$\mathrm{H}_{\infty}$ Control of One Class of Bilinear Positive Continuous-Time Systems

This paper focuses on the quadratic stabilization of one class of bilinear positive continuous-time systems subject to system disturbances. Using the quadratic Lyapunov function, new control design conditions are proposed as a set of linear matrix inequalities, which reflect the principle of diagonal stabilization and guarantee the quadratic stability and positivity of the closed-loop system. Taking into account an upper bound for the
$\mathrm{H}_{\infty}$
norm of the disturbance transfer function matrix, the task is redefined as a convex problem and it is shown that quadratic stabilization of the bilinear systems can be achieved by state feedback control. The key properties are illustrated by the detailed construction of the required system parameter structures in the example.