On linear parameter varying control of time‐delayed positive systems

Abstract

The original purpose of this article is to tackle the stabilizing controller design for linear parameter varying positive systems with multiple delays, in which the considered delays are time-invariant. The proposed control strategy consists of a parameter varying state-feedback controller. A complete solution to the design problem is proposed by utilizing a Lyapunov approach. In order to decrease the conservativeness of the controller design, we define a slack variable to prevent the multiplication of the Lyapunov matrices with the system matrices. Moreover, the slack variable is considered as a parameter varying matrix. Finally, the criteria are transformed into linear matrix inequalities. Therefore, sufficient conditions stabilize the closed-loop system and preserve the positive property. A special case for the mentioned system is the deep brain stimulation system used in the area of Parkinson's treatment. It enables neuroscientists to control abnormal brain activity by implanting electrodes that generate electrical impulses. This system is provided as a practical example to guarantee the effectiveness of the obtained outcomes.