DLQE

The discrete-time linear quadratic estimator (DLQE) calculates the optimal estimator gain matrix L for discrete-time systems with process and measurement noise.

It minimizes the expected squared error of the state estimate x_e[n] using sensor measurements y[n].

Excel Usage

=DLQE(A, G, C, QN, RN, NN)

A (list[list], required): The discrete state dynamics matrix A (NxN).
G (list[list], required): The discrete process noise input matrix G (NxNw).
C (list[list], required): The output matrix C (Ny x N).
QN (list[list], required): Process noise covariance matrix (Nw x Nw).
RN (list[list], required): Sensor noise covariance matrix (Ny x Ny).
NN (list[list], optional, default: null): Optional cross-covariance matrix (Nw x Ny).

Returns (list[list]): The discrete Kalman estimator gain matrix L.

Example 1: Discrete LQE design

Inputs:

A		G		C		QN		RN
0.5	0	1	0	1	1	0.1	0	1
0	0.3	0	1			0	0.1

Excel formula:

=DLQE({0.5,0;0,0.3}, {1,0;0,1}, {1,1}, {0.1,0;0,0.1}, {1})

Expected output:

Result
0.0514686
0.0260185

Example 2: Single-state discrete estimator

Inputs:

A	G	C	QN	RN
0.6	1	1	0.2	0.5

Excel formula:

=DLQE({0.6}, {1}, {1}, {0.2}, {0.5})

Expected output:

0.206232

Example 3: Coupled-state measurement model

Inputs:

A		G		C		QN		RN
0.9	0.2	1	0	1	-1	0.03	0	0.4
0.1	0.8	0	1			0	0.03

Excel formula:

=DLQE({0.9,0.2;0.1,0.8}, {1,0;0,1}, {1,-1}, {0.03,0;0,0.03}, {0.4})

Expected output:

Result
0.273274
0.0608544

Example 4: Partially observed discrete estimator

Inputs:

A		G		C		QN		RN
0.7	0.1	1	0	1	0	0.05	0	0.3
0	0.6	0	1			0	0.02

Excel formula:

=DLQE({0.7,0.1;0,0.6}, {1,0;0,1}, {1,0}, {0.05,0;0,0.02}, {0.3})

Expected output:

Result
0.151352
0.00438577

Python Code

Show Code

import control as ct
import numpy as np

def dlqe(A, G, C, QN, RN, NN=None):
    """
    Linear quadratic estimator (Kalman filter) for discrete-time systems.

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

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

    Args:
        A (list[list]): The discrete state dynamics matrix A (NxN).
        G (list[list]): The discrete process noise input matrix G (NxNw).
        C (list[list]): The output matrix C (Ny x N).
        QN (list[list]): Process noise covariance matrix (Nw x Nw).
        RN (list[list]): Sensor noise covariance matrix (Ny x Ny).
        NN (list[list], optional): Optional cross-covariance matrix (Nw x Ny). Default is None.

    Returns:
        list[list]: The discrete Kalman estimator gain matrix L.
    """
    try:
        def to_np(x):
            if x is None:
                return None
            if not isinstance(x, list):
                return np.array([[float(x)]])
            if len(x) > 0 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)
        g_np = to_np(G)
        c_np = to_np(C)
        qn_np = to_np(QN)
        rn_np = to_np(RN)
        nn_np = to_np(NN)

        if nn_np is None:
            L, P, E = ct.dlqe(a_np, g_np, c_np, qn_np, rn_np)
        else:
            L, P, E = ct.dlqe(a_np, g_np, c_np, qn_np, rn_np, nn_np)

        return L.tolist()
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

A *

The discrete state dynamics matrix A (NxN).

G *

The discrete process noise input matrix G (NxNw).

C *

The output matrix C (Ny x N).

QN *

Process noise covariance matrix (Nw x Nw).

RN *

Sensor noise covariance matrix (Ny x Ny).

Optional cross-covariance matrix (Nw x Ny).