Journal of Control Science and Engineering

Volume 2008, Article ID 289391, 6 pages

http://dx.doi.org/10.1155/2008/289391

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

^{1}Automated Production Department, École de Technologie Supérieure, Université du Québec, Montréal, QC, Canada H3C 1K3^{2}Department 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 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 [5–8]. 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 [10–13].
Recent work [14–16] 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, 14–16] 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
where *t* is
the time and the subscript *k* is the
cycle number (). Matrices *A*, *B*, and *C* are time-invariant. The state vector in cycle *k* is ,
the (constant over one cycle) input vector is ,
and the output vector is .

The control task is to update the control input (heater temperature setpoints) after each
cycle such that the terminal output (sheet surface temperatures) converges to a
desired terminal value at time *T*. From linear system theory, one can write the
solution of (1) at as

From this terminal state, we calculate the corresponding terminal output as

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
where , .
Matrix is defined by
and matrix by

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

The -transform of (4) is where , , and are the -transforms of , , and , respectively.

The following assumptions are made in this paper.
(A1)The initial state repeats itself. Thus, 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*: .
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 from the nominal values contained in . In thermoforming machines, a large part of the uncertainties in the perturbed system is due to process nonlinearities.

Figure 1 shows the nominal system with the TILC controller and the weighting functions and in the -domain. The output disturbance signal in the block diagram contains the term in (4) due to the initial state.

From the closed-loop system shown in Figure 1, we obtain the output sensitivity matrix and the complementary sensitivity matrix

Furthermore, we can define the input sensitivity matrix

The mixed-sensitivity design for the system is based on finding a stabilizing controller that minimizes the norm

This norm uses two weighting functions.

(i) is a diagonal weighting function on the
process error signal, with each term on the main diagonal defined as
where . To establish a connection to the more widely
used Laplace-domain weighting functions in continuous-time H_{∞} control, was obtained from a bilinear transformation of
the continuous-time first-order lag .
The DC gain of is , its high-frequency gain is , and is a parameter that can shift the frequency
range at which the lag is applied. The
inverse of provides an upper bound for the shape of the
sensitivity function *S*. The
parameters shaping are shown in Figure 2.

A minimal state-space realization of is given by

All state-space matrices are diagonal. The entries on the main diagonal of the state-space matrices are given by where .

The parameter is the DC gain of , that is, . To minimize the steady-state error, we need this value to be smaller than 1. is the high-frequency gain of , that is, . This value is chosen to be larger than one. As stated earlier, is a parameter to adjust the crossover frequency of the weighting function.

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

Since the TILC algorithm must be causal, we need to obtain a controller with a strictly proper transfer function. To do so, the controller is decomposed in two parts, as shown in Figure 3. A proper controller , designed with the mixed-sensitivity analysis, is combined with a delay (). The minimal state-space realization of the delay is given by

In Figures 1 and 3, the nominal system 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 and is used for the minimization of (11).

To simplify the analysis, we can rewrite plant with its own state-space realization as follows:

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

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

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 are different from 0. Then, all of the
eigenvalues of lie strictly inside the unit circle.*

*Proof. *Since is diagonal, its eigenvalues are equal to the entries
on the diagonal. Thus, the eigenvalues of are equal to ,
with .
Because both and are greater than 0 for all *i*, we have and .
This implies that the eigenvalues as given by (14) are strictly lower than 1,
for all .
Hence, all eigenvalues of 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 has a solution if the following four conditions
[17] hold:
*(1)

*is stabilizable;*(2)

*is detectable;*(3)

*is full column rank for all ;*(4)

*is full row rank for all .*

Lemmas 1 and 2 lead to the following theorem.

Theorem 1. *The H _{∞} mixed-sensitivity problem
of the system 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 are strictly inside the unit circle as shown
in Lemma 1.

The third condition in Lemma 2 concerns the matrix

The column rank of this matrix must be equal to the rank of plus the number of inputs *m*. Since matrix is Hurwitz with all eigenvalues having their amplitude
below 1, and and are full rank, the rank of (24) is equal to for all .
Thus, the third condition is satisfied.

The fourth condition is related to the rank of

The row rank of this matrix must be
equal to the rank of plus the number of outputs *p*. For the same reasons as above, the
rank of (25) is equal to ,
for all ,
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

We choose as design parameter for the weighting function , and for . 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 the parameters , .
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 ): where

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.

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 (IR_{T2} and IR_{T5}) and on the bottom surface (IR_{B2} and IR_{B5}),
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.

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).

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 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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