Process industries differ significantly from other industries. Their production processes are continuous, and the characteristics of processes and equipment are often highly complex. The controlled objects frequently exhibit uncertainty, nonlinearity, large time delays, and strong coupling between variables. Obtaining an accurate mathematical model of the production process is virtually impossible. All these factors make conventional PLD control and other traditional control methods ineffective. Modern control theory, which emerged and developed in the 1960s, has achieved remarkable results in many fields, but when applied to process control in the process industry, a significant gap remains between theory and practice. The main reason is that it is difficult to obtain accurate mathematical models for most process industry processes. Therefore, it is necessary to consider using advanced control strategies to solve problems in engineering practice. Predictive control is the most widely used advanced control technology in the process industry. Richareet proposed the MAC (Mode AIgorithmic Control) algorithm as early as 1977, marking the emergence of predictive control. These algorithms describe the dynamic behavior of a process using process responses directly detected from the production site. They do not require prior knowledge of the process model's structure and parameters, nor do they necessitate complex identification to establish a mathematical model. They can design a control system based on a specific optimization index, determining a sequence of control variables to minimize a certain error index between the controlled variable and the softened desired trajectory over a future period. This algorithm employs a continuous online rolling optimization approach, constantly using feedback correction based on the error between the measured system output and the predictive model output during optimization. Therefore, it can overcome the influence of model errors and certain uncertainties to a certain extent, enhancing the system's robustness. It is highly suitable for controlling complex industrial production processes. In 1978, Richalet first elaborated on the motivations, mechanisms, and application effects of this type of algorithm in industrial control, and gave this new type of computer control algorithm a unified name—Predictive Control (MAC). The MAC algorithm is based on the impulse response model and is used for long-term time-domain prediction. It mainly consists of three parts: ① predictive model; ② reference trajectory; ③ rolling optimization. Predictive control arose from the needs of industrial control practice and has undergone significant development. As an optimization control algorithm, regardless of its form or changes, predictive control must include three basic principles: predictive model, rolling optimization, and feedback correction. It is precisely this rolling optimization principle of predictive control that reduces the dependence of the control system on the mathematical model of the controlled object, adapting to the requirements of complex industrial process control and possessing very broad development and application prospects. This paper will briefly outline the research and development status of predictive control, as well as research results on topics such as the combination of predictive control with other control strategies and predictive models in nonlinear control. 1. Development of Predictive Control Since Richart proposed the MAC algorithm, new predictive control algorithms have emerged continuously. The following is a brief review of the main algorithms. Dynamic Matrix Control (DMC), proposed by Richart in 1980, is based on a step response model and is suitable for asymptotically stable objects. For weakly nonlinear objects, it can be linearized near the operating point. For unstable objects, conventional PID control can be used to stabilize them first, and then the DMC algorithm can be used. Compared with MAC, the biggest advantage of DMC is that it has no steady-state error. In 1986, Morshedl proposed a generalized dynamic matrix control (UDMC) to primarily address nonlinear optimization problems. Bruqn published two papers proposing a predictive control algorithm (PCA), which, like MAC, is based on an impulse response model, but solves the instability problem of MAC for non-minimum phase systems because PCA adds constraints to the increment of the control variable. In 1988, Richailet proposed a predictive function control (PFC) method based on predictive control principles and successfully applied it to the fast and high-precision tracking control of industrial robots, achieving excellent results. Predictive function control is also a promising direction for the development of predictive control, and there has been much research in this area. 2. Adaptive Predictive Control Adaptive control is suitable for systems with a certain degree of uncertainty and has less dependence on the model. Therefore, it has long received attention and has a mature theoretical foundation. However, adaptive algorithms have poor robustness. In 1984, Ydstie proposed EHAC (Extended Horizon Adaptive Control), which was based on the ARMAX model and was not applicable to non-minimum phase systems. In 1985, De Keyser proposed EPSAC (Extended Predictive Self-Adaptive Control), also using the ARMAX model. Through long-horizon prediction, this algorithm reduced the accuracy requirements of the model compared to adaptive control and enhanced robustness. In 1987, Clarke, building on EHAC and EPSAC, proposed Generalized Predictive Control (GPC). Based on the CARIMA model, it solved the problem of poor error elimination between measured variables and setpoints when using the ARMAX model under large load disturbances. GPC signifies the organic combination of adaptive and predictive control, giving the control system excellent complementarity. It not only improves the adaptability of predictive control to uncertain environments but also enhances the robustness of adaptive control. GPC has been well applied in practice, with many successful examples. The research on adaptive predictive control is only for linear systems. The main method for dealing with nonlinear systems is to make a linear equivalent transformation of the nonlinear system, that is, to make the nonlinear system equivalent to a time-varying linear system. There is a lack of qualitative analysis and effective equivalent transformation methods. In short, there are still many problems to be solved in applying adaptive predictive control to nonlinear systems. 3 Internal Model Predictive Control In 1982, Garcia et al. studied a new type of control structure - internal model control (IMC). In their research, they studied the robustness, stability and parameter selection of predictive control algorithms from the perspective of IMC structure, and believed that predictive control algorithms and internal model control have an inherent connection and can be classified into a unified structure. Based on the principle of internal model control, there is a lot of research on predictive control. References [5, 6] analyzed the stability and robustness of the closed-loop system by transforming predictive control to the internal model control structure, and provided guidance on the parameter selection of the controller from a methodological point of view. Reference [7] proposed a new method for designing FIR type internal model inverse dynamic controllers using matrix OR decomposition. Reference [8] applied the principle of internal model control to analyze the GPC system. The quantitative expression of predictive control under the internal model structure is derived, and the robustness is analyzed based on the mismatch between the model and the object, making it possible to quantitatively study the relationship between design parameters and robustness. Reference [9] studied the controller equations, closed-loop system input-output and error equations of various predictive control algorithms (MAC, DMC, GPC, GPP) based on the internal model control principle, and summarized a unified formula. At the same time, the DMC algorithm was described under the internal model control framework, and its stability and robustness were analyzed. Starting from the internal model control structure and the minimization implementation form, the closed-loop performance of its predictive control system was analyzed, and the analytical relationship between the system dynamic response, anti-interference, robustness and design coefficients was given. This work has certain significance in the research of predictive control, and has established an intuitive connection between internal model control and predictive control. Reference [10] used the internal model control structure to analyze the defects of GPC in dynamic robustness when the system is not modeled, and proposed to use a mismatch filter to enhance the robustness of the system. At the same time, a suboptimal mismatch filter design method was proposed for the characteristics of GPC. Due to the inherent natural connection between internal model control and predictive control, the analysis and research of predictive control from the perspective of internal model control structure has great potential and broad prospects. More research is needed in this regard. 4 Predictive Control of Nonlinear Systems As mentioned above, most controlled processes in the process industry have nonlinear characteristics. Therefore, the control of nonlinear systems is the main research problem. The traditional process control method is to design the controller using a linearized model near the operating point. However, by using the characteristics of predictive control, better control effect can be achieved for nonlinear systems. The method used is to use a nonlinear predictive model to predict the future dynamics of the system. There are many ways to obtain nonlinear predictive models, but there are four main types: mechanism-based predictive models, experiment-based predictive models, intelligent predictive models, and linearized predictive models. (1) Mechanism-based predictive models Based on the differential and difference equations established according to the physical or chemical characteristics of the controlled process, the model that solves the high requirements of adaptive dead zone controller for lag accuracy is called a mechanism model. Obviously, to establish a mechanism model, one must have a thorough understanding of the controlled process (i.e., the object). As mentioned earlier, mechanistic models are almost impossible to obtain in the process industry. The research on the stability and robustness of the nonlinear MPC (Mode I Predictive Control) method is mainly based on mechanistic models, so it has little practical value and only provides some guidance for theoretical analysis. (2) Experiment-based prediction models Experimental models refer to models with a defined structure but whose parameters need to be identified through experiments. Experiments can be conducted offline or online. Commonly used experimental models include the Volterra model, Hammerstein model, and Wiener model. The Volterra model is an impulse response model for nonlinear systems. The accuracy of describing the dynamic process of the system depends on the order of the Volterra sequence. The higher the order, the higher the accuracy of the description, but high-order Volterra sequences require a large number of experiments to obtain the coefficients. For a class of nonlinear systems that can be divided into static nonlinear and dynamic linear, the Hammerstein model can be used to describe them. The Hammerstein model has a simple structure. It can be used for PH processes and processes with nonlinear characteristics such as dead zones, switching characteristics, and power functions. At the same time, when appropriate performance indicators are selected, the control problem can be decomposed into a dynamic optimization problem of a linear model and a static root-finding problem of a nonlinear model. The Wiener model can also describe a class of static nonlinear and dynamic linear separable nonlinear systems, but the linear dynamic element needs to be in front of the nonlinear static gain. (3) Predictive models based on intelligent means With the development of control strategies such as fuzzy control, neural networks, and artificial intelligence, intelligent control has increasingly demonstrated its superiority in solving nonlinear system control problems compared to other control methods. Combining predictive control with intelligent control is an effective way to improve control performance. An important aspect of the combination is that the nonlinear model of predictive control is described by an intelligent model. For example, the Fuzzy model is used as the predictive model in reference [11]. Reference [12] also gives a GPC algorithm that combines fuzzy control and predictive control. In addition, neural network models are also frequently used. Many documents have shown that a multi-layer feedforward neural network can arbitrarily and accurately approximate a continuous function. Some documents have further proven that a feedforward network with only one hidden layer can arbitrarily and accurately approximate a continuous function and its derivatives by using the Sigmoid function or other types of nonlinear functions. For this reason, when controlling nonlinear systems, the combination of predictive control and neural networks has led to the MPC method based on neural networks. Early research in this area mainly used neural networks to build models and then optimized them. Reference [18] used a three-layer BP neural network to build a model of a nonlinear object for multi-step prediction, then used numerical methods for optimization, and used quadratic programming to solve the objective function, thus forming a predictive controller. It was then applied to chemical processes. There have been many studies on this topic since then. However, due to the problems of local minima and slow convergence speed in neural network learning algorithms, many studies have focused on improving neural network algorithms. Reference [19] proposed to address the shortcomings of neural networks by using a rate gradient algorithm on the basis of a two-layer feedforward network and using a multi-level step response to build a global linear model to achieve rolling optimization, thus realizing DMC control of nonlinear systems. Reference [20] gave an improved global optimization adaptive fast BP algorithm, which was used in the generalized predictive control algorithm and solved the problem of speed in GPC real-time control. Reference [21] proposed a method based on BP networks to reduce the network size by using prior knowledge, improve the learning speed, and then use a genetic algorithm to optimize the control trajectory, thus overcoming the problems of local minima and slow convergence speed. In summary, the current research on this topic mainly focuses on solving the problems of neural network learning algorithms themselves. Genetic algorithms do not demand a specific form of expression for the problem and are globally optimized. Therefore, combining genetic algorithms with predictive control is a feasible optimization technique. The main approach is to use genetic algorithms as an optimization technique for the design of nonlinear model predictive controllers. This ensures the global optimality of the control law within the limited range of control input while improving the real-time performance of the control system. It has certain applications in this area. However, it mainly targets linear systems and has little research on nonlinear systems. There should be some room for development. (4) Predictive Model Based on Linearization The linearization model is the earliest method used to solve nonlinear problems. Its advantages are that nonlinear MPC optimization calculation is simple and real-time performance is good. A representative example is a nonlinear 0DMC method, which linearizes the nonlinear model at the sampling point to form a predictive controller and has been successfully applied in practice. The multi-model method is a commonly used method for dealing with nonlinear systems. Its characteristic is that multiple linear models are used to approximate the nonlinear object. Reference [25] discusses a linearized multi-model representation of a nonlinear system and provides a multi-model reference trajectory for the linearized sub-model, thus obtaining a nonlinear multi-model predictive control method. 5. Problems with Predictive Control Predictive control lacks in-depth theoretical analysis. Because it requires online rolling optimization to predict the output of the controlled process over a large area, the structure of the control system is very complex, making quantitative analysis difficult. Furthermore, there is limited research on the stability and robustness of predictive control algorithms. At the same time, the predictive control problem for nonlinear systems has not been well solved. Improvements in these areas require research on algorithms that closely adhere to the three mechanisms of predictive control: model prediction, rolling optimization, and feedback correction. In solving the control problems of nonlinear systems, predictive control should be used in conjunction with other intelligent control strategies, which is also an important direction for the development of predictive control.