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

Abstract

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.

Novel Proportional–Integral–Derivative Control Framework on Continuous-Time Positive Systems Using Linear Programming

“…Remark 4. In [27], a PID controller was proposed for discrete-time positive systems by virtue of the co-positive Lyapunov function associated with the linear programming approach. However, it is not feasible to develop the presented PID control approach for continuous-time positive systems.…”

“…In Theorem 1, the positivity and stability of the closed-loop system ( 12) cannot be directly guaranteed by using a similar method to [27]. A three-stage strategy is presented in Theorem 1 to construct a PID control framework on continuous-time positive systems by virtue of linear programming.…”

“…A systematic proportional-derivative control approach was proposed in [25] and [26] for positive systems by introducing a low past filter. By employing the copositive Lyapunov function and linear programming, a PID control framework was constructed in [27] for discrete-time positive systems. For the PID control of continuous-time positive systems, there has not been a systematic design approach.…”

“…It cannot handle the case that the derivative of the state or output is chosen as the derivative component of the PID controller. The discrete-time PID control framework in [27] transformed the PID control problem into the control of a time-delay system. Thus, the derivative component is dealt with by virtue of a time-delay approach.…”

Regional PID Control of Switched Positive Systems With Multiple Equilibrium Points

“…In [32], the proportional-derivative controller was designed for PSs. The PID control design of PSs was considered in [33] by utilizing CLFs and LP. This leaves much room to explore the PID of SPSs.…”