The eigenvalues of a state-space system representation can be described with the following equation
$$eig(A)=det(sI-A)=0$$
The eigenvalues of the state-space system can tell us about the stability of the system.
| Eigenvalue | Effect |
| Positive real number | Driven away from a steady state (unstable) |
| Negative Real Number | Driven toward a steady state (stable) |
| 0 | Remains at disturbed position |
| Identical to another eigenvalue | effects cannot be determined |
| Complex Positive Real Number | Oscillates around steady state with increasing amplitude (unstable) |
| Complex Negative Real Number | Oscillates around a steady state with decreasing amplitude (stable) |
| Imaginary Number | Oscillates around steady-state value with constant amplitude |
Your estimator gain matrix should have eigenvalues that are at least 5-10 times as fast as the controller eigenvalues. Slower eigenvalues provide sensor filtering.
[[Hopf Bifurcations]] – if two eigenvalues of a state-space system cross the imaginary axis, it can generate a hopf bifurcation.
[[Limit Points]] – generated as eigenvalues pass the imaginary axis
[[Advantages of the QR algorithm]] – can solve for all eigenvalues of a system
[[Tip Jet Dynamics]] – shows table of eigenvalues of tip-jet rotor
[[Root-Locus Plot of PI Pitch Rate Tracking Controller]] – shows eigenvalue of open/closed loop systems.
[[Modified Mu-Synthesis]] – ensure that any eigenvalues that are not within spec are actual physical states and not just controller states.
[[AC45 3DOF Model]] – shows eigenvalues of the state-space models
Sources
- bevlyMECH4420Lecturea
- DynamicalSystemsAnalysis
- bevlyMECH4420Lectureb
Backlinks
[[Complex Numbers]]
Eigenvalue
F-22 Flight Control System
[[Root Locus Plots]]
[[Stability Analysis of Systems]]
State-Space Estimators
[[State-Space Model]]