Exact Decomposition of Optimal Control Problems via Simultaneous Block Diagonalization of Matrices

Area of Science:

• Control Theory
• Applied Mathematics
• Systems Engineering

Background:

• Optimal control problems (OCPs) are crucial for large-scale linear dynamical systems.
• Existing methods often rely on system symmetries for decomposition.
• These methods may not always yield the most efficient subproblems.

Purpose of the Study:

• To develop an exact decomposition method for large-scale OCPs.
• To reduce high-dimensional OCPs into independent, lower-dimensional problems.
• To improve computational efficiency compared to existing techniques.

Main Methods:

• Implementation of the algebraic method of simultaneous block diagonalization (SBD).
• Decomposition of large-scale linear dynamical systems into independent subproblems.
• Preservation of all system and objective function information during decomposition.

Main Results:

• SBD provides advantages in subproblem dimensions.
• Significant reduction in computation time is achieved.
• Demonstrated benefits of SBD over group symmetry-based decomposition using networked systems.

Conclusions:

• Simultaneous block diagonalization is an effective method for decomposing large-scale OCPs.
• SBD offers superior efficiency in terms of subproblem size and computation time.
• The method is practical and beneficial for complex systems, including networked ones.