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Journal of Control Science and Engineering
Volume 2008 (2008), Article ID 289391, 6 pages
http://dx.doi.org/10.1155/2008/289391
Research Article

Robust Design of Terminal ILC with H Mixed Sensitivity Approach for a Thermoforming Oven

1Automated Production Department, École de Technologie Supérieure, Université du Québec, Montréal, QC, Canada H3C 1K3
2Department of Electrical and Computer Engineering, McGill University, Montréal, QC, Canada H3A 2A7

Received 9 November 2007; Accepted 2 May 2008

Academic Editor: Kevin Moore

Copyright © 2008 Guy Gauthier and Benoit Boulet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a robust design approach for terminal iterative learning control (TILC). This robust design uses the H mixed-sensitivity technique. An industrial application is described where TILC is used to control the reheat phase of plastic sheets in a thermoforming oven. The TILC adjusts the heater temperature setpoints such that, at the end of the reheat cycle, the surface temperature map of the plastic sheet will converge to the desired one. Simulation results are included to show the effectiveness of the control law.

1. Introduction

Up till now, the cycle-to-cycle temperature control of industrial thermoforming machines has been rather primitive. The in-cycle temperature control is performed with traditional PID control of heater temperatures [1, 2]. However, the cycle-to-cycle adjustments of the heater temperature setpoints are made manually by trial and error. Typically, this causes financial losses for thermoformers because of bad setpoint adjustments leading to wasted parts.

For the thermoforming application, the terminal iterative learning control (TILC) algorithm is an efficient cycle-to-cycle control technique to adjust the setpoint temperatures of heaters in the thermoforming oven [3, 4]. Infrared temperature sensors are placed in the oven to measure the surface temperature of the plastic sheet. The TILC adjusts the heater temperature setpoints to control the terminal surface temperature to a desired temperature profile at the end of the cycle.

TILC was first addressed in [5]. There have been various applications of TILC, notably in rapid thermal processing for chemical vapor deposition [58]. TILC is derived from iterative learning control (ILC), an approach that has attracted a lot of attention. The main difference between TILC and ILC is that TILC has access to a measurement of the process output only at the end of the cycle, whereas ILC uses output measurements throughout the cycle. Many papers have been written about ILC as shown in a survey by Moore [9].

This paper is about the use of H mixed-sensitivity analysis as a tool to design robust TILC controllers. The mixed-sensitivity approach was successfully used by other researchers for ILC, see [1013]. Recent work [1416] addressed the robustness problem in the cycle domain using H optimal iterative learning control based on a super vector approach.

While robust ILC design has been studied extensively, robust TILC has attracted less attention. However, high-order TILC has been proposed in [5, 6, 8, 1416] to improve the robustness.

Section 2 presents the system to be controlled by TILC. Section 3 introduces H concepts necessary to carry out the robust design such as weighting functions and their parameters. Section 4 presents the H mixed-sensitivity method to design a controller. Simulation results, using a TILC designed by H mixed-sensitivity on a model of a thermoforming machine are given in Section 5. Section 6 concludes.

2. Problem of Set Up

The TILC algorithm is applied to a continuous, linear time-invariant system. This system may be the linearized model of a thermoforming machine [3, 4] and is represented by ̇𝑥𝑘(𝑡)=𝐴𝑥𝑘(𝑡)+𝐵𝑢𝑘,𝑦𝑘(𝑡)=𝐶𝑥𝑘(𝑡),(1) where t is the time and the subscript k is the cycle number (𝑘). Matrices A, B, and C are time-invariant. The state vector in cycle k is 𝑥𝑘(𝑡)𝑛, the (constant over one cycle) input vector is 𝑢𝑘𝑚, and the output vector is 𝑦𝑘(𝑡)𝑝.

The control task is to update the control input 𝑢𝑘 (heater temperature setpoints) after each cycle such that the terminal output 𝑦𝑘(𝑇) (sheet surface temperatures) converges to a desired terminal value 𝑦𝑑 at time T. From linear system theory, one can write the solution of (1) at 𝑡=𝑇 as 𝑥𝑘(𝑇)=𝑒𝐴𝑇𝑥𝑘(0)+𝑇0𝑒𝐴(𝑇𝜏)𝑢𝐵𝑑𝜏𝑘.(2)

From this terminal state, we calculate the corresponding terminal output as 𝑦𝑘(𝑇)=𝐶𝑒𝐴𝑇𝑥𝑘𝐶(0)+𝑇0𝑒𝐴(𝑇𝜏)𝑢𝐵𝑑𝜏𝑘.(3)

Now, we change the notation to emphasize the fact that for the cycle-to-cycle control, cycle k is equivalent to the time argument of a discrete-time system. Equation (3) is rewritten as 𝑦𝑇[𝑘]=Γ0𝑥0[𝑘]+Ψ0𝑢[𝑘],(4) where 𝑦𝑇[𝑘]=𝑦𝑘(𝑇), 𝑥0[𝑘]=𝑥𝑘(0). Matrix Γ0𝑝×𝑛 is defined by Γ0=𝐶𝑒𝐴𝑇,(5) and matrix Ψ𝑝×𝑚 by Ψ0=𝐶𝑇0𝑒𝐴(𝑇𝜏)𝐵𝑑𝜏.(6)

Thus, we can apply discrete-time control on system (4) which will appear like cycle-to-cycle control to the system (1).

The 𝑧-transform of (4) is ̂𝑦𝑇(𝑧)=Ψ0̂𝑢(𝑧)+Γ0̂𝑥0(𝑧),(7) where ̂𝑢(𝑧), ̂𝑦𝑇(𝑧), and ̂𝑥0(𝑧) are the 𝑧-transforms of 𝑢[𝑘], 𝑦𝑇[𝑘], and 𝑥0[𝑘], respectively.

The following assumptions are made in this paper. (A1)The initial state repeats itself. Thus, 𝑥𝑘(0)=𝑥0[𝑘]=𝜒0 is a constant vector for all cycles. This corresponds to the assumption that all plastic sheets are at the same initial temperature before being heated.(A2)The desired terminal output is constant for all cycles k: 𝑦𝑑[𝑘]=𝛾. Since we want to thermoform plastic sheets to obtain a desired part in a repetitive way, the desired temperature must remain constant.

3. Preliminaries on the Mixed-Sensitivity Problem

The mixed-sensitivity approach is used to design a robust discrete-time controller. This controller has to keep the system stable despite perturbations on the entries of matrix Ψ from the nominal values contained in Ψ0. In thermoforming machines, a large part of the uncertainties in the perturbed system Ψ is due to process nonlinearities.

Figure 1 shows the nominal system with the TILC controller 𝐶(𝑧) and the weighting functions 𝑊1(𝑧) and 𝑊2(𝑧) in the 𝑧-domain. The output disturbance signal 𝑤[𝑘] in the block diagram contains the term Γ0𝑥0[𝑘] in (4) due to the initial state.

289391.fig.001
Figure 1: Block diagram of the system.

From the closed-loop system shown in Figure 1, we obtain the output sensitivity matrix 𝑆=𝐼+Ψ0𝐶1,(8) and the complementary sensitivity matrix 𝑇=Ψ0𝐶𝐼+Ψ0𝐶1.(9)

Furthermore, we can define the input sensitivity matrix 𝑈=𝐶𝐼+Ψ0𝐶1.(10)

The mixed-sensitivity design for the system is based on finding a stabilizing controller 𝐶(𝑧) that minimizes the norm 𝑊𝑑=1𝑆𝑊2𝑈.(11)

This norm uses two weighting functions.

(i) 𝑊1 is a diagonal weighting function on the process error signal, with each term on the main diagonal defined as 𝑊1𝑖𝑖1(𝑧)=𝑀1𝑖2+𝑀1𝑖𝜔1𝑖𝑧2𝑀1𝑖𝜔1𝑖2+𝜀1𝑖𝜔1𝑖𝑧2𝜀1𝑖𝜔1𝑖,(12) where 𝑖{1,2,,𝑝}. To establish a connection to the more widely used Laplace-domain weighting functions in continuous-time H control, 𝑊1𝑖𝑖(𝑧) was obtained from a bilinear transformation of the continuous-time first-order lag 𝑊𝑐(𝑠)=(1/𝑀1𝑖)((𝑠+𝑀1𝑖𝜔1𝑖)/(𝑠+𝜀1𝑖𝜔1𝑖)). The DC gain of 𝑊𝑐(𝑠) is 𝜀11𝑖, its high-frequency gain is 𝑀11𝑖, and 𝜔1𝑖 is a parameter that can shift the frequency range at which the lag is applied. The inverse of 𝑊1(𝑧) provides an upper bound for the shape of the sensitivity function S. The parameters shaping 𝑊1 are shown in Figure 2.

289391.fig.002
Figure 2: Frequency response magnitude of 𝑊11𝑖(𝑒𝑗𝜔) and its parameters.

A minimal state-space realization of 𝑊1(𝑧) is given by 𝑊1𝐴=[𝑊1𝐵𝑊1𝐶𝑊1𝐷𝑊1]=𝐷𝑊1+𝐶𝑊1𝑧𝐼𝐴𝑊11𝐵𝑊1.(13)

All state-space matrices are diagonal. The (𝑖,𝑖) entries on the main diagonal of the state-space matrices are given by 𝐴𝑊1𝑖𝑖=2𝜀1𝑖𝜔1𝑖2+𝜀1𝑖𝜔1𝑖,(14)𝐵𝑊1𝑖𝑖=1,(15)𝐶𝑊1𝑖𝑖=4𝜔1𝑖𝑀1𝑖𝜀1𝑖𝑀1𝑖2+𝜀1𝑖𝜔1𝑖2,(16)𝐷𝑊1𝑖𝑖=2+𝑀1𝑖𝜔1𝑖𝑀1𝑖2+𝜀1𝑖𝜔1𝑖,(17)𝑖{1,2,,𝑝}𝜀1𝑖𝑊11𝑖(𝑧) where 𝑊11𝑖(𝑒𝑗0)=𝑊11𝑖(1)=𝜀1𝑖.

The parameter 𝑀1𝑖 is the DC gain of 𝑊11𝑖(𝑧), that is, 𝑊11𝑖(𝑒𝑗𝜋)=𝑊11𝑖(1)=𝑀1𝑖. To minimize the steady-state error, we need this value to be smaller than 1. 𝜔1𝑖 is the high-frequency gain of 𝑊2, that is, 𝑊2𝑖𝑖=𝑀12𝑖,(18). This value is chosen to be larger than one. As stated earlier, 𝑀2𝑖+ is a parameter to adjust the crossover frequency of the weighting function.

(ii) 𝑖{1,2,,𝑚} is a diagonal weighting function on the process input signal, with each term on the main diagonal defined as 𝑊2 where 𝑊2 and 𝑊2. The inverse of Ψ gives an upper bound for the magnitude of the input sensitivity function U on the unit circle. Note that there are no dynamics in 𝐶(𝑧). Weighting function 𝐶(𝑧) also expresses the uncertainty level in the system subject to real parametric perturbations in the entries of 𝑧1𝐼𝑚.

Since the TILC algorithm must be causal, we need to obtain a controller with a strictly proper transfer function. To do so, the controller 𝑊delay=𝑧1𝐼𝑚 is decomposed in two parts, as shown in Figure 3. A proper controller 𝑊delay𝐴=[𝑊delay𝐵𝑊delay𝐶𝑊delay𝐷𝑊delay0]=[𝐼𝑚𝐼𝑚0].(19), designed with the mixed-sensitivity analysis, is combined with a delay (𝐶(𝑧)). The minimal state-space realization of the delay 𝑧1𝐼 is given by Ψ0

289391.fig.003
Figure 3: Block diagram with the controller 𝑧1𝑧2𝑒=𝑊1𝑊1Ψ0𝑊delay0𝑊2𝑊delay𝐼Ψ0𝑊delay𝑦𝑑𝑢0𝑥002𝑑𝑐𝑦=𝑃𝑑𝑢0𝑥002𝑑𝑐(20) and the delay 𝑃.

In Figures 1 and 3, the nominal system 𝐴𝑃=[𝑊delay0𝐵𝑊1Ψ0𝐶𝑊1𝐴𝑊10𝐵𝑊1𝐵𝑊delay𝐵𝑊1Ψ0𝐷𝑊delay𝐷𝑊1Ψ0𝐶𝑊delay𝐶𝑊1𝑊2𝐶𝑊delay0Ψ0𝐶𝑊delay0𝐷𝑊10𝐷𝑊1Ψ0𝐷𝑊delay𝑊2𝐷𝑊delay𝐼𝑝Ψ0𝐷𝑊delay].(21) is a full-rank real matrix and, therefore, has no dynamics.

4. Mixed-Sensitivity Problem of TILC

To solve the mixed-sensitivity problem we have to minimize the norm in (11) to a value smaller than 1. But first, we must check whether the problem is well posed [17].

From Figure 3, we can construct the open-loop generalized plant between the inputs and the outputs assuming that the controller has been removed. This generalized plant is 𝑃=[00Ψ0𝐴𝑊10𝐼𝑝𝐼𝑚0𝐷𝑊1Ψ0𝐶𝑊1𝑊20Ψ00𝐷𝑊1000𝐼𝑝0].(22) and is used for the minimization of (11).

To simplify the analysis, we can rewrite plant 𝐴𝑃=[𝐵1𝐵2𝐶1𝐶2𝐷11𝐷12𝐷21𝐷22].(23) with its own state-space realization as follows: 𝜀1𝑖

With the definitions given in the previous section, we can write 𝐴𝑊1

To simplify the notation in the remaining part of the paper, we can rewrite (22) as 𝐴𝑊1

It is important to note that all matrices appearing in (22) are full rank.

Before going further, we need the following lemma.

Lemma 1. Suppose all 𝐴𝑊1 are different from 0. Then, all of the eigenvalues of 𝐴𝑊1𝑖𝑖 lie strictly inside the unit circle.

Proof. Since 1𝑖𝑝 is diagonal, its eigenvalues are equal to the entries on the diagonal. Thus, the eigenvalues of 𝜀1𝑖 are equal to 𝜔1𝑖, with 2𝜀1𝑖𝜔1𝑖<2. Because both 2+𝜀1𝑖𝜔1𝑖>2 and 𝐴𝑊1𝑖𝑖 are greater than 0 for all i, we have 𝑖=1,2,,𝑝 and 𝐴𝑊1. This implies that the eigenvalues 𝛾>0 as given by (14) are strictly lower than 1, for all (𝐴,𝐵2). Hence, all eigenvalues of (𝐶2,𝐴) are strictly inside the unit circle.

We also need the following standard lemma on the suboptimal discrete-time H problem [17].

Lemma 2. The suboptimal H problem corresponding to the generalized plant in (20) and consisting of computing a controller yielding a norm (11) less than some desired [𝐴𝐼𝑛𝑒𝑗𝜃𝐵2𝐶1𝐷12] has a solution if the following four conditions [17] hold: (1)𝜃[0,2𝜋] is stabilizable;(2)[𝐴𝐼𝑛𝑒𝑗𝜃𝐵1𝐶2𝐷21] is detectable;(3)𝜃[0,2𝜋] is full column rank for all Ψ0;(4)𝐴𝑊1 is full row rank for all 𝐴𝐼𝑛𝑒𝑗𝜃𝐵2𝐶1𝐷12=𝐼𝑚𝑒𝑗𝜃0𝐼𝑚Ψ0𝐴𝑊1𝐼𝑝𝑒𝑗𝜃0𝐷𝑊1Ψ0𝐶𝑊10𝑊2.00(24).

Lemmas 1 and 2 lead to the following theorem.

Theorem 1. The H mixed-sensitivity problem of the system 𝐴𝐼𝑛𝑒𝑗𝜃 controlled by a TILC algorithm with the mixed-sensitivity function defined by (8) and (10) can be solved.

Proof. Since the matrix A is Hurwitz, the first two conditions in Lemma 2 are fulfilled. The eigenvalues of the delay are equal to 0 and the eigenvalues of 𝐴 are strictly inside the unit circle as shown in Lemma 1.
The third condition in Lemma 2 concerns the matrix 𝐶𝑊1
The column rank of this matrix must be equal to the rank of 𝐷𝑊2 plus the number of inputs m. Since matrix 𝑚+𝑛 is Hurwitz with all eigenvalues having their amplitude below 1, and 𝜃[0,2𝜋] and 𝐴𝐼𝑛𝑒𝑗𝜃𝐵1𝐶2𝐷21=𝐼𝑚𝑒𝑗𝜃00Ψ0𝐴𝑊1𝐼𝑛𝑒𝑗𝜃𝐼𝑝Ψ00𝐼𝑝.(25) are full rank, the rank of (24) is equal to 𝐴𝐼𝑛𝑒𝑗𝜃 for all 𝑛+𝑝. Thus, the third condition is satisfied.
The fourth condition is related to the rank of 𝜃[0,2𝜋]
The row rank of this matrix must be equal to the rank of Ψ0=.0.03360.33400.01210.12020.33400.03360.12020.01210.01210.12020.03360.33400.12020.01210.33400.0336(26) plus the number of outputs p. For the same reasons as above, the rank of (25) is equal to 𝑊1𝑀1𝑖=10, for all 𝜔1𝑖=0.1, the number of rows of (25). Thus, the fourth condition is fulfilled.
In conclusion, because all conditions of Lemma 2 are satisfied, the H mixed-sensitivity approach can be used to design a robustly stable TILC algorithm.

Theorem 1 shows that the H mixed-sensitivity approach can be used to compute the TILC controller and this can be done using design tools like Matlab’s “dhinflmi” or “dhinfric” functions.

The next section presents a TILC design for a thermoforming machine. This design was tested on a thermoforming oven model.

5. Simulation Results

To test the effectiveness of the mixed-sensitivity approach, we take as an example a design based on a model of a thermoforming machine.

Only the heating phase of the thermoforming process is considered here. Molding is not considered since the purpose of the TILC algorithm is to heat the plastic sheet up to a desired surface temperature map, before the molding phase.

Linearizing the model of a thermoforming oven [3] configured with four heater zones and four infrared temperature sensors around the operating point, we obtain 𝜀1𝑖=0.01

We choose as design parameter for the weighting function 𝑖=1,,4, 𝑊2 and 𝑀2𝑖=10 for 𝑖=1,,4. This will shape the sensitivity function of the closed-loop system such that the system responds relatively quickly, that is, in a few cycles. For a thermoforming machine, we want to get acceptable heater temperature setpoints as fast as possible in order to limit the number of wasted parts.

We select for 𝑑=0.5305 the parameters 𝑘𝐶(𝑧)=1(𝑧)𝑘2(𝑧)𝑘3(𝑧)𝑘4𝑘(𝑧)2(𝑧)𝑘1(𝑧)𝑘4(𝑧)𝑘3𝑘(𝑧)3(𝑧)𝑘4(𝑧)𝑘1(𝑧)𝑘2𝑘(𝑧)4(𝑧)𝑘3(𝑧)𝑘2(𝑧)𝑘1𝑚(𝑧)1(𝑧),(27), 𝑘1×𝑧(𝑧)=0.7417𝑧+0.1288𝑧0.04412,𝑘0.2309𝑧+0.01342×,𝑘(𝑧)=2.3929𝑧+0.1262𝑧0.1184𝑧0.1009𝑧0.15593×,𝑘(𝑧)=0.6750𝑧+0.1297𝑧0.1402𝑧0.1033𝑧0.01614×𝑧(𝑠)=1.1938𝑧+0.1271𝑧0.11752,×.0.2067𝑧+0.0114𝑚(𝑧)=𝑧0.9993𝑧0.1301𝑧0.1074𝑧0.0949𝑧+0.1255(28). This weighting function will shape the U function to limit the actuator efforts and reduce the risk of large overshoot and saturation. On a thermoforming machine, an overshoot in heater temperature can cause the plastic sheet to sag too much and damage the thermoforming oven.

The mixed-sensitivity approach leads us to the following controller (with Ψ0): 𝜇 where H

This discrete-time controller corresponds to a fifth-order TILC to control the thermoforming oven heater temperature setpoints.

This controller was implemented and tested on the nonlinear model of the thermoforming machine. In this model, the initial sheet temperature is 27°C and subject to a slow variation of 10°C. The measurement noise in the simulation was a Gaussian white noise with standard deviation equal to 1°C, which is representative of infrared sensor noise.

Figure 4 shows the behavior of the infinity norm (maximum component) of the terminal surface temperature error. The desired terminal temperatures are 150°C and 160°C at the end of a 3-minute heating cycle. The error becomes smaller than 5°C at the 7th iteration (or cycle) and seems to remain within this bound thereafter.

289391.fig.004
Figure 4: Error on terminal surface temperature (1st simulation).

For high-density polyethylene (HDPE) thermoplastic sheets, a terminal temperature kept inside a ±10°C margin of the desired temperature is acceptable for forming and the risk of getting a defective part is low [1, 2].

The terminal temperatures on the top surface of the sheet (IRT2 and IRT5) and on the bottom surface (IRB2 and IRB5), shown in Figure 5, converge smoothly to the desired terminal temperatures, despite the variation of the initial condition. This variation combined with the noise in the measurements of the surface temperature explains the slight variations in the temperature measurements.

289391.fig.005
Figure 5: Sheet terminal surface temperatures (1st simulation).

The evolution of heater zone temperatures is shown in Figure 6. Note that the oven model has 6 independent heater zones on top that can be grouped together, for example, here zones 1, 2, and 3 are controlled as a single zone and similarly for zones 4, 5, and 6. The same applies for the bottom heater zones. We can see a smooth behavior of the zone setpoints. The system has to compensate for variation in initial condition (here the initial surface temperature of the sheet).

289391.fig.006
Figure 6: Heater temperature setpoints (1st simulation).

A second simulation was performed with the same parameters, except for the ambient air temperature which was set 10°C higher. The rate of the convergence in the second simulation was nearly the same as that obtained in the first simulation. The heater temperature setpoints converged to lower values, since the ambient air temperature is higher. The simulation results showed that the TILC controller can adapt to seasonal changes in temperature and slow variation in the initial temperature of the plastic sheet.

Convergence to the desired terminal temperature is slower than the so-called deadbeat response provided by the TILC in [4]. On the other hand, results have shown that the controller presented here is more robust to perturbations than the deadbeat TILC. This is an illustration of the tradeoff between performance and robustness in feedback control systems.

6. Conclusion

The simulation results of the previous section demonstrate the effectiveness of the TILC algorithm for sheet reheat. We can argue that the TILC controlled thermoforming machine will keep the surface temperature profile of the plastic sheet to the desired one and adapt to slow temperature variations that inevitably happen in a thermoforming facility. For a given plastic sheet and temperature map, the heater temperature setpoints will be different on a hot summer day than on a cold winter day. Even when there is a long delay between the processing of two successive batches of sheets, the system will learn again new heater setpoints and converge to the desired temperature profile, as shown experimentally in [4].

In future work, we will test the system with nonfeasible temperature map for nonsquare or rank-deficient H to see how the TILC law can manage. Finally, on a more theoretical aspect of TILC, we will analyze the robust performance of the system with -analysis.

Acknowledgments

The authors would like to acknowledge the financial support for this research from the Natural Sciences and Engineering Research Council of Canada, the Government of Québec, and the Industrial Materials Institute of the National Research Council of Canada.

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