NON-LINER CONROL SREGY FOR SRIP CSING PROCESSES V.. Oliveira, R. R. Nascimento, N. S. D. rrifano E. Gesualo, J. P. V. osetti Departamento e Engenharia Elétrica, USP São Carlos CP 359, CEP 3560-970 São Carlos, SP, Brasil Instituto e Pesquisas ecnológicas o Estao e São Paulo, Divisão e Metalurgia CEP 05508-90 Ciae Universitária, São Paulo, SP, Brasil bstract his paper present a non-linear control strategy to regulate the molten steel level of a strip-casting process. he molten steel level along with the separation force is consiere the most critical to the prouction of high quality steel strips. he molten steel level may be controlle using the tunish output flow or the casting spee. However, the casting spee is usually use to control the roll force separation. o improve the strip thickness uniformity, we propose the introuction of an intermeiary tunish submerse into the pool between the rotating rolls. he molten steel level is thus controlle by the intermeiary tunish output flow. In this paper, we consier a feeback linearization controller with a selective fuzzy control law for moel-estimation error compensation. Stability analysis of the fuzzy Mamani controller using the Lyapunov irect metho is also inclue. Simulation results are presente consiering the real system parameters an both the stopper actuator ynamics an the rolling mill DC motor ynamics. For comparison we inclue results with the conventional PID. Keywors Strip-casters, twin roll, feeback linearization, fuzzy control, Mamani moels.. INRODUCION he twin roll strip-casting process belongs to a new generation of casting processes, the calle near-net-shape processes. he twin roll strip-casting process was first conceive by Henry Bessemer in the mile of last century []. twin roll casting process is essentially a two rolling mill equippe with three main control loops: the molten steel level control loop, the separation force control loop an the casting spee control loop. he molten steel level along with the separation force is consiere the most critical to the prouction of high quality steel strips. In [] an aaptive fuzzy controller for the molten steel level in a strip-casting process is propose. hey combine an aaptive fuzzy controller an a switching control strategy to compensate for moel-estimation errors an eliminate isturbances. s the feeing of the molten steel into the pool forme between the two rotating rolls is a source of isturbance in the molten steel level, in this paper we consier the use of an intermeiary tunish which is submerse into the pool to reuce the steel level fluctuations [3]. he intermeiary tunish consists of a refractary recipient with holes to irect the molten steel to the pool forme between the two rotating rolls. Differing from [], we use a feeback linearization controller an a selective.fuzzy controller to take care of the moelestimation errors instea of an aaptation law. s in [] we use a switching control term to eliminate isturbances. We also inclue the intermeiary tunish in the system moelling. he propose fuzzy control sheme is simple an follows a conventional fuzzy moeling. strip-caster pilot plant installe at IP São Paulo is shown in Figure. he main control units are the mill rive, the cooling an the coiler control units [4]. he plant is equippe with a set of Programmable Logic Control (PLC) units to perform the measurements an control.. SYSEM MODELING he molten steel level system may be escribe as a nonlinear system base on the continuity equation of the steel flow an on the Bernoulli equation. Following, we escribe the various components of the casting process.. Intermeiary unish Molten Steel Level Qi Q o he ynamic moel of the steel level in the intermeiary tunish for input flow rate an output flow rate is given as h ( Qi Qo) () t
h where Qi c f ; Q o K h with an the steel height an area of the intermeiary tunish, respectively; c f the flow coefficient, the actuator position; K n f f g, the area of the holes, f π r, nf an r the number an raius of the holes an g the acceleration ue to gravity [5]. f Lale unish Hyraulic servo Submerse tunish Stopper Sensor C Motor PLC acogenerator PLC acogenerator PLC Coiler M Power amplifier Power amplifier Belt Figure : Schematic layout of the strip-caster pilot plant installe at IP São Paulo.. Mill Drive Molten Steel Level Qo Q o he ynamic moel of the input an output flow in the mill rive is escribe as V Qo Qo t () where V is the volume of the molten steel between the rolls, Qo the output flow from the rolls an Q o the inflow from the intermeiary tunish. In terms of the level in the mill rive, () can be written as [] h t [ Qo Qo ] (3) M ( xg, h ) where M x g : ( + R) R h L an Q o Lx g vr, with v r the casting spee. he nominal parameters for both the mill rive an intermeiary tunish are liste in able. able : Process parameters. 6.75E-3m c f 3.5E-m 3 /s K 0.3E-m 3 s -/ R,L 3.75E-m x g E-3m n f 6 5.0E-5m g 9.8 m/s f r 4E-3m v r 0.093m/s
.3 Hyraulic ctuator he hyraulic servo system consiere to rive the stopper valve can be escribe by v K K (4) a p eqo 3 as + as + a3s + a4 where a ( MaV t /4βe), a ( MaKco + BVt / 4 βe), a3 ( BpKco+ p ) an a4 KKapKeqo with M a, V t, β e, K co, B p, p, K, K eqo an Ka as in able. he block iagram of the hyraulic servo system is illustrate in Figure. Stopper Reference v Proporcional controler Hyraulic actuator Stopper position Position measurement Fig. : Hyraulic servo actuator. able : Hyraulic parameters. M a 50Kg V t 7.9E-5m 3 β e 7.8E8Pa K co 3.3E-(m 3 /s)/v p 0E-4m K 40V/m B p 00Ns/m K a 30 K eq0 E-5.3 System Equations In the space state form for x h, x h x x x x x, 3, 4 3, 5 3 we have the complete system equation x f ( x) + Bu (5) where f( x) ( cf x3 K x ) 0 ( K x Q ) 0, B 0, u v an b x5 a a3 a a 4 x5 x4 x 3 a a a 0 M( x ) x4 M ( x ) [ xg + R R x ] L. 3
3. NONLINER MOLEN SEEL LEVEL CONROL he molten steel level between the twin rolls may be regulate using as control input the inflow Q or the casting spee v. However, the casting spee v r is usually use to control the roll separation force ue to the system constraints. herefore, as we introuce an intermeiary tunish, here the molten steel level control is pursue by controlling the level of the intermeiary tunish with an inner control loop for the stopper actuator using a servo-valve. he purpose of the inner control loop is to avoi abrupt changes in the valve position. he inclusion of the intermeiary tunish is very important to the quality of the final prouct, as it is etaile in Section. However, ue to its inclusion the molten steel level in the intermeiary tunish nees to be monitore an consiere in the controller to avoi possible overflow, guaranteeing a goo working conition for the process. In this section we explore the use of the Mamani fuzzy control an the feeback linearization control to regulate the molten steel level. he fuzzy control approach has emerge as an alternative to control plants that exhibit complex non-linear behaviour. i r 3. Mamani Fuzzy Control he fuzzy logic systems use is formulate using the Mamani s metho, which has been successfully applie to a variety of inustrial processes an consumer proucts [6]. he first fuzzy controller configuration consiere is systematize in Figure 3. Defining the error e as e : x, in this configuration, the fuzzy controller is forme by three input fuzzy sets, the error e, the error erivative e an the level in the intermeiary tunish actuator input voltage name v. x x, an one output fuzzy set corresponing to the stopper he fuzzy controller use the singleton fuzzifier, the center average efuzzifier, the prouct inference rule an a fuzzy rule base, which consists of a collection of fuzzy IF-HEN rules of the following form. R( ): IF e is F an e is F an x is F 3 HEN the stopper reference for is G,,,, (6) v 4 r F F F 3 where r 7, the number of linguistic rules;, an are the input fuzzy sets an G the output fuzzy set; ( e, e ) U, x V in an v H o, with U in, Vin an H o input an output universes of iscourse, respectively. he rule bases are shown in able 3 where the first marke cell can be interprete as follows: IF the error e is Negative Small (NP) an the error erivative e is Positive Small (PP) with the intermeiary tunish height Zero HEN the stopper actuator input voltage is Zero. ll the membership functions are shown in Figure 4. v Let w: [ e e x ] an v: v. fuzzy implication for w an v enote µ R ( w, v) is a fuzzy set in the prouct space U in V in H o, that is the fuzzy relation inuce by the rules with a membership function an a membership grae. When no rule involves the association of the input linguistic terms, F, F an F 3 with the output G 4, the fuzzy implication is simply assigne to zero. he fuzzy implication rule is given by the prouct-operation rule 3 4 F F F G µ ( F, F, F, G ) µ ( e) µ ( e ) µ ( x ) µ ( v ) (7) R 3 4 4 in Level reference Fuzzy controller v FLC Hyraulic servo actuator u Flow ynamics an plant h h x x ] Figure 3. Schematic of the steel level control unit using the Mamani fuzzy controller with [. : 4
Figure 4: Membership functions. 3.. Stability analysis he stability of the Mamani fuzzy controller can be analyze using the Lyapunov s irect metho by forming an input-output mapping for the feeback fuzzy system an consiering the fuzzy base rules [7]. Figure 5 shows the complete non-linear system uner fuzzy control. For the stability analysis it is consiere the hyraulic servo actuator in series with the flow ynamics an plant. he linearize flow ynamics an plant about the operating point ), ( ξ h is of the form z y u B z z + (8) where 0, 0 0 f c B x M K x K 5
with z : h h an u :. he hyraulic servo actuator ynamics is of the form ξ ξ + B v y C ξ FLC (9) with 0 0 0 0 0, B 0, a a3 a4 b [ a a a a 0 0] an ξ : [ x3x4x5 ]. able 3: Rule bases for the fuzzy controller. N: negative, P: positive, M: meium, Z: zero; G: big an PP: positive small Error Error Error Error erivative NG NM NP Z PP PM PG NG NG NG NG NG NM NP Z NM NG NG NG NM NP Z PP NP NG NG NM NP Z PP PM Z NG NM NP Z PP PM PG PP NM NP Z PP PM PG PG PM NP Z PP PM PG PG PG PG Z PP PM PG PG PG PG Z Z Z Z Z Z Z Height of the intermeiary tunish Error erivative NG NM NP Z PP PM PG NG NG NG NG NG NM NM NP NM NG NG NG NM NM NP Z NP NG NG NM NM NP Z PP Z NG NM NM NP Z PP PM PP NM NM NP Z PP PM PM PM NM NP Z PP PM PM PG PG NP Z PP PM PM PG PG PP PP PP PP PP PP PP Height of the intermeiary tunish Error erivative NG NM NP Z PP PM PG NG NG NG NG NG NG NG NM NM NG NG NG NG NG NM NM NP NG NG NG NG NM NM NM Z NG NG NG NM NM NM NM PP NG NG NM NM NM NM NP PM NG NM NM NM NM NP NP PG NM NM NM NM NP NP NP PG PG PG PG PG PG PG Height of the intermeiary tunish 6
heorem. he feeback system (5) with a control law constructe via the IF-HEN rules (6), rule base given in able 3 an membership functions given in Fig. 4, is locally asymptotically stable. Proof. Consier the Lyapunov function caniate for the complete linearize system V ( z) z Pz (0) where P is a positive efinite symmetric matrix. Since y z.he time erivative of V is given as V ( z) z ( P + P) z + z PB u. () he linearize flow ynamics an plant is a stable system in the sense of Lyapunov as it contains a pure integrator. For this system there exists a positive efinite symmetric matrix P an a positive semiefinite matrix Q such that P + P Q [7-8]. he hyraulic actuator is a feeback system with faster time response than the flow system an therefore its ynamic is not consiere in the stability analysis. y + - e t e e FLC v FLC Plant an actuator x x Figure 5: Cascae fuzzy control system structure. he local stability of the feeback system is guarantee if the fuzzy control law v is such that FLC V z ( P + P ) z + y P Bu 0. () Now, the linearize flow ynamics an plant yiel 8 0 0.3593 0 3.778 5.433 5.85 Q,, P, B. 0 0 0.848 0 5.433 0 0 s u an v has the same sign in the vicinity of the operating point, V ( z) 0 is assure if FLC ( + FLC < 7.4397x 6.668x ) v 0. (3) he stability checking can be carrie out consiering the fuzzy rule base built for the control law. he possible control actions must be analyze consiering the change of variables to the operating point. he error is e y x. Note that the error e has the opposite sign of x an that x has the same sign of x as the intermeiary tunish output correspons to the input of the mill rive. We consier now the fuzzy controller actions corresponing to the peripherical cases presente in able 3. hese actions can be represente by the rules R, R, R3, R4 given below R: IF e > 0 an e > 0 an x 0 HEN v > 0 > FLC R: IF e > 0 an e < 0 an x 0 HEN v > 0 > FLC 7
R3: IF e < 0 an e > 0 an x 0 HEN < 0 < v FLC R4: IF e < 0 an e < 0 an x 0 HEN < 0. < v FLC Substituting the results of the rules R, R, R3 an R4 in (4) it is verifie that V < 0 is satisfie in the peripherical cases assuring a feeback system stable by an extension of La Salle invariance principle [9]. In aition, the system trajectory can not be stuck at a value other than the equilibrium ue to the structure of the rule bases, which assures the asymptotically stability of the equilibrium. Fig. 6 illustrates the behavior of a V for the process LC-RD for a typical simulation case, consiering the non-linear equations. V V t t Figure 6: Derivative of the Lyapunov function. On the right, a zoom near the passage by zero is shown. 3. Feeback Linearization Control In this section the metho of feeback linearization with compensation for moeling errors is use to regulate the molten steel level at the esire value y. In the metho of feeback linearization, the nonlinearities in a nonlinear system are cancele to yiel a close-loop linear system. he compensation for moeling errors is accomplishe by incluing a fuzzy control law. In what follows we consier the feeback linearization metho applie to the flow ynamics an plant. he flow ynamics an plant ynamic are obtaine from (5) as K x x x [ K x Qo M ( x) + u ] 0 (4) x an x ssuming in (4) that, are measure, a control law xg Q i α ( h) + β ( h) v (5) with α : R R, R β : R, h [ x x ] an v an equivalent control, can be foun so that the nonlinear system ynamics is transforme into an equivalent linear ynamics of a simpler form as follows [8]. z z z v (6) using ( h) [ z L f z an solving the following system for z z ] z a z a 0 f f g 0 g 0. (7) 8
where z h L f z an a f g [ f, g] gf f, which is the Lie brackets. x Solving (7) we obtain z x an Now, using z z x t x + x x we obtain z z [ K x Qo ]. M ( x ) z K x x M ( x ) an Q i α ( h) + β ( h) v with x[ K x Qo ] + x R x M ( x ) o] 4x x [ K x Q α ( x) K x +, K R x M ( x ) x M ( x) β ( x). K (8) Base on the equivalent linear system, a tracking controller for the level can be obtaine. We efine the error e : z y, with y the reference level. hen, the linear control can be given by v Ke Kz. Figure 7 presents the feeback control system. x y v K z u u( h, v) Hyraulic actuator Flow ynamics an plant h Linearization loop z z(h) Figure 7: Feeback linearization control system. o guarantee e 0 as t a control term calle supervisory control u, is ae to Q. he supervisory control is of the form u s : sgn ( e), with a esign parameter an sgn the sign function s i sgn( e) + if e > 0 if e < 0. he term supervisory control is inspire in the variable structure with sliing moe technique [7-8]. Moreover, in orer to compensate moeling errors that inevitably occur in the strip-casting system, a fuzzy control term u is ae to in (5). he fuzzy control term u is forme by two input fuzzy sets, the error e an the error erivative e, an one output fuzzy set corresponing to an aitional voltage for v, whereas the cascae fuzzy controller is forme by three input fuzzy sets, the error c e, the error erivative e an the molten steel level v in the intermeiary tunish, an one output fuzzy set corresponing to the stopper actuator input voltage v. he fuzzy control term consists of a collection of fuzzy IF-HEN rules of the following form. x c Q i Fuzzy control term 9
R( ): IF e is M an e is M HEN the aitional stopper reference for is K,,..., r (9) v 3 where r 3, the number of linguistic rules; M, M are the input fuzzy sets an K3 the output fuzzy set; (e, e ) Uin an, with U an input an output universes of iscourse, respectively. he control law thus becomes v Q i H o in H o α (x) + β ( x )[ K e K z + sgn( e)]. he rule base, the membership functions an the molten steel level basic control configuration with the fuzzy control term are shown in able 4, Figures 8 an 9, respectively. he stability of the feeback control system can be verifie in a similar manner as in Section 3. for suitable K, K an. able 4: Rule base for the selective fuzzy unit. Error Error erivative N Z P N P P Z Z P Z N P Z N N Figure 8: he membership functions for the selective fuzzy unit. 0
Reference level Input-state linearization + - + Position control Stopper actuator u Flow ynamics an Plant h Figure 9. Schematic of the feeback linearization fuzzy control system. 4. SIMULION RESULS ll results were obtaine consiering the stopper actuator ynamics. he intermeiary tunish height is h 0., the stopper valve position maximum is 0.05[m]. he molten steel levels in normal operation are in the interval 3 h [0. 0.4] an the nominal inflow is Q. which is in accorance with the esign of the intermeiary o 307e 3[ m / s] tunish. he esire values of gap an level are set as xg 0.00[m] an h 0.3[m], respectively. In the esign of the Manami fuzzy control law, we use seven linguistic rules (r 7) to the input fuzzy set error e an error erivative e an three linguistic rules (r 3) to the input fuzzy set of the molten steel level in the intermeiary tunish. he latter is neee to avoi possible over flow. For the output fuzzy set, the stopper actuator input voltage, we use seven linguistic rules (r 7). o verify the performance of the control strategies consiere, Figures 0 an show the process responses to a step reference with the esire molten steel level chosen as h 0.3m. In Figure 0, a 0% outflow Q o isturbance is introuce after 50s. In Figure, a 0% roll gap isturbance containing up to the thir harmonic of the roll velocity is introuce (Lee. et al). CONCLUSION x g In this paper fuzzy control strategies for the molten steel level in a strip-caster plant installe at the IP São Paulo are propose. Different fuzzy control strategies are combine in orer to achieve a high performance regulation. n important feature of the fuzzy controllers is their flexibility to consier the control esign constraints in the process an control variables. he simulation results show the superiority performance of the feeback linearization controller with a fuzzy control term for compensation of moeling errors as compare to conventional PID an stan alone cascae fuzzy controllers. he cascae fuzzy controller can be tune to respon to moeling errors but at the cost of loosing in aaptability to real operation conitions. he avantage of the Mamani s fuzzy moel is the simplicity of its control loop, which follows the same structure as the PID control. Since the aim of the process is to prouce a soliifie strip of constant thickness uner a constant roll separation force, the main control unit of the strip-caster system must inclue the control of the separation force an this will be consiere in future work. Fuzzy Control
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