# The Design of 2 DOF IMC-PID Controller in Biochemical Reaction Networks
###### Abstract  
DNA molecules can be adopted to design biomolecular control circuits that can effectively control biochemical reaction processes. However, the leak reaction in actual biochemical reactions causes a significant uncertainty for reactions. In this paper, the first-order time-delay system is selected as the controlled object. A two-degree-of-freedom internal model PID controller (2 DOF IMC-PID) is constructed for the first time within the framework of chemical reaction networks (CRNs). Under this control strategy, the set-point tracking and disturbance suppression are tuned with individual controllers, respectively. The controller parameters are determined by two filtering parameters that affect the controller's performance, so the parameter tuning is simpler and more targeted. Then, the 2 DOF IMC-PID controller is implemented in DSD reaction networks, with less overshoot in the 2 DOF IMC-PID control system than the traditional PID control system and the 2 DOF PID control system. Finally, a 2 DOF IMC-PID division gate control system is established to effectively inhibit the impacts of leak reactions on the computation results. Although the leak reaction occurs at the division gate, the ideal output can be produced by the 2 DOF IMC-PID division gate control system. 2 DOF IMC-PID controller; chemical reaction networks (CRNs); DSD reaction networks; leak reaction; division gate +
Footnote †: journal: _Sci._  
Footnote †: journal: _Sci. systems [17; 18; 19; 20; 21]. The division gate, especially, can be built by using the three basic calculation methods of addition, subtraction, and multiplication to realize the function of calculating the ratio of two chemical species [22]. However, leak reactions of biochemical reactions make studying chemistry circuits difficult. Leak reactions refer to the fact that, for some reasons (e.g., some base pairs of the DNA strand can be temporarily interrupted [23]), the concentrations of the initial species are different from the ideal set concentrations. For example, the initial species concentrations drop due to the leak reaction in the division gate. The calculation of the division gate is incorrect due to the influence of the leak reaction. Many biomolecular controllers have been designed by using CRNs as the programming language. A linear I/O system is implemented through ideal chemical reactions by Oishi et al. [24]. Yordanov et al. [25] built a PI controller using CRNs and designed DSD reactions for catalysis, degradation, and annihilation. A QSM feedback controller is constructed with DSD by Sawlekar et al. [26]. Mathias et al. [27; 28] devised and realized a signal cycle controller and feedforward controller based on DSD. Paulino et al. [29] modeled a delay time system and PID controller using CRNs. Yuan et al. [30] evidenced that the 2 DOF PID controller performs better than the 1 DOF PID controller at the molecular level. Wang et al. [31] accomplished the PI control of the three-dimensional oscillatory chaotic system at the biomolecular level. A state feedback controller is designed by using CRNs to deal with the problem of the leak reaction in the addition gate [32]. Previous research shows that the 2 DOF PID biomolecular controller is operated by setting five parameters [30; 33], but the complex parameter adjustment problem is ignored. The 2 DOF IMC-PID control strategy is indicated to provide better control characteristics and leads to less complexity in parameter tuning [34]. Currently, the internal model control (IMC) strategy is used in many industrial situations [35; 36; 37; 38], but there are few studies and applications in biochemistry. Traditional PID controllers are capable of stabilizing linear systems without a time delay and provide excellent control performance. In fact, the time delay is unavoidable in biochemical systems; for example, due to the temperature, pH, and other factors, it creates a time delay when establishing molecular devices by using DSD [30].