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Explicit Model Predictive Controller Design for Thickness and Tension Control in a Cold Rolling Mill
17th IFAC Symposium on Control, Optimization and Automation in 17th IFAC Symposium on Control, Optimization and Automation in 17th IFAC Symposium on Control, Optimization and Automation in Mining, Mineral and Metal Processing 17th IFAC Symposium on Optimization Mining, Mineral and Metal Processing 17th IFAC Symposium on Control, Control, Optimization and and Automation Automation in in Mining, Mineral Processing Vienna, Austria. and AugMetal 31 - Sept 2, 2016Available online at www.sciencedirect.com Mining, Mineral and Metal Processing Vienna, Mineral Austria. and AugMetal 31 - Sept 2, 2016 Mining, Processing Vienna, Austria. Aug 31 -- Sept 2, 2016 Vienna, Vienna, Austria. Austria. Aug Aug 31 31 - Sept Sept 2, 2, 2016 2016
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IFAC-PapersOnLine 49-20 (2016) 126–131
Explicit Model Predictive Controller Explicit Model Predictive Controller Explicit Model Predictive Controller Explicit Model Predictive Controller Design for Thickness and Tension Control Design for Thickness and Tension Control Design for Thickness and Tension Control Design forinThickness and Tension Control a Cold Rolling Mill in a Cold Rolling Mill in a Cold Rolling Mill in a Cold Rolling ∗∗Mill ∗ ∗∗∗
Tomoyoshi Ogasahara ∗ Morten Hovd ∗∗ Kazuya Asano ∗∗∗ Tomoyoshi Ogasahara ∗ ∗ Morten ∗∗ Kazuya ∗∗∗ Tomoyoshi Ogasahara Morten Hovd Hovd ∗∗ Kazuya Asano Asano ∗∗∗ Tomoyoshi Ogasahara Tomoyoshi Ogasahara ∗ Morten Morten Hovd Hovd ∗∗ Kazuya Kazuya Asano Asano ∗∗∗ ∗ ∗ Instrument and Control Engineering Research Dept., Steel Research ∗ Instrument and Control Engineering Research Dept., Steel Research ∗ Instrument and Control Engineering Research Dept., Steel Research ∗ Instrument Engineering Research Steel Research Lab., JFE and SteelControl Corp., Fukuyama, Hiroshima, 721-8510, Instrument and Control Engineering Research Dept., Dept., Steel Japan Research Lab., Corp., Fukuyama, Lab., JFE JFE Steel Steel Corp.,[email protected]). Fukuyama, Hiroshima, Hiroshima, 721-8510, 721-8510, Japan Japan Lab., JFE Steel Corp., Fukuyama, Hiroshima, 721-8510, Japan (e-mail: Lab., JFE Steel Corp., Fukuyama, Hiroshima, 721-8510, Japan (e-mail: [email protected]). ∗∗ (e-mail: [email protected]). (e-mail: [email protected]). Engineering Cybernetics, Norwegian University of ∗∗ Department of (e-mail: [email protected]). of Engineering Cybernetics, Norwegian University of ∗∗ ∗∗ Department Department of Engineering Cybernetics, Norwegian University of ∗∗ Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Science and and Technology, N-7491 Trondheim, Trondheim, Norway Norway Science Technology, N-7491 (e-mail: [email protected]). Science and Technology, N-7491 Trondheim, Norway (e-mail: [email protected]). ∗∗∗ (e-mail: [email protected]). (e-mail: [email protected]). Lab., JFE Steel Corp., Kawasaki, Kanagawa, ∗∗∗ Steel Research (e-mail: [email protected]). Research Lab., JFE Steel Corp., Kawasaki, Kanagawa, ∗∗∗ ∗∗∗ Steel Research Lab., JFE Steel Corp., Kawasaki, Kanagawa, ∗∗∗ Steel Steel Research Lab., JFE Steel Corp., Kawasaki, 210-0855, Japan Steel Research Lab., 210-0855, JFE Steel Japan Corp., Kawasaki, Kanagawa, Kanagawa, 210-0855, Japan 210-0855, Japan (e-mail: [email protected]) 210-0855, Japan (e-mail: [email protected]) (e-mail: [email protected]) [email protected]) (e-mail: (e-mail: [email protected]) Abstract: This paper addresses modeling of cold rolling mill and controller design based on Abstract: This This paper paper addresses addresses modeling modeling of of aaa cold cold rolling rolling mill mill and and controller controller design design based based on on Abstract: Abstract: This paper modeling of rolling mill and design based on explicit model control(explicit The control are to track the exit Abstract: Thispredictive paper addresses addresses modelingMPC). of aa cold cold rolling millobjectives and controller controller design based on explicit model predictive control(explicit MPC). The control objectives are to track the exit explicit model predictive control(explicit MPC). The control objectives are to track the exit explicit MPC). The control are to thickness to the reference with high accuracy with minimized strip tension In the explicit model predictive control(explicit MPC). The control objectives objectives are deviation. to track track the the exit thicknessmodel to the thepredictive reference control(explicit with high high accuracy accuracy with minimized strip tension tension deviation. In exit the thickness to reference with with minimized strip deviation. In the thickness to the reference with high accuracy with minimized strip tension deviation. In the simulation, good control is obtained when the disturbance entering the system is small enough. thickness togood the reference with highwhen accuracy with minimized strip tension deviation. In the simulation, control is obtained the disturbance entering the system is small enough. simulation, good control is obtained when the disturbance entering the system is small enough. simulation, good control is when the disturbance the system enough. However, the thickness the offset from reference in case of because the simulation, good controlshows is obtained obtained when the the disturbance entering theacceleration, system is is small small enough. However, the the thickness shows the offset offset from the referenceentering in case case of of acceleration, because the However, thickness shows the from the reference in acceleration, because the However, the thickness shows the offset from the reference in case of acceleration, because the approximation error of the linearized model increases and it is different from the model in However, the thickness shows the offset fromincreases the reference iniscase of acceleration, because the approximation error of the linearized model and it different from the model in approximation error of the linearized model increases and it is different from the model in the approximation error model and it the in design of the controller. In order to compensate for the offset, an additional approximation error of of the the linearized model increases increases andcontrol it is is different different from the model modelintegral in the the design of of the the controller. controller. In linearized order to to compensate compensate for the the control offset, from an additional additional integral design In order for control offset, an integral design of controller. In order to compensate for the offset, an additional integral logic using the estimation of exit thickness is proposed. The validity this approach was design of the the controller. Inerror order to compensate for the control control offset, anof additional integral logic using the estimation error of exit thickness is proposed. The validity of this approach was logic using the estimation estimation error offree exitcontrol thickness isachieved. proposed. The The validity of of this approach approach was logic using the exit thickness proposed. verified by simulation and offset was logic using the estimation error offree exitcontrol thickness isachieved. proposed. The validity validity of this this approach was was verified by simulation simulation and error offsetof was is verified by and offset free control control was was achieved. verified by simulation and offset free achieved. verified by simulation and offset free control was achieved. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: process control, process models, predictive control, optimal control, Keywords: process control, control, process process models, models, predictive predictive control, control, optimal optimal control, control, Keywords: process Keywords: process control, control,system, processsteel models, predictive control, control, optimal optimal control, control, multi-input/multi-output industry. Keywords: process process models, predictive multi-input/multi-output system, steel industry. multi-input/multi-output system, steel industry. multi-input/multi-output system, steel industry. multi-input/multi-output system, steel industry. 1. ware and high computer tends to be high. 1. INTRODUCTION INTRODUCTION ware and and high high performance performance computer tends to be high. 1. ware performance computer tends to to be be high. 1. INTRODUCTION INTRODUCTION ware and high performance computer tends high. Furthermore, the time of solving an optimization problem 1. INTRODUCTION ware and high performance computer tends to be high. Furthermore, the time of solving an optimization problem Furthermore, the time of solving an optimization problem Furthermore, the time of solving an optimization problem possibly exceeds the sampling period. Consequently, the Furthermore, the time of solving an optimization problem possibly exceeds the sampling period. Consequently, the In aa cold strip mill, as shown in Fig. 1, the preceding possibly exceeds the sampling period. Consequently, the possibly exceeds the sampling period. Consequently, the applications are limited. In cold strip mill, as shown in Fig. 1, the preceding possibly exceeds the sampling period. Consequently, the applications are limited. In a cold strip mill, as shown in Fig. 1, the preceding In cold stripreceding mill, as asstrip shown inwelded Fig. 1, 1,together the preceding preceding strip and the are and it applications are are limited. limited. In aa cold strip mill, shown Fig. the strip and the the receding strip arein welded together and it it applications applications are limited. strip and receding strip are welded together and In explicit MPC(Bemporad et al. (2000)), the strip and the receding strip are welded together and it is rolled continuously. The exit thickness of each of the In explicit MPC(Bemporad et al. al. (2000)), (2000)), the the optimization optimization strip andcontinuously. the receding The stripexit are thickness welded together and it In explicit MPC(Bemporad et is rolled of each of the optimization is rolled continuously. The exit thickness of each of the In explicit MPC(Bemporad et al. (2000)), the optimization problem in the MPC for the linear time-invariant system is rolled continuously. The exit thickness of each of the rolling stands should be controlled with high accuracy. It is In explicit MPC(Bemporad et al. (2000)), the optimization problem in the MPC for the linear time-invariant system is rolled continuously. The exit thickness of each of the rolling stands should be controlled with high accuracy. It is problem in the MPC for the linear time-invariant system rolling stands should be controlled with high accuracy. It is problem in the MPC for the linear time-invariant system such as a quadratic programming problem is converted rolling stands should be controlled with high accuracy. It is also necessary to minimize strip tension deviations during in quadratic the MPC programming for the linear time-invariant system such as a problem is converted rolling stands should be controlled with high accuracy. It is problem also necessary to minimize strip tension deviations during such as aa quadratic problem is also necessary to minimize strip tension deviations during such asmulti-parametric quadratic programming programming problem is converted converted into programming problem, and then also necessary to in minimize strip tension deviations during thickness control order prevent rolling trouble such as such a quadratic programming problem is converted into aaaas multi-parametric programming problem, and then then also necessary to minimize strip tension deviations during thickness control in order to to prevent rolling trouble such as into multi-parametric programming problem, and thickness control in order to prevent rolling trouble such as into a multi-parametric programming problem, and then the solution is obtained as a piecewise affine function thickness control in order to prevent rolling trouble such as strip breakage. These variables are controlled by adjusting into a multi-parametric programming problem, and then the solution is obtained as a piecewise affine function thickness control in order to prevent rolling trouble such as the solution is obtained as a piecewise affine function strip breakage. These variables are controlled by adjusting strip breakage. These variables are controlled by adjusting the solution is obtained obtained asin piecewise affine function of the state of the plant offline. Hence it does not strip breakage. These variables are controlled bypoint, adjusting the roll gap and the roll speed. At the welding the solution is piecewise affine function of the the state of of the plant plant as in aaoffline. offline. Hence it does does not strip breakage. variables by adjusting the roll roll gap and andThese the roll roll speed.are At controlled the welding welding point, the the of state the in Hence it not the gap the speed. At the point, the of the state of the plant in offline. Hence it does not require online optimization and can potentially reduce the roll gap and the roll speed. At the welding point, the roll gap and the roll speed are synchronized to achieve the state ofoptimization the plant inand offline. Hence it does not require online can potentially reduce the roll gap and the roll speed. At the welding to point, the of roll gap and the roll speed are synchronized achieve require online and can reduce roll gap roll speed are to achieve require online optimization optimization and can potentially potentially reduce the implementation cost and overcome the limitation of roll gap and and the the roll These speed actions are synchronized synchronized to by achieve the thickness change. are executed feed online optimization can potentially reduce the implementation implementation cost and andand overcome the limitation limitation of roll gap and the roll speed are synchronized to achieve the thickness thickness change. These actions are executed executed by feed require the cost overcome the of the change. These actions are by feed the implementation cost and overcome the limitation of normal MPC. Therefore, explicit MPC is applied in this the thickness change. These actions are are executed by feed the forward control. When the welding point is not passing implementation cost and overcome theapplied limitation of normal MPC. Therefore, Therefore, explicit MPC is is in this the thickness change. These actions executed by feed forward control. When the welding point is not passing normal MPC. explicit MPC applied in this forward control. When the welding point is not passing normal MPC. Therefore, Therefore, explicit explicit MPC MPC is is applied applied in paper. forward control. When the control welding is point isfor notthickness passing normal through the mill, feedback used in this this paper. MPC. forward the welding not passing through control. the mill, mill,When feedback control ispoint usedis for thickness paper. through the feedback control is used for thickness paper. through the mill, feedback control is used for thickness and tension control. through the control. mill, feedback control is used for thickness paper. The outline of this paper is as follows. In section 2, the and tension The outline outline of of this this paper paper is is as as follows. follows. In In section section 2, 2, the the and and tension tension control. control. The and tension control. The outline of this paper is as follows. In section 2, the state space model of a cold rolling mill is presented. In secThis paper deals with thickness and tension control probThe outline of this paper is as follows. In section In 2, secthe state space model of a cold rolling mill is presented. This paper deals with thickness and tension control probspace model aa cold mill is In secThis paper with thickness and tension control probstate space model of of cold rolling rolling mill is presented. presented. In section 3, the design procedure of the explicit MPC is given. This paper deals deals withoperation thickness for and two tension controlrolling prob- state lem during normal adjacent state model of a cold rolling mill is presented. In section 3, 3,space the design design procedure of the explicit MPC is given. This paper deals with thickness and tension control problem during during normal operation for two adjacent rolling tion the procedure of the explicit MPC is given. lem normal operation for two adjacent rolling tion 3, the design procedure of the explicit MPC is given. Next, the simulation results and controller modifications lem during normal operation for two adjacent rolling stands in a rolling mill, as depicted in Fig. 2. tion 3, the design procedure of the explicit MPC is given. Next, the simulation results and controller modifications lem during normalmill, operation for in two adjacent rolling Next, the simulation results and controller modifications stands in a rolling rolling as depicted Fig. 2. stands Next, the simulation simulation resultsperformance and controller controller modifications for improving the tracking of the thickness stands in in a rolling mill, mill, as as depicted depicted in in Fig. Fig. 2. 2. Next, the results and for improving improving the tracking tracking performance of modifications the thickness thickness stands in aa rolling mill, as depicted in Fig. 2. Model Predictive Control (MPC) is well known for its for the performance of the for improving the tracking performance of the thickness are shown in section 4. Model Predictive Control (MPC) is well known for its for improving the tracking performance of the thickness are shown in section 4. Model Predictive Control (MPC) is well known for its Model Predictive Control (MPC)to isaccount well known known forconits are control performance and ability for the are shown shown in in section section 4. 4. Model Control (MPC) well its control Predictive performance and ability ability to is account for the thefor conare shown in section 4. control performance and to account for concontrol performance and ability to account for the constraints of the plant. Therefore it has been regarded as control performance and ability to account for the constraints of the plant. Therefore it has been regarded as straints of plant. Therefore it been regarded as straints of the the plant.and Therefore it has has in been regarded as a practical solution is widespread the process instraints of the plant. Therefore it has been as practical solution and is widespread widespread in theregarded process ininaaa practical solution and is in the process practical solution and is widespread in the process industry. For example, an application of nonlinear MPC to adustry. practical is widespread in the process For solution example,and an application application of nonlinear nonlinear MPC into 2. DYNAMICAL MODEL OF A ROLLING MILL dustry. For example, an of MPC to 2. DYNAMICAL DYNAMICAL MODEL MODEL OF OF A A ROLLING ROLLING MILL MILL dustry. For example, an application of nonlinear MPC to a cold rolling mill is reported by Ozaki et al. (2009) and dustry. For example, an application of nonlinear MPCand to 2. a cold rolling mill is reported by Ozaki et al. (2009) 2. DYNAMICAL MODEL OF A ROLLING MILL aait cold rolling mill is reported by Ozaki et al. (2009) and 2. DYNAMICAL MODEL OF A ROLLING MILL cold rolling mill is reported by Ozaki et al. (2009) and shows good results under acceleration and deceleration ait cold rolling mill is reported by Ozaki et al. (2009) and shows good results results under under acceleration acceleration and and deceleration it shows it shows good good results it under acceleration and deceleration deceleration conditions. However requires online optimization and In this section, nonlinear dynamical model of cold it shows good results acceleration and deceleration conditions. However itunder requires online optimization optimization and In In this this section, section, aaa nonlinear nonlinear dynamical dynamical model model of of aa a cold cold conditions. However it requires online and conditions. However it requires online optimization and In this section, a nonlinear dynamical model of a cold the implementation cost related to the optimization softrolling mill and its linear state space model are described. conditions. Howevercost it requires online optimization softand In this mill section, a nonlinear dynamical model of a cold the implementation related to the optimization rolling and its linear state space model are described. the the implementation implementation cost cost related related to to the the optimization optimization softsoft- rolling rolling mill mill and and its its linear linear state state space space model model are are described. described. the implementation cost related to the optimization softrolling mill and its linear state space model are described. Copyright © 2016, 2016 IFAC 130Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 130 Copyright © 2016 IFAC 130 Copyright © 2016 IFAC 130 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 130Control. 10.1016/j.ifacol.2016.10.108
IFAC MMM 2016 Vienna, Austria. Aug 31 - Sept 2, 2016 Tomoyoshi Ogasahara et al. / IFAC-PapersOnLine 49-20 (2016) 126–131
127
• Thickness Model hi = Si + • Transport Delay Model Hi+1 = hi exp −
Fig. 1. Layout of a Continuous Cold Rolling Mill
• Roll Speed Model
Vi (s) =
Pi . Mi
L s . (1 + fi )Vi
(6)
(7)
1 V ref . Tv s + 1 i
(8)
1 S ref . Ts s + 1 i
(9)
• Roll Gap Model Si (s) =
Fig. 2. Adjacent Rolling Stands in a Rolling Mill 2.1 Rolling Models and Actuator Models The model consists of rolling models and actuator models. Each model is shown below. Here, the variables in the models are presented in Table 1. Note that the suffix i denotes the i th stand. • Rolling Force Model Pi =w(ki − 0.7σi−1 − 0.3σi ) Ri × 1.08 + 1.79ri µi − 1.02ri Hi × R (Hi − hi ) × 1000. where, 16(1 − ν 2 ) Pi , Ri = Ri 1 + Es π w(Hi − hi ) Hi − h i ri = . Hi • Forward Slip Model 1 Hi − hi 2 −1 tan fi = tan 2 hi 1 Hi ki−1 (1 − σi ) − . log 4µi hi ki (1 − σi−1 )
(1)
The rolling force model given in (1) and forward slip model given in (2) are reported by Misaka (1967). The friction coefficient between strip and work roll is dependent on the roll speed and it is modeled as (3). The deformation resistance model given in (4) is based on the effectiveness of strain hardening. Strip tension results from the elastic deformation of the strip. Therefore, it is proportional to the difference of the strip velocity between the exit of i th stand and the entry of i + 1 th stand. It is modeled as (5). The thickness model shown in (6) indicates that it is determined by the roll gap Si and elastic deformation of the work roll, which is represented as Pi /Mi . The exit thickness of the i th stand is transported to the i + 1 th stand. Thus this system includes time delay. This relationship is represented in (7). The actuators of the mill are electric motors and hydraulic cylinders. The former are for controlling the roll speed, and latter for controlling the roll gap position. The motors are controlled by the Automatic Speed Regulator(ASR) and the hydraulic cylinders are controlled by the position control system. Thanks to these minor feedback control, these actuators can be modeled as first order transfer function described in (8) and (9). 2.2 Linear State Space Model
(2)
• Friction Model µi = µa exp(−µb × Vi ) + µc . (3) • Deformation Resistance Model n H0 ki = 1.15l m + 1.15 log . (4) hi • Strip Tension Model Es hi+1 dσi = (1 + fi+1 )Vi+1 − (1 + fi )Vi . (5) dt L Hi+1 131
The linear state space model is derived from the nonlinear model shown above. First, total derivative of the thickness model is expressed as (10) ∆hi = ∆Si + ∆Pi /Mi . where, ∂Pi ∂Pi ∆hi + ∆Hi ∆Pi = ∂hi ∂Hi (11) ∂Pi ∂Pi ∂Pi + ∆σi−1 + ∆σi + ∆Vi . ∂σi−1 ∂σi ∂Vi By substituting (11) for (10), the thickness model is linearized as
∆hi =
1 ∂Pi ∆Hi Mi ∆Si + Mi + Q i ∂Hi ∂Pi ∂Pi ∂Pi + ∆σi−1 + ∆σi + ∆Vi . ∂σi−1 ∂σi ∂Vi (12)
IFAC MMM 2016 128 Vienna, Austria. Aug 31 - Sept 2, 2016 Tomoyoshi Ogasahara et al. / IFAC-PapersOnLine 49-20 (2016) 126–131
exp −
where, Qi = −
∂Pi . ∂hi
It is assumed that the entry tension of i th stand (σi−1 ) and the exit tension of i + 1 th stand (σi+1 ) is constant. Therefore, the exit thickness of each of the stands is represented as follows. • Exit thickness of i th stand ∆hi =
1 ∂Pi ∆Hi Mi ∆Si + Mi + Q i ∂Hi ∂Pi ∂Pi + ∆σi + ∆Vi + dhi . ∂σi ∂Vi
(13)
• Exit thickness of i + 1 th stand 1 ∂Pi+1 ∆hi+1 = ∆Hi+1 Mi+1 ∆Si+1 + Mi+1 + Qi+1 ∂H i+1 ∂Pi+1 ∂Pi+1 ∆σi + ∆Vi+1 + dhi+1 . + ∂σi ∂Vi+1 (14) where, dhi and dhi+1 are unmeasurable output disturbance. • Entry thickness of i + 1 th stand Next, the transport delay model shown as (7) is expressed as a 3rd order Pad´e approximation function represented as Table 1. Parameters in the Model Symbol
Unit
Pi w ki σi Ri Ri Hi hi hi+1 ri Es ν µi µa µb µc fi l m n H0 L Vi Viref Si Siref s Tv Ts Mi
kN m MPa MPa m m m m m MPa m m m/s m/s m mm s s kN/m
Description Rolling force Strip Width Deformation resistance Strip tension between stands Work roll radius Flattened work roll radius Entry thickness Exit thickness of i th stand Exit thickness of i + 1 th stand Reduction rate Young’s modulus of strip Poisson’s ratio of work roll Friction coefficient Parameter in friction model Parameter in friction model Parameter in friction model Forward slip ratio Parameter of ki Parameter of ki Parameter of ki Thickness before rolling Length between stands Roll speed Reference of roll speed Roll gap Reference of roll gap Laplace operator Time constant of ASR Time constant of roll gapr Mill modulus
Value 1160×10−3 − 346 180×10−3 0.75×10−3 0.488×10−3 0.285×10−3 205800 0.3 0.0303 0.1974 0.024 100.3 0.017 0.164 1.5×10−3 6.0 1/60 0.01 500
132
b 2 s 2 + b1 s + b0 + d0 . + a2 s2 + a1 s + a0 (15) where, a0 , a1 , a2 , b0 , b1 , b2 , and d0 are parameters which are dependent on time delay. By using (15), the state space representation of it is obtained as 0 1 0 x1 0 d x1 0 1 x2 = 0 x2 + 0 ∆hi , dt x −a0 −a1 −a2 x3 1 3 x1 ∆Hi+1 = [b0 b1 b2 ] x2 + d0 ∆hi . (16) x3 L s (1 + fi )Vi
≈
s3
where, x1 , x2 , and x3 , are the state variables in the Pad´e approximation. Then by substituting (13) into (16), ∆Hi+1 can be expressed by the state variables of ∆Vi , ∆Si , ∆σi , x1 , x2 , x3 , ∆Hi , and dhi . • Tension model In order to linearize the tension model, the total derivative of (5) is calculated as Es hi+1 dσi ∆ = ∆ (1 + fi+1 )Vi+1 − (1 + fi )Vi . dt L Hi+1 d∆σi Es 1 ↔ = (1 + fi+1 )Vi+1 ∆hi+1 dt L Hi+1 hi+1 2 (1 + fi+1 )Vi+1 ∆Hi+1 Hi+1 hi+1 + Vi+1 ∆fi+1 Hi+1 hi+1 + (1 + fi+1 )∆Vi+1 Hi+1 −
− Vi ∆fi − (1 + fi )∆Vi
+ vT .
(17) where, vT is unmeasurable tension disturbance. In (17), ∆fi and ∆fi+1 need to be expanded. Now, it is assumed that the output disturbance of each of the stands does not affect the forward slip ratio. Then ∆fi and ∆fi+1 are expanded as ∂fi ∂fi ∂fi ∂fi ∆Vi + ∆σi + ∆Hi + (∆hi −dhi ), ∆fi = ∂Vi ∂σi ∂Hi ∂hi (18) and ∂fi+1 ∂fi+1 ∆fi+1 = ∆Vi+1 + ∆σi ∂Vi+1 ∂σi ∂fi+1 ∂fi+1 + ∆Hi+1 + (∆hi+1 − dhi+1 ). ∂Hi+1 ∂hi+1 (19) Here, ∆hi in (18) and ∆hi+1 in (19) correspond to (13) and (14) respectively, and ∆Hi+1 in (19) is given by (16). • Roll speed model The reference of the roll speed of ith stand is dependent on the reference of the downstream stand to minimize the imbalance of the mass flow. Hence Viref is defined as the additional term of the reference of i + 1 th stand. Thereby, the dynamics of the roll speed of i th stand is represented as
IFAC MMM 2016 Vienna, Austria. Aug 31 - Sept 2, 2016 Tomoyoshi Ogasahara et al. / IFAC-PapersOnLine 49-20 (2016) 126–131
href i+1 (1 + fi+1 ) ref Vi+1 . + href i (1 + fi ) (20) ref where, href are the reference of the exit and h i+1 i thickness. fi means the nominal value of forward slip ratio calculated by the set value. The roll speed of i + 1 th stand is represented as 1 1 ref d∆Vi+1 = − ∆Vi+1 + V . (21) dt Tv Tv i+1 • Roll gap The roll gap model given by (9) is converted to the state space model realized as 1 1 d∆Si = − ∆Si + Siref , (22) dt Ts Ts and d∆Si+1 1 1 ref = − ∆Si+1 + Si+1 . (23) dt Ts Ts • Rolling force The folling force satisfies (10). Therefore it can be calculated by following equation. ∆Pi = M (∆hi − ∆Si ). (24) Here, ∆hi corresponds to (13). 1 1 d∆Vi = − ∆Vi + dt Tv Tv
Viref
In this system, the measurable variables are roll speed, roll gap, strip tension, exit thickness, and rolling force.
Ax
+Bu + Ed,
Cx
+F d.
(25)
where, A, B, C, E, and F are matrix with proper size and T
x = [∆Vi ∆Vi+1 ∆Si ∆Si+1 ∆σi x1 x2 x3 ] , ref T , u = Viref Siref Si+1 T ref d = Vi+1 ∆Hi vT dhi dhi+1 ,
At each timestep, the MPC solves an optimization problem. The decision variables in the optimization problem are the future inputs, while the objective function reflects the predicted future bahaviour of the system. The first element of the calculated future inputs is applied, and the optimization is executed anew for each timestep. The objective function is shown below. N −1 min (˜ x(k) − x ˜ref )T Q(˜ x(k) − x ˜ref ) ∆u(0),··· ,∆u(N −1)
k=0
xT uT
T
+ ∆u(k)T R∆u(k) .
, Q ≥ 0, and R > 0. N means T prediction horizon. x ˜ = (xref )T (uref )T is the target state. This objective function is minimized subject to the upper and lower constraints as below and the model in (26).
where, x ˜ =
ref
xmin ≤ x(k) ≤ xmax ,
umin ≤ u(k) ≤ umax ,
∆umin ≤ ∆u(k) ≤ ∆umax ,
ymin ≤ y(k) ≤ ymax .
Thus, this MPC problem is a quadratic programming because the objective function is quadratic and the constraints are linear function. Bemporad et al. (2000) show that it can be expressed as a multi parametric quadratic programming (mp-QP) and the solution is organized as a piecewise affine function parametrized by the state. Therefore the solution can be calculated explicitly and on-line optimization is not required. This algorithm is referred to as explicit MPC.
Finally, the state space representation is obtained by integrating (13), (14), (16), (17), (20), (21), (22), (23), and (24) and summarized as below. dx = dt y=
129
T
For achieving offset free control in the case of disturbance entering the system, the effects of the disturbance need to be counteracted. At the steady state, the system given as (26) satisfies the following relationships because xref = x(k + 1) = x(k), uref = u(k + 1) = u(k), and ∆u(k) = 0. Ed d I − Ad Bd xref = C O uref y ref − F d
y = [∆Vi ∆Vi+1 ∆Si ∆Si+1 ∆σi ∆hi ∆hi+1 ∆Pi ∆Pi+1 ]. Therefore if we specify some of the reference of the output and state variables, the target of xref and uref which achieve offset free control can be calculated. In this case, 3. EXPLICIT MPC DESIGN ref the disturbance except for Vi+1 is unmeasurable. Hence, In this section, the MPC formulation is considered. The they should be estimated by a observer. The objective here is to track the exit thickness with minimized tension discretized system of (25) is given by deviation. Therefore, ∆hi , ∆hi+1 , and ∆σi are set to 0. The unspecified states are calculated by solving below. x(k + 1) = Ad x(k) +Bd u(k) + Ed d(k), xt E d B A d d y(k) = Cx(k) +F d(k). = −F d C O uref Then the system is augmented by introducing the change where, ref T = u(k + rate of input(∆u(k) = ∆Viref ∆Siref ∆Si+1 T 1) − u(k)). Then the system is represented as xt = [∆Vi ∆Vi+1 ∆Si ∆Si+1 x1 x2 x3 ] ,
Ad Bd Ed x(k) O I O u(k) O O I d(k) x(k) y(k) = [C O F ] u(k) . d(k)
x(k + 1) u(k + 1) = d(k + 1)
O + I ∆u(t), O (26)
A : After elimination of 5th column of I − A
C , F : After elimination of 5th column and
extraction of 6th and 7th row of C and F The block diagram of the control system is illustrated in Fig. 3.
133
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Fig. 3. Block Diagram of the Control System 4. SIMULATION The simulation results are presented here. It is executed R with the Multi-Parametric Toolby utilizing MATLAB box 3.0 developed by Herceg et al. (2013) and YALMIP developed by Lofberg (2004). The rolling condition of the simulation is summarized in Table 1. In addition to this, the initial roll speed of i + 1 th stand is 5[m/s]. The sampling period Ts is 10ms and prediction horizon N is set to 2. The constraints are as follows.
Fig. 4. Time Chart of the Outputs and Inputs
ref ≤ 1.0, −1.0 ≤ ∆Siref , ∆Si+1
ref /Ts ≤ 1.0, −1.0 ≤ ∆Siref /Ts , ∆Si+1
−1.0 ≤ Viref ≤ 1.0,
−0.1 ≤ ∆Viref /Ts ≤ 0.1.
Subsequently, the weight matrices in the objective function is set as below . Q = diag(1, 1, 1000, 1000, 1, 1000, 1000, 1000, 1, 1000, 1000), R = diag(0, 0, 0). As a result of explicit MPC design, the number of polyhedral partition of the feedback gain is 352. The control results are depicted in Fig. 4 and the disturbance and its estimated value by an observer is illustrated in Fig. 5. Here, the disturbance enters into the entry thickness of i th stand, the strip tension, the exit thickness of each of the stands, and the roll speed of i + 1 th stand. In this case, the disturbance can be estimated precisely and the thickness and the tension are well controlled to the reference. On the other hand, results when accelerating the roll speed depicted in Fig. 6 show offset in the thickness. The estimation accuracy of the disturbance depicted in Fig. 7 gets worse in this case. This is because the system is dependent on the time delay between stands, however, the observer utilizes the linearized model around the initial roll speed. Therefore the approximation error in the linearized model becomes significant. In order to achieve offset free control, an integral logic is introduced. It is based on the fact that the estimation error of the thickness is nearly 0 at the steady state in spite of the existence of offset in thickness. And then it modifies the reference of the thickness. The modification value hm i (z) is calculated as 134
Fig. 5. Time Chart of the Disturbance and its Estimation Value by an Observer kTs ref ˆ i − hi | < 5µm (h − hi ) if |h hm (z) = . z−1 i i z −1 hm ˆ i − hi | ≥ 5µm if |h i
where, k is integral gain, z is discrete time operator, and ˆ i is the estimation value of the thickness. h
The control system is shown in Fig. 8. Note that in designing the controller, the reference of the each of the stands is introduced to the state vector of the augmented model given by (26). The control results are illustrated in Fig. 9 and Fig. 10. The logic works well and tracking performance is improved. REFERENCES Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E. (2000). The explicit solution of model predictive control via multiparametric quadratic programming. In American Control Conference, 872–876. Chicago, USA.
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Fig. 6. Time Chart of the Outputs and Inputs in the case of Acceleration
131
Fig. 9. Time Chart of the Disturbance and its Estimation Value in case of Acceleration with Integral Logic
Fig. 10. Modified Reference of the Thickness
Fig. 7. Time Chart of the Disturbance and its Estimation Value in the case of Acceleration
Fig. 8. The Control System with Integral Logic
135
Herceg, M., Kvasnica, M., Jones, C., and Morari, M. (2013). Multi-Parametric Toolbox 3.0. In Proc. of the European Control Conference, 502–510. Z¨ urich, Switzerland. URL http://control.ee.ethz.ch/~mpt. Lofberg, J. (2004). Yalmip : A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference. Taipei, Taiwan. URL http:// users.isy.liu.se/johanl/yalmip. Misaka, Y. (1967). Control equations for cold tandem mills. Journal of Japan Society for Technology of Plasticity, 8(75), 188–200. Ozaki, K., Ohtsuka, T., Fujimoto, K., Kitamura, A., and Nakayama, M. (2009). Nonlinear receding horizon control of a tandem cold millin acceleration and deceleration conditions. ICCAS-SICE 2009, 2158–2163.
ScienceDirect
IFAC-PapersOnLine 49-20 (2016) 126–131
Explicit Model Predictive Controller Explicit Model Predictive Controller Explicit Model Predictive Controller Explicit Model Predictive Controller Design for Thickness and Tension Control Design for Thickness and Tension Control Design for Thickness and Tension Control Design forinThickness and Tension Control a Cold Rolling Mill in a Cold Rolling Mill in a Cold Rolling Mill in a Cold Rolling ∗∗Mill ∗ ∗∗∗
Tomoyoshi Ogasahara ∗ Morten Hovd ∗∗ Kazuya Asano ∗∗∗ Tomoyoshi Ogasahara ∗ ∗ Morten ∗∗ Kazuya ∗∗∗ Tomoyoshi Ogasahara Morten Hovd Hovd ∗∗ Kazuya Asano Asano ∗∗∗ Tomoyoshi Ogasahara Tomoyoshi Ogasahara ∗ Morten Morten Hovd Hovd ∗∗ Kazuya Kazuya Asano Asano ∗∗∗ ∗ ∗ Instrument and Control Engineering Research Dept., Steel Research ∗ Instrument and Control Engineering Research Dept., Steel Research ∗ Instrument and Control Engineering Research Dept., Steel Research ∗ Instrument Engineering Research Steel Research Lab., JFE and SteelControl Corp., Fukuyama, Hiroshima, 721-8510, Instrument and Control Engineering Research Dept., Dept., Steel Japan Research Lab., Corp., Fukuyama, Lab., JFE JFE Steel Steel Corp.,[email protected]). Fukuyama, Hiroshima, Hiroshima, 721-8510, 721-8510, Japan Japan Lab., JFE Steel Corp., Fukuyama, Hiroshima, 721-8510, Japan (e-mail: Lab., JFE Steel Corp., Fukuyama, Hiroshima, 721-8510, Japan (e-mail: [email protected]). ∗∗ (e-mail: [email protected]). (e-mail: [email protected]). Engineering Cybernetics, Norwegian University of ∗∗ Department of (e-mail: [email protected]). of Engineering Cybernetics, Norwegian University of ∗∗ ∗∗ Department Department of Engineering Cybernetics, Norwegian University of ∗∗ Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway Science and and Technology, N-7491 Trondheim, Trondheim, Norway Norway Science Technology, N-7491 (e-mail: [email protected]). Science and Technology, N-7491 Trondheim, Norway (e-mail: [email protected]). ∗∗∗ (e-mail: [email protected]). (e-mail: [email protected]). Lab., JFE Steel Corp., Kawasaki, Kanagawa, ∗∗∗ Steel Research (e-mail: [email protected]). Research Lab., JFE Steel Corp., Kawasaki, Kanagawa, ∗∗∗ ∗∗∗ Steel Research Lab., JFE Steel Corp., Kawasaki, Kanagawa, ∗∗∗ Steel Steel Research Lab., JFE Steel Corp., Kawasaki, 210-0855, Japan Steel Research Lab., 210-0855, JFE Steel Japan Corp., Kawasaki, Kanagawa, Kanagawa, 210-0855, Japan 210-0855, Japan (e-mail: [email protected]) 210-0855, Japan (e-mail: [email protected]) (e-mail: [email protected]) [email protected]) (e-mail: (e-mail: [email protected]) Abstract: This paper addresses modeling of cold rolling mill and controller design based on Abstract: This This paper paper addresses addresses modeling modeling of of aaa cold cold rolling rolling mill mill and and controller controller design design based based on on Abstract: Abstract: This paper modeling of rolling mill and design based on explicit model control(explicit The control are to track the exit Abstract: Thispredictive paper addresses addresses modelingMPC). of aa cold cold rolling millobjectives and controller controller design based on explicit model predictive control(explicit MPC). The control objectives are to track the exit explicit model predictive control(explicit MPC). The control objectives are to track the exit explicit MPC). The control are to thickness to the reference with high accuracy with minimized strip tension In the explicit model predictive control(explicit MPC). The control objectives objectives are deviation. to track track the the exit thicknessmodel to the thepredictive reference control(explicit with high high accuracy accuracy with minimized strip tension tension deviation. In exit the thickness to reference with with minimized strip deviation. In the thickness to the reference with high accuracy with minimized strip tension deviation. In the simulation, good control is obtained when the disturbance entering the system is small enough. thickness togood the reference with highwhen accuracy with minimized strip tension deviation. In the simulation, control is obtained the disturbance entering the system is small enough. simulation, good control is obtained when the disturbance entering the system is small enough. simulation, good control is when the disturbance the system enough. However, the thickness the offset from reference in case of because the simulation, good controlshows is obtained obtained when the the disturbance entering theacceleration, system is is small small enough. However, the the thickness shows the offset offset from the referenceentering in case case of of acceleration, because the However, thickness shows the from the reference in acceleration, because the However, the thickness shows the offset from the reference in case of acceleration, because the approximation error of the linearized model increases and it is different from the model in However, the thickness shows the offset fromincreases the reference iniscase of acceleration, because the approximation error of the linearized model and it different from the model in approximation error of the linearized model increases and it is different from the model in the approximation error model and it the in design of the controller. In order to compensate for the offset, an additional approximation error of of the the linearized model increases increases andcontrol it is is different different from the model modelintegral in the the design of of the the controller. controller. In linearized order to to compensate compensate for the the control offset, from an additional additional integral design In order for control offset, an integral design of controller. In order to compensate for the offset, an additional integral logic using the estimation of exit thickness is proposed. The validity this approach was design of the the controller. Inerror order to compensate for the control control offset, anof additional integral logic using the estimation error of exit thickness is proposed. The validity of this approach was logic using the estimation estimation error offree exitcontrol thickness isachieved. proposed. The The validity of of this approach approach was logic using the exit thickness proposed. verified by simulation and offset was logic using the estimation error offree exitcontrol thickness isachieved. proposed. The validity validity of this this approach was was verified by simulation simulation and error offsetof was is verified by and offset free control control was was achieved. verified by simulation and offset free achieved. verified by simulation and offset free control was achieved. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: process control, process models, predictive control, optimal control, Keywords: process control, control, process process models, models, predictive predictive control, control, optimal optimal control, control, Keywords: process Keywords: process control, control,system, processsteel models, predictive control, control, optimal optimal control, control, multi-input/multi-output industry. Keywords: process process models, predictive multi-input/multi-output system, steel industry. multi-input/multi-output system, steel industry. multi-input/multi-output system, steel industry. multi-input/multi-output system, steel industry. 1. ware and high computer tends to be high. 1. INTRODUCTION INTRODUCTION ware and and high high performance performance computer tends to be high. 1. ware performance computer tends to to be be high. 1. INTRODUCTION INTRODUCTION ware and high performance computer tends high. Furthermore, the time of solving an optimization problem 1. INTRODUCTION ware and high performance computer tends to be high. Furthermore, the time of solving an optimization problem Furthermore, the time of solving an optimization problem Furthermore, the time of solving an optimization problem possibly exceeds the sampling period. Consequently, the Furthermore, the time of solving an optimization problem possibly exceeds the sampling period. Consequently, the In aa cold strip mill, as shown in Fig. 1, the preceding possibly exceeds the sampling period. Consequently, the possibly exceeds the sampling period. Consequently, the applications are limited. In cold strip mill, as shown in Fig. 1, the preceding possibly exceeds the sampling period. Consequently, the applications are limited. In a cold strip mill, as shown in Fig. 1, the preceding In cold stripreceding mill, as asstrip shown inwelded Fig. 1, 1,together the preceding preceding strip and the are and it applications are are limited. limited. In aa cold strip mill, shown Fig. the strip and the the receding strip arein welded together and it it applications applications are limited. strip and receding strip are welded together and In explicit MPC(Bemporad et al. (2000)), the strip and the receding strip are welded together and it is rolled continuously. The exit thickness of each of the In explicit MPC(Bemporad et al. al. (2000)), (2000)), the the optimization optimization strip andcontinuously. the receding The stripexit are thickness welded together and it In explicit MPC(Bemporad et is rolled of each of the optimization is rolled continuously. The exit thickness of each of the In explicit MPC(Bemporad et al. (2000)), the optimization problem in the MPC for the linear time-invariant system is rolled continuously. The exit thickness of each of the rolling stands should be controlled with high accuracy. It is In explicit MPC(Bemporad et al. (2000)), the optimization problem in the MPC for the linear time-invariant system is rolled continuously. The exit thickness of each of the rolling stands should be controlled with high accuracy. It is problem in the MPC for the linear time-invariant system rolling stands should be controlled with high accuracy. It is problem in the MPC for the linear time-invariant system such as a quadratic programming problem is converted rolling stands should be controlled with high accuracy. It is also necessary to minimize strip tension deviations during in quadratic the MPC programming for the linear time-invariant system such as a problem is converted rolling stands should be controlled with high accuracy. It is problem also necessary to minimize strip tension deviations during such as aa quadratic problem is also necessary to minimize strip tension deviations during such asmulti-parametric quadratic programming programming problem is converted converted into programming problem, and then also necessary to in minimize strip tension deviations during thickness control order prevent rolling trouble such as such a quadratic programming problem is converted into aaaas multi-parametric programming problem, and then then also necessary to minimize strip tension deviations during thickness control in order to to prevent rolling trouble such as into multi-parametric programming problem, and thickness control in order to prevent rolling trouble such as into a multi-parametric programming problem, and then the solution is obtained as a piecewise affine function thickness control in order to prevent rolling trouble such as strip breakage. These variables are controlled by adjusting into a multi-parametric programming problem, and then the solution is obtained as a piecewise affine function thickness control in order to prevent rolling trouble such as the solution is obtained as a piecewise affine function strip breakage. These variables are controlled by adjusting strip breakage. These variables are controlled by adjusting the solution is obtained obtained asin piecewise affine function of the state of the plant offline. Hence it does not strip breakage. These variables are controlled bypoint, adjusting the roll gap and the roll speed. At the welding the solution is piecewise affine function of the the state of of the plant plant as in aaoffline. offline. Hence it does does not strip breakage. variables by adjusting the roll roll gap and andThese the roll roll speed.are At controlled the welding welding point, the the of state the in Hence it not the gap the speed. At the point, the of the state of the plant in offline. Hence it does not require online optimization and can potentially reduce the roll gap and the roll speed. At the welding point, the roll gap and the roll speed are synchronized to achieve the state ofoptimization the plant inand offline. Hence it does not require online can potentially reduce the roll gap and the roll speed. At the welding to point, the of roll gap and the roll speed are synchronized achieve require online and can reduce roll gap roll speed are to achieve require online optimization optimization and can potentially potentially reduce the implementation cost and overcome the limitation of roll gap and and the the roll These speed actions are synchronized synchronized to by achieve the thickness change. are executed feed online optimization can potentially reduce the implementation implementation cost and andand overcome the limitation limitation of roll gap and the roll speed are synchronized to achieve the thickness thickness change. These actions are executed executed by feed require the cost overcome the of the change. These actions are by feed the implementation cost and overcome the limitation of normal MPC. Therefore, explicit MPC is applied in this the thickness change. These actions are are executed by feed the forward control. When the welding point is not passing implementation cost and overcome theapplied limitation of normal MPC. Therefore, Therefore, explicit MPC is is in this the thickness change. These actions executed by feed forward control. When the welding point is not passing normal MPC. explicit MPC applied in this forward control. When the welding point is not passing normal MPC. Therefore, Therefore, explicit explicit MPC MPC is is applied applied in paper. forward control. When the control welding is point isfor notthickness passing normal through the mill, feedback used in this this paper. MPC. forward the welding not passing through control. the mill, mill,When feedback control ispoint usedis for thickness paper. through the feedback control is used for thickness paper. through the mill, feedback control is used for thickness and tension control. through the control. mill, feedback control is used for thickness paper. The outline of this paper is as follows. In section 2, the and tension The outline outline of of this this paper paper is is as as follows. follows. In In section section 2, 2, the the and and tension tension control. control. The and tension control. The outline of this paper is as follows. In section 2, the state space model of a cold rolling mill is presented. In secThis paper deals with thickness and tension control probThe outline of this paper is as follows. In section In 2, secthe state space model of a cold rolling mill is presented. This paper deals with thickness and tension control probspace model aa cold mill is In secThis paper with thickness and tension control probstate space model of of cold rolling rolling mill is presented. presented. In section 3, the design procedure of the explicit MPC is given. This paper deals deals withoperation thickness for and two tension controlrolling prob- state lem during normal adjacent state model of a cold rolling mill is presented. In section 3, 3,space the design design procedure of the explicit MPC is given. This paper deals with thickness and tension control problem during during normal operation for two adjacent rolling tion the procedure of the explicit MPC is given. lem normal operation for two adjacent rolling tion 3, the design procedure of the explicit MPC is given. Next, the simulation results and controller modifications lem during normal operation for two adjacent rolling stands in a rolling mill, as depicted in Fig. 2. tion 3, the design procedure of the explicit MPC is given. Next, the simulation results and controller modifications lem during normalmill, operation for in two adjacent rolling Next, the simulation results and controller modifications stands in a rolling rolling as depicted Fig. 2. stands Next, the simulation simulation resultsperformance and controller controller modifications for improving the tracking of the thickness stands in in a rolling mill, mill, as as depicted depicted in in Fig. Fig. 2. 2. Next, the results and for improving improving the tracking tracking performance of modifications the thickness thickness stands in aa rolling mill, as depicted in Fig. 2. Model Predictive Control (MPC) is well known for its for the performance of the for improving the tracking performance of the thickness are shown in section 4. Model Predictive Control (MPC) is well known for its for improving the tracking performance of the thickness are shown in section 4. Model Predictive Control (MPC) is well known for its Model Predictive Control (MPC)to isaccount well known known forconits are control performance and ability for the are shown shown in in section section 4. 4. Model Control (MPC) well its control Predictive performance and ability ability to is account for the thefor conare shown in section 4. control performance and to account for concontrol performance and ability to account for the constraints of the plant. Therefore it has been regarded as control performance and ability to account for the constraints of the plant. Therefore it has been regarded as straints of plant. Therefore it been regarded as straints of the the plant.and Therefore it has has in been regarded as a practical solution is widespread the process instraints of the plant. Therefore it has been as practical solution and is widespread widespread in theregarded process ininaaa practical solution and is in the process practical solution and is widespread in the process industry. For example, an application of nonlinear MPC to adustry. practical is widespread in the process For solution example,and an application application of nonlinear nonlinear MPC into 2. DYNAMICAL MODEL OF A ROLLING MILL dustry. For example, an of MPC to 2. DYNAMICAL DYNAMICAL MODEL MODEL OF OF A A ROLLING ROLLING MILL MILL dustry. For example, an application of nonlinear MPC to a cold rolling mill is reported by Ozaki et al. (2009) and dustry. For example, an application of nonlinear MPCand to 2. a cold rolling mill is reported by Ozaki et al. (2009) 2. DYNAMICAL MODEL OF A ROLLING MILL aait cold rolling mill is reported by Ozaki et al. (2009) and 2. DYNAMICAL MODEL OF A ROLLING MILL cold rolling mill is reported by Ozaki et al. (2009) and shows good results under acceleration and deceleration ait cold rolling mill is reported by Ozaki et al. (2009) and shows good results results under under acceleration acceleration and and deceleration it shows it shows good good results it under acceleration and deceleration deceleration conditions. However requires online optimization and In this section, nonlinear dynamical model of cold it shows good results acceleration and deceleration conditions. However itunder requires online optimization optimization and In In this this section, section, aaa nonlinear nonlinear dynamical dynamical model model of of aa a cold cold conditions. However it requires online and conditions. However it requires online optimization and In this section, a nonlinear dynamical model of a cold the implementation cost related to the optimization softrolling mill and its linear state space model are described. conditions. Howevercost it requires online optimization softand In this mill section, a nonlinear dynamical model of a cold the implementation related to the optimization rolling and its linear state space model are described. the the implementation implementation cost cost related related to to the the optimization optimization softsoft- rolling rolling mill mill and and its its linear linear state state space space model model are are described. described. the implementation cost related to the optimization softrolling mill and its linear state space model are described. Copyright © 2016, 2016 IFAC 130Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 130 Copyright © 2016 IFAC 130 Copyright © 2016 IFAC 130 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 130Control. 10.1016/j.ifacol.2016.10.108
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127
• Thickness Model hi = Si + • Transport Delay Model Hi+1 = hi exp −
Fig. 1. Layout of a Continuous Cold Rolling Mill
• Roll Speed Model
Vi (s) =
Pi . Mi
L s . (1 + fi )Vi
(6)
(7)
1 V ref . Tv s + 1 i
(8)
1 S ref . Ts s + 1 i
(9)
• Roll Gap Model Si (s) =
Fig. 2. Adjacent Rolling Stands in a Rolling Mill 2.1 Rolling Models and Actuator Models The model consists of rolling models and actuator models. Each model is shown below. Here, the variables in the models are presented in Table 1. Note that the suffix i denotes the i th stand. • Rolling Force Model Pi =w(ki − 0.7σi−1 − 0.3σi ) Ri × 1.08 + 1.79ri µi − 1.02ri Hi × R (Hi − hi ) × 1000. where, 16(1 − ν 2 ) Pi , Ri = Ri 1 + Es π w(Hi − hi ) Hi − h i ri = . Hi • Forward Slip Model 1 Hi − hi 2 −1 tan fi = tan 2 hi 1 Hi ki−1 (1 − σi ) − . log 4µi hi ki (1 − σi−1 )
(1)
The rolling force model given in (1) and forward slip model given in (2) are reported by Misaka (1967). The friction coefficient between strip and work roll is dependent on the roll speed and it is modeled as (3). The deformation resistance model given in (4) is based on the effectiveness of strain hardening. Strip tension results from the elastic deformation of the strip. Therefore, it is proportional to the difference of the strip velocity between the exit of i th stand and the entry of i + 1 th stand. It is modeled as (5). The thickness model shown in (6) indicates that it is determined by the roll gap Si and elastic deformation of the work roll, which is represented as Pi /Mi . The exit thickness of the i th stand is transported to the i + 1 th stand. Thus this system includes time delay. This relationship is represented in (7). The actuators of the mill are electric motors and hydraulic cylinders. The former are for controlling the roll speed, and latter for controlling the roll gap position. The motors are controlled by the Automatic Speed Regulator(ASR) and the hydraulic cylinders are controlled by the position control system. Thanks to these minor feedback control, these actuators can be modeled as first order transfer function described in (8) and (9). 2.2 Linear State Space Model
(2)
• Friction Model µi = µa exp(−µb × Vi ) + µc . (3) • Deformation Resistance Model n H0 ki = 1.15l m + 1.15 log . (4) hi • Strip Tension Model Es hi+1 dσi = (1 + fi+1 )Vi+1 − (1 + fi )Vi . (5) dt L Hi+1 131
The linear state space model is derived from the nonlinear model shown above. First, total derivative of the thickness model is expressed as (10) ∆hi = ∆Si + ∆Pi /Mi . where, ∂Pi ∂Pi ∆hi + ∆Hi ∆Pi = ∂hi ∂Hi (11) ∂Pi ∂Pi ∂Pi + ∆σi−1 + ∆σi + ∆Vi . ∂σi−1 ∂σi ∂Vi By substituting (11) for (10), the thickness model is linearized as
∆hi =
1 ∂Pi ∆Hi Mi ∆Si + Mi + Q i ∂Hi ∂Pi ∂Pi ∂Pi + ∆σi−1 + ∆σi + ∆Vi . ∂σi−1 ∂σi ∂Vi (12)
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exp −
where, Qi = −
∂Pi . ∂hi
It is assumed that the entry tension of i th stand (σi−1 ) and the exit tension of i + 1 th stand (σi+1 ) is constant. Therefore, the exit thickness of each of the stands is represented as follows. • Exit thickness of i th stand ∆hi =
1 ∂Pi ∆Hi Mi ∆Si + Mi + Q i ∂Hi ∂Pi ∂Pi + ∆σi + ∆Vi + dhi . ∂σi ∂Vi
(13)
• Exit thickness of i + 1 th stand 1 ∂Pi+1 ∆hi+1 = ∆Hi+1 Mi+1 ∆Si+1 + Mi+1 + Qi+1 ∂H i+1 ∂Pi+1 ∂Pi+1 ∆σi + ∆Vi+1 + dhi+1 . + ∂σi ∂Vi+1 (14) where, dhi and dhi+1 are unmeasurable output disturbance. • Entry thickness of i + 1 th stand Next, the transport delay model shown as (7) is expressed as a 3rd order Pad´e approximation function represented as Table 1. Parameters in the Model Symbol
Unit
Pi w ki σi Ri Ri Hi hi hi+1 ri Es ν µi µa µb µc fi l m n H0 L Vi Viref Si Siref s Tv Ts Mi
kN m MPa MPa m m m m m MPa m m m/s m/s m mm s s kN/m
Description Rolling force Strip Width Deformation resistance Strip tension between stands Work roll radius Flattened work roll radius Entry thickness Exit thickness of i th stand Exit thickness of i + 1 th stand Reduction rate Young’s modulus of strip Poisson’s ratio of work roll Friction coefficient Parameter in friction model Parameter in friction model Parameter in friction model Forward slip ratio Parameter of ki Parameter of ki Parameter of ki Thickness before rolling Length between stands Roll speed Reference of roll speed Roll gap Reference of roll gap Laplace operator Time constant of ASR Time constant of roll gapr Mill modulus
Value 1160×10−3 − 346 180×10−3 0.75×10−3 0.488×10−3 0.285×10−3 205800 0.3 0.0303 0.1974 0.024 100.3 0.017 0.164 1.5×10−3 6.0 1/60 0.01 500
132
b 2 s 2 + b1 s + b0 + d0 . + a2 s2 + a1 s + a0 (15) where, a0 , a1 , a2 , b0 , b1 , b2 , and d0 are parameters which are dependent on time delay. By using (15), the state space representation of it is obtained as 0 1 0 x1 0 d x1 0 1 x2 = 0 x2 + 0 ∆hi , dt x −a0 −a1 −a2 x3 1 3 x1 ∆Hi+1 = [b0 b1 b2 ] x2 + d0 ∆hi . (16) x3 L s (1 + fi )Vi
≈
s3
where, x1 , x2 , and x3 , are the state variables in the Pad´e approximation. Then by substituting (13) into (16), ∆Hi+1 can be expressed by the state variables of ∆Vi , ∆Si , ∆σi , x1 , x2 , x3 , ∆Hi , and dhi . • Tension model In order to linearize the tension model, the total derivative of (5) is calculated as Es hi+1 dσi ∆ = ∆ (1 + fi+1 )Vi+1 − (1 + fi )Vi . dt L Hi+1 d∆σi Es 1 ↔ = (1 + fi+1 )Vi+1 ∆hi+1 dt L Hi+1 hi+1 2 (1 + fi+1 )Vi+1 ∆Hi+1 Hi+1 hi+1 + Vi+1 ∆fi+1 Hi+1 hi+1 + (1 + fi+1 )∆Vi+1 Hi+1 −
− Vi ∆fi − (1 + fi )∆Vi
+ vT .
(17) where, vT is unmeasurable tension disturbance. In (17), ∆fi and ∆fi+1 need to be expanded. Now, it is assumed that the output disturbance of each of the stands does not affect the forward slip ratio. Then ∆fi and ∆fi+1 are expanded as ∂fi ∂fi ∂fi ∂fi ∆Vi + ∆σi + ∆Hi + (∆hi −dhi ), ∆fi = ∂Vi ∂σi ∂Hi ∂hi (18) and ∂fi+1 ∂fi+1 ∆fi+1 = ∆Vi+1 + ∆σi ∂Vi+1 ∂σi ∂fi+1 ∂fi+1 + ∆Hi+1 + (∆hi+1 − dhi+1 ). ∂Hi+1 ∂hi+1 (19) Here, ∆hi in (18) and ∆hi+1 in (19) correspond to (13) and (14) respectively, and ∆Hi+1 in (19) is given by (16). • Roll speed model The reference of the roll speed of ith stand is dependent on the reference of the downstream stand to minimize the imbalance of the mass flow. Hence Viref is defined as the additional term of the reference of i + 1 th stand. Thereby, the dynamics of the roll speed of i th stand is represented as
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href i+1 (1 + fi+1 ) ref Vi+1 . + href i (1 + fi ) (20) ref where, href are the reference of the exit and h i+1 i thickness. fi means the nominal value of forward slip ratio calculated by the set value. The roll speed of i + 1 th stand is represented as 1 1 ref d∆Vi+1 = − ∆Vi+1 + V . (21) dt Tv Tv i+1 • Roll gap The roll gap model given by (9) is converted to the state space model realized as 1 1 d∆Si = − ∆Si + Siref , (22) dt Ts Ts and d∆Si+1 1 1 ref = − ∆Si+1 + Si+1 . (23) dt Ts Ts • Rolling force The folling force satisfies (10). Therefore it can be calculated by following equation. ∆Pi = M (∆hi − ∆Si ). (24) Here, ∆hi corresponds to (13). 1 1 d∆Vi = − ∆Vi + dt Tv Tv
Viref
In this system, the measurable variables are roll speed, roll gap, strip tension, exit thickness, and rolling force.
Ax
+Bu + Ed,
Cx
+F d.
(25)
where, A, B, C, E, and F are matrix with proper size and T
x = [∆Vi ∆Vi+1 ∆Si ∆Si+1 ∆σi x1 x2 x3 ] , ref T , u = Viref Siref Si+1 T ref d = Vi+1 ∆Hi vT dhi dhi+1 ,
At each timestep, the MPC solves an optimization problem. The decision variables in the optimization problem are the future inputs, while the objective function reflects the predicted future bahaviour of the system. The first element of the calculated future inputs is applied, and the optimization is executed anew for each timestep. The objective function is shown below. N −1 min (˜ x(k) − x ˜ref )T Q(˜ x(k) − x ˜ref ) ∆u(0),··· ,∆u(N −1)
k=0
xT uT
T
+ ∆u(k)T R∆u(k) .
, Q ≥ 0, and R > 0. N means T prediction horizon. x ˜ = (xref )T (uref )T is the target state. This objective function is minimized subject to the upper and lower constraints as below and the model in (26).
where, x ˜ =
ref
xmin ≤ x(k) ≤ xmax ,
umin ≤ u(k) ≤ umax ,
∆umin ≤ ∆u(k) ≤ ∆umax ,
ymin ≤ y(k) ≤ ymax .
Thus, this MPC problem is a quadratic programming because the objective function is quadratic and the constraints are linear function. Bemporad et al. (2000) show that it can be expressed as a multi parametric quadratic programming (mp-QP) and the solution is organized as a piecewise affine function parametrized by the state. Therefore the solution can be calculated explicitly and on-line optimization is not required. This algorithm is referred to as explicit MPC.
Finally, the state space representation is obtained by integrating (13), (14), (16), (17), (20), (21), (22), (23), and (24) and summarized as below. dx = dt y=
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T
For achieving offset free control in the case of disturbance entering the system, the effects of the disturbance need to be counteracted. At the steady state, the system given as (26) satisfies the following relationships because xref = x(k + 1) = x(k), uref = u(k + 1) = u(k), and ∆u(k) = 0. Ed d I − Ad Bd xref = C O uref y ref − F d
y = [∆Vi ∆Vi+1 ∆Si ∆Si+1 ∆σi ∆hi ∆hi+1 ∆Pi ∆Pi+1 ]. Therefore if we specify some of the reference of the output and state variables, the target of xref and uref which achieve offset free control can be calculated. In this case, 3. EXPLICIT MPC DESIGN ref the disturbance except for Vi+1 is unmeasurable. Hence, In this section, the MPC formulation is considered. The they should be estimated by a observer. The objective here is to track the exit thickness with minimized tension discretized system of (25) is given by deviation. Therefore, ∆hi , ∆hi+1 , and ∆σi are set to 0. The unspecified states are calculated by solving below. x(k + 1) = Ad x(k) +Bd u(k) + Ed d(k), xt E d B A d d y(k) = Cx(k) +F d(k). = −F d C O uref Then the system is augmented by introducing the change where, ref T = u(k + rate of input(∆u(k) = ∆Viref ∆Siref ∆Si+1 T 1) − u(k)). Then the system is represented as xt = [∆Vi ∆Vi+1 ∆Si ∆Si+1 x1 x2 x3 ] ,
Ad Bd Ed x(k) O I O u(k) O O I d(k) x(k) y(k) = [C O F ] u(k) . d(k)
x(k + 1) u(k + 1) = d(k + 1)
O + I ∆u(t), O (26)
A : After elimination of 5th column of I − A
C , F : After elimination of 5th column and
extraction of 6th and 7th row of C and F The block diagram of the control system is illustrated in Fig. 3.
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Fig. 3. Block Diagram of the Control System 4. SIMULATION The simulation results are presented here. It is executed R with the Multi-Parametric Toolby utilizing MATLAB box 3.0 developed by Herceg et al. (2013) and YALMIP developed by Lofberg (2004). The rolling condition of the simulation is summarized in Table 1. In addition to this, the initial roll speed of i + 1 th stand is 5[m/s]. The sampling period Ts is 10ms and prediction horizon N is set to 2. The constraints are as follows.
Fig. 4. Time Chart of the Outputs and Inputs
ref ≤ 1.0, −1.0 ≤ ∆Siref , ∆Si+1
ref /Ts ≤ 1.0, −1.0 ≤ ∆Siref /Ts , ∆Si+1
−1.0 ≤ Viref ≤ 1.0,
−0.1 ≤ ∆Viref /Ts ≤ 0.1.
Subsequently, the weight matrices in the objective function is set as below . Q = diag(1, 1, 1000, 1000, 1, 1000, 1000, 1000, 1, 1000, 1000), R = diag(0, 0, 0). As a result of explicit MPC design, the number of polyhedral partition of the feedback gain is 352. The control results are depicted in Fig. 4 and the disturbance and its estimated value by an observer is illustrated in Fig. 5. Here, the disturbance enters into the entry thickness of i th stand, the strip tension, the exit thickness of each of the stands, and the roll speed of i + 1 th stand. In this case, the disturbance can be estimated precisely and the thickness and the tension are well controlled to the reference. On the other hand, results when accelerating the roll speed depicted in Fig. 6 show offset in the thickness. The estimation accuracy of the disturbance depicted in Fig. 7 gets worse in this case. This is because the system is dependent on the time delay between stands, however, the observer utilizes the linearized model around the initial roll speed. Therefore the approximation error in the linearized model becomes significant. In order to achieve offset free control, an integral logic is introduced. It is based on the fact that the estimation error of the thickness is nearly 0 at the steady state in spite of the existence of offset in thickness. And then it modifies the reference of the thickness. The modification value hm i (z) is calculated as 134
Fig. 5. Time Chart of the Disturbance and its Estimation Value by an Observer kTs ref ˆ i − hi | < 5µm (h − hi ) if |h hm (z) = . z−1 i i z −1 hm ˆ i − hi | ≥ 5µm if |h i
where, k is integral gain, z is discrete time operator, and ˆ i is the estimation value of the thickness. h
The control system is shown in Fig. 8. Note that in designing the controller, the reference of the each of the stands is introduced to the state vector of the augmented model given by (26). The control results are illustrated in Fig. 9 and Fig. 10. The logic works well and tracking performance is improved. REFERENCES Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E. (2000). The explicit solution of model predictive control via multiparametric quadratic programming. In American Control Conference, 872–876. Chicago, USA.
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Fig. 6. Time Chart of the Outputs and Inputs in the case of Acceleration
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Fig. 9. Time Chart of the Disturbance and its Estimation Value in case of Acceleration with Integral Logic
Fig. 10. Modified Reference of the Thickness
Fig. 7. Time Chart of the Disturbance and its Estimation Value in the case of Acceleration
Fig. 8. The Control System with Integral Logic
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Herceg, M., Kvasnica, M., Jones, C., and Morari, M. (2013). Multi-Parametric Toolbox 3.0. In Proc. of the European Control Conference, 502–510. Z¨ urich, Switzerland. URL http://control.ee.ethz.ch/~mpt. Lofberg, J. (2004). Yalmip : A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference. Taipei, Taiwan. URL http:// users.isy.liu.se/johanl/yalmip. Misaka, Y. (1967). Control equations for cold tandem mills. Journal of Japan Society for Technology of Plasticity, 8(75), 188–200. Ozaki, K., Ohtsuka, T., Fujimoto, K., Kitamura, A., and Nakayama, M. (2009). Nonlinear receding horizon control of a tandem cold millin acceleration and deceleration conditions. ICCAS-SICE 2009, 2158–2163.