Linear programming-based proportional-integral-derivative control of positive systems

This paper investigates the proportional-integral-derivative (PID) control of positive systems in the discrete-time case. First, a tuning parameter is introduced to construct the proportional-integral-derivative control framework of positive systems. The proportional-integral-derivative control problem is thus transformed into the control synthesis of positive systems with time-delay. By virtue of a matrix decomposition approach, the gain matrices of proportional-integral-derivative controller are designed. Using co-positive Lyapunov functions and linear programming, the positivity and stability of positive systems are achieved under the designed proportional-integral-derivative controller. Then, the proportional-integral-derivative control of positive systems with reference signal is solved. The main contribution of this paper lies in that linear programming associated to the matrix decomposition approach is first presented to handle the proportional-integral-derivative control of positive systems. Finally, two illustrative examples are provided to verify the effectiveness of the proposed design.