143 1.4.4: More KF examples for state estimation of a linear system

143.1 Open-loop estimation

In this lesson, we explore some additional examples, comparing with results from the KF in Lesson 1.4.3.
The first example implements an open-loop state estimator, for comparison with KF.
To perform open-loop (OL) estimation, the main program loop is modified as follows:

for k = 2:length(t)
    % KF Step 1a: State estimate time update
    xhat = Ad*xhat + Bd*u(k-1); % use prior value of "u"

    % KF Step 1b: Error covariance time update
    SigmaX = Ad*SigmaX*Ad' + SigmaW;

    % For open-loop state estimation, there is no measurement feedback.
    % So, delete steps 1c, 2a, 2b, and 2c.

    % [Store information for evaluation/plotting purposes]
    xhatstore(:,k) = xhat;
    boundstore(:,k) = 3*sqrt(diag(SigmaX));
end

143.2 Sample open-loop results: States and estimates

Plot shows OL output for the spring-mass-damper example, using simulation data from earlier this week.
Comparing with the KF result for the same data in Lesson 1.4.3, we immediately notice that the confidence bounds for OL are much wider than they were for KF.
We also notice that the position estimate (especially) of the OL estimator is far less accurate than it was for KF.
The unknown influence of process noise w_k has caused the OL estimator to be less accurate and less confident than the KF.

143.3 Sample open-loop results: Estimation errors

Plot shows an alternate way of viewing the OL output, looking directly at the state-estimation errors, x_k - \hat{x}_k^+.
Comparing with the corresponding results from Lesson 1.4.3, we notice far more structure (correlation) in the OL prediction errors than the KF estimation errors.
This is because w_k causes the states to evolve according to the system’s dynamics in a way that cannot be predicted OL.
Hence, the confidence bounds are also much wider because of this uncertainty:
- About 0.1 versus 0.01 for position, and
- 0.02 versus 0.01 for velocity.
OL predictions become even worse if we don’t know x_0 exactly.

143.4 What if the KF is poorly initialized?

How well does the KF perform if we don’t know x_0 exactly?
Lesson 1.4.2 simulated the spring-mass-damper system with a true initial state x_0 = 0.
Lesson 1.4.3 initialized the KF with \hat{x}_0^+ = 0 and \Sigma_{x,0}^+ = 0, indicating perfect knowledge of x_0 (i.e., no uncertainty).
Suppose that we don’t know x_0 exactly, which is the typical case.
- Then, we must enter a different \hat{x}_0^+ and \Sigma_{x,0}^+ in the code.

% System's true initial state is x = [0;0]. But, what if we don't know that?
xhat = [0.25; -0.05];
xhatstore(:,1) = xhat;

% Assume that xhat is a 2-sigma sample from the true distribution of x(0).
SigmaX = diag(xhat/2).^2;
boundstore(:,1) = 3*sqrt(diag(SigmaX));

% Continue with standard KF code for the main simulation loop...
for k = 2:length(t)

143.5 KF results for “bad” initialization

Plots show KF output when the state is initialized poorly.
After a few seconds, the results are nearly the same as when the KF was initialized with exact knowledge of x_0. Feedback works!

143.6 KF results for “bad” initialization (zoom)

Zooming in on the first portions of the KF output, we see how effective the feedback mechanism of the KF actually is.
Position uncertainty plummets in a single timestep (we measure this state directly, albeit with noise v_k); velocity uncertainty converges in only a few seconds also.

143.7 Summary

In this lesson we implemented different scenarios to help gain intuition into how well the KF works.
- I recommend that you also try different things—this is a great way to learn!
Specifically, we first compared KF state estimation to open-loop state prediction.
- The OL predictor has no feedback, so cannot know the impact of w_k on the state.
- So, its predictions are inferior to KF estimates; its confidence bounds are looser.
We then looked at the KF output when we don’t know x_0 exactly.
- We found that the feedback mechanism built in to the KF allows the filter state estimates to converge to the true states quite quickly for this example.
- Convergence rate depends on relative sizes \Sigma_w and \Sigma_v, as you will learn later.
- OL predictors have no way of knowing if their \hat{x}_0 is incorrect, so wouldn’t converge.
So, KF can work very well if all the assumptions made in their derivation are met.
- What if they are not? We explore this in the next lesson.

--- title: "1.4.4: More KF examples for state estimation of a linear system" description: "In this lesson, we will explore some additional examples of KF state estimation, comparing with results from an open-loop state predictor. We will also see how the KF performs when it is initialized with a poor guess of the initial state." date: "2026-04-22" keywords: [Kalman filter, state estimation, open-loop prediction, initialization] --- ## Open-loop estimation - In this lesson, we explore some additional examples, comparing with results from the KF in Lesson 1.4.3. - The first example implements an open-loop state estimator, for comparison with KF. - To perform open-loop (OL) estimation, the main program loop is modified as follows: ```octave for k = 2:length(t) % KF Step 1a: State estimate time update xhat = Ad*xhat + Bd*u(k-1); % use prior value of "u" % KF Step 1b: Error covariance time update SigmaX = Ad*SigmaX*Ad' + SigmaW; % For open-loop state estimation, there is no measurement feedback. % So, delete steps 1c, 2a, 2b, and 2c. % [Store information for evaluation/plotting purposes] xhatstore(:,k) = xhat; boundstore(:,k) = 3*sqrt(diag(SigmaX)); end ``` ## Sample open-loop results: States and estimates - Plot shows OL output for the spring-mass-damper example, using simulation data from earlier this week. - Comparing with the KF result for the same data in Lesson 1.4.3, we immediately notice that the confidence bounds for OL are much wider than they were for KF. - We also notice that the position estimate (especially) of the OL estimator is far less accurate than it was for KF. - The unknown influence of process noise $w_k$ has caused the OL estimator to be less accurate and less confident than the KF. ![](images/L.1.4.4-1.png) ## Sample open-loop results: Estimation errors - Plot shows an alternate way of viewing the OL output, looking directly at the state-estimation errors, $x_k - \hat{x}_k^+$. - Comparing with the corresponding results from Lesson 1.4.3, we notice far more structure (correlation) in the OL prediction errors than the KF estimation errors. - This is because $w_k$ causes the states to evolve according to the system's dynamics in a way that cannot be predicted OL. - Hence, the confidence bounds are also much wider because of this uncertainty: - About 0.1 versus 0.01 for position, and - 0.02 versus 0.01 for velocity. - OL predictions become even worse if we don't know $x_0$ exactly. ![](images/L.1.4.4-2.png) ## What if the KF is poorly initialized? - How well does the KF perform if we don't know $x_0$ exactly? - Lesson 1.4.2 simulated the spring-mass-damper system with a true initial state $x_0 = 0$. - Lesson 1.4.3 initialized the KF with $\hat{x}_0^+ = 0$ and $\Sigma_{x,0}^+ = 0$, indicating perfect knowledge of $x_0$ (i.e., no uncertainty). - Suppose that we don't know $x_0$ exactly, which is the typical case. - Then, we must enter a different $\hat{x}_0^+$ and $\Sigma_{x,0}^+$ in the code. ```octave % System's true initial state is x = [0;0]. But, what if we don't know that? xhat = [0.25; -0.05]; xhatstore(:,1) = xhat; % Assume that xhat is a 2-sigma sample from the true distribution of x(0). SigmaX = diag(xhat/2).^2; boundstore(:,1) = 3*sqrt(diag(SigmaX)); % Continue with standard KF code for the main simulation loop... for k = 2:length(t) ``` ## KF results for "bad" initialization - Plots show KF output when the state is initialized poorly. - After a few seconds, the results are nearly the same as when the KF was initialized with exact knowledge of $x_0$. Feedback works! ![](images/L.1.4.4-3.png) ## KF results for "bad" initialization (zoom) - Zooming in on the first portions of the KF output, we see how effective the feedback mechanism of the KF actually is. - Position uncertainty plummets in a single timestep (we measure this state directly, albeit with noise $v_k$); velocity uncertainty converges in only a few seconds also. ![](images/L.1.4.4-4.png) ## Summary - In this lesson we implemented different scenarios to help gain intuition into how well the KF works. - I recommend that you also try different things---this is a great way to learn! - Specifically, we first compared KF state estimation to open-loop state prediction. - The OL predictor has no feedback, so cannot know the impact of $w_k$ on the state. - So, its predictions are inferior to KF estimates; its confidence bounds are looser. - We then looked at the KF output when we don't know $x_0$ exactly. - We found that the feedback mechanism built in to the KF allows the filter state estimates to converge to the true states quite quickly for this example. - Convergence rate depends on relative sizes $\Sigma_w$ and $\Sigma_v$, as you will learn later. - OL predictors have no way of knowing if their $\hat{x}_0$ is incorrect, so wouldn't converge. - So, KF can work very well if all the assumptions made in their derivation are met. - What if they are not? We explore this in the next lesson.