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Learning the Extended Kalman Filter

version (2.1 KB) by Yi Cao
An implementation of Extended Kalman Filter for nonlinear state estimation.


Updated 23 Jan 2008

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Editor's Note: This file was selected as MATLAB Central Pick of the Week

This is a tutorial on nonlinear extended Kalman filter (EKF). It uses the standard EKF fomulation to achieve nonlinear state estimation. Inside, it uses the complex step Jacobian to linearize the nonlinear dynamic system. The linearized matrices are then used in the Kalman filter calculation.

The complex step differentiation seems improving the EKF performance particularly in accuracy such that the optimization and NN training through the EKF are better than through the UKF (unscented Kalman filter, Other complex step differentiation tools include the CSD Hessian available at

Cite As

Yi Cao (2021). Learning the Extended Kalman Filter (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (97)

Jay shankar Nanda

I am facing a problem while running this code
error is like Maximum recursion limit of 500 reached. Use set(0,'RecursionLimit',N) to change the limit. Be aware that exceeding your
available stack space can crash MATLAB and/or your computer.

Error in ekf>create@(x)[x(2);x(3);0.05*x(1)*(x(2)+x(3))]

Can anyone help how to fix this

Veronica Echeverria

Dear Yi Cao. Thank you very much for this great explanation. I have a question regarding the measurement mapping matrix.
My system is -k grad(T) =q , I want to study the heat conductivity as a function of the temperature change. Therefore, I discretize my system with Finite Differences and obtain let's say 10 temperature nodes+the variable "k", which I want to predict as well. If I measure all of my discretized nodes, but I don't measure k, because it's physically infeasible, how can I do the mapping function h()?

yw liu


Mushahid Hussain

hi sir im having trouble in understanding JASSCAD function can you explain it?

Nauman Ali Larik


Zihao Zhou

Great code, thank you.
However, I just modified the state equations with the inclusion of function F(t,x) whereby F is the Hodgkin-Huxley model function
f=@(x) x + F(k,x)+ B*I; % nonlinear state equations
h=@(x) C * x; % measurement equation
s=[-54.3 rand rand rand]';

then I always have the following error notice:
Error using chol
Matrix must be positive definite.

Error in ekf (line 60)
R=chol(H*P12+R); %Cholesky factorization

May I know how should I resolve this situation and plot the result?

Miguel Budag Becker

Hey Nathan Ellis,

I looked into your problem and found that both equations are right, because P is symmetric. So P = P', and then P12'=HP'=HP.

Frank Owusu

want to implement the EKF on a time series data... but I'm really struggling with errors for just trying to run this function.. anyone to guide me on this? It's very urgent

Muhammad Zahid

Muhammad Zahid

How to use EKF for Freeway Traffic Density Estimation ? Please somebody Guide mee

Abdulrahman Alabrash

Kathy Manson

Why didn't you use the EKF simulink function block?

Fajar Pratama

Where is the Taylor progression ? Is there no need taylor progression in this ekf function ? because in other paper that i read there is require the taylor regression. pls help me out . Thankyou

yu jun

Kaju Bubanja

Fabrizio Schiano

Is there a tutorial about this? Where can I find it? Thanks

Zhiwen Chen

Omar Aljanaideh

For those who got Error in ==> elf at 51
Just make sure that the MATLAB directory is same as the file location.

Nathan Ellis

I think there might be a mistake in the commented out alternative covariance update equation:
% P = P-K*P12';

P12 was previously defined as P12=PH'
Update equation is meant to be:
P = P-KHP or P = (I-KH)P

As far I can figure:
P12' =/= HP (P12'=HP')
and so;
% P = P-K*P12'; is incorrect?

Raja Sekhar Bandaru

Mohammad Hoque

I am trying to estimate SOC of lithium-ion battery cell of 3.7 V_nominal, and 15.5 Ah in real time using extended Kalman filter (EKF). I am facing the problem to write MATLAB code for EKF with the noise covariance and other measurement and observation noises terms. How can I solve this problem in estimating SOC of li-ion battery. I am using MATLAB R2014a.

Hu Liping

Hi Ali,

Please have a look at the measurement equation. It only meatures the first state.

najib valiyff

Hi, I still get the Maximum recursion limit of 500 reached. Use
set(0,'RecursionLimit',N) to change the limit. Be
aware that exceeding your available stack space
can crash MATLAB and/or your computer.

Error in

Message. Please help I've tried everything mentioned below including copy and pasting into command window

ailu wang

Simon Levy

Nice work, Yi Cao! I used your code to understand the EKF, and I linked to it in an interactive online tutorial:


I want to learn how to use EKF

usdb usdb1

hello; I have to do EKF with Matlab/simulink;could someone help me?


Somehow only the first state is tracked correctly. The estimate of x(2) and x(3) is always almost zero no matter how I change the model. Clearly there is a bug...


Hi everybody!
I really have not understood this code yet. In my case, I also study on EKF for GPS data that I want to apply EKF to due with noise and missing data in GPS data. I have one GPS data columm with more than 2000 of length. Who could show me how to do it?
Thank you so much for your kinds

Ahmed Smati

sehr gut!


Lines 51 and 53 are given by:


Why is x1 = fstate(x) used as the input for calculating the jacobian of the measurement equation? It makes more sense if the jacobian of the measurement equation is also evaluated at the current state x. Am I interpreting that part incorrectly?

usdb usdb1

when i highlith error between variable and its estimate (by adding a new variable err=x-xestimate) i plot err. a cycle limit (oscillation )is in this figure.and a gap appear between variable and its estimate .
is it an explanation and solution to this.


Hello everybody,
i have more general question about the extended kalman filter usage. what is not clear to me why EKF uses non-linear functions f and h for state prediction and estimate, while in other places the Jacobian of these functions is used.
Why the following is never used?
first calculate the liniarized state and measurements models at previous estimate point using Jacobian. Use the liniearized state transition and measurements matrix everywhere instead of non-linear in this specific iteration.
I would really appreciate your help
Thank you


Great submission, thanks!
One question though: in the parameter explanation you define inputs x and P as "a priori" state estimate and "a priori" estimated state covariance. In my understanding this is not right, as "a priori" values are only available right after the prediction step of the filter.

So, in my opinion x and P are the "a posteriori" values of the previous time step. The "a priori" values of x and P of the current time step are available after the prediction step of your filter (vals x1 and P in lines 51 and 52).
Do you agree?



Input argument "wstate" is undefined.

Error in ==> ekf at 51
[x1,A]=jaccsd(wstate,x); %nonlinear update and linearization at current state


Hi guys, i need some help please. I Use Matlab R2012b to try to run the code/example. I usually copy the whole code,place a new editor,highlight the example,right click,left click 'evaluate selection'(as i don't see any 'run').But on Matlab's command window, it shows the highlighted example and says "Undefined function 'ekf' for input arguments of type 'function_handle'." Please who knows what could be wrong? What could i be doing wrong? Thank you. John

usdb usdb1

the states are well estimated by EKF,but if chaging in extended state variable at middle of the simulation EKF converdge always to the initial value one.
please i want explication.

Masiyang Islam

Maria Perdomo


Just one question:

why is the nonlinear update at line 51 done with out a numerical integration but just making the new x_k=f(x_k-1)?

W. Chong


please someone explain what these lines do from line 51 to line 78

Maria Perdomo

f is the nonlinear differential ecuation to be integrated, thats why the @ so that the ekf funtion can call it, an withing the ekf solve this ecuation to get the first estimation, the h is the measurement equation, also as handle function (@) so that it can be called by the ekf to calculate the kalman gain.


can someone explain what these lines do:




what does the graph of 2nd and 3rrd state represent here


^^ please check your e-mail

Khan 954

fstate in line 51 represents the non-linear state equations, which are function of x
in the example fstate is f, which is in line 26


what is fstate in line 51?

Khan 954

the program is working well
copy the program from line 21 to line 46 and run it, its working


hi, im experiencing an error at line 51 and the program is not running due to that
can u please provide guidance in that regard

Olaf Gerritse


run the code as below in command window:


has an error:
??? Undefined function or method 'ekf' for input arguments of type
how to run these codes?

Simon Omekanda

Simon Omekanda

Can you go over the steps to properly run this function please?! I am still getting error that have been mentioned above in some comments, mainly:
??? Maximum recursion limit of 500 reached. Use set(0,'RecursionLimit',N)
to change the limit. Be aware that exceeding your available stack space can
crash MATLAB and/or your computer.

Error in ==> ekf>create@(x)[x(2);x(3);0.05*x(1)*(x(2)+x(3))]



I downloaded the file and ran it on R2006b. Got the following error. Could someone tell me what I am doing wrong? I guess I have to uncomment a few things and run in some sequence, but unable to figure out what

??? Input argument "fstate" is undefined.

Error in ==> ekf at 51
[x1,A]=jaccsd(fstate,x); onlinear update and linearization at current state


I get the error message "??? Undefined function or method 'ekf' for input arguments of type 'function_handle'"


thanks you



Is there possible to use the code in a case where I have different rates of prediction and correction steps. (e.g., could I have 10 predictions before make one correction...). Is it easy to do this in the current version of the code?

George Lim

Hi! This is a nice code for EKF. I have a question though: In your example if we assume that the value 0.05 is unknown parameter and we want simultaneous state and parameter estimation can we augment the state as with the parameter as:
n=4; %number of state
q=0.1; %std of process
r=0.1; %std of measurement
Q=q^2*eye(n); % covariance of process
% or Q=diag[Q 0]; % if no process noise is included in the parameter
R=r^2; % covariance of measurement
f=@(x)[x(2);x(3);x(4)*x(1)*(x(2)+x(3));x(4)]; % nonlinear state equations
h=@(x)x(1); % measurement equation
s=[0;0;1;0.1]; % initial state
x=s+q*randn(4,1); %initial state % initial state with noise
P = eye(n); % initial state covraiance
N=20; % total dynamic steps
xV = zeros(n,N); %estmate % allocate memory
sV = zeros(n,N); %actual
zV = zeros(1,N);
for k=1:N
z = h(s) + r*randn; % measurments
sV(:,k)= s; % save actual state
zV(k) = z; % save measurment
[x, P] = ekf(f,x,P,h,z,Q,R); % ekf
xV(:,k) = x; % save estimate
s = f(s) + q*randn(3,1); % update process

Yi Cao


Yes, it is possible. Please look at the submission:

Maria Perdomo

Hi, Is it possible to use your code for parameter identfication?


hi Yi
could you suggest an examplar definition of the function "f" together with initial state "s" for some real life example of a system? this would help me and other inexperienced guys to better understand this example. thank you very much in advance.

Big Andy


when i run this program..the following error is displayed in the command window:
"??? Input argument "fstate" is undefined.

Error in ==> ekf at 51
[x1,A]=jaccsd(fstate,x); %nonlinear update and linearization at current state"

please rectify my problem asap...
Also suggest to me how to implement this code for multi sensor data fusion wherein the input is in the form of signals from n different in how do I express my input here in terms of f and h.


Hi, how should I modify the m-file if I want to change the measurement- and process noise to:
w ~ N(u,Q)
v ~ N(e,R)

Matthew Coleman

Matthew Coleman

Congratulations. This is the first EKF library I manged to get working at all.

The example takes measurements go in the s matrix, not the x. Now that is fixed everything is good.

It is a long time since I did Kalman or Matlab. You clever guys underestimate how dumb you need to make your comments to get us newbies started.

Joao Henriques

Yi Cao


Yes, it is possible. For example, see


Aeimit Lakdawala

Is it possible to do constrained nonlinear optimization with EKF?

iasri icar

hi yi, would like to know if its appropriate to use EKF for forecasting of agricultural yields like fish ,rice etc. i am planning to use EXPAR with EKF for the problem stated above and would you kindly be able to give some of your ideas regarding the same.thankyou, with regards bishal.


i wrote a very simple compound pendulum code, and some how this ekf algorithm does not work for that. only change that i had to do to that example file was change the states to 2 and rest
this should give me sinusoidal waveform but it does not.
can you point out to me what could be wrong.

tim Heights

if in EKF i have to add state noise compensation, any good example or guidance here. how can i add to the example given by Yi Cao.
secondly, any one who could also recommend some book about it.


Yi Cao

For continuous-time EKF, please look at

Rohit Hippalgaonkar

Hi I am looking for an example where the EKF is applied to a continuous-time non-linear system with non-zero inputs (say measurements are taken at regular time samples through a non-linear (even linear would do) measurement process. I have looked around for this kind of example in the standard texts but haven't found any. Also a good source showing the implementation of the EKF wherein we linearize about a single operating point (as against linearizing about the predicted state every time) would be really helpful! Thanks in advance! Rohit

Dapat Chawah

Sorry, this comment is meant to be in the unscented kalman filter file discussion

Dapat Chawah

This code is working good for N<=150
but when N exceeds this limit, a nonsense happens
Is there any improvement to the code considering this error?

V. Poor

Yi Cao

This error occurs because you run the example incorrectly so that ekf calls itself more than 500 times. To run the example, you need copy contents between "%{" and "%}" then past it on matlab command window to execute the example.

It also could be because your MATLAB version is too old to support block comments. If that is the case, you can comment out all line by adding "%" at the begining of each line between "%{" and "%}" to solve the problem.

Zhongjie Chen

Hi, I still get the error

??? Maximum recursion limit of 500 reached. Use set(0,'RecursionLimit',N)
to change the limit. Be aware that exceeding your available stack space
crash MATLAB and/or your computer.

Error in ==> ekf>create@(x)[x(2);x(3);0.05*x(1)*(x(2)+x(3))]

after following your instructions. How do I correctly run the code?

maureen clerc

Beware: function ekf changes the value of the measurement covariance matrix R. It shouldn't be the case. Otherwise the code is nice and efficient.

Lihong Wang

Nice code. But if there is load disturbance on the state, why the estimate from the EKF almost ignores the load disturbance?

Tim Davis

Excellent! Nice use of CHOL instead of INV (as can be seen in two other Kalman-filter codes on the File Exchange). Nice to see good numerics at work.

I see that "K=P12*inv(...)" is commented out; that's perfect. It gives the math behind what the CHOL and backslashes are doing.

Saroj Pandey

it is very good and helpful in my project.

Neric Lau

Very helpful for learning Kalman Filter Implementation.

Dmitry Sh

Nicely made, and very intuitive if one has an idea how a linear Kalman Filter works. However I found that numerically solving the Jacobian is not always the best form of linerisation, especially for simpler cases when an analytic Jacobian can be computed by hand.
In my experiments (with simple non-linear models) an analytic Jacobian usually gave a significant improvement of fit when compared to its numeric counterpart. Maybe you could add an option on how it should be solved

hu aijun

feng yu

meng jun

very well.

Reza Baghaei


Yi Cao

Dear Edwin,

As I expected, this error is due to your way to run the example because the error message shows that the error occures at line 19, which is a commented line to begin the example.

To correctly run the example, you can follow the following steps:

1. select the example lines correctly
2. press control-t to uncomment the selection
3. right-click to run the selection
4. click un-do to recover the file. (DO NOT click the save button.)

For you and other users' convenient, I updated the file with block-comment lines for the example. Now, you just need to select and right-click to run the example without change the file. The update will appear a few hours later.

Hope this help.

edwin de Vries

??? Error: File: ekf.m Line: 19 Column: 10
Expression or statement is incomplete or incorrect.

??? Maximum recursion limit of 500 reached. Use set(0,'RecursionLimit',N)
to change the limit. Be aware that exceeding your available stack space can
crash MATLAB and/or your computer.

Error in ==> ekf>create@(x)[x(2);x(3);0.05*x(1)*(x(2)+x(3))] at 25
f=@(x)[x(2);x(3);0.05*x(1)*(x(2)+x(3))]; % nonlinear state equations

sayed mohammad mousavi gazafrudi

MATLAB Release Compatibility
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