Kinetic feedback design for polynomial systems

• Polynomial feedback controllers are proposed for the stabilization of polynomial systems with linear input structure that results in a closed loop system with a complex balanced or weakly reversible realization.
• The feedback resulting in a complex balanced closed loop system having a prescribed equilibrium point can be computed using linear programming (LP).
• The robust version of the problem is also solvable with an LP solver.
• The feedback computation for rendering a polynomial system to deficiency zero weakly reversible form can be solved in the mixed integer linear programming (MILP) framework.
• New monomials (complexes) into the feedback does not improve the solvability of the problems.

New computational methods are proposed in this paper to construct polynomial feedback controllers for the stabilization of polynomial systems with linear input structure around a positive equilibrium point. Using the theory of chemical reaction networks (CRNs) and previous results on dynamical equivalence, a complex balanced or weakly reversible zero deficiency closed loop realization is achieved by computing the gain matrix of a polynomial feedback using optimization. It is shown that the feedback resulting in a complex balanced closed loop system having a prescribed equilibrium point can be computed using linear programming (LP). The robust version of the problem, when a convex set of polynomial systems is given over which a stabilizing controller is searched for, is also solvable with an LP solver. The feedback computation for rendering a polynomial system to deficiency zero weakly reversible form can be solved in the mixed integer linear programming (MILP) framework. It is also shown that involving new monomials (complexes) into the feedback does not improve the solvability of the problems. The proposed methods and tools are illustrated on simple examples, including stabilizing an open chemical reaction network.