How do I calculate the static Kalman gain (= steady-state Kalman gain)?

Dear Matlab users,

how do I calculate the static Kalman gain in advance? Is there any functions that do this for me if I provide the system matrices and covariance data Q and R of the process and measurement noises respectively? Please inform me ways to do this.

Thank you!

Answer 1

[Nitin Khola]

A popular FileExchange submission "Kalman Filter Package" might have some functionality coded for you. The following is the link to the submission: http://www.mathworks.com/matlabcentral/fileexchange/38302-kalman-filter-package

I encourage you to use the trial version of our product, "Control System Toolbox" which offers a wide array of functionalities related to analyzing, designing, and tuning linear control systems. Below is the link to the homepage of this product: http://www.mathworks.com/products/control/

Answer 2

[Roger Labbe]

There are two accepted ways to do this.

First, write a numerical simulation and just see what value it converges to.

Second, any of several textbooks will give you the procedure to compute it directly. I am not going to try to type several pages of instruction here, not the least because I am not solid on the theory itself. Crassidis' "Optimal Estimation of Dynamic Systems" gives the clearest presentation that I know of in section 5.3.4. Dan Simon's book talks about computing this for different filters (Crassidis is for a linear, discrete filter), and sites the research that discusses how to determine if your filter will converge or not.

Me, since I'm not sending rockets into space or doing other life-critical work, I just do the numerical simulation!

Answer 3

[Bill Tubbs]

For the record, I found a way to do it using the idare function (see this material online: hw5sol.pdf from S. Boyd).

This is for the "prediction form" of the Kalman Filter where the correction gain is used as follows:

The steady-state correction gain is found by solving this equation for the steady-state error covariance:

% Example system matrices
A = [0.7 1;
     0   1];
B = [1; 0];
C = [0.3 0];
D = 0;

% Error covariance matrices
Q = diag([0.0001 0.01]);
R = 0.01;

% Solve discrete-time algebraic Riccati equation
[P,K] = idare(A',C',Q,R,[],[]);
K = K';

% Check results satisfy eqn.s
assert(all(A*(P - P*C'*(C*P*C' + R)^-1*C*P)*A' + Q - P < 1e-14, [1 2]))
assert(all(K - A*P*C'*(C*P*C' + R)^-1 < 1e-14, [1 2]))

K

K =

1.7674

0.7514

You can also check this by using the built-in Kalman filter object:

% Construct model
n = 2; ny = 1; Ts = 1;
N = zeros(n, ny);
G = eye(n);  % apply process noises to all states
H = zeros(ny, n);  % no direct transmission of noises
Gmodel = ss(A, [B G], C, [D H], Ts);
% Use MATLAB's Kalman filter function
[~, K, P] = kalman(Gmodel, Q, R, N, 'delayed');
K

K =

1.7674

0.7514

If anyone knows a better way to do this I would be interested.

Answer 4

[Bill Tubbs]

In addition to my answer above, I just discovered a special function for Kalman estimator design called dlqe in the Control System Toolbox:

G = eye(2);
[Kf,P,~] = dlqe(A,G,C,Q,R);
Kf

Kf =

1.4515

0.7514

I can't find the documentation page for this function online but the help text says:

[M,P,Z,E] = dlqe(A,G,C,Q,R) returns the gain matrix M such

that the discrete, stationary Kalman filter with observation

and time update equations

x[n|n] = x[n|n-1] + M(y[n] - Cx[n|n-1] - Du[n])

x[n+1|n] = Ax[n|n] + Bu[n]

produces an optimal state estimate x[n|n] of x[n] given y[n] and

the past measurements.

I think Kf here is the gain for the "filtering" version of the Kalman filter:

So, then you can calculate the prediction form gain simply as follows:

K = A * Kf

K =

1.7674

0.7514

I think this function may be obsolete though:

>> which dlqe

/Applications/MATLAB_R2021b.app/toolbox/control/ctrlobsolete/dlqe.m