What Is the Kalman Controllability & Observability Rank Calculator?
Controllability tests whether actuators can drive the system to any arbitrary state; observability tests whether sensors can reconstruct all states.
How Does the Kalman Controllability & Observability Rank Calculator Work?
Forms the algebraic controllability and observability matrices from state-space matrices (A, B, C) and checks determinant non-zero status.
Kalman Controllability & Observability Rank Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
mathcal{C} = [B quad AB], quad mathcal{O} = egin{bmatrix} C \ CA end{bmatrix}, quad ext{Rank} = n = 2
Kalman rank criterion matrices for 2-state continuous LTI systems.
How to Use the Kalman Controllability & Observability Rank Calculator
- Input elements of 2x2 system matrix A, 2x1 input matrix B, and 1x2 output matrix C.
Step-by-Step Example Calculation
Damped Mass-Spring-Damper System
Input Values:
a11:0
a12:1
a21:-3
a22:-4
b1:0
b2:1
c1:1
c2:0
Worked Steps: Det(C) = -1 (Rank 2, Controllable), Det(O) = -1 (Rank 2, Observable) -> Fully Controllable & Observable.
Understanding Your Result
Reports determinants and states whether system is controllable, observable, or rank deficient.
Factors That Affect the Result
- Actuator/sensor placement relative to dynamic mode eigenvectors.
When Should You Use This Calculator?
- State-feedback pole placement design (Ackermann formula) and Luenberger observer / Kalman filter synthesis.
Assumptions & Limitations
- Applies to 2nd-order linear time-invariant (LTI) continuous dynamical systems.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Exact matrix determinant rank test.