*3.2. Structure in Which Fast-Mode Outputs Follow the Slow-Mode Output*

To make the temperature difference between two points close to zero, the output of the slow response mode is used as the reference value for the fast response modes, so that the fast mode outputs follow the slow mode, as is shown in Figure 4.

**Figure 4.** Fast mode output following the slow mode.

#### *3.3. Structure for Obtaining a Fast-Mode Reference Value*

In the proposed control structure, the Smith predictor method is introduced for not only dead time compensation but for reference generation, as shown in Figure 5. The model output without dead time, which is indicated by *y*1*n*, is used as the reference value *r*<sup>2</sup> for the fast mode. This structure allows avoidance of a further delay of the fast mode response due to the delay of the slow-mode output. In this proposal, the model based method has been introduced for both Smith predictor and feed-forward compensation, and the model is approximated by coupled SISO systems with time delay.

**Figure 5.** Block diagram of Smith predictor compensation.

#### *3.4. Structure in Which Decoupling Compensation Is Introduced to Compensate the Coupling Effect*

The coupling influence affects the temperature of two points; thus, decoupling compensation was introduced into the system to reduce the influence of the coupling term between the two points. The structure of the decoupling compensation is shown in Figure 6, where the *C*<sup>21</sup> and *C*<sup>12</sup> are the decoupling compensators. The decoupling compensators were designed as (2) and (3), respectively. In this proposal, the decoupling compensators consider the transient response between the two points using only the high-frequency gains.

$$C\_{12} = P\_{11} P\_{12}^{-1} \tag{2}$$

$$C\_{21} = P\_{21} P\_{22}^{-1} \tag{3}$$

**Figure 6.** Block diagram of decoupling compensation.

#### *3.5. Structure of Feed-Forward Compensation to Follow Fast-Mode Reference Value*

A feed-forward path was added to compensate for the dynamic delay of the fast-mode system, so that the fast-mode output can follow the slow-mode output without delay. This compensation makes the temperature differences between slow mode and fast modes extremely small. Its structure is shown in Figure 7.

**Figure 7.** Structure of feed-forward compensation.

#### *3.6. Structure of Compensation for Dead-Time Difference between Fast and Slow Modes*

After the compensations described above are made, a temperature difference still remains in the outputs of both modes due to the dead-time difference between the fast and slow modes. To avoid this problem, when the dead time of the slow mode (*L*1) is larger than that of the fast mode (*L*2), the reference value of the fast mode is delayed by the difference (*L*1− *L*2), as indicated by (*d*1) in Figure 2. This compensation causes both outputs to have the same dead time, that is, *L*1. Conversely, when *L*<sup>1</sup> is smaller than *L*2, the delay of the slow mode is included as (*L*<sup>2</sup> − *L*1), as indicated by (*d*2) in Figure 2, so that both outputs have the same dead time, that is, *L*2. This effectively reduces the temperature difference between the two modes. The dead-time difference compensation of both the fast and slow modes is shown in Figure 8.

**Figure 8.** Dead-time difference compensation.

## *3.7. Structure Enabling Control of the Output Ratio of Two Modes*

In the proposed SMBC method, the output ratio can be controlled even when the reference temperatures at multiple points are different. The gain block (*r*2/ *r*1), as indicated by (*e*) in Figure 2, which is located after the reference (*r*2), corresponds to this part. The fast-mode output follows different references from the slow-mode reference while maintaining a constant ratio. As a result, the setting time and ratio of each output's trajectory can be precisely matched in each mode.

#### **4. Controlled Objects Identification**

In this proposal, to precisely verify the control efficiency of the SMBC method, the controlled object was based on a real plant system; thus, a step response experiment needed to be performed to obtain the transfer function of the controlled objects. Figure 9 shows the experimental setup for the multi-point temperature control system with strong coupling effect, which use a DSP as the temperature controller. The system has four coupling channels, and each channel has two independent heaters and one temperature sensor. The temperature can be controlled through controlling the duty ratio of the PWM signal.

**Figure 9.** Experimental setup.

In this study, the two channels Ch1 and Ch2 were used as the controlled objects to apply the SMBC control method. A step response method was introduced to these two channels for controlled object identification, and the identification results for the paired coupling terms are shown in Figure 10.

**Figure 10.** System identification results.

Thus, the identification results can be finalized as (4). The system controller parameters design, system simulation, and experiments are all based on the identified plant transfer functions.

$$P(s) = \begin{bmatrix} P\_{11}(s) & P\_{12}(s) \\ P\_{21}(s) & P\_{22}(s) \end{bmatrix} = \begin{bmatrix} \frac{4.34}{16398+1}e^{-30s} & \frac{1.91}{5848s+1}e^{-125s} \\ \frac{1.67}{5984s+1}e^{-150s} & \frac{4.52}{1211s+1}e^{-25s} \end{bmatrix} \tag{4}$$

#### **5. System Simulation Results**

To verify the control efficiency of the proposed control SMBC method, a simulation was carried out in the MATLAB/SIMULINK environment with the controlled objects described above. The results of the simulation focused on the transient response and the temperature difference between the two channels. Control efficiency was verified by comparison with the conventional PI and gradient control methods. In simulation, the controllers *C*<sup>1</sup> and *C*<sup>2</sup> were calculated as (5) and (6), respectively. Hence, one of the most important factors is the stability of the controller, and there have already existed several methods for controller stability analysis [26,27], however, as the PID controller in this proposed system is designed based on the Ziegler–Nichols method (step response tuning method), the stability has been ensured [28]. The parameters *Kp*, *Ti*, and *Td* are decided by the plant parameters *K*, *T*, and *L* which have already been defined.

$$C\_1 = \frac{1639\text{s} + 1}{178\text{s}}\tag{5}$$

$$C\_2 = \frac{1211s + 1}{178s} \tag{6}$$

Also, as already mentioned before, a decoupling compensation was added by considering only the high-frequency gain between the two channels, where the decoupling controllers *C*<sup>21</sup> and *C*<sup>12</sup> are as (7) and (8), respectively.

$$C\_{21} = 0.1580\tag{7}$$

$$C\_{12} = 0.0875\tag{8}$$

The feed-forward compensation of the fast mode reference delay time was designed as (9) and the dead time difference compensation between the slow mode and the fast mode were designed as (10). Hence, in this simulation the reference temperature of the two channels stays the same, so that the temperature difference ratio is set as 1, as shown in (11).

$$C\_{FF} = \frac{1211s + 1}{4.52s + 4.52} \tag{9}$$

$$L\_2 - L\_1 = \text{5s} \tag{10}$$

$$r\_2/r\_1 = 1\tag{11}$$

To realize the SMBC control structure, the simulation was divided into two phase. Phase 1 was the SMBC control system with Smith predictor compensation. Phase 2 was the SMBC control system with Smith, feed-forward, and decoupling compensation. A step signal was applied to the slow-mode reference (Ch1). Simulation results for the two phases are shown in Figures 11 and 12, respectively. To evaluate the proposed SMBC control method, a simulation of the conventional PI control method and gradient control method with decoupling compensation were also carried out. Simulation results for the two methods are shown in Figures 13 and 14 respectively.

**Figure 11.** SMBC with Smith predictor compensation.

**Figure 12.** SMBC with Smith, feed-forward, and decoupling compensation.

**Figure 13.** Conventional proportional integral (PI) control system.

**Figure 14.** Gradient temperature control system.

The analysis of the simulation results can be devided into two phases. Phase 1: The transient response, the response time of both PI control method and gradient control system is about 60 *s*, similar to the proposed SMBC method, however, the conventional PI control system has an overshoot as 0.3 *deg C* (30% of the reference value) and the gradient control system has an overshoot as 0.5 *deg C* (50% of the reference value). For the SMBC with Smith preditive control system, there is no overshoot but with an asynchronous response between two channels, and it can be compensated by introducing feed-forward and decoupling compensation. Thus, the transient response has been improved. Phase 2: The temperature difference between the controlled two channels, as shown in the simulation results, yielded the maximum temperature difference of the conventional PI control system as 0.18 *deg C* (18% of the reference value) and that of gradient control system is about 0.05 *deg C* (5% of the reference value). However, the temperature difference of SMBC control system is only 0.01 *deg C* (1% of the reference value) and quickly dropped to 0 *deg C*, where we can observe that the temperature difference has been reduced. Thus, the control efficience of the proposed SMBC method has been evaluated.

#### **6. Experimental Results**

Experiments with the proposed SMBC control method were carried out using parameter values identical to those used in the simulation, and the experimental setup was the same as that shown in Figure 9. According to the identified plant transfer function, Ch1 was the slow response mode due to its larger time constant and Ch2 was the fast response mode with a smaller time constant. The experiments were carried out by using Ch1 output as the fast mode (Ch2) reference and making the Ch2 output follow the output of Ch1. The experiments were divided into the same two phases used in the simulation. Also, the step signal was applied to the slow-mode reference (Ch1). The experimental results for both phases are shown in Figures 15 and 16, respectively. To evaluate the proposed SMBC

control method, experiments with a conventional PI control system were also carried out. The results are shown in Figure 17.

**Figure 15.** SMBC with Smith predictor compensation.

**Figure 16.** SMBC with Smith, feed-forward, and decoupling compensations.

**Figure 17.** Conventional PI and decoupling control.

The analysis of the experimental results also can be divided into two phases. Phase 1: Transient response, from the results, the transient response of all control systems is simular as about 60 *s* rising time of each channel. However, the conventional PI control system has an overshoot of 0.3 *deg C* (30% of the reference value), while the proposed SMBC control system had no overshoot. Although the SMBC with Smith predictor compensation has an asynchronous response between the two channels, it can be improved by introducing feed-forward and decoupling compensation by which we could state that the transient response of both channel has been improved. Phase 2: The temperature

difference. The conventional PI control system has a maximum temperature difference of 0.3 *deg C* (30% of the reference value), while that of proposed SMBC with Smith, feed-forward and decoupling compensation system is about 0.21 *deg C* (21% of the reference value) and quickly drops to 0 *deg C*, where the temperature difference has been reduced. As a result, the simulation and experimental results are shown to be similar, indicating a positive evaluation for the proposed SMBC control method.
