| Parameter | Example value | Meaning |
|---|---|---|
| A | 0.9 0.1; -0.2 0.8 | State transition. Captures internal dynamics. |
| B | 0.1; 0.05 | Control influence. Maps input into state change. |
| C | 1 0 | Observation mapping. Converts state to measured output. |
| D | 0 | Direct input-to-output path. Often zero. |
| x0 | 1; 0 | Initial condition. Starting state vector. |
| u | 1 | Constant input. Kept fixed across simulation steps. |
| Steps | 10 | Number of discrete updates to compute. |
| dt | 0.1 | Euler step size for continuous discretization. |
yk = C xk + D uk
Bd = B·dt
xk+1 = Ad xk + Bd u
- Choose the model type, then set steps and dt if needed.
- Enter A, B, C, and optionally D using the given separators.
- Provide x0 and u as vectors matching the matrix dimensions.
- Press Submit to compute trajectories and outputs above.
- Use CSV or PDF export to store results for documentation.
System structure and dimensions
A state space model represents a system with a compact set of matrices. This calculator validates that A is n by n, B is n by m, C is p by n, and D is p by m. Dimension checks prevent silent mistakes when you switch from single input to multi input designs. The simulation table reports n, m, and p so you can confirm the configuration before exporting.
Use consistent units and scaling before entering matrices, because poorly scaled states can hide issues. Try different u vectors to represent impulses, steps, or controls. For identification tasks, adjust C to match measured channels and compare outputs against sequences to tune A and B.
Discrete and continuous workflows
For discrete models the update uses x_{k+1}=A x_k + B u and y_k=C x_k + D u. For continuous models the tool applies a forward Euler discretization, forming A_d=I + A·dt and B_d=B·dt, then simulating with the discrete update. Euler is fast and interpretable, but accuracy depends on dt, so you should compare runs with smaller dt when dynamics are stiff.
Trajectory interpretation for learning
Each row shows step k, state vector x_k, and output vector y_k. Trends in x indicate stability, oscillation, or drift, while y captures what a sensor would measure. If C selects a subset of states, outputs may look smooth even when internal states vary, which is common in partially observed systems used in filtering and control.
Stability screening and caveats
When the system is 1×1 or 2×2 the calculator computes eigenvalues and flags likely stability. Discrete stability typically requires eigenvalue magnitudes below one, while continuous stability often requires negative real parts. Larger systems need numerical routines not included here, so treat this check as a quick screen, not a proof.
Export ready reporting
CSV export preserves the exact vectors displayed for reproducible analysis in spreadsheets or notebooks. PDF export produces a compact table suitable for audit trails, model documentation, or classroom handouts. Keep step counts modest to avoid overly large files, and include your matrix assumptions alongside exports for traceability.
What formats are accepted for matrices?
Enter rows separated by semicolons or new lines. Separate columns with commas or spaces. Brackets are ignored. The tool requires rectangular matrices with numeric values only.
How do I choose n, m, and p?
n is the number of state variables, m is the number of inputs, and p is the number of outputs. Pick them from your model definition, then ensure A is n×n, B is n×m, and C is p×n.
Why does the continuous option change my matrices?
Continuous dynamics use ẋ=Ax+Bu, so the calculator discretizes with Euler to simulate steps. It forms A_d=I+A·dt and B_d=B·dt. Smaller dt generally improves the approximation.
What does the stability badge mean?
For 1×1 and 2×2 systems the calculator estimates eigenvalues. Discrete models are likely stable when all magnitudes are below one. Continuous models are likely stable when real parts are negative. Larger systems are not evaluated.
Can I model time-varying inputs?
This version uses a constant input vector u across all steps, which fits step responses and steady forcing. To emulate time-varying inputs, rerun the calculator with different u values and stitch the exported tables.
Why are my outputs zero or flat?
Check C and D first. If C selects only certain states, other state changes will not appear in y. If D is zero and u is constant, y may converge quickly. Also verify x0 is not all zeros.