OBSV

This function builds the observability matrix for a linear state-space system with state matrix A and output matrix C.

\mathcal{O} = \begin{bmatrix} C \\ C A \\ C A^2 \\ \vdots \\ C A^{n-1} \end{bmatrix}

A system is observable when this matrix has full rank, meaning the internal state can be reconstructed from output measurements over finite time. The matrix is commonly used when checking whether observer design or state estimation is feasible.

Excel Usage

=OBSV(A, C)

• A (list[list], required): State dynamics matrix A.
• C (list[list], required): Output matrix C.

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

Example 1: Observability of a second order system

Inputs:

A		C
0	1	1	0
-2	-3

Excel formula:

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

Expected output:

Result
1	0
0	1

Example 2: Single-state observability matrix

Inputs:

A	C
2	3

Excel formula:

=OBSV({2}, {3})

Expected output:

3

Example 3: Two-state system with two output channels

Inputs:

A		C
1	0	1	0
0	2	0	1

Excel formula:

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

Expected output:

Result
1	0
0	1
1	0
0	2

Example 4: Double integrator observability matrix

Inputs:

A		C
0	1	1	0
0	0

Excel formula:

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

Expected output:

Result
1	0
0	1

Python Code

Show Code

import control as ct
import numpy as np

def obsv(A, C):
    """
    Compute the observability matrix.

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

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

    Args:
        A (list[list]): State dynamics matrix A.
        C (list[list]): Output matrix C.

    Returns:
        list[list]: The observability 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)
        c_np = to_np(C)

        O = ct.obsv(a_np, c_np)
        return O.tolist()
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

A *

State dynamics matrix A.

C *

Output matrix C.