables and measured variables [118]. It considers how much of a change in one input variable affects a particular output variable, compared to how much it affects other outputs. Selecting optimal pairings through RGA analysis leads to improved control system performance and stability and reduced coupling effects. To calculate it, first the steady-state transfer function matrix of the MIMO system is obtained, which represents the gain ratio between each input and output. Next, it is multiplied elementwise by the inverse of the transpose of itself. A good pairing is achieved when the obtained matrix resembles the identity matrix by rearranging the order of the files or columns. In what follows, pairing rules suggested by [119] are followed: Pairing rule 1. Prefer pairings such that the rearranged system, with the selected pairings along the diagonal, has an RGA matrix close to identity at frequencies around the closed-loop bandwidth. Pairing rule 2. Avoid (if possible) pairing on negative steady-state RGA elements. Therefore, the first task is to determine the desired closedloop bandwidth. Looking at the Bode plots in Figure 3.9, this 4.1. Control properties 67 bandwidth is estimated to be 0.4 rad/s, as beyond that point, too much phase lag is accumulated in the power-to-temperature transfer function. Next, Figure 4.2 shows the magnitude of frequency dependent RGA elements for different choices of the second measuring point, in particular x2=−0.02, −0.03, −0.04, −0.05m. It can be seen that the option providing the best values at the bandwidth frequency, represented in Figure 4.2 with a dashed vertical line, is x2=−0.04 m. In this case, the RGA matrix becomes Λ(j0.4)=  0.7784 +0.1708j0.2216 −0.1708j 0.2216 −0.1708j0.7784 +0.1708j  , (59) which suggests that the temperature at x1=−0.01 mshould be paired with the power input, whereas the temperature at x2= −0.04 mshould be paired with the velocity input. To check the second pairing rule, the steady state RGA is calculated for x2=−0.04 m, and its values are found to be non-negative. It is interesting to notice how the static RGA gets closer to the identity as x2moves away. However, the price of this coupling reduction are large delays in the open-loop response, which in turn implies a very limited closed-loop bandwidth. After selecting the points in the material where the temperature is going to be read —see Figure 3.6—, the system to control 68 Chapter 4. Decentralized MIMO control of the 1-D case a) b) x2=−0.05 x2=−0.04 x2=−0.03 x2=−0.02 Relative Gain Frequency (rad/s) Relative Gain 10-3 10-2 10-1 100101 0 0.5 1 1.5 2 10-3 10-2 10-1 100101 0 0.2 0.4 0.6 0.8 1 FIGURE 4.2: RGA plots for points x2=− 0.02, −0.03, −0.04, −0.05, when x1=−0.01: a) Diagonal Relative Gain, b) Off-diagonal Relative Gain is defined by the transfer matrix G(s) =   g11(s)g12(s) g21(s)g22(s)  =   ∆Tq(x1,s) ∆Q(s) ∆Tv(x1,s) ∆V(s) ∆Tq(x2,s) ∆Q(s) ∆Tv(x2,s) ∆V(s)   , (60) where x1=−0.01 mand x2=−0.04 m. 4.2 Decentralised multivariable control 4.2.1 Benchmark problem statement As explained before, the objective is to reach and maintain certain temperatures in two points of the moving piece, therefore 4.2. Decentralised multivariable control 69 controlling the cooling curve and ensuring good mechanical properties in the material. More specifically, the proposed benchmark consists in a tracking control problem: beginning at the equilibrium conditions defined by Table 3.2, i.e. T(x1) = 1397 and T(x2) = 389, manipulate the input power and speed in order to reach a new cooling curve in which T(x1) = 1600 and T(x2) = 500. 4.2.2 Controller design This chapter focuses on testing the model rather than on controller synthesis. Therefore, the simplest approach for multivariable control design is adopted: independent decentralised control. As a consequence, the system is governed by a diagonal controller whose elements are tuned taking into account only the diagonal elements of the transfer matrix (60). The objective is to obtain a working control scheme whose response can be compared with others provided by more sophisticated techniques. In spite of the fact that the transfer functions (37) and (41) are irrational, their frequency response can be used to adjust a controller just like for rational functions. This is a great advantage of the frequency domain approach. In this case, the loop shaping in the Bode plot seeks to maximise the system bandwidth while keeping a phase margin of 40◦in both loops. The non-minimum phase nature of the plant makes these goals easily attainable for 76 Chapter 5. Multivariable QFT control of the 3D case the welding pool by adequately manipulating the power and speed of the heat source. A benchmark system is established to evaluate the effectiveness of the proposed control system. The results demonstrate significant improvement in temperature control, leading to enhanced layer construction quality and reduced need for height corrections or cooling pauses. Optimising the path for new layers in additive manufacturing (AM) is crucial for several reasons. It can: 1. Improve quality: A well-planned path minimises defects like warping, cracking, and poor surface finish. 2. Reduce time: By minimising travel distances and optimising tool movements, printing time can be significantly reduced. 3. Save material: Optimised paths can minimise wasted material and improve material utilisation. Path planning strategies are key aspects of optimising the path for new layers in AM. Specific software tools are used in order to translate a 3D model into layer information and generate toolpaths based on user-defined parameters and chosen path planning strategies. Among the parameters, thermal considerations are important: Heat distribution is critical in AM. Optimising the path helps manage heat buildup to prevent warping and ensure consistent material properties. But this can impact the time needed to create the part because back-and-forth Chapter 5. Multivariable QFT control of the 3D case 77 FIGURE 5.1: Illustration showing a part created by layers following a path in a single direction and with back-and-forth movements. movements are avoided. If the heat buildup can be controlled by other means and direction flips are allowed, the time to create the parts is reduced. Figure 5.1 illustrates an additively manufactured part with 3 layers. In one case, the layers are deposited in the same forward direction (arrow with the solid line) where the tools are repositioned without depositing following the path of the arrow with the dashed line. In the second case, the first layer is deposited in the forward direction, the second is deposited in the backward direction and the third again in the forward direction. A flip direction occurs each time the tool reaches the end of the layer. The second approach is obviously much faster than the first, but if other measures are not taken into account, it can create heat build-up problems at the edges. 78 Chapter 5. Multivariable QFT control of the 3D case 5.1 The flip of direction problem Control in additive manufacturing of metals is more challenging than one pass welding, as temperature profiles vary over time, due to repeated heat input from the torch. However, compensating for this constant heat build-up is not a difficult challenge in controller design as the dynamics are very slow. On the other hand, compensating for specific irregularities in the base metal, such as small holes or cracks through which the torch passes, is extremely difficult, since sensors only detect such situations too late for any corrected action. In this way, if a situation that may cause a sudden disturbance in the system is known in advance, feedforward control can be applied, as in the case of a direction flip manoeuvre. In this type of problem, where there is a sudden change of the state of the system, pulling it out of its equilibrium, the use of feedforward is of great help, and the QFT technique is known and stands out for its use. A case that occurs quite frequently is the contribution of several cords or when a wall is being built by such additive manufacturing. When the travel direction changes, material and energy begin to be added to the newly heated area, so if control is not applied, the temperature rises rapidly producing anisotropy [120]. Unlike welding systems under steady state conditions, this case does present a challenge for control design. It is also a case that, although specific, occurs relatively frequently in the 5.1. The flip of direction problem 79 paths followed by additive manufacturing systems [121]. When direction flip occurs, the points are moved to their symmetric positions as in the case where a robot tool reverses with the sensors attached. This produces a sudden disturbance in the system, as the state of the points ahead of the torch is replaced by that of the points behind it. The primary goal of the control system is to restore the temperature at the designated points to their predefined values by manipulating the travel speed vand power Q. This stable temperature profile is crucial for maintaining the desired mechanical properties throughout the process, as shown in [11]. This chapter addresses precisely the regulation of the HAZ after the direction flip of the movement in the fabrication of a piece using additive manufacturing. To this aim, control is used to return as quickly as possible to the previous equilibrium state. 5.1.1 Direction flip modelled as an impulse disturbance As mentioned in 3.5.2, the linear state space model obtained from the nonlinear Finite Element model will be used for the design. However, this model needs to be enhanced to adequately represent the flip of direction manoeuvre. In the linearized model, this motion is equivalent to a sudden change of state of the system, in which temperatures are instantly replaced by that of the points symmetrically located with respect to the x axis. 80 Chapter 5. Multivariable QFT control of the 3D case 0 5 10 15 0 500 1000 1500 2000 0 5 10 15 1000 1500 2000 2500 3000 FIGURE 5.2: Open loop time response to the impulse disturbance in the linear and nonlinear models. Considering that in state space initial conditions can be represented by impulse inputs reaching directly each state with the corresponding value, a change of state can be modelled by a difference between the new and the previous state. This allows the flip of direction to be modelled by an impulse input whose amplitude is the difference between the equilibrium in the forward direction (+) and the equilibrium in the backward direction (−). The resulting state space system is therefore ˙ x=Ax+[BQBvx0−−x0+]∆Q(t) ∆v(t) δ(t). (66) Notice that the input matrix Bis decomposed into its constituent vectors BQ,Bv, and now includes a third column x0−−x0+that reflects the direction flip. 5.2. Control approach 81 To the best of the author’s knowledge, this method of deriving a linear model directly from the FEM method, as well as the representation of a sudden manoeuvre as an impulse disturbance, has not been used or documented before. Once the linear control model has been obtained, we can proceed to design a control system that attenuates the effects of the direction flip impulse disturbance. However, the fact that such disturbance is known in advance allows the introduction of feedforward compensation. The next section presents the followed approach in the design of the controller. The design is undertaken in section 5.4. 5.2 Control approach The system derived in section 3.5.2 is of the same order as the number of points of the FEM model. To proceed with the control design this system is reduced to the minimum order model keeping the same frequency response. Figure 5.2 shows the time response comparison between the linear system obtained with the explained procedure and reduced to the 40th order, and the nonlinear FEM model. A MATLAB script that performs this reduction using the matrices obtained before is shown in Appendix A.3. 82 Chapter 5. Multivariable QFT control of the 3D case 5.2.1 Benchmark system To test different control strategies, a benchmark system is defined using the developed model. The piece used for the benchmark is the one described in subsection 3.5. The system works with a nominal power of Q=7500 Watts and a travel speed of v=5 mm/s. The measurement points have been placed in the 1200 degree isotherm when the system is in a stationary situation [11]. Table 5.1 shows the parameters used in the proposed control problem. It must be taken into account that the dimensions of the piece have been chosen in such a way that the model is equivalent to a semi-infinite space, that is, X∈(−∞,∞), Y∈(−∞,∞)and Z∈[0, ∞), to avoid edge effects. TABLE 5.1: Parameters of the piece used in the benchmark. Symbol Unit Value Description Xmm 120 Dimension in X axis Ymm 40 Dimension in Y axis Zmm 10 Dimension in Z axis ρkg/m37870 Material density CpJ/(Kg K) 2719 Specific heat capacity KJ/(m s K) 60 Thermal conductivity hW/(m2K)0 Convection coefficient Q0W 7500 Steady state heat input v0mm/s (-5,0,0) Steady state travel speed Pimm (0,0,0) Heat application point P1mm (14.53,0,0) Measurement point 1 P2mm (0,2.35,0) Measurement point 2 T1re f ◦C 1200 P1temperature reference T2re f ◦C 1200 P2temperature reference 5.2. Control approach 83 5.2.2 Analysis Following [11], the measurement points selected are on the right of the heat input and behind it, in the points crossing the same isotherm. Several isotherms have been considered, trying to strike a balance between responsiveness and the measurement difficulty near the heat source. A higher isotherm makes the system response faster but presents the problem of having to measure the temperature very near the heat source, where infrared sensors are affected by radiation. Looking for this compromise, the 1200 degree isotherm was selected. Taking into account the selected benchmark parameters, the first measured temperature T1belongs to the point in the movement axis P1, and temperature T2corresponds to a point P2in the perpendicular axis, as indicated in Table 5.1. For the input-output pairing, an RGA analysis has been performed, yielding similar results as in the 1D case [122] i.e. to recommend pairing T1with Qand T2with v. In the 3D case as in the 1D case, the point whose temperature is used to control the speed input is the one closest to the heat source. 5.2.3 Control strategy To address the disturbance rejection problem, a control structure with two degrees of freedom is employed, by incorporating a prefilter at the input of the known disturbance. Figure 5.3 84 Chapter 5. Multivariable QFT control of the 3D case − E(s)G(s)ΔQ(s) Δv(s)P(s) M(s)F(s) δ(s) D(s) ΔT1(s) ΔT2(s) FIGURE 5.3: Control structure. depicts the controller architecture, where the elements to be designed are the 2 ×2 diagonal matrix G(s)and the 2 ×1 vector F(s).P(s)and D(s)represent, respectively, the modelled plant and disturbance dynamics. Finally, δ(s)serves as the input to this control configuration, generating an impulse when a disturbance occurs and remaining zero otherwise. The outputs are the two temperatures that the controller aims to maintain at the equilibrium values. M(s)is the vector of transfer functions that encapsulates the desired response of the system to an impulsive disturbance. Employing a model for matching the disturbance response demands an enhancement in existing MIMO QFT techniques. This development is elaborated on in Subsection 5.3.3. The desired system dynamics M(s) are specified in subsection 5.4.1. 5.3 The QFT method The selected controller design method is Quantitative Feedback Theory (QFT) [123]. The rationale behind the QFT technique is 5.3. The QFT method 85 to minimise the use of feedback, shifting the responsibility to the feedforward controller. The fundamental principle is that feedback should be limited to compensating for system uncertainties, including unknown disturbances. It is a robust control design method that focuses on ensuring system performance over a specified range of plant uncertainties. The technique requires the definition of a desired result and a maximum allowable tolerance, known as a specification. Then a set of plants are defined for which this specification must be met. A discrete set of relevant frequencies is selected to translate the specifications into frequency-domain constraints, which are represented as QFT bounds in the Nichols chart. The controller is designed by manipulating its parameters to shape the nominal open-loop response in the Nichols chart, avoiding the QFT bounds at all design frequencies. Controllers and filters are designed to meet the specifications in this frequency list, but if the frequencies are wisely chosen, fulfilment of the specifications can be expected in the whole spectrum and by the whole set of plants. The final design using this method usually involves several iterations of the mentioned steps and can be defined through the following algorithm that the designer must follow: 1. Model the uncertain plant and the nominal plant P(s),P0(s) 2. Until a satisfactory design is reached (a) Define/tune specifications 92 Chapter 5. Multivariable QFT control of the 3D case  ˆ p21d1+ˆ p22d2 ˆ p22 +g22 ≤x2b2, (79)  ˆ p21 ˆ p22 +g22 b1≤(1−x2)b2, (80) where x1∈(0,1)and x2∈(0, 1)are design parameters that must be adjusted to find the point that favours less bandwidth in g11 and g22. Unlike the sequential method, using the non-sequential procedure, the loops can be designed in any order, as the designed controller is not applied in the design of subsequent loops. Nevertheless, an iterative redesign of loops generally takes place, as reducing the tolerances b1and b2, when the design allows, can benefit the performance of the other loops. This can permit further reduction of tolerances in other loops, and so on. In order to use the sequential method, first, both sides of (68) are multiplyed on the left by the matrix h1 0 −ˆ p21 ˆ p11+g11 1i, (81) giving hˆ p11+g11 ˆ p12 0ˆ p2 22+g22 iy1 y2=hˆ p11 ˆ p12 ˆ p2 21 ˆ p2 22 ihd1 d2i(82) where ˆ p2 21 =ˆ p21 −ˆ p21 ˆ p11 ˆ p11 +g11 =ˆ p21g11 ˆ p11 +g11 (83) and ˆ p2 22 =ˆ p22 −ˆ p21 ˆ p12 ˆ p11 +g11 . (84) 5.3. The QFT method 93 From the previous equations, y1and y2can be expressed, respectively, as y1=ˆ p11d1+ˆ p12d2−ˆ p12y2 ˆ p11 +g11 , (85) y2=ˆ p2 21d1+ˆ p2 22d2 ˆ p2 22 +g22 . (86) Due to the Cauchy-Schwarz inequality we know that y1and y2 are upper bounded by y1≤|ˆ p11d1+ˆ p12d2|+|ˆ p12|y2 |ˆ p11 +g11|, (87) y2≤ ˆ p2 21d1+ˆ p2 22d2 ˆ p2 22 +g22 . (88) Taking advantage of this, we can design the MIMO 1DOF diagonal controller as two SISO controllers. First, we define upper bound specifications, b1and b2, for each loop. Then, we can use these specification to calculate g11 and g22, respectively, ensuring they comply with the following inequalities: |ˆ p11d1+ˆ p12d2| |ˆ p11 +g11|≤x1b1, (89) |ˆ p12| |ˆ p11 +g11|b2≤(1−x1)b1, (90) and  ˆ p2 21d1+ˆ p2 22d2 ˆ p2 22 +g22 ≤b2, (91) where a division of tolerances through parameter x1is applied in (87), in the same manner as in the non-sequential method. 94 Chapter 5. Multivariable QFT control of the 3D case u(s) F(s) − G(s)P(s) y(s) − e(s)G(s)P(s) w(s) F(s)D(s) y(s) ——– ——- ——— − e(s)G(s)u(s)P(s) d(s) y(s) 1 FIGURE 5.5: Block representation of a disturbance rejection 1DOF MIMO problem. u(s) F(s) − G(s)P(s) y(s) − e(s)G(s)P(s) w(s) F(s)D(s) y(s) ——– ——- ——— − e(s)G(s)u(s)P(s) d(s) y(s) 1 FIGURE 5.6: Block representation of a measured disturbance rejection 2DOF MIMO problem. From these equations the decoupling QFT bounds are created and the design process can proceed with the method as described in the previous subsection. 5.3.2 Measured disturbance rejection When the disturbance can be measured, the solution can be greatly improved using an 2DOF structure with a pre-filter F(s)as illustrated in Figure 5.6. This method was developed in [128]. The transfer matrix from the disturbance to the error Tew is given by Tew =[1+PG]−1[−D−PF], (92) 5.3. The QFT method 95 that can be rearranged as hP−1+GiTew =−P−1D−F, (93) and using again the notation P−1= [ ˆ pij], the element-wise expression becomes tew ij =1 ˆ pii +gii "− n ∑ k=1 ˆ pikdkj −fij −∑ k=1 ˆ piktew kj #. (94) Due to overbounding, the disturbance rejection specification tew ij ≤bij is guaranteed if a solution gij,fij is found to the SISO-equivalent problems −fij/ˆ pii −∑n k=1ˆ pikdkj/ˆ pii 1+gii/ˆ pii ≤bij(1−xij) and ∑k=1|ˆ pik/ˆ pii|bkj |1+gii/ˆ pii|≤bijxij. with xij ∈(0,1)being used to divide the tolerance bij between both expressions. This division is usually done manually for each design frequency so that the bound requirement after the distribution has been applied is balanced. In this work, a method has been created to automate the choice of this distribution, which is described in Subsection 5.3.4. 96 Chapter 5. Multivariable QFT control of the 3D case The sequential method is developed applying Gauss elimination in (5.3.2) with gives rise to the element-wise expression tew ij =1 ˆ pi ii +gii "− n ∑ k=1 ˆ pikdkj +∑ k<1 ˆ pk ik fk kj ˆ pk kk +gkk −fij −∑ k>1 ˆ pi iktew kj #, where ˆ pr ij and fr ij incorporate the previously designed elements of Gand F, according to the recursive formulas ˆ pr+1 ij =ˆ pr ij −ˆ pr ir ˆ pr rj ˆ pr rr +grr , (95) fr+1 ij =fr ij −ˆ pr ir fr rj ˆ pr rr +grr , (96) where r≥1, ˆ p1 ij =ˆ pij and f1 ij =fij. With this arrangement for the sequential method, the SISO-equivalent problems are −fi ij/ˆ pi ii −∑n k=1ˆ pi ikdkj/ˆ pi ii +∑k<1ˆ pk ik fk kj/(ˆ pk kk +gkk) 1+gii/ˆ pi ii ≤bij(1−xij) and ∑k>1|ˆ pi ik/ˆ pi ii|bkj |1+gii/ˆ pi ii|≤bijxij. 5.3.3 Measured disturbance rejection with model matching To address the direction flip problem, we employ a two-degreeof-freedom control structure (as described in Subsection 5.2.3). This structure includes a prefilter at the input of the known disturbance to improve disturbance rejection. Figure 5.3 illustrates the controller architecture, where the design parameters are the 5.3. The QFT method 97 2×2 diagonal matrix G(s)and the 2 ×1 vector F(s). The vector of transfer functions M(s)encapsulates the desired system response to an impulsive disturbance, i.e., the specification. This is a refinement of the problem that was not present in the original methodology. To match the disturbance response using a model, existing MIMO QFT techniques are enhanced. To obtain the functions defining the bounds for the current problem, a mathematical derivation has been carried out based on the work of [128]. In the cited work, the QFT methodology is extended to the problems of model matching and disturbance rejection for n×nmultivariable plants. Following the same procedure, a derivation for disturbance rejection with an ideal response to match will be shown. According to Figure 5.3, the closed loop response between the disturbance and the error Teffi is given by [1+PG]Teffi =[M−PF −D], (97) and multiplying by ˆ P=P−1, we get ˆ P+GTeffi =ˆ PM −F−ˆ PD. (98) Considering a 2 ×2 case, if this expression is premultiplied by the matrix Mg=1 0 −b p1 21 b p1 11+g11 1, (99) it is possible to perform Gaussian elimination and get the upper diagonal matrix characteristic of the sequential method, whose 98 Chapter 5. Multivariable QFT control of the 3D case element-wise expression for a 2 ×2 system is teδ 11 =1 b p1 11 +g11 b p1 11m11 +b p1 12m21 −f1 11 −b p1 11d11 −b p1 12d21 −b p1 12teδ 21 and teδ 21 =1 b p2 22 +g22 b p2 21m11 +b p2 22m21 −f2 21 −b p2 21d11 −b p2 22d21, (100) where b p2 21 =b p1 21 −b p1 21 b p1 11 b p1 11 +g11 , b p2 22 =b p1 22 −b p1 21 b p1 12 b p1 11 +g11 , and f2 21 =f1 21 −b p1 21 f1 11 b p1 11 +g11 . (101) Thanks to overbounding, the disturbance rejection specification teδ ij ≤bij is guaranteed if a solution gij,fij is found to the SISO-equivalent problems  1 1+g11 ˆ p1 11 m11 +ˆ p1 12 ˆ p1 11 m21 −1 ˆ p1 11 f11 −d11 −ˆ p1 12 ˆ p1 11 d21 ≤b11 (1−x11), (102) 5.3. The QFT method 99  1 1+g11 ˆ p1 11 ˆ p1 12 ˆ p1 11 b21≤b11x11, (103) and  1 1+g22 ˆ p2 22 ˆ p2 21 ˆ p2 22 m11 +m21 1 ˆ p2 22 f21 −1 ˆ p2 22 f2 11 −ˆ p2 21 ˆ p2 22 d11 −d21 ≤b21,(104) where x11 expresses the division of tolerances between the performance equation (102) and the coupling equation (103) in the first loop. The performance equations (102) and (104) correspond to 2 DOF SISO problems for which a nonconservative solution and a bound-building procedure is provided in [128]. Once again, the coefficient x11 balances the requirements of decoupling and performance in the first loop, and its value is adjusted to make the bounds arising from each condition lay together. Thanks to the sequential method, the division of tolerances is unnecessary in the second loop. 5.3.4 Obtaining an optimal division of tolerances To determine the optimal vector of coefficients x11, an optimization algorithm is employed. The optimum value for the coefficients is achieved when the bounds of performance and coupling are as close as possible. Previously to this work, optimization of this parameter was performed manually, requiring the engineer to visually inspect plots of the bounds and adjust the 100 Chapter 5. Multivariable QFT control of the 3D case coefficient by hand until the bounds aligned for each design frequency. This manual process was time-consuming and inefficient, especially when iterating through multiple specifications and design modifications. To address this, the current work employs an optimization algorithm implemented in MATLAB. The implementation utilises a cost function that calculates the difference between the bounds and employs fminbnd to minimise this difference. The algorithm employed is as follows: 1. For ωi=ω1..ωn (a) f=@(x)cost_f unction(calculate_bounds(ωi,...)) (b) x11(i) = f minbnd(f,0,1) The cost function (cost_f unction) computes the difference between the bounds, being zero when the bounds coincide perfectly. Figure 5.7 shows the bounds obtained by the algorithm for the optimal x11 that resulted in the minimal value of the cost function. Without the algorithm, the designer has to try different x11 values and decide when the bounds are most alike in the Nichols plot. 5.4 Design Before the controller synthesis, the linearized plant is scaled to avoid numerical calculation problems. The operation is performed following Skogestad [119] recommendations with error and input scaling matrices (Deand Du), resulting in the scaled 5.4. Design 101 -260 -240 -220 -200 -180 -160 -140 -120 -100 -8 -6 -4 -2 0 2 4 FIGURE 5.7: Bounds b11 and b21 of frequency 0.5 rad/s at the optimal x11 value. plant P=De−1¯ PDu. (105) where Pis the scaled plant, Deis the error scaling matrix, Duis the input scaling matrix and ¯ Pis the original plat. 5.4.1 Specifications Disturbance rejection control typically aims to return to the equilibrium state existing before the disturbance. In this case, the disturbance implies a sudden change in the state of the system due to the instantaneous repositioning of the measurement points. The ideal output would be an immediate return to previous values, but it is not reasonable nor practical to demand an arbitrarily quick return to the equilibrium. Instead, a trajectory 108 Chapter 5. Multivariable QFT control of the 3D case -360 -315 -270 -225 -180 -135 -90 -45 0 -80 -60 -40 -20 0 20 40 FIGURE 5.12: Loop shaping of f21. 5.5 Results 5.5.1 Specification compliance Figure 5.13 presents the magnitude of the Bode diagram representing the error between the uncertain plant disturbance response and the desired response. The error remains in all cases below the specified tolerance, indicated in red. This verifies that the QFT methodology, which employs a discrete number of frequencies, also complies with the specified tolerance along the whole frequency spectrum. 5.5.2 Time response Figure 5.15 displays the time response of the system. The plot shows the controlled response of the linear system plants used in the design (in blue) and the response of the simulation using 5.5. Results 109 10-2 10-1 100101102 -80 -40 0 40 80 10-2 10-1 100101102 -80 -40 0 40 80 FIGURE 5.13: Magnitude of the Bode diagram of the error of plants with respect to the specification. the nonlinear finite elements model (in green) along with the limits of the desired responses established in the specification (in red). The improvement with respect to the open loop system (Figure 5.2) is noticeable for all the plants because the return to the stationary state is produced more than 10 seconds earlier. The nonlinear response simulated with the FEM model is also compliant with the specification defined in subsection 5.4.1. The control actions are checked to ensure that their values fall within acceptable bounds. Figure 5.16 shows the power (∆Q) and speed (∆v) changes required by the system to achieve the results shown in Figure 5.15. The designed control actions for the plants are shown in blue, while the results of the control actions obtained from the simulation using the nonlinear FEM model are presented in green. 110 Chapter 5. Multivariable QFT control of the 3D case In both cases, the nonlinear simulation falls within the results of the different plants used in the design, demonstrating the achievement of robust performance. 5.5.3 Nonlinear simulation In order to check the validity of the designed control, a closed loop nonlinear simulation is performed. The simulation loop follows the algorithm depicted in subsection 3.5.1, where the speed and power are updated in each iteration. To do that, the filters and controllers of the control structure M,Fand G(see Figure 5.3) are discretized using Matlab’s function c2dwith a time period of 0.02 seconds. Equation (112) shows the Z-transform of the g11 filter, and Figure 5.14 shows a Bode diagram of the continuous and discrete filter versions. Gd11(z) = 0.0001818z2+0.0003636z+0.0001818 z2−1.818z+0.8182 . (112) The coefficients of the resulted Z-transformed functions are used to implement IIR filters in the simulation loop and feed the values needed to perform the next iteration. The inputs and outputs are stored in each iteration and saved as a Matlab matrix at the end of the simulation to be visualised in the plot diagrams shown (Figures 5.15 and 5.16). 5.5. Results 111 -100 -80 -60 -40 -20 0 Magnitude (dB) 10-1 100101102 -180 -135 -90 Phase (deg) Frequency (rad/s) FIGURE 5.14: Bode diagrams of the continuous and discrete versions of G11. 0 5 10 15 20 25 30 -1500 -1000 -500 0 500 0 5 10 15 20 25 30 0 500 1000 1500 2000 FIGURE 5.15: Time response of the linear plants and the nonlinear nominal plant. 112 Chapter 5. Multivariable QFT control of the 3D case 0 5 10 15 20 25 30 -4000 -3000 -2000 -1000 0 0 5 10 15 20 25 30 -1 0 1 2 3 FIGURE 5.16: Control actions of the linear design and nonlinear simulation. 5.5.4 Temperature profiles As stated in the introduction, the metallographic characteristics of the resulting parts are largely determined by the heatingcooling profiles of the molten zone [131–133]. In this subsection, the temperature profiles obtained with the developed system are considered. To evaluate the improvements obtained with the control system, the surface temperature of the part in the case of a 180 degree direction change in the absence of any control system is compared against the designed solution. Figure 5.17 shows the area between the temperatures of points P1and P2for two simulations. The solid blue line represents the temperature over time after the direction flip occurs in the absence of any control. The 5.5. Results 113 dashed red line represents the temperature in the case of a simulation where the designed control system is active. The area of the difference of temperatures for the both cases is represented in yellow. It can be seen that in the case of the system under control the temperatures remain closer to the reference value of 1200 ◦C. That means that the heat affected zone shape returns faster to this nominal value defined by the isotherms (see Figure 3.4). The stabilisation time of the system under control is 2 seconds whereas the uncontrolled system needs 12 seconds to return to equilibrium. This improvement is further reinforced by a substantial reduction in the squared error of temperatures calculated by SE =Z20 0(T(t)−Tre f )2dt. (113) The first point exhibits a 5-fold reduction, while the second point achieves a 4-fold improvement. These results are promising, but experimental tests are necessary to quantify the actual quality benefits. 114 Chapter 5. Multivariable QFT control of the 3D case 0 2 4 6 8 10 12 14 16 18 20 1000 1500 2000 2500 3000 0 2 4 6 8 10 12 14 16 18 20 1000 1500 2000 2500 3000 FIGURE 5.17: Difference in temperature profiles with and without control. 115 Chapter 6 Conclusion Welding is a technique used to permanently join pieces of metal. By applying heat, these pieces are melted and then allowed to solidify, forming a strong bond. In some instances, a filler material is introduced to create a bead of metal that connects the pieces. Records indicate that the first experiment related to what we now know as arc welding dates back to 1880. A century and a half after its invention, the significance of welding processes in modern manufacturing is hard to overstate. Welding is a precise, safe, and cost-effective method for joining materials in manufacturing industries. It’s estimated that over fifty percent of manufactured products contain welded joints, making this technology the predominant method for joining metallic materials. Welding processes must adapt to diverse operational conditions such as the materials to be joined, thickness ranges, joint 116 Chapter 6. Conclusion types, etc., to meet increasingly stringent performance and dimensional requirements. As a result, proper weld control remains an unsolved technological challenge. The purpose of weld control is to obtain a weld with adequate mechanical properties in any scenario. Strong joints with low residual stresses and free of distortions are sought. These metallurgical characteristics are largely determined by the heating and cooling profiles of the molten zone, so it makes sense to try to control these profiles. Despite its importance as a manufacturing process, the use of control loops to improve metal joining results has not been the subject of as much research by the control community as other areas. There seems to be a need for a framework in which control engineers can work more comfortably. This work aimed to develop models that relate controllable parameters to temperature, as well as control systems to maintain the process in a desired state. To achieve this goal, two applications were considered. First, a long-duration welding process, typically found in large structures, is analyzed. A simple decentralized control system adjusts power and travel speed based on a 1D model to maintain a desired cooling curve. Second, an additive manufacturing process is investigated. Due to the layered nature of additive manufacturing and heat accumulation, a multivariable controller designed using the sequential QFT technique is employed to regulate the surface temperature profile by adjusting power and travel speed. Special consideration is given to the changes in Chapter 6. Conclusion 117 direction that the torch has to make to start new layers. In heat transfer, a critical challenge for engineers, particularly in welding, is to model moving heat sources. In the early 20th century, welding engineers began to investigate this topic, employing both empirical and theoretical approaches to understand how heat behaves during welding and Rosenthal published the theory of heat flow from a moving source to arc welding, where the heat distribution of the weld under quasi-stationary conditions was expressed in an equation where power an travel speed are parameters. In order to design controllers, we need not only the steadystate expressions but also equations that indicate how temperature values change with changes in the power and speed parameters. One of the objectives of this thesis has been to find these equations. Transfer functions that indicate how temperature changes at any point of the part when there are changes in the power of the heat source have been presented for the 1-D, 2-D, and 3-D cases. In the case of changes in temperature when the travel speed changes, the transfer functions are presented for the 1-D and 2-D cases. The transfer functions were applied to create a 1-D welding control 2 ×2 model. The model included heat power and torch traveling speed as control inputs, leading to a multivariable system. The output variables being the temperatures of two positions along the piece. Their locations were determined using RGA analysis. A simple decentralized PI control was proposed 124 Chapter 7. Resumen historia sorprendentemente larga, aunque recientemente está ganando más atención. Las raíces de la WAAM se remontan a la década de 1920, cuando surgió la idea de utilizar un arco eléctrico para fundir y depositar material capa por capa. Si bien la idea existe desde hace tiempo, la WAAM se está volviendo cada vez más atractiva hoy en día debido a sus ventajas, que incluyen la rentabilidad para piezas grandes, tiempos de construcción más rápidos en comparación con otras técnicas de fabricación aditiva y un uso eficiente de los materiales. En general, la WAAM representa un puente interesante entre las técnicas tradicionales, como la soldadura, y los avances modernos en la fabricación aditiva. Su larga historia y su desarrollo continuo la posicionan como una herramienta potencialmente valiosa para diversas aplicaciones industriales. La fabricación aditiva de metales se está convirtiendo en una alternativa competitiva a la fabricación por mecanizado tradicional. Aunque el potencial para ahorrar material se ha reconocido desde los inicios, la aparición de defectos de deformación, porosidad y agrietamiento ha obstaculizado su adopción generalizada en la industria. Afortunadamente, la combinación de las tecnologías de soldadura actuales y los sistemas de control pueden transformar los robots de soldadura por arco en impresoras 3D para piezas metálicas. Con ello se obtienen las ventajas de la fabricación aditiva, sin la necesidad de grandes inversiones. 7.1. Introducción 125 La automatización de los procesos de soldadura en la industria local es dispar y depende en gran medida del sector industrial en el que se centre el análisis. Así, sectores como el naval siguen contando hoy en día con una elevada presencia de soldadores que realizan trabajos manuales muy variados, en situaciones complejas desde el punto de vista de la seguridad, la higiene y la ergonomía laboral. En esta línea tradicional, no sólo se identifican sectores, sino también operaciones, destacando el marcado carácter manual del proceso de reparación de componentes, siendo en la mayoría de los casos procesos de reparación manuales. En el otro extremo del desarrollo hacia la robotización y digitalización del proceso de soldadura al arco se encontrarían industrias con un claro enfoque tecnológico, como el sector de la automoción o el aeronáutico. Los procesos de soldadura deben adaptarse a diversas condiciones operativas como los materiales a unir, rangos de espesores, tipos de unión, etc. para cumplir con requisitos de rendimiento y dimensionales cada vez más estrictos, por lo que el control adecuado de la soldadura sigue siendo un reto tecnológico sin resolver. El propósito del control de la soldadura es obtener una soldadura con propiedades mecánicas adecuadas en cualquier escenario. Se buscan uniones fuertes, con pocas tensiones residuales y libres de distorsiones. Estas características metalográficas están determinadas en gran medida por los perfiles de calentamiento-enfriamiento de la zona fundida, por lo que tiene sentido tratar de controlar estos perfiles. 126 Chapter 7. Resumen Entre las variables de soldadura destacan el movimiento del robot y en consecuencia el movimiento de la antorcha de soldadura (ubicación de la fuente de energía), los parámetros de soldadura (parámetros eléctricos para el caso de soldadura por arco) o los defectos ocasionados en las uniones soldadas. Estos parámetros dinámicos, como la intensidad de corriente, el aporte térmico o la velocidad de avance de la soldadura, entre otros, están interrelacionados influyendo en la calidad final de la soldadura y haciendo de ésta un proceso complejo. En consecuencia, con el objetivo de maximizar la calidad de las uniones, mejorar la estabilidad del proceso, reducir costes y aumentar la productividad, se han impulsado procesos de automatización del proceso de soldadura y sistemas de monitorización y control en lazo cerrado en tiempo real. A pesar de su importancia como proceso de fabricación, el uso de lazos de control en la soldadura no ha sido objeto de tantos trabajos de investigación por parte de la comunidad de control como otros ámbitos. Parece existir la necesidad de un marco en el que los ingenieros de control puedan trabajar más cómodamente. Este trabajo tiene como objetivo desarrollar modelos que relacionen parámetros controlables con la temperatura, así como sistemas de control para mantener el proceso en el estado deseado. Con ese objetivo se consideran dos casos de uso. En primer lugar, un proceso de unión soldada de larga duración (caso de un barco, puente, contenedor de grandes dimensiones, etc.) en 7.2. Estado del arte 127 el que, utilizando un modelo unidimensional, se ajusta la potencia y la velocidad de avance para mantener una curva de enfriamiento deseada, utilizando un control simple descentralizado. El segundo caso de uso se centra en un proceso de fabricación aditiva. Se ha seleccionado este caso porque es un proceso más largo y se puede beneficiar más que un simple cordón de soldadura de unos pocos centímetros. En la fabricación aditiva, las piezas se construyen en capas y el calor se acumula, especialmente al cambiar la dirección de desplazamiento de la antorcha de soldadura. En ese caso, utilizando la técnica secuencial QFT para diseñar un controlador multivariable, se ajusta la potencia y la velocidad de avance para regular el perfil de temperatura de la superficie de la pieza. 7.2 Estado del arte La mayoría de los trabajos de investigación publicados sobre soldadura se centran en la búsqueda de parámetros óptimos para procesos concretos y nuevos materiales para facilitar las uniones o mejorar sus propiedades mecánicas. Muchos trabajos también buscan la reducción de porosidad, grietas o defectos mediante ajuste de parámetros, aditivos en el gas de protección o aditivos en los materiales de aportación. En las últimas décadas se ha hecho un gran esfuerzo para comprender mejor las técnicas de soldadura, lo que ha llevado 128 Chapter 7. Resumen al desarrollo de modelos matemáticos para el proceso de soldadura, que se dividen en dos enfoques principales. El primero consiste en modelos analíticos que resuelven la ecuación de calor bajo supuestos simplificados sobre formas de entrada de calor, pérdidas térmicas y configuraciones geométricas. A pesar de su simplicidad, estos modelos proporcionan una comprensión más general de los procesos de soldadura. La contribución seminal en este campo la da Rosenthal [14], quien resuelve el caso de estado estacionario cuando el metal base se mueve a velocidad constante, las propiedades térmicas permanecen constantes y la fuente de calor es una fuente puntual. Posteriormente se han intentado superar estas limitaciones. La fusión del metal base ha sido introducida por [56], se han considerado modelos de fuente de calor distribuida, incluyendo distribuciones gaussianas [57,58], gaussianas doble-elipsoidales [59] y pseudo-gaussianas [11]. Incluso se ha estimado el tamaño del baño de soldadura y la penetración [60,61]. El modelo analítico se resuelve utilizando las funciones de Green en [62]. Por otro lado, los métodos numéricos, como FEM o FVM [63], permiten realizar simulaciones complejas de soldadura, prestando atención a los detalles específicos de cada aplicación, esto es, modelo de fuente de calor, geometría y fijación de la pieza, materiales y sus transformaciones de fase, etc. Evidentemente, esta flexibilidad tiene un coste, ya que cada aspecto de la aplicación debe considerarse en el modelo [64,65], lo que a menudo implica resolver problemas de transmisión inversa de calor [66], 7.2. Estado del arte 129 o incluso ensayo y error [67], para obtener resultados de simulación precisos. Lindgren en su libro "Computational Welding Mechanics" [68] con el mismo título que su mentor ([64]) desarrolla aún más el trabajo haciendo hincapié en los métodos FEM que incluían dinámica de fluidos (Figura 2.3). Además, la discretización espacial simplifica la resolución, pero la convergencia a la solución exacta solo está garantizada para modelos de orden superior, y se han propuesto nuevos métodos adaptativos [69] para reducir los costes computacionales. En [78] se utiliza el método de migración de isotermas para modelar la dinámica del baño de fusión. Este método consiste en modificar la ecuación de calor de tal manera que la variable independiente sea el espacio en lugar de la temperatura. Utilizando la discretización y el método de diferencias finitas, se obtiene un modelo en espacio de estados del sistema cuasiestacionario dada la fuente de calor puntual y la parte móvil. Utilizando el modelo desarrollado, en [79] se diseña un controlador PI que ajusta la potencia para regular la posición de la isoterma de cambio de fase en una posición específica. Con este enfoque, es posible ajustar el tamaño de la zona afectada por el calor por encima de la temperatura específica, pero no la forma de la zona. Más recientemente, se han desarrollado técnicas de control para procesos de fabricación aditiva basados en arco eléctrico. Algunos trabajos se han centrado en el desarrollo de modelos de los procesos físicos [80–82]. Entre los trabajos que utilizan el 130 Chapter 7. Resumen control, la mayoría se centran en la parte geométrica de la capa resultante [83–86]. En algunas otras fuentes [87,88], el control de temperatura se realiza mediante enfriamiento forzado, pero no actúan sobre la potencia de la fuente. También han surgido técnicas de control que emplean inteligencia artificial [73–77], que ofrecen soluciones adaptativas e independientes del modelo para el control de la soldadura. Estudios recientes se han centrado en mitigar el problema de los cambios de dirección en la construcción de WAAM de paredes delgadas. En [93], los autores investigan la viabilidad de utilizar mediciones de temperatura entre capas para monitorizar y controlar las propiedades geométricas y metalúrgicas de paredes delgadas durante WAAM. Si bien esta investigación proporciona información valiosa, no propone una estrategia de control específica. Varios estudios, entre ellos [94–96], se han centrado en determinar los parámetros WAAM óptimos para fabricar piezas específicas. En [97], los autores comparan la eficacia de dos enfoques de enfriamiento para la gestión térmica de estructuras de paredes delgadas. Concluyen que se pueden lograr geometrías similares utilizando técnicas de enfriamiento activo o pasivo, pero el enfriamiento pasivo da como resultado tiempos de deposición más prolongados. En [98] se propone un controlador PID monovariable para ajustar la potencia térmica utilizando el ancho del baño de fusión 7.3. Modelado 131 en lugar de utilizar la temperatura de la superficie. [99] también implementa un controlador PID para abordar la geometría resultante, utilizando la altura de la capa depositada como entrada y la velocidad de alimentación del hilo del sistema WAAM como salida de control. La cuestión de la inversión de la dirección de las capas en WAAM se ha explorado en [100]y[101]. Estos estudios proponen variaciones de lazo abierto en la velocidad al principio y al final de las piezas delgadas para mitigar la acumulación de calor. Hasta la fecha, ningún trabajo de investigación ha abordado el control de la temperatura de la superficie durante las inversiones de la dirección de las capas en WAAM utilizando un enfoque multivariable de lazo cerrado basado en una técnica de control robusta. 7.3 Modelado La calidad de una soldadura depende de muchos factores. Algunos se pueden controlar directamente en tiempo real, tal como la cantidad de material de aportación, la potencia calorífica y su punto de aplicación. Otros, en cambio, son factores que no se pueden controlar, como las impurezas en los materiales o las imperfecciones geométricas. Durante el proceso de soldadura, cada punto espacial de los materiales implicados experimenta 132 Chapter 7. Resumen una variación de temperatura a lo largo del tiempo. Una soldadura de calidad requiere que cada punto alcance una temperatura suficientemente alta y una velocidad de enfriamiento suficientemente lenta. Por ello, se considera que regular la temperatura ajustando los parámetros controlables puede mejorar la calidad de las soldaduras. En esta sección se presentan modelos de evolución de la temperatura de una pieza soldada, adecuados para su uso en el diseño de sistemas de control. La pieza soldada se puede modelar en un espacio unidimensional, bidimensional o tridimensional, dependiendo de la forma de la pieza. Para los casos unidimensional y bidimensional, se presenta un modelo puramente analítico. Para el caso tridimensional, el enfoque seguido es una linealización de un modelo numérico basado en el método de elementos finitos. En ambos casos el punto de partida es la ecuación de transferencia de calor. 7.3.1 Obtención de funciones de transferencia Las funciones de transferencia se utilizan ampliamente en ingeniería de control como modelos de sistemas, ya que permiten la extracción de propiedades en el dominio de la frecuencia. Se obtienen a partir de ecuaciones lineales dinámicas utilizando la transformada de Laplace. En el caso de la ecuación diferencial 7.3. Modelado 133 parcial en (6), la función de transferencia se calcula resolviendo T(s) = 1 ρCpZ∞ 0e−st(k∇2T(t) + δQ(t)−v(t)∇T(t))dt. Las funciones de Green G(x,x′)proporcionan una forma general de describir la respuesta de una solución de ecuación diferencial a un término fuente arbitrario. Representan la respuesta impulsional en el punto x′de un operador diferencial lineal no homogéneo (L) definido en un dominio (x∈X⊂R3) con condiciones iniciales y de contorno especificas: LGx,x′=δ(x−x′). La expresión (17) es adecuada para modelar una antorcha de soldadura como una fuente de calor puntual en el origen, g(r,t)=δ(r)Q(t), donde δ(·)representa la función delta de Dirac, y además se supone que las condiciones iniciales y de contorno son cero. En este caso la ecuación (17) se reduce a (18). Las funciones de transferencia dinámicas de los procesos de soldadura se pueden obtener en general como T(r,s) Q(s)=α kG(r,s|0,0), donde G(r,s|0,0)oG(r,s)es directamente la transformada de Laplace de la función de Green G(r,t|0,0), y el factor α k=1 ρCp introduce las propiedades físicas del material base. Este resultado no depende del planteamiento del problema