Linear Quadratic Control of Positive Systems: A Projection-Based Approach

This technical article investigates the linear quadratic regulator (LQR) design for continuous-time positive linear systems. Based on positive systems theory and Lyapunov theory, the solvability and optimality of the positivity-preserving LQR problem are analyzed through the lens of optimization, and two projection theorems are derived for single-input and multi-input positive systems, respectively, which paves the way for developing a projected gradient descent algorithm. The proposed results fill the literature gap by considering both the optimality and positivity in LQR design, and numerical simulations are provided to verify the effectiveness of the results.