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

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 [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 𝑥𝑦̇_𝑘𝑘(𝑡)=𝐴𝑥_𝑘(𝑡)+𝐵𝑢_𝑘,(𝑡)=𝐶𝑥_𝑘(𝑡),(1) 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 𝑥_𝑘(𝑡)∈ℝ^𝑛, the (constant over one cycle) input vector is 𝑢_𝑘∈ℝ^𝑚, and the output vector is 𝑦_𝑘(𝑡)∈ℝ^𝑝.

The control task is to update the control input 𝑢_𝑘 (heater temperature setpoints) after each cycle such that the terminal output 𝑦_𝑘(𝑇) (sheet surface temperatures) converges to a desired terminal value 𝑦_𝑑 at time T. From linear system theory, one can write the solution of (1) at 𝑡=𝑇 as 𝑥_𝑘(𝑇)=𝑒^𝐴𝑇𝑥_𝑘(0)+^𝑇0𝑒^{𝐴(𝑇−𝜏)}𝐵𝑑𝜏𝑢_𝑘.(2)

From this terminal state, we calculate the corresponding terminal output as 𝑦_𝑘(𝑇)=𝐶𝑒^𝐴𝑇𝑥_𝑘(0)+𝐶^𝑇0𝑒^{𝐴(𝑇−𝜏)}𝐵𝑑𝜏𝑢_𝑘.(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 𝑦_𝑇[𝑘]=Γ₀𝑥₀[𝑘]+Ψ₀𝑢[𝑘],(4) where 𝑦_𝑇[𝑘]=𝑦_𝑘(𝑇), 𝑥₀[𝑘]=𝑥_𝑘(0). Matrix Γ₀∈ℝ^𝑝×𝑛 is defined by Γ₀∶=𝐶𝑒^𝐴𝑇,(5) and matrix Ψ∈ℝ^𝑝×𝑚 by Ψ₀∶=𝐶^𝑇0𝑒^{𝐴(𝑇−𝜏)}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 𝑦_𝑇̂(𝑧)=Ψ₀𝑢(̂𝑧)+Γ₀𝑥̂₀(𝑧),(7) where 𝑢(̂𝑧), 𝑦_𝑇̂(𝑧), and 𝑥̂₀(𝑧) are the 𝑧-transforms of 𝑢[𝑘], 𝑦_𝑇[𝑘], and 𝑥₀[𝑘], respectively.

The following assumptions are made in this paper. (A1)The initial state repeats itself. Thus, 𝑥_𝑘(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: 𝑦_𝑑[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.

Figure 1 

Block diagram of the system.

From the closed-loop system shown in Figure 1, we obtain the output sensitivity matrix 𝑆=𝐼+Ψ₀𝐶⁻¹,(8) and the complementary sensitivity matrix 𝑇=Ψ₀𝐶𝐼+Ψ₀𝐶⁻¹.(9)

Furthermore, we can define the input sensitivity matrix 𝑈∶=𝐶𝐼+Ψ₀𝐶⁻¹.(10)

The mixed-sensitivity design for the system is based on finding a stabilizing controller 𝐶(𝑧) that minimizes the norm 𝑑∶=‖𝑊¹𝑊₂𝑆𝑈‖_∞.(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 𝑊_1𝑖𝑖(𝑧)∶=1𝑀_1𝑖2+𝑀^1𝑖𝜔_1𝑖𝑧−2−𝑀^1𝑖𝜔_1𝑖2+𝜀^1𝑖𝜔_1𝑖𝑧−2−𝜀^1𝑖𝜔_1𝑖,(12) where 𝑖∈{1,2,…,𝑝}. To establish a connection to the more widely used Laplace-domain weighting functions in continuous-time H_∞ control, 𝑊_1𝑖𝑖(𝑧) was obtained from a bilinear transformation of the continuous-time first-order lag 𝑊_𝑐(𝑠)=(1/𝑀_1𝑖)((𝑠+𝑀_1𝑖𝜔_1𝑖)/(𝑠+𝜀_1𝑖𝜔_1𝑖)). The DC gain of 𝑊_𝑐(𝑠) is 𝜀^−11𝑖, its high-frequency gain is 𝑀^−11𝑖, and 𝜔_1𝑖 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 of 𝑊−11𝑖(𝑒𝑗𝜔) and its parameters.

A minimal state-space realization of 𝑊₁(𝑧) is given by 𝑊₁=[𝐴^𝑊1𝐵_𝑊1𝐶_𝑊1𝐷_𝑊1]∶=𝐷_𝑊1+𝐶_𝑊1𝑧𝐼−𝐴^𝑊₁⁻¹𝐵_𝑊1.(13)

All state-space matrices are diagonal. The (𝑖,𝑖) entries on the main diagonal of the state-space matrices are given by 𝐴_{𝑊1𝑖𝑖}=2−𝜀^1𝑖𝜔_1𝑖2+𝜀_1𝑖𝜔_1𝑖,(14)𝐵_{𝑊1𝑖𝑖}=1,(15)𝐶_{𝑊1𝑖𝑖}=4𝜔^1𝑖𝑀_1𝑖−𝜀_1𝑖𝑀_1𝑖2+𝜀_1𝑖𝜔_1𝑖²,(16)𝐷_{𝑊1𝑖𝑖}=2+𝑀^1𝑖𝜔_1𝑖𝑀_1𝑖2+𝜀_1𝑖𝜔_1𝑖,(17)𝑖∈{1,2,…,𝑝}𝜀_1𝑖𝑊^−11𝑖(𝑧) where 𝑊^−11𝑖(𝑒^𝑗0)=𝑊^−11𝑖(1)=𝜀_1𝑖.

The parameter 𝑀_1𝑖 is the DC gain of 𝑊^−11𝑖(𝑧), that is, 𝑊^−11𝑖(𝑒^𝑗𝜋)=𝑊^−11𝑖(−1)=𝑀_1𝑖. To minimize the steady-state error, we need this value to be smaller than 1. 𝜔_1𝑖 is the high-frequency gain of 𝑊₂, that is, 𝑊_2𝑖𝑖=𝑀^−12𝑖,(18). This value is chosen to be larger than one. As stated earlier, 𝑀_2𝑖∈ℝ₊ 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 𝑧⁻¹𝐼_𝑚.

Since the TILC algorithm must be causal, we need to obtain a controller with a strictly proper transfer function. To do so, the controller 𝑊_delay=𝑧⁻¹𝐼_𝑚 is decomposed in two parts, as shown in Figure 3. A proper controller 𝑊_delay=[𝐴^𝑊delay𝐵_𝑊delay𝐶_𝑊delay𝐷_𝑊delay]=[0𝐼_𝑚𝐼_𝑚0].(19), designed with the mixed-sensitivity analysis, is combined with a delay (𝐶(𝑧)). The minimal state-space realization of the delay 𝑧⁻¹𝐼 is given by Ψ₀

Figure 3 

Block diagram with the controller ⎡⎢⎢⎢⎣𝑧1𝑧2𝑒⎤⎥⎥⎥⎦=⎡⎢⎢⎢⎣𝑊1−𝑊1Ψ0𝑊delay0𝑊2𝑊delay𝐼−Ψ0𝑊delay⎤⎥⎥⎥⎦𝑦𝑑𝑢0𝑥002𝑑𝑐∶=𝑃𝑢𝑦𝑑0𝑥002𝑑𝑐(20) and the delay 𝑃.

In Figures 1 and 3, the nominal system 𝑃=[𝐴^𝑊delay−𝐵_𝑊10−𝐷_𝑊1Ψ₀𝐶_𝑊1𝐴_𝑊1𝐵0_𝑊1𝐵_𝑊1𝐵Ψ₀𝐷^𝑊delay_𝑊delayΨ₀𝐶_𝑊delay𝐶_𝑊1𝑊−₂Ψ𝐶_𝑊delay00𝐶_𝑊delay0𝐷0^𝑊1𝐷^𝑊1Ψ₀𝐷_𝑊delay𝐼_𝑝𝑊₂𝐷_𝑊delayΨ₀𝐷_𝑊delay].(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=[−0Ψ⁰0−𝐷_𝑊1𝐴_𝑊1𝐼0_𝑝𝐼0^𝑚Ψ₀𝐶_𝑊1𝑊−₂Ψ₀00𝐷𝐼0^𝑊1_𝑝000].(22) and is used for the minimization of (11).

To simplify the analysis, we can rewrite plant 𝑃=∶[𝐴𝐶₁𝐵¹𝐵₂𝐶₂𝐷¹¹𝐷₁₂𝐷₂₁𝐷₂₂].(23) with its own state-space realization as follows: 𝜀_1𝑖

With the definitions given in the previous section, we can write 𝐴_𝑊1

To simplify the notation in the remaining part of the paper, we can rewrite (22) as 𝐴_𝑊1

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 𝐴_𝑊1 are different from 0. Then, all of the eigenvalues of 𝐴_{𝑊1𝑖𝑖} lie strictly inside the unit circle.

Proof. Since 1≤𝑖≤𝑝 is diagonal, its eigenvalues are equal to the entries on the diagonal. Thus, the eigenvalues of 𝜀_1𝑖 are equal to 𝜔_1𝑖, with 2−𝜀_1𝑖𝜔_1𝑖<2. Because both 2+𝜀_1𝑖𝜔_1𝑖>2 and 𝐴_{𝑊1𝑖𝑖} are greater than 0 for all i, we have 𝑖=1,2,…,𝑝 and 𝐴_𝑊1. 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 [^{𝐴−𝐼𝑛𝑒𝑗𝜃𝐵2}_𝐶1𝐷12] has a solution if the following four conditions [17] hold: (1)𝜃∈[0,2𝜋] is stabilizable;(2)[^{𝐴−𝐼𝑛𝑒𝑗𝜃𝐵1}_𝐶2𝐷21] is detectable;(3)𝜃∈[0,2𝜋] is full column rank for all Ψ₀;(4)𝐴_𝑊1 is full row rank for all 𝐴−𝐼^𝑛𝑒^𝑗𝜃𝐵₂𝐶₁𝐷₁₂=⎡⎢⎢⎢⎢⎢⎣−𝐼^𝑚𝑒^𝑗𝜃0𝐼_𝑚−𝐷−Ψ_0𝑊1𝐴_𝑊1−𝐼_𝑝𝑒^𝑗𝜃0𝑊₂Ψ₀𝐶_𝑊1000⎤⎥⎥⎥⎥⎥⎦.(24).

Lemmas 1 and 2 lead to the following theorem.

Theorem 1. The H_∞ mixed-sensitivity problem of the system 𝐴−𝐼_𝑛𝑒^𝑗𝜃 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 𝐶_𝑊1
The column rank of this matrix must be equal to the rank of 𝐷_𝑊2 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 𝐴−𝐼^𝑛𝑒^𝑗𝜃𝐵₁𝐶₂𝐷₂₁=⎡⎢⎢⎢⎣−𝐼^𝑚−Ψ𝑒^𝑗𝜃₀00𝐴_𝑊1−𝐼_𝑛𝑒^𝑗𝜃𝐼_𝑝−Ψ₀0𝐼_𝑝⎤⎥⎥⎥⎦.(25) are full rank, the rank of (24) is equal to 𝐴−𝐼_𝑛𝑒^𝑗𝜃 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=⎡⎢⎢⎢⎢⎢⎣0000....03013312342060120000....30103321430206210000....01031233203412600000....1030213302432106⎤⎥⎥⎥⎥⎥⎦.(26) plus the number of outputs p. For the same reasons as above, the rank of (25) is equal to 𝑊₁∶𝑀_1𝑖=10, for all 𝜔_1𝑖=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 𝜀_1𝑖=0.01

We choose as design parameter for the weighting function 𝑖=1,…,4, 𝑊₂ and 𝑀_2𝑖=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 𝐶(𝑧)=⎡⎢⎢⎢⎢⎢⎣𝑘¹𝑘(𝑧)𝑘₂(𝑧)𝑘₃(𝑧)𝑘₄(𝑧)₂𝑘(𝑧)𝑘₁(𝑧)𝑘₄(𝑧)𝑘₃(𝑧)₃𝑘(𝑧)𝑘₄(𝑧)𝑘₁(𝑧)𝑘₂(𝑧)₄(𝑧)𝑘₃(𝑧)𝑘₂(𝑧)𝑘₁(𝑧)⎤⎥⎥⎥⎥⎥⎦𝑚⁻¹(𝑧),(27), 𝑘₁(𝑧)=×−0.𝑧7417𝑧+0.1288𝑧−0.0441₂𝑘−0.2309𝑧+0.0134₂,𝑘(𝑧)=×2.3𝑧92−90𝑧.1+0009.12𝑧62−0𝑧.1−5509₃.1,184𝑘(𝑧)=×0.6𝑧75−00𝑧.1+0303.12𝑧97−0𝑧.0−1601₄.1,402(𝑠)=×−1.𝑧1938𝑧+0.1271𝑧−0.1175₂𝑚(𝑧)=×𝑧𝑧−−0−.09.090.992340967𝑧𝑧𝑧−++00.0.10.3110211545,𝑧.−0.1074(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.

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 (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.

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).

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.