**1. Introduction**

In recent years, having benefited from the rapid progress in industrial process and its control system, thermal processing systems, especially those with temperature control, are attracting more attention for achieving high-quality and high-performance processing. In thermal processing, the temperature uniformity is one of the most basic and important requirements. For example, most of the applications of thermal processes are centered on the food industry, where requirements of temperature uniformity and faster transient response are strongly in demand.

Due to these requirements, various control methods have been proposed and introduced into temperature control systems. Among the various thermal processing techniques, the conventional proportional–integral–derivative (PID) controller has become the most commonly used control method benefiting from its simplicity and wide applicability [1–6]. However, due to the nonlinearity and the coupling influence of multi-point systems, it is difficult to realize precise temperature control based only on the conventional PID control method. Thus, the series the series connected fuzzy-proportional integral (SF-PI) control method is proposed to improve the control performance [7].

For the coupling difficulties as stated above, a novel decoupling compensation or quasidecoupling method has been investigated in multi-input, multi-output (MIMO) temperature control systems by building an equivalent model of multiple SISO (single-input, single-output) systems [8–13]. However, if we cannot obtain a precise plant model, that decoupling compensation method would not work as efficiently as expected. Thus, the system identification methods have been applied to the temperature control system, and the most popular method is the step response method [14–16]. With the mathematical model of the system, the Model Predictive Control (MPC) method has been proposed on the heating system to provide precise control [17].

Moreover, although a precise plant model has been obtained and the decoupling compensation method has been applied, there remain some difficulties, such as: dead-time difference between the multiple points and the large time delay in the plant. These difficulties have a significant impact on the transient response and temperature difference. Thus, some advanced compensation methods have been proposed: feed forward compensation, data-driven tuning, and gradient temperature control [18–21].

Furthermore, for the excessively complex thermal process system, the mathematic model could not be obtained precisely. Thus, the data-driven approaches have been proposed as a potential method for controlling that system [22], and some optimization methods have also been proposed for these excessively complex processes [23].

Although after all the above methods have been introduced into the temperature control system in the thermal process, there are still some problems with the transient response and the closely controlled temperature of the multi-point temperature control system with dead-time difference and strong coupling effects. For example, in MIMO heating systems the temperature difference between each point should be under 5% of the reference temperature, no matter whether the transient response is fast or slow. Until now, the abovementioned method still cannot achieve the effect we expected in multi-point temperature control systems that have a large time constant and big dead-time difference. For example, if the system is composed of only a Smith predictive control method, the temperature difference may not be prevented. Also, if the system is composed of only a gradient control method, the transient response may not be drastically improved. Our previous work has proposed a pole-zero cancellation with reference model method for such a class of temperature control system [24]. This paper focuses on the multi-point temperature heating system, improving the transient response of each channel and reducing the temperature difference between each point, while also providing a systematic design method for the controller design, and a novel slow-mode-based control method is proposed for multi-point temperature control systems [25]. In the proposed method, the temperature differences and transient characteristics of all points can be controlled by making the output of the fast modes follow the output of the slow mode. In this paper, the SMBC method combined with decoupling and dead-time compensation is applied to the MIMO temperature control system, in addition to the theoretical analysis of the system being introduced, and simulations and experiments to verify the control efficiency of the proposed SMBC control method are described. The results are then compared to the conventional PI control and gradient temperature control methods.

#### **2. Difficulties in Multi-Point Temperature Control**

From the viewpoint of practical application of multi-point temperature control, well-tuned PID controllers with decoupling and dead-time compensations are used. The Ziegler–Nichols ultimate gain method and the constant heating rate method are representative heuristic methods based on experimental data [1,2]. These methods attempt to make the actual temperatures follow the reference values while minimizing the temperature differences between multiple points. However, in many cases, the following difficulties still exist: (1) The PID controller must be designed considering the effect of the dead-time difference of the controlled objects. (2) Although the temperature difference is somewhat decreased due to this consideration, a significant improvement in the transient response cannot be expected. (3) The transient response needs to be improved. (4) When the reference values of multiple points are different, it is difficult to control the individual setting times with the same temperature ratio.

#### **3. Configuration of SMBC System**

This section describes the proposed SMBC system. For convenience, the slow mode is defined as the plant system with the largest time constant, while the fast mode is defined as systems with smaller time constants. The block diagram of the proposed SMBC control system is shown in Figure 1. In the figure, *rs* is the reference value; *ys* is the output temperature of the slow mode; *rf* <sup>1</sup> and *rf n*, which are equal to *ys*, are the reference values of the fast modes. In this proposal, the reference value of the fast modes is set to be the output value of the slow mode, and *yf* <sup>1</sup> and *yf n* are the outputs of the fast modes. *Cs* is defined as the controller of the slow mode, *Cf* <sup>1</sup> and *Cf n* are indicate as the controllers of the fast modes.

**Figure 1.** Block diagram of slow-mode-based control (SMBC) control system.

For simplicity, the proposed SMBC control method was applied to a simplified multi-point temperature control system with two inputs and two outputs. The block diagram is shown in Figure 2, where *r* and *y* indicate the reference and the output temperatures respectively. The subscripts 1 (Ch1: channel 1) and 2 (Ch2: channel 2) refer to the slow mode and the fast mode, respectively. The steady-state gain, time constant, and dead time of the plant are expressed by *K*, *T*, and *L* respectively. The proposed SMBC structure can be divided into seven blocks.

**Figure 2.** Overview of SMBC structure.

#### *3.1. Structure in Which the Controlled Objects Are under Strong Coupling Effectiveness*

Since the controlled plant has been simplified to a two inputs, two outputs system, the two channels are strongly under coupled effectiveness. The diagram of the two channels is shown in Figure 3, where *P*<sup>11</sup> and *P*<sup>22</sup> indicate the controlled objects of Ch1 and Ch2, respectively, and *P*<sup>12</sup> and *P*<sup>21</sup> indicate the respective coupling terms between these two channels.

**Figure 3.** Block diagram of controlled objects.

The transfer function of both the controlled objects and the coupling terms are expressed as a first-order plus time delay system as (1), where the plant parameters *K*, *T*, and *L* have already been defined.

$$P(\mathbf{s}) = \frac{\mathbf{K}}{\mathbf{T}\mathbf{s} + \mathbf{1}} \mathbf{e}^{-L\mathbf{s}} \tag{1}$$
