Data-Driven Identification and Control of Positive Systems

This paper considers the data-driven identification and control of positive dynamic systems. Since such systems appear in diverse applications whereby analytical models may not be easily available, the input-output data are collected for identification and control. Direct application of subspace system identification (SSID) to collected data of positive system does not guarantee the positivity of the resulting state space parameters. The proposed data-driven identification offers a procedure to obtain the state space representation of a positive system by applying nonnegative matrix factorization (NMF) to the Hankel matrix. The identified positive system is used for subsequent positive stabilization and performance requirements. This approach resembles the indirect adaptive control strategy with the understanding that the classical system identification based adaptive control cannot guarantee positivity constraint. Subsequently, we also address a direct approach in which data-driven positive control can be achieved. Using formulas for data-driven control, we formulate the problem of positive stabilization of linear systems and provide a solution for it directly from collected input output data. This allows to obtain state feedback gain matrices without the intermediate subspace identification. With the aid of available stability results of positive systems, the solution of data-driven positive stabilization can be obtained using data-dependent LMI. Finally, numerical examples are provided to support the theoretical results.