adaptive coordinated control for hot strip finishing mills

8/6/2019 Adaptive Coordinated Control for Hot Strip Finishing Mills

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JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2011, 18(4): 36-43

Adaptive Coordinated Control for Hot Strip Finishing M illsJ IAO Xiao-hong' , S H A O Li-ping' , PENG Yan'

(1. Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, Hebei, China;2. Institute of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, Hebei, China)

Abstract: To further improve the control accuracy for s tri p dimension of hot s tri p mills, an adaptive control scheme isinvestigated for a hot strip finishing mill based on the decentralization and coordination among the controllers ofgauge, tension and looper. Consequently, the adaptive controller designed can regulate simultaneously the st rip exitthickness, the st rip tension and the looper angle to ensure bette r performance of the str ip quality of finishing mills.Moreover, the control scheme is proposed in considerat ion of the essential nonlinearity and the unavoidable frictionphenomena in the mechanical system, so the controller can be efficient in a wider range of working situations. Th esimula tion resu lts of a model obtained from a real hot str ip finishing mill show the effectiveness of the proposed con-trol scheme in comparison with the conventional control method.Key words: hot stri p finishing mills; exist gauge; tension; looper angle; adaptive control

A hot strip mill is composed of a roughing mill, afinishing mill and a coileg']. In the finishing rollingprocess which extremely influences product quality,control is accomplished to ensure that the productmeets its required dimensions at the mill exit andthat the strip is rolled in a sta ble mannerC2'. Th eschematic of two st and s in the finishing mill and theinter-stand gap can be seen in Ref. [Z]. A looper in-stalled at inter-stand can control the strip tensionand the mass flow of t he two stands. Th e mass flowout of t he upstream and into the downstream standsis controlled to avoid any mass flow mismatch anddisturbance to the inter-stand strip tension. To avoid abuild up or reduction in the length of s tri p betweenthe stands, the speed of the upstream stand is mod-ulated to maintain the looper at a reference angle.The looper pivot torque is also modulated accordingto the measured looper angle to keep the strip ten-sion at it s se tup reference. The stand gap is modulatedto keep the exit gauge (thickness) at its setup reference.Conventionally, each of exit thickness, tension andlooper angle is controlled independentlyC3'. However, itshould be noted that the system is highly interactive orcoupled, there exi sts a mutua l interaction among thestrip gauge, st rip tension and looper angle. Henc e,many multivariable control techniques have been

proposed for the finishing mill to consider the inter-action between the looper angle and the strip ten-s ion, s im u l t ane~u s ly~"~~ .ut in fact, when conven-tional automatic gauge control ( AGC) works, themass flow between stands is changed, and thenlooper angle and stri p tension are disturbed. There-fore, the multivariable control, which takes into ac-count the interactions among gauge, tension andmass flow, looks more effective such as Ref. [S ] andRef. [ l o ] . However, these methods are based onthe linearization of system at the operation point andthe neglect of nonlinear character istic of sys tem.

Motivated by th e analysis above, under the con-sideration of th e essenti al nonlinearity of system andthe presence of t he unavoidable friction phenomena ,an adaptive control scheme is investigated for a hotstrip finishing mill. The control scheme proposedcan realize decentralization and coo rdination amongthe controlle rs of gauge , tension and looper. Conse-quently, the adaptive controller designed can regu-late simultaneously the strip exit thickness, the striptension and the looper angle to ensure better coordi-nation among thickness, tension and mass flow.Moreover, the control design is based on the essen-tial nonlinearity model of syst em and t he unavoida-ble friction phenomena in the mechanical system is

Foundation Item: I tem Sponsored by Nat ional Natural Science Foundat ion of China ( 60774018)Biography:JIAO Xiao- hong( l966-) , Female , Doc to r , P r o f esso r ; E-mail: j i a o x h a y s u . e d u. cn ; Received Date: April 18 , 2010


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Issue 4 Adaptive Coordinated Control for Hot Strip Finishing Mills ' 37 '

under consideration, so the controller can be effi-cient in a wider range of working situa tions.1 Modeling Hot Strip Finishing Mill

In Fig. 1 , geometrical quanti ties of t he looperand inter-st and are defined for a pair of consecutiverolling standsL8'. In the following subsections, thedynamic models of each part of the system will bedeveloped.

Fig. 1 Looper and interstand geometry1 . 1 Looper dynamics

The looper dynamics is modeled as follows.. .J 8 =Tu- f w, (1)where, 8 denotes the looper angle; J is the inertia;ru denotes the torque applied to the looper; w, epre-sents the unmodeled phenomena such as the torquenecessary for bending of t he st rip over the looperroll; and q o a d denotes the total load acting on thelooper , which is composed of thre e main compo-

( 2 )The torque r,, by strip tension o (for a strip of

width B and thickness h ) is modeled as:r ,=oBhl[~in(B+8~)-sin(8-8~ >I ( 3 )

where 81 and 82 are evaluated as:11 =arctan 1 lsinsfr-d]f l c o d , 82 =arctan[ L -a - codlsin8f r- diwhere, 1 , r , a , d , L are defined as looper length,

looper roll radius, the distance between looper andstand, the height from actuator shaft to stand shaft,the distance between two stands respectively. Theyare shown in Fig. 1.

The torque rs by strip weight (for a strip densi-ty p> is :

(4)where 1, (81, l Z 8 ) denote the loop length betweenthe ith stand, the ( i f 1 ) th stand and the looperroll, respectively

r, =pghB[i , (0 ) +iz (~)]icOse

1, ( 8 ) = 2 / ( l ~ i n 8 f r - - d ) ~ f ( a f l c o s 8 > ~1, ( 8 ) = / ( l ~ i n 8 f r - d ) ~ f ( L - a - l c 0 s 8 ) ~

The torque rl, by mass of the looper arm is:

r,,= M , +-Ma glcos0 (5)I : Iwhere M , and Maare mass of the looper roll and armrespectively.

The torque ru is provided through a hydrauliccylinder controlled by a servo valve, for simplicity,which can be represented as a first-order system.

(rU- ,Fr )f ,u = - - ( 6 )where, T , - , F r , 1, and u, represent the time con-st an t, friction force occurring in hydraulic valve, ac-tuator arm in electro-hydraulic servo and control sig-nal , respectively. The friction force is repres entedas the well-known LuGre :

dz

1Ti,

( 7 )where, oo is the stiffness coefficient; o1 is the damp-ing coefficient; 0 , represents the viscous frictionterm; and u is the relative velocity between the sur-face, namely, the servo valve spool motion velocity.From the schematic geometry in Fig. 1, it follows u=(1fr)BsinB. z denotes the average deflection of thebristles and is modeled as :

Fr = g o z+ 1 - z udt

( 8 )

where, the arbitrary steady-state behavior g ( u ) > Omodels the Stribec k effect with a Gaussia n function;F , is the dynamical friction; F , is the static friction;and us is the Stribeck velocity.

1. 2 Strip tensionTh e strip tension o is approximately proportion-

al to both the Young's Modulus E and the stripstretc h according to the following equationE8':

[L i ( 8 ) l z (8)-L-f(t>] ( 9 )"(")=L+E(t)where f ( t ) denotes the deviation of the inter-standstrip length with respect to L and it s dynamic behav-ior is described by the differential equation:i(t>=v'(t>-v+'(t)fw;(t) ( 1 0 )

where v ' ( t ) is the exit speed from the i th stand andV'+' ( t> he entry speed to the ( i f l ) th stand; w:(t>denotes unmodeled dynamics phenomena like, forinstance, the effect of the temperature, the plasticdeformation of th e steel strip , the uncertainty of thegeometric quantities, etc. The relation between v' ( t )and work roll angular speed W* ( t ) and the relationbetween V t l ( t ) and w'" ( t ) can be defined as:

(11)L t )= 1f f" W' ( t )R ',V + ' ( t ) = ( l - y ) w ' + ' (t)R'+'


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38 * Journal of Iron and Steel Research. International- Vol. 18where, f v is the forward slip; and p is the backwardslip.

F ur th er mo re , a s is ~ s u a l [ ~ * * ~ ,t should be notedthat:1) f ( t ) can be assumed to be very small in nor-mal operation as compared with L , and consequently,it can often be simplified assuming L+f( t )=L.

2 ) w '+ l ( t ) can be assumed to be invariable. Butf and 9 would be changed by following the strip ten-sion. Henc e, t heir influence on the strip tension isrepresented as a term with the uncertain parameter.

3) There is the following relative rep rese ntat ion:

Moreover, the dynamics of the work roll angularspeed w ( t ) can be described by a first-order system:

(13)where, T, is the time constant; and u, is the controlsignal.

1'w ( t >= - w ( t > +U,T,

1 .3From Fig. 2 , it can be obtained that h ( t ) So +2x,

with exit thickness of the strip h ( t ) , the work rolldisplacement x , ( t ) and the initial roll gap S o . By theNewton's second law, there exists

Stand model and exit thickness

Fig. 2 Single stand and gauge geometry

( 1 4 )where f ( t ) is the stress force provided by the hy-draulic cylinder; c , is the damping coefficient; w, e-notes unmodeled phenomena such as the gravity ofthe work roll, backup roll and mill housing; and P ( t )is the rolling force and can be approximately evalua-ted according to the following equation:

..m, = ~ ( t )c s & -f(t> f w ,

P ( t ) = JR [H ( t ) - h ( t ) ]B (15)where Y,s the constrained yield stress; Q is a cor-recting coefficient; 1 / R L H ( t ) h ( t ) J is the contactarc between the work rolls and the strip; and Z ( t ) is

the entry strip tension.

ing first-order system with the time constant TfMoreover, f ( t ) can be described as the follow-

( 1 6 )Tif ( t )= - f ( t > + U f2 Control Problem Formulation

From the analysis above, the dynamic descriptionof t he finishing mill can be represented as follows:

c , 0 . 2 1- h - - f f - w ,m, m, ms( 1 7 )

Lsidfr-d) -sin(e-L--a-LcosBhere f l ( 8 )=L[sin(B+arctan


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I ssue 4 Adapt ive Coordinated Cont rol for Hot Strip Finishing Mills * 39

P = O

1 E 1J L msw1= -w, w, -we, w3 =-ws.

-P11 PI, 0 0 0 -Pl, P,, 0 0 00 0 0 P,4 P,,

0 1 0 0 ( 2 1 )- 0 0 0 P,5 P,,

Th e parameters pl , p 2 , p3 are unknown due tothe uncertain parameters E , Y , z, , of the finishingmill, and wl , w 2 , 3 represent the external disturb-ances. Moreover, it should be noted that the re ex-ists the friction force, while the stat e z of the dy-namic model of friction force is unmeasurable.

Theref ore, the design specifications of control-ler for the finishing mills are classified as follows:

1 ) Stable mill operation is ensured, i. e. thestability of the closed-loop system is guaranteed un-der interactions, parameter uncertainties and exter-nal disturbances.

2 ) Proper strip quality, namely proper closed-loop performance is obtained by the decentralizationand coordination for th e control loops of looper an-gle, tension and gauge, which means that the striptension, the looper angle and the exit thickness canbe regulated promptly to their desired values.

3 ) The friction force affecting the behaviormode of the looper mechanisms should be controlledto avoid looper's stick-slip motion.

Based on the design specifications above, an adap-tive control problem is formulated as follows:

For the Eqn. (181, it is obvious the equilibriumpoint is at X = O < x = [ x l x2 5 3 4 xs x6 x7 s 8 l T ) .Th en , the control objective is to find a smooth adap-

A i At ive coordinated controller u = a ( x , S ) , 6= j? (x ,S >with the system state z and the adaptive parameterestimates S= pl p z p 3 p 23 such that for all admis-sible uncertain parameter ties pl , pz , p 3 , frictionforce Ff (z ,u ) and external disturbances w=[wl ze,,w 3 I T , he resulting closed loop system satisfies thefollowing performances :

1) When w=O , the resulting closed-loop systemis globally stable in the sense of Lyapunov, and thesystem states x can converge to the origin, i. e. x-0 as t -m.

2 ) When wfO, the resulting closed-loop system

A A A A A A

1-x3 Ir"


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- 40 Journal of Iron and Steel Research, International Vol. 18~

( 2 5 )renders the derivative of V 1 o satisfy the followinginequality

1Yk.154+k ,x s+> (P , , x , +Pj5X5)]

v,< k , P12xT - k , P,, - ,, )x; &xi - k , P45i-f(k,p,5-p45)xg +,WTW+a3 (PlZx1 Pz, xz )z6 -

62 5 3 7 -c1 (P45 2 4 f p55 x5 ) 6 f i ( p l - 1 ) f(Ptjx4 +P,ss~j F4 ( 5 4 ) (p2 -pz 1- P45~4

(26)where k 3 > 0 , k z , k , should be chosen to satisfy thecondition k z P ,, - ,, > O f k , P,, - ,, > . Thus , i tfollows from the Eqn. (23) that the positive definitematrix P is chosen as :PlZ =Pzz =P4,=P,, =1, P,, = k , f k , ,

AP55x51x5 p3 - 3 1

P44=k,+k5, ( k l ,k,>O, k , , k , > l )tion is constructed as t he following form

2 ) For the Eqn. (181, a positive definite func-1 1 12 2 2z ( x ) = v 1 ( x l , x , ,x3 1x4 ,x:,>+-z;+-z;+-z~

(27)then , considering the Eqn. (26) and Eqn. (IX), andchoosing the control law

(28 )

(29 )

(30)with any given positive constants k 6 , k 7 , k 8 rendersthe derivative of V z x) to satisfy the inequality:

A aa 3

i z 6

p l )+ (x4+x5) - 2 8 a x ,} F4 (x4 ) (pZ - $ Z ) f(31)

Moreover, it should be noted that the friction forceis described as the form Eqn. (7) -Eqn. ( 9 ) , then, theestimate fo r the friction force is constructed a s

aa 3 A- 2 4 +x5) }x, p 3- 3 1

A A( p Z - p 2 ) + { 2 8 a a 3 - 5 4 f x 5 1)x5 p3 - p 3 )(33)

3) For the whole closed-loop system, a positivedefinite function is constructed a sA A (34)with 6= [ p l pz p3 2IT and r =d i a g { r , r 2 , r 3 , 4 ,r,>O ( i = l , 2 , 3 , 4 ) , and the adaptive laws arechosen as follows

1V ( X , ~ )vzx ) +T ( 6 - 6 ) T r ( 6 - 6 )

the following inequality can be obtained* AV ~ x , 6 ~ < - ~ , x ~ - ~ ( k z - l ~ 5 ; - k 3 x i - k 4 5 ~ - ~ k , -

I U I A1 xg -k 6 z: -k7 z; k 8 z i -r4 ~ (2- 2)' +g ( u >(36)

4 ) Choose the penalty output signal y = [ p l x zpZx3 p 3 x 5 1 with the coefficients p i p z , p , > ,then the Eqn. (36) is rewritten asV ( x )- , IvI z- ) -- Tyf wT w

(37)

Y 2- WTW2

A 1 r"g ( u > 2 2* A


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I ssue 4 Adapt ive Coordinated Cont rol for H o t Strip Finishing Mills * 4 1 *

k , x;+ k ,- -+pi 1.x: f k , *; f k 7 z ; k , z;and the controller parameters k , ( i = l , 2 , *.. , 8 ) arechosen to satisfyk z > l + g P i , k 3 > y & , k , > l f ~ & ,k l , k , 9k6 tk 7 ,k ,>O.

Thus, Lyapunov stability of the closed-loopsys tem and the convergence of s tate s x and z - z aswell as L2 disturbance attenuation are guaranteed.

When w = O , it follows from Eqn. (37) that(38)( x , 6 > < - Q ( x > - r , vI(z- Z > ~ < O

thus, by Eqn. (34) and Eqn. (38), the closed-loop sys-tem consisting of (18) with Eqn. (28)-Eqn. (30 ) andEqn. (35) is globally stable in the sense of Lyapunova t x = 0 , 6- = 0 . Furthermore, with the Eqn.(38 , it follows from LaSalle invariance principlethat x-0, z - z+O as t - a . Then, by the defini-tion of z 6 , z 7 , z , , there exists x - 0 , x 7+ 0 , x8+0 , as t - a . Hence, the result is concluded that thesystem states x can converge to the origin and thestat e of friction force observer z approaches to z.

When wf 0, t follows from Eqn. (3 7) thatEqn. ( 1 9 1 , i. e. th e closed-loop system from the dis-turbance input w to the penalty output y has L2 gainnot larger than y.

A

I AAg ( u >

A

A

A

4 Simulation ResultsTh e simulation research is done on a mode l of a

hot strip finishing mill to show the effectiveness ofthe proposed control scheme. T h e dynamic simula-tion model is constructed according to the integrateddynamical description of the finishing mill presen tedin the previous sections. T h e physical parameters ofthe hot strip mill s ystem are chosen as follows:J = 7 kg m 2, E=50000 MPa , L=3 . 5 m , a=

2.216 m, 1=0. 376 m , r=O. 083 m , d=O. 067 m,B=O. 359 m, R=O. 2 m , L,=O. 2 m , M, =49 kg ,H=O. 0136 m , Ma =16 8 kg, ,0=7800 kg/ m3,g z 9 . 8 N /k g, f=O. 0 8 2 , rns=400 kg,T1.=0.0232, T,=O. 1592, Tr=O. 0032.

An adaptive coordinated controller is designedas Eqn. ( 2 8 1- qn. ( 3 0 with Eqn. (35 ) accordingto the design procedure presented in the former sec-tion, and the tuning parameters of the control laware chosen as:

k7=5 , & =5 , p1=pz=p3=o. 1 , y = o . 5 , r l = l O ,k 1 = 4 O , k Z = l O , k l = 2 0 , k4=4O, k5-15, k 6 - 5 ,r 2 = 4 , r 3 = 1 0 , r 4 = l .

And assume that t he desired values of the looperangular position, str ip tension and e xit thick ness inthe rolling process are constant and here they arechosen as B d = 0. 436 3 rad , bd = 3.44 MPa, hd =0 . 0 1 1 1 m.

In simulation, the uncertain physical parame-ters, mechanical friction phenomena and externaldisturbance will be considered. Moreover, severalsources of characteristic disturbances will be includ-ed in the simulation analysis: disturbance on th estrip tension, disturbance on the st rip entry thick-ness, discretization of the looper angular positionmeasure, friction torque present in the looper mech-anism (t he LuGre model is used). T o this end, thesimulations in the following five operating cases arepresented.

1) System is operating f rom th e initial conditionto the desired equilibrium, and in the operatingprocess, there exists uncertain external disturbancebesides LuGr e friction phenomena. Simulation re-sul ts of looper angula r, strip tension and exit thick-ness are shown in Fig. 3.

2) System is operating at the equilibrium butthe physical parameters perturbation occurs at t = l s.Simulation results of looper ang ula r, str ip tensionand exit thickness are shown in Fig. 4.

3) System is operating at the equilibrium but thelooper angular perturbation occurs for t>1 s and there

0 1 2 3 4 . 5 6 0 1 2 3 4 5 6(a ) Response of the looper angular posi t ion; ( b ) Response of t he s t r i p t ens ion ;

System responses for two control structures in case 1( c ) Response of the exi t th ickness.

Fig. 3


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4 2 Journal of Iron and Steel Research, International Vol. 18

-.-0.1 0 1 2 n 4 5 6

g 4p 3d5*0 2

1m 0 1 2 4 5 0Tinids(a ) Response of the looper angular position; ( b ) Response of the strip tension; (c) Response of the exit thickness,

Fig. 4 System responses for two control structures in case 2exis j viscous friction. Simulation results of looper Simulation result s of looper angular, stripangular , strip tension and exit thickness are shown and exit thickness are shown in Fig. 7.

:nsion

in Fig. 5 .4) System is operating at the equilibrium but

the strip tension perturbation occurs for t > 1 s .Simulation res ults of looper angul ar, str ip tensionand exit thickness a re shown in Fig. 6.

5 ) System is operating at the equilibrium butthe exit thickness perturbation occurs for t>1 s.

Moreover, to show the effectiveness of the pro-posed controller clearly, the comparison with theconventional PID control scheme is also given as thedash curves in Fig. 3 to Fig. 7.

From the simulation res ults , it can be illustra-ted that the proposed adaptive coordinated controllercan not only regulate promptly exit thickness, strip

$ 0.4 ..2 0.3 *bwJ 0.2 - PID control

0.1 0.0110 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6Tiniefs

(a) Response of the looper angular position; ( b ) Response of the strip tension;System responses for two control structures in case 3

(c) Response of the exit thickness.Fig. 5

I ! I I I I I l l I I I I I0 1 2 3 4 5 6 0 1 2 3 4 5Tirnels

(a) Response of the looper angular position; ( b ) Response of the strip tension;(c) Response of the exit thickness.

Fig. 6 System responses for two control structures in case 4

0 1 2 3 4 5 6 0 1 2 3 4 5 (i 0 1 2 3 4 5 6Timefs

(a) Response of the looper angular position; ( b ) Response of the strip tension;System responses for two control structures in case 5

(c) Response of the exit thickness.Fig. 7


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I s s u e 4 A d a p t i v e C o o r d i n a te d C o n t r o l for H o t S t r i p F i n i s h in g Mills ' 4 3 'tension and looper angle to the desired values in thepresence of parameter variations, but also reject thestick slip motion from the friction force and attenu-ate external disturbance and the characteristic dis-turbances to a desired level.5 Conclusion

For t he enhancement of the stability and properdynamical performance of hot st rip finishing mills,an adaptive control scheme based on the decentrali-zation and coordination is proposed for the tension,looper and gauge control systems via utilizing thenonlinear adaptive recursive design technique. Th esystem with the proposed adaptive controller has anadvantage both in regulating promptly strip tension,looper angle and exit thickness and in attenuatingthe unavoidable friction phenomena and external dis-turbance during rolling operation process. Irrespec-tive of th e rolling conditions with dis turbances oflooper angle, stri p tension and exit thickness as wellas additional external disturbance, the resultingclosed-loop system can be rendered to convergequickly to the desired operating point.References:[ l ] Choi I S, Ross i t e r J A , F l em i ng P J. Looper and Tension Con-

t rol in Hot Rol l ing Mil ls: A Survey [J]. Jour na l of Pr ocess

C o n t r o l , 2 0 07 , 1 7 ( 6 ) : 5 0 9.H e a m s G, R e ev e P , S m i t h P, et al . Ho t Str ip Mil l Mult ivar i-ab l e Mass F low Con t r o l [J]. I E E P r oc-Con t ro l T heo r y App l ,2 0 0 4 , 1 5 1 ( 4 ) : 3 86 .I m a n ar i H I M o r i m a t su Y, ekiguchi K. Looper H-InfinityControl for Hot-Str ip Mil ls [J]. I E E E T r a n s a c t i o n s on Indus-t r y App l i ca t i ons, 1 9 9 7 , 3 3 ( 3 ) : 7 9 0 .Asada H , Ki t amur a A , Ni sh ino S, et al. Adapt ive and RobustControl M ethod Wi th Est imat ion of Rol ling Character ist ics forL ooper Ang le Con t r o l a t Ho t S t r i p Mil l [J]. ISIJ Internat ional ,2 0 0 3 , 4 3 ( 3 ) : 358.H e s k e t h T , J i ang Y, lements D, et al. Control ler Design forHot Str ip Finishing Mil ls [J]. I E E E T r a n s C o n t r o l S y s t e mT echno logy , 1 9 9 8 , 6 ( 2 ) : 2 0 8 .Kazuya A , Kazuh i r o Y , T akssh i K , e t a l. Ho t S t r i p Mil l T en-sion-Looper Control Based on Decentralization and Coordination[J]. Cont r o l E ng P r ac t i ce , 2000, 8(3) : 337 .Janabi-Sharifi F. A Neuro-Fuzzy Sys tem for Looper TensionControl in Rol l ing Mills [J]. Control Eng Pract ice, 2005, 13( 1 ) : 1.Riccardo F, Fr ancesco A C, T homas P. Frict ion Compensat ionin the Inte rstan d Looper of H ot Str ip Mil ls: A Sl iding-ModeCon t r o l Appr oach [J]. Cont r o l E ng P r ac t ice , 2 0 0 8 , 1 6 ( 2 ) :214.O k a d a M , M u r a y a m a K , U r a n o A , e t al . O p t im a l C o n t ro l Sy s-tem for H ot Str ip Finishing Mil l [J]. Control Eng Pract ice,1 9 9 8 , 6 ( 8 ) : 1 0 2 9 .

H e a r n s G , Gr imble M J . Robus t Mul t i va r i abl e Con t r o l f o rHo t S t r i p Mi l ls [J]. ISIJ Internat ional , 2000, 40(10) : 995 .C a n u d as D e W i t C , O l ss o n H , A s t r o m K J , et al . A Ne wModel for Control of Sys tem s with Fr ict ion [J]. I E E E T r a n sAutomat ion Con t r o l , 1995 , 40(3 ) : 419.

adaptive coordinated control for hot strip finishing mills

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