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Multivariable identification of a winding process by subspace methods for tension control
Control Engineering Practice 6 (1998) 1077—1088
Multivariable identification of a winding process by subspace methods for tension control T. Bastogne*, H. Noura, P. Sibille, A. Richard Centre de Recherche en Automatique de Nancy (CRAN), CNRS ESA 7039, Universite& Henri Poincare& , Nancy 1, BP 239, F-54506 Vandœuvre Cedex, France Received 1 August 1997
Abstract The challenging problem of multivariable tension control in winding systems is addressed in this paper. Two control configurations, the SS model and the ST model configurations (SS: speed-speed and ST: speed-torque), are considered and applied to a pilot winding plant. The relevance of using linear models in such multivariable systems is examined. Tension simulations based on a proposed linear ST model yield interesting results in both open and closed-loops. Emphasis is also laid on the applicability of subspace methods to winding tension state-space model identification. Such methods simplify the problem of multivariable identification by reducing the associated polynomial model structural identification to the easier problem of order estimation. It is shown that the design of a Linear Quadratic Gaussian (LQG) controller capable of ensuring efficient performance over the entire winding zone can be based on an ‘average’ state-space model identified around a nominal operating point. 1998 Elsevier Science ¸td. All rights reserved. Keywords: Multivariable identification; subspace methods; LQG control; metal; pulp and paper processing
1. Introduction Winding systems are in general continuous, nonlinear processes. They are encountered in a wide variety of industrial plants such as rolling mills in the steel industry (Parant et al., 1992; Walker and Wyatt-Mair, 1995), plants involving web conveyance (Sievers et al., 1988, Ebler et al., 1993; Braatz et al., 1996) including coating, papermaking and polymer film extrusion processes. In all cases, a web is a long, thin sheet of material, stiff or flexible, which can be coated or uncoated. The main role of a winding process is to control the web conveyance in order to avoid the effects of friction and sliding, as well as the problems of material distortion, which can, not only slow down production and thereby reduce productivity, but can also damage the quality of the final product. The solution consists of maintaining a traction effort on the strip by controlling the tension at different points along the web. A web under insufficient tension cannot track properly, and may wrinkle or prevent accurate cutting of the strip. In Ebler et al. (1993); Hearns et al. (1996), load cells, loopers or dancer rolls are among common mech-
* Corresponding author. E-mail: [email protected]
anical devices used in handling web tension. In the present application, another strategy is developed, which consists of directly adjusting the unwinding and rewinding tensions by controlling the torque or the angular speed of the reels. This investigation is built around a pilot plant, a subsystem of an industrial plant, which very well illustrates the problems encountered in larger processes. Rewinding and unwinding reel radii changes, which are not measured, constitute a major factor which makes the design of a multivariable winding tension controllers difficult. Most of the multivariable models reported in the literature for winding systems adopt a modeling approach for the design. The latter proposes a large number of tension equations, expressed under a wide variety of structures ranging from the simple relation of Hooke’s law to the more-involved, nonlinear relations proposed in Ebler et al. (1993). The present study is a further development of previous work published in Bastogne et al. (1997); Noura and Bastogne (1997). The first problem addressed in this paper is to analyse the relevance of identifying, rather than modeling, a multivariable linear time-invariant (LTI) model, with a view to controlling the unwinding and rewinding tensions of a plastic web. Two different control strategies, entitled the SS and ST configurations
0967-0661/98/$ — see front matter 1998 Elsevier Science Ltd. All rights reserved PII: S 0 9 6 7 - 0 6 6 1 ( 9 8 ) 0 0 0 6 9 - 0
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(SS: speed-speed and ST: speed-torque), are examined. The first approach is inspired by the majority of the theoretical equations of a strip tension, and consists of using both angular speeds as the manipulated inputs. The second control technique, called the ST configuration, proposes to handle the torque instead of the angular speed of a reel in order to drive the tension. The question of linearity about the relationship between the outputs (winding tensions) and the previously defined inputs (speed and/or torque) is discussed in both configurations. With this aim in view, experiments, followed by an identification approach based on subspace methods (4SID: Subspace based State-Space System IDentification) and prediction error methods (PEM), have been realized to estimate a linear multi-input/multi-output (MIMO) model of the tensions in each configuration. The simulation errors for each model (SS or ST type) in open-loop conditions are computed and analysed to accept or reject the approximation of the tension behavior by a linear relationship. The performances of a LQG tension controller applied to the pilot plant are analysed to check the previous conclusions. The second main goal of this paper is to illustrate the applicability of the recent subspace identification methods (Van Overschee and De Moor, 1996) to winding processes. The output errors in both open-loop and closed-loop conditions are examined and used as an estimation quality criterion. The 4SID and ARX (AutoRegressive polynomial structure of the PEM method) estimation results are also compared. The second part of the paper deals with a description of the pilot plant. In the third section, devoted to the modeling of the process, some basic theoretical relations on the winding tensions are recalled. The details of the experiment design that precedes the identification study are exposed in Section 4. The basic principle of the subspace identification methods is described in Section 5. Sections 6 and 7 are devoted to the identification of multivariable SS and ST models. The design and the application of a LQG controller, based on the previously estimated ST model, to the winding pilot plant is presented in Section 8. Finally, concluding remarks and future works are given in Section 9.
2. Process description A diagram of the plant is presented in Fig. 1. The main part of this pilot plant is a winding process composed of a plastic web and three reels, respectively called the unwinding, pacer and rewinding reels. Each reel is coupled with a DC-motor via gear reduction. The angular speed of each reel (S , S , S ) and both the tensions between the reels (¹ , ¹ ) are measured by tachometers and tension meters. At a second level, each motor is driven by a local controller composed of one or two PI
Fig. 1. Winding pilot system.
controllers. The first control loop adjusts the motor current (I , I , I ), and its integration time constant is about 40 ms, while the second loop controls the angular speed with an integration time constant equal to approximately 200 ms. The setpoints of those controllers (I* /S*, S* , I* /S*) are computed by a programmable logic con troller (PLC) in order to control both tensions and the linear velocity of the strip. For the application of new multivariable control structures, a real-time development environment (Simulink Real-Time Workshop#dSPACE) based on a PC computer is used instead of the PLC.
3. Modeling of the winding tensions 3.1. Wide variety of tension equations Several modeling studies have been proposed to describe the tension behavior of a web in different winding processes (Sievers et al., 1988; Parant et al., 1992; Ebler et al., 1993). All those theoretical models are, in large measure, based on the Hooke’s equation, given by Eq. (1), which expresses the linear relationship between the traction evolution, d¹ , and the elongation, de, of an G elastic stick. EA d¹ " de, G l with l E A
distance between reels axes (m), Young’s modulus (N/m), cross-section of the strip (m).
(1)
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In such a winding system, the elastic film moves from an unwinding reel to a rewinding reel. Consequently, the elongation is time-variant, and can be expressed in terms of a linear velocity difference, as defined by Eq. (2): d¹ EA G" (» (t)!» (t)), G> G dt l
(2)
with i index of the reel (cell), » reel peripheral velocity (m/s), G ¹ tension between reels i and i#1 (N). G Based on this last expression, an empirical tension equation, is proposed in Parant et al. (1992): EA d(D¹ (t)) » G # D¹ (t)" D» (t) , G G l l dt
(3)
D¹ (t)"¹ (t)!¹ (t) , (4) G G G\ D» (t)"» (t)!» (t), (5) G G> G where » is the average linear velocity of the strip. Around the nominal operating condition, » is generally constant. Accordingly, Eq. (3) can be approximated as a first-order differential equation with constant coefficients. The interest of this formula in comparison to Eq. (2) is in pointing out the tension interaction between the cell i and its precedent by introducing the previous tension ¹ in the G\ computation of ¹ . By taking account of other physical G phenomena, such as the invariability of mass and the balance of the momentum, Ebler et al. propose a nonlinear equation defined by Eq. (4)
d E E E l " » (t)! » (t), dt E#¹ (t) E#¹ (t) G\ E#¹ (t) G G G\ G (6) where E is the related modulus of elasticity. 3.2. Nonlinear effects due to the winding radii An important characteristic of the pilot plant, described in Section 2, resides in the variation of the radii, r and r respectively, of the unwinding and rewinding reels. The peripheral velocity of a reel is linked to its angular speed through the equation: » (t)"g ) S (t) ) r (t), (7) G G G G S angular speed of reel i (rd/s), G r radius of reel i (m), G g sliding factor3[0; 1]. G With Eq. (2) in mind, the above tension equation can be seen as representing either a bilinear relation or a quadratic relation. To the time-variations in reel radii correspond variations in the moments of inertia J
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and J of the outer reels, as expressed by the following equation: r (t) J (t)"M (t) G . G G 2
(8)
In Eq. (8), the mass M (t) is a time-variant parameter; G a more convenient way to express J (t) is then given by G Eq. (9) by taking account of the web density, o. J (t)"a ) (r (t)!r )#J , G G ? ? noj a" i 3 +1, 3, , 2
(9) (10)
where j is the strip width, J the moment of inertia of the ? driving shaft and r its radius. ? By way of summary, one encounters a wide range of theoretical models of the web tension in winding processes. The models encountered differ in their type and their complexity. All consider either the peripheral velocity or the angular reel speeds as degrees of freedom, through which the linear velocity and the tensions in a strip can be manipulated. Regarding the pilot plant, the first problem that must be settled is the choice of an adequate equation to describe the behavior of the tension. Where a linear relation correctly reproduces the web tension, there is no reason for resorting to a nonlinear model. But the question is whether it is realistic to use a linear equation for the control of such a system. Section 6 discusses the relevance of using a linear model to express the relation between web tension and reel angular speeds. In the absence of a priori knowledge about physical parameters, the present study proposes a multivariable identification approach, the details of which are explained in the next two sections.
4. Design of the experiment 4.1. Control objectives Driving a winding process essentially comes down to controlling the web’s linear velocity » and web tensions ¹ and ¹ about a given operating point. Since the pacer reel radius r is constant, one can control the strip’s linear velocity » by manipulating the angular speed S of the central motor using a ‘classical’ PID controller. However, the time-variations of r and r and the presence of important interactions make the design of the tension control more difficult. This justifies the identification of a multivariable tension model. 4.2. Pairings of controlled and manipulated variables The theoretical equations, presented in Section 3 to express the tension, show that angular speeds, S and S provide two degrees of freedom which can be used as
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Fig. 2. SS (speed-speed) web-control configuration. Fig. 4. Tension behavior with current excitations on all the winding conditions.
take account of the interaction between the central motor and the tensions, the angular speed (S ) is considered as a measured perturbation input. Estimation dataset. A portion of the estimation dataset measured over the identification zone is displayed in the diagrams of Fig. 5. The sampling rate ¹ was set at 0.1 s, Q and the input was generated by a random procedure based on 1000 samples. Fig. 3. Tension behavior with angular speed excitations.
5. Estimation algorithms control inputs. The associated control scheme is referred here to as the SS (speed speed) control configuration. One drawback of this strategy is illustrated by Fig. 3. As can be seen, exciting the two speed variables gives rise to increases in the tensions, which quickly produce effects of saturation in the tension meters, making the measurements go beyond the pre-set limits of the winding zone. Moreover, the resulting dataset does not contain sufficient sampled data to allow an accurate estimation of the model. This explains the use of the current setpoints (I*) and (I*) as the new manipulated inputs, instead of the angular speed setpoints. This second strategy consists of manipulating the torques T and T of the unwinding and rewinding reels. The associated control sheme is referred to as the ST (speed torque) control configuration, the details of which are given in Section 7 below. This new technique has the advantage of doing away with the slowest control loop (the angular control loop with ¹ "200 ms), thereby reducing the major time-constant G of the global system. Fig. 4 shows the tension behavior when the current setpoints are kept constant during the whole operating zone. In that case, a dotted window defines the nominal conditions and the identification area in which the tension trends are assumed to be linear and are removed before the model estimation step. To
Both state-space and input-output representations are considered for the winding model estimation. Most of the design of the identification procedure is based on the 4SID algorithms. An ARX-structure polynomial representation (PEM method) is then used to examine the ‘quality’ of the results of the 4SID estimation. 5.1. Subspace identification methods: basic principles The choice of state-space representation is motivated by the need to take account of the tension interaction during the winding process. The identification of the model is based on subspace methods (Larimore, 1990; Verhaegen, 1994; Van Overschee and De Moor, 1996). The major advantage of these associated algorithms resides in the reduction of the problem of structural estimation to the simpler problem of order estimation. The subspace algorithms use the input-output measurements directly to estimate discrete-time state-space models for MIMO systems. The relevant relations are given by: x "Ax #Bu #w I> I I I y "Cx #Du #v , (11) I I I I where x 3 1L is the state vector at the instant k, y 31K I I is the output vector, u 31N the input vector, A31L;L I
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Fig. 5. Estimation dataset. (a) Tension ¹ ; (b) Tension ¹ ; (c) Angular speed S ; (d) Angular speed S ; (e) Angular speed S ; ( f ) Current setpoint I * ; (g) Angular speed setpoint S * ; (h) Current setpoint I * .
the matrix of the dynamic system, B3 1L;N the input matrix, C3 1K;L the output matrix and D 3 1K;N the direct feed-through matrix. v 3 1K and w 3 1L denote I I unobservable signal vectors representing sequences of white noise. The hypotheses about the variables w and I v may be summarized as follows: I w Q S N (w2 v2 ) " E d *0. (12) O O v S2 R NO N The matrices Q 3 1L;L, S31L;K, R 31K;K denote the covariance matrices of the w and v vector sequences. All I I 4SID methods are geometric in their approach, and involve the subspaces spanned by the rows and/or columns of matrices constructed from the input/output data. As De Moor pointed out, the general algorithm of the subspace methods involves three major steps.
(1) Unlike the more ‘classical’ estimation methods based on a curve-fitting to derive polynomial models, the 4SID technique relies on subspace fitting strategies
for the approximation of the extended observability matrix C or the state matrix X, defined by: G C
C" G
A $
X"(x x 2x ). I I> I>H
(13)
CAG\
(2) Secondly, a singular value decomposition of the previously estimated matrix is computed to estimate the order n of the state-space model. (3) The final step computes the matrices AK , BK , CK and DK by solving over-determined sets of linear equations owing to least-squares or total least-squares computation techniques. The algorithm developed for the present investigation is based on De Moor and Van Overschee’s N4SID method which exploits the Matlab functions subid.m (Van Overschee and De Moor, 1996) and n4sid.m (Ljung, 1995).
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5.2. Multivariable ARX model The identification of a multivariable model for a winding process classically uses a polynomial description of an ARX type of structure. The relevant equation is given by: A(q\) y(t)"B(q\)u(t!n )#e(t) I with:
(14)
A(q\)"+a (q\), ; , GH K K a (q\)"d #a (q\)#2#aL?GH (q\L?GH), GH GH GH GH B(q\)"+b (q\), ; , GH K N GH b (q\)"b (q\LI )#2#bL@GH(q\LIGH\L@GH>) , GH GH GH where na 3 RK;K, nb 31K;N, and d denotes GH Kronecker delta symbol.
(15) (16) (17) (18) the
6. Identification of an SS linear model for the pilot winding plant The input and output vectors of the SS configuration are given by: y "(¹ (k) ¹ (k))2 I u "(» (k) » (k) » (k))2. I
velocities. A second model can then be estimated by using »K , »K and »K as inputs. Estimation of the order (n). The theoretical expressions for the tension suggest n"2 as a first approximation. Examination of the simulation errors computed with the 4SID algorithms roughly confirms this estimation, with nL "3. For the ARX polynomial structure, several observability indices have been chosen for each output, from l"[1 1] to l"[2 2]. Results of 4SID and ARX estimations. The main point here concerns the size of the estimation errors. On the basis of Figs. 6—9, giving the results of validation, the assumption of a linear relationship between tension and angular speeds or linear velocities cannot be validated. In addition, replacing the angular speeds in the input vector by linear velocities produces no significant decrease in the simulation error. Furthermore, compared to the ARX model, the results obtained with the 4SID methods enhanced the simulation error by 5%. Even though it is difficult to isolate the factors accounting for the difference in performance, one can emphasize the robustness of the 4SID methods in relation to real data.
(19) (20)
Approximating radii and linear velocities. Unlike the angular speeds, the radii and the linear velocities are not measured, the only exception being the radius r , which is kept constant at a value of 35 mm. A theoretical model was proposed for the radii in Eq. (21). The proposed model is validated by a large number of measurements on the initial and final values of each of the radii. More explicitly, one has d e r "! ) S , dt 2n e d r " )S , dt 2n
(21) Fig. 6. Simulated tension ¹K with 4SID.
(22)
where e"0.2 mm is the strip thickness. In the discussion that follows, no sliding effect is assumed to exist between the strip and the pacer reel, i.e. g "1, ∀i3[1; 3]. The G estimated variations of the radii during the estimation dataset are given by: Dr +0.95 cm and Dr +1.7 cm. It can be observed that the largest radius variation does not exceed 11%. In the estimation of the linear model, one can reasonably ignore the influence of the radii on the two tensions. Thus, S can be used in place of » as G G the input to the model (i"1, 2, 3) . However, to check the validity of the last assumption, Eq. (7) was used with the measurements of the three angular velocities and the estimated radii, rL and rL , to approximate the linear
Fig. 7. Simulated tension ¹K with 4SID.
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Fig. 10. ST (speed-torque) web control configuration. Fig. 8. Simulated tension ¹K with ARX.
The problem here is to check the relevance of the linearity assumption. As in the preceding section, the method consists of first applying the 4SID techniques to the dataset of Section 4 to estimate a linear state-space model, and then examining the associated simulation errors. 7.1. Order estimation
Fig. 9. Simulated tension ¹K with ARX.
7. Identification of a linear ST model The conclusion of the preceding section suggests another control strategy which is capable of reasonably supporting a linear law relating the tension to the input variables. The key to this novel approach is the control of a tension point via the reel torque. The schematic diagram of the new control strategy, referred to as the ST control configuration, is presented in Fig. 10. With reference to the diagram, it can be seen that the torque T is G> directly related to the motor torque and, as can be seen, when the radius is approximately constant during the test, the two torques are similar. The DC motors of the winding process are of permanent magnet material, i.e., they are characterized by a constant flux. Consequently, in each of the motors, the torque is linearly related to the induced current I , driven by two local controllers R and G R . One can therefore assume the existence of a linear tension model with the current setpoints serving as inputs to the model. In the ST control strategy, the associated input u , is given by: I u "(IK * (k)IK * (k)SK * (k))2. I
(23)
The most ambiguous aspect of the subspace algorithms has to do with selecting the order of the model. The decision is based on the detection of a gap in the spectrum of the singular values of the oblique projection matrix denoted O . The subspace spanned by the column G or row vectors of O coincides with either Im (C ) or G G Im (X). The first selection procedure consists of picking a singular value (p ), and considering other singular L values, p (p , as caused by noise. This means that only O L the first n singular values are characteristics of the dynamic behavior of the system, which values correspond to the poles of the model. More formally,
R 0 O "(U U ) Q G Q L 0 R L
V2 Q , V2 L
(24)
R "diag(p , 2 , p ) , Q L
(25)
nL "rank(R ). Q
(26)
As Fig. 11 illustrates, in the case of a finite set of data samples or data corrupted by noise, the results of application of this strategy become subjective and the decision as regards the order of the model is an arbitrary one. A more practical procedure is to choose the value n which minimizes the simulation error as shown in Fig. 12. This procedure in the 4SID identification methods allows more computational shortcuts than using ’classical’ techniques such as the PEM methods, which need to determine several structural indices. As shown in Fig. 12, the theoretical order, in the sense of a minimization of the simulation error, should be n"8.
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Fig. 11. Singular value spectrum.
Fig. 12. Simulation error versus order.
Nevertheless, this choice does not bring enough improvement in comparison with a reduced order given by n"3, which is the final estimation. 7.2. 4SID estimated state-space models The numerical values of the model corresponding to nL "3 are given by the following equations: x "AK x #BK u #BK S , I> I I I y "CK x , I I
(27)
in which:
0.8430 !0.0494 !0.0610
AK " !0.1464 !0.0244
!0.6535
0.6720 !0.4750 0.0513 0.2425
0.9011
Fig. 13. ¹ with its estimations.
(28)
0.0478
0.5628 , BK " !0.0799 BK " !0.5235 !0.0449 !0.0660 0.0068
CK "
0.5165 !0.0830 !0.1294 0.2222
0.7257
0.3086
(29)
,
with u "(I * (k)I * (k))2. I Figs. 13 and 14 present the real tensions ¹ and ¹ (represented by the solid lines), with their estima tions using 3rd and 8th-order models (dotted and dashed lines). As can be seen, the third-order model gives a sufficiently close illustration of the tension behavior. Compared to the SS model, the simulation errors in the present case are much lower: e (¹ )"34.7% and Q e (¹ )"42.8%, compared to the average of 85% for the Q SS model. These results are close to the ARX performances defined by: e (¹ )"35.4% and e (¹ )"44.8% Q Q with n "n "2 ) ones(2), n "ones(2), and ones(2) is ? @ I
Fig. 14. ¹ with its estimations.
a 2;2 matrix of ones. On the basis of these results, it is clear that in choosing between the SS and ST linear models, the preference must go to the ST control scheme. The next section considers the relevance of such models to the tension control application.
T. Bastogne et al. / Control Engineering Practice 6 (1998) 1077—1088
8. Optimal tension controller The advantages of the state-space formulation are especially apparent when designing controllers for MIMO systems. The first step in any multivariable control design should be an attempt either to find an approximate model consisting of two or more single input-output models, or else to decouple the control law matrix K and the estimator law matrix ¸. The application of such a decoupling technique is not treated in this study. Indeed, the objective here is not to compare several control strategies of the tensions, but to discover whether the multivariable state-space model estimated by the 4SID identification technique is sufficiently relevant for a given multivariable control application. Among the design methods of digital control systems using state-space models, the linear quadratic regulator, commonly abbreviated as LQR, plays a key role. It is also the mother of many current, systematic control design procedures for MIMO systems, such as the LQG approach. The control law considered here is only based on a LQG scheme. The latter enables the decomposition of the problem into both a feedforward control of S , and a state feedback control of the tensions ¹ and ¹ .
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BK BK , B " B " (34) ? ? 0 ; 0 ; K N K 0; L K B " C "(C 0 ). (35) ? ? K !¹ I Q K I is the m;m identity matrix. p"2 and m"2 give the K number of inputs and controlled outputs in that order. Classically, the state-feedback control law LQ#(I) is given by:
x u "!KX "!(K K ) I , I I V O q I 1 ,\ E (X2 Q X #u2R u ) K"arg min H A H H A H N , H
(36)
, (37)
where E+ ) , denotes the value of the mathematical expectation of the expression in braces. The weighting matrices Q and R are respectively non-negative definite symmetA A ric and positive definite symmetric matrices. In the present application, Q has been chosen so as to make the A ordered couple (A, Q ) detectable. R is a tuning matrix A A of the LQ controller. More explicitly:
CK 2CK Q" A 0 ; K L
8.1. LQG controller design
r 0 R " A A 0 r A
The design of a LQG controller associated with an integral error is presented next. The idea is to highlight the validity of the linear ST tension model defined in Eq. (27).
8.1.2. Optimal state estimation In the multivariable model given in Eq. (27), the state variables are unknown, and are estimated using the following Kalman filter
8.1.1. Optimal state-feedback control An infinite-horizon LQG controller is developed in this section. An integral error (q ) is added for the purI pose of eliminating the static error. The relevant relation is given by: q !q I> I"y !r , I I ¹ Q
¸"arg min
X "A X #B u #B S * #B r , I> ? I ? I ? I ? I
(31)
y "C X , I ? I
(32)
where
yL "CK xL , I I where
(30)
where (r ) denotes the reference vector. The associaI tion of the system state vector (x ) and the integral I error (q ) yields a composite open-loop system govI erned by the following augmented state-space representation:
x X" I I q I
xL "(AK !¸CK )xL #(BK BK ¸) I> I
AK 0; L K , A" ? ¹C I Q K
(33)
0; L K . I K
(38)
u I S* , I y I
,\ 1 lim E (x !xL )2(x !xL ) I I I I N , H
(39)
(40)
. (41)
The solution of this quadratic problem is obtained by solving the algebraic Riccati equation, and depends on the covariance matrices Q and R of the state and output. The estimation of the matrices R and Q are chosen such that:
r 0 R " D Q "BK BK 2, (42) D D 0 r D with (A, Q ) stabilisable, and where r and r are the D D D tuning parameters of the Kalman filter. The LQG control law is therefore based on the Certainty Equivalence
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Fig. 15. Tension multivariable controller.
Principle, that is, on replacing x by its estimation xL in I I Eq. (36), given by:
xL u "!KXK "!(K K ) I . I I V O q I
(43)
8.1.3. Weighted integral-error In addition to the four controller design parameters (r , r , r , r ), a weighting matrix (K ) is introduced in A A D D G order to allow direct and independent adjustment of the influence of the integral error on the control law for each output. The integral error is given by: q !q I> I"K (y !r ), (44) G I I ¹ Q where K 3 1K is a diagonal positive definite matrix. G Thus, the matrices of the new augmented state-space system become:
AK 0; 0; L K , L K . A" B " ? ? ¹KC I !¹ K Q G K Q G The choice of the design parameters is given by:
1 0 R" , A 0 1
1 0 R " , D 0 1
(45)
3 0 K" . (46) G 0 10
Fig. 15 shows the block diagram of the LQG#(I) control law that was first tested through simulation before the real-time application to the pilot plant using the dSPACE board associated with the Matlab/Simulink software. 8.2. Control implementation and results Figs. 16—19 present the output results of the implemented LQG controller for square reference signals ¹ * and ¹ * around a nominal operating point. The curve in the dotted line corresponds to the real measurement of
Fig. 16. ¹ tracking reference results: beginning of winding processing.
the controlled tension, while the solid line corresponds to the controlled tension simulated with the multivariable ST model determined with the 4SID algorithms. The similarity between the real and the simulated responses is a clear indication, not only of the applicability of the 4SID techniques for the identification of MIMO systems, but also of the relevance of the linear ST model for the control of the web tensions. Secondly, one also observes that the transient strip tension remains a correct behavior from the beginning of the winding (r : big) to the end (when r becomes small), and that this does not only apply over the small region on which the identification of the state-space model has been estimated. Analysis of performance. To understand the quasi-constant performance over the entire winding interval, the 4SID identification method has been applied to eight
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Fig. 20. Evolution of frequency characteristics.
Fig. 17. ¹ tracking reference results: end of winding processing.
datasets, extracted from eight different spots spread along the entire winding interval. Fig. 20 presents the frequency responses of the singular values GK ( ju) of the eight models thus estimated. A significant steadystate evolution of the model conditioning factor c( ju) can be noted: c "1.3)c(0))c "5. The radii changes are the main causes of this model evolution. The linear controller is able to handle this kind of change, which is essentially related to the static characteristics of the model. This explains why the constant level of control is maintained throughout the entire winding process.
9. Conclusion and perspectives
Fig. 18. ¹ tracking reference results: beginning of winding processing.
Fig. 19. ¹ tracking reference results: end of winding processing.
Two identification procedures based on SS and ST control configurations are proposed for web tension control in a winding pilot plant. The relevance of an identified multivariable linear time-invariant ST model as means of such control is weighed against classical solutions based on the modeling approach exploiting theoretical equations of strip tension, from which the proposed SS control scheme is inspired. This latter scheme uses reel angular speeds as inputs. The linear ST model identification results have shown a significant reduction (about 50% in comparison with the SS model) in the simulation errors, and thus confirmed the applicability of the linear description technique in correctly approximating the winding tension behavior. A further advantage of the ST model over the SS model resides in the avoidance of an additional local control loop for the adjustment of the angular speeds. This makes the ST control scheme faster than the SS configuration. The ST model identification has been achieved with the 4SID subspace methods in order to obtain a statespace representation that is able to take account of the tension interactions of the winding process. The
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simplicity of the 4SID methodology has been illustrated in this paper. Moreover, the size of the simulation errors highlights the advantage of the subspace algorithms in comparison to an ARX polynomial structure associated with the PEM technique, for which the need for structural identification constitutes another drawback. The identified linear ST model has facilitated the development of an LQG controller implemented in a dSPACE board connected to the local controllers. Measured and simulated controlled tensions have shown significant similarity over the entire winding interval. The small size of the simulation errors under closed-loop conditions further confirmed the relevance of the 4SIDtechnique-based linear state-space model for the control application envisaged. As final conclusion, an LTI model of such a nonlinear process is not a utopia. One only needs to choose the convenient control configuration, i.e. the ST scheme. Moreover, it is shown and explained that despite the non-measured radii changes, the identified LTI MIMO model is sufficient for the development of an efficient LQG control of the tension on the entire winding zone. The influence of the variation of reel radii have hitherto been neglected in short interval estimation datasets. The authors’ conviction is that improvements could be expected in the quality of the models if such variations were to be taken into account in the identification stage. It is therefore intended to orient future investigations in this direction, in order to determine a MIMO model able to describe the overall tension behavior during the whole winding processing, and not only around a local operating point. A bilinear extension of subspace identification methods (Favoreel et al., 1996) would be interesting in this case.
References Bastogne, T., Noura, H., Richard, A., & Hittinger, J.M. (1997). Application of subspace methods to the identification of a winding process. In: Proc. of the 4th European Control Conference. Vol. 5. Bruxelles. Braatz, R.D., Ogunnaike, B.A., & Featherstone, A.P. (1996). Identification, estimation and control of sheet and film processes. In: Proc. of the 13th ¹riennial IFAC ¼orld Congress. San Francisco, pp. 319—324. Ebler, N.A., Arnason, R., Michaelis, G., & D’Sa, N. (1993). Tension control: Dancer rolls or load cells. IEEE ¹rans. on Industry applications, 29(4), 727—739. Favoreel, W., De Moor, B., & Van Overschee, P. (1996). A bilinear extension of subspace identification for systems subject to white inputs. In: Proc. of the American Control Conference. Vol. 1, Albuquerque, New Mexico, pp. 607—611. Hearns, G., Katebi, M.R., & Grimble, M.J. (1996). Robust control of a hot strip mill looper. In: Proc. of the 13th ¹riennial IFAC ¼orld Congress. San Francisco, USA, pp. 445—450. Larimore, W.E. (1990). Canonical variate analysis in identification, filtering, and adaptative control. Proc. of the 29th Conference on Decision and Control. Honolulu, Hawai, pp. 596—604. Ljung, L. (1995). System identification toolbox user’s guide. The Math Works, Inc. Noura, H., & Bastogne, T. (1997). Tension optimal control of a multivariable winding process. In: Proc. of the American Control Conference. Vol. 3, Albuquerque, New Mexico, pp. 2499—2503. Parant, F., Coeffier, C., & Iung, C. (1992). Modeling of web tension in a continuous annealing line. Iron and Steel Engineer, pp. 46—49. Sievers, L., Balas, M.J., & Von Flotow, A. (1988). Modeling of web conveyance systems for multivariable control. IEEE ¹rans. on Automatic Control, 33(6), 524—531. Van Overschee, P., & De Moor, B. (1996). Subspace identification for linear systems — ¹heory, implementation, applications. Kluwer Academic Publishers. Verhaegen, M. (1994). Identification of the deterministic part of the MIMO state space models given in innovations form from inputoutput data. Automatica, 30(1), 61—74. Walker, Jr. N.D., & Wyatt-Mair, G.F. (1995). Sensor signal validation using analytical redundancy for an aluminium cold rolling mill. Control Eng. Practice, 3(6), 753—760.
Multivariable identification of a winding process by subspace methods for tension control T. Bastogne*, H. Noura, P. Sibille, A. Richard Centre de Recherche en Automatique de Nancy (CRAN), CNRS ESA 7039, Universite& Henri Poincare& , Nancy 1, BP 239, F-54506 Vandœuvre Cedex, France Received 1 August 1997
Abstract The challenging problem of multivariable tension control in winding systems is addressed in this paper. Two control configurations, the SS model and the ST model configurations (SS: speed-speed and ST: speed-torque), are considered and applied to a pilot winding plant. The relevance of using linear models in such multivariable systems is examined. Tension simulations based on a proposed linear ST model yield interesting results in both open and closed-loops. Emphasis is also laid on the applicability of subspace methods to winding tension state-space model identification. Such methods simplify the problem of multivariable identification by reducing the associated polynomial model structural identification to the easier problem of order estimation. It is shown that the design of a Linear Quadratic Gaussian (LQG) controller capable of ensuring efficient performance over the entire winding zone can be based on an ‘average’ state-space model identified around a nominal operating point. 1998 Elsevier Science ¸td. All rights reserved. Keywords: Multivariable identification; subspace methods; LQG control; metal; pulp and paper processing
1. Introduction Winding systems are in general continuous, nonlinear processes. They are encountered in a wide variety of industrial plants such as rolling mills in the steel industry (Parant et al., 1992; Walker and Wyatt-Mair, 1995), plants involving web conveyance (Sievers et al., 1988, Ebler et al., 1993; Braatz et al., 1996) including coating, papermaking and polymer film extrusion processes. In all cases, a web is a long, thin sheet of material, stiff or flexible, which can be coated or uncoated. The main role of a winding process is to control the web conveyance in order to avoid the effects of friction and sliding, as well as the problems of material distortion, which can, not only slow down production and thereby reduce productivity, but can also damage the quality of the final product. The solution consists of maintaining a traction effort on the strip by controlling the tension at different points along the web. A web under insufficient tension cannot track properly, and may wrinkle or prevent accurate cutting of the strip. In Ebler et al. (1993); Hearns et al. (1996), load cells, loopers or dancer rolls are among common mech-
* Corresponding author. E-mail: [email protected]
anical devices used in handling web tension. In the present application, another strategy is developed, which consists of directly adjusting the unwinding and rewinding tensions by controlling the torque or the angular speed of the reels. This investigation is built around a pilot plant, a subsystem of an industrial plant, which very well illustrates the problems encountered in larger processes. Rewinding and unwinding reel radii changes, which are not measured, constitute a major factor which makes the design of a multivariable winding tension controllers difficult. Most of the multivariable models reported in the literature for winding systems adopt a modeling approach for the design. The latter proposes a large number of tension equations, expressed under a wide variety of structures ranging from the simple relation of Hooke’s law to the more-involved, nonlinear relations proposed in Ebler et al. (1993). The present study is a further development of previous work published in Bastogne et al. (1997); Noura and Bastogne (1997). The first problem addressed in this paper is to analyse the relevance of identifying, rather than modeling, a multivariable linear time-invariant (LTI) model, with a view to controlling the unwinding and rewinding tensions of a plastic web. Two different control strategies, entitled the SS and ST configurations
0967-0661/98/$ — see front matter 1998 Elsevier Science Ltd. All rights reserved PII: S 0 9 6 7 - 0 6 6 1 ( 9 8 ) 0 0 0 6 9 - 0
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(SS: speed-speed and ST: speed-torque), are examined. The first approach is inspired by the majority of the theoretical equations of a strip tension, and consists of using both angular speeds as the manipulated inputs. The second control technique, called the ST configuration, proposes to handle the torque instead of the angular speed of a reel in order to drive the tension. The question of linearity about the relationship between the outputs (winding tensions) and the previously defined inputs (speed and/or torque) is discussed in both configurations. With this aim in view, experiments, followed by an identification approach based on subspace methods (4SID: Subspace based State-Space System IDentification) and prediction error methods (PEM), have been realized to estimate a linear multi-input/multi-output (MIMO) model of the tensions in each configuration. The simulation errors for each model (SS or ST type) in open-loop conditions are computed and analysed to accept or reject the approximation of the tension behavior by a linear relationship. The performances of a LQG tension controller applied to the pilot plant are analysed to check the previous conclusions. The second main goal of this paper is to illustrate the applicability of the recent subspace identification methods (Van Overschee and De Moor, 1996) to winding processes. The output errors in both open-loop and closed-loop conditions are examined and used as an estimation quality criterion. The 4SID and ARX (AutoRegressive polynomial structure of the PEM method) estimation results are also compared. The second part of the paper deals with a description of the pilot plant. In the third section, devoted to the modeling of the process, some basic theoretical relations on the winding tensions are recalled. The details of the experiment design that precedes the identification study are exposed in Section 4. The basic principle of the subspace identification methods is described in Section 5. Sections 6 and 7 are devoted to the identification of multivariable SS and ST models. The design and the application of a LQG controller, based on the previously estimated ST model, to the winding pilot plant is presented in Section 8. Finally, concluding remarks and future works are given in Section 9.
2. Process description A diagram of the plant is presented in Fig. 1. The main part of this pilot plant is a winding process composed of a plastic web and three reels, respectively called the unwinding, pacer and rewinding reels. Each reel is coupled with a DC-motor via gear reduction. The angular speed of each reel (S , S , S ) and both the tensions between the reels (¹ , ¹ ) are measured by tachometers and tension meters. At a second level, each motor is driven by a local controller composed of one or two PI
Fig. 1. Winding pilot system.
controllers. The first control loop adjusts the motor current (I , I , I ), and its integration time constant is about 40 ms, while the second loop controls the angular speed with an integration time constant equal to approximately 200 ms. The setpoints of those controllers (I* /S*, S* , I* /S*) are computed by a programmable logic con troller (PLC) in order to control both tensions and the linear velocity of the strip. For the application of new multivariable control structures, a real-time development environment (Simulink Real-Time Workshop#dSPACE) based on a PC computer is used instead of the PLC.
3. Modeling of the winding tensions 3.1. Wide variety of tension equations Several modeling studies have been proposed to describe the tension behavior of a web in different winding processes (Sievers et al., 1988; Parant et al., 1992; Ebler et al., 1993). All those theoretical models are, in large measure, based on the Hooke’s equation, given by Eq. (1), which expresses the linear relationship between the traction evolution, d¹ , and the elongation, de, of an G elastic stick. EA d¹ " de, G l with l E A
distance between reels axes (m), Young’s modulus (N/m), cross-section of the strip (m).
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In such a winding system, the elastic film moves from an unwinding reel to a rewinding reel. Consequently, the elongation is time-variant, and can be expressed in terms of a linear velocity difference, as defined by Eq. (2): d¹ EA G" (» (t)!» (t)), G> G dt l
(2)
with i index of the reel (cell), » reel peripheral velocity (m/s), G ¹ tension between reels i and i#1 (N). G Based on this last expression, an empirical tension equation, is proposed in Parant et al. (1992): EA d(D¹ (t)) » G # D¹ (t)" D» (t) , G G l l dt
(3)
D¹ (t)"¹ (t)!¹ (t) , (4) G G G\ D» (t)"» (t)!» (t), (5) G G> G where » is the average linear velocity of the strip. Around the nominal operating condition, » is generally constant. Accordingly, Eq. (3) can be approximated as a first-order differential equation with constant coefficients. The interest of this formula in comparison to Eq. (2) is in pointing out the tension interaction between the cell i and its precedent by introducing the previous tension ¹ in the G\ computation of ¹ . By taking account of other physical G phenomena, such as the invariability of mass and the balance of the momentum, Ebler et al. propose a nonlinear equation defined by Eq. (4)
d E E E l " » (t)! » (t), dt E#¹ (t) E#¹ (t) G\ E#¹ (t) G G G\ G (6) where E is the related modulus of elasticity. 3.2. Nonlinear effects due to the winding radii An important characteristic of the pilot plant, described in Section 2, resides in the variation of the radii, r and r respectively, of the unwinding and rewinding reels. The peripheral velocity of a reel is linked to its angular speed through the equation: » (t)"g ) S (t) ) r (t), (7) G G G G S angular speed of reel i (rd/s), G r radius of reel i (m), G g sliding factor3[0; 1]. G With Eq. (2) in mind, the above tension equation can be seen as representing either a bilinear relation or a quadratic relation. To the time-variations in reel radii correspond variations in the moments of inertia J
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and J of the outer reels, as expressed by the following equation: r (t) J (t)"M (t) G . G G 2
(8)
In Eq. (8), the mass M (t) is a time-variant parameter; G a more convenient way to express J (t) is then given by G Eq. (9) by taking account of the web density, o. J (t)"a ) (r (t)!r )#J , G G ? ? noj a" i 3 +1, 3, , 2
(9) (10)
where j is the strip width, J the moment of inertia of the ? driving shaft and r its radius. ? By way of summary, one encounters a wide range of theoretical models of the web tension in winding processes. The models encountered differ in their type and their complexity. All consider either the peripheral velocity or the angular reel speeds as degrees of freedom, through which the linear velocity and the tensions in a strip can be manipulated. Regarding the pilot plant, the first problem that must be settled is the choice of an adequate equation to describe the behavior of the tension. Where a linear relation correctly reproduces the web tension, there is no reason for resorting to a nonlinear model. But the question is whether it is realistic to use a linear equation for the control of such a system. Section 6 discusses the relevance of using a linear model to express the relation between web tension and reel angular speeds. In the absence of a priori knowledge about physical parameters, the present study proposes a multivariable identification approach, the details of which are explained in the next two sections.
4. Design of the experiment 4.1. Control objectives Driving a winding process essentially comes down to controlling the web’s linear velocity » and web tensions ¹ and ¹ about a given operating point. Since the pacer reel radius r is constant, one can control the strip’s linear velocity » by manipulating the angular speed S of the central motor using a ‘classical’ PID controller. However, the time-variations of r and r and the presence of important interactions make the design of the tension control more difficult. This justifies the identification of a multivariable tension model. 4.2. Pairings of controlled and manipulated variables The theoretical equations, presented in Section 3 to express the tension, show that angular speeds, S and S provide two degrees of freedom which can be used as
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Fig. 2. SS (speed-speed) web-control configuration. Fig. 4. Tension behavior with current excitations on all the winding conditions.
take account of the interaction between the central motor and the tensions, the angular speed (S ) is considered as a measured perturbation input. Estimation dataset. A portion of the estimation dataset measured over the identification zone is displayed in the diagrams of Fig. 5. The sampling rate ¹ was set at 0.1 s, Q and the input was generated by a random procedure based on 1000 samples. Fig. 3. Tension behavior with angular speed excitations.
5. Estimation algorithms control inputs. The associated control scheme is referred here to as the SS (speed speed) control configuration. One drawback of this strategy is illustrated by Fig. 3. As can be seen, exciting the two speed variables gives rise to increases in the tensions, which quickly produce effects of saturation in the tension meters, making the measurements go beyond the pre-set limits of the winding zone. Moreover, the resulting dataset does not contain sufficient sampled data to allow an accurate estimation of the model. This explains the use of the current setpoints (I*) and (I*) as the new manipulated inputs, instead of the angular speed setpoints. This second strategy consists of manipulating the torques T and T of the unwinding and rewinding reels. The associated control sheme is referred to as the ST (speed torque) control configuration, the details of which are given in Section 7 below. This new technique has the advantage of doing away with the slowest control loop (the angular control loop with ¹ "200 ms), thereby reducing the major time-constant G of the global system. Fig. 4 shows the tension behavior when the current setpoints are kept constant during the whole operating zone. In that case, a dotted window defines the nominal conditions and the identification area in which the tension trends are assumed to be linear and are removed before the model estimation step. To
Both state-space and input-output representations are considered for the winding model estimation. Most of the design of the identification procedure is based on the 4SID algorithms. An ARX-structure polynomial representation (PEM method) is then used to examine the ‘quality’ of the results of the 4SID estimation. 5.1. Subspace identification methods: basic principles The choice of state-space representation is motivated by the need to take account of the tension interaction during the winding process. The identification of the model is based on subspace methods (Larimore, 1990; Verhaegen, 1994; Van Overschee and De Moor, 1996). The major advantage of these associated algorithms resides in the reduction of the problem of structural estimation to the simpler problem of order estimation. The subspace algorithms use the input-output measurements directly to estimate discrete-time state-space models for MIMO systems. The relevant relations are given by: x "Ax #Bu #w I> I I I y "Cx #Du #v , (11) I I I I where x 3 1L is the state vector at the instant k, y 31K I I is the output vector, u 31N the input vector, A31L;L I
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Fig. 5. Estimation dataset. (a) Tension ¹ ; (b) Tension ¹ ; (c) Angular speed S ; (d) Angular speed S ; (e) Angular speed S ; ( f ) Current setpoint I * ; (g) Angular speed setpoint S * ; (h) Current setpoint I * .
the matrix of the dynamic system, B3 1L;N the input matrix, C3 1K;L the output matrix and D 3 1K;N the direct feed-through matrix. v 3 1K and w 3 1L denote I I unobservable signal vectors representing sequences of white noise. The hypotheses about the variables w and I v may be summarized as follows: I w Q S N (w2 v2 ) " E d *0. (12) O O v S2 R NO N The matrices Q 3 1L;L, S31L;K, R 31K;K denote the covariance matrices of the w and v vector sequences. All I I 4SID methods are geometric in their approach, and involve the subspaces spanned by the rows and/or columns of matrices constructed from the input/output data. As De Moor pointed out, the general algorithm of the subspace methods involves three major steps.
(1) Unlike the more ‘classical’ estimation methods based on a curve-fitting to derive polynomial models, the 4SID technique relies on subspace fitting strategies
for the approximation of the extended observability matrix C or the state matrix X, defined by: G C
C" G
A $
X"(x x 2x ). I I> I>H
(13)
CAG\
(2) Secondly, a singular value decomposition of the previously estimated matrix is computed to estimate the order n of the state-space model. (3) The final step computes the matrices AK , BK , CK and DK by solving over-determined sets of linear equations owing to least-squares or total least-squares computation techniques. The algorithm developed for the present investigation is based on De Moor and Van Overschee’s N4SID method which exploits the Matlab functions subid.m (Van Overschee and De Moor, 1996) and n4sid.m (Ljung, 1995).
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5.2. Multivariable ARX model The identification of a multivariable model for a winding process classically uses a polynomial description of an ARX type of structure. The relevant equation is given by: A(q\) y(t)"B(q\)u(t!n )#e(t) I with:
(14)
A(q\)"+a (q\), ; , GH K K a (q\)"d #a (q\)#2#aL?GH (q\L?GH), GH GH GH GH B(q\)"+b (q\), ; , GH K N GH b (q\)"b (q\LI )#2#bL@GH(q\LIGH\L@GH>) , GH GH GH where na 3 RK;K, nb 31K;N, and d denotes GH Kronecker delta symbol.
(15) (16) (17) (18) the
6. Identification of an SS linear model for the pilot winding plant The input and output vectors of the SS configuration are given by: y "(¹ (k) ¹ (k))2 I u "(» (k) » (k) » (k))2. I
velocities. A second model can then be estimated by using »K , »K and »K as inputs. Estimation of the order (n). The theoretical expressions for the tension suggest n"2 as a first approximation. Examination of the simulation errors computed with the 4SID algorithms roughly confirms this estimation, with nL "3. For the ARX polynomial structure, several observability indices have been chosen for each output, from l"[1 1] to l"[2 2]. Results of 4SID and ARX estimations. The main point here concerns the size of the estimation errors. On the basis of Figs. 6—9, giving the results of validation, the assumption of a linear relationship between tension and angular speeds or linear velocities cannot be validated. In addition, replacing the angular speeds in the input vector by linear velocities produces no significant decrease in the simulation error. Furthermore, compared to the ARX model, the results obtained with the 4SID methods enhanced the simulation error by 5%. Even though it is difficult to isolate the factors accounting for the difference in performance, one can emphasize the robustness of the 4SID methods in relation to real data.
(19) (20)
Approximating radii and linear velocities. Unlike the angular speeds, the radii and the linear velocities are not measured, the only exception being the radius r , which is kept constant at a value of 35 mm. A theoretical model was proposed for the radii in Eq. (21). The proposed model is validated by a large number of measurements on the initial and final values of each of the radii. More explicitly, one has d e r "! ) S , dt 2n e d r " )S , dt 2n
(21) Fig. 6. Simulated tension ¹K with 4SID.
(22)
where e"0.2 mm is the strip thickness. In the discussion that follows, no sliding effect is assumed to exist between the strip and the pacer reel, i.e. g "1, ∀i3[1; 3]. The G estimated variations of the radii during the estimation dataset are given by: Dr +0.95 cm and Dr +1.7 cm. It can be observed that the largest radius variation does not exceed 11%. In the estimation of the linear model, one can reasonably ignore the influence of the radii on the two tensions. Thus, S can be used in place of » as G G the input to the model (i"1, 2, 3) . However, to check the validity of the last assumption, Eq. (7) was used with the measurements of the three angular velocities and the estimated radii, rL and rL , to approximate the linear
Fig. 7. Simulated tension ¹K with 4SID.
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Fig. 10. ST (speed-torque) web control configuration. Fig. 8. Simulated tension ¹K with ARX.
The problem here is to check the relevance of the linearity assumption. As in the preceding section, the method consists of first applying the 4SID techniques to the dataset of Section 4 to estimate a linear state-space model, and then examining the associated simulation errors. 7.1. Order estimation
Fig. 9. Simulated tension ¹K with ARX.
7. Identification of a linear ST model The conclusion of the preceding section suggests another control strategy which is capable of reasonably supporting a linear law relating the tension to the input variables. The key to this novel approach is the control of a tension point via the reel torque. The schematic diagram of the new control strategy, referred to as the ST control configuration, is presented in Fig. 10. With reference to the diagram, it can be seen that the torque T is G> directly related to the motor torque and, as can be seen, when the radius is approximately constant during the test, the two torques are similar. The DC motors of the winding process are of permanent magnet material, i.e., they are characterized by a constant flux. Consequently, in each of the motors, the torque is linearly related to the induced current I , driven by two local controllers R and G R . One can therefore assume the existence of a linear tension model with the current setpoints serving as inputs to the model. In the ST control strategy, the associated input u , is given by: I u "(IK * (k)IK * (k)SK * (k))2. I
(23)
The most ambiguous aspect of the subspace algorithms has to do with selecting the order of the model. The decision is based on the detection of a gap in the spectrum of the singular values of the oblique projection matrix denoted O . The subspace spanned by the column G or row vectors of O coincides with either Im (C ) or G G Im (X). The first selection procedure consists of picking a singular value (p ), and considering other singular L values, p (p , as caused by noise. This means that only O L the first n singular values are characteristics of the dynamic behavior of the system, which values correspond to the poles of the model. More formally,
R 0 O "(U U ) Q G Q L 0 R L
V2 Q , V2 L
(24)
R "diag(p , 2 , p ) , Q L
(25)
nL "rank(R ). Q
(26)
As Fig. 11 illustrates, in the case of a finite set of data samples or data corrupted by noise, the results of application of this strategy become subjective and the decision as regards the order of the model is an arbitrary one. A more practical procedure is to choose the value n which minimizes the simulation error as shown in Fig. 12. This procedure in the 4SID identification methods allows more computational shortcuts than using ’classical’ techniques such as the PEM methods, which need to determine several structural indices. As shown in Fig. 12, the theoretical order, in the sense of a minimization of the simulation error, should be n"8.
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Fig. 11. Singular value spectrum.
Fig. 12. Simulation error versus order.
Nevertheless, this choice does not bring enough improvement in comparison with a reduced order given by n"3, which is the final estimation. 7.2. 4SID estimated state-space models The numerical values of the model corresponding to nL "3 are given by the following equations: x "AK x #BK u #BK S , I> I I I y "CK x , I I
(27)
in which:
0.8430 !0.0494 !0.0610
AK " !0.1464 !0.0244
!0.6535
0.6720 !0.4750 0.0513 0.2425
0.9011
Fig. 13. ¹ with its estimations.
(28)
0.0478
0.5628 , BK " !0.0799 BK " !0.5235 !0.0449 !0.0660 0.0068
CK "
0.5165 !0.0830 !0.1294 0.2222
0.7257
0.3086
(29)
,
with u "(I * (k)I * (k))2. I Figs. 13 and 14 present the real tensions ¹ and ¹ (represented by the solid lines), with their estima tions using 3rd and 8th-order models (dotted and dashed lines). As can be seen, the third-order model gives a sufficiently close illustration of the tension behavior. Compared to the SS model, the simulation errors in the present case are much lower: e (¹ )"34.7% and Q e (¹ )"42.8%, compared to the average of 85% for the Q SS model. These results are close to the ARX performances defined by: e (¹ )"35.4% and e (¹ )"44.8% Q Q with n "n "2 ) ones(2), n "ones(2), and ones(2) is ? @ I
Fig. 14. ¹ with its estimations.
a 2;2 matrix of ones. On the basis of these results, it is clear that in choosing between the SS and ST linear models, the preference must go to the ST control scheme. The next section considers the relevance of such models to the tension control application.
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8. Optimal tension controller The advantages of the state-space formulation are especially apparent when designing controllers for MIMO systems. The first step in any multivariable control design should be an attempt either to find an approximate model consisting of two or more single input-output models, or else to decouple the control law matrix K and the estimator law matrix ¸. The application of such a decoupling technique is not treated in this study. Indeed, the objective here is not to compare several control strategies of the tensions, but to discover whether the multivariable state-space model estimated by the 4SID identification technique is sufficiently relevant for a given multivariable control application. Among the design methods of digital control systems using state-space models, the linear quadratic regulator, commonly abbreviated as LQR, plays a key role. It is also the mother of many current, systematic control design procedures for MIMO systems, such as the LQG approach. The control law considered here is only based on a LQG scheme. The latter enables the decomposition of the problem into both a feedforward control of S , and a state feedback control of the tensions ¹ and ¹ .
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BK BK , B " B " (34) ? ? 0 ; 0 ; K N K 0; L K B " C "(C 0 ). (35) ? ? K !¹ I Q K I is the m;m identity matrix. p"2 and m"2 give the K number of inputs and controlled outputs in that order. Classically, the state-feedback control law LQ#(I) is given by:
x u "!KX "!(K K ) I , I I V O q I 1 ,\ E (X2 Q X #u2R u ) K"arg min H A H H A H N , H
(36)
, (37)
where E+ ) , denotes the value of the mathematical expectation of the expression in braces. The weighting matrices Q and R are respectively non-negative definite symmetA A ric and positive definite symmetric matrices. In the present application, Q has been chosen so as to make the A ordered couple (A, Q ) detectable. R is a tuning matrix A A of the LQ controller. More explicitly:
CK 2CK Q" A 0 ; K L
8.1. LQG controller design
r 0 R " A A 0 r A
The design of a LQG controller associated with an integral error is presented next. The idea is to highlight the validity of the linear ST tension model defined in Eq. (27).
8.1.2. Optimal state estimation In the multivariable model given in Eq. (27), the state variables are unknown, and are estimated using the following Kalman filter
8.1.1. Optimal state-feedback control An infinite-horizon LQG controller is developed in this section. An integral error (q ) is added for the purI pose of eliminating the static error. The relevant relation is given by: q !q I> I"y !r , I I ¹ Q
¸"arg min
X "A X #B u #B S * #B r , I> ? I ? I ? I ? I
(31)
y "C X , I ? I
(32)
where
yL "CK xL , I I where
(30)
where (r ) denotes the reference vector. The associaI tion of the system state vector (x ) and the integral I error (q ) yields a composite open-loop system govI erned by the following augmented state-space representation:
x X" I I q I
xL "(AK !¸CK )xL #(BK BK ¸) I> I
AK 0; L K , A" ? ¹C I Q K
(33)
0; L K . I K
(38)
u I S* , I y I
,\ 1 lim E (x !xL )2(x !xL ) I I I I N , H
(39)
(40)
. (41)
The solution of this quadratic problem is obtained by solving the algebraic Riccati equation, and depends on the covariance matrices Q and R of the state and output. The estimation of the matrices R and Q are chosen such that:
r 0 R " D Q "BK BK 2, (42) D D 0 r D with (A, Q ) stabilisable, and where r and r are the D D D tuning parameters of the Kalman filter. The LQG control law is therefore based on the Certainty Equivalence
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Fig. 15. Tension multivariable controller.
Principle, that is, on replacing x by its estimation xL in I I Eq. (36), given by:
xL u "!KXK "!(K K ) I . I I V O q I
(43)
8.1.3. Weighted integral-error In addition to the four controller design parameters (r , r , r , r ), a weighting matrix (K ) is introduced in A A D D G order to allow direct and independent adjustment of the influence of the integral error on the control law for each output. The integral error is given by: q !q I> I"K (y !r ), (44) G I I ¹ Q where K 3 1K is a diagonal positive definite matrix. G Thus, the matrices of the new augmented state-space system become:
AK 0; 0; L K , L K . A" B " ? ? ¹KC I !¹ K Q G K Q G The choice of the design parameters is given by:
1 0 R" , A 0 1
1 0 R " , D 0 1
(45)
3 0 K" . (46) G 0 10
Fig. 15 shows the block diagram of the LQG#(I) control law that was first tested through simulation before the real-time application to the pilot plant using the dSPACE board associated with the Matlab/Simulink software. 8.2. Control implementation and results Figs. 16—19 present the output results of the implemented LQG controller for square reference signals ¹ * and ¹ * around a nominal operating point. The curve in the dotted line corresponds to the real measurement of
Fig. 16. ¹ tracking reference results: beginning of winding processing.
the controlled tension, while the solid line corresponds to the controlled tension simulated with the multivariable ST model determined with the 4SID algorithms. The similarity between the real and the simulated responses is a clear indication, not only of the applicability of the 4SID techniques for the identification of MIMO systems, but also of the relevance of the linear ST model for the control of the web tensions. Secondly, one also observes that the transient strip tension remains a correct behavior from the beginning of the winding (r : big) to the end (when r becomes small), and that this does not only apply over the small region on which the identification of the state-space model has been estimated. Analysis of performance. To understand the quasi-constant performance over the entire winding interval, the 4SID identification method has been applied to eight
T. Bastogne et al. / Control Engineering Practice 6 (1998) 1077—1088
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Fig. 20. Evolution of frequency characteristics.
Fig. 17. ¹ tracking reference results: end of winding processing.
datasets, extracted from eight different spots spread along the entire winding interval. Fig. 20 presents the frequency responses of the singular values GK ( ju) of the eight models thus estimated. A significant steadystate evolution of the model conditioning factor c( ju) can be noted: c "1.3)c(0))c "5. The radii changes are the main causes of this model evolution. The linear controller is able to handle this kind of change, which is essentially related to the static characteristics of the model. This explains why the constant level of control is maintained throughout the entire winding process.
9. Conclusion and perspectives
Fig. 18. ¹ tracking reference results: beginning of winding processing.
Fig. 19. ¹ tracking reference results: end of winding processing.
Two identification procedures based on SS and ST control configurations are proposed for web tension control in a winding pilot plant. The relevance of an identified multivariable linear time-invariant ST model as means of such control is weighed against classical solutions based on the modeling approach exploiting theoretical equations of strip tension, from which the proposed SS control scheme is inspired. This latter scheme uses reel angular speeds as inputs. The linear ST model identification results have shown a significant reduction (about 50% in comparison with the SS model) in the simulation errors, and thus confirmed the applicability of the linear description technique in correctly approximating the winding tension behavior. A further advantage of the ST model over the SS model resides in the avoidance of an additional local control loop for the adjustment of the angular speeds. This makes the ST control scheme faster than the SS configuration. The ST model identification has been achieved with the 4SID subspace methods in order to obtain a statespace representation that is able to take account of the tension interactions of the winding process. The
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simplicity of the 4SID methodology has been illustrated in this paper. Moreover, the size of the simulation errors highlights the advantage of the subspace algorithms in comparison to an ARX polynomial structure associated with the PEM technique, for which the need for structural identification constitutes another drawback. The identified linear ST model has facilitated the development of an LQG controller implemented in a dSPACE board connected to the local controllers. Measured and simulated controlled tensions have shown significant similarity over the entire winding interval. The small size of the simulation errors under closed-loop conditions further confirmed the relevance of the 4SIDtechnique-based linear state-space model for the control application envisaged. As final conclusion, an LTI model of such a nonlinear process is not a utopia. One only needs to choose the convenient control configuration, i.e. the ST scheme. Moreover, it is shown and explained that despite the non-measured radii changes, the identified LTI MIMO model is sufficient for the development of an efficient LQG control of the tension on the entire winding zone. The influence of the variation of reel radii have hitherto been neglected in short interval estimation datasets. The authors’ conviction is that improvements could be expected in the quality of the models if such variations were to be taken into account in the identification stage. It is therefore intended to orient future investigations in this direction, in order to determine a MIMO model able to describe the overall tension behavior during the whole winding processing, and not only around a local operating point. A bilinear extension of subspace identification methods (Favoreel et al., 1996) would be interesting in this case.
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