CTRB

This function builds the controllability matrix for a linear state-space system with state matrix A and input matrix B.

\mathcal{C} = [B, AB, A^2 B, \dots, A^{n-1} B]

A system is controllable when this matrix has full rank, meaning every state can be reached from a suitable input over finite time. The result is commonly used as a prerequisite check before pole placement, LQR design, and related controller synthesis tasks.

Excel Usage

=CTRB(A, B)

• A (list[list], required): State dynamics matrix A.
• B (list[list], required): Input matrix B.

Returns (list[list]): The controllability matrix.

Example 1: Controllability of a second order system

Inputs:

A		B
0	1	0
-2	-3	1

Excel formula:

=CTRB({0,1;-2,-3}, {0;1})

Expected output:

Result
0	1
1	-3

Example 2: Single-state controllability matrix

Inputs:

A	B
2	3

Excel formula:

=CTRB({2}, {3})

Expected output:

3

Example 3: Two-state system with two control channels

Inputs:

A		B
1	0	1	0
0	2	0	1

Excel formula:

=CTRB({1,0;0,2}, {1,0;0,1})

Expected output:

Result
1	0	1	0
0	1	0	2

Example 4: Double integrator controllability matrix

Inputs:

A		B
0	1	0
0	0	1

Excel formula:

=CTRB({0,1;0,0}, {0;1})

Expected output:

Result
0	1
1	0

Python Code

Show Code

import control as ct
import numpy as np

def ctrb(A, B):
    """
    Compute the controllability matrix.

    See: https://python-control.readthedocs.io/en/latest/generated/control.ctrb.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        A (list[list]): State dynamics matrix A.
        B (list[list]): Input matrix B.

    Returns:
        list[list]: The controllability matrix.
    """
    try:
        def to_np(x):
            if x is None:
                return None
            if not isinstance(x, list):
                return np.array([[float(x)]])
            if x and not isinstance(x[0], list):
                x = [x]
            return np.array([[float(v) if v is not None and str(v) != "" else 0.0 for v in row] for row in x])

        a_np = to_np(A)
        b_np = to_np(B)

        C = ct.ctrb(a_np, b_np)
        return C.tolist()
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

A *

State dynamics matrix A.

B *

Input matrix B.