Cross Directional Control

Transcription

1 Cross Directional Control Graham C. Goodwin Day 4: Lecture 4 16th September 2004 International Summer School Grenoble, France

2 1. Introduction In this lecture we describe a practical application of receding horizon control to a common industrial problem, namely web-forming processes. Web-forming processes represent a wide class of industrial processes with relevance in many different areas such as paper making, plastic film extrusion, steel rolling, coating and laminating.

3 In a general set up, web processes (also known as film and sheet forming processes) are characterised by raw material entering one end of the process machine and a thin web or film being produced in (possibly) several stages at the other end of the machine. The raw material is fed to the machine in a continuous or semi-continuous fashion and its flow through the web-forming machine is generally referred to as the machine direction [MD].

4 Sheet and film processes are effectively two-dimensional spatially distributed processes with several of the properties of the sheet of material varying in both the machine direction and in the direction across the sheet known as the cross direction [CD].

5 actuators cross direction sensors machine direction Figure: Generic web-forming process.

6 The main objective of the control applied to sheet and film processes is to maintain both the MD and CD profiles of the sheet as flat as possible, in spite of disturbances such as variations in the composition of the raw material fed to the machine, uneven distribution of the material in the cross direction, and deviations in the cross-directional profile.

7 In order to control the cross-directional profile of the web, several actuators are evenly distributed along the cross direction of the sheet. The number of actuators can vary from only 30 up to as high as 300. The film properties, on the other hand, are either measured via an array of sensors placed in a downstream position or via a scanning sensor that moves back and forth in the cross direction. The number of measurements taken by a single scan of the sensor can be up to 1000.

8 Difficulties the high dimensionality of the cross-directional system; the high cross-direction spatial interaction between actuators; the uncertainty in the model; the limited control authority of the actuators.

9 2. Problem Formulation It is generally the case that web-forming processes can be effectively modelled by assuming a decoupled spatial and dynamical response. This is equivalent to saying that the effect of one single actuator movement is almost instantaneous in the cross direction whilst its effect in the machine direction shows a certain dynamic behaviour.

10 These observations allow one to consider a general model for a cross-directional system of the form where q 1 is the unitary shift operator. y k = q d h(q) Buk + d k, (1)

11 It is assumed that the system dynamics are the same across the machine and thus h(q) can be taken to be a scalar transfer function. In addition, h(q) is typically taken to be a low order, stable and minimum-phase transfer function. A typical model is a simple first-order system with unit gain: h(q)= (1 α) q α. (2) A transport delay q d accounts for the physical separation that exists between the actuators and the sensors in a typical cross-directional process application.

12 The matrix B is the normalised steady state interaction matrix and represents the spatial influence of each actuator on the system outputs. In most applications it is reasonably assumed that the steady state cross-directional profile generated by each actuator is identical. As a result, the interaction matrix B usually has the structure of a Toeplitz symmetric matrix.

13 The main difficulties in dealing with cross-directional control problems are related to the spatial interaction between actuators and not so much to the complexity of dynamics, which could reasonably be regarded as benign. A key feature is that a single actuator movement not only affects a single sensor measurement in the downstream position but also influences sensors placed in nearby locations. Indeed, the interaction matrix B is typically poorly conditioned in most cases of practical importance.

14 The poor conditioning of B can be quantified via a singular value decomposition B= USV T (3) where S, U, V R m m. S= diag{σ 1,σ 2,...,σ m } is a diagonal matrix with positive singular values arranged in decreasing order, and U and V are orthogonal matrices such that UU T = U T U= I m and VV T = V T V= I m, where I m is the m m identity matrix. If B is symmetric then U= V.

15 If B is poorly conditioned then the last singular values on the diagonal of S are very small compared to the singular values at the top of the chain{σ i } m. This characteristic implies that the control i=1 directions associated with the smallest singular values are more difficult to control than those associated with the biggest singular values, in the sense that a larger control effort is required to compensate for disturbances acting in directions associated with smallσ i.

16 This constitutes not only a problem in terms of the limited control authority usually available in the array of actuators, but it is also an indication of the sensitivity of the closed loop to uncertainties in the spatial components of the model.

17 The control objective in cross-directional control systems is usually stated as the requirement to minimise the variations of the output profile subject to input constraints. This can be stated in terms of minimising the following objective function: subject to input constraints V = y k 2 2 k=0 u k u max. (4)

18 Another type of constraint typical of CD control systems is a second-order bending constraint defined as 1 u i+1 k u i k b max for i= 1,...,m, (5) where u i k = ui k ui 1 is the deviation between adjacent actuators k in the input profile at a given time instant k. 1 The superscript indicates the actuator number.

19 3. Example 1 To illustrate the ideas involved in cross-directional control, we consider a 21-by-21 interaction matrix B with a Toeplitz symmetric structure and exponential profile: b ij = e 0.2 i j for i, j= 1,...,21, (6) where b ij are the entries of the matrix B.

20 Cross-directional index Figure: Cross-directional profile for a unit step in actuator number 11.

21 We consider the transfer function h(q)= 1 e 0.2 q e 0.2, (7) which is a discretised version of the first-order system ẏ(t)= y(t)+u(t) with sampling period T s = 0.2 sec.

22 The next figure shows the singular values of the interaction matrix B. We observe that there exists a significant difference between the largest singular valueσ 1 and the smallest singular valueσ 21, indicating that the matrix is poorly conditioned. Dealing with the poor conditioning of B is one of the main challenges in CD control problems as we will see later.

23 Singular values index Figure: Singular values of the interaction matrix B.

24 In order to estimate the states of the system and the output disturbance d k, a Kalman filter is implemented as described for an extended system that includes the dynamics of a constant output disturbance: x k+1 = Ax k + Bu k, d k+1 = d k, y k = Cx k + d k,

25 In our case A= diag{e 0.2,...,e 0.2 }, B= (1 e 0.2 ) B, C= I m.

26 The state noise covariance [ ] Im 0 Q n =, 0 100I m and output noise covariance R n = I m, were considered in the design of the Kalman filter.

27 We will consider the finite horizon quadratic objective function with both prediction and control horizons set equal to one, that is V 1,1 = 1 2 (yt 0 Qy 0+ u T 0 Ru 0+ x T 1 Px 1). (8) Q= I m, R= 0.1I m. (9)

28 We assume the system is subject to physical constraints on the inputs of the form: u i k 1 for all k, i= 1,...,21.

29 Design 1 The first control strategy that we try on the problem is a linear quadratic Gaussian [LQG] controller designed with the same weighting matrices as above. This design clearly does not take into consideration the constraints imposed on the input profile. As might be expected, the application of such a blind (or serendipitous) approach to the problem would, in general, not achieve satisfactory performance.

30 Input clipping Input clipping disturbance Cross-directional index (a) Input profile Cross-directional index (b) Output profile. Figure: Input-output steady state profiles with input clipping.

31 The above results illustrates a phenomenon that is well known in the area of cross-directional control, namely alternate inputs across the strip converge to large alternate values, that is, input picketing occurs. We will see below, when we test alternative design methods, that this picketing effect can be avoided by careful design leading to significantly improved disturbance compensation.

32 Design 2 We next try RHC considering initially only input constraints. The achieved steady state input and output profiles are presented below where we have also included, for comparison, the profiles obtained with the input clipping approach.

33 Input clipping RHC 1 Input clipping RHC disturbance Cross-directional index Cross-directional index (a) Input profile. (b) Output profile. Figure: Input-output steady state profiles using RHC (square-dashed line) and input clipping (circle-solid line).

34 We observe, perhaps surprisingly, that the steady state response achieved with RHC does not seem to have improved significantly compared with the result obtained by just clipping the control in the LQG controller. In addition, the input profile obtained with RHC seems to be dominated by the same high spatial frequency modes as those that resulted from the input clipping approach.

35 However, this is a reasonably well understood difficulty in CD control systems: The picket fence profile in the input arises from the controller trying to compensate for the components of the disturbance in the high spatial modes which, in turn, require bigger control effort, driving the inputs quickly into saturation.

36 Design 3 (Using the Singular Value Structure of the Hessian The commonly accepted solution to this inherent difficulty is to let the controller seek disturbance compensation only in the low spatial frequencies.

37 When the prediction horizon chosen is N= 1 then in the vector formulation of the quadratic optimisation problem we can write Γ=B and the Hessian of the objective function is simply H= B T QB+ R = BT B+ R. This implies that the singular values of the Hessian are simply the singular values squared of the interaction matrix B shifted by the weighting in the input R.

38 We next limit the optimization to the first few singular vectors. This has two potential advantages: (i) It avoids chasing hard to control high spatial frequency distances. (ii) It may improve robustness since the gain associated with high spatial frequencies may be poorly defined.

39 SVD RHC strategy RHC 1 SVD RHC strategy RHC disturbance Cross-directional index Cross-directional index (a) Input profile. (b) Output profile. Figure: Input and output steady state profiles using the SVD RHC strategy (circle-solid line) and RHC (square-dashed line).

40 We observe that in this case the picket fence profile has disappeared from the input whilst the output profile has not changed significantly. Clearly a slight degradation of the output variance is to be expected owing to the suboptimality of the strategy. The steady state profiles obtained with RHC have been repeated in the above figure for comparison purposes.

41 4. Example 2 We conclude this lecture with a second example having 11 actuators and 11 sensors. Here we will run the simulations in real time so that you can see the picketing develop when the disturbance is applied.

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45 5. Conclusions This lecture has presented a realistic application of constrained control to a difficult industrial control problem, namely cross directional control. This system has high complexity due to: i. the large number of actuators (several hundred), ii. large interactions, and iii. severe constraints on the actuator signals.