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## TCM Modeling and Control in Literature

According to [Bry73], cold rolling mills have been used since the 15th century in the form of manual rolling with very low industrial performance and weak producti-vity. During this time, metal sheets were obtained by means of hot rolling. Referring to [Rob78] hot mills for iron rolling was established in 1798. However there are limi-tations to obtain thin metal sheets using hot rolling. This is due to metal melting at high temperatures, which leads to a sever strip deformation and possible rupture o the sheets when rolled to thin layers. A rolling chain was needed based on metal pres-sing under force with no heating, to produce thin metal layers. A good work led to the built of the Lauth mill in the 1850, motivating the industry to process in its deve-lopment. These mills were developed continuously till the first cold tandem was built in Czechoslovakia [Bry73] in 1892. Meanwhile this advancement was moving on in Germany [Ged98], where the first mill was built in 1923 and the second one did not finish until 1932. Roberts in [Rob78] concentrated on the development of cold mills in the United States of America with the beginning of the 19th century. In 1926, a 4 stand mill was finished. The early cold tandems were functioning with loosen strip between stands. Later on, mills were designed such that the strip between each two stands is tighten due to a tension force. This force plays a role in the rolling mechanism.

Since that time, the increasing requests on metal sheets due to the industrial evolu-tion in the manufacturing of automobiles, trains, jets: : :etc, have motivated the deve-lopment of this process to enhance its productivity and improve the final strip quality. One of these sophisticated improvements is the automation of the process, such step has opened a new research door between the fields of automation and steelmaking in-dustry. Many reasons have played a role in this progress such as the necessity of larger and faster mills, improvements done in the domains of actuator driving, instrumen-tation and high level control technology. A brief overview about the existing studies related to the control of tandem cold mill will be given to enrich us on the road to our objective.

**Roll Force and Torque Mathematical models**

Since 1950, the research efforts were put in developing mathematical models that demonstrate the evolution of physical variables during the rolling process. These mo-dels are either related to the whole tandem setup such as state space model for a single stand mills, or for single physical quantities such as roll force, torque, yield stress : : :etc. One of the most complicated variables are the rolling force and torque, where they are applied by the work rolls on the strip to obtain the desired thickness reduction. These variables are the resultant contribution of several interactions among strip thickness and hardness, work roll radius and speed, entry and exit tensions. The majority of the rolling theories [PL91] are based on a specified cause-effect chronological order among the physical variables. In 1943, E. Orowan [Oro43] developed a detailed systematic cal-culation of the roll pressure for hot and cold rolling. This was the fundamental work for the existence of new generation of models in cold rolling. In 1948, R. Bland and H. Ford [BF48] presented the calculation of roll force and torque in cold strip rolling in the presence of tensions and assuming that the work rolls are deformed in an elliptical form according to Hitchcock’s study [Hit35,Hil98]. It is one of the classical models that are generally used for the analogical calculation, though it possesses high complexity in its equations. It is composed of set of algebraic equations with integrals for the cal-culation of rolling force and torque needed to be applied to the strip to obtain a desired percentage reduction. The calculation of the force in this model version was based on the assumption that the rolling zone situated between the work rolls is divided into three regions as shown in Figure 2.4. The entry region, where the applied effort is the entry elastic force. The middle region where the applied effort is the plastic force and the exit region where the exit elastic force is applied. The rolling force is the sum of the three applied efforts. However the specialists in [BF48] have seen that the entry and exit elastic forces are mush less than the plastic one, and thus they can be negligible. After that, a second version was presented in [BF52] where these elastic efforts are no more negligible and they are important in the aim to have a more precise total force values. Thus algebraic expressions were assigned to these elastic quantities. It was followed by another study [BS53] which was given under a note related to the last developed roll force. This note states that the strip thickness reaches a minimum thickness wi-thin the rolling bite less than its exit thickness. This was interpreted by explaining that the strip recovers a little bit of its entry thickness after being deformed at the moment it is released by the work rolls. An additional algebraic expression was given to the minimal thickness, which was considered as a corrective parameter in the expression of the total roll force. The final obtained expressions of the rolling force and torque were algebraic non-linear coupled equations were used widely such as in [Ged98]. A simplified model was given in [FEB51] assuming some approximations for the entry and exit elastic forces which are summed in a single expression. Also the integrals are approximated by non-integral algebraic expressions, but this only reduces calculation effort. A noticeable coupling between the roll force and deformed radius was obtained. The number of algebraic expressions are less than the number of physical intermediate variables. For that and after some time, Bryant and Osborn in [Bry73,FPS08,ACMH12] have developed a set of algebraic non-coupled equations with much more approxima-tions and much less calculative effort. Though it includes many assumptions and mean values replacing the integral expressions, acceptable results were obtained. This model was used in all Pittner’s studies [PS04,Pit06,PS06,PS07a,PS07b] in proposing different multivariable control strategies for cold rolling.

These models are sets of analogical equations, but each one has a different level of complexity and non-linearity. Another point of view for calculating these variables was to use numerical finite element methods. Such methods [DMS+13, Cao15, Cao16] were proposed recently and aim to obtain highly precised values.

### Tandem Cold Mill Modeling and Control

The aim of cold rolling is to produce flat sheets or high quality strips. Before star-ting, the mill setup configuration is important in the operation to fix some factors. For example the user has to determine the percentage reduction that should be achieved across each mill stand. This leads to estimate the rolling load needed to be submitted to the strip within each pair of work rolls. Also the configuration of the interstand ten-sions as well as the rolling speed references has to be precised. In ideal cold rolling where the strip does not encounter any physical or mechanical errors, these references would be sufficient to perform the objective. However ideal rolling does not exist in reality and strip errors show up continuously as long as the steel coil is not finished. Thus to arrive to the desired product quality, certain modifications applied to the strip by process actuators are needed. Moreover controlling these inputs also permit us to improve chain productivity and its performance. Another reasons revealing the essen-tial role of control in tandem rolling mill, are the presence of system-entry strip uncer-tainties as well as the internal and external disturbances result form the environment of the complex system.

Tandem cold rolling is a complex multivariable process, where the internal physi-cal quantities interact among each other. The passage of the strip between stands leads to the coupling of stand variables in the mill. The ability to regulate and manage these couplings is essential and difficult in the same time. This point becomes more and more critical due to the increasing performance requisites. In [Bry73], Bryant has accompli-shed a basic rich study in the control of cold rolling process in 1973, treating general cases of tandem mill including the phase of modeling, simulation and control. For the control part, he followed a strategy based on the concept of non-interactive control structure. This strategy means that the set of actuators are designed to be able to apply independent changes to the strip at any location in the mill. Such strategy permits to separate the interstand tension control from the strip thickness control. This decom-position enables control methods based on single-input single-output form to be ap-plied directly. Moreover the similarity between actuators promote the idea of identical controllers for the tension loops. Bryant proposed another control view which depends on adding the effect of feedforward to the feedback action. He had considered several structures based on this combination, the thickness variation measurement in the first stand was used to apply changes to the strip by feedforward. Also in [Edw78], the laser gauge thickness measurement just after the first stand is used as a feedforward for the next stands. In general, in this control loops the tension is controlled by the mill actuator through variating the work roll gap. Whereas the strip thickness control is achieved through variating the rolling speed via the stand motor. The control of thickness is affected by the number of mill stands with respect to the available num-ber of laser gauge measurements. The mill structure is different from one to another regarding the available actuators and sensors. Different mill structures can be found in [Rob78, FS83, Smi87, Gin93a, Gin93b].

As the multivariable control [SP07,BEGFB94] design started to rise up in the field of process control, it attempts to manage the interaction between process variables. The application of such control view has been transferred to the rolling problem by many authors. In [HK93], a wide view of control objectives in the steel making industry has been given. In this study, more than one control strategies are presented such as bar gauge control, spindle torsional vibration control and others. These strategies are either model based or model free control synthesis.

In [HMF+88], a multivariable state space control technique is applied to the cold rolling of aluminum in 2 stand mill. The author tried to improve the results of the single-input single-output control approach done in [YK79]. The rolling force model have been linearized [HSM84] and used to construct the linear state space model. They focused on regulating the errors on the thicknesses, since they are the crucial points in his point of view. The disturbances in the system are modeled as constant ones, and they are estimated with the states by a linear estimator. A feed-forward compensating the effect of the disturbance is combined with the estimated state feedback. In the fra-mework of a small system (actuators and motors are considered systems of first order) size, the gains of the controller are chosen manually by moving the poles of the system to the left hand side of the complex plane. The parameter uncertainty and transport de-lays are not mentioned in this study and play no role in the controller synthesis. Ano-ther study in [HSDB+11] has used the combination between the feed-forward and feed-back loops. The rolling force model is based on the Orowan’s equations [Oro43,Bry73]. The aim was to improve the performance of the control loops during the Flying Gauge Change, i.e. passage of inter-coil welding.

#### Time Delay Systems in Brief

Time delay systems (TDSs) also called retarded differential equations (RDEs) belong to the class of functional differential equations [Hal71,HL93] which are infinite dimensio-nal systems. These systems are also called delayed differential equation (DDE) [NVDD98, KNG99, Kha98]. The instantaneous time derivative of the system trajectory x is given in terms of the function values at previous times. These systems are also called as he-reditary systems or differential-difference equation systems [Ric03]. The aftereffect of these systems have motivated their usage and increased their appearance in process mathematical modeling. In the fields of industry, numerous processes contain delay effects such as communication networks with transport delays or such as in chemical process with dead delay effects. Recently TDS show more and more interest especially in the field of system control. The effects of time delays in the process usually can’t be easily managed using the classical control methods, whereas some control studies proposed to approximate or neglect these delays. This might change the system model with respect to the real process especially in the case of time dependent delays. So lets start introducing some basic notions about this class of systems. The general form of TDS can be given by x(t) = f(t; xt); t t0; (3.11) x( ) = ( ); 2 [t0 ; t0] .

where x 2 Rn is the trajectory of the system. Also xt( ) = x(t )nis the delayed state vector with > 0 the time delay of the system. f : R C ;n ! R is continuous in its both arguments and locally Lipschitz continuous in the second one. The function f indicates that the derivative of the state variable x at a time t depends on t and x( ) for 2 [t0 ; t0]. Thus to determine the future evolution of the state, it is necessary to specify the initial value of the state variable along an interval of length . This leads us to complete the definition of a TDS by providing its initial condition over [t0 ; t0]. Usually in ordinary differential equation (ODE), the initial condition is referred to the single initial value x0 = x(t0). However in TDS, the initial condition refers to the func-tion 2 C ;n = C ([t0 ; t0] ; Rn) which is the minimum data necessary to construct the solution of the system (3.11). A fundamental issue in the study of both ODE and DDE is the existence and uniqueness of a solution. For the case of ODE, the necessary argu-ments responding to this issue can be found in [Kha02]. On the other hand, it is little more complicated for TDS, referring to [Ric03, MVAN05, Fri14] we give the following definition assuming that the delay is constant.

**Table of contents :**

**Chapter 1 General Introduction **

**Chapter 2 Cold Rolling of Steel **

2.1 Introduction

2.2 Process Overview

2.3 TCM Modeling and Control in Literature

2.3.1 Roll Force and Torque Mathematical models

2.3.2 Tandem Cold Mill Modeling and Control

2.4 Conclusion

**Chapter 3 Mathematical Preliminaries **

3.1 Introduction

3.2 Linear Quadratic Regulation

3.3 Stability

3.3.1 Basic Definitions

3.3.2 Lyapunov’s Direct Method for Stability

3.4 Stability of Time Delay Systems

3.4.1 Time Delay Systems in Brief

3.4.2 Stability Concept for General TDS

3.4.2.1 Lyapunov-Krasovskii Direct Method

3.4.3 Stability of Linear Systems with Time-Varying Delays

3.4.4 Stabilization of Linear Systems with Time-Varying Delays

3.5 Stability of Singular Systems

3.5.1 Singular Systems in Brief

3.5.2 Preliminaries on Singular Systems

3.5.3 Stability of Linear Singular Systems

3.5.4 Stabilization of Linear Singular Systems

3.6 Stability of Time Delay Singular Systems

3.6.1 Preliminaries of Time Delay Singular Systems

3.6.2 Stability of Time Delay Singular System

3.6.3 Stabilization of Time Delay Singular System

3.7 Conclusion

**Chapter 4 Tandem Cold Mill : Model and Current Control Strategy **

4.1 Introduction

4.2 Description of the Tandem Stand

4.2.1 Stand Actuator

4.2.2 Stand Motor

4.2.3 Yield Stress Model

4.2.4 Sims’ Gauge Relation

4.2.5 Mass Flow Conservation

4.2.6 Strip Exit Speed

4.2.7 Interstand Tension

4.2.8 Interstand Transport Delay

4.2.9 Rolling Force Model

4.3 TCM Multivariable Model

4.3.1 Exit Thickness Integral Action

4.3.2 Multivariable Non-linear Descriptor Model

4.4 TCM Linearized Model

4.4.1 TCM Time Delay Singular System

4.5 Current Multi-loop PID Control for Tandem Process

4.6 Conclusion

**Chapter 5 State Feedback Control for TCM Process **

5.1 Introduction

5.2 Existence and Uniqueness of Solution

5.3 Controller Synthesis Paradigm

5.3.1 Formulation of Controller Synthesis as a Constrained Optimization Problem

5.3.2 Process Performance translated as a Quadratic Cost

5.3.3 Stability and Performance Optimization via Lyapunov-Krasovskii Functional

5.3.4 Control solution of the Optimization Synthesis Problem

5.4 State Feedback Synthesis

5.4.1 Controller Synthesis via LMI Problem

5.4.2 Majorant Minimization

5.4.3 Numerical example

5.5 TCM Controller Synthesis

5.5.1 Controller Tuning

5.6 Conclusion

**Chapter 6 Dynamic Output Feedback Control for TCM Process **

6.1 Introduction

6.2 Output Feedback Synthesis

6.2.1 Problem Formulation

6.2.2 Closed Loop System Model

6.2.3 Control Design : Synthesis Analysis

6.2.4 LMI formulation of the Synthesis Problem

6.3 Dynamic Output Feedback Feasibility Problem

6.3.1 Controller Synthesis via LMI Feasibility Problem

6.3.2 Numerical example

6.4 DOF Synthesis for the TCM Process

6.5 Conclusion

**Chapter 7 General Conclusion **

**References**