Gramians & state-space analysis

[jamestjsp]

Summary

Implement controllability/observability gramians and matrix-form analysis tools.

Functions

• Gram(sys, type) — compute controllability or observability gramian (continuous/discrete)
• Ctrb(A, B) — controllability matrix [B, AB, A²B, ...]
• Obsv(A, C) — observability matrix [C; CA; CA²; ...]
• CtrbF(A, B, C) — controllability staircase form (extends existing ControllabilityStaircase)
• ObsvF(A, B, C) — observability staircase form

Implementation Notes

• Gram solves Lyapunov equations: controllability gramian Wc satisfies A*Wc + Wc*A' + B*B' = 0
• Ctrb/Obsv build full rank matrices — warn about numerical issues for large n
• Staircase forms already partially implemented via ControllabilityStaircase

Depends on

• Lyapunov & Riccati equation solvers #3 Lyapunov & Riccati equation solvers (for Gram)

Blocks

• Balanced realization & advanced model reduction
• System norms