Optimal L₁-controller synthesis for positive systems and its robustness properties

In a recent study, we introduced an L₁-induced norm (L ₁ gain in short) as a performance index for linear time-invariant positive systems, where the L₁-norm of the disturbance input and the performance output signals is evaluated with positive weighting vectors. Moreover, we showed that the L₁ gain with the weighting vectors plays an essential role in stability analysis of interconnected positive systems. Aiming at extending this result for stabilization of interconnected positive systems, in this paper, we study L₁-optimal feedback controller synthesis for positive systems with given weighing vectors. In particular, we will show that an L₁-optimal state-feedback gain designed for a fixed positive system and a fixed pair of weighing vectors is robustly optimal against variations on the input matrix, the direct feedthrough matrix of the controlled positive system as well as variations on the weighting vector for the disturbance input signal. This property is of course promising for robust L₁-optimal control of uncertain positive systems. We illustrate the robust optimality property by numerical examples.