Calculator · Dynamics & Control
State-Space Analysis Calculator
Enter the matrices of a continuous-time, real, single-input single-output model x' = A x + B u, y = C x + D u. The calculator returns its modes, its structural properties and whether the realization is minimal.
Model (A, B, C, D)
A matrix
B vector
C vector
Modes and stability
| Eigenvalues of A | -1, -2 |
|---|---|
| Characteristic polynomial det(sI − A) | s^2 + 3s + 2 |
| Stability | Asymptotically stable — every eigenvalue in the open left half-plane |
| Displayed precision | 6 significant digits |
Structural properties
| Controllability rank | 2 of 2 |
|---|---|
| Controllable | Yes |
| Observability rank | 2 of 2 |
| Observable | Yes |
| Minimal realization | Yes — controllable and observable |
| Controllability condition number | 10.908 |
| Observability condition number | 1 |
Eigenvalue map
| Root | Re(s) | Im(s) | Multiplicity |
|---|---|---|---|
| Pole × | -1 | 0 | 1 |
| Pole × | -2 | 0 | 1 |
Technical reference
ẋ = A·x + B·u, y = C·x + D·u
Controllability matrix: C_ctr = [B, AB, A²B, …, A^(n−1)B]
Observability matrix: O = [C; CA; CA²; …; CA^(n−1)]
- x
- state vector; physical units depend on the chosen states
- u, y
- scalar input and output (SISO); units are model-dependent
- A, B, C, D
- real, constant model matrices/vectors/scalar; units follow the state, input and output definitions
- n
- state dimension[–]
Controllability and observability are decided from the rank of C_ctr and O (via SVD) and cross-checked with the Popov–Belevitch–Hautus test; the two properties are reported separately because full rank of one does not imply the other.
These rank conditions characterize this linear, time-invariant, continuous-time model only — they say nothing about a nonlinear or time-varying system, for which A, B, C, D are not even constant.
Minimality (controllable and observable together) means every internal mode of A appears in the input–output transfer function; a non-minimal realization can still be perfectly stable while hiding modes from the output.
Assumptions
- · Continuous-time, finite-dimensional, real-valued, single-input single-output.
- · If the SVD rank test and the PBH cross-check disagree, the result is reported as uncertain rather than resolved silently.
- · Minimality means controllable AND observable — a non-minimal model has modes the transfer function cannot show.
- · Pole placement and full state trajectories belong to the State-Space Lab, not to this summary.
Continue
A, B, C and D travel with you — nothing is replaced by defaults.