49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA
Feedback control of the two-phase Stefan problem, with an application to the continuous casting of steel
Bryan Petrus, Joseph Bentsman, Member, IEEE, and Brian G. Thomas
Abstract—A full-state feedback control law is derived that Since these goals can be achieved via temperature stabilizes the two-phase Stefan problem with respect to a regulation, an automated control strategy based on the steel reference solution using control of the Neumann boundary temperature would be greatly beneficial to the industry. condition. Stability and convergence are shown via a Lyapunov However, most casters use open-loop methods to control the functional on the error system with moving boundaries. A flow rates of the water sprays that cool the surface. A key second control law is also derived, for which stability is proved and convergence is conjectured due to the clearly convergent obstacle to the design of such control is the strong inherent simulation results. A simple Dirichlet controller is also nonlinearity of the solidification evolution equations. The considered, and is used to design a boundary-output-based many results in recent years for control of linear or even estimator that, in combination with full-state feedback semi-linear distributed parameter systems are not applicable. controllers, yields a plausible output feedback control law with Previous work on control of processes governed by the boundary sensing and actuation. The performance of the Stefan problem can be generally divided into three control laws is demonstrated using numerical simulation. categories: numerical optimization methods [2, 3], solutions of the inverse Stefan problem [4-6], and feedback control I. INTRODUCTION methods [7-10]. The numerical optimization methods in [2] HE temperature of a solidifying, pure material can be and [3] can take into account realistic metallurgical Td escribed by a non-linear partial differential equation constraints and quality conditions. However, since the (PDE) of the form commonly known as a Stefan problem. simulation involved is highly complex and nonlinear, they This problem divides the domain into two or more time- cannot realistically run in real-time. The inverse methods and varying subdomains separated by moving boundaries. In a feedback control methods, with the exception of [10], use the model of an industrial casting process, these domains Stefan problem as a model and focus on control of the correspond to the solid and liquid phases of the material. boundary position, which would ensure whale prevention, The movement of the boundary between the phases is but not necessarily the steel quality. The inverse problem, as described by the Stefan condition, a differential equation solved in [4] and [5] directly and in [6] by minimizing a cost derived from an energy balance at the boundary that is a functional, is very numerically complex and thus limited to function of the left and the right spatial derivatives of the design of open-loop controllers. The feedback control temperature at the boundary[1]. methods are better suited for real-time control, but the The Stefan problem can be used to model a variety of control in [7] and [8] is simplified to the “on-off” thermostat- industrial casting applications. The present paper focuses on style one. In [9] and [10], PI controllers are designed based the continuous casting of steel. Key quality and safety goals on a discretized form of the solidification evolution for this process can be achieved by matching an ideal equations. However, neither controls the full temperature temperature history. For example, certain cracks can be distribution. [9] only considers the solidification boundary, prevented by keeping the temperature in a specific range, while [10] focuses on the steel surface temperature. since the ductility of steel is a function of temperature and Our goal in this paper is to stabilize the solution to the time. In a continuous caster, the steel is contained within Stefan problem relative to a reference solution. The latter is rolls that drive the steel downwards. If the steel is not fully assumed to be safe and provide good metallurgical quality solidified when it exits the last of these rolls, a severe bulge, under nominal conditions, so that the process goals are met known as a “whale,” is formed due to the pressure of the by reducing the reference error to zero. Our key tool is a liquid steel. This can be prevented by ensuring that the Lyapunov functional on solutions of the Stefan problem with temperature through the entire cross-section is below the a moving boundary. This allows for the construction of a solidification temperature before the end of containment. control law that stabilizes the error, and shows convergence of the error to zero asymptotically. B. Petrus (e-mail: [email protected]), J. Bentsman (phone: 217-244- 1076; fax: 217-244-6534; e-mail: [email protected]), and B. G. In Section II, we give a brief description of the problem. Thomas (e-mail: [email protected]) are with the Department of In Section III, we provide two control laws for the Neumann Mechanical Sciences and Engineering, University of Illinois at Urbana- boundary condition of the Stefan problem and the associated Champaign, IL 61801 USA. J. Bentsman is the corresponding author. proof of convergence for one of them. In Section IV, we This work was supported by NSF grant CMMI-0900138 and the examine the relevance of these assumptions for the Continuous Casting Consortium at the University of Illinois.
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application considered. To address controller The assumptions, respectively, ensure that the equations are implementability, in Section V we offer a simple Dirichlet well defined at t = 0 and that solutions have sufficient control law, which allows for the design of an estimator regularity. Throughout this paper, we deal with the case in based on boundary temperature measurements, that, when which ε
The solidifying steel in a continuous slab caster, called the u (t) with initial conditions T (x,0) = T0 (x) and s(0) = s0 strand, is rectangular. The outer solid shell of the strand satisfying assumptions (A1) and (A2). This reference encloses the liquid core. The relative size of the latter temperature profile should satisfy the metallurgical goals and decreases as the strand is cooled, giving rise to the internal constraints of the process, and could, for example, be moving solid/liquid interface. The temperature evolution calculated for the continuous caster using previous results, equations for the strand are three dimensional, and must e.g [2-6]. That is, matching the reference temperature should account for, at the least, heat diffusion, advection at the result in safe operation and good quality steel. We add one casting speed, and the phase change. However, through more assumption on the reference profile: symmetry and scaling arguments, the problem reduces to a (A3) sɺ (t) > 0 for all t ≥ 0 . one-dimensional “slice” that moves through the caster at the ɶ casting speed. A more detailed discussion of this modeling We denote the errors as T ( x,,,t) =T ( x t ) −T ( x t) , and setting can be found in [13]. sɶ (t) =s(t) − s (t) . Also, we denote uɶ (t) =u (t) − u (t). We denote the temperature within the slice as T ( x,t) , for Subtracting the PDEs yields 0 ≤x ≤ L and t ≥ 0 , where x = 0 and x = L correspond to ɶ ɶ Tt ( x,,,t) = aTxx ( x t) x ∈(0, L) \{s,. s} (3) the outer surface and the center of the strand, respectively. Also, since solutions to (1)-(2) are twice spatially The boundary between solid and liquid phases is denoted differentiable outside of the solidification front, they must s(t) . Then the following partial differential equation (PDE) have continuous first spatial derivatives. Thus, if models the evolution of temperature within the slice: + − s (t) ≠ s(t) , then Tx(s ()t,, t) = T x (s ()t t) , and so Tt ( x,,,t) =aTxx ( x t) 0 1732 the reference error Tɶ ( x,t) converges uniformly to 0 as Then (6) simplifies to ɶ + 2Tx (s) s t → ∞ . u t= u t − Tɶ x . () () ɶ ɶ xx () s− Proof: Consider the Lyapunov functional TT()()0+ xx 0 1L 2 1 s 2 V TTTɶ:= ɶ2 + ɶdx = 1 TT ɶ 2 + ɶ dx Using these relationships and (3), we can again take the time () 2∫0( x) 2 ∫ 0 ( x ) derivative of (7), which in the degenerate case only has a (7) s 2L 2 single boundary. After integrating by parts, 2 ɶ2 ɶ ɶ 2 ɶ +TT +dx +TT + dx , + ∫()()x ∫ x L s s1 s 2 VɺT ɶ,t= − a TT ɶ2 + ɶ 2 dx + aT ɶs T ɶ x . (12) ()()∫0 x xx x()() xx s− s s s s s s where 1 := min{ , } and 2 := max{ , } . Note that ɶ ɶ If Tx (s) = 0 , then (10) clearly holds. If Tx (s) > 0 , then for V Tɶ is equivalent to the square of the Sobolev norm, ( ) ɶ + all ε > 0 sufficiently small, T (s +ε ) > 0 . If Txx (s ) > 0 , ɶ ɶ ɶ TTT:= + x , (8) ɶ 1,2 2 2 then by (3), Tt (s +ε ) >c > 0 for all ε > 0 sufficiently small. in the sense that This means Tɶ (s(t) +ε, t +δ ) > 0 for all δ > 0 sufficiently 1ɶ2 ɶ 1 ɶ 2 T ≥V ()TT ≥ . (9) small. But, by assumption (A3), within the degenerate time 21,2 4 1,2 interval, Since solutions of the Stefan problem are twice differentiable ⇒ except at the boundary, the first weak derivative exists and s(t +δ) =s (t + δ) >s (t ) = s(t) s(t + δ ) =s(t) + ε such solutions are in the Sobolev space W 1,2 (0, L) . for some ε > 0 . This means, taking δ small enough, ɶ ɶ Assuming that s (t) ≠ s(t) , and ignoring the degenerate 0 =T (s(t +δ),, t + δ ) =T (s(t) +ε t +δ ) > 0. ɶ + ɶ − case for now, taking the time derivative of (7) using the PDE By contradiction, then, Txx (s ) ≤ 0. Similarly, Txx (s ) ≥ 0. (3) yields: Therefore, s+ s + 2 12 2 + ɺ ɶ1ɺ ɶ2 ɶ 1 ɺ ɶ 2 ɶ s V T,t = −s1TT +x −s 2 TT + x ɶ ɶ () () −() − aTx()()s T xx x − ≤ 0 2s1 2 s 2 s s s and (10) follows from (12). The same argument holds under +1TTTTɶ ɶ + ɶ ɶdx + 2 TT ɶ ɶ + T ɶ T ɶ dx ∫0 ()()xx x xxx ∫s xx x xxx ɶ 1 reversed signs in the case Tx (s) < 0 . Thus, in the degenerate L +TTɶ ɶ + T ɶ T ɶ dx. case, under the given control law, the estimate (10) is still ∫s ()x x xxx 2 valid. We note here that the expression above contains the third As an immediate conclusion of (9) and (10), under this spatial derivative. Since T and T are solutions to the control law the reference error Tɶ is bounded in the parabolic heat equation on the time-varying domains W 1,2 (0, L) Sobolev norm. 0,,s∪ s L and 0,,s∪ s L , respectively, they will be ( ) ( ) ( ) ( ) We now apply an invariance principle for general at least three times differentiable. This can be shown using evolution equations from [15]. Define the spaces an appropriate change of variables and Theorem 3.10, p. 72 X = W 1,2 (0, L) and Y = C 0 (0, L) , and let f ( x) be an ɶ in [14]. Therefore, T will also have the third spatial admissible initial value for the reference error. That is, derivative except at the boundary points. f x =TT − where T and T satisfy assumptions (A1) Now, integrating by parts, applying the boundary ( ) 0 0 0 0 conditions from (1) and combining like terms gives and (A2). Define G ::=γ ( f ) = ∪{S (t) f } where S (t) f is L t≥0 VɺT ɶ,t = −a TT ɶ2 + ɶ 2 dx −auɶ 0TT ɶ 0 + ɶ 0 ( ) ∫0 ( x xx) ()()()( xx ) the solution to the error equations under the given control s+ s + s+ s + law. Since solutions to the Stefan problem are continuous ɶ ɶ ɶ1 ɶ ɶ ɶ 2 1 ɶ21 1 ɶ 2 2 −aTTT + −aTTT + −sɺT − s ɺ T . 2 x() xx− x() xx − 1 x− 2 x − and piecewise- C , G ⊂ X . Further, it can be shown using a s1 s 2 2s1 2 s 2 slight extension of the Rellich-Kondrachov theorem in [16] u t Hence, if the control ( ) satisfies (6), then that X is compactly embedded in Y . Therefore, G is L ɺ ɶ ɶ2 ɶ 2 ɶ compactly embedded in Y and, as noted above, G is X - V (T,t) = −a∫ (TTx + xx )dx =:W (T ) ≤ 0. (10) 0 bounded. Define Now, we consider the degenerate case, in which L L Vˆ ()y :=TTɶ2 + ɶ 2 dx and Wˆ ()y :=TTɶ2 + ɶ 2 dx s (t) = s(t) for some time interval of length greater than ∫0 ( x ) ∫0 ( x xx ) zero. This means Tɶ (s (t), t) = 0 in this interval, and since to be, respectively, the extensions of V and W (defined in (10)) to Cl G , the closure of G in the supremum norm. the boundaries move as governed by (2), Y Since functions in G will be twice differentiable almost T s−−T s + =T s − −T s + x( ) x( ) x( ) x ( ) everywhere, both of these functionals are well defined, (11) ɶ+ ɶ − ɶ positive semi-definite, and lower semi-continuous on Cl G . ⇒ Tx()()s =T x s =:T ()s . Y 1733 Thus, all the conditions of Theorem 6.3, p. 195, in [15] are where the initial conditions satisfy (A1) and (A2). Then, the met, giving the following result: reference error Tɶ ( x,t) is bounded in the L2 norm. lim dy S (t) f , M 3 = 0 (13) t→∞ ( ) Proof: Consider the Lyapunov functional 1L 1 2 where V ()TTɶ:=∫ ɶ2dx = T ɶ 20 2 2 M ::= y ∈ Cl G Wˆ ()y = 0 . (15) 3 { Y } 1s 1 s 1 L =1Tɶ2dx + 2 T ɶ 2dx + T ɶ 2dx, ɶ ɶ ɶ ∫0 ∫s ∫ s In general, M 3 ::={T ()()()x Txx ≡0 ≡ T xx x } , that is 2 21 2 2 where s:= min s, s and s:= max s, s . As in Theorem T ( x) =T ( x) + C for some constant C . So, consider any 1 { } 2 { } ɶ 1, we take the time derivative and integrate by parts, constant element T ( x) ≡ C in G . If C ≠ 0 , then s≠ s , but substituting in the PDEs and boundary conditions where since T is continuously differentiable except at s , appropriate. The result is + + − − L T s =T s =T s = T s . 2 1 1 x( ) x( ) x( ) x ( ) VɺT ɶ,t = −a T ɶdx −uɶt T ɶ0 +s ɺT ɶs − sɺT ɶ s . () ∫0 x ()() () () Then by (2), b b If the control law satisfies (14), sɺ b T s +T s − = −( x( ) − x ( )) = 0. L VɺT ɶ,t = −a T ɶ 2dx ≤ 0. (16) This contradicts assumption (A3). This means that ( ) ∫0 x In the degenerate case s= s , control law (14) reduces to M 3 ∩ G = {0} , and since M 3 ⊂ ClY G , M 3 = {0} . u= u . Again taking the time derivative and integrating by Therefore, (13) is equivalent to ɶ parts gives (16), where the boundary terms drop out because lim T ()x,t = 0. □ ɶ ɶ ɶ t→∞ ∞ uɶ = 0 and T (s) =T (s ) = 0 . Therefore, V (T ) , and Remark 1. It does not follow from Theorem 1 that the ɶ solidification front position converges as well. If the consequently T , is bounded over time. □ 2 temperature gradient in the reference profile is small, the Although the proof does not guarantee convergence, the solidification front position error may be arbitrarily large for control law in simulation has shown convergent behavior. small temperature errors. For practical applications, though, Therefore, we formulate the following conjecture. this gradient is not small, and the solidification front Conjecture 1. Let the system (1)-(2) be controlled such that converges to the reference position as illustrated in the u(t) is given by (14) where the initial conditions satisfy (A1) simulation in Section VI. and (A2) and the reference temperature history satisfies Remark 2. The well-posedness of the 1-D Stefan problem Tɶ x t Lp has been examined in depth, e.g. in [1, 17, 18], typically (A3). Then, the reference error ( , ) converges in the requiring boundedness of the boundary conditions and their norm, p ≤ 2 , to an ε-neighborhood of zero reference error. time derivatives. The control law (6) may be unbounded, and A plausible proof could be based on the results in [15] as therefore it may be necessary to regularize it in order to in Theorem 1, or use Barbalat’s Lemma. Either method prove the general well-posedness of the closed-loop system. ɶ would require showing that Tx is bounded along In the simulations, some regularity is attained by bounding 2 the control, which does not result in the loss of convergence. trajectories of the error system under control (14). A rigorous analysis of this issue will be carried out in subsequent work. IV. APPLICABILITY OF THE CONTROL LAW Remark 3. The presence of the second spatial derivative of Assumptions (A1) and (A2) will be true for all physically the temperature error in the control law (6) ensures error possible initial conditions. Assumption (A3) is generally true 1,2 convergence by inducing the relatively strong W (0, L) for any practical reference profile. An alternative to (A3) Sobolev norm topology, but it also places additional ensuring convergence to the reference system is: smoothing requirements on the measurements. Relaxing the (A4) The initial conditions satisfy T0 ( x) = Tf for topology and removing the second spatial derivative yields a x ≥ s , and T x = T for x ≥ s . second control law given below that only depends on the first 0 0 ( ) f 0 spatial derivative. However, it is only proven to be stable Under this assumption, from the boundary conditions at relative to the reference temperature, with the convergence x = s (t) and x = L , it follows that Tɶ ( x) = 0 for x ≥ s and conjectured based on given simulation results. 2 ∩ Theorem 2. Let the system (1)-(2) be controlled such that all t ≥ 0 . This means M 3 G = {0} in the proof of Theorem 1 1, and the conclusion still holds. u ()t =u ()t +sɺTɶ()s − sɺT ɶ ()s bTɶ ()0 There are two ways in which the model given by (1)-(2) (14) significantly differs from the physical system. First, we 1 s+ s + =u t −Tɶs T ɶx −T ɶs T ɶ x assume boundary heat flux, when in fact control is limited to () ɶ ()()xs−()() x s − T ()0 the cooling water sprays that have fixed, spatially varying 1734 footprints. Moreover, the water flow rates are strictly limited TABLE I by the spray piping system. Although Section III does not THERMODYNAMIC PROPERTIES USED IN SIMULATIONS investigate saturation, we placed bounds on the control Symbol Description Value signals in the simulations discussed in Section IV and -6 conjecture that the controlled system converges for initial a thermal diffusivity 3.98 x 10 W/m K b Stefan condition constant 1.102 x 10-8 m2/K s conditions close to zero reference error. Tf melting temperature 1783 K Second, we have assumed full state feedback is available. L half-thickness of strand 0.1 m It is clear that in the real process the temperature at any point below the surface cannot be measured. An important area for 1850 future improvement of this work, then, is in output feedback (K) 1800 design, which is briefly addressed in the next section. 0 1750 V. DIRICHLET CONTROL AND ESTIMATOR DESIGN 1700 First, we consider a controller in which the boundary surface temperature can be set exactly equal to the reference. 1650 Initial temperature, T reference Theorem 3. Let the reference and the actual temperatures actual satisfy assumptions (A1), (A2), and (A4). In addition, 1600 0 0.02 0.04 0.06 0.08 0.1 assume that T0 ( x) < Tf for x < s0 and T0 ( x) < Tf for Distance from strand surface, x (m) Fig. 1. Initial temperature profiles for reference temperature T0 and x < s0 , and that T (0,t ) < Tf for all time. If the system (1)- actual temperature T . (2) is controlled such that 0 T 0,t = T 0,t (17) ( ) ( ) Corollary 1. Define the feedback-based estimates Tˆ ( x,t) ɶ for all time, then the reference error T ( x,t) is bounded in and sˆ(t) to be a solution to (1)-(2) with the Dirchlet 2 the L norm. boundary condition, based on boundary measurement of the Proof: Under these assumptions, applying the maximum plant, Tˆ (0,t) = T (0,t) . Then, if T and Tˆ satisfy principle for parabolic equations, T ( x,t) < T for all f assumptions (A1), (A2), and (A4), and for all times t , x < s t . This means that T s− < 0 , and noting the signs 2 ( ) x ( ) T (0,t) < Tf , the estimation error is bounded in the L norm. in (2), sɺ > 0 for all time. The same holds for T and s . This leads us to the following conjecture for an output- Again we use (15) as a Lyapunov functional candidate, feedback controller design. and take the time derivative. Integrating by parts and Conjecture 2. Let Tˆ ( x,t) and sˆ(t) be the estimates of the applying the boundary conditions, plant T x,t and s t using the output injection described L ( ) ( ) ɺ ɶ ɶ2 1ɺ ɶ 1 ɺ ɶ V T,.t = −a ∫ Tx dx +sT ()s − sT ()s in Corollary 1. Let the plant be controlled using the certainty () 0 b b equivalence method, i.e. calculating control law (6) or (14) Under assumption (A4), if s> s , then T s =T s = T , ɶ ( ) ( ) f based on the estimates. Then the reference error T ( x,t) ɺ ɶ and from above, T (s ) 1735 60 60 40 40 20 20 0 0 −20 0 −20 0 0 200 0 200 0.05 400 Reference temperature error (K) 600 0.05 400 Reference temperature error (K) 0.1 800 600 Time, t (s) 0.1 800 Time, t (s) Distance from strand surface, x (m) Distance from strand surface, x (m) a) Reference temperature error Tɶ (x,t ) a) Reference temperature error Tɶ (x,t ) 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 reference actual 0.02 reference Solidification front position, s (m) 0.01 actual 0 100 200 300 400 500 600 700 800 Solidification front position, s (m) Time, t (s) 0.01 0 100 200 300 400 500 600 700 800 b) Solidification front positions s (t) and s(t ) Time, t (s) 4 x 10 b) Solidification front positions s (t) and s(t ) 10 Fig. 2. Simulation results for system (1)-(2) with no control action. 8 6 position appear to converge to constant, non-zero values. 4 This approximates the current spray cooling state-of-the-art in most continuous casters, in which spray practices do not 2 account for changes in superheat or mold heat removal. 0 Neumann boundary control, u (K/s) 0 100 200 300 400 500 600 700 800 Figure 3 shows simulation results using control law (6). Time, t (s) Although not shown here, results were similar using control c) Neumann boundary control u (t) law (14) and the output-feedback control method described Fig. 3. Simulation results for system (1)-(2) under control law (6). in Conjecture 2. Although convergence was not proved for bounded control values, the reference temperature error [9] B. Furenes and B. Lie, "Solidification and control of a liquid appears to converge to 0. metal column," Simulation Modelling Practice and Theory, vol. 14, pp. 1112-1120, 2006. REFERENCES [10] K. Zheng, B. Petrus, B.G. 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