Controllability Check Calculator Online Calculator
Determine if your linear system is controllable using Kalman's rank condition. Enter state-space matrices below — free, instant, and accurate for control engineering, aerospace, robotics, and academic studies in the USA.
Controllability Check Tool (Kalman Rank Condition)
Use the controllability check calculator online calculator to test controllability for any LTI system: ẋ = Ax + Bu. Enter matrix A (n×n) and matrix B (n×m). The tool computes the controllability matrix C = [B, AB, A²B, ..., Aⁿ⁻¹B] and its rank. System is controllable if rank(C) = n.
What is a Controllability Check? (Kalman’s Criterion)
In modern control theory, a system is controllable if we can drive the state from any initial condition to any desired final state using admissible inputs within finite time. The controllability check calculator online calculator applies the Kalman rank condition: for a state-space model (A,B), form the controllability matrix C = [B, AB, A²B, …, Aⁿ⁻¹B]. If rank(C) equals the number of states (n), the system is fully controllable.
This concept is fundamental for pole placement, optimal control, and observer design. Engineers in the US aerospace and automotive industries rely on these tests daily.
📊 Comparison Table: Controllable vs. Uncontrollable Systems
| System (A, B) | n (states) | Controllability Matrix Rank | Controllable? | Insight |
|---|---|---|---|---|
| A = [[1,1],[0,1]], B = [[0],[1]] | 2 | 2 | ✅ Yes | Canonical controllable pair (double integrator) |
| A = [[1,0],[0,2]], B = [[0],[0]] | 2 | 0 | ❌ No | Zero input matrix, no control influence |
| A = [[0,1],[-2,-3]], B = [[0],[1]] | 2 | 2 | ✅ Yes | Standard controllable state-space form |
| A = [[1,0],[0,1]], B = [[1],[0]] | 2 | 1 | ❌ No | Second state unreachable |
🔍 Step-by-Step Controllability Analysis Checklist (2026)
- Step 1: Write the system in state-space form: ẋ = Ax + Bu. Determine dimension n.
- Step 2: Build controllability matrix C = [B | AB | A²B | … | Aⁿ⁻¹B] (n × n*m).
- Step 3: Compute rank(C) via Gaussian elimination or SVD.
- Step 4: Compare rank(C) with n. If equal → fully controllable.
- Step 5: Use this online tool to instantly verify results, especially before controller design.
Pro Tip: For MIMO systems (multiple inputs), the rank condition still holds: rank(C) = n ensures each state is reachable via some input combination.
Milestones in Controllability Theory
Rudolf Kalman introduces state-space controllability and rank condition.
Kalman, Ho, and Narendra formalize realization theory.
Robust control & H-infinity methods rely on controllability for synthesis.
AI-based control design tools integrate automated controllability checks for autonomous systems.
Understanding these milestones helps grasp why controllability check remains a cornerstone of modern control engineering.
Exclusive Insights: 5 Common Pitfalls When Testing Controllability
- Numerical rounding: Use tolerance (1e-10) when computing rank. Our tool uses robust elimination with epsilon.
- Modal decomposition: A system can be uncontrollable if an eigenvalue has no corresponding input influence (Hautus test).
- MIMO misconception: More inputs doesn’t guarantee controllability; rank must still be n.
- Incorrect B dimension: B must have n rows. Our parser validates size mismatch.
- Time-varying systems: This calculator addresses LTI systems; for time-variant, different criteria exist.
Quick Hack: For 2nd-order systems, try our controllability check calculator online calculator to verify if both position and velocity are reachable.
Frequently Asked Questions
It constructs the controllability matrix (C) and computes its rank. If rank = number of states, the system is controllable.
Yes, the script works for any square A matrix and matching B. Up to 6x6 works efficiently. For higher dimensions, numerical precision remains good.
Controllability is a core concept in control systems, frequently tested in Fundamentals of Engineering (FE) and Professional Engineer (PE) control exams.
Uncontrollable states indicate that some dynamics cannot be influenced by inputs. Solutions: redesign actuators, change sensors, or use state feedback with controllable subspace decomposition.
Yes, we refreshed all algorithms for 2026 with updated matrix handling and example use cases relevant to modern control labs.
Absolutely. Enter B as a matrix (e.g., "1,0;0,1") to test multi-input controllability.
We use row reduction with tolerance 1e-8, providing high accuracy for practical engineering matrices.