CARE

This function computes the stabilizing solution matrix X for the continuous-time algebraic Riccati equation used in linear quadratic regulator design and continuous-time state estimation.

A^T X + X A - X B R^{-1} B^T X + Q = 0

The matrix X balances state penalty terms from Q against control effort penalties from R. For stabilizable systems with appropriate weighting matrices, the stabilizing solution defines the optimal quadratic cost-to-go and is a standard building block for LQR feedback gains.

Excel Usage

=CARE(A, B, Q, R)

A (list[list], required): State dynamics matrix A.
B (list[list], required): Input matrix B.
Q (list[list], required): State weighting matrix Q.
R (list[list], optional, default: null): Input weighting matrix R.

Returns (list[list]): The solution matrix X.

Example 1: Simple Riccati solution

Inputs:

A		B	Q		R
0	1	0	1	0	1
0	0	1	0	1

Excel formula:

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

Expected output:

Result
1.73205	1
1	1.73205

Example 2: Scalar continuous Riccati equation

Inputs:

A	B	Q	R
-1	1	2	1

Excel formula:

=CARE({-1}, {1}, {2}, {1})

Expected output:

0.732051

Example 3: Omitted R uses identity weighting

Inputs:

A		B	Q
0	1	0	2	0
0	0	1	0	1

Excel formula:

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

Expected output:

Result
2.7671	1.41421
1.41421	1.95664

Example 4: Diagonal stable system with anisotropic state cost

Inputs:

A		B		Q		R
-1	0	1	0	3	0	2	0
0	-2	0	1	0	1	0	4

Excel formula:

=CARE({-1,0;0,-2}, {1,0;0,1}, {3,0;0,1}, {2,0;0,4})

Expected output:

Result
1.16228	0
0	0.246211

Python Code

Show Code

import control as ct
import numpy as np

def care(A, B, Q, R=None):
    """
    Solve the continuous-time algebraic Riccati equation.

    See: https://python-control.readthedocs.io/en/latest/generated/control.care.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.
        Q (list[list]): State weighting matrix Q.
        R (list[list], optional): Input weighting matrix R. Default is None.

    Returns:
        list[list]: The solution matrix X.
    """
    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)
        q_np = to_np(Q)
        r_np = to_np(R)

        X, L, G = ct.care(a_np, b_np, q_np, r_np)
        return X.tolist()
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

A *

State dynamics matrix A.

B *

Input matrix B.

Q *

State weighting matrix Q.

Input weighting matrix R.