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Linear Dynamical Systems
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Zero eigenvalues
Marginal stability leading to constant but non-zero velocity of state vector
Damping ratio
The system's rate of convergence to steady state
Stability of system
Persistent behavior over time, without unbounded growth
Diagonalizable matrix
System can be decoupled into independent modes of behavior
Non-diagonalizable matrix
Presence of Jordan blocks requiring more complex analysis
Eigenvalues with positive real parts
System deviates from equilibrium, potentially leading to unbounded growth
Constant coefficient linear differential equation
Predictable and structured system response
BIBO Stability
Bounded input leads to bounded output
System controllability
The ability to move the system from any initial state to any desired final state within finite time
High condition number of the system matrix
Sensitive system behavior to parameter changes or noise
Eigenvalues with negative real parts
System returns to equilibrium after disturbance
Purely imaginary eigenvalues
Neutral stability with continuous oscillation
System observability
The ability to deduce any system state from outputs in finite time
Symmetric system matrix
Conservative system with all real eigenvalues
Complex conjugate eigenvalues
Oscillatory behavior
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