Robust Optimal Reference-Tracking Design Method for Stochastic Synthetic Biology Systems: T–S Fuzzy Approach

At present, the development in the nascent field of synthetic gene networks is still difficult. Most newly created gene networks are nonfunctioning due to intrinsic parameter fluctuations, uncertain interactions with unknown molecules and external disturbances of intra and extracellular environments on the host cell. How to design a completely new gene network, that is to track some desired behaviors under these intrinsic and extrinsic disturbances on the host cell, is the most important topic in synthetic biology. In this study, the intrinsic parameter fluctuations, uncertain interactions with unknown molecules and environmental disturbances, are modeled into the nonlinear stochastic systems of synthetic gene networks in vivo. Four design specifications are introduced to guarantee the stochastic synthetic gene network, which can achieve robust optimal tracking of a desired reference model in spite of these intrinsic and extrinsic disturbances on the host cell. However, the robust optimal reference-tracking design problem of nonlinear synthetic gene networks is still hard to solve. In order to simplify the design procedure of the robust optimal nonlinear stochastic-tracking design for synthetic gene networks, the Takagi-Sugeno (T-S) fuzzy method is introduced to solve the nonlinear stochastic minimum-error-tracking design problem. Hence, the robust optimal reference-tracking design problem under four design specifications can be solved by the linear matrix inequality (LMI)-constrained optimization method using convex optimization techniques. Further, a simple design procedure is developed for synthetic gene networks to meet the four design specifications to achieve robust optimal reference tracking. Finally, an eigenvalue-shifted design method is also proposed as an expedient scheme to improve the stochastic optimal-tracking design method of synthetic gene oscillators.