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This paper presents a robust design approach for terminal iterative learning control (TILC). This robust design uses the

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 [

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 [

TILC was first addressed in [

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 [_{∞} 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
[

Section _{∞} concepts necessary to carry out the robust design such as weighting functions
and their parameters. Section _{∞} 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

The TILC algorithm is applied to a continuous, linear time-invariant
system. This system may be the linearized model of a thermoforming machine [

The control task is to update the control input

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

Thus, we can apply discrete-time control on system (

The

The following assumptions are made in this paper.

The initial state repeats itself. Thus,

The desired terminal output is constant for
all cycles

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

Figure

Block diagram of the system.

From the closed-loop system shown in Figure

Furthermore, we can define the input
sensitivity matrix

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

This norm uses two weighting functions.

(i) _{∞} control,

Frequency response magnitude of

A minimal state-space realization of

All state-space matrices are diagonal. The

The parameter

(ii)

Since the TILC algorithm must be causal, we need to obtain a
controller with a strictly proper transfer function. To do so, the controller

Block diagram with the controller

In Figures

To solve the mixed-sensitivity problem we have to minimize
the norm in (

From Figure

To simplify the analysis, we can rewrite plant

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 (

It is important to note that all matrices appearing in (

Before going further, we need the following lemma.

Suppose all

Since

We also need the following standard lemma on the suboptimal
discrete-time H_{∞} problem [

The suboptimal H_{∞} problem
corresponding to the generalized plant in (

Lemmas

The H_{∞} mixed-sensitivity problem
of the system

Since the matrix

The third condition in Lemma

The column rank of this matrix must be equal to the rank of

The fourth condition is related to the rank of

The row rank of this matrix must be
equal to the rank of

In conclusion, because all conditions
of Lemma _{∞} mixed-sensitivity approach can be
used to design a robustly stable TILC algorithm.

Theorem _{∞} 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.

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 [

We choose as design parameter for the weighting function

We select for

The mixed-sensitivity approach leads us to the following controller
(with

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 ^{°}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.

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 [

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

Sheet terminal surface temperatures (1st simulation).

The evolution of heater zone temperatures is shown in
Figure

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 [

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 [

In future work, we will test the system with nonfeasible
temperature map for nonsquare or rank-deficient

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.