Linear Quadratic Optimal Control of Time-Invariant Linear Networks With Selectable Input Matrix

Area of Science:

• Control theory
• Network science
• Systems engineering

Background:

• Traditional linear quadratic regulator (LQR) assumes a fixed input matrix (B) for time-invariant linear systems.
• Designing the input matrix (B) offers potential for cost reduction but is not covered by conventional LQR.
• Riccati differential equations (RDEs) are central to LQR but obtaining explicit solutions for a variable B is challenging.

Purpose of the Study:

• Investigate the impact of a designable input matrix (B) on minimizing network control costs.
• Develop analytical methods to optimize the input matrix (B) for reduced quadratic cost functions.
• Explore network control strategies through input matrix design.

Main Methods:

• Formulated an equivalent expression for the quadratic cost function with respect to the input matrix (B).
• Derived the analytical gradient of the cost function concerning the matrix variable B.
• Proposed optimization problems and gradient-based algorithms for input matrix design.

Main Results:

• Demonstrated significant cost reduction in controlling LTI systems by designing the input matrix (B).
• Identified that optimal input matrix design involves sparse and evenly distributed input sources.
• Established inequalities for cost functions to guide optimization.

Conclusions:

• Designing the input matrix (B) is a viable strategy for substantially lowering control costs in LTI systems.
• Optimal network control requires careful placement of input sources, favoring sparse and uniform distribution.
• Findings provide insights into effective control strategies through input matrix design for LTI systems.