1. Introduction

Many of the industrial processes are MIMO system with interaction. One typical application is heating and/or cooling plate for semiconductor wafer fabrication. The plate usually consists of multiple zones and each zone temperature should be controlled accurately. The requirements to the temperature control are

(1) The temperature should reach the set-point variable (SV) as quick as possible to realize high throughput.

(2) Temperature uniformity or temperature gradient is kept constant both transient and steady state response.

In spite of these requirements, temperature control is very challenging due to the interactions between the neighbour zones and many studies have been conducted [1-3]. Model-following servo (MFS) control [4, 5] is suitable for the MIMO system with interaction because, 1) it has the advantage of the optimal control such as stability and robustness, 2) we need not consider decoupling design explicitly. As shown in Figure 1, the MFS control consists of plant, reference model, integrator, state feedback gain K₁ from the plant, feedforward gain K₂ from the reference model, and gain K₃ for the integrator. A low order system that has a desired dynamics is usually chosen as a reference model. The process variable, y_p, follows the step response of the reference model, y_r.

Figure 1. The structure of the model following servo control

For the multi-zone heating system, if we use same reference model, all process variables (PV) follow the same reference trajectory, which realizes the above requirement (2). Control systems in real world however usually contain constraint such as actuator saturation. Applying MFS control to such system worsens its control performance. We have to find low order system as a reference model that satisfies desired response speed without saturating manipulated variable (MV). However, in order to satisfy the above requirement (1) the MV should be maximum (saturated). As a result, some of the PVs can not follow reference trajectory and temperature uniformity worsens. Although the MFS control has been studied for long time [6-12], no studies show the negative effect of saturation on MFS control for MIMO system. Other than the MFS control, reference governor [13, 15] and model predictive control [16, 17] are able to consider constraint explicitly. The reference governor reshapes the reference signal so that its state and/or MV always stay within the constraints. It is very effective method since we can design reference governor and feedback controller independently. However this method only guarantees constraint fulfilment but not response speed specification. In order to guarantee both constraint fulfilment and response speed, two different optimization problems must be solved iteratively [14]. Model predictive control is a powerful control method that can easily handle dead-time system, non-minimum phase system, and system with constraints. However it requires several parameter tunings such as the number of the step of the each horizon and weight for the cost functions.

In this paper we propose a simple yet practical design method of reference model for the MFS control. It is suitable for the MIMO system with interaction and input saturation. The method provides ideal reference model that enables near time optimal response and uniformity of PVs. This paper consists of five sections. In section 2, we explain disadvantage of conventional MFS control by showing some experimental results. In section 3, proposed design method is explained. In section 4, the effectiveness of our proposed method is shown by an experimental result. And section 5 concludes this study.

2. Disadvantage of Conventional MFS Control

Figure 2 illustrates test plant for the temperature control of an aluminium plate which size is 400 x 150 x t4. Three Peltier modules for heating and cooling are allocated unsymmetrically in a longitudinal direction of the plate. Plate temperatures are measured by three Type-K thermocouples. Each of them is allocated near the Peltier module. Zone number is assigned as zone 1, 2, and 3 from the left.

We designed conventional MFS controller for this test plant. In order to obtain a plant model we conducted step response experiment for one zone at a time. Each response data is approximated by a first order plus dead time model. Whole test plant model is expressed by transfer function matrix as shown in eq. (1).

(1)

Replacing dead time to a first order Pade approximation, eq. (1) is converted to a discrete time state space model with its sampling time 0.1 sec. At the same time balanced realization and reduced order realization are applied to the model.

By these realizations, the model originally eighteenth order is reduced to the fourth order as shown eq. (2) where its coefficient A_p, B_p, and C_p are (3).

(2)

(3)

We chose second order plus dead time system as a reference model. It’s undamped natural frequency, ω₀ is 1/45 red/sec, damping ratio, ζ is 0.9, and dead time is 2 sec. The MV does not saturate in its response. Applying this to all zone, the reference model, G_r(s), is

(4)

where

(5)

Replacing dead time to a first order Pade approximation, eq. (4) is converted to a discrete time state space model (6) with its sampling time 0.1 sec.

(6)

where

(7)

Figure 2. Test plant for the temperature control of aluminium plate

With eq. (2) and eq. (6), an augmented system is derived from appendix.

(8)

The optimal control law minimizing the performance index (9)

(9)

is given by

(10)

where we set weight matrices Q, R as

(11)

Gain K₁, K₂, and K₃ are calculated as

(12)

We added integrator anti-windup and state estimator for the feedback gain K₁. Figures 3 show result of set-point response experiment. Figure 3(a) shows zone temperature (PV1, PV2, and PV3), reference trajectory y_r, and temperature uniformity, dPV defined by eq. (13).

(13)

figure 3(b) shows the MV of each zone. As the figures indicate, all zone temperatures follow reference trajectory quite close and temperature uniformity is less than 0.35 deg C. All MVs do not saturate as we designed.

In order to speed up the response, we changed undamped natural frequency of the reference model from 1/45 rad/sec to 1/30 rad/sec. The experimental result is shown in Figure 4. Graph legends are the same as figure 3. The response speed of the PVs becomes faster while the MV of zone 3 saturates. However, the PV of zone 3 cannot follow reference trajectory because the saturation limits its maximum speed. As a result the temperature uniformity during the transient response worsens by 2.1 deg C. From these experimental results it is clear that the conventional MFS control cannot maintain uniformity when its MV saturates. Conventional MFS control uses low order system as a reference model whose step response satisfies desired response specification. Therefore its response speed is the same as the step response of the reference model but not its maximum speed.

Figure 3. Experimental result of set-point response. Reference model parameter ω0=1/45 rad/sec, ζ=0.9

Figure 4. Experimental result of set-point response. Reference model parameter ω0=1/30 rad/sec, ζ=0.9

3. Proposed Design Method

Figure 5 shows a staircase MV pattern that achieves time optimal response. The MV should be maximum until PV reaches to a point X. After passing the point X the MV should be changed to minimum to decelerate the speed. When the PV reaches to the SV, the MV is switched to a steady state value, MVss that can keep PV at SV. There are two ways to observe switching point, one is to watch PV and switch MV at the optimal point X, another is to time and switch MV at the optimal time t₁ and t₂.

Figure 5. Ideal manipulate variable pattern for the time optimal control

We employ this idea to design reference model as illustrated in Figure 6. Unlike the conventional method shown in Figure 6(a), proposed method in Figure 6(b) first converts SV into corresponding MV pattern that satisfies time optimal response. The optimal MV pattern is then applied to the plant model to produce reference trajectory. Although this idea is suitable for the SISO (single input single output) system it has a difficulty for the MIMO system. For the MIMO system, we have to design MV patterns for all inputs. However the interaction of the plant model makes design MV pattern difficult.

Figure 6. Comparison of the reference model

To cope with this difficulty, we simplified the plant model from an experimental result and designed MV pattern by simulation. The design procedure is explained below.

1) Master-Slave step response experiment

In order to find fastest response that can maintain uniformity of PVs, one of the zones which has the slowest response speed is chosen and assigned as a master zone. Maximum MV is applied to the master zone and the rest of the zones are made follow the master zone. This experiment can be realized by master-slave PID control.

Figure 7. Experimental result for the master-slave response

2) System identification

A first order plus dead time model, G_m(s) can be build from the response data of the master zone. Dead time is replaced to a first order Pade approximation.

3) Seeking staircase MV pattern

The staircase MV pattern that realizes time optimal response of the G_m(s) is sought by iterative simulation. Here we set t₂=t₁ to shorten the simulation time. Although the response obtained becomes near time optimal the rising time up to 99% is almost the same as the time optimal case. We evaluate optimal MV pattern by overshoot of PV and IAE (Integral Absolute Error) criteria expressed eq. (14).

(14)

4) Building a reference model

The reference model G_r(s) is obtained by assigning G_m(s) to all zones. With this method, all reference trajectories are the same, which guarantees PV’s uniformity.

4. Experiment for Validation

In order to validate the effectiveness, the proposed method was applied to the test plant controller and the same experiment shown in section 2 was conducted.

1) Master-Slave step response experiment

Zone 3 was the slowest zone and assigned as master zone. Under the master-slave PID control configuration, maximum MV was set to the master zone and made rest of the zones follow. Figure 7 shows the result. Since MV3 for the master zone is maximum, this response is the fastest that can maintain uniformity of PVs. The uniformity after 50 seconds form the beginning is less than 0.1 deg C.

2) System Identification

A first order plus dead time model was identified as (15).

(15)

Replacing dead time to a first order Pade approximation, we obtained (16).

(16)

Figure 8. MV pattern and reference trajectory

3) Seeking staircase MV pattern

Iterative simulation to seek the optimal switching time, t₁ was executed by using G_m(s). Temperature width form the equilibrium to the SV was 25 deg C. The t₁ was incremented 0.1 sec in each iteration. The optimal switching time obtained was 148 sec and the MV at steady state was MVss=25/44.2=0.56. Figure 8(a) shows MV pattern obtained and figure 8(b) shows reference model output.

4) Building a reference model

Assigning eq. (16) to all zones, the reference model was obtained. Discrete time state space model is the same equation as eq. (6) where

(17)

Using this reference model and MV pattern we conducted set-point response experiment which was the same as the one shown in section 2. The SV was 51.5 deg C, 25 deg C above the equilibrium temperature. Figure 9 show the result. Graph legends are the same as figure 3. Compare to the conventional method shown in figure 3 and 4, both response speed and temperature uniformity are improved.

Figure 9. Experimental result of proposed method

5. Conclusions

We proposed a simple yet practical design method of reference model for the MFS control. It is suitable for the MIMO system with interaction and input saturation. In this paper we first claimed disadvantages of conventional method by showing experimental result. It indicates input saturation worsens both uniformity of PVs and response speed. In our proposed method we introduced unique reference model design and input signal design. The reference model is obtained by master-slave step response test. Using this model, staircase input signal that realizes near time optimal response is obtained by simulation. We showed the effectiveness of our method by experimental result. The result showed improvement of both response speed and temperature uniformity compare to the conventional method. Future work is to create method applicable to both heating and cooling that have different dynamics.

Appendix

Discrete time MFS control

Consider an m-inputs m-outputs plant

(A1a)(A1b)

where and are states, manipulated variable, and process variable respectively. We assume (A_p, B_p) is controllable, and (C_p, A_p) is observable. A reference model is expressed m-inputs and m-outputs system shown eq. (A2).

(A2a)(A2b)

where and x_r, r, y_r are states, reference, and reference output respectively. We assume (A_r, B_r) is controllable, (C_r, A_r) is observable, and the model is asymptotically stable. Considering difference equation of eq. (A1a) and (A2a), we obtain

(A3)

and

(A4)

An error signal is defined as (A5).

(A5)

Using equations from (A1) to (A5), first order difference equation of the error signal is

(A6)

From Eq. (A3), (A4), and (A6), and assuming Δr(k)=0, the following augmented system is obtained.

(A7a)(A7b)

Where

The optimal control law minimizing the performance index (A8)

(A8)

is given by

(A9)

With initial condition x_p(0)=0, x_r(0)=0, and u_p(0)=0, the eq. (A9) yields

(A10)

where

(A11)

and P is the solution of the steady state Riccati equation (A12).

(A12)