DARE

This function computes the stabilizing solution matrix X for the discrete-time algebraic Riccati equation used in optimal control and discrete-time filtering problems.

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

The solution defines the quadratic cost-to-go for a discrete-time regulator and is used to derive optimal state feedback gains. When R is omitted, the underlying library uses an identity input weighting matrix.

Excel Usage

=DARE(A, B, Q, R)

A (list[list], required): Discrete state dynamics matrix A.
B (list[list], required): Discrete input matrix B.
Q (list[list], required): State weighting matrix Q.
R (list[list], optional, default: null): Input weighting matrix R. If omitted, the identity matrix is used.

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

Example 1: Discrete Riccati solution

Inputs:

A		B	Q		R
1	1	0.5	1	0	1
0	1	1	0	1

Excel formula:

=DARE({1,1;0,1}, {0.5;1}, {1,0;0,1}, {1})

Expected output:

Result
2.3671	1.11803
1.11803	2.58748

Example 2: Scalar discrete Riccati equation

Inputs:

A	B	Q	R
0.8	1	1	1

Excel formula:

=DARE({0.8}, {1}, {1}, {1})

Expected output:

1.36995

Example 3: Omitted R uses identity weighting in discrete time

Inputs:

A	B	Q
0.9	1	2

Excel formula:

=DARE({0.9}, {1}, {2})

Expected output:

2.584

Example 4: Two-state discrete regulator design

Inputs:

A		B	Q		R
1	0.1	0	1	0	0.5
0	0.9	1	0	2

Excel formula:

=DARE({1,0.1;0,0.9}, {0;1}, {1,0;0,2}, {0.5})

Expected output:

Result
15.8497	1.74312
1.74312	2.53848

Python Code

Show Code

import control as ct
import numpy as np

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

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

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

    Args:
        A (list[list]): Discrete state dynamics matrix A.
        B (list[list]): Discrete input matrix B.
        Q (list[list]): State weighting matrix Q.
        R (list[list], optional): Input weighting matrix R. If omitted, the identity matrix is used. 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.dare(a_np, b_np, q_np, r_np)
        return X.tolist()
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

A *

Discrete state dynamics matrix A.

B *

Discrete input matrix B.

Q *

State weighting matrix Q.

Input weighting matrix R. If omitted, the identity matrix is used.