Reduction of state variables based on regulation and filtering performances

This paper provides a principal component analysis of linear discrete-time systems on the basis of optimal control and estimation. The analysis is to reveal the important state components which remain necessary for reducing performance degradation under dimensional constraints on control and estimation laws. The trade-off relations between the dimension and performance degradation are expressed as system invariants representing the importance of each principal component, which are characterized as the eigenvalues of matrices depending on the solutions of both Lyapunov and Riccati equations. Based on the analysis, the paper also provides model reduction techniques for the systems generating the optimal input and estimate with the desirable properties of stability, reachability, and observability being preserved in the reduced systems.