Event-triggered asynchronous sliding mode control of CSTR based on Markov model
Introduction
As the core device in the process of nonferrous metal smelting, the control design and optimization of metallurgical reactor are of great significance to the improvement of production performance, energy conservation and consumption reduction [1], [2]. Especially, as a typical metallurgical reactor device, continuous stirred tank reactor (CSTR) has been widely used in the actual production process due to its advantages such as simple operation, economy and high efficiency. However, on account of the strong nonlinearity, strong correlation coupling and susceptibility to be affected by disturbances, it may bring some challenges during the control optimization of CSTR. At present, some relevant researches on the design of appropriate control methods are gaining wide attentions by academia and industry, such as, classical PID control algorithm [3], [4], adaptive algorithm [5], model predictive control combined with advanced algorithms [6], [7], etc.
It should be pointed that the some remarkable results mentioned above only consider the single-mode CSTR, with the cascade relationship of the multi-modes is ignored. During real industrial process, there exists multi-mode switching phenomenon of CSTR. Firstly, the reactor device is composed of multiple reactor cascades, and the fluctuation of the temperature, product qualities and other parameter in previous reactor may lead to different operating characteristics in the next reactor. In addition, external disturbance and environmental sudden changes will cause the mode transfer phenomenon in industrial process. In the recent decade, some modeling and design approaches have been proposed recently to deal with the multimodal behaviors of CSTR. In [8], Markov model is introduced to describe the switching among normal, abnormal and fault modes in the reaction process of CSTR, and the real-time monitoring of multi-mode processes has been studied. In [9], the CSTR is regarded as a networked jump system, and Markov probability matrix is developed to describe the mode switch of the system. However, most of the existing CSTR models ignore the temperature fluctuation of the reactor, which will make the reaction rate and reaction equilibrium point change and mode transfer. Once the mode switching phenomenon is not addressed properly, it may affect the quality of non-ferrous metal products, and in severe cases the system may become unstable or even explode. Thus, it is of great significance to accurately model and control the mode switching caused by temperature change in CSTR. Owing to the strong modeling ability for multi-mode switching phenomena, Markov model has been widely used in power systems, aircraft systems, and industrial process [10], [11], [12], [13], etc. This motivates our current work to develop an effective model of CSTR by the Markov approach.
Sliding mode control (SMC), which is featured with excellent transient performance and strong robustness to perturbations or disturbances [14], [15], has found wide-ranging applications in diverse engineering fields. However, industrial systems face complex working conditions, and above mentioned traditional SMC method is difficult to obtain good control effects. In order to overcome this problem, some novel forms have been proposed, such as observer-based sliding mode [16], [17], [18], terminal sliding mode [19], [20], [21], high-level Sliding mode [22], [23]. On the other hand, due to transmission delay and pocket drop in network environment, the modes of plant systems are not exactly known, which inevitably leads to asynchronization between the plant and the sliding mode. Such phenomenon has received increasing attentions in the past years, and some great efforts have been made on asynchronous control and design for Markov jump systems (MJSs) [24], [25], [26], [27].
Compared with the time triggered mechanism, the event-triggered strategy can save communication resources and reduce computing costs by pre-designing event-triggered conditions. Recently, the event-trigger strategy has been aroused wide concern and employed to solve bandwidth limitation and network congestion in communication [28], [29], [30], [31]. In the actual non-ferrous metallurgical industry environment, the information transmission capacity of CSTR is confined, and small fluctuations of the material parameters are not expected to be updated to the controller. Therefore, an event-triggered strategy is introduced to determine whether the current data of CSTR should be released. In addition, considering the product concentration in the CSTR reaction system cannot be measured online in real time, a state observer is designed based on event-triggered to achieve effective estimation of state information.
In this paper, a discrete asynchronous sliding control method based on event-triggered strategy is proposed for the CSTR in the non-ferrous metallurgical industry. The main contributions of this paper are summarized as follows: i) The Markov model is developed to modelling the multi-mode switching of the CSTR caused by temperature fluctuation, and a more accurate description of the operation modes of the CSTR is achieved. ii) A state observer combined with the event-triggered strategy is developed to achieve effective estimation of the unmeasurable system states in CSTR which can save signal transmission resources and computing costs. iii) A discrete asynchronous sliding mode controller is synthesized to guarantee the stability of the sliding mode dynamic system and the finite time reachability of the sliding dynamics.
Section snippets
Model dynamics
The normal process of the CSTR is shown schematically in Fig. 1. To guarantee the safety and product quality of industrial production, it is very vital to control the temperature of the reaction process to ensure that the system works in equilibrium. Without loss of generality, we assume that the reactants in CSTR are well mixed, that is, the material concentration and reaction temperature in the reactor are equal to those in the outlet.
According to material balance and energy balance in the
Design and stability analysis of sliding mode surface
This part will focus on an integral type sliding surface design first, where its definition is given bywhere constant matrix satisfies that is non-singular, and is the gain of controller that will be designed later. In addition, it is assumed that is satisfied.
In order to proof the reachability of the sliding dynamics, the observer is synthesized to ensure the sliding dynamics trajectories to converge toward the designed
Simulation
The dimensionless parameters of CSTR are . And the other parameters are given as follows:to ensure the matrix is invertible,
The Markov transition matrix of the system is and the model detection matrix is :
Assume that the external disturbance is
Conclusion
In this paper, a Markov model was adopted, which can effectively describe the mode switching phenomenon of CSTR, and a discrete asynchronous sliding mode control algorithm was proposed. The designed asynchronous state observer based on the event-triggered mechanism can save the communication resources effectively and solve the unmeasurable problem of the system states. The sliding mode controller was constructed to ensure the dynamic reachability of the closed-loop system. Experimental
Declaration of Competing Interest
We declare that we have no financial or personal relationship with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature kind in any product, service or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (61973319), the Excellent Youth Natural Science Foundation of Hunan Province under Grant 2019JJ30032, the State Key Laboratory of Robotics and Systems(HIT) (SKLRS-2020-KF-14), and the Fundamental Research Funds for the Central Universities of Central South University (2019zzts565), the Project of State Key Laboratory of High Performance Complex Manufacturing, Central South University, (Grant No. ZZYJKT2019-14).
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