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

Authors: Guy Gauthier; Benoit Boulet
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

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


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## Body

## 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.Section2 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
(1)x˙k(t)=Axk(t)+Buk,yk(t)=Cxk(t),
where t is
the time and the subscript k is the
cycle number (k∈ℕ). Matrices A, B, and C are time-invariant.  The state vector in cycle k is xk(t)∈ℝn,
the (constant over one cycle) input vector  is uk∈ℝm,
and the output vector is yk(t)∈ℝp.The control task is to update the control inputuk (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
(2)xk(T)=eATxk(0)+(∫0TeA(T−τ)Bdτ)uk.From this terminal state, we calculate the corresponding
terminal output as(3)yk(T)=CeATxk(0)+(C∫0TeA(T−τ)Bdτ)uk.Now, we change the notation to emphasize the fact that for the
cycle-to-cycle control, cyclek is
equivalent to the time argument of a discrete-time system. Equation (3) is
rewritten as
(4)yT[k]=Γ0x0[k]+Ψ0u[k],
where yT[k]=yk(T), x0[k]=xk(0).
Matrix Γ0∈ℝp×n is defined by
(5)Γ0:=CeAT,
and matrix Ψ∈ℝp×m by
(6)Ψ0:=C∫0TeA(T−τ)Bdτ.Thus, we can apply discrete-time control on system (4) which will
appear like cycle-to-cycle control to the system (1).Thez-transform of (4) is
(7)y^T(z)=Ψ0u^(z)+Γ0x^0(z),
where u^(z), y^T(z),
and x^0(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 cyclesk: 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 matrixΨ 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.Figure1 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.Figure 1
Block diagram of the system.From the closed-loop system shown in Figure1, we obtain the
output sensitivity matrix
(8)S=(I+Ψ0C)−1,
and the complementary sensitivity matrix
(9)T=Ψ0C(I+Ψ0C)−1.Furthermore, we can define the input
sensitivity matrix(10)U:=C(I+Ψ0C)−1.The mixed-sensitivity
design for the system is based on finding a stabilizing controllerC(z) that minimizes the norm
(11)d:=∥[W1SW2U]∥∞.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
(12)W1ii(z):=1M1i{(2+M1iω1i)z−(2−M1iω1i)(2+ε1iω1i)z−(2−ε1iω1i)},
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 ε1i−1, its high-frequency gain is M1i−1, 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.Figure 2
Frequency response magnitude ofW1i−1(ejω) and its parameters.A minimal state-space realization ofW1(z) is given by
(13)W1=[AW1BW1CW1DW1]:=DW1+CW1(zI−AW1)−1BW1.All state-space matrices are diagonal. The(i,i)
entries on the main diagonal of the state-space matrices are given by
(14)AW1ii=2−ε1iω1i2+ε1iω1i,(15)BW1ii=1,(16)CW1ii=4ω1i(M1i−ε1i)M1i(2+ε1iω1i)2,(17)DW1ii=(2+M1iω1i)M1i(2+ε1iω1i),
where i∈{1,2,…,p}.The parameterε1i is the DC gain of W1i−1(z), that is, W1i−1(ej0)=W1i−1(1)=ε1i. To minimize the steady-state error, we need this value to be smaller
than 1. M1i is the high-frequency gain of W1i−1(z), that is, W1i−1(ejπ)=W1i−1(−1)=M1i. This value is chosen to be larger than one. As stated earlier, ω1i is a parameter to adjust the crossover
frequency of the weighting function.(ii)W2 is a diagonal weighting function on the
process input signal, with each term on the main diagonal defined as
(18)W2ii=M2i−1,
where M2i∈ℝ+ and i∈{1,2,…,m}.  The inverse of W2 gives an upper bound for the magnitude of the input
sensitivity function U on the unit
circle. Note that there are
no dynamics in W2.
Weighting  function W2 also expresses the uncertainty level in the
system subject to real parametric perturbations in the entries of Ψ.Since the TILC algorithm must be causal, we need to obtain a
controller with a strictly proper transfer function. To do so, the controllerC(z) is decomposed in two parts, as shown in Figure 3. A proper
controller C˜(z),
designed with the mixed-sensitivity analysis, is combined with a delay (z−1Im). The minimal state-space realization of the
delay Wdelay=z−1Im is given by
(19)Wdelay=[AWdelayBWdelayCWdelayDWdelay]=[0ImIm0].Figure 3
Block diagram with the controllerC˜(z) and the delay z−1I.In Figures1 and 3, the nominal system Ψ0 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 Figure3, 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
(20)[z1z2e]=[W1−W1Ψ0Wdelay0W2WdelayI−Ψ0Wdelay][ydu˜]:=P[ydu˜]
and is used for the minimization of (11).To simplify the analysis, we can rewrite plantP with its own state-space realization
as follows:
(21)P=[[AWdelay0−BW1Ψ0CW1AW1][0BW1][BWdelayBW1Ψ0DWdelay][−DW1Ψ0CWdelayCW1W2CWdelay0][−Ψ0CWdelay0][DW10][DW1Ψ0DWdelayW2DWdelay]IpΨ0DWdelay].With the definitions given in the previous section, we can
write(22)P=[[00−Ψ0AW1][0Ip][Im0][−DW1Ψ0CW1W20][−Ψ00][DW10][00]Ip0].To simplify the notation in the remaining part of the paper,
we can rewrite (22) as
(23)P=:[A[B1B2][C1C2][D11D12D21D22]].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 allε1i are different from 0. Then, all of the
eigenvalues of AW1 lie strictly inside the unit circle.Proof.
SinceAW1 is diagonal, its eigenvalues are equal to the entries
on the diagonal. Thus, the eigenvalues of AW1 are equal to AW1ii,
with 1≤i≤p.
Because both ε1i and ω1i are greater than 0 for all i, we have 2−ε1iω1i<2 and 2+ε1iω1i>2.
This implies that the eigenvalues AW1ii as given by (14) are strictly lower than 1,
for all i=1,2,…,p.
Hence, all eigenvalues of AW1 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 γ>0 has a solution if the following four conditions
[17] hold:
(1)
(A,B2) is stabilizable;(2)
(C2,A) is detectable;(3)
[A−InejθB2C1D12] is full column rank for all θ∈[0,2π];(4)
[A−InejθB1C2D21] is full row rank for all θ∈[0,2π].Lemmas1 and
2 lead to the following theorem.Theorem 1.
The H∞ mixed-sensitivity problem
of the system Ψ0 controlled by a TILC algorithm with the mixed-sensitivity
function defined by (8) and (10) can be solved.Proof.
Since the matrixA is Hurwitz, the first two conditions in Lemma 2 are fulfilled. The eigenvalues of the delay are equal
to 0 and the eigenvalues of AW1 are strictly inside the unit circle as shown
in Lemma 1.
The third condition in Lemma2 concerns the matrix
(24)[A−InejθB2C1D12]=[−Imejθ0Im−Ψ0AW1−Ipejθ0−DW1Ψ0CW10W200].
The column rank of this matrix must be equal to the rank ofA−Inejθ plus the number of inputs m. Since matrix A is Hurwitz with all eigenvalues having their amplitude
below 1, and CW1 and DW2 are full rank, the rank of (24) is equal to m+n for all θ∈[0,2π].
Thus, the third condition is satisfied.
The fourth condition is related to the rank of(25)[A−InejθB1C2D21]=[−Imejθ00−Ψ0AW1−InejθIp−Ψ00Ip].
The row rank of this matrix must be
equal to the rank ofA−Inejθ plus the number of outputs p. For the same reasons as above, the
rank of (25) is equal to n+p,
for all θ∈[0,2π],
the number of rows of (25). Thus, the fourth condition is fulfilled.
In conclusion, because all conditions
of Lemma2 are satisfied,
the H∞ mixed-sensitivity approach can be
used to design a robustly stable TILC algorithm.Theorem1 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
(26)Ψ0=[0.03360.33400.01210.12020.33400.03360.12020.01210.01210.12020.03360.33400.12020.01210.33400.0336].We choose as design parameter for the weighting functionW1:M1i=10, ω1i=0.1 and ε1i=0.01 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 forW2 the parameters M2i=10, i=1,…,4.
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
(withd=0.5305):
(27)C(z)=[k1(z)k2(z)k3(z)k4(z)k2(z)k1(z)k4(z)k3(z)k3(z)k4(z)k1(z)k2(z)k4(z)k3(z)k2(z)k1(z)]m−1(z),
where
(28)k1(z)=−0.7417(z+0.1288)(z−0.0441)×(z2−0.2309z+0.0134),k2(z)=2.3929(z+0.1262)(z−0.1184)×(z−0.1009)(z−0.1559),k3(z)=0.6750(z+0.1297)(z−0.1402)×(z−0.1033)(z−0.0161),k4(s)=−1.1938(z+0.1271)(z−0.1175)×(z2−0.2067z+0.0114),m(z)=(z−0.9993)(z−0.1301)(z−0.1074)×(z−0.0949)(z+0.1255).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.Figure4 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.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.Figure 5
Sheet terminal surface temperatures
(1st simulation).The evolution of heater zone temperatures is shown in
Figure6. 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).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-deficientΨ0 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 μ-analysis.



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*Source: 289391-2008-06-29.xml*