Stabilization of positive linear discrete-time systems by using a Brauer's theorem.

Area of Science:

• Control Theory
• Systems Engineering
• Applied Mathematics

Background:

• Positive linear discrete-time systems (PLDS) are crucial in various fields, including economics and biology.
• Stabilizing these systems while maintaining positivity is a significant challenge in control theory.
• Existing methods may alter multiple system eigenvalues, complicating targeted stabilization.

Purpose of the Study:

• To develop a method for stabilizing PLDS using linear state feedback.
• To enable modification of specific system eigenvalues without affecting others.
• To provide sufficient conditions for the stability and positivity of the closed-loop system.

Main Methods:

• A novel method based on Brauer's theorem is proposed for eigenvalue modification.
• The method is applied to both single-input single-output (SISO) and multi-input multi-output (MIMO) systems.
• Sufficient conditions for closed-loop stability and positivity are derived.

Main Results:

• The proposed method successfully stabilizes PLDS by selectively modifying eigenvalues.
• Sufficient conditions for stability and positivity are proven for both SISO and MIMO cases.
• The method's efficacy is demonstrated through numerical examples.

Conclusions:

• The Brauer's theorem-based method offers a precise approach to stabilizing PLDS.
• This technique is applicable to complex MIMO systems and can be extended to stochastic systems.
• The findings contribute to advanced control strategies for positive dynamical systems.