Controllability Grammian:

Observability Matrix: O = [C^T A^T C^T ... (A^T) ^n-1 C^T]

Observability Grammian:

Solution to Discrete State Equation:

Transient Specs. continuous: discrete:

Controller Form:

Where each l_i is the largest integer such that T_cf has full rank

Separation Principle: The eigenvalues for the combined observer/controller are:

Reduced-Order Observer:

Block Diagram:

where:

Markov Parameters: , Hankel Matrix

Lyapunov Theorem 1: Origin is asymptotically stable eq. pt. if

Lyapunov Theorem 2: Origin is ISL stable eq. pt. if

Lyapunov Corollary 1: Origin is asymptotically stable eq. pt. if

Lyapunov Theorem 3: Origin is GLOBALLY asymptotically stable eq. pt. if

Lyapunov Equations: Continuous: Discrete:

Lie Algebra Tools: , , ,

Liebnitz's Formula:

Nonlinear Phase Variable (Controllable Canonical) Form: